Foundations of High-Energy-Density Physics: Physical Processes of Matter at Extreme Conditions 1107124115, 9781107124110

Table of contents :
Contents
Preface
1 Introduction
1.1 High-Energy-Density on Earth
1.2 Some Connections to Prior Work
1.3 Outline
1.4 Notation, Variables, and Units
2 Characteristics of High-Energy-Density Matter
2.1 Landscape of High-Energy-Density Matter
2.2 Compressing Atoms
2.3 Electron Degeneracy
2.4 Equation of State
2.5 Collisionality and Equilibrium
2.6 Radiation
2.7 Magnetic Fields
2.8 Warm Dense Matter
2.9 Scaling from Astrophysics to the Terrestrial Laboratory
3 Fundamental Microphysics of Ionized Gases
3.1 Kinetic Theory
3.1.1 The Distribution Function
3.1.2 Evaluation of the Collision Term
3.1.3 Boltzmann’s H-Theorem
3.1.4 The Maxwell-Boltzmann Distribution Function
3.2 Statistical Mechanics
3.2.1 The Distribution Functions
3.2.2 The Maxwell-Boltzmann Distribution Function (Again)
3.3 Thermodynamics
3.3.1 First Law of Thermodynamics
3.3.2 Second Law of Thermodynamics
3.3.3 Helmholtz Free Energy
3.3.4 Maxwell’s Relations and Thermodynamic Consistency
3.4 The Fermi Gas
3.4.1 The Chemical Potential
3.4.2 The Grand Canonical Ensemble
3.5 Debye Shielding and Quasineutrality
3.6 Fluid Conservation Equations
3.7 Electron Plasma Frequency and Plasma Waves
3.8 Coulomb Collisions
3.8.1 The Scattering Angle
3.8.2 Scattering Cross Section
3.8.3 Energy Loss
3.8.4 Coulomb Logarithm
3.9 Multiple Coulomb Scattering
3.9.1 Velocity Decrements and Diffusion
3.9.2 Relaxation Times
3.10 Radiation as a Fluid
3.10.1 Planck’s Law
3.10.2 Stefan’s Law
3.10.3 Thermodynamics of Equilibrium Radiation
3.11 One-Electron Atom
3.11.1 Bohr’s Hypothesis
3.11.2 Bohr-Sommerfeld Quantization
3.11.3 Quantum Theory of Atomic Structure
3.12 Plane Electromagnetic Waves
3.12.1 Plane Electromagnetic Waves in a Good Conductor
3.12.2 Field Energy in a Dispersive Medium
3.13 Permittivity and Electrical Conductivity
4 Ionization
4.1 Saha
4.2 Thomas-Fermi
4.3 Pressure Ionization and Continuum Lowering
4.3.1 Debye-Hückel
4.3.2 Ion-Sphere
4.3.3 Stewart-Pyatt
4.3.4 Ecker-Kröll
4.4 Collisional-Radiative
4.5 Screened Hydrogenic Average-Atom
4.6 Time-Dependent Non-LTE Average-Atom
4.6.1 Population Rate Equations
4.6.2 Steady-State Non-LTE
4.6.3 Dielectronic Recombination
4.6.4 Reconstruction of Ionic States
4.7 Other Models
5 Entropy and the Equation of State
5.1 Two-Temperature Thermodynamics
5.2 Perfect Gas
5.3 Realistic Gas
5.4 Debye-Hückel
5.4.1 Thermodynamic Properties
5.4.2 Nonequilibrium Debye-Hückel
5.4.3 Density Fluctuations
5.5 Strongly Coupled Plasma
5.5.1 Static Properties
5.5.2 Scattering Experiments
5.5.3 Dynamic Properties
5.5.4 Thermodynamic Properties
5.5.5 Ornstein-Zernike
5.5.6 Hypernetted-Chain
5.5.7 One-Component Plasma (OCP)
5.6 Thomas-Fermi
5.7 Density Functional Theory
5.8 Solids
5.8.1 Debye Model
5.8.2 Mie-Grüneisen Model
5.8.3 Lindemann Melt Law
5.9 Quotidian Equation of State (QEOS)
5.9.1 Electronic EOS
5.9.2 Chemical Bonding Correction
5.9.3 Ionic EOS
5.10 Screened Hydrogenic Average-Atom
5.11 Tabular EOS
6 Hydrodynamics
6.1 Frames of Reference
6.1.1 Eulerian and Lagrangean Derivatives
6.1.2 Reynolds Transport Theorem
6.2 Conservation Equations for Ideal Fluids
6.2.1 The Equation of Continuity
6.2.2 The Equation of Momentum
6.2.3 The Equations of Energy
6.2.4 Bernoulli’s Equation
6.2.5 Vorticity
6.3 Method of Characteristics
6.3.1 Isothermal Expansion
6.3.2 Adiabatic Expansion
6.4 Acoustic Disturbances
6.5 Shocks
6.5.1 Nonlinear Acoustic Waves
6.5.2 Rankine-Hugoniot Equations
6.5.3 Jump Relations in a Polytropic Gas
6.5.4 The Shock Tube
6.5.5 Shock Reflection
6.5.6 Multiple Shock Reflections
6.5.7 Shocks Moving from Heavy into Light Media
6.5.8 Sedov-Taylor Blast Wave
6.6 Viscous and Heat Conducting Fluids
6.6.1 Damping of an Acoustic Wave
6.6.2 Structure of the Shock Front
6.6.3 The Relaxation Layer
6.7 Elastic-Plastic Behavior of Solids
6.7.1 Hooke’s Law
6.7.2 Homogeneous Deformations
6.7.3 Thermal Deformations
6.7.4 Elastic Deformations
6.7.5 Plastic Flow
6.7.6 Yield Strength
6.8 Transitioning from Planar to Elliptical Flow
6.9 Fluid Instabilities
6.9.1 Rayleigh-Taylor
6.9.2 Kelvin-Helmholtz
6.9.3 Richtmyer-Meshkov
7 Thermal Energy Transport
7.1 Linear Heat Conduction
7.2 Nonlinear Heat Conduction
7.3 The Heat Flux
7.4 Thermal and Electrical Conductivities
7.4.1 Onsager Relations
7.4.2 Transport Coefficients
7.4.3 Relaxation Times
7.4.4 Electron-Ion Coulomb Logarithm
7.5 Electron-Ion Energy Exchange
7.6 Electron Degeneracy Effects
7.7 Inhibited Thermal Transport
7.8 Nonlocal Heat Transport
8 Radiation and Radiative Transfer
8.1 The Radiation Field
8.1.1 Specific Intensity
8.1.2 Radiation Energy Density and Mean Intensity
8.1.3 Radiation Flux and Momentum Density
8.1.4 Radiation Pressure Tensor
8.2 Interaction of the Radiation Field with Matter
8.2.1 Absorption
8.2.2 Emission
8.2.3 Kirchhoff’s Law
8.2.4 Scattering
8.2.5 Stimulated Emission
8.3 The Equation of Transfer
8.3.1 Boundary Conditions on the Transfer Equation
8.3.2 Equation of Transfer in Various Coordinate Systems
8.4 Moments of the Transfer Equation
8.5 Optical Depth
8.6 Approximate Descriptions of the Radiative Transfer Equation
8.6.1 Free-Streaming Approximation
8.6.2 Diffusion Approximation
8.6.3 Telegrapher’s Equation
8.6.4 Eddington’s Approximation
8.6.5 Equilibrium Diffusion
8.6.6 Higher-Order Approximations
8.6.7 Multigroup Approximation
8.7 Opacity Averaging
8.8 Steady-State Transfer
8.8.1 Formal Solution
8.8.2 Slab Geometry
8.8.3 Milne’s Equation
8.8.4 Eddington-Barbier Relation
8.9 The Comoving Frame Representation
8.9.1 Doppler and Aberration Transformations
8.9.2 Transforming the Specific Intensity, Emissivity, and Absorptivity
8.9.3 Transforming the Transfer Equation
8.9.4 Transforming the Moment Equations
8.9.5 Comoving-Frame Transfer Equation
8.9.6 Comoving-Frame Moment Equations
8.9.7 Transforming the Radiation-Matter Coupling Terms
8.9.8 Diffusion in the Comoving Frame
8.10 View Factors
9 Transition Rates and Optical Coefficients
9.1 Radiative Transitions
9.1.1 Einstein Relations
9.1.2 Bound-Bound Optical Coefficients
9.1.3 Quantum Mechanics of Radiative Processes
9.1.4 Einstein-Milne Relations
9.1.5 Bound-Free Optical Coefficients
9.1.6 Free-Free Optical Coefficients
9.1.7 Thomson Scattering
9.1.8 Maximum Opacity Theorem
9.2 Collisional Transitions
9.2.1 Excitation and De-excitation
9.2.2 Ionization and Three-Body Recombination
10 Radiation Hydrodynamics
10.1 Incorporating Radiation in Euler’s Equations
10.1.1 Fixed-Frame Equations
10.1.2 Comoving-Frame Equation
10.1.3 Consistency of the Equations in Both Coordinate Frames
10.1.4 Equilibrium Diffusion
10.1.5 Nonequilibrium Diffusion
10.1.6 Flux Limiting
10.2 Thermodynamic Relations in the Presence of Radiation
10.2.1 Equilibrium Radiation and a Perfect Gas
10.2.2 Equilibrium Radiation and an Ionizing Gas
10.3 Marshak Waves
10.4 Radiating Shock Waves
10.4.1 Radiative Precursors
10.4.2 Rankine-Hugoniot Relations and Jump Conditions
10.4.3 Fluid Dynamics of Radiating Shocks
10.4.4 Radiative Cooling of a Thin Layer
10.4.5 Shocks in Optically Thin Material
10.4.6 Optically Thick Shocks in the Flux Regime
10.4.7 Shocks in Optically Thin Upstream Material
10.4.8 Radiation-Dominated Shock Waves
10.5 Shock Structure
10.5.1 Subcritical
10.5.2 Supercritical
10.6 Ionization Fronts
11 Magnetohydrodynamics
11.1 Plasma Electrodynamics
11.2 Equations of Magnetohydrodynamics
11.2.1 Momentum Equation
11.2.2 Induction Equation
11.2.3 Electron Thermal Equation
11.2.4 Electron Degeneracy
11.2.5 Ion Thermal Equation
11.2.6 Bohm Diffusion
11.2.7 High-Frequency Plasma Oscillations
11.2.8 Magnetic Energy
11.2.9 Generalized Ohm’s Law
11.2.10 Hall Effect
11.2.11 One-Dimensional Cylindrically Symmetric Equations
11.3 Magnetic Reconnection
11.3.1 Biermann Battery
11.3.2 Sweet-Parker Reconnection
11.3.3 Hall MHD Reconnection
11.3.4 Plasmoid Formation
11.4 Magnetic Confinement
12 Electromagnetic Wav-Material Interactions
12.1 Electromagnetic Wave Propagation in Homogeneous Medium
12.1.1 Interaction of Free Electrons with an Electromagnetic Wave
12.1.2 Longitudinal Waves and Spatial Dispersion
12.2 Propagation in Inhomogeneous Isotropic Medium
12.2.1 Exact Solution in a Linear Density Gradient
12.2.2 Reflectivity and Phase Shift
12.2.3 Geometrical Optics Approximation
12.2.4 Weak Reflection
12.2.5 Oblique Incidence
12.2.6 Ponderomotive Force and Momentum Deposition
12.2.7 Ray-Trace “Equation of Motion”
12.3 Reflection at an Interface
12.4 Density Profile Modification
12.5 Absorption of Electromagnetic Energy
12.5.1 Collisional (Inverse Bremsstrahlung)
12.5.2 Nonlinear Inverse Bremsstrahlung
12.5.3 Resonance
12.6 Dielectric Permittivity (Revisited)
12.6.1 Plasma Conductivity
12.6.2 Near-Free Electron Metals
12.7 Kinetic Instabilities
12.7.1 Matching Conditions
12.7.2 Damping
12.7.3 Instability Threshold
12.7.4 Parametric Decay and Two-Stream Instabilities
12.7.5 Stimulated Raman Scattering
12.7.6 Two-Plasmon Decay
12.7.7 Stimulated Brillouin Scattering
12.7.8 Nonlinear Aspects
12.8 Magnetic Fields
12.8.1 Spontaneous Generation of Magnetic Fields
12.8.2 Faraday Rotation
12.9 Relativistic Considerations
12.9.1 Electromagnetic Wave Propagation
12.9.2 Interaction with Transparent Matter
12.9.3 Nonlinear Dynamics at the Vacuum Boundary
References
Index

Citation preview

FOUNDATIONS OF HIGH-ENERGY-DENSITY PHYSICS Physical Processes of Matter at Extreme Conditions High-energy-density physics explores the dynamics of matter at extreme conditions. This encompasses temperatures and densities far greater than those we experience on Earth. It applies to normal stars, exploding stars, active galaxies, and planetary interiors. High-energy-density matter is found on Earth in the explosion of nuclear weapons and in laboratories with high-powered lasers or pulsed-power machines. The physics explored in this book is the basis for large-scale simulation codes needed to interpret experimental results, whether from astrophysical observations or laboratory-scale experiments. The key elements of high-energy-density physics covered are gas dynamics, ionization, thermal energy transport, radiation transfer, intense electromagnetic waves, and their dynamical coupling. Implicit in all this is a fundamental understanding of hydrodynamics, plasma physics, atomic physics, quantum mechanics, and electromagnetic theory. Beginning with a summary of these topics and then exploring the major ones in depth, this book is a valuable resource for research scientists and graduate students in physics and astrophysics. jon larsen is the founder and President of Cascade Applied Science, Inc., the company that developed a suite of simulation codes for the high-energy-density physics (HEDP) research community. He is an expert in radiation hydrodynamic simulations and is particularly well known for a code known as HYADES, which is widely used by universities and by the National Laboratories.

FOUNDATIONS OF HIGH-ENERGY-DENSITY PHYSICS Physical Processes of Matter at Extreme Conditions JON LARSEN Cascade Applied Sciences Inc., Colorado

University Printing House, Cambridge CB2 8BS, United Kingdom One Liberty Plaza, 20th Floor, New York, NY 10006, USA 477 Williamstown Road, Port Melbourne, VIC 3207, Australia 4843/24, 2nd Floor, Ansari Road, Daryaganj, Delhi – 110002, India 79 Anson Road, #06–04/06, Singapore 079906 Cambridge University Press is part of the University of Cambridge. It furthers the University’s mission by disseminating knowledge in the pursuit of education, learning, and research at the highest international levels of excellence. www.cambridge.org Information on this title: www.cambridge.org/9781107124110 10.1017/9781316403891 © Jon Larsen 2017 This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 2017 Printed in the United Kingdom by Clays, St Ives plc A catalogue record for this publication is available from the British Library Library of Congress Cataloging-in-Publication Data Names: Larsen, Jon, 1942- author. Title: Foundations of high-energy-density physics / Jon Larsen, Cascade Applied Sciences Inc., Colorado. Description: Cambridge, United Kingdom ; New York, NY : Cambridge University Press, [2017] | Includes bibliographical references and index. Identifiers: LCCN 2016035536| ISBN 9781107124110 (Hardback ; alk. paper) | ISBN 1107124115 (Hardback ; alk. paper) Subjects: LCSH: High pressure physics. | Matter–Properties. | Physics–Mathematical models. | Plasma density. | Hydrodynamics. Classification: LCC QC280.2. L37 2017 | DDC 539.7/6–dc23 LC record available at https://lccn.loc.gov/2016035536 ISBN 978-1-107-12411-0 Hardback Cambridge University Press has no responsibility for the persistence or accuracy of URLs for external or third-party Internet Web sites referred to in this publication and does not guarantee that any content on such Web sites is, or will remain, accurate or appropriate.

Contents

Preface

page xiii

1 Introduction 1.1 High-Energy-Density on Earth 1.2 Some Connections to Prior Work 1.3 Outline 1.4 Notation, Variables, and Units

1 3 4 5 8

2 Characteristics of High-Energy-Density Matter 2.1 Landscape of High-Energy-Density Matter 2.2 Compressing Atoms 2.3 Electron Degeneracy 2.4 Equation of State 2.5 Collisionality and Equilibrium 2.6 Radiation 2.7 Magnetic Fields 2.8 Warm Dense Matter 2.9 Scaling from Astrophysics to the Terrestrial Laboratory

9 10 11 14 16 17 20 21 23 25

3 Fundamental Microphysics of Ionized Gases 3.1 Kinetic Theory 3.1.1 The Distribution Function 3.1.2 Evaluation of the Collision Term 3.1.3 Boltzmann’s H-Theorem 3.1.4 The Maxwell-Boltzmann Distribution Function 3.2 Statistical Mechanics 3.2.1 The Distribution Functions 3.2.2 The Maxwell-Boltzmann Distribution Function (Again)

27 28 29 33 34 35 36 37 43

v

vi

Contents

3.3

3.4

3.5 3.6 3.7 3.8

3.9

3.10

3.11

3.12

3.13 4

Thermodynamics 3.3.1 First Law of Thermodynamics 3.3.2 Second Law of Thermodynamics 3.3.3 Helmholtz Free Energy 3.3.4 Maxwell’s Relations and Thermodynamic Consistency The Fermi Gas 3.4.1 The Chemical Potential 3.4.2 The Grand Canonical Ensemble Debye Shielding and Quasineutrality Fluid Conservation Equations Electron Plasma Frequency and Plasma Waves Coulomb Collisions 3.8.1 The Scattering Angle 3.8.2 Scattering Cross Section 3.8.3 Energy Loss 3.8.4 Coulomb Logarithm Multiple Coulomb Scattering 3.9.1 Velocity Decrements and Diffusion 3.9.2 Relaxation Times Radiation as a Fluid 3.10.1 Planck’s Law 3.10.2 Stefan’s Law 3.10.3 Thermodynamics of Equilibrium Radiation One-Electron Atom 3.11.1 Bohr’s Hypothesis 3.11.2 Bohr-Sommerfeld Quantization 3.11.3 Quantum Theory of Atomic Structure Plane Electromagnetic Waves 3.12.1 Plane Electromagnetic Waves in a Good Conductor 3.12.2 Field Energy in a Dispersive Medium Permittivity and Electrical Conductivity

Ionization 4.1 Saha 4.2 Thomas-Fermi 4.3 Pressure Ionization and Continuum Lowering 4.3.1 Debye-Hückel 4.3.2 Ion-Sphere 4.3.3 Stewart-Pyatt 4.3.4 Ecker-Kröll

46 47 48 49 50 51 53 56 58 61 65 69 70 74 75 76 78 78 87 89 90 93 95 96 97 102 105 112 114 116 118 122 127 135 141 146 147 150 155

Contents

4.4 4.5 4.6

Collisional-Radiative Screened Hydrogenic Average-Atom Time-Dependent Non-LTE Average-Atom 4.6.1 Population Rate Equations 4.6.2 Steady-State Non-LTE 4.6.3 Dielectronic Recombination 4.6.4 Reconstruction of Ionic States 4.7 Other Models

vii

157 161 169 170 174 175 180 182

5 Entropy and the Equation of State 5.1 Two-Temperature Thermodynamics 5.2 Perfect Gas 5.3 Realistic Gas 5.4 Debye-Hückel 5.4.1 Thermodynamic Properties 5.4.2 Nonequilibrium Debye-Hückel 5.4.3 Density Fluctuations 5.5 Strongly Coupled Plasma 5.5.1 Static Properties 5.5.2 Scattering Experiments 5.5.3 Dynamic Properties 5.5.4 Thermodynamic Properties 5.5.5 Ornstein-Zernike 5.5.6 Hypernetted-Chain 5.5.7 One-Component Plasma (OCP) 5.6 Thomas-Fermi 5.7 Density Functional Theory 5.8 Solids 5.8.1 Debye Model 5.8.2 Mie-Grüneisen Model 5.8.3 Lindemann Melt Law 5.9 Quotidian Equation of State (QEOS) 5.9.1 Electronic EOS 5.9.2 Chemical Bonding Correction 5.9.3 Ionic EOS 5.10 Screened Hydrogenic Average-Atom 5.11 Tabular EOS

185 187 189 191 192 194 196 199 202 204 207 209 212 213 214 216 219 226 229 229 233 236 237 238 238 239 245 248

6 Hydrodynamics 6.1 Frames of Reference 6.1.1 Eulerian and Lagrangean Derivatives 6.1.2 Reynolds Transport Theorem

251 252 252 253

viii

Contents

6.2

6.3

6.4 6.5

6.6

6.7

6.8 6.9

7

Conservation Equations for Ideal Fluids 6.2.1 The Equation of Continuity 6.2.2 The Equation of Momentum 6.2.3 The Equations of Energy 6.2.4 Bernoulli’s Equation 6.2.5 Vorticity Method of Characteristics 6.3.1 Isothermal Expansion 6.3.2 Adiabatic Expansion Acoustic Disturbances Shocks 6.5.1 Nonlinear Acoustic Waves 6.5.2 Rankine-Hugoniot Equations 6.5.3 Jump Relations in a Polytropic Gas 6.5.4 The Shock Tube 6.5.5 Shock Reflection 6.5.6 Multiple Shock Reflections 6.5.7 Shocks Moving from Heavy into Light Media 6.5.8 Sedov-Taylor Blast Wave Viscous and Heat Conducting Fluids 6.6.1 Damping of an Acoustic Wave 6.6.2 Structure of the Shock Front 6.6.3 The Relaxation Layer Elastic-Plastic Behavior of Solids 6.7.1 Hooke’s Law 6.7.2 Homogeneous Deformations 6.7.3 Thermal Deformations 6.7.4 Elastic Deformations 6.7.5 Plastic Flow 6.7.6 Yield Strength Transitioning from Planar to Elliptical Flow Fluid Instabilities 6.9.1 Rayleigh-Taylor 6.9.2 Kelvin-Helmholtz 6.9.3 Richtmyer-Meshkov

Thermal Energy Transport 7.1 Linear Heat Conduction 7.2 Nonlinear Heat Conduction 7.3 The Heat Flux

254 254 255 256 258 259 260 262 264 266 270 271 275 277 284 287 292 293 295 300 304 306 312 314 317 318 319 321 323 324 326 330 332 338 341 344 346 348 353

Contents

7.4

7.5 7.6 7.7 7.8

Thermal and Electrical Conductivities 7.4.1 Onsager Relations 7.4.2 Transport Coefficients 7.4.3 Relaxation Times 7.4.4 Electron-Ion Coulomb Logarithm Electron-Ion Energy Exchange Electron Degeneracy Effects Inhibited Thermal Transport Nonlocal Heat Transport

8 Radiation and Radiative Transfer 8.1 The Radiation Field 8.1.1 Specific Intensity 8.1.2 Radiation Energy Density and Mean Intensity 8.1.3 Radiation Flux and Momentum Density 8.1.4 Radiation Pressure Tensor 8.2 Interaction of the Radiation Field with Matter 8.2.1 Absorption 8.2.2 Emission 8.2.3 Kirchhoff’s Law 8.2.4 Scattering 8.2.5 Stimulated Emission 8.3 The Equation of Transfer 8.3.1 Boundary Conditions on the Transfer Equation 8.3.2 Equation of Transfer in Various Coordinate Systems 8.4 Moments of the Transfer Equation 8.5 Optical Depth 8.6 Approximate Descriptions of the Radiative Transfer Equation 8.6.1 Free-Streaming Approximation 8.6.2 Diffusion Approximation 8.6.3 Telegrapher’s Equation 8.6.4 Eddington’s Approximation 8.6.5 Equilibrium Diffusion 8.6.6 Higher-Order Approximations 8.6.7 Multigroup Approximation 8.7 Opacity Averaging 8.8 Steady-State Transfer 8.8.1 Formal Solution 8.8.2 Slab Geometry 8.8.3 Milne’s Equation

ix

357 358 360 367 368 371 374 379 387 391 392 393 395 395 396 398 398 398 399 400 401 402 405 406 409 412 412 413 414 418 419 422 424 427 428 430 431 433 437

x

Contents

8.8.4 Eddington-Barbier Relation The Comoving Frame Representation 8.9.1 Doppler and Aberration Transformations 8.9.2 Transforming the Specific Intensity, Emissivity, and Absorptivity 8.9.3 Transforming the Transfer Equation 8.9.4 Transforming the Moment Equations 8.9.5 Comoving-Frame Transfer Equation 8.9.6 Comoving-Frame Moment Equations 8.9.7 Transforming the Radiation-Matter Coupling Terms 8.9.8 Diffusion in the Comoving Frame 8.10 View Factors 8.9

9

Transition Rates and Optical Coefficients 9.1 Radiative Transitions 9.1.1 Einstein Relations 9.1.2 Bound-Bound Optical Coefficients 9.1.3 Quantum Mechanics of Radiative Processes 9.1.4 Einstein-Milne Relations 9.1.5 Bound-Free Optical Coefficients 9.1.6 Free-Free Optical Coefficients 9.1.7 Thomson Scattering 9.1.8 Maximum Opacity Theorem 9.2 Collisional Transitions 9.2.1 Excitation and De-excitation 9.2.2 Ionization and Three-Body Recombination

10 Radiation Hydrodynamics 10.1 Incorporating Radiation in Euler’s Equations 10.1.1 Fixed-Frame Equations 10.1.2 Comoving-Frame Equation 10.1.3 Consistency of the Equations in Both Coordinate Frames 10.1.4 Equilibrium Diffusion 10.1.5 Nonequilibrium Diffusion 10.1.6 Flux Limiting 10.2 Thermodynamic Relations in the Presence of Radiation 10.2.1 Equilibrium Radiation and a Perfect Gas 10.2.2 Equilibrium Radiation and an Ionizing Gas 10.3 Marshak Waves

440 441 442 444 445 446 447 449 450 451 456 460 461 463 466 467 476 478 485 490 494 497 497 500 508 509 510 513 515 516 519 522 523 523 526 528

Contents

xi

10.4 Radiating Shock Waves 10.4.1 Radiative Precursors 10.4.2 Rankine-Hugoniot Relations and Jump Conditions 10.4.3 Fluid Dynamics of Radiating Shocks 10.4.4 Radiative Cooling of a Thin Layer 10.4.5 Shocks in Optically Thin Material 10.4.6 Optically Thick Shocks in the Flux Regime 10.4.7 Shocks in Optically Thin Upstream Material 10.4.8 Radiation-Dominated Shock Waves 10.5 Shock Structure 10.5.1 Subcritical 10.5.2 Supercritical 10.6 Ionization Fronts

540 542 543 545 549 552 553 555 557 560 564 567 569

11 Magnetohydrodynamics 11.1 Plasma Electrodynamics 11.2 Equations of Magnetohydrodynamics 11.2.1 Momentum Equation 11.2.2 Induction Equation 11.2.3 Electron Thermal Equation 11.2.4 Electron Degeneracy 11.2.5 Ion Thermal Equation 11.2.6 Bohm Diffusion 11.2.7 High-Frequency Plasma Oscillations 11.2.8 Magnetic Energy 11.2.9 Generalized Ohm’s Law 11.2.10 Hall Effect 11.2.11 One-Dimensional Cylindrically Symmetric Equations 11.3 Magnetic Reconnection 11.3.1 Biermann Battery 11.3.2 Sweet-Parker Reconnection 11.3.3 Hall MHD Reconnection 11.3.4 Plasmoid Formation 11.4 Magnetic Confinement

576 578 580 580 582 589 591 593 594 595 596 597 598 599 600 602 604 607 608 611

12 Electromagnetic Wav-Material Interactions 12.1 Electromagnetic Wave Propagation in Homogeneous Medium 12.1.1 Interaction of Free Electrons with an Electromagnetic Wave 12.1.2 Longitudinal Waves and Spatial Dispersion

625 626 628 629

xii

Contents

12.2 Propagation in Inhomogeneous Isotropic Medium 12.2.1 Exact Solution in a Linear Density Gradient 12.2.2 Reflectivity and Phase Shift 12.2.3 Geometrical Optics Approximation 12.2.4 Weak Reflection 12.2.5 Oblique Incidence 12.2.6 Ponderomotive Force and Momentum Deposition 12.2.7 Ray-Trace “Equation of Motion” 12.3 Reflection at an Interface 12.4 Density Profile Modification 12.5 Absorption of Electromagnetic Energy 12.5.1 Collisional (Inverse Bremsstrahlung) 12.5.2 Nonlinear Inverse Bremsstrahlung 12.5.3 Resonance 12.6 Dielectric Permittivity (Revisited) 12.6.1 Plasma Conductivity 12.6.2 Near-Free Electron Metals 12.7 Kinetic Instabilities 12.7.1 Matching Conditions 12.7.2 Damping 12.7.3 Instability Threshold 12.7.4 Parametric Decay and Two-Stream Instabilities 12.7.5 Stimulated Raman Scattering 12.7.6 Two-Plasmon Decay 12.7.7 Stimulated Brillouin Scattering 12.7.8 Nonlinear Aspects 12.8 Magnetic Fields 12.8.1 Spontaneous Generation of Magnetic Fields 12.8.2 Faraday Rotation 12.9 Relativistic Considerations 12.9.1 Electromagnetic Wave Propagation 12.9.2 Interaction with Transparent Matter 12.9.3 Nonlinear Dynamics at the Vacuum Boundary

631 632 636 637 640 642 651 658 659 664 668 673 676 681 683 684 687 689 691 694 697 698 701 704 705 709 711 711 714 717 718 722 724

References Index

726 733

Preface

The material covered in this book is the outgrowth of the development of the radiation hydrodynamics simulation codes HYADES and h2d (h2d is the twodimensional version of HYADES) that I developed. HYADES began its life in 1988, and much as a toddler does, it struggled to stand on its own for a couple of years. During those early years, the code development did just fine, but the documentation consisted of handwritten notes on scraps of paper stored in a “filing cabinet” that was housed in one cardboard box. By the time HYADES was out of diapers, a more formal organization of documentation was developing. One portion of that documentation is the physics principles used in the codes. During its adolescent years, that document went through fits and starts. Some chapters were easily completed, while others remained little more than crude outlines, unshaped by any parental guidance. Occasionally, “mom” (my conscience or my users) would get on my case and spur me to clean up my room. Translation: another chapter got written, albeit far from polished. But as adulthood approached, there came a realization that some maturity needed to be developed. Thus, this book was initiated to take that earlier work and transform it into a thriving adult, something that might endure for one or two decades and be of use to society. It is my sincere hope that at least some of the objectives are met and that this book is of more use than for propping up the corner of one’s desk. My interest in high-energy-density physics began in early 1970, when I began work at the University of California’s Lawrence Radiation Laboratory (soon to change its name to the Lawrence Livermore National Laboratory [LLNL]). I was a part of the effort to investigate the feasibility of inertial fusion. Those were heady times, as the first thermonuclear fusion reactions were observed in small-scale laboratory experiments. The race was on to produce increasing numbers of thermonuclear neutrons (the neutron derby). But the euphoria soon wore off as the number of neutrons produced hit a limit. It was then into the trenches to understand the detailed physics of those early inertial fusion experiments through xiii

xiv

Preface

a vibrant experimental program augmented by theoretical and computational efforts. I became intrigued with the physics elements upon which radiation hydrodynamics codes are based and how those elements are encapsulated into a simulation code. My role in the fusion program was using the computational resources to develop an understanding of the experimental results. For the better part of the 1970s and 1980s, there was one recurring issue: why isn’t there a “simple” radiation hydrodynamics simulation code that can be used by an experimentalist; a code that doesn’t require significant training and a full-time dedication to running the code to obtain meaningful results? Then the light came on: why not construct such a simulation code, but develop it from the point of view of the experimentalist? I made the decision to strike out on my own as an entrepreneur. And thus HYADES came into being. All of this would not have been possible without the absorption of knowledge from the experts on the subject. During my tenure at LLNL, my nearly daily interaction with John Nuckolls, George Zimmerman, John Lindl, Mordy Rosen, Bill Kruer, Kent Estabrook, Bruce Langdon, Dick Lee, Ray Kidder (occasionally), and many others provided me with insight to the principles of high-energy-density physics. I moved to Ann Arbor, Michigan, in the early 1980s to continue the research at KMS Fusion, Inc. There I met a number of outstanding scientists, including Phil Campbell, Roy Johnson, Fred Mayer, Dick Berger, Linda Powers, Doug Drake, and Walter Fechner. As I traveled further down the independent path, I had the chance to interact with many others, including Steve Rose, Paul Drake, John Castor, Dimitri Mihalas, Gerry Pomraning, Justin Wark, Nels Hoffmann, Alexander Rubenchik, and Mike Feit, to name just a few. I am especially indebted to Bruce Remington (LLNL) for his encouragement to take on the solo effort and for his continued support over these many years. I am grateful to the editors at Cambridge University Press, including Róisín Munnelly and Simon Capelin, who saw the potential of this book. I am indebted to my friend and editor Amy Collette Casey, who transformed a ragged manuscript into a readable treatise. And of course I extend my deepest gratitude to my wife and children for their patience during my “absence”; I am sure they often wondered about my sanity in taking on this endeavor.

1 Introduction

Our terrestrial environment is, for the most part, a benign environment. Even catastrophic natural events such as severe hurricanes, explosive volcanic eruptions, and massive earthquakes are rather tame compared to physical events beyond our rocky planet. The stars and the galaxies have environments far too hostile for our liking. Even when nature unleashes a violent thunderstorm, the energy may be quite large, but the energy density is not. However, if Thor hurls a thunderbolt our way, we take notice of the explosive release of energy in a small volume and we see firsthand evidence of high-energy density. Generally, we are not aware of everyday instances of high-energy density that are all around us. That is, we don’t realize that the energy that holds matter together involves high-energy-density physics. Early man undoubtedly experienced the violent and often destructive nature of a lightning strike, but had no comprehension of its energy density and plasma physics on display. Not until near the end of the nineteenth century did people realize that larger energy densities existed in nature beyond the uncontrollable lightning events. Early experiments on the structure of atoms suggested far greater energies were responsible for keeping atoms whole. The binding of the electron to the nucleus of the hydrogen atom is 13.6 electronvolts (eV), with even greater binding for higher atomic number elements. In uranium, for example, the outermost electron is bound with an energy of 5.8 eV, while the removal of the last electron requires 132 kilo-electronvolts (keV) to form the bare nucleus. Even greater energies, due to the nuclear force, were later found when nuclear structure experiments were performed; for example, the energy released in the fission of one uranium nucleus is about 180 mega-electronvolts (MeV). The structure of our surroundings is dictated by Coulomb forces between individual atoms, which in the case of a carbon–carbon molecular bond is about 4 eV. Larger assemblies of carbon atoms are held together by even weaker forces. For example, in graphite, adjacent planes of carbon atoms are bound together by 1

2

Introduction

bonds of about 0.05 eV per carbon atom. For reference, this energy corresponds to a temperature of about 600 Kelvin (K), about twice that of room temperature. By comparison, iron has a melting point of about 1,800 K.1 Life on Earth is dependent on solar radiation. While the surface (photosphere) of the Sun has a temperature of roughly 5,800 K, the major portion of the radiant energy reaching the surface of the Earth has photon energies in the range 0.06–0.4 eV (wavelengths between 300 and 2,000 nanometers [nm]). It is this matching of photon energy with molecular bonding energies that allows formation of new molecules, and thus the rise of biological structures; that is, the creation of life itself. The wide distribution of binding/bonding energies is better cast in terms of energy density, which is just another expression for pressure. For example, the covalent bond energy of the hydrogen molecule is a bit less than 5 eV, which corresponds to an internal energy density of about 1011 Pascals (Pa) or 1 megabar (Mbar). We would expect that if an assembly of hydrogen molecules is subjected to an external pressure in excess of this internal energy density, the structure of the molecules would be altered. In fact, at these elevated conditions, hydrogen (as well as for all matter) exhibits a new property: the neutrally charged atom becomes ionized. This more energetic state is often referred to as the fourth state of matter, the plasma. Indeed, metals that exhibit a high electrical conductivity have conduction electrons that are loosely bound to individual atoms, and so tend to wander through the lattice. These metals, while solid, exhibit many of the characteristics of plasma. We are surrounded by plasmas. For example, the radiation from the Sun, which warms us, is produced by plasma; the “neon” advertising signs light up storefronts; the welder’s torch assembles useful structures; light from fluorescent electrical fixtures lets us see in the dark; and so on. Traditional plasma theory arose from experiments on low-density, high-temperature gases, where ionization had transformed the neutral atoms into negatively charged electrons and positively charged ions. The study of this new form of matter focused on the “individual” behavior of the constituent particles. But for higherdensity plasma, this approach is nearly impossible because of the extreme complexity of the dynamics of ions and electrons in close proximity to one another. It is far more advantageous to treat the plasma as two fluids: one for the free electrons and one for the heavy ions. This is the basic approach for describing high-energydensity matter. Consider air at its normal density, about 10–3 grams per cubic centimeter (g-cm‒3), at a pressure of 1 Mbar. At this pressure, the temperature is about 10 keV (~108 K). 1

This reference is a bit careless about equating energy to a temperature. The examples are basically “singleparticle” energies, while temperature implies averaging energies over some distribution function.

1.1 High-Energy-Density on Earth

3

As the density is further decreased, but still has a pressure of 1 Mbar, the temperature becomes hot enough that relativistic plasma is established. This regime, too, is beyond the realm of traditional plasma physics. Dense plasma also exhibits characteristics of condensed matter. Because of the close proximity of ions, they are highly correlated and form structures not unlike that encountered in the solid state. Thus, high-energy-density physics concepts also draw on the wealth of theory developed in condensed matter physics. While highenergy-density matter is loosely defined as matter with energy density greater than 1 Mbar, recent research has addressed cooler, but still dense, conditions, known as warm dense matter. This book does not explicitly address the physics specific to those environmental conditions, but much of what is discussed is directly applicable to that topic.

1.1 High-Energy-Density on Earth Perhaps the earliest demonstration of high-energy-density phenomenon in the laboratory was conducted by Martinus van Marum in Amsterdam in 1790. He employed a 1 kilojoule (kJ) capacitive energy store composed of one hundred Leyden jars, which was discharged into a wire 1 meter (m) long, causing its explosion and vaporization. A later, independent discovery in Australia (Pollock and Barraclough, 1905) arose from the radial collapse of a copper tube used at the Hartley Vale kerosene refinery as a lightning conductor; the outcome of the experiment is shown in Figure 1.1. The researchers correctly interpreted the results as being due to the force arising from the lightning’s current interacting with its own magnetic field. By the mid-twentieth century, the power of nuclear weapons had been demonstrated. Based on the exploitation of the energy released in the fission of uranium, the uncontrolled release of energy created energy densities in excess of 104 Mbar. Scientists used underground nuclear explosions to explore the physics of matter at extreme conditions. They quickly realized that if this energy source could be controlled, it offered a laboratory source for continuing experiments on the nature of materials at high-energy densities. Thus began the exploration and development

Figure 1.1. A copper tube suffering the effects of a lightning strike (Credit: Brian James, School of Physics, University of Sydney).

4

Introduction

of magnetic fusion energy (MFE) as a possible source of energy production. In the later decades of the twentieth century, technological developments saw the creation of pulsed electrical devices and high-power lasers. These two types of machines made possible the laboratory study of matter at extreme conditions. Even with the construction of large Z-pinch machines and multiterrawatt, shortwavelength optical lasers, it wasn’t until well into the 1980s that the instrumentation achieved the spatial and temporal resolutions needed for meaningful experiments to be performed. With flexible, high-power sources and sophisticated diagnostics available in the laboratory, the ability to address fundamental physics questions about the structure and behavior of matter at extreme conditions became a reality. Not only was the field of fusion research advanced, but important experiments relevant to questions of interest in astrophysics and planetary science caught the attention of researchers worldwide. 1.2 Some Connections to Prior Work Since the mid-1960s, the “bible” of high-energy-density physics has been the twovolume set by Ya. Zel’dovich and Yu. Raizer (1966), Physics of Shock Waves and High-Temperature Hydrodynamic Phenomena. Their work was complemented by an exquisite treatise on radiation hydrodynamics by D. Mihalas and B. Mihalas (1984), Foundations of Radiation Hydrodynamics. These two books are augmented by many physics texts on specific topics, including statistical mechanics, electrodynamics, hydrodynamics, equation of state, plasma physics, atomic physics and spectroscopy, and radiative transfer, to name but a few. It is important to note that many of the topics we will address were first explored by astronomers and astrophysicists in their attempts to explain the structure of the stars and the formation of the galaxies. Much of this work was pioneered by S. Chandraskehar and other preeminent physicists of the twentieth century. In the late 1990s and the first years of the twenty-first century, numerous books appeared addressing specific aspects of high-energy-density physics. The first comprehensive work is the textbook by R. P. Drake (2006), appropriately named High-Energy-Density Physics, and more recently, my book with J. Colvin, Extreme Physics (Colvin & Larsen, 2014). As I proceed into the depths of highenergy-density physics in this book, I will note numerous references to prior work to consult for additional information. There is always a dilemma in preparing a comprehensive work: what to include, what to develop in some detail, what to hint at, and what to ignore. Since this work is the outgrowth of the development of the radiation hydrodynamics simulation code HYADES, I have included those subjects of most interest needed to achieve

1.3 Outline

5

that goal. There remain a number of topics that should have been included, but page limitations prevailed. Preparation of the manuscript for this book relied heavily on handwritten notes dating back to the 1980s, many of them rather sketchy. Unfortunately, some attributions have been lost over time and have not been readily relocated; I take full responsibility for this deficiency. 1.3 Outline High-energy-density physics draws on many physics disciplines, from classical, to quantum, to relativistic topics. Central to high-energy-density physics is the bulk motion of matter and its associated energy content and transport, especially that of radiative transfer. As high-energy-density matter is ionized plasma, the free electrons and ionized atoms (the ions) are given equal importance. As we shall see, thermal radiation is an essential ingredient of high-energy-density matter. This requires that the radiation be treated as a third “fluid,” on an equal footing with the electron and ion fluids, taking account of the mass difference between the photons and material particles. There is one additional aspect of high-energy-density physics that is not found in traditional plasma physics, and that is electron degeneracy. This condition arises when plasma is at high density and low temperature. Degenerate electrons have a profound effect on nearly all aspects of high-energy-density matter. There are aspects of the physics that should be treated in addition to those discussed here. The list includes gravity, relativistic effects, etc., but they are of little or no concern in the terrestrial laboratory. Further, I omit any discussion of atomic spectroscopy, fusion applications, and diagnostic instruments and nearly all mention of experimental results. The objective of this book is to provide a comprehensive collection of the most important physical phenomena encountered in high-energy-density matter. Chapter 2 is a brief summary of the topics that define high-energy-density matter and possible ways to characterize it from matter in other states. Chapter 3 is a collection of many fundamental physics concepts that are often covered in undergraduate and first-year graduate physics courses. I refer back to this chapter many times throughout the text. The reader well versed in basic physics can skip this chapter; I include it for reference for those individuals who are a bit “rusty.” The real substance of this book begins with Chapter 4. A most important quantity in high-energy-density plasma is the ionization level, that is, the average number of unbound electrons per ion. In traditional plasma, this is determined by binary collisions among the particles, which are controlled by the temperature. In high-energy-density matter, the close proximity of one atom to another causes a

6

Introduction

change in the overlap of electron orbitals. The outermost electrons become pressure ionized even for very low temperatures. Another important ionization mechanism is that due to the radiation field – photoionization. Higher-energy photons can interact with the deeper-lying atomic levels, causing those electrons to be removed from the atom. In fact, the combination of processes leads to timedependent, nonequilibrium ionization. Chapter 5 discusses the properties of matter at extreme conditions as characterized by the equation of state (EOS). We are familiar with the EOS for a perfect gas, but this simple model is of limited use in high-energy-density physics because it is an “isolated atom” model. The problem of calculating the internal states of matter (pressure and specific energy) as functions of environmental variables (density and temperature) is exceedingly difficult for strongly coupled plasma. Realistic models need to account for more than just “closest neighbor” atoms; there are several computational approaches that yield sensible results. The basis for some of this work comes from solid-state theory. Chapter 6 develops the equations of material response. Based on the conservation equations of mass, momentum, and energy, and using an EOS for a particular material, the basic model of gas dynamics is developed. Because we are dealing with high-energy-density matter, shock waves are likely to form. Their behavior is easily found from the hydrodynamic equations. A better approximation includes the effects of viscosity and thermal conductivity, which leads to a picture of the structure within the shock front. A further improvement for modeling the response of “solid” materials is based on an elastic-plastic description of deformation. Finally, we briefly address the complex topic of fluid instabilities. This chapter does not include the effects of the radiation field; that topic is left to Chapter 10. Chapter 7 discusses the heat flow in high-energy-density matter. For the most part, this conductive transfer of energy is due to the electron fluid. After considering the basics of nonlinear thermal conductivity, I address the thermal and electrical transport coefficients that are determined by density and temperature gradients. Electron degeneracy effects play a significant role in determining the rate of heat flow. The issue is complicated even further by the existence of steep temperature gradients, which result in a nonequilibrium electron velocity distribution, leading to thermal energy transport inhibition. Chapter 8 is devoted to the development of radiation transport theory. Because photons can have a wide range of mean free paths and are moving at the speed of light, the theory is complex, and we must begin from the basic concepts, then turn to the fundamental issue of how radiation interacts with the electrons by their absorption, emission, and scattering. The full set of radiation transport equations is “impossible” to solve, even with advanced computational resources, so I reduce

1.3 Outline

7

the set to several approximations, each of which is relevant under certain conditions. I also find that even in the most basic conditions, certain aspects of the transport theory must include relativistic effects, and that there is a preferred frame of reference for calculating radiation transport. Lastly, I address the topic of view factors. Chapter 9 focuses on atomic theory and the calculation of the radiative absorption, emission, and scattering coefficients. My approach is to treat the electron quantum mechanically, in the hydrogenic approximation, but the radiation classically. Rate coefficients are also developed for collisional processes that necessarily include electron degeneracy effects. Chapter 10 begins the discussion of combining the basic hydrodynamic theory with radiation transport theory. This makes for some very interesting situations, and I can only lightly touch the subject; several excellent books on this topic have been published in recent years. I consider a number of situations that might be encountered in the laboratory or in the astrophysical environment, developing the essence of each. Chapter 11 examines the magnetic fields that are known to play a prominent role in some astrophysical phenomena. They also occur in the laboratory setting, but usually are not given a position of importance. Magnetic fields affect plasma in nearly all aspects, from altering the hydrodynamic response, to modifying the thermal conductivity and associated transport coefficients, and to creating an environment where collisions between magnetic field geometries can cause energetic mass ejections, which, for example, are readily observed in the solar environment. An additional equation describing the diffusion of magnetic fields in plasma is introduced. Further, magnetic fields can be produced by pulsed-power devices to create high-energy-density plasma; the most common configuration is that of the Z-pinch. I discuss some simple models of how the Z-pinch forms as a result of the magnetic pressure created by high-current flow. The last chapter, Chapter 12, focuses on the interaction of electromagnetic waves with plasma. This topic is the outgrowth of the investigation of atmospheric propagation and heating (absorption) of the ionosphere in the mid-twentieth century. The most important aspect of this topic is oblique incidence of the wave on a stratified planar medium. If the polarization is such that the electric vector lies in the plane of incidence, a number of phenomena can occur at the so-called critical-density point. These include profile modification due to the ponderomotive force and formation of plasma instabilities that can create very energetic electrons. I also consider the spontaneous creation of magnetic fields and the effects of magnetic fields on changing the identity of the electromagnetic wave. I conclude with a brief discussion about very high-intensity electromagnetic waves, which produce relativistic effects.

8

Introduction

1.4 Notation, Variables, and Units As the material presented in this book is drawn from a number of disciplines and different authors, using the notation from those sources results in a less than uniform description. I have made every attempt to keep the notation consistent within this book, yet retain some of the original description. Consequently, the reader will find more than one mathematical symbol is used to describe the same quantity; for example, the radiation energy density is designated both by ur and E r . And of course, a specific symbol is used for multiple quantities, such as σ for a cross section, for the electrical conductivity, and for the Stefan-Boltzmann constant. I make clear the particular use as the book progresses. The mathematical expression of the physics, for the most part, is written as vector differential equations. I assume the reader is familiar with the mathematical symbols used, such as the differential operators: r, r, and r. Variables appear as scalars, such as ω for frequency; vectors, such as v for the velocity (with the scalar representation v); and tensors, such as P for pressure. I introduce four vectors in Chapter 8, where it is natural to discuss certain aspects of radiation transport with this formalism. For the most part, I work in a Cartesian coordinate system. Occasionally I find it appropriate to use some other system, such as spherical. I prefer (and use) the Gaussian cgs (centimeter-gram-second) system of units, with temperatures expressed in keV (thousands of electronvolts), and occasionally eV. At times it is more convenient to use other units common to a specific instance, especially when noting values for real quantities; for example, for the “modest” temperatures of the outer layers of the Sun, I use degrees Kelvin. Another measure employed for measuring pressure and energy density is the megabar (Mbar), which is commonly used in the industry, but is not consistent with the cgs system. And then there is laser intensity, for which the practitioners use the measure watts-cm‒2, which is really a flux (the true use of “intensity” implies a solid angle). Also, they measure the laser wavelength in micrometers.

2 Characteristics of High-Energy-Density Matter

The early part of the twentieth century saw the beginnings of plasma physics with research on gaseous discharges done by Langmuir, Penning, and others. These lowdensity discharges produced plasma with a great number of particles per Debye sphere and thus define the term “ideal plasma.” Ideal plasma is characterized as being quasineutral, spatially uniform, and having equilibrium (Maxwell-Boltzmann) particle distributions. For the most part, these “early” plasmas were treated as ionized hydrogen gas, even though other (higher atomic number) gases were studied. As research progressed, many phenomena were observed that required an expansion of the basic definition of plasma. Most notable were the discovery of instabilities that could only be explained by nonequilibrium particle distributions and fluctuations in the distributions. The complexity of the ideal plasma now required additional descriptive terms to characterize these energetic gases adequately. In particular, the role of unbound and weakly bound charged particles provides one way to distinguish this state of matter from those of a solid, liquid, or gas. In ideal plasma, the presence of a distribution of ion stages is treated as a perturbation of the thermodynamic properties. These “traditional” plasmas are analyzed assuming that individual particles interact in pairs and the interaction energies are small compared to thermal energies. In contrast to the relatively benign ideal plasma, high-energy-density plasma is much more than just an extension of the basic ideas of ideal plasma physics. One defining quantity of high-energy-density physics is that now there are (quite likely) very few particles in a Debye sphere, and for materials other than hydrogen, the matter will be undergoing ionization, thus forming ions with different electronic configurations, even though they are of the same atomic number. Indeed, high-energy-density materials rarely consist of hydrogen. But that is not to say that hydrogen is unimportant, because experiments involving hydrogen can lead to improved theoretical models, which become the basis for high-energy-density physics. 9

10

Characteristics of High-Energy-Density Matter

The first theories about high-energy-density physics originated with investigations about the structure of stars. Eddington, Chandrasekhar, Schwarzschild, and others developed theories about strongly compressed, ionized matter within which the transport of radiant energy was essential to explain astrophysical observations. In turn, these theories were essential to the development of nuclear weapons, which became the foundation for modern high-energy-density investigations. Formation of a galaxy involves the gravitational interaction among the billons of stars, while high-energy-density matter relies on the Coulomb interaction between the charged particles. There is a close similarity between a gas of charged particles interacting by electrostatic forces and galactic dynamics where the gas of particles is replaced by a “gas of stars” interacting by gravitational forces. Both forces have potentials that are proportional to the inverse of the distance between their constituent particles. 2.1 Landscape of High-Energy-Density Matter As noted in the previous chapter, the simplistic definition of high-energy-density matter having a minimum pressure of 1 Mbar greatly limits the discussion of important physical regimes. Figure 2.1 sketches an expansive density-temperature space in which several regions of conditions are indicated. The 1 Mbar pressure contour crosses this space and does not really separate high-energy-density matter from ideal plasma or condensed matter regions. The region of the figure for temperatures greater than 511 keV is the realm of relativistic plasma, which we pay little attention to because it is much more complex than we have space to deal with in this book. Of greater interest is the lower portion, but largely excluding the lower-left corner where “normal” matter resides. For low density, ionization begins when the temperature is about 1 eV. The line separating ionized and un-ionized gas increases in temperature as the density increases, reflecting the increasing density of electrons that can recombine with the ions. As the density approaches 1 g-cm‒3, hydrogen is completely ionized for any temperature; this occurs because of pressure ionization/continuum lowering. Excluding the region marked “warm dense matter,” the remainder of the space can be roughly divided into “ideal” plasma and high-energy-density plasma. A convenient division for the two is the line for the coupling parameter Γ ¼ 1, even though the traditional plasma has Γ  1. The most important characteristic of high-energy-density plasma is that it has less than about one particle per Debye sphere. The second line, labeled Ψ ¼ 0, defines the boundary where the electrons are no longer considered “classical” but exhibit Fermi degeneracy. The heavy solid line, the 1 Mbar pressure contour, is vertical at low temperature because matter is Fermi degenerate there, while at high temperature the pressure is

11

2.2 Compressing Atoms Ion density (cm–3) 1020

1021

1022

1023

1024

1025

1026

1028

1027

102

1

Temperature (keV)

101

1

Gb

Hot dense matter

ar

M

ba

y=

0

103 109 108

r

r-T track for sun

Pr = 1 Gbar 100

107

Pr = 1 Mbar

G=0

10–1 10–2 10–3

Ionized d Un-ionize

Warm dense matter

105

y ar et s n a e Pl cor

104 103

10–4 10–5 10–5

106

Temperature (keV)

103

1019

10–4

10–3

10–2

10–1

100

101

102

103

104

102 105

Density (g-cm–3)

Figure 2.1. Density-temperature space of interest to high-energy-density physics. While off the scale of the figure, supernova progenitors occupy the region where the density is greater than about 107 g-cm‒3 and temperatures are hundreds of keV. At the extreme low-density side, but with similar temperatures, is the realm of gamma-ray bursters.

linearly proportional to density and temperature as described by a perfect gas equation of state. For comparison, the other heavy dashed line indicates pressures of 1 gigabar (Gbar). These pressure lines omit the thermal radiation contributions. On the left-hand side are two dashed lines, one for 1 Mbar and the other for 1 Gbar radiation pressures. When sufficient thermal radiation is present, it too can exert a force on matter, and for high enough equilibrium temperatures, the matter becomes radiation-dominated. Upon moving to higher density, at some point the matter pressure exceeds the radiation pressure, and the two pressure lines descend with increasing density to join the Fermi degenerate pressure. We now explore in more depth some of the features just discussed.

2.2 Compressing Atoms The hallmark of high-energy-density plasma is that, whether solid or liquid, matter can be compressed to densities higher than that of their normal state. In doing so, adjacent atoms are forced together so that their electronic wave functions overlap.

12

Characteristics of High-Energy-Density Matter

R0 >>lD

R0 ª lD

R0 k B T. Thus, μ ¼ 0 is a convenient dividing line between classical and degenerate behavior. As illustrated in Figure 3.2b, at zero temperature the distribution function is a step function; for E < μ (i.e., ðE  μÞ=kB T ¼ ∞), it is one, and if E > μ (i.e., ðE  μÞ=kB T ¼ þ∞), it is zero. Here, some of the states become empty with those electrons moving into the higher-energy tail. The energy of the state with 50 percent occupancy (N i =gi ¼ 1=2) is just the Fermi energy. For the gas at nonzero temperature, but for kB T EF , μ
EF , and thus empty states are available only above (or within ∽kB T) of the Fermi energy. Ignoring the variation in the density of states, we may assume gðE Þ
gðE F Þ. Together with the observation that the function f FD ðE Þ is symmetric about E F , the number of electrons being raised to energy ε is gðEF Þf FD ðE ÞdE. Each electron has increased its energy by a factor of 2E. Using (3.99), the increase in the internal energy (per unit volume) is ð∞ ð∞ u  u0 ¼ 2gðE F Þ Ef FD ðE ÞdE
2gðE F Þ ¼

2π 3

3




0

2me h2

3=2

E eE=kB T

þ1

dE

0 1= ðkB T Þ2 E F2 :

(3.105)

Relaxing the condition k B T E F , the electron number density for nonzero temperature is now written
 ð∞ 4π p2 4 1 μ p ffiffiffi dp ¼ ne ¼ 2 3 ðEμÞ=k T F 1= , B π λ3de 2 k B T þ1 h e

(3.106)

0

where the function F j ðxÞ is a Fermi-Dirac integral (Cody & Thatcher, 1967) ð∞ F j ðxÞ ¼

tj dt: exp ðt  xÞ þ 1

(3.107)

0

The density of states per unit volume is thus gðpÞ ¼

8π p2 : 3 eðEμÞ=kB T þ 1 h

(3.108)

Figure 3.4 shows the distribution of particles as a function of energy for nonzero temperature. We see that particles within energy kB T of the Fermi energy are excited to higher energy levels; these particles have an energy gain of 3kB T=2.

55

3.4 The Fermi Gas

Figure 3.4. Electron number density as a function of energy, measured in units of E F , for different temperatures: 0:01T F (solid line), 0:1T F (short-dashed line), and 0:5T F (long-dashed line).

For small degeneracy (large, negative μ=kB T), F j
e μ=kB T Γð j þ 1Þ, where Γ is pffiffi the Gamma function. Specifically, F 1=2
2π e μ=kB T , and (3.106) becomes ne ¼

2 μ=kB T e : λ3de

(3.109)

The chemical potential as computed by eliminating the electron density from (3.98) and (3.106) is shown in Figure 3.5. For small degeneracy, the chemical potential can be found from (3.109)
3 μ ne λde ¼ log : (3.110) kBT 2 Comparing this expression to (3.53) for Maxwell-Boltzmann particles, we see they are the same to within a constant, apart from the masses. For an ideal Fermi gas, for which kB T EF , we can show with some labor that the chemical potential may be approximated by " #

 π2 T 2 π4 T 4 μ
kBT F 1   þ  : (3.111) 12 T F 80 T F

56

Fundamental Microphysics of Ionized Gases 102 Degenerate

Classical

lµ/kBTel

101

100

10–1 10–2

10–1

100

101

102

103

Te /TF

Figure 3.5. Chemical potential, in units of kB T e , as a function of temperature, in units of T F . The degenerate case is shown by the solid line, and the classical (nondegenerate) case by the short-dashed line, whose values are negative. Expression (3.110) extended into the degenerate region is shown by the long-dashed line whose values are positive. 1 In the limit of large degeneracy (large, positive μ=kB T), F j
jþ1 3  =2 F 1=2
23 kBμT and the electron density is


3= 8π 2me 2 3=2 ne ¼ μ , 3 h2



μ kB T

jþ1 ; thus

(3.112)

which is independent of temperature. Finally, we note that for the conditions of high-energy-density plasma of interest to us, the ions are never degenerate. 3.4.2 The Grand Canonical Ensemble We worked with the canonical ensemble to develop the partition function for MaxwellBoltzmann particles. The canonical ensemble allows the subsystem to exchange energy, but not particles, with the remainder of the system. For particles with spin, it is inconvenient to impose the restriction that the number of particles in the subsystem be held constant. In situations where the fluctuation in the number of particles is small,

3.4 The Fermi Gas

57

we might just as well let the number fluctuate and speak only of the mean number of particles. The grand canonical ensemble is well suited to this discussion. Proceeding in a fashion similar to that for the Maxwell-Boltzmann distribution, we find that the chemical potential acts like a partition function for the electrons. Without laboring through the details, we summarize the results. (The reader should consult the texts on statistical mechanics mentioned at the opening of this chapter.) The probability density for Maxwell-Boltzmann particles is given by the canonical ensemble ρðEÞ ¼ AeE=kB T , which is the Boltzmann factor. For the grand canonical ensemble, we have ρðℰÞ ¼ eðΩþμNℰÞ=kB T ,

(3.113)

where Ω is the grand potential and the classical grand partition function is ~ ¼ eΩ=kB T : Z

(3.114)

F ¼ ℰ  TS ¼ Ω þ μN ,

(3.115)

The Helmholtz free energy is

where N is the mean number of particles. Then from (3.115), dΩ ¼ dℰ  TdS  SdT  μdN  N dμ ¼ PdV  SdT  N dμ,

(3.116)

and thus




 ∂Ω ∂Ω ∂Ω , S¼ , and N ¼  : P¼ ∂V T , μ ∂T V , μ ∂μ V , T

For the classical perfect gas, we have for the grand partition function ! ∞ X V μN =k T μ=k T B ~ ~ ¼ Z e Z N ¼ exp e B 3 , λd N¼0

(3.117)

(3.118)

where Z~ N is given by (3.58). (The function eμ=kB T is also known as the fugacity.) We then find V ~ ¼ kB Te μ=kB T , Ω ¼ kB T log Z λ3d V N ¼ eμ=kB T 3 , λd
3 N λd μ ¼ k B T log : (3.119) V So far, we have treated particles as being independent of one another. However, when we have electrically charged particles present, their collective behavior becomes important.

58

Fundamental Microphysics of Ionized Gases

3.5 Debye Shielding and Quasineutrality A high-temperature gas is composed of an almost electrically neutral mixture of ions and electrons. (An ion is the nucleus of the atom, which may have some, or none, of its electrons in bound states.) It is almost neutral because departures from neutrality give rise to electric fields that tend to restore the neutrality. The shielding of the exposed charges is limited somewhat by the thermal motion of the particles. Competition between the electrostatic forces and thermal forces gives rise to a distance characteristic of the shielding; this is the Debye length. For plasmas composed of elements heavier than hydrogen, there may be more free electrons than ions, and neutrality may be characterized by the requirement eðZ ni  ne Þ ¼ 0, where Z is the number of free electrons per atom, and ne , ni are the number densities of the free electrons and ions, respectively. We can easily show this by means of Poisson’s equation r  D ¼ 4πρe , where the electric displacement vector is D and ρe is the charge density. Writing this in spherical coordinates with the constitutive relation D ¼ ϵE, ϵ being the permittivity or dielectric constant, we obtain an equation for the electrostatic potential Φ (recall E ¼ rΦ)
 1 d 4π 2 dΦ r (3.120) ¼  eðZ ni  ne Þ: 2 r dr dr ϵ Separately the ions and electrons are in thermal equilibrium and are assumed to have Maxwell-Boltzmann distributions characterized by the temperature T i and T e . Each Maxwellian distribution now has an added term to account for the electrostatic potential


1 2 f ðvÞ ¼ A exp  mv þ qΦ =kB T , (3.121) 2 where A is a normalizing term and q is the charge. Integrating over the velocity and using the boundary condition that nðΦ ! 0Þ ¼ n∞ gives a Boltzmann factor for each specie ni ¼ ni0 eZ



eΦ=k B T i

and ne ¼ ne0 eþeΦ=kB T e ,

(3.122)

where nj0 is the mean concentration of charges of species j. We assume there are sufficient particles of both types in the spherical shell of volume 4πr 2 Δr so that a statistical treatment is valid, and that the potential does not vary appreciably over Δr. That is, eðdΦ=dr ÞΔr ðkB T e ; k B T Þ=Z . Then we have setting ϵ ¼ 1,  d2 Φ 2 dΦ Z eΦ=k B T i þeΦ=kB T e þ n e  n e ¼ 4πe Z : i0 e0 dr 2 r dr

(3.123)

3.5 Debye Shielding and Quasineutrality

59

Assuming a hot plasma, so that jqΦj=kB T 1, expanding the exponentials, keeping only the linear terms and using the neutrality requirement, results in " # d2 Φ 2 dΦ 4πe2 ðZ Þ2 ni0 ne0 ¼ Φ  4πeðZ ni0  ne0 Þ: þ þ (3.124) dr 2 r dr kB Ti Te The last term is zero, since the system is assumed electrically neutral. Defining the Debye length λD to be " # 1 e2 ðZ Þ2 ni ne , (3.125) ¼ 4π þ kB Ti Te λ2D and using the boundary condition Φ ! 0 as r ! ∞, the solution to Poisson’s equation yields the radially symmetric potential Z e r=λD e : (3.126) r This equation has the same form as the Yukawa potential encountered in nuclear physics. Figure 3.6 exhibits the radial dependence of the electrostatic potential. In the presence of plasma, the potential falls off more rapidly than 1=r. Owing to Φð r Þ ¼

Figure 3.6. Radial dependence of the Debye-sphere electrostatic potential is shown by the solid line. For comparison, the potential from a point charge is shown by the dashed line.

60

Fundamental Microphysics of Ionized Gases

the exponential factor, the potential drops by 1=e at the shielding distance λD ; alternately, the field of a static point charge Z e is screened beyond a distance of order λD . Whenever local concentrations of charge arise or external potentials are introduced in the system, these are shielded out in a distance that is short compared to λD , leaving the bulk of the plasma free of large electric potentials or fields. Outside of the sheath, r2 Φ is very small and Z ni ¼ ne to a factor of less than one part in 106; it takes only a small charge imbalance to give rise to potentials of order kB T=e. An example for hydrogen plasma with ni ¼ 1021 cm‒3 gives rise to an electric force of 160 V-cm‒1 when its neutrality is violated by 1 percent over a distance of 1 micrometer. The conclusion that no potential differences can be maintained is not justified, because temperature fluctuations oppose the restoring force. Another way to describe screening is to say that charge-density fluctuations on a scale size greater than or equal to λD are suppressed in comparison to the fluctuations that would exist in a neutral gas of the same density and temperature. Thus, plasma provides a very effective shielding mechanism against electric fields. For very short time scales, the inertia of the ions prevents them from moving 2 and we have the electron Debye length λ2 De ¼ 4πe ne =k B T e . For plasma with Z 1, ion screening is dominant and λD
λDi λDe . This length is the smallest natural scale in plasma.4 Electron degeneracy effects may be included in (3.125) via the Fermi temperature T F 0 1 λ2 D ¼

4πe2 B 1 Z C ne @qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ A: kB T2 þ T2 Ti e

4

(3.127)

F

For real plasma, the ion Debye length needs to be modified to include the distribution of charges, such as described by the Saha ionization model. Then

λDi ¼


2X 1=2 4πe nj Z 2j , kB T

where the sum is over the ion states, with nj and Z j being the number density of ions and the charge of the jth state. Often, one defines the Debye length in terms of the effective charge P 2 P 2 nj Z j nj Z j ¼ , Z eff P ne nj Z j and thus

λD ¼

 2 1=2 4πe2 ne 1 þ Z eff kB T

We note that Z eff is the appropriate average charge for calculations involving binary collisions, while Z ¼ ne =ni is used for particle counting.

61

3.6 Fluid Conservation Equations 101

20.0 10.0

0.1 0.05

100 5.0

Temperature (keV)

0.02 2.0 1.0 0.5

0.01

10–1 0.2

10–2

10–3 1019

1020

1021 1022 Ion density (cm–3)

1023

1024

Figure 3.7. The number of particles in a Debye sphere depends upon the density, temperature, and ionization level of the plasma. Contour values are shown for aluminum.

Our derivation for the potential distribution is based on the assumption that there are many particles within the Debye sphere ND ¼

4π ðni þ ne Þλ3D : 3

(3.128)

Plasma is considered “ideal” if the number of particles in the Debye sphere approaches infinity. However, this is generally not the case in high-energy-density plasma, as shown in Figure 3.7. In the portion of the figure where the density is low and the temperature large, there may be a few tens of particles in the Debye sphere. In contrast, for a low-temperature, dense plasma, there may be only a small fractional number of particles present. This is the region where electron degeneracy further reduces the ability of the plasma to shield the ions. Obviously, high-energydensity plasma is never an “ideal” plasma. Even though the Debye length has little meaning for high-energy-density plasma, it is still a useful quantity.

3.6 Fluid Conservation Equations In the introduction to this chapter, the argument was made that high-energy-density matter can (and should) be thought of as a fluid. Whether the matter is considered

62

Fundamental Microphysics of Ionized Gases

as a single fluid, two fluids, or even three fluids, the fluid approximation is centrally important. We develop the conservation equations of gas (fluid) dynamics, which arise from the Boltzmann equation (3.11); we consider a nonneutral gas of a single species, such as the electron fluid. The equation of conservation of mass is the zero-th moment obtained by integrating (3.11) over velocity. For the present, we take the external force to be the Lorentz force
 1 F¼q Eþ vB : (3.129) c Then ð

 ð
 ð ð
∂f q 1 ∂f dv: dv þ v  rfdv þ E þ v  B  rv fdv ¼ ∂t m c ∂t coll

The first term can be written as ð ð ∂f ∂ ∂n dv ¼ fdv ¼ , ∂t ∂t ∂t and the second term ð ð v  rfdv ¼ r  vfdv ¼ r  ðnhviÞ ¼ r  ðnuÞ,

(3.130)

(3.131)

(3.132)

since v is an independent variable and therefore is not affected by the gradient operator. By definition, the average velocity of the fluid is u. The third integral in (3.130) is evaluated in two parts: the first, with the electric field, vanishes since ð ð ð E  rv fdv ¼ rv  ðfE Þdv ¼ fE  dS ¼ 0: (3.133) S∞

The divergence is integrated to give the value of fE on the surface at v ¼ ∞. This vanishes if f ! 0 faster than v2 as v ! ∞, as is necessary for any distribution with finite energy. The v  B term in (3.130) may be written as ð ð ð (3.134) ðv  BÞ  rv fdv ¼ rv  ð f v  BÞdv  f rv  ðv  BÞdv ¼ 0: The first integral on the right-hand side of (3.134) may be converted to a surface integral, and if f falls faster than any power of v as v ! ∞, the integral vanishes; this is certainly true for a Maxwellian distribution. The second integral also vanishes because v  B is perpendicular to rv .

3.6 Fluid Conservation Equations

63

The collision term in (3.130) vanishes because collisions cannot change the total number of particles (recombination is not considered here). Using expressions (3.131) through (3.134) results in the equation of continuity ∂n þ r  ðnuÞ ¼ 0: ∂t

(3.135)

The first moment of the Boltzmann equation is obtained by multiplying (3.11) by mv and integrating over velocity. Then  ð ð
ð ∂f 1 m v dv þ m vðv  rÞfdv þ q v E þ v  B  rv fdv ∂t c ð
 ∂f dv: (3.136) ¼m v ∂t coll The right-hand side of this equation represents the change of momentum due to collisions. The first term in (3.136) gives ð ð ∂f ∂ ∂ m v dv ¼ m vf dv ¼ m ðn; uÞ: (3.137) ∂t ∂t ∂t The term in (3.136) with the Lorentz force may be written 


ð ð
1 1 v E þ v  B  rv fdv ¼ rv  f v E þ v  B dv c c
  ð ð
1 1  f vrv  E þ v  B dv  f E þ v  B  rv vdv: c c

(3.138)

The first two integrals on the right-hand side of this equation vanish for the same reasons as before, and rv v is just the identity tensor I. Thus,   ð
ð
1 1 q v E þ v  B  rv fdv ¼ q E þ v  B fdv c c
 (3.139) 1 ¼ qn E þ u  B : c Again, v is an independent variable not related to the gradient operator, so that the second integral in (3.136) becomes ð ð ð vðv  rÞfdv ¼ r  ð fvvÞdv ¼ r  f vvdv: (3.140) Since the average of a quantity is 1=n times its weighted integral over velocity, we have ð r  f vvdv ¼ r  nhvvi: (3.141)

64

Fundamental Microphysics of Ionized Gases

We may separate the velocity v into the fluid velocity and the thermal velocity (fluctuations) v ¼ u þ w. Since u is already an average, we have r  ðnhvviÞ ¼ r  ðnuuÞ þ r  ðnhwwiÞ þ 2r  ðnuhwiÞ,

(3.142)

but the average hwi is obviously zero. The second term on the right-hand side of (3.142) defines the stress tensor T ¼ mnhwwi,

(3.143)

and the first term in (3.142) becomes r  ðnuuÞ ¼ ur  ðnuÞ þ nðu  rÞu :

(3.144)

Collecting the pieces together (3.136) becomes
 ∂ 1 m ðnuÞ þ mur  ðnuÞ þ mnðu  rÞu þ r  T  qn E þ u  B ∂t c ð
 ∂f ¼m v dv, (3.145) ∂t coll then combining the first two terms using (3.135) yields the equation of motion

∂u mn þ ðu  rÞu ∂t
 ð
 1 ∂f ¼ qn E þ u  B  r  T þ m v dv: (3.146) c ∂t coll This equation describes the flow of momentum. The last term is the collision term, which allows for the coupling between different fluids. The second moment of the Boltzmann equation is obtained by multiplying (3.11) by mvv=2 and integrating over velocity:  ð ð ð
m ∂f m q 1 vv dv þ vvðv  rÞfdv þ vv E þ u  B  rv fdv 2 ∂tð
2 2 c m 2 ∂f (3.147) v ¼ dv: 2 ∂t coll Using algebra similar to that used in developing the momentum equation, the total energy equation is



∂ 1 2 1 2 m nhwwi þ nu þ r  m nhwwi þ nu u ∂t 2 2

ð
 1  2  m 2 ∂f (3.148) þr  mnhwwiu þ mn w w  qnu  E ¼ dv: v 2 2 ∂t coll

3.7 Electron Plasma Frequency and Plasma Waves

65

The quantity mnhwwi is just the thermal energy density. The next-to-last term, on the left-hand side, will be addressed in detail in Chapter 6. We see the first part of that term contains the stress tensor T. Observe that as we move from the continuity equation to the energy equation, each moment brings in the next-higher one. The continuity equation (3.135) involves the mean velocity. Then the momentum equation (3.146) brings in the pressure (stress tensor), and the total energy equation (3.148) involves the heat flow qnu  E. We could continue with even higher moments coupling all of the equations together, but we always have one more unknown than the number of equations. At some point, we must truncate the series of equations by making assumptions about the new term being introduced. If we make a simple assumption about the heat flow, we obtain the equation of state; the equation of state is a broad topic and will be addressed in Chapter 5. A particular simple assumption makes use of the isothermal equation of state P ¼ nk B T, where the temperature is taken as constant. This is appropriate when ω=k vth , where ω and k are the frequency and wave number characteristic of the physical process being considered, and vth is the thermal velocity of the particles. (We will expand our discussion about these quantities in the next section.) In the opposite limit, ω=k vth , we can neglect the heat flow; this leads to an adiabatic equation of state. The momentum conservation equation (3.146) explicitly references the mass and charge of the volume element under consideration. Plasma is composed of electrons and ions, and thus the response of the lighter electrons will be different from that of the heavier ions. For example, the electric and magnetic fields act differently for the two species. In addition, the coupling term, which we identify as collisions between the two fluids, affects each differently. At very low density and/ or high temperature, the coupling becomes small and the electrons act independently of the ions, but the electric and magnetic fields still have major effects. The electrons move faster than the ions, and thus transport energy better, while the heavier ions transport the momentum. Further, as we will see in the next section, the momentum equation may give rise to plasma oscillations. Therefore, we arrive at the conclusion that separate conservation equations are needed for the electron fluid and the ion fluid, but they are not independent. 3.7 Electron Plasma Frequency and Plasma Waves The two-fluid description of plasma has the characteristic features of being able to support waves or collective modes of interaction. There are a number of these modes of increasing complexity. We shall address two of the simplest modes. The first is a high-frequency wave known as an electron plasma wave and the second is

66

Fundamental Microphysics of Ionized Gases

a low-frequency ion-acoustic wave. These waves play central roles in plasma instabilities (Kruer, 1988), and in turn, instabilities play an important role in plasma physics. However, they are not central to our discussion of high-energydensity physics. Consider an environment where particle motion is coupled with an electromagnetic field. Particle motion gives rise to fields, and fields result in particle motion. For the moment, consider cold, uniform plasma absent any external fields. If a few electrons are displaced slightly from their equilibrium position, the resulting electrostatic field pulls electrons from the higher-density region back to their previous, now depleted, region. The electrostatic force would restore the original charge neutrality, were it not for the electron inertia, which keeps them moving in an oscillatory motion about the equilibrium position. For this simple idea, the ions with their much larger mass may be assumed to remain nearly stationary. Further, we assume that there is no magnetic field and no thermal motion; that the plasma is infinite in extent; and that the electrons move in one direction only, the direction of the electric field. The motion of the electrons is governed by the equation of continuity (3.135) ∂ne ∂ þ ðne ue Þ ¼ 0, ∂x ∂t

(3.149)

and momentum (3.146), ignoring the pressure gradient and collision terms
 ∂ ∂ e (3.150) þ ue ue ¼  ne E: ne ∂t ∂x me Taking the time derivative of (3.149) and the spatial derivative of (3.150) and eliminating the term ∂2 ðne ue Þ=∂t∂x, we obtain ∂2 ne ∂2  2  e ∂  2 ne ue  ðne E Þ ¼ 0: 2 me ∂x ∂t ∂x

(3.151)

Poisson’s equation relates the electric field to the charge density ∂E ¼ 4πeðne  Z ni0 Þ, ∂x

(3.152)

where Z is the average charge state of the ions and ni0 is the uniform background ion density. Assuming small-amplitude perturbations in density, velocity, and electric field, we linearize expressions (3.151) and (3.152). Let ne ¼ ne0 þ δne , ue ¼ ue0 þ δue and E ¼ E 0 þ δE. The equilibrium quantities, identified by subscript “zero,” express the state of the plasma in the absence of oscillations, thus ∂ne0 ∂ne0 ∂ue0 ∂E0 ¼ ue0 ¼ E0 ¼ 0, and ¼ ¼ ¼ 0: ∂x ∂t ∂t ∂t

(3.153)

3.7 Electron Plasma Frequency and Plasma Waves

67

Poisson’s equation (3.152) becomes ∂ðδEÞ ¼ 4πeðδne Þ, ∂x

(3.154)

∂2 ðδne Þ e ∂ðδEÞ ¼ ne0 : ∂t2 me ∂x

(3.155)

and (3.151) becomes

Combining (3.154) and (3.155) yields a wave equation describing the small-amplitude fluctuations in electron density
2  ∂ 2 þ ωpe ðδne Þ ¼ 0: (3.156) ∂t 2 The electron plasma frequency is
ωpe ¼

4πe2 ne0 me

1=2 (3.157)

with background electron density ne0 ¼ Z ni0 . In this first-order approximation, the density fluctuations are localized and do not propagate. This fact is expressed by the absence of spatial derivatives and also the vanishing of the group velocity vg ¼ jdω=dk j. There is an important connection between the electron plasma frequency and the electron Debye length. From (3.125), but for electrons only, we have
1=2
1= kB T kBT 2 1 vth λDe ¼ ¼
: (3.158) 2 4πne e me ωpe ωpe Thus the Debye length represents the distance traveled by a thermal electron during the oscillation period of one plasma wave. We now redo the preceding analysis, but include the electron pressure gradient term, so (3.150) is now written as ∂ ∂  2 e 1 ∂Pe ne ue ¼  ne E  , ðne ue Þ þ ∂t ∂x me me ∂x

(3.159)

with the electron pressure given by Pe ∽n3e , assuming an adiabatic equation of state under the assumption that the wave has a phase velocity vph ¼ ω=k vth . Again taking the time derivative of (3.149) and the spatial derivative of (3.159), and eliminating the ∂2 ðne ue Þ=∂t∂x term, we arrive at ∂2 ne ∂2  2  e ∂ 1 ∂2 Pe   n u ð E Þ  ¼ 0: n e e e me ∂x me ∂x2 ∂t 2 ∂x2

(3.160)

68

Fundamental Microphysics of Ionized Gases

Using Poisson’s equation (3.152) and linearizing the variables, with the additional fact Pe ¼ ne0 k B T e þ δPe , and δPe ¼ 3me v2th ðδne Þ, (3.160) becomes ∂2 ðδne Þ e ∂ðδE Þ 1 ∂2 ðδPe Þ   n ¼ 0: e0 ∂t2 me ∂x me ∂x2

(3.161)

Combining (3.161) and (3.154) gives a wave equation for the density fluctuations
2  2 ∂ 2 ∂ 2  3vth 2 þ ωpe ðδne Þ ¼ 0: (3.162) ∂t 2 ∂x We seek a solution to (3.162) of the form δne ∽eiðkxωtÞ from which we find the Bohm-Gross dispersion relation for electron plasma oscillations ω2 ¼ ω2pe þ 3k2 v2th :

(3.163)

The frequency is nearly that of the electron plasma frequency, plus a small thermal correction, which depends on the wave number. Because the phase velocity of these waves is very large, the result is restricted to small values of k. The group velocity, vg ¼

3kB T e k 3v2th , ¼ me ω vph

(3.164)

has more physical significance since it describes the velocity with which perturbations propagate; it is much smaller than the thermal velocity. Charge-density oscillations in plasma will also occur at a much lower frequency determined by ion inertia. One now considers the motion of both the electron and the ion fluids. However, we can neglect the inertia of the electrons since the frequency of these oscillations is much less than the characteristic frequency to which the electrons respond, ωpe . An analysis similar to the preceding produces a dispersion relation for ion-acoustic waves ωia ¼ kcs , where pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi cs ¼ ðZ k B T e þ 3k B T i Þ=mi is the ion sound speed. These low-frequency waves are the analogue of sound waves in an ordinary gas. The ions provide the inertia, and the fluctuations in pressure provide the restoring force. The electron pressure fluctuations are transmitted to the ions via the electrostatic field. If kinetic effects are included, there will also be a small damping or growth depending on the details of the electron distribution function for velocities near the phase velocity of the wave (Kruer, 1988). Electron plasma waves are not very important in dense plasma, in most cases. In ideal plasma, these waves play a role in plasma instabilities, which can occur in low-density, but high-temperature, regimes. Wave dispersion relations are complicated because of the strong influence of magnetic fields on single-particle motion,

3.8 Coulomb Collisions

69

and the wave amplitudes can grow much larger than thermal-equilibrium amplitudes. However, in dense plasma the collision rate is quite large and this tends to maintain wave populations at the thermal level. In addition, a high collision rate reduces the effects of magnetic fields. The preceding approach is the simplest approximation. The proper way to proceed in this analysis is to replace the equation of motion with the collisionless Boltzmann equation. However, the procedure is quite tedious and yields complex integrals, which necessitate messy contour integrals.

3.8 Coulomb Collisions The previous discussion regarding the Boltzmann equation (3.11) mentions that the right-hand side of the equation accounts for the effect of interacting particles. The simplest term is for binary collisions between particles; it is approximated by the Krook collision term
 ∂f f  f0 ¼ , (3.165) ∂t coll τc where f 0 is the symmetric part of the distribution function and τ c is a “deflection” time. Recalling classical mechanics, a simple binary collision is governed by the conservation of mass and energy. Consider two particles with masses m1 and m2 moving with initial velocities u1 and u2 prior to the collision, and u1 0 and u2 0 following the collision. Conservation of energy requires  1  1 m1 u21 þ m2 u22 ¼ m1 u1 02 þ m2 u2 02 : 2 2

(3.166)

~ ¼ m1 m2 =ðm1 þ m2 Þ is the If the total mass of the system is M ¼ m1 þ m2 and m reduced mass, then  1  1 ~ 2 , m1 u21 þ m2 u22 ¼ MV 2 þ mv 2 2

(3.167)

 1  1 ~ 02 : m1 u1 02 þ m2 u2 02 ¼ MV 2 þ mv 2 2

(3.168)

and

From (3.166) through (3.168), we see v ¼ v0 , which shows that the relative velocity of the two particles is changed only in direction but not in magnitude by the collision.

70

Fundamental Microphysics of Ionized Gases

Coulomb collisions are binary elastic collisions between two charged particles interacting through their mutual electric field. As with any inverse-square law, the resulting trajectories of the colliding particles are hyperbolic Kepler orbits. In plasma, where there is a high density of particles, the typical kinetic energy of the particles is too large to produce a significant deviation from the initial trajectories of the colliding particles, and the cumulative effect of many collisions must be considered. First we consider a single interaction between particles one and two. Take two particles having charges q1 and q2 interacting via the Coulomb force F ¼ q1 q2 =r2 , r being the separation distance of the two charges. (The force is repulsive when q1 and q2 have the same sign; otherwise, it is attractive.) This is an extremely long-range force compared to that between neutral particle collisions (hard-sphere collisions). A consequence of this long range is that while the force exerted on a test particle falls as r2 , the number of field particles in the range r to r þ dr rises as r2 , and therefore there will be roughly equal contributions to the collisional interaction from particles at all distances. In fact, the dominant effect comes from a multitude of weak collisions with particles at large distances.

3.8.1 The Scattering Angle Switching to the notation of Chandrasekhar (1942), let us assume that particle 1, referred to as the test particle, is so fast that during the encounter particle 2, the field particle, may be considered at rest. Alternatively, we can assume the mass of the field particle is much larger than that of the test particle, as would be the case with the test particle being an electron and the field particle an ion. Referring to Figure 3.8, the test particle has an initial velocity v0 such that the trajectory would carry it with a minimum separation distance b, the impact parameter, past the field particle. For fast incoming particles, the interaction time is short and the deflection is small. We may make simple estimates of the effects of the interaction using the unperturbed trajectory, that is, a straight line. However, let us consider the exact treatment of the trajectory. The Coulomb force (or any central force) may be decomposed into two equations for the test particle: one for the radial motion,   qt qf mt r€  rθ_ 2 ¼ 2 , r and the other for the angular motion,   mt rθ€ þ 2r_ θ_ ¼ 0:

(3.169)

(3.170)

71

3.8 Coulomb Collisions

Figure 3.8. Trajectory of a charged test particle passing a stationary charged particle with opposite charge is a hyperbola. The impact parameter is b; the scattering angle χ is given by (3.175).

Expression (3.170) is an expression of a central force having no angular force component. Integration of (3.170) gives an equation for the conservation of angular momentum L ¼ mt r 2 θ_ ¼ const ¼ mt v0 b:

(3.171)

If we define K ¼ qt qf , then for a given energy and angular momentum, the turning point (periapsis or apoapsis) for a repulsive force (K > 0) occurs at a larger value of r than if there is no force (K ¼ 0) for which the trajectory would be a straight line. For an attractive force (K < 0), with L ¼ 0 and positive total energy E (see 3.173, later in this section), the turning point occurs at a smaller value of r. The trajectories for the three cases are indicated in Figure 3.9. For K < 0, the center of force lies at the interior focus of the hyperbola, and for K > 0 it lies at the exterior focus. The nature of the orbits for an inverse-square force law is measured by the effective potential K L2 0 0 V ðr Þ ¼ þ : (3.172) r 2mt r2 Figure 3.10 shows this potential for K > 0, K ¼ 0, and K < 0. For the repulsive case, there are no periodic motions in r; only positive total energies are possible. For the attractive case, K < 0, and ðqt qf Þ2 mt =2L2 < E < 0, the coordinate oscillates between two turning points. _ and integrating over time produces Using (3.170) in (3.169), multiplying by r, the equation for conservation of energy of the test particle  V ðr Þ E 1 1 ¼ r_ 2 þ r2 θ_ 2 þ ¼ const ¼ v20 : mt 2 mt 2

(3.173)

72

Fundamental Microphysics of Ionized Gases

K>0

K=0

K0

'V '(r)

K=0

0

K < 0, L π 0 K < 0, L = 0

r

Figure 3.10. Effective potentials for the central inverse-square law force.

3.8 Coulomb Collisions

73

The potential energy is a function of only the distance between the test and field particles: F ðrÞ ¼ dV ðrÞ=dr. Eliminating θ_ by use of (3.171) gives the integral ð∞
θ¼ r

2qt qf b2  1 mt v20 r r 2

1=2

b dr: r2

(3.174)

Now define s ¼ b=r, then ð=r


b

θ¼

2qt qf s 1s  mt v20 b

1=2

2

0

ds:

(3.175)

The apsides of the orbit are the points for which dr=dθ ¼ 0. This occurs when r ¼ r 0 ¼ b=s0 , where s0 is the largest root of the equation 1  s20 

2qt qf s0 ¼ 0: mt v20 b

(3.176)

The angle for which dr=dθ ¼ 0, measured from an axis parallel to the initial asymptote of the trajectory, is θ0 , and the angle of deflection χ is the supplement of twice θ0 . That is, χ ¼ π  2θ0 , as shown in Figure 3.8. The integral (3.175) may be evaluated to give χ qt qf : (3.177) tan ¼ 2 mt v20 b For example, in hydrogen plasma, with the test particle being the electron, ¼ ð3kB T=me Þ. Taking the impact parameter to be the interparticle spacing, b
6  108 cm, which corresponds to ne ¼ ni
1021 cm‒3, and a temperature of 1 keV, then χ
2e2 =3kB Tb ¼ :0016 radians. For such small deflections, it is appropriate to use

v20

χ
2

qt qf : mt v20 b

(3.178)

Suppose the test particle is an electron and the field particle an ion with charge Z e and number density ni , and set the reduced mass to that of the electron. The number of collisions suffered by the electron in a time δt with the ions having impact parameters in the range b to b þ db is 2πni v0 bdbδt. Then for random processes the mean-square angle of the cumulative deflection angle is

bð bð max max X ðZ Þ2 e4 ni db 2 2 χ ðv0 ; bÞbdb ¼ 8π δt χ
2πni v0 δt : b m2e v30 bmin

bmin

(3.179)

74

Fundamental Microphysics of Ionized Gases

This is an important quantity when considering multiple scattering events, as we shall take up that topic shortly. Since successive collisions are independent events, the central-limit theorem of statistics can be used to show that for a large number of collisions, the distribution in angle will be approximately Gaussian around the forward direction with a mean-square angle given by (3.179). The integral over impact parameter is referred to as the Coulomb logarithm and is written logΛ ¼ log ðbmax =bmin Þ. We will address this term shortly.

3.8.2 Scattering Cross Section The probability of scattering is usually expressed in terms of a cross section. The differential cross section for scattering a particle through an angle between χ and χ þ dχ can be thought of as the effective scattering area surrounding the scattering center dσ ¼ 2πbdb:

(3.180)

This is the area of an annulus with radius b and width db. From (3.177)   χ q q  t f , (3.181) b¼ cot 2 mt v20 and   q q  1 t f   dχ: db ¼ 2 mt v0 2 sin 2 2χ

(3.182)

Then the differential scattering cross section is
dσ ¼

qt qf 2mt v20

2

1   dΩ ¼ σdΩ, sin 4 2χ

(3.183)

where dΩ ¼ 2π sin χdχ is the differential solid angle. The force F is defined as the total time derivative of the momentum, therefore the change in momentum can be calculated as the time integral of the force. The component of the force in the direction of the trajectory, F ∥ , is antisymmetric and the time integral vanishes. The perpendicular force is F⊥ ¼

qt qf sin θ: r2

(3.184)

The change in momentum of the energetic particle is easily calculated by a change of integration variable from t to θ using the fact that the velocity is

3.8 Coulomb Collisions

75

essentially constant, which is to say that the angular deflection is small. Using conservation of angular momentum (3.171) yields dt ¼

b dθ: v0 sin 2 θ

(3.185)

3.8.3 Energy Loss The momentum change during a single encounter at impact parameter b is found from þ∞ ð

Δp ¼ ∞

  jF ⊥ jdt ¼ 2qt qf 

θð0

0

  q q  sin 3 θ b t f ð1  cos θ0 Þ, (3.186) dθ ¼ 2 bv0 b2 v0 sin 2 θ

where θ0 is the angle at the periapsis. Using (3.177) gives 8 9 2 !2 31=2 >  < 2 q q  > mt v0 b 5 = t f Δp ¼ 2 : 1  41 þ > bv0 > qt qf ; :

(3.187)

The momentum lost by the test particle is gained by the field particle. Hence the kinetic energy gained by the field particle is ΔE


q2t q2f 1 ðΔpÞ2
2 : 2mf mf v20 b2

(3.188)

For the aforementioned hydrogen plasma, ΔE=E
2:6  106 : Both the amount of energy transferred and the magnitude of the deflection angle during a single collision are minuscule. Now let us consider the effects of multiple random collisions. The energetic test particle will interact with many “stationary” field particles, approaching each with various impact parameters. The energy loss rate involves adding up the effects of all interactions that may occur during a time interval dt, or alternately along a path-length element ds. Let the number density of the field particles be nf , so that the number of interactions per path-length element is dN ¼ 2πnf bdbdx. Then ð dE dN 1 2 2 nf ¼ ΔE ¼ 4πqt qf db: (3.189) 2 dx dx mt v0 b The energy loss rate is easily calculated from dE dE ¼ v0 : dt dx

(3.190)

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Fundamental Microphysics of Ionized Gases

Collisions between particles of similar mass can result in the equilibrium distribution being achieved after only a few collisions. The reason for this is that, because of the similarity of the masses, each collision can transfer energy comparable to their initial kinetic energies. 3.8.4 Coulomb Logarithm The term within the integral of (3.179) diverges for both small and large impact parameters. Thus, “cutoff” values must be used for the integral’s limits. The divergence for a small impact parameter is somewhat spurious and results from the approximation (3.178), which allows χ ! ∞ as b ! 0, instead of (3.177), which guarantees that χ remains bounded. One possible choice for bmin is the value that results in a 90 deflection. Using (3.177), we see   q q  t f bmin ¼ : (3.191) mt v20 Equivalently, this is the distance of closest approach (a head-on collision), the point at which the kinetic and potential energies are equal. A second limitation is set by the uncertainty principle bmin ¼ h=4πmt v0 , which is essentially equivalent to the de Broglie wavelength of the particle λd ¼ h=mt v0 . In nearly all situations, (3.191) is small compared to the uncertainty principle value. If the test particle is an electron and the field particle an ion, then for nondegenerate electrons v20 ¼ 3k B T e =me and thus bmin ¼

Z e2 : 3kB T e

(3.192) qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Electron degeneracy effects are included by replacing T e with T 2e þ T 2F . In the case of the upper limit of the integral, the divergence arises because the long-range nature of the Coulomb force extends to infinity. Even at very large impact parameters, there is some scattering. Only if the force is “cut off” at some distance will the scattering cross section be finite. The charge neutrality requirement naturally sets the upper limit to the Debye length (3.125), so that for the electron–ion collision

bmax

1=2 kB T e T i ¼ : 4πne e2 ðZ T e þ T i Þ

(3.193)

The correction for electron degeneracy is the same as in the preceding; see (3.127). In dense plasma, the Debye-Hückel treatment of screening is inappropriate since strong ion–ion correlations become dominant, and the screening length would be less than the interatomic spacing R0 . In practice, use the larger of the two.

77

3.8 Coulomb Collisions

log DE(b)

DEmax

bmax

bmax log b

Figure 3.11. Incremental energy transfer as a function of impact parameter. Expression (3.186) is given by the dashed line.

We can visualize the energy loss of a test particle as a function of the impact parameter in Figure 3.11. Within the interval bmin < b < bmax , the energy transfer is approximately given by (3.188), but for impact parameters outside this interval, the energy transfer is considerably less. As the low-energy bound is approached, expression (3.188) should be replaced by the more accurate one: ΔE ¼ 2

q2t q2f mf v20

b þ 2

q2t q2f 1 1 :  q2 q2 2
2 m v2 2 2 f 0 b þ bmin t f

(3.194)

mf v20

At the other extreme of very distant collisions, (3.188) is in error because of the binding of the atomic electron. We have assumed that the electrons are free, whereas they may actually be (loosely) bound to the ion. As long as the collision time is short compared to the orbital period of motion, it may be expected that the collision will be sudden enough to treat the electron as free. If, on the other hand, the collision time is very long compared to the orbital period, the electron will make many cycles of motion as the incident particle passes slowly by and will be influenced adiabatically by the fields with no net transfer of energy. The impact parameter bmax is the point for which the collision time and orbital period are comparable. If ω is a characteristic atomic frequency of motion, then δt∽1=ω, so that bmax ∽v0 =ω, and we can expect for impact parameters larger than bmax the

78

Fundamental Microphysics of Ionized Gases

energy transfer falls below that given by (3.188), going rapidly to zero. A simple estimate for the angular frequency can be found by assuming the electron is moving in a circular orbit about a nuclear charge. This is Kepler’s third law, which balances the centrifugal force against the attractive Coulomb force. More on this topic will be presented shortly in the section on the one-electron atom. We note that much ink has been spent discussing the Coulomb logarithm in the published literature. Even to this day there continues to be additional development of this topic. Finally, we arrive at the concept of “collision time.” A simple measure is the time required for a test particle to undergo a 90 deflection. Then we use (3.177) and let bmin ¼ b90 be the impact parameter for the deflection. The cross section using this impact parameter is σ ¼ πb290 . If there are nf field particles per cubic centimeter, the 90 deflection time is τ 90 ¼

1 : πnf v0 b290

(3.195)

3.9 Multiple Coulomb Scattering Coulomb collisions cause plasma to “relax” toward equilibrium conditions. The time required for this to be accomplished is found by calculating averages from multiple scatterings using distribution functions for the particles. The “relaxation time” is a somewhat ambiguous term used to denote the time in which collisions produce a large alteration in some original velocity distribution. As noted in the previous section, there is the 90 deflection time τ 90 , but there are other times of importance, such as the energy equilibration time and the self-collision time. Before we develop expressions for these, we need to examine the details of multiple collisions of test particles with field particles. As the test particle passes through plasma, it will experience many deflections, mostly small, and there will be a cumulative loss of velocity Δv. The importance of relaxation times will become apparent in subsequent chapters. Therefore, we make an effort to establish a solid mathematical development of those quantities. 3.9.1 Velocity Decrements and Diffusion Let PðvÞ be the probability that a particle changes its velocity from v to v þ Δv in the time interval Δt due to multiple small-angle collisions. The spatially uniform distribution function at time t is ð f ðv; tÞ ¼ f ðv  Δv; t  Δt ÞPðv  ΔvÞd3 ðΔvÞ : (3.196)

3.9 Multiple Coulomb Scattering

79

Assuming Δv is small if Δt is small, expanding (3.196) becomes  ð ∂f f ðv; t Þ ¼ f ðv; t ÞPðvÞ  Δt PðvÞ  Δvrv ½ f ðv; tÞPðvÞ d3 ðΔvÞ ∂t  ð 1 ∂2 (3.197) þ ½ f ðv; t ÞPðvÞ d 3 ðΔvÞ þ    : Δvi Δvj 2 ∂vi ∂vj Ð The probability of some transition taking place is PðvÞd 3 ðΔvÞ ¼ 1. We define the average change in velocity per unit time as ð 1 PðvÞΔvd3 ðΔvÞ, (3.198) hΔvi ¼ Δt and also

ð   1 Δvi Δvj ¼ PðvÞΔvi Δvj d 3 ðΔvÞ: Δt

(3.199)

Neglecting higher-order terms gives us an expression for the Fokker-Planck collision term in (3.11)
   ∂f ∂ 1 ∂2  ½hΔvi if ðv; tÞ þ ¼ Δvi Δvj f ðv; t Þ : ∂t coll ∂vi 2 ∂vi ∂vj This equation may be recast in the form of an equation of continuity
 ∂f þ rv  g ¼ 0, ∂t coll where

1 g ¼ hΔvif  rv  ½hΔvΔvif : 2

(3.200)

(3.201)

(3.202)

The effect of multiple small-angle collisions may be thought of as causing a continuous flow of phase points in velocity space, described by the flow vector g. To understand the physical significance of hΔvi and hΔvΔvi, consider the behavior of a stream of test particles with velocity v passing through plasma. In a short time interval, they will have slowed as a result of collisions to approach the average field particle velocity. This average change in velocity is hΔvi. (The quantity mt hΔvi is the dynamical friction.) The spreading of the points is characterized by the diffusion term hΔvΔvi. This last term is a tensor whose off-diagonal components are zero, while the D E D E D E diagonal components are ðΔv∥ Þ2 , ðΔv⊥ Þ2 =2, and ðΔv⊥ Þ2 =2. The subscripts refer to the components of the velocity parallel and perpendicular to the incident direction of the test particle. Assuming that the field particles have a Maxwell-Boltzmann distribution, then we need only three quantities: the

80

Fundamental Microphysics of Ionized Gases

D E D E slowing-down term hΔv∥ i, and the two spreading terms ðΔv∥ Þ2 and ðΔv⊥ Þ2 . Calculation of these quantities can be tedious (as we shall show later), but for now we assume mt mf and that the field particle is at rest. This last assumption lets the integral over the velocity distribution function of the field particles drop out. A simple estimate of the slowing-down coefficient (per unit time) is found from rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi D E (3.189), with ΔE
mt v0 ðΔv∥ Þ
mt v0 ðΔv∥ Þ2 , with hΔv∥ i ¼ 4π


 qt qf 2 nf logΛ: mt v20

(3.203)

Referring to Figure 3.8, χ ¼ π  2θ, and the perpendicular spreading term is found from Δv⊥ ¼ v0 sin χ ¼ 2v0 sin θ cos θ: (3.204) Using (3.177) and letting bmin ¼ b90 be the impact parameter for a 90 deflection (3.191), then  2 4v20 bb90 ðΔv⊥ Þ2 ¼ (3.205)  2 2 : b 1 þ b90 Averaging over all the test particles, the number of encounters with an impact parameter between b and b þ db (per second) is 2πbnf v0 db. This gives D

ðΔv⊥ Þ

2

E

bmaxð=b90

¼

8πnf v30 b290 0

x3 ð1 þ x2 Þ2

dx:

(3.206)

The integral evaluates to approximately log ðbmax =b90 Þ, assuming bmax =b90 1. As before, we must truncate the integral to avoid the divergence at infinity; we use D E log ðbmax =b90 Þ ¼ logΛ. The three diffusion coefficients hΔv∥ i, ðΔv⊥ Þ2 , and D E ðΔv∥ Þ2 are defined as a rate, that is, measured “per second.” D E A simple expression for the spreading term in the parallel direction, ðΔv∥ Þ2 , D E is not easily determined. We know that at zero incident velocity ðΔv⊥ Þ2 is twice D E ðΔv∥ Þ2 , while in the opposite limit, when the test particle velocity is greater than the field particle’s velocity, the diffusion in velocity space is primarily perpendicular to the original velocity. If successive encounters are truly random, it is impossible to predict what the precise value will be. But if we consider many test particles with the same initial velocity, we may use statistical methods to find the average velocity change. For an

3.9 Multiple Coulomb Scattering

81

isotropic distribution of velocities, the component perpendicular to the axis will have hΔv⊥ i ¼ 0 because of symmetry requirements. However, hΔv∥ i 6¼ 0. The mean-square value of Δv⊥ will not vanish. Detailed computations of these three diffusion coefficients must include motion of the center of mass of the colliding particles. The original solution is due to Chandrasekhar (1942, 1943a,b), who developed the theory for the dynamics of star clusters. Since the gravitational potential has the same 1=r dependence as the Coulomb potential, we can easily modify his theory for the discussion at hand. If the test particle has relative velocity u with respect to the field particle distribution f f ðvÞd 3 v, then the probability that the collision results with the test particle being injected into the solid angle dΩ per unit time is PðvÞ ¼ uf f ðvÞσ ðu; ΩÞdΩd3 v,

(3.207)

where σ ðu; ΩÞ is the collision cross section. For the Coulomb force, we use (3.183) ~ with the modification that the test particle mass is replaced by the reduced mass m. We can now rewrite (3.198) in Cartesian component form as ðð (3.208) hΔvi i ¼ Δvi uf f ðvÞσ ðu; ΩÞdΩd 3 v, and (3.199) as 



ðð

Δvi Δvj ¼

Δvi Δvj uf f ðvÞσ ðu; ΩÞdΩd3 v:

(3.209)

The particle velocities are vt and vf before the collision and vt 0 and vf 0 after. The corresponding relative velocities are u and u0 , with the vector representation being u0 ¼ u þ Δu. Assuming that each scattering event results largely in a change of direction of the particle’s motion, then ju0 j
juj. For a test particle scattered through angle χ, then Δu
2u sin ð χ=2Þ. The change in the relative velocity in the direction parallel to the incident velocity is thus χ : (3.210) Δu∥ ¼ 2u sin 2 2 Integrating (3.208) over the solid angle yields ð fΔu∥ g ¼

¼

Δu∥ uσdΩ ¼  q q 2 t f

~ mu

χð max

4π χ min

 q q 2 t f

~ 2 2mu

cos sin

χ χ2 2

d

2

4πu

χ : 2

ð sin 2 χ sin 4

2χ

sin χdχ

2

(3.211)

82

Fundamental Microphysics of Ionized Gases

The integral in (3.211) is h χ iχ max χ 
 log min , log sin  2 χ min 2

(3.212)

since 1 χ max χ min . Then using (3.177) gives the result χ qt qf χ min b90 1 ¼ ¼ ,
tan min ¼ ~ 2 bmax bmax Λ 2 2 mu

(3.213)

leading to fΔu∥ g ¼ 4π

q q 2 1 t f logΛ: ~ u2 m

(3.214)

We note that by symmetry,fΔu⊥ g ¼ 0. Thus, the vector fΔug can be expressed in the laboratory frame as fΔug ¼ 4π

q q 2 u t f logΛ: ~ u3 m

(3.215)

But we need the change of test particle velocity Δv∥ , not Δu∥ . The transformation between the two coordinate systems is vt ¼ V þ

mf u, mt þ mf

(3.216)

where V is the center of mass velocity, and thus Δvt ¼

mf Δu: mt þ mf

(3.217)

Consequently, ð

q2t q2f u logΛ fΔvt g ¼ Δvt uσdΩ ¼ 4π ~ t u3 mm

(3.218)

is the average rate of change of velocity of test particles due to collisions with field particles. Now integrating over the field particles,
 ð   vt  vf 3 mt qt qf 2 logΛ f f vf  (3.219) hΔvt i ¼ 4π  d vf : vt  vf 3 ~ mt m There is a noticeable resemblance between (3.219) and the equation expressing the electrostatic field  at the point vt resulting from a space-charge distribution proportional to f f vf .

3.9 Multiple Coulomb Scattering

  If f f vf is a Maxwell-Boltzmann distribution, then (3.61) becomes   nf l3f l2 v2 f f vf ¼ 3= e f f , π2

83

(3.220)

with l2f ¼ mf =2kB T f , and the parallel component of (3.219) becomes vð0
  v2f nf l3f mt q t q f 2 1 logΛ 2 4π 3= exp l2f v2f dvf : hΔv∥ i ¼ 4π ~ mt m v0 π2

(3.221)

0

The integral in (3.221) can be evaluated by parts vð0

 lf vf 2l2f vf exp l2f v2f dvf 0

h

¼ lf vf exp



l2f v2f

iv0 0

vð0

 þ lf exp l2f v2f dvf :

(3.222)

0

Let x ¼ lf v0 and y ¼ lf vf , then (3.221) becomes
 qq 2 mt hΔv∥ i ¼  4π t f logΛ ~ m mt 2 3
2 ðx ðx     2 lf 4 d nf pffiffiffi exp y2 dy  x exp y2 dy5: π x dx 0

(3.223)

0

The quantity lf v0 is simply the ratio of the test particle velocity to the root-meansquare two-dimensional velocity of the field particles. The integrals are just the Ðx 2 error function φðxÞ ¼ p2ffiffiπ ey dy and its derivative with respect to x, φ0 ðxÞ. Finally, 0

we arrive at the dynamical friction coefficient (Chandrasekhar, 1943a,b)
   mt G lf v0 , (3.224) hΔv∥ i ¼ AD l2f 1 þ mf where GðxÞ ¼

φðxÞ  xφ0 ðxÞ , 2x2

(3.225)

and
 qt qf 2 AD ¼ 8π nf logΛ: mt

(3.226)

84

Fundamental Microphysics of Ionized Gases

In the limit of high-speed test particles lf v0 ¼ x ! ∞, φðxÞ ! 1, GðxÞ ! 1=2x2 , and (3.224) is in agreement with (3.203).   The calculation of the components of the diffusion tensor Δvi Δvj , (3.209) proceeds along similar Ð 1 lines as for hΔvi, with one additional assumption: all terms not containing χ dχ
logΛ will be considered small and neglected. This allows considerable simplification and is justified in that the neglected terms are roughly an order of magnitude smaller than the others. As mentioned earlier, the off-diagonal components are zero. In the laboratory frame, the tensor has the form q q 2 1    ui uj t f δij  2 logΛ: Δui Δuj ¼ 4π ~ u u m

(3.227)

  ∂2 u ∂ uj uδij  ui uj =u ¼ , ¼ u2 ∂vi ∂vj ∂vi u

(3.228)


   qt qf 2 ∂2 u Δvi Δvj ¼ 4π logΛ : ∂vi ∂vj mt

(3.229)

Now using

in (3.227), we find

Carrying out the integration over the field particles yields 






qt qf Δvi Δvj ¼ 4π mt

2 logΛ

∂2 G , ∂vi ∂vj

where the “potential” of the field particles is ð    G ðvt Þ ¼ f f vf vt  vf d 3 vf : A second potential can be introduced ð   mt f f vf   d 3 vf , H ðvt Þ ¼ ~ vt  vf  m

(3.230)

(3.231)

(3.232)

and thus (3.219) becomes


qt qf hΔvt i ¼ 4π mt

2 logΛ

∂H : ∂vt

(3.233)

3.9 Multiple Coulomb Scattering

85

The potentials G and H are the Rosenbluth potentials (Shkarofsky et al., 1966). There is a close similarity with electrostatics, and thus that formalism can be exploited for calculating the dynamical friction due to various distributions of field particles.5 For (3.219) and (3.233) to hold, (3.232) must obey “Poisson’s equation” mt   r2v H ¼ 4π f f vf : (3.234) ~ m The Rosenbluth potentials are related by 2H ¼

mt 2 r G: ~ v m

(3.235)

To find the components of the diffusion tensor, we note that (3.232) can be written 
  mt φ lf v0 H ¼ nf lf 1 þ : (3.236) l f v0 mf Expression (3.224) becomes


qt qf hΔv∥ i ¼ 4π mt

2 logΛ

∂H : ∂v

(3.237)

Applying Gauss’ theorem to (3.235) results in ~ 1 ∂G m ¼2 ∂v mt v20

vð0

H v2f dvf :

(3.238)

0

Then (3.230) becomes   1 ∂ Δvi Δvj ¼ AD 2 v0 ∂vf

vð0

  φ lf vf 2 vf dvf vf

(3.239)

0

or 

5

E 1D E 1D E A   D  D Δvi Δvj ¼ ðΔv∥ Þ2 þ ðΔv⊥ Þ2 þ ðΔv⊥ Þ2 ¼ φ l f v0 : v0 2 2

(3.240)

The Fokker-Planck collision term (3.200) can be expressed in terms of the Rosenbluth potentials. In vector notation, we have 1 mt 4π qt qf

!2 1 log Λ




∂f 1 ¼ r  f rH  r  f rrG : ∂t coll 2

86

Fundamental Microphysics of Ionized Gases

The parallel and perpendicular components are thus D E A   D ðΔv∥ Þ2 ¼ G lf v0 v0 and D E A      D 2 ðΔv⊥ Þ ¼ φ lf v0  G lf v0 : v0

(3.241)

(3.242)

The functions GðxÞ and φðxÞ  GðxÞ as a function of x ¼ lf v0 are shown in Figure 3.12. In the limit of zero test particle velocity, Gðx ! 0Þ ! 0, and there is no dynamical friction, as expected. However, the diffusion coefficients do not vanish pffiffiffi pffiffiffi pffiffiffi since GðxÞ=x ! 2=3 π , φðxÞ=x ! 2= π , and ½φðxÞ  GðxÞ=x ! 4=3 π . D E D E Hence, ðΔv⊥ Þ2 ¼ 2 ðΔv∥ Þ2 , which is reasonable, since for the zero-energy

1.0

2.00

0.9

1.75

0.8

1.50

0.6

1.25

0.5

1.00

0.4

0.75

0.3

0.50

0.1

0.25

tD /tE

G(x), j(x)–G(x)

test particles, all directions in velocity space are equivalent. For very fast test particles, when x ! ∞, φðxÞ ! 1, GðxÞ ! 1=2x2 , and we see 2 the dynamical frictionD decreases E as 1=v0 ; the D parallel E diffusion coefficient 2 3 decreases even faster ðΔv∥ Þ ∽1=v0 , while ðΔv⊥ Þ2 ∽1=v0 . Consequently, very fast particles diffuse mostly sideways. If we increase the test particle mass, keeping everything else the same, the dynamical friction tends to dominate, although all coefficients diminish, since AD ∽1=m2t . In the limiting case of the test particle of D velocity E exceeding the random velocity D E 2 2 the field particles, as  is often the case, ðΔv⊥ Þ varies as AD =v0 , while ðΔv∥ Þ is less by a factor 1= 2l2f v20 .

0.0 0.00 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 x

Figure 3.12. Functions GðxÞ (solid line) and φðxÞ  GðxÞ (dashed line). Also shown is the ratio of the deflection time τ D to the energy-exchange time τ E .

3.9 Multiple Coulomb Scattering

87

3.9.2 Relaxation Times Each encounter of a test particle with a field particle results in deflections that are random and small. As such, one can introduceDseveral E relaxation D times Erelated to the slowing term hΔv∥ i and the diffusion terms ðΔv∥ Þ2 and ðΔv⊥ Þ2 . Following Spitzer (1967), the slowing-down time is defined as τs ¼ 

v0 v  i 0 ¼h  : hΔv∥ i 1 þ mt AD l2 G lf v0 mf

(3.243)

f

pffiffiffi For small velocities, τ s is nearly constant, since GðxÞ=x ! 2=3 π pffiffiffi 3 π : τs
 2 1 þ mmft AD l3f

(3.244)

The mean velocity of the test particles will approach zero exponentially, with a time constant τ s . For large velocities, τ s ∽v30 , since GðxÞ ! 1=2x2 . The deflection time is defined as τD ¼ D

v20 ðΔv⊥ Þ2

E¼

v3   0   : AD φ lf v0  G lf v0

(3.245)

  When the field particle mass is large and thus lf v0 is large, φðxÞ ! 1 and GðxÞ ! 0, and thus τD


v30 m2t v30 : ¼ AD 8πq2t q2f nf logΛ

(3.246)

Comparing this to the simple estimate for a 90 deflection (3.195), τ D is less than τ 90 by a factor 1=ð8logΛÞ; recall that τ 90 was based upon single encounters. The energy-exchange time is E2 τE ¼  2 , ΔE where in a single encounter i 1 mt h ðv0 þ Δv∥ Þ2 þ ðΔv⊥ Þ2  mt v20 ΔE ¼ 2 2 i mt h 2 2v0 Δv∥ þ ðΔv∥ Þ þ ðΔv⊥ Þ2
mt v0 Δv∥ , ¼ 2

(3.247)

(3.248)

having retained only the dominate terms. Consequently, τE ¼

m2t v40 v30 D E¼  : 4AD G lf v0 4m2t v20 ðΔv∥ Þ2

(3.249)

88

Fundamental Microphysics of Ionized Gases

For large velocities, τ D =τ E ∽1=v20 and deflections dominate over energy exchange. The ratio τ D =τ E is shown in Figure 3.12. Note that τ E is a minimum for lf v0 ¼ 1 and becomes much larger for particles moving at speeds much above or much below the average speed of the field particles. Consider the situation where the average particle in a thermal distribution is interacting with other particles of its own species. p If ffiffiffiffiffiffiffi the ffi average particle travels with roughly the mean thermal speed, then lf v0 ¼ 3=2
1:225, φ  G
4G, and so τ D
τ E . More exactly, from Figure 3.12 we see τ D =τ E
1:14. This relaxation time is a good estimate of the time required to isotropize the velocity distribution and for the distribution to approach equilibrium. This special value of τ D
τ E ¼ τ c is designated the self-collision time. A detailed numerical solution to the Fokker-Planck equation shows that in the neighborhood of the average energy, the time required for relaxation of a distribution that initially is strongly nonMaxwellian is quite close to τ c . However, the time needed for the distribution to become Maxwellian for energies ranging from zero to several times the average energy is about 10τ c . Thus τ c provides a semiquantitative estimate only. Energy exchange between the electron fluid and the ion fluid through collisions is relatively slow, because of the large mass ratio. Let us assume the temperatures of the two species are T e and T i . Then an electron with velocity ve changes its energy per unit time according to (3.248), which can be rewritten as


 me me AD   2 2AD li 1 þ φ lf ve , (3.250) ve Gðli ve Þ þ hΔEe i ¼ 2 mi ve where (3.224), (3.241), and (3.242) have been used. The rate of change of electron energy is then ð ne l3 2 2 3 dT e : (3.251) hΔE e i 3=e ele ve 4πv2 dv ¼ ne k B 2 2 dt π This may be written as dT e Te  Ti ¼ , dt τ eq

(3.252)

where the equilibration time is found from the integral in (3.251), resulting in τ eq


3= kB T e kB T i 2 ¼ pffiffiffiffiffi 2 þ : mi 8 2π ðZ Þ e4 ni logΛ me 3me mi

(3.253)

The discussion in this section addressed conventional scattering of free electrons by the ion core; it is characterized by simple Rutherford scattering in a Coulomb potential. However, in dense plasma the scattering becomes sensitive to the ion

3.10 Radiation as a Fluid

89

core structure and to the arrangement of nearby ions. We address some of these issues in Chapter 5. 3.10 Radiation as a Fluid We developed the two-fluid description of high-energy-density matter previously in Section 3.6. The appreciable mass difference between an electron and an ion, together with the collisions between the two populations, requires separate descriptions for each fluid. There is, however, a third “fluid” to be considered, namely radiation. The properties of radiation may be treated with methods not much different from those for particles with mass. We treat the radiation as a gas of point, massless particles. Each photon carries an energy E with associated frequency ν, where E ¼ hν. These massless particles also carry momentum p ¼ E=c ¼ hν=c. In the absence of matter, the photons travel in straight lines at the speed of light. Hence they can transport energy and momentum long distances very rapidly. Even in the presence of matter, the mean free path of the photons may be much longer than the collisional mean free path of the particles having mass; the speed of photons is much greater than thermal velocities of the massive particles. Radiation also exhibits pressure and is thus capable of doing work. As we shall see later in this book, the interaction of radiation with matter has a special importance. Because the radiation energy is propagated at the speed of light, relativistic effects must be considered, although the effects are important only in certain respects. The properties of the radiation field can be developed using Maxwell’s equations, which yield a wave picture. However, we employ a quantum mechanical approach. This is the proper prescription when considering the interaction of the radiation with the electron fluid. (The interaction with the more massive ion fluid is essentially nonexistent.) In Section 3.1, we developed the velocity distribution function for particles with mass. Radiation, too, has a distribution function characterized by frequency rather than particle velocity. We defer much of the discussion about the transport of radiation to Chapter 8. Our present interest is the “fluid-like” nature of radiation. Consider the nature of the radiation inside an enclosed cavity; the cavity is also known by the name hohlraum, German for “hollow area.” This cavity is void of matter; only the radiation is present. The walls of the cavity are completely opaque and of constant and uniform temperature, and the dimensions of the cavity are large compared to the wavelengths of the photons. Since the cavity’s walls are in thermal equilibrium, the emitted radiation is of necessity also thermal. An important property of the frequency-dependent (that i.e., spectral) intensity, I ν , is that it is independent of the properties of the cavity’s walls

90

Fundamental Microphysics of Ionized Gases

and depends only on the temperature. This is demonstrated by taking a second cavity, whose walls are at the same temperature as the first cavity. Each cavity has a small hole through the wall, and the two holes are placed facing one another, with a filter between the two. This filter passes a single frequency but no others. If I ν emitted by the first cavity into the second is not the same as I ν of the second cavity emitted into the first, energy will flow spontaneously between the two enclosures. But both cavities are at the same temperature, so this violates the second law of thermodynamics. Therefore, we may define this equilibrium radiation as blackbody radiation. Thermal radiation becomes blackbody radiation only for optically thick material. A blackbody is an idealized physical body that absorbs all incident electromagnetic radiation. Because of this perfect absorptivity at all wavelengths, a blackbody is also the best possible emitter of thermal radiation, which it radiates incandescently in a characteristic, continuous spectrum that depends only on the body’s temperature. At Earth-ambient temperatures, this emission is in the infrared region of the electromagnetic spectrum and is not visible. The object appears black, since it does not reflect or emit any visible light. A corollary of this emissivity statement is that it is not possible for a body at equilibrium to radiate more energy than a blackbody at equilibrium. Early attempts to describe radiation with classical electromagnetic theory resulted in a significant discrepancy when compared to experiments. If each Fourier mode of the equilibrium radiation within the cavity, having perfectly reflective walls, was considered as a degree of freedom, and if all those degrees of freedom could freely exchange energy, then according to the equipartition theorem of classical physics, each degree of freedom would have one and the same quantity of energy. This theory leads to the paradox known as the ultraviolet catastrophe, or ultraviolet divergence, which results in an infinite amount of energy in any continuous field. 3.10.1 Planck’s Law Near the turn of the previous century, Max Planck devised a law that now bears his name. He hypothesized that radiation comes in discrete quanta of energy, and the radiation in the hohlraum is a superposition of quantized oscillations or modes. Since photons have zero spin, the more elegant way to derive Planck’s law is to use Bose-Einstein statistics; this was done earlier in this chapter (see equation (3.47), with α ¼ 0). We consider here the less formal semiclassical approach. There are two parts to this: first, we find the density of photon states within the cavity, and second, we calculate the average energy per photon state. ^ inside a box of Consider a photon of frequency ν traveling in the direction n dimensions Lx , Ly , and Lz . The wave vector of the photon is k ¼ ð2πν=cÞ^ n:

3.10 Radiation as a Fluid

91

Provided that each dimension is large compared to the wavelength, the photon can be thought of as a standing wave. The number of nodes in the wave in each direction is, for example, nx ¼ k x Lx =2π, since there is one node for each integral number of wavelengths in given orthogonal directions. The wave can be said to have changed states in a distinguishable manner when the number of nodes in a given direction changes by one or more. The number of node changes in a wave number interval is, for example, Δnx ¼ Lx Δkx =2π. Thus, the change in the number of states in the three-dimensional wave vector element Δk x Δky Δkz ¼ d 3 k is ΔN ¼ Δnx Δny Δnz ¼

Lx Ly Lz ð2π Þ

3

d 3 k:

(3.254)

This is a critical point in Planck’s thinking: radiation occurs only when the oscillators in the wall of the cavity change from one possible state to another; this implies the oscillators cannot radiate energy continuously, but only in discrete packets, or quanta. Now the volume of the cavity is Lx Ly Lz , and noting that the photons may have two independent polarizations (the electric vectors are orthogonal to each other and to the wave vector k), the number of states per unit volume per unit three-dimensional wave number is 2=ð2π Þ3 . Now d3 k ¼ k 2 dkdΩ ¼

ð2π Þ3 ν2 dνdΩ, c3

(3.255)

so that the density of states (the number of states per solid angle per volume per frequency) is ρν ¼

2ν2 : c3

(3.256)

We now need the average energy of each state. From quantum theory, each photon has energy hν. If a particular state has n photons, the energy of the state is En ¼ nhν. According to statistical mechanics, the probability of finding a state of energy En is proportional to exp ðEn =kB T Þ. (The temperature is that of the radiation field, not of the matter.) The average energy is thus ! P∞ ∞ En =kB T X ∂ E =k T 0 En e n B : (3.257) e hE i ¼ P∞ E =k T ¼   log n B 0e ∂ kB1T 0 If we let x ¼ ehν=kB T , then the denominator of (3.257) is 1 þ x þ x2 þ    ¼

1 1 , ¼ 1  x 1  ehν=kB T

(3.258)

92

Fundamental Microphysics of Ionized Gases

while the numerator is  d  1 þ x þ x2 þ    hνðx þ 2x2 þ 3x3 þ   Þ ¼ hνx dx
 d 1 hνx hνehν=kB T ¼ hνx ¼ : ¼ dx 1  x ð1  xÞ2 ð1  ehν=kB T Þ2

(3.259)

Thus hE i ¼

hνehν=kB T hν ¼ hν=k T : hν=k T B e B 1 1e

(3.260)

This is the standard expression for Bose-Einstein statistics with a limitless number of particles (the chemical potential is zero). Finally, the energy per solid angle per volume per frequency is the product of the average energy (3.260) and the density of states (3.256). The spectral intensity is I ν ¼ 4πcρv hE i ¼ 4πBðνPÞ ðT Þ, which brings us to Planck’s law BðνPÞ ðT Þ ¼

2hν 3 1 : 2 hν=k c e BT  1

(3.261)

This is the isotropic specific intensity in thermal equilibrium, the blackbody radiation emitted by a perfect radiator. Figure 3.13 plots the Planck function for several temperatures. There are several properties and consequences of Planck’s law: 1. For hν kB T, we obtain the Rayleigh-Jeans law. In this case, a series expansion for the exponential can be employed to give Bν ! I ðνRJÞ ðT Þ ¼

2ν 2 k B T: c2

(3.262)

Notice that the Rayleigh-Jeans law does not contain Planck’s constant. It was originally derived assuming that hEi ¼ kB T, the classical equipartition value for the energy of an electromagnetic wave. If (3.262) is applied to all frequencies, the integral over frequency diverges; this is the ultraviolet catastrophe mentioned earlier. 2. For hν kB T, we obtain the Wien law: Bν ðT Þ ! I ðνWÞ ðT Þ ¼

2hν 3 hν=kB T e : c2

(3.263)

Figure 3.14 shows these two limiting functions together with the Planck function for a specific temperature.

93

3.10 Radiation as a Fluid 1014 1013

1.0

Bn(T) (erg-s–1-cm–2-sr–1-keV–1)

1012 1011 1010

0.1

109 108 107

0.01

106 105 104

0.001

103 102 101 100 10–7 10–6 10–5 10–4 10–3 10–2 10–1

100

101

102

hn (keV)

Figure 3.13. Equilibrium radiation distribution function, the Planck function, for temperatures 0.001, 0.01, 0.1, and 1.0 keV.

3. Of two blackbody curves, the one with higher temperature lies entirely above the other: ∂BðνPÞ ðT Þ 2h2 ν4 ehν=kB T >0 : ¼ 2 ∂T c kB T 2 ðehν=kB T  1Þ2

(3.264)

Note that BðνPÞ ! 0 as T ! 0 and BðνPÞ ! ∞ as T ! ∞. 4. Wien’s displacement law states that the peak of the blackbody function for a specific temperature is given by νmax
2:82

kB T : h

(3.265)

The peak frequency of the Planckian increases linearly with temperature. 3.10.2 Stefan’s Law The spectral intensity for blackbody radiation is given by (3.261). Integrating this expression over the solid angle and over all frequencies gives the energy density ð 1 8πh 3  hν=kB T ur ¼ 3 ν e 1 dν : (3.266) c

94

Fundamental Microphysics of Ionized Gases 1014 1013

Bn(T) (erg-s–1-cm–2-sr–1-keV–1)

1012 1011 1010 109 108 107 106 105 104 103 102 101 100 10–7 10–6 10–5 10–4 10–3 10–2 10–1

100

101

102

hn (keV)

Figure 3.14. At low frequency, the Planck function (solid line) is approximated by the Rayleigh-Jeans law (dashed line), while at high frequency it is approximated by the Wien law (dot-dashed line). The temperature is 0.1 keV. The ultraviolet catastrophe occurs when the Rayleigh-Jeans law is extended to high frequencies.

If we let x ¼ hν=kB T, we obtain ur ¼ 8π

ðkB T Þ4 ðchÞ

ð∞

3

  x3 ex 1 þ ex þ e2x þ    dx:

(3.267)

0

  Integrating the series in parenthesis term by term gives 6 1 þ 24 þ 34 þ    ¼ π 4 =15. This leads to Stefan’s law ur ¼

8π 5 k 4B 4 T ¼ aT 4 : 15 c3 h3

(3.268)

Upon integrating the Planck function (3.261) over frequency, we find BðPÞ ðT Þ ¼

c aT 4 : 4π

(3.269)

The Stefan-Boltzmann constant σ is defined such that πBðPÞ ðT Þ ¼ σT 4 ,

(3.270)

3.10 Radiation as a Fluid

95

and we see σ ¼ ac=4. The rationale for this definition follows from calculating the radiation flux emerging from a blackbody, F ðνPÞ

ð1

¼ 2π BðνPÞ ðT Þμdμ ¼ πBðνPÞ ðT Þ,

(3.271)

0

where μ is the cosine of the angle; the frequency-integrated flux is F ðPÞ ¼ σT 4 .

3.10.3 Thermodynamics of Equilibrium Radiation Radiation that is in equilibrium with the walls of the cavity has both an energy density and a stress (pressure) associated with it. When energy is inserted into or withdrawn from the cavity, the radiation field can do mechanical work. The energy density was mentioned previously; it will be developed in more detail along with the stress in Chapter 8. Blackbody radiation, like any system in thermodynamic equilibrium, may be treated by thermodynamic methods. Consider a cavity, filled with equilibrium radiation, which may be compressed by moving the walls inward. In Section 3.3, we showed the combined first and second laws of thermodynamics are TdS ¼ dℰ þ PdV. Because thermal radiation is isotropic, the pressure is related to the energy density (letting u ¼ ur ) by P ¼ u=3. Now ℰ ¼ uV, where V is the volume under consideration, thus dS ¼

V du u 1u dT þ dV þ dV: T dT T 3T

Since dS is a perfect differential,

 ∂S V du ∂S 4u ¼ ¼ and , ∂T V T dT ∂V T 3 T

(3.272)

(3.273)

we obtain ∂2 S 1 du 4 u 4 1 du ¼ ¼ 2þ , ∂T∂V T dT 3T 3 T dT

(3.274)

and thus du 4u ¼ , dT T

du dT ¼ 4 , and log u ¼ 4 log T þ log a, u T

(3.275)

where a is a constant of integration. Thus, we obtain Stefan’s law (3.268).

96

Fundamental Microphysics of Ionized Gases

The entropy of blackbody radiation is 4 S ¼ aT 3 V, 3

(3.276)

and the law of adiabatic expansion for blackbody radiation is PV =3 ¼ const. The adiabatic index for radiation is γ ¼ 4=3. The heat capacity of radiation at constant volume, from Section 3.3, is
 ∂ℰ ∂  4  aT V ¼ 4aT 3 V: CV ¼ ¼ (3.277) ∂T V ∂T 4

However, the heat capacity at constant pressure is not C P ¼ γCV , as it is for a perfect gas. In fact, C P is infinite. To understand this result, consider introducing heat into the cavity while holding the pressure constant. Now P ¼ aT 4=3 , so the temperature remains constant while the volume increases to accommodate the increase in energy. Since dQ ¼ CdT and dQ > 0 while dT ¼ 0, we see C ¼ ∞. 3.11 One-Electron Atom During the latter part of the nineteenth century and the early years of the twentieth century, numerous experiments were performed that probed matter on a very fine scale. The observations required explanation and thus began the development of atomic theory. Experiments using the scattering of charged particles (as well as electromagnetic radiation) demonstrated that the atom has structure. Rutherford’s experiment established that the mass of an atom is concentrated in a very small nucleus, while its volume and its physical and chemical properties are determined by a comparatively loose surrounding structure of electrons. Other experiments, such as the conduction of electricity through rarified gases, showed the existence of free electrons. Milliken’s oil drop experiment provided direct confirmation that the electron’s charge-to-mass ratio is a constant, and that the charge on a droplet can be measured in integral amounts. Numerous explanations were advanced for these observations, some with success, many without. One of the more impressive results from the Lorentzian theory of the electron was the determination of the classical radius of the electron. To explain his results, Rutherford advanced a planetary model for the atom, that of planetary electrons orbiting a solar nucleus. However, this model encountered serious difficulties. The laws of classical mechanics predict that the electron would release electromagnetic radiation while orbiting the nucleus. Because the electron would lose energy, it would gradually spiral inward, collapsing onto the nucleus. The classical model predicts that all atoms are unstable. Also, as the electron spirals inward, the photon emission would increase in frequency as the orbit got

97

3.11 One Electron Atom

smaller and faster, and one would see a smear (in frequency) of electromagnetic radiation. However, experiments show light is emitted from atoms only at discrete frequencies.

3.11.1 Bohr’s Hypothesis Some of the experiments performed near the dawn of the twentieth century suggested the existence of light quanta. The classical theory of electromagnetic radiation interacting with small metallic particles could not explain the photoelectric effect. That is, the atom does not behave like a classical mechanical system, which can absorb energy in amounts that are arbitrarily small. The existence of sharp emission and absorption lines and Einstein‘s light quantum hypotheses suggest the atom can exist only in well-defined discrete stationary states with energies E 1 ,E 2 ,E3 ,    . In 1913, Bohr introduced the hypothesis that only those spectral lines can be absorbed for which the energy of the photon has exactly such a value that it can raise the atom from one stationary state to a higher one. That is, the absorption lines are given by the equations E2  E 1 ¼ hν1 , E 3  E1 ¼ hν2 , and so on. A schematic of Bohr’s thinking is shown in Figure 3.15. Confirmation of Bohr’s theory on the existence of discrete energy levels in the atom is given by the experiments of Franck and Hertz using bombarding electrons. n ¥

E¥

4

E4

3

E3 Ha

Hb

2

E2

1

E1

Figure 3.15. Observation of the discrete emission lines from hydrogen led to the “term” diagram describing the energy level structure of the isolated atom. The H α and H β lines of the Balmer series are shown.

98

Fundamental Microphysics of Ionized Gases

The modern theory of atomic structure has its roots in Bohr’s model of the atom. Prior to his work, the classical mechanical picture of the atom has the orbit of an electron being an ellipse about the heavy nucleus; this is just Kepler’s first law of planetary motion. For the special case of a circular orbit of radius r and the electron revolving about the stationary nucleus with angular frequency ω, the centrifugal force is balanced by the attractive Coulomb force, and we can write r 3 ω2 ¼

Ze2 , me

(3.278)

where Ze is the nuclear charge; this is just Kepler’s third law. The energy of the revolving electron is the sum of the kinetic and potential energies E¼

me 2 2 Ze2 r ω  : 2 r

(3.279)

In this context, the normalized E denotes the work needed to remove the electron to infinite radius and to bring it to rest, that is, rω ! 0. Combining (3.278) with (3.279) gives
1= me 2 2 Ze2 me Z 2 e4 ω2 3 E¼ r ω ¼ , ¼ 2 2r 8

(3.280)

from which follows jEj3 me Z 2 e4 ¼ ¼ const: ω2 8

(3.281)

According to classical ideas, the electron can be at any orbital radius, and hence can have any angular velocity that corresponds to a definite value for the energy. In contrast is Bohr’s model, which hypothesizes that the electrons in the atom can exist only in definite discrete energy states, and thus only certain orbital radii are stable. Early spectroscopy experiments by Balmer on the hydrogen atom showed the frequency of the emission lines follow
 1 1 ν ¼ R∞ c 2  2 , with m > n, (3.282) n m where R∞ is the Rydberg wave number (constant) for hydrogen, and n and m are integers representing the discrete states. Recall that Bohr’s hypothesis is hν ¼ En  E m . The energy scale is such that E n is the work needed to remove the electron from state n to rest at infinity; this is the ionization energy from state n (note E∞ ¼ 0Þ:

3.11 One Electron Atom

99

The determination of the Rydberg constant is found from Bohr’s correspondence principle, which asserts that the higher the Bohr state of the atom, the more closely the atom obeys the laws of classical mechanics. As n increases, the intervals between the individual levels become smaller and smaller, the levels become closer in energy, and the atom approaches asymptotically the state of motion required by classical mechanics. For the case where the initial and final states are highly excited states, then n  m is small compared to m and n, and (3.282) becomes ν


2Rc ðm  nÞ, n3

(3.283)

while for the lowest frequency emitted, taking m  n ¼ 1 ν1


2Rc : n3

(3.284)

For m  n ¼ 2, we get a frequency twice as high, for m  n ¼ 3 it is three times as high, and so on. According to classical theory, the spectrum has the same character as that of an electrically charged particle vibrating with angular frequency ω ¼ 2πν and the associated harmonics. Then for m ! ∞, (3.282) gives the Balmer term
1=3  ν 2=3 hRc Rch3 ω2 1 E∽  2 ¼ hRc ¼ , n 2Rc 16π 2

(3.285)

where we have used (3.284). This expression has the same form as (3.280) for the energy of a revolving electron. We may write (3.285) as jE j3 Rch3 ¼ ¼ const, ω2 16π 2

(3.286)

and in the limiting case of very large n, (3.281) and (3.286) must be equal. Hence R ¼ R∞ Z 2 with R∞ ¼

2π 2 me e4 : ch3

(3.287)

This value for R∞ , the Rydberg constant, agrees well with the experimentally determined value. Even better agreement is obtained when the motion of the nucleus is included. Bohr’s explanation of the structure of the atom is based on laboratory observations that led to the empirical relation given by (3.282). Even though this simple explanation for the Balmer spectrum of hydrogen is in excellent agreement with experiment, we have no explanation of why the special assumption E n ¼ R∞ hc=n2 is correct. The answer is found in the quantum theory of the atom.

100

Fundamental Microphysics of Ionized Gases

Before we proceed with the development of the quantum theory, we collect here a few formulas that are of great importance for Bohr’s quantum theory. For an infinitely heavy nucleus and circular orbits, the radius of the orbit corresponding to the nth state, from (3.280), (3.285), and (3.287) is rn ¼ 

Ze2 Ze2 n2 n2 ¼ a0 , ¼ 2 2E Z 2hcR∞ Z

(3.288)

where the Bohr radius is a0 ¼ h2 =4π 2 me e2 . Similarly for the angular velocity ωn ¼

4πcR∞ Z 2 Z2 ¼ ω , 0 n3 n3

(3.289)

where ω0 ¼ 4πcR∞ . Of special importance is the angular momentum of the electron about the nucleus. From (3.288) and (3.289), this is Ln ¼ me r2n ωn ¼ me a20 ω0 n ¼ n

h : 2π

(3.290)

In the Bohr atom, the angular momentum of a discrete state is an integral multiple of h=2π. This is called the quantum condition for angular momentum. This elementary theory of Bohr deals only with circular orbits. As we have seen in the preceding, the classical Kepler problem allows elliptical orbits. Consider the one-electron atom with the electron moving in a plane defined by its orbit; this is a two-dimensional problem, and the trajectory of the electron is, in general, an ellipse. The equation of an ellipse is r¼

að1  ϵ 2 Þ , 1 þ ϵ cos θ

(3.291)

where a is one-half the largest diameter (major axis) and ϵ is the eccentricity. If b is one-half of the smallest diameter (minor axis), then sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2 b ϵ ¼ 1 ; (3.292) a for a circular orbit ϵ ¼ 0. The area of the ellipse is A ¼ πab. The general equation of a conic section is 1 ¼ B þ A cos θ; r

(3.293)

for an ellipse B > A, and so B¼

1 ϵ and A ¼ : 2 að1  ϵ Þ að1  ϵ 2 Þ

(3.294)

3.11 One Electron Atom

Then

101

   B  A :  ϵ¼ and a ¼  2 jBj A  B2 

(3.295)

We have already established, in Section 3.8, that for an inverse-square law force, the angular momentum L is a constant as is the energy E, which must be negative for an elliptical orbit (K < 0, where K ¼ Ze2 , and me K 2 =2L2 < E < 0). The roots of (3.172) are the turning points of the radial motion 1 r 1, 2

me K ¼ 2  L

"


me K L2

2

2me E þ 2 L

#1=2 :

Upon comparing (3.293) with (3.296), we see rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi me K 2me E B ¼  2 and A ¼ B2 þ 2 : L L

(3.296)

(3.297)

From the last part of (3.295), with K ¼ Ze2 , we have the semi-major axis of   Ze2  :  (3.298) a¼ 2E  It is curious that this relation does not involve the eccentricity or the angular momentum; the energy depends only on the length of the semi-major axis. The eccentricity is then
ϵ¼

2L2 E 1þ me Z 2 e4

1=2 :

(3.299)

The orbit of the electron can be determined from conservation of energy. The sum of the kinetic and potential energies is  Ze2 me  2 r_ þ r2 θ_ 2  ¼ E: 2 r We can write r_ 2 ¼ θ_ 2 ðdr=dθÞ2 , which gives "
 # dr 2 L2 Ze2 ¼ E: þ r2  dθ 2me r 4 r

(3.300)

(3.301)

The variables in (3.301) can be separated, yielding dθ L , ¼  dr r 2m Er2 þ 2m Ze2 r  L2 1=2 e e

(3.302)

102

Fundamental Microphysics of Ionized Gases

which integrates to " θ ¼ sin

1

# me Ze2 r  L2    θ0 : r 2me L2 E þ m2e Z 2 e4

(3.303)

Choosing θ0 ¼ π=2, we solve for the orbital radius r¼

L2 =me Ze2 L2 1 ¼ :  2   1= 2 2 1  2L E=me Z 2 e4 þ 1 cos θ me Ze ð1  ϵ cos θÞ

(3.304)

The problem now is to determine if elliptical orbits can be quantized. Using the Hamiltonian theory, Ehrenfest proved that if an action variable is an adiabatic invariant (e.g., angular momentum), then that variable may be quantized. A proper discussion of this topic is lengthy and not very relevant to our purpose here, so we summarize the results in simplified form.

3.11.2 Bohr-Sommerfeld Quantization

Þ The quantization rule, with J being the adiabatic invariant, is J ¼ pdq, which is just the area of the ellipse, and the quantum postulate states that the area of the closed curve, in one period, is an integral multiple of h. This is just de Broglie’s hypothesis, that to every moving particle there corresponds a wave such that the linear momentum and the wavelength are related by λ ¼ h=p. Rather than performing the integral in the p  q plane, the proper way is to consider a p  q cylinder of circumference 2π. Then, in general, in complicated systems with many degrees of freedom, we find the Bohr-Sommerfeld quantization rule: þ þ J ¼ pdq ¼ p dq ¼ 2πp ¼ nh: (3.305) We can thus imagine a revolving wave rather than a revolving electron; this is an important point of the quantum theory. This rule may be generalized to multiply periodic systems. These are systems with several degrees of freedom whose motions can be described by a sequence of functions p1 ðq1 Þ, p2 ðq2 Þ,    . The trajectories in phase space are such that the momentum pi is a function of the coordinate qi only, to the exclusion of all other variables. Each function pi ðqi Þ represents a periodic motion of frequency νi . Systems of this kind are said to be separable and thus are multiply periodic. The motion can be resolved into the superposition of simple periodic vibrations and their harmonics (Lissajous’ figures are an example).

103

3.11 One Electron Atom

For the Kepler problem, the momenta pθ and pr are functions of their respective conjugate coordinates and the phase integrals for the system are þ J θ ¼ 2πpθ and J r ¼ pr dr: (3.306) The integral in (3.306) can be evaluated most easily by observing that
 dr dr 1 dr 2 dθ, pr dr ¼ me θ dθ ¼ pθ dθ dθ r dθ

(3.307)

and using (3.304), we find 2π ð

J r ¼ pθ 0

"

ϵ 2 sin 2 θ ð1  ϵ cos θÞ2

dθ ¼ J θ

#

1 1=

ð1  ϵ 2 Þ 2

1 :

(3.308)

Eliminating ϵ between (3.308) and (3.299) gives E ¼ 2π 2

me Z 2 e4 ðJ r þ J θ Þ2

:

(3.309)

Now the Bohr-Sommerfeld quantization rule requires J θ ¼ kh and J r ¼ nr h, so the quantized energy values are E n ¼ 2π 2

me Z 2 e4 Z2 ¼ R ch , ∞ n2 n2 h2

(3.310)

where n ¼ nr þ k is the principal quantum number. The expression for the eccentricity (3.299) can be written ϵ2 ¼ 1 

k2 : n2

(3.311)

Table 3.2 Eccentricities and Semi-axes Lengths n

k

ϵ

a

b

1 2

1 1 2 1 2 3

0pffiffiffiffiffiffiffiffi 3=4 0pffiffiffiffiffiffiffiffi p8=9 ffiffiffiffiffiffiffiffi 5=9 0

1 4 4 9 9 9

1 2 4 3 6 9

3

104

Fundamental Microphysics of Ionized Gases k=2 k=1 n=1 k=1

n=2

k=2

k=3

k=1

n=3

Figure 3.16. Electron orbits for n  3 according to the Bohr-Sommerfeld theory. The eccentricity and lengths of the major and minor semi-axes of the orbits, in units of a0 =Z, are listed in Table 3.2.

Comparing this expressions with (3.292) shows that the ratio of the minor semiaxis to the major semi-axis of the ellipse is ðb=aÞ ¼ ðk=nÞ. The quantized energy (3.310) depends only upon the sum of the quantum numbers nr and k. This property, which is characteristic of the Coulomb potential, derives from the fact that the azimuthal and radial frequencies are equal. To the energy E n , there correspond n quantized orbits, defined respectively by the values of k ¼ 1,2,    , n (we eliminate the value k ¼ 0, which states there is no angular momentum and thus the electron would pass through the nucleus). This is a consequence of the fact that the energy of the system depends upon the phase integrals only through their sum J θ þ J r as seen in (3.309). This sum is, in turn, an expression of the circumstance that the frequencies associated with the periodic motions of r and θ are the same, so that the classical orbit is a closed curve. For all of the orbits having a given n, there is always one that is circular, namely for nr ¼ 0 and thus k ¼ n. This explains why all the energy levels were found by consideration of the circular case alone. Figure 3.16 draws the semiclassical orbits for three values of the principal quantum number. For the situation where there are several different states of motion that correspond to the same total energy, these states are said to be degenerate with respect to that energy. Our simple picture does not permit the elimination of degeneracy, but when a more realistic formulation of the hydrogen-like atom is employed, the effects of neighboring electrons as well as relativistic effects and the inherent spin of the electron remove the degeneracy.

3.11 One Electron Atom

105

The next level of sophistication is to treat the one-electron atom in a fully quantum mechanical framework. 3.11.3 Quantum Theory of Atomic Structure Since we seek to develop a quantum mechanical description of the motion of an electron in discrete states, we can begin with the wave function for a free particle Ψ ¼ eiðkxωtÞ ¼ e h ðpxEtÞ : 2πi

(3.312)

Differentiating (3.312) with respect to x and t, we find the momentum and energy operators h ∂ h ∂ ¼ p and ¼ E: 2πi ∂x 2πi ∂t

(3.313)

If the particle moves along a circle of circumference l, we can replace x with l ¼ 2πr, with r being the radius of the circle. An increase of x by l brings us back to the same point, and Ψ must be a single-valued quantity on the circle. From (3.312) 2πi

e h pl ¼ 1 ¼ e2πin ,

(3.314)

or p ¼ pn ¼

nh nh ¼ : l 2πr

(3.315)

This signifies that in the case of circular motion, (3.314) does not possess solutions for all values of momentum but only for the discrete values 1h=l, 2h=l, 3h=l,    . The formalism of quantum theory is based on the energy functionH ðp; qÞ of the h ∂ Hamiltonian theory. The momentum is replaced in the operator H 2πi ∂q ; q and h ∂ the energy is replaced by the operator 2πi ∂t . Then the energy equation H ðp; qÞ  E ¼ 0 becomes Schrödinger’s equation


h ∂ h ∂ H ;q þ Ψ ¼ 0: (3.316) 2πi ∂q 2πi ∂t Since we seek stationary solutions, that is, those in which the wave function consists of an amplitude function independent of the time and a factor periodic in time (standing vibrations), we make the assumption that Ψ∽eð2πi=hÞEt and (3.316) may be written


h ∂ ; q  E Ψ ¼ 0: (3.317) H 2πi ∂q This is an eigenvalue problem, and we need to find those finite values of E that satisfy (3.317) with the condition of single-valuedness.

106

Fundamental Microphysics of Ionized Gases

Using this simple picture of quantum theory to guide us, we turn to finding a quantum mechanical description of the Kepler problem for which the electron orbits may be ellipses. Let us consider the three-dimensional problem of the oneelectron atom in spherical polar coordinates for a Coulomb force. Schrödinger’s equation becomes
 ~ 8π 2 m Ze2 2 r Ψþ 2 Eþ Ψ ¼ 0, (3.318) r h ~ is the reduced mass of the single electron and the nucleus. In this context, where m r is the radial distance from the center of mass, not from the nucleus. The boundary condition is the behavior of the wave function at infinity. The natural condition to impose is that the wave function should vanish at infinity “more strongly” than 1=r. This follows from the statistical interpretation of the square of the amplitude of the wave function as the probability of the electron being found at a definite point of space. That is, the electron must always be at a finite distance. Expression (3.318) may be split into three equations since it is separable by Ψ ¼ RðrÞΘðθÞΦðϕÞ. The detailed development of the following can be found in many modern physics texts such as Leighton (1959). The differential equation for Φ is d2 Φ þ m2 Φ ¼ 0: dϕ2

(3.319)

The solution is simply Φ ¼ eimϕ , where m must be an integer to keep Φ a singlevalued function of position. The differential equation for Θ is a bit more complex:


1 d dΘ m2 sin θ þ lðl þ 1Þ  Θ ¼ 0: (3.320) sin θ dθ dθ sin 2 θ Writing this equation slightly differently yields the differential equation for the associated Legendre functions Pm l ðμÞ, where μ ¼ cos θ . The normalized solution to (3.320) is

1= ð2l þ 1Þðl  mÞ! 2 m Θlm ðθÞ ¼ Pl ðμÞ: 2ðl þ mÞ!

(3.321)

The orbital quantum number is l; it can take on negative and positive integer values including zero. Hence, there are 2l þ 1 independent spherical harmonics, and jmj  l. Up to this point neither the energy eigenvalue En nor the potential energy function has entered the discussion. This is because the potential energy is a function of only r, and the preceding results will apply to any system involving a central force.

3.11 One Electron Atom

107

We must now evaluate RðrÞ and find the permissible energy eigenvalues En . The differential equation for the radial wave function is 2


 ~ d 2 d 8π 2 m Ze2 lðl þ 1Þ R ¼ 0: (3.322) þ Eþ þ 2  r2 dr2 r dr r h The solution must be finite and continuous for all values of r, from zero to infinity. We are interested in solutions with E < 0, for they correspond to elliptical orbits, and energy must be supplied in order to remove the electron to infinite distance. (The case where E > 0 would correspond to hyperbolic orbits.) To simplify things, we shall measure the radius in multiples of a Bohr radius and the energies in multiples of the ground state energy of the Bohr atom: a0 0 ¼

~ 2 e4 h2 2π 2 mZ 0 and E ¼  : 0 ~ 2 4π 2 mZe h2

That is, we set r ¼ a0 0 ρ and E ¼ ξE 0 0 . Then (3.322) becomes 2

d 2 d 2 lðl þ 1Þ ξ þ  þ R ¼ 0: ρ ρ2 dρ2 ρ dρ

(3.323)

(3.324)

At very large distances from the center of mass, this becomes
2  d  ξ Rð∞Þ ¼ 0, (3.325) dρ2  pffiffiffi which has the solutions Rð∞Þ ¼ exp ρ ξ ; we chose the negative sign since the positive sign would increase the wave function exponentially beyond all bounds as ρ ! ∞. A second special region is that at the origin. Omitting terms in (3.324), which tend to infinity more slowly than 1=ρ2 as ρ ! 0, we obtain 2

d 2 d lðl þ 1ÞÞ Rð0Þ ¼ 0: (3.326) þ ρ2 dρ2 ρ dρ The possible solutions are Rð0Þ ¼ ρl and Rð0Þ ¼ ρl1 . The second of these is not permitted as Rð0Þ ! ∞ as ρ ! 0. Hence the radial wave function has the form pffiffi R∽eρ ξ f ð ρÞ: (3.327) The function f ðρÞ must behave regularly at the origin and at infinity. Using (3.327) in (3.324), we get i pffiffiffi df 2 h pffiffiffi d2 f 2ðl þ 1Þ df þ  2 ξ þ 1  ξ ð l þ 1 Þ f ¼ 0: dρ2 ρ dρ dρ ρ

(3.328)

108

Fundamental Microphysics of Ionized Gases

∞  pffiffiffij pffiffiffi P A power series solution for f ðρÞ in 2ρ ξ may be used; that is f ¼ bj 2ρ ξ . 0

These polynomials are related to the associated Laguerre polynomials. Using these in (3.328) gives 

∞  pffiffiffi j2 X pffiffiffi j1
1 bj 2ρ ξ jð j þ 2l þ 1Þ  bj 2ρ ξ j þ l þ 1  pffiffiffi ¼ 0: (3.329) ξ j¼0 pffiffiffi Since this series must vanish identically for each power of 2ρ ξ , we obtain the recursion relation for the coefficients:


 1 bjþ1 ð j þ 1Þð j þ 2l þ 2Þ ¼ bj j þ l þ 1  pffiffiffi : ξ

(3.330)

The function f ð ρÞ is finite at the origin and thus f ð0Þ ¼ b0 . At infinite distance the function becomes infinite. If however, the series does terminate, Rð∞Þ ¼ 0 despite the fact that f ð∞Þ becomes infinite. We know R vanishes at infinity by pffiffi ρ ξ virtue of the exponential factor e . The series ends after n  l terms. The condition that the series shall terminate is obtained from (3.330), when pffiffiffi pffiffiffi nr þ l þ 1 ¼ 1= ξ . That is, 1= ξ must be a positive integer or ξ ¼ 1=n2 , where n ¼ nr þ l þ 1; n is the principal quantum number, while nr is the radial quantum number. The radial quantum number is just the number of nodes in the radial wave function, not counting the zeros for r ¼ 0 (in the case where l > 0) and r ¼ ∞. We see that solutions of the differential equation that satisfy the conditions of finiteness, continuity, and one-valuedness can be found only for certain values of the parameter ξ; namely ξ ¼ 1=n2 . Hence certain definite energy levels alone are possible, namely En 0 ¼ ξE0 0 ¼ 

~ 2 e4 2π 2 mZ , n2 h2

(3.331)

in agreement with the Bohr theory (3.309). In our development of the quantum mechanical picture of the one-electron atom, we have used the reduced mass rather than the electron’s mass. The connection 0 between these two is found using (3.318), a0 0 ¼ mm~e a0 and En ¼ mm~e En02 . The normalized radial wave function, for x ¼ 2Zr=na0 0 , is Rnl ðxÞ ¼

1=
3= 2 Z 2 ðn  l  1Þ! 2 l x=2 2lþ1 xe Lnl1 ðxÞ, n2 a0 0 ðn þ lÞ!3

(3.332)

3.11 One Electron Atom

109

where Lkj ðxÞ are the associated Laguerre polynomials.6 Letting ρ ¼ Zr=a0 0 , the first few wave functions are
3=2 Z eρ , R10 ¼ 2 a0 0 pffiffiffi
3=2
 1 2 Z 1  ρ eρ=2 , R20 ¼ 2 2 a0 0 pffiffiffi
3=2 6 Z ρeρ=2 , R21 ¼ 12 a0 0 pffiffiffi
3=

2 3 Z 2 2 2 R30 ¼ 1  ρ þ ρ2 eρ=3 , 9 a0 0 3 27 pffiffiffi
3=2
 8 6 Z 1 R31 ¼ ρ 1  ρ eρ=3 , 81 a0 0 6 pffiffiffiffiffi
3=2 2 30 Z R32 ¼ ρ2 eρ=3 : (3.333) 1215 a0 0 Figure 3.17 graphs these radial wave functions. The quantum mechanical description of the one-electron atom, using the Schrödinger theory, is similar in many respects to the Bohr theory. However, there are important differences between these theories, mainly having to do with the question of the exact state of motion of the electron within the atom. In Bohr’s theory, the electron is pictured as executing certain elliptical orbits where size, eccentricity, and orientation are determined by quantization rules, but are otherwise described by classical mechanics. In contrast, in quantum mechanical theory, the electron cannot be pictured as being in a definite orbit but can only be described as being at a given location with a certain probability. Figure 3.18 shows the radial probability density for a few wave functions. The mean value of the radius (expectation value) can be found by using (3.332) in Ð 2 2  a0 0 rR r dr 1  : (3.334) hriqm ¼ Ð 2nl 2 ¼ 3n2  lðl þ 1Þ 2 Z Rnl r dr

6

The associated Laguerre polynomials are defined by Lkn ðxÞ ¼

n X m¼0

ð1Þm

ðn þ kÞ! xm : ðn  mÞ!ðk þ mÞ!m!

Some authors designate by Lkj the polynomial equal to ð1Þk Lkj-k in our notation.

110

Fundamental Microphysics of Ionized Gases 2.0

1.5

R(a0/Z )3/2

R10 1.0 R20 0.5

R31 R21 R32

0.0 R30 –0.5 0.0

2.0

4.0

6.0

8.0

Zr/a0

Figure 3.17. Graphs of the radial wave function Rnl ðr Þ for n ¼ 1, 2, 3, and l ¼ 0, 1, 2.

0.6

(rR)2(a0/Z )

0.5

(rR10)2

0.4 (rR21)2 (rR )2 20 0.4 (rR32)2 (rR31)2

(rR30)2

0.1

0.0 0.0

5.0

10.0

15.0

20.0

25.0

Zr/a0

Figure 3.18. Graphs of the charge-density probability r 2 ½Rnl ðr Þ2 for the wave functions of Figure 3.17.

3.11 One Electron Atom

111

(We omit the messy algebra in getting the last part of this expression.) Similarly, the mean value of the square of the radius is a0 0 2  2 n2  2 : r qm ¼ 5n þ 1  3lðl þ 1Þ Z 2

(3.335)

  0 When l takes its maximum value of n  1, the mean value is hr i ¼ n n þ 12 aZ0 , and  2    2 r ¼ n2 n þ 12 ðn þ 1Þ aZ0 0 , from which we obtain the root-mean-square radial deviation qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n pffiffiffiffiffiffiffiffiffiffiffiffiffiffi a0 0 hri 2n þ 1 (3.336) Δr ¼ hr 2 i  hri2 ¼ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi : 2 Z 2n þ 1 For very large values of n, Δr=r becomes very small and the electron remains practically localized in the vicinity of a sphere of radius n2 a0 0 =Z, whereas the energy of the level, Ze2 =2n2 a0 0 , is the same as that of a classical electron describing a circular orbit of radius n2 a0 0 =Z. We note that states of maximum l correspond to the classical circular orbits of the Bohr theory, where k ¼ n. Because of the exponential behavior of the radial wave function outside the classical region, the quantity jRnl ðrÞj2 has its largest values within the region of the Bohr orbit. For the case l ¼ n  1, which corresponds to the circular orbit, the wave function has just one maximum in this region (see Figure 3.17), and as n becomes large, the position of this maximum can be shown to approach the corresponding classical radius. This is an example of the correspondence principle. Further similarities to the Bohr orbits are made evident by the calculation of other expectation values. Within the framework of the Bohr-Sommerfeld quantization, the period for an _ electron to complete one rotation is found from the angular momentum L ¼ me r2 θ, with the orbital radius given by (3.304): L3 τ¼ me Z 2 e 4

2ðπ

0

1 L3 2π dθ ¼ : 2 4 1  ϵ cos θ me Z e ð1  ϵ 2 Þ3=2

(3.337)

The classical mean value of the radius is found from  3=2 2π ðτ ð 1 L2 1  ϵ 2 1 dθ hricl ¼ rdt ¼ 2 τ me Ze 2π ð1  ϵ cos θÞ3 0

¼ k2



a0 1  ϵ Z 2π

3 2 =2

0

π ð2 þ ϵ 2 Þ 5= 2

ð1  ϵ 2 Þ

¼

 a0 1 2 3n  k 2 , 2 Z

(3.338)

112

Fundamental Microphysics of Ionized Gases

which is identical to (3.334) if we make the association of the classical angular momentum quantum k 2 to the quantum mechanical value of lðl þ 1Þ. Similarly, we can calculate the mean value of the square of the radius ðτ  2 a0 2 1 2 n2  2 5n þ 1  3k 2 r cl ¼ r dt ¼ , τ 2 Z

(3.339)

0

in agreement with (3.335). Note that not all averages over powers of the orbital radius yield the correspondence k2 ! lðl þ 1Þ. There are two significant omissions from both the semiclassical and quantum mechanical theories. First, according to the Bohr theory, the tangential velocity of an electron in a bound orbit may be found from (3.288) and (3.289), vn ¼ 2πZe2 =hn ¼ αcZ, where α is the fine-structure constant. Upon inserting numbers, we see for high-atomic-number atoms that the velocity is a significant fraction of the speed of light, especially for the innermost Bohr orbits. Therefore, relativistic effects must be included in the description of the one-electron atom. Second, we have ignored the effects of electron spin. The next level of sophistication is to redo the one-electron atom using Dirac’s formalism; both of these effects are addressed therein. When the relativistic change in mass of the electron is taken into account, the velocity changes in different parts of the Coulomb potential lead to a small splitting of the levels with l. Excellent discussions on this topic may be found in the literature; see Born (1962).

3.12 Plane Electromagnetic Waves A central feature of Maxwell’s equations for the electromagnetic field is the existence of traveling wave solutions that represent the transport of energy from one point to another. The simplest and most fundamental electromagnetic waves are transverse plane waves. We first consider the waves in a simple nonconducting medium with constant permittivity and permeability. We use CGS Gaussian units throughout. The basic equations governing electromagnetic waves are Maxwell’s equations: Faraday’s law:

rE¼

1 ∂B , c ∂t

(3.340)

where E is the electric field and B is the magnetic induction Ampere’s law:

rH¼

1 ∂D 4π þ J, c ∂t c

(3.341)

3.12 Plane Electromagnetic Waves

where H is the magnetic intensity, D is the displacement, and J ¼

113

P

nj qj uj is the

j

current density with the sum over particle density, charge, and velocity

with ρe ¼

P

Poisson’s equation:

r  D ¼ 4πρe ,

(3.342)

nj qj being the charge density; and the expression for the absence of

j

magnetic monopoles: r  B ¼ 0:

(3.343)

These four fundamental relations are augmented by the constitutive relations D ¼ ϵE and B ¼ μH,

(3.344)

where ϵ is the permittivity and μ is the susceptibility (permeability). If the medium has no free charges ( ρ ¼ 0) and no electric current (J ¼ 0) – that is, vacuum – the wave equations for the electric field and magnetic induction are easily found from (3.340) and (3.341). Anticipating the solutions have the wellknown form eiðk  xωtÞ , where k is the wave vector and ω is the field frequency, and removing the fast time dependence gives rE¼i

ω H, c

(3.345)

with μ ¼ 1 as we consider only nonmagnetic materials, and if the free space permittivity is ϵ 0 r  H ¼ i

ω ϵ 0 E: c

(3.346)

The transverse electric field is found by eliminating H in (3.345) and (3.346), along with r  E ¼ 0, to give r2 E þ

ω2 ϵ 0 E ¼ 0: c2

(3.347)

A field equation for the magnetic intensity may be found in a similar fashion by eliminating E in (3.345) and (3.346). Alternately, the magnetic intensity may be found by differencing (3.345) with respect to the coordinates H¼

c ðk  EÞ: ω

(3.348)

c ðk  HÞ : ω ϵ0

(3.349)

In a similar fashion, (3.346) yields E¼

114

Fundamental Microphysics of Ionized Gases

Eliminating E and H between (3.348) and (3.349) yields the relation between the pffiffiffiffiffi magnitude of the wave vector and the electromagnetic field’s frequency k ¼ ϵ 0 ω=c. The general relation between E and H is given by (3.348) or (3.349). In particular, we can take the scalar product of these expressions with k to obtain k  E ¼ 0 and k  H ¼ 0. Then squaring either, we find E2 ¼ H2 =ϵ 0 . It must be remembered, however, that because all three vectors k, E, and H are complex, these expressions do not in general have the same significance as when we have pure real vectors.

3.12.1 Plane Electromagnetic Waves in a Good Conductor As “plasma” is composed of “free” electrons and ions, there will be an electrical conductivity σ, and Ohm’s law relates the electric field to the current J ¼ σE:

(3.350)

Redoing the preceding analysis, including the current, gives a revised (3.346)
 4π ω ω σ  i ϵ 0 E ¼ i ϵE: (3.351) rH¼ c c c We can define the complex dielectric function (permittivity) ϵ ¼ ϵ0 þ i

4π σ; ω

(3.352)

the conductivity itself may be complex. Expression (3.349), with ϵ 0 ¼ 1 for free space, becomes
 4π ω σi E ¼ iðk  HÞ: (3.353) c c Expressions (3.348) and (3.353) imply, for ϵ 6¼ 0, all three vectors E, H, and k are mutually perpendicular, and the wave is transverse. If ϵ ¼ 0, the equations admit the existence of longitudinal, purely electric (electrostatic) waves with H ¼ 0 and E parallel to k; these are plasma waves (see Section 3.7). Elimination of either H or E from (3.353) with (3.348) yields
2 

 ω 4πω E 2 k  ϵþi 2 σ ¼ 0; (3.354) 2 H c c the propagation vector is now complex:
 ω2 4πσ k2 ¼ 2 ϵ 1 þ i : ωϵ c

(3.355)

3.12 Plane Electromagnetic Waves

115

The first term corresponds to the displacement current and the second to the conduction current contribution. Taking the square root of (3.355), assuming σ is real, we can write k ¼ a þ ib, where 2qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 31=2 4πσ 2

 1 þ ωϵ  1 pffiffiffi ω a 5 : (3.356) ¼ ϵ 4 b 2 c For a poor conductor ð4πσ ωϵ Þ, we find approximately rffiffiffi pffiffiffi ω 2π 1 k ¼ a þ ib
ϵ þ i σ: (3.357) c c ϵ In this limit, ℜeðkÞ ℑmðkÞ, and the attenuation of the wave (ℑmðkÞ) is independent of frequency, aside from the possible frequency variation of the conductivity. For a good conductor ð4πσ ωϵ Þ, a and b are approximately equal, so pffiffiffiffiffiffiffiffiffiffiffi 2πωσ k
ð1 þ iÞ : (3.358) c Waves propagating as eiðk  xωtÞ are damped transverse waves. The fields can be written as

  E E0 b^n  x iða^n  xωtÞ e , (3.359) ¼ e H0 H ^ is a unit vector in the direction of k. Expression (3.342) (with ρ ¼ 0) where n ^ ¼ 0, while (3.348) gives shows that E0  n H0 ¼

c ða þ ibÞ^ n  E0 : ω

(3.360)

This demonstrates that E and H are out of phase in a conductor. Defining the magnitude and phase of k to be "
2 #1=4 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi ω 4πσ , (3.361) 1þ jkj ¼ a2 þ b2 ¼ ϵ c ωϵ with phase angle ϕ ¼ tan

1



 b 1 1 4πσ ¼ tan : a 2 ωϵ

(3.362)

Expression (3.360) may now be written in the form "
 #1=4 pffiffiffi 4πσ 2 ^  E0 : eiϕ n H0 ¼ ϵ 1 þ ωϵ

(3.363)

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Fundamental Microphysics of Ionized Gases

This equation is interpreted so that H lags E in time by the phase angle ϕ and has relative amplitude: "
 #1=4 4πσ 2 jH0 j pffiffiffi ¼ ϵ 1þ : ωϵ jE0 j

(3.364)

In very good conductors, the magnetic field is very large compared to the electric field and lags in phase by almost 45o; the field energy is almost entirely magnetic in nature. The waves given by (3.359) show an exponential damping with distance. That is, the electromagnetic wave entering a conductor is damped to 1=e of its initial amplitude in a distance, referred to as the skin depth, δ¼

1 c
pffiffiffiffiffiffiffiffiffiffiffi : b 2πωσ

(3.365)

Thus, there is a rapid attenuation of waves penetrating into a good conductor, which implies that for high frequencies the current flows only on the surface.

3.12.2 Field Energy in a Dispersive Medium The penetration of an electromagnetic wave into a medium is frequently accompanied by the deposition of some of that energy. The oscillation of the electrons of the medium in the electric field and subsequent collisions with the heavy particles causes heat to be generated. Multiplying Faraday’s law (3.340) by H, Ampere’s law (3.341) by E, and summing the two gives
 1 ∂D ∂B E þH þ 4πJ  E ¼ E  r  H  H  r  E, (3.366) c ∂t ∂t which upon rearranging gives Poynting’s theorem: JE ¼ 

c 1 ∂ r  ðE  HÞ  ðE  D þ H  BÞ: 4π 8π ∂t

(3.367)

Integrating this expression over a finite volume shows that the left-hand side is the total rate of work done by the fields. This power represents the conversion of electromagnetic energy into thermal and/or mechanical energy. It must be balanced by a decrease in the electromagnetic field within the volume. The time derivatives in (3.367) are the change in the electrostatic and magnetic energy densities. Then the total energy density is

3.12 Plane Electromagnetic Waves

ur ¼

 1 1  2 ðE  D þ H  BÞ ¼ E þ H2 : 8π 8π

117

(3.368)

The last part is true if the volume element is free space (ϵ ¼ 1). Expression (3.367) can now be written as J  E ¼ r  S þ

∂u : ∂t

(3.369)

The quantity S¼

c EH 4π

(3.370)

represents energy flow (energy-flux density) and is called Poynting’s vector. For plane waves propagating in the z-direction in homogeneous material, using k ¼ a þ ib, the electric field obeys E ¼ E0 eiðkzωtÞ ¼ E0 ebz eiðazωtÞ :

(3.371)

Using (3.371) and averaging over one period of the electromagnetic wave gives hc i c 1 c c ℜeðE  HÞ ¼ ℜe ða þ ibÞjE 0 j2 e2bΔz ¼ ajE0 j2 eαΔz ; hSi ¼ 4π 2 8π ω 8π (3.372) the energy absorption coefficient is α ¼ 2b. That is, the flux of energy decreases by a factor e over a distance α1 . In steady state, the Joule energy dissipated per second per unit volume is r  S ¼ 

∂S ¼ J  E, ∂z

and using Ohm’s law after averaging is   hJ  Ei ¼ σ E2 ¼ αhSi:

(3.373)

(3.374)

For a wave propagating in the z-direction (3.371), the phase velocity is given by vph ¼ ω=k, which depends on the frequency. If two waves of slightly different frequencies, and thus different phase velocities, are added together, then the  “packet” propagates as eiðkzωtÞ 1 þ eiðΔkzΔωtÞ . The modulation term (in square brackets) propagates with the group velocity vg ¼ Δω=Δk ! dω=dk. The product of these two velocities gives vph vg ¼ c2 =ϵ. The group velocity is the rate of propagation of a signal. It also has significance as the rate of energy transfer, which is defined as the ratio of the time-averaged flux hSi to the mean energy density hur i. Averaging the electromagnetic energy density (3.368) gives

118

Fundamental Microphysics of Ionized Gases



1 ∂ðωϵ Þ EE þ HH : hur i ¼ huE i þ huH i ¼ 16π ∂ω

(3.375)

In the absence of absorption (3.348) becomes H ¼ η^ n  E and ϵ ¼ η2 , where η is the index of refraction. Then hur i ¼

η dðωηÞ E  E : 8π dω

(3.376)

The time-averaged Poynting’s vector is hSi ¼ ðcη=8π ÞE  E , and the velocity of energy propagation is ven ¼

c dω hSi ¼ ¼ vg , ¼ hui dðωηÞ=dω dk

(3.377)

which is just the group velocity. In the case where absorption is present, the picture is more complicated, since the concept of a group velocity is no longer meaningful. The dielectric function is a function of frequency, so the group velocity may be written vg ¼

c : ½ηðωÞ þ ωðdη=dωÞ

(3.378)

For η > 1 and normal dispersion ðdη=dωÞ > 0, the velocity of energy flow is less than the phase velocity and also less than c. In regions of anomalous dispersion, however, dη=dω can become large and negative, and the group velocity is greatly different from the phase velocity, often becoming larger than c; clearly, in this case, the concept of group velocity is no longer meaningful. In a dispersive medium, this leads to the spreading of the wave as it propagates through the medium. Dispersion of electromagnetic waves is of significance in high-energydensity physics; we consider this in more detail in Chapter 12. In many of the preceding equations, we hint that the electrical conductivity plays an important role in the propagation of electromagnetic energy. We now address the physics behind the conductivity.

3.13 Permittivity and Electrical Conductivity Equation (3.352) expresses the relation between the permittivity and the conductivity. In certain instances, it is the permittivity that is easiest to use, while in others it is the conductivity. The permittivity is determined by the motion of the free electrons as well as the heavy particles. For frequencies that are high compared to the natural frequencies of the material, the polarization vector P is found by

3.13 Permittivity and Electrical Conductivity

119

regarding the electrons as free while neglecting their interaction with one another and with the heavy particles. Consider a specific electromagnetic frequency ω and that the velocities of the electrons are nonrelativistic and thus move distances comparable to the interatomic spacing R0 , which is small compared to the wavelength of the electromagnetic radiation. If we allow that the electromagnetic field is approximately uniform, then the equation of motion for an electron is simply me €x ¼ eE, and the resulting displacement of the electron is xðt Þ ¼ eEðt Þ=me ω2 . The polarization of a body is the dipole moment per unit volume, so summing over all electrons within the volume, we obtain X e2 ne P¼e x¼ E: me ω2

(3.379)

The electric displacement is related to the polarization by D ¼ ϵE ¼ E þ 4πP, from which we find ϵ ¼ 1  4π

ω2pe e2 ne 1 ¼ 1  , me ω2 ω2

(3.380)

where ωpe is the electron plasma frequency discussed in Section 3.7. (We have assumed that the permittivity of free space is infinitestimally different from one.) In reality, the motion of the heavy particles must be accounted for. Then the equations of motion result in ! 4πe2 ne X nj : (3.381) þ ϵ ¼1 2 ω me mj j This equation contains no information about the conductivity of the material; some additional physics is needed. The simplest model for electrical conductivity, originally due to Drude, is that in a metal there is a density of electrons free to move under the action of an applied electric field but subject to damping forces by collisions. The equation of motion for a single electron is x þ νeff x_ ¼ €

e e E¼ E0 eiωt , me me

(3.382)

eE : me ωðω þ iνeff Þ

(3.383)

which has the solution x¼

In the preceding equations, νeff is an effective collision frequency, in which each collision of an electron with a heavy particle gives the heavy particle an average

120

Fundamental Microphysics of Ionized Gases

_ As these are Coulomb collisions, the material discussed momentum of order me x. in Sections 3.8 and 3.9 are relevant. The effective collision rate is related to the inverse of a collision time. For the present time, we suggest that νeff is the electron– ion collision frequency. In the absence of collisions, (3.383) gives x ¼ eE=me ω2 . The time derivative of this quantity gives the electron’s oscillating (also known as the quiver) velocity: vosc ¼

eE0 : me ω

(3.384)

Proceeding along the same lines as for the development of (3.380), the complex plasma permittivity is ϵ ¼1

 ω2pe ω2pe νeff  1  i ¼1 2 : ωðω þ iνeff Þ ω ω þ ν2eff

(3.385)

This formula is often called the Drude dielectric function. Comparing (3.379) with (3.347), the Drude conductivity is σ¼

ω2pe 1   ðνeff þ iωÞ: 4π ω2 þ ν2eff

(3.386)

The DC electric conductivity, defined for ω ¼ 0, is σ dc ¼ ω2pe =4πνeff . The complex dielectric permittivity may be expressed in terms of the familiar indices of refraction and absorption, respectively η and χ: ϵ ¼ ϵ r þ iϵ i ¼ ðη þ iχ Þ2 ¼ 1 þ i

4π σ: ω

(3.387)

The relations ϵ r ¼ η2  χ 2 and ϵ i ¼ 2ηχ are obvious. The two parts of (3.387) may be rewritten to provide the indices in terms of the real and imaginary parts of the dielectric permittivity, η¼

ϵr þ

pffiffiffiffiffiffiffiffiffiffiffiffiffiffi!1=2 ϵ 2r þ ϵ 2i , 2

(3.388)

and χ¼

ϵ r þ

pffiffiffiffiffiffiffiffiffiffiffiffiffiffi!1=2 ϵ 2r þ ϵ 2i , 2

(3.389)

where the sign of the interior root was chosen so as to keep χ a real number.

3.13 Permittivity and Electrical Conductivity

121

Looking at (3.387) through (3.389), we see that for the case of no conductivity, pffiffiffi pffiffiffi pffiffiffi η ¼ ϵ ¼ 1 and χ ¼ 0. If jϵ j 4πσ=ω,pηffiffiffiffiffiffiffiffiffiffiffiffiffi
ϵffi
1 and χ
2πσ=ω ϵ . In the opposite case of jϵ j 4πσ=ω, η
χ
2πσ=ω. Returning to the dispersion relation k2 ¼ ω2 ϵ ðωÞ=c2 , since ω is real, the wave vector is, in general, complex. The difference in the directions kr and ki correspond to the difference in the planes of equal phase and equal amplitude. Such planes are said to be inhomogeneous. For homogeneous plane waves, the planes of equal phase and amplitude are parallel, and the wave vector may be written k¼

ω ðη þ iχ Þ^ n, c

(3.390)

^ is a unit vector. From (3.348) where n H ¼ ðη þ iχ Þ^ n  E: This may be compared to (3.360). Writing in polar coordinates pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðη þ iχ Þ ¼ η2 þ χ 2 eiϕ ¼ η2 þ χ 2 ð cos ϕ þ i sin ϕÞ,

(3.391)

(3.392)

and again we see that H lags E by the phase angle ϕ ¼ tan 1 ð χ=ηÞ; the relative amplitude is pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi jH0 j c ¼ jkj ¼ η2 þ χ 2 , jE0 j ω

(3.393)

which should be compared to (3.364). We have finished with the preliminaries and next turn to discussions of some of the important aspects of high-energy-density physics.

4 Ionization

Ionization is the process of removing one or more bound electrons from the eigenstates of an atom into a continuum of levels, with the liberated electron having a finite energy at infinite distance from the atom. (We assume that the temperature and/or density conditions of the matter are sufficient so that all molecules have been dissociated into the constituent atoms.) The continuum begins at the ionization potential of the atom above the ground state; this is depicted in Figure 4.1. Every atom has a sequence of ionization stages, each with a progressively higher ionization potential as successive electrons are removed. With the Bohr model of the atom in mind, the ionization potentials increase because each subsequent electron being removed is closer to the nuclear charge as well as being less screened from the nuclear charge than the previously removed electron. We denote the successive ionization potentials by I m such that I 1 is the energy required to remove the first (outermost) electron from the neutral atom, I 2 is the energy required to remove the second electron from the singly ionized atom, and so on. To help visualize this, consider the carbon atom: with one electron removed, it resembles a boron atom; with two electrons removed, it resembles a beryllium atom; and so on. The energy required to completely strip the atom of its electrons is the sum of these single-particle ionization energies I m . Figure 4.2 shows the successive ionization potentials for a few elements of the periodic table; these data were obtained from a relativistic Hartree-Fock calculation for isolated atoms (Scofield, 1975). At low density, individual atoms/ions interact only slightly, and they may be considered as isolated. As the density is increased, neighboring particles perturb the states so that an isolated atom description is no longer appropriate. In the absence of ionization, the particle interactions must be taken into account for densities greater than about one-tenth of the normal solid density. However, when ionization is considered, the long-range Coulomb force enhances the interaction. This becomes important when the Coulomb potential energy at the average 122

123

Ionization

Em+1,k

Em,j Em+1,2 Em,2

Em+1,1 Im+1

Em,1

Figure 4.1. An atom in the mth ionization stage is further ionized to the ðm þ 1Þth stage by addition of energy. The difference in the ground state energies of the two stages is the ionization potential. 102

Lonization potential (keV)

101

100

10–1

10–2 C

Fe

Al 10–3

0

16

Au

Ag

32

48

64

80

Number of bound electrons

Figure 4.2. Ionization potentials as a function of the number of bound electrons for selected elements.

interatomic separation is greater than about one-fiftieth of the thermal energy. The effect of the free-electron sea is to change the boundary conditions at large distances for the potential well in which the bound charges move. This causes some of the higher-bound states to merge into the free-electron continuum. The new boundary also shifts all of the bound-level energies.

124

Ionization

Neutral atoms, ions, and free electrons in thermal equilibrium obey the laws of statistical mechanics; the particular distribution function is proportional to the Boltzmann factor eI=kB T , where I is a potential of some form. We see that the ionization process depends upon temperature and begins at kinetic temperatures much lower than the ionization potential. The reason for ionization beginning at low temperature is that the statistical weight of the free electron is very large. Ionization begins sooner the lower the ionization potential. Successive stages of ionization proceed until all bound electrons are liberated, but the onset of the next stage of ionization begins before the previous stage ends. Thus, the atoms in a volume of gas have a distribution of ionization stages, ranging from the neutral atom to the fully stripped atom. In practice, there are only a few stages distributed about some average value. The ensemble of the distribution is characterized by an average ionization level Z  . If the “material” consists of a mixture of elements, then it will contain differently charged ions of each element. We use the term “material” in a broad sense; most commonly, we think of a gas, but it also applies to warm, dense matter even though that state may exhibit characteristics of a liquid or solid-state matter. Our discussion proceeds, assuming we have a simple gas consisting of atoms of a single element. The ionization level Z  is the effective number of free electrons per ion. Even though our discussion, in Section 3.11, about the Bohr atom and its quantization addressed a single atom with just one electron, as we shall see, Z  need not be an integer when an ensemble of ions in thermal equilibrium is considered. The level of ionization is central to much of the theory of high-energy-density matter, from hydrodynamic response (governed by the equation of state), to conduction of thermal energy, to the transport of radiation, and so on. For example, the plasma pressure is linear in Z  , the electron thermal conductivity contains a factor ðZ  Þ2 , the bremsstrahlung emission rate is proportional to ðZ  Þ2 , and photoelectric absorption cross section is proportional to ðZ  Þ4 . There are a number of processes that can produce an ionization event. These may be, for the most part, grouped into three types: charged particle collisions, radiationinduced ionization, and intense electromagnetic field ionization. In turn, each of these three categories has a processes that depend upon the energy/intensity and type of ionizing particle. In this chapter, we initially focus on the ionization physics produced by thermal distributions of electrons that were discussed in Chapter 3. Ionization resulting from the impact of an electron (or from the absorption of a photon) may be viewed as a chemical reaction. The electron collisional ionization reaction for an atom in its ground state A0 is represented as e þ A0 ! A1 þ 2e , which is an endothermic reaction; the energy of the incident electron must be greater than the ionization potential. The inverse process is an exothermic reaction and referred to as three-body recombination.

Ionization

125

The rates governing these two reactions are rather complex as described in expressions for the cross sections, but for now we can express them in the forms Rðe þ A0 ! A1 þ 2e Þ ¼ αðeÞ ne n0 ,

(4.1)

where n0 is the density of ions before the ionization event, and RðA1 þ 2e ! e þ A0 Þ ¼ βðeÞ n1 n2e ,

(4.2)

with n1 being the density of ions in the ionized stage; ne is of course the number density of free electrons. The cross section information is contained in the coefficients αðeÞ and βðeÞ . The dimensions of αðeÞ are L3 T1, and for βðeÞ they are L6 T1. We see that both reaction rates depend upon the number densities of the initiating particles. In a nonequilibrium state, the rate of change in free-electron density is determined by the competition between (4.1) and (4.2): dne (4.3) ¼ αðeÞ ne n0  βðeÞ n1 n2e : dt For this example, particle conservation requires n1 ¼ ne . Under equilibrium conditions, the principle of detailed balance requires dne =dt ¼ 0, and thus
ne ¼

αðeÞ n0

1=2

βðeÞ

;

(4.4)

we see that only the ground state of the atom need be considered. At temperatures so low that αðeÞ may be neglected, the initial electron density decays according to dne ¼ βðeÞ n3e , dt since n1 ¼ ne , and the solution is ne ðtÞ ¼ h

ne ð0Þ 1þ

2βðeÞ n2e ð0Þt

(4.5)

i1=2 :

(4.6)

If the ionizing reaction from the ground state involves the absorption of a photon of energy hν, where hν > I, then hν þ A0 ! A1 þ e ; the inverse process is radiative recombination, which is also referred to as two-body recombination. The two reaction rates are Rðhν þ A0 ! A1 þ e Þ ¼ αðrÞ n0 ,

(4.7)

RðA1 þ e ! hν þ A0 Þ ¼ βðrÞ ne n1 :

(4.8)

and

The units of αðrÞ are T1 and for βðrÞ , L3 T1.

126

Ionization

Some of the ionization models we address in this chapter assume a simple picture: that of the Bohr atom. Only principal quantum numbers n are of interest. Previously, in Section 3.11, we noted the degeneracy of the nth eigenstate is given by 2

n1 X

ð2l þ 1Þ ¼ 2n2 ,

(4.9)

l¼0

providing there are no interactions with neighboring ions. The isolated atom has an infinite number of bound states, but Figure 4.1 shows a maximum level with a finite but large quantum number, say n  10, which defines the zero of energy E∞ ¼ 0. The energies of the levels are measured against this reference point. The simple Bohr model places these eigenstate energies at Z2 , (4.10) n2 where Z is the atomic number of the atom and I H ¼ 2π 2 me e4 =h2 ¼ 0.0136 keV is the ionization potential of the hydrogen atom. This choice of zero energy allows us to define the free electrons to have positive energy. The E ∞ ¼ 0 level defines the “bottom” of the free-electron continuum. There is an important distinction in terminology used here: ionization potentials, I n , are the negative of the level energies, that is, I n ¼ E n . Bound states of the atom have level energies less than zero, while electron states in the continuum have positive energy. Plasma in local thermodynamic equilibrium (LTE) exists under ideal conditions as required by equilibrium statistical mechanics. The ionization state of the material is defined by the local temperature and electron number density. Establishment of a level of ionization is governed by the competition between two processes: collisional ionization and its inverse, and radiative ionization and its inverse. In many cases, the departure from LTE is severe, especially at low density, and is largely driven by radiative processes that deviate from equilibrium values; the matter may be in steady state but not in LTE! At these low densities, collisional ionization is balanced by radiative recombination; this is the so-called coronal equilibrium, for which the solar corona is an excellent example. At high densities, the balance is between collisional ionization and three-body (collisional) recombination. In general, the following are the conditions for LTE: E n ¼ I H

1. Collisional processes dominate radiative processes. This is not true for low-density plasma where photoionization rates and radiative recombination rates vastly exceed collisional rates. Dielectronic recombination must also be taken into account.1 1

Dielectronic recombination is the mechanism by which a free electron is captured by an ion and the excess energy of the recombination is taken up by a bound electron, which then occupies an excited state. Thus, the

4.1 Saha

127

2. All transition processes are in detailed balance. Collisional processes are described by a Maxwellian (or perhaps a Fermi-Dirac) distribution function, and a transition is exactly balanced by its inverse; therefore, elastic collisions dominate. In addition, radiative processes take place at their equilibrium rates as characterized by a Planckian distribution function. The radiative processes are closely coupled to the collisional processes. The plasma is not radiation-dominated. 3. Temperature and density gradients are shallow, so that the rate of change of equilibrium distributions is small compared to the rate constants for the associated atomic processes. That is, the plasma is locally steady state. Clearly, the general treatment of ionization is highly dependent on the details of atomic structure. Calculations incorporating the important atomic physics features can be exceedingly complicated and not at all suited for inclusion in numerical simulations of high-energy-density physics. We seek, therefore, simple approximations to the ionization of LTE plasma. For a sufficiently dense material, the two types of particles with mass, ions, and free electrons can be thought of as two distinctly separate fluids; the two-fluid description will be discussed in subsequent chapters. The ion fluid plays no essential role in ionization processes. That is not to say the ions are not important, as we shall see. Our attention is drawn to a single atom, but we will be mindful of the effects neighboring ions bring to the model atom/ion.

4.1 Saha The Saha ionization model was developed by Meghnad Saha in 1920 as a theory to explain the spectral classification of stars and was later applied to modeling their interiors. This equation, which assumes nondegenerate free electrons, gives the ratio between densities of ions in successive ionization stages. This model is valid only for weakly ionized plasma, where the screening of the ion charge by other ions and the free electrons is negligible. The Saha model has its foundation in statistical mechanics (Cox and Giuli, 1968), of which we discussed some aspects in Section 3.2. The correct Boltzmann counting for indistinguishable particles, which resolves Gibbs’ paradox, leads to the partition function X gj eEj =kB T , (4.11) Z~ ¼ j

initial recombination is radiationless. The doubly excited ion then relaxes either by autoionization or via radiative cascades. Dielectronic recombination competes with radiative recombination.

128

Ionization

where gj is the degeneracy of the microstate with energy Ej . The ionization process, whether by electron impact or photon absorption, takes an atom in the mth ionization stage and transforms it to the ðm þ 1Þth stage while promoting the bound electron into the continuum. Then, for the three types of particles present in an ionizing event, the total probability is the product of the individual probabilities:  nm  nmþ1  ne Y   Z~ m Z~ mþ1 Z~ e   : (4.12) W¼ W nj ¼ nm ! nmþ1 ! ne ! j As we are interested in the most probable state, (4.12) should be maximized, but it is more convenient to maximize logW since it will have a maximum at the same place as W. For nm large, we can approximate the factorials using Stirling’s formula to arrive at ~ m þ nmþ1 log Z ~ mþ1 þ ne log Z ~ e  nm ð log nm  1Þ logW ¼ nm log Z nmþ1 ð log nmþ1  1Þ  ne ð log ne  1Þ: (4.13) This relation is subject to the conservation of particles, so that a change in the number density of one type results in a change of the others according to δnm ¼ δnmþ1 ¼ δne . Taking the variation of (4.13), with respect to the change in the number of m-ions arising from the ionization, gives the result ~ mþ1 þ log Z ~ e  log Z ~ m  log nmþ1  log ne þ log nm ¼ 0, log Z

(4.14)

which yields the fundamental relation nmþ1 ne Z~ mþ1 Z~ e ¼ : nm Z~ m

(4.15)

The partition function for each type of particle is the product of its translational and structural partition functions. The electron’s partition function has only the translational component, while the two ion states have both translational and structural terms. The translational partition functions follow from the Boltzmann law; they are integrals of the distribution function over phase space ðð 4πg 2 ~ (4.16) Z trans ¼ 3 p2 ep =2mkB T dpd3 x, h where g is the degeneracy of the particle; for the electron, g ¼ 2 to account for the two spins as required by the Pauli exclusion principle. Performing the integral results in gV = Z~ trans ¼ 3 ð2πmkB T Þ 2 : h 3

(4.17)

4.1 Saha

129

The structural partition function for an ion with a hydrogen-like configuration is 
 X X IH Z2 1 wn =kB T 2 ~ Z struc ¼ (4.18) gn e ¼ 2n exp  1 2 : n kB T n n Since we are considering only principal quantum numbers, the degeneracy of the nth level is 2n2 , as given by (4.9), and the excitation energy of the level is wn ¼ E 1  En ¼ I H Z 2 ð1  1=n2 Þ, with I H being the ionization potential of hydrogen. Using the partition functions in (4.15), we arrive at the Saha equation for ionization nmþ1 g 2 ne ¼ mþ1 3 eI mþ1 =kB T , nm gm λde

(4.19)

where we have used the expression for the thermal de Broglie wavelength of the pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi electron λde ¼ h= 2πme k B T . We have assumed the mass of the ion in the mth ionization stage is the same as that of the ion in the ðm þ 1Þth stage. Rather than use the hydrogen-like expressions for the ionization potentials, as seen in (4.18), we allow for more realistic values in (4.19). Note that the ground stage in (4.19) is for m ¼ 0. In Section 3.4, for low-density plasma, we found an expression for the electron number density in terms of the chemical potential: ne ¼

2 μ=kB T e : λ3de

(4.20)

Using this in the Saha equation (4.19) gives nmþ1 gmþ1 ðμþI mþ1 Þ=kB T ¼ e : nm gm

(4.21)

  From (4.20), we can define μ ¼ k B Tξ, where ξ ¼ log 2=ne λ3de . The quantity ne λ3de is the number of electrons in a cubic de Broglie wavelength, typically, ξ ¼ 10  30. The ratio of the stage degeneracies in (4.21) is of order unity, and therefore the exponential term dominates. This means that the two stages are populated equally when μ ¼ I mþ1 , which implies that the thermal energy at this transition temperature satisfies kB T  I mþ1 =ξ, which is a good order of magnitude less than a first guess of kB T  I mþ1 . That is, at a temperature of kB T ¼ I mþ1 =ξ, there is a substantial energy penalty to be paid by ionizing an electron, but the electron then has more phase space to occupy. In addition, the transition between ionization stages is usually quite sharp, on the order of ΔT  T=ξ, so for fixed density, the stage most populated changes abruptly with temperature; that is, there is usually one ionization stage that dominates.

130

Ionization

12

24

10

20 lonization level

(b) 28

lonization level

(a) 14

8 6

16 12

4

8

2

4

0 10–3

10–2

10–1

100

0 10–3

101

Temperature (keV)

10–2

10–1

100

101

Temperature (keV)

Figure 4.3. Ionization levels from the Saha ionization model: (a) for aluminum at densities 0.00027 (solid line), 0.027 (short-dashed line), 2.7 (long-dashed line), 270.0 (dot-dashed line) g cm3; (b) for iron at densities 0.00079 (sold line), 0.079 (short-dashed line), 7.9 (long-dashed line), 790.0 (dot-dashed line) g-cm3.

The ionization stage predicted by (4.19) exhibits plateaus associated with gaps in the ionization potentials between inner shells, as seen in Figure 4.2; that is, the ionization potentials are not continuous functions. For example, helium-like ions are stable over a wide range of temperatures, especially at low temperatures. Figure 4.3 shows the ionization level predicted by the Saha equation for aluminum and iron at several densities. At low temperatures, where the atom is single ionized, the second ionization potential is approximately twice that of the first ionization potential, and we can simplify (4.19) to one equation for m ¼ 0 y2 g 2 I 1 =kB T e , ¼ 1 1  y g0 ni λ3de

(4.22)

where y ¼ ye ¼ y1 ¼ 1  y0 , with ni ¼ n0 þ n1 being the heavy particle number density. For hydrogen gas, g0 ¼ 2 and g1 ¼ 1, and we can write (4.22) as 3=

y2 θ2 ¼ 158:5 e:0136=θ , 1y ρ

(4.23)

where θ is the temperature (measured in keV) and ρ is the mass density. Equation (4.23) is easily solved for y, which is the ionization level Z  . For I 1 =kB T  1, Z  e ρ1=2 eI 1 =2kB T ; the ionization level increases rapidly with temperature and very slowly with decreasing density. A sequence of Saha equations (4.19) shows that there is a distribution of ion stages about some average ionization value. Let us consider the distribution of

4.1 Saha

131

stages to be a continuous function, and we may then replace the system of equations with a differential equation. For a specific density and temperature, using the differential nmþ1 ¼ nm þ

dn Δm, dm

(but Δm ¼ 1) in (4.19) gives   1 dn 1þ ne ¼ AeI mþ1 =kB T , nðmÞ dm

(4.24)

(4.25)

where nðmÞ is a continuous function. For m  0, the ratio of the statistical weights gmþ1 =gm is of order unity and may be omitted. Using particle and charge conservation, the average number of free electrons per ion is Ð mnðmÞdm ne ¼ : m¼ Ð (4.26) ni nðmÞdm At the peak of the distribution, dn=dm ¼ 0, and thus A ¼ mni eþI =kB T :

(4.27)

The ionization potential corresponding to the average ionization stage m is I . The resulting Saha equation for the first stage is then n1 ¼ eðI 1 I Þ=kB T , n0

(4.28)

with similar expressions for the following stages. Combining these gives the ratio of the number of ions in the mth stage to the ground stage m ¼ 0: m nm Y ¼ eðI i I Þ=kB T : n0 i¼1

(4.29)

This can be reduced to "
2 # nm 1 Im  I , ¼ exp  2 kB T n0

(4.30)

which has a Gaussian shape. Treating the ionization potential as a continuous function also, and expanding about I , gives I ð mÞ ¼ I þ and using this in (4.30) gives

dI ðm  m Þ, dm

(4.31)

Ionization

n/n0

132

1

6

11

16

21

26

Charge state

Figure 4.4. Distribution of charge stages has an approximately Gaussian distribution. The material is iron at a particle density of 8:5  1020 cm3 and temperature 0.1 keV. Nearly all of the ions have a charge within a few units of the average ionization level.

"
# nm mm 2 , (4.32)  exp  Δ n0

where Δ2 ¼ 2k B T= dI=dm . Figure 4.4 shows the distribution of relative populations of ionization stages for iron. Nearly all of the ions have a charge within a few units of the average ionization level m. From the figure, the peak ionization value is m  13:8, while the solution to the Saha equation gives essentially the same value for Z  . Let us return, for the moment, to the expression governing the detailed balance between collisional ionization and three-body recombination (4.3). In equilibrium, we have αðeÞ ne n0 ¼ βðeÞ n1 n2e , and using (4.19) we see the ratio of the ionization to recombination coefficients is βðeÞ g0 λ3de I 1 =kB T ¼ : e αðeÞ g1 2

(4.33)

If the charged ion and electron particle concentrations are small compared to the neutral atom values, then from (4.3) recombination is not significant and ionization

4.1 Saha

133

proceeds by electron impact, much as that of an electron avalanche. Assuming the temperature is constant in time and that the number density of the ground stage is approximately constant, then dne =dt ¼ ne =τ e , where the “relaxation” time for equilibrium is τ e ¼ 1=αðeÞ n0 . This leads to the temporal dependence of the electron number density ne ðtÞ ¼ ne ð0Þet=τ e . The typical dependence of the ionization cross section σ e on the electron velocity increases from the ionization threshold E e ¼ I, to a maximum at an electron energy several times the threshold energy, and then slowly decreases. Only those electrons in the tail of the Maxwell-Boltzmann distribution function possess sufficient energy for ionization; the number of such electrons is exponentially small, proportional to exp ðme v2 =2k B T Þ  1. Therefore, the dominant electrons are those whose energies only slightly exceed the ionization potential. We may assume the cross section has the dependence σ e ðvÞ  C ðE e  I Þ, where C is a constant. The rate constant for ionization from the ground stage is ðeÞ

α

ð∞ ¼ vk


 I σ e ðvÞvf MB ðvÞdv ¼ σ e hvi þ 2 eI=kB T , kBT

(4.34)

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where the mean thermal velocity is hvi ¼ 8kB T=πme , and σ e is an average value of the velocity-dependent cross section, the value of which corresponds precisely to the electron energy E e ¼ I þ kB T; thus σ e ¼ Ck B T. Using (4.33), the rate constant for three-body recombination from the first ionization stage is
 g0 h3 I ðeÞ þ 2 σe: β ¼ (4.35) g1 2π 2 m2e kB T k B T The Saha equation may also be applied to the excitation/de-excitation of atoms/ ions. The ionization process may be viewed as a limiting case of electronic excitation, and rate equations similar to (4.1), (4.2), (4.7), and (4.8) can be used. The number of excited atoms is usually much fewer than the number of atoms in the ground stage. The role of ionization from excited states is not insignificant, since ionization of excited atoms is caused by collisions with particles of lower energies. A more in-depth discussion about this topic can be found in Zel’dovich and Raizer (1966). The Saha ionization model cannot treat high-density conditions where interatomic energies become dominant; the model is formulated in terms of exact energy levels of the isolated ion. In quantum mechanical terms, the Hamiltonian is diagonal in that electrons are localized on specific ions. There are additional terms in the Hamiltonian that are not diagonal. These represent wave functions centered on one ion overlapping states centered on an adjacent ion. This overlap leads to nonzero matrix elements of the kinetic and potential energy operators.

134

Ionization

In low-density plasma, the off-diagonal terms of the Hamiltonian are exponentially small. In contrast, in high-density plasma the off-diagonal terms are important. The result is that the concept of hydrogen-like ions is not tenable and the Saha model is unsatisfactory. We can demonstrate this, for hydrogen, by looking at the electron–ion coupling parameter Γei ¼

Z  e2 : R0 kB T

(4.36)

This parameter is a measure of the electrostatic repulsion energy to the thermal energy evaluated at the ion-sphere boundary R0 . One sees that Γei < 1 for low density and/or high temperature. Using the expressions for the ion-sphere radius, the electron de Broglie wavelength, and the ionization potential of the first stage, the coupling parameter can be written Γ3ei


3=2 8 ni λ3de  3 I 1 ðZ Þ ¼ pffiffiffi : 3 π 2 kBT

(4.37)

Now, using the Saha equation (4.22), with y ¼ Z  , to replace ni λ3de , we conclude Γ3ei


3=2 8  I1  g1 ¼ pffiffiffi Z ð1  Z Þ eI 1 =kB T : 3 π g0 kB T

(4.38)

One sees that Γ3ei  0:154=gm (gm > 1) because Z  ð1  Z  Þ  1=4 for all Z  and x3=2 ex  0:41 for all x. In addition, g1 =g0 < 1, thus Γei < 0:54 for all densities and temperatures. We close our discussion of the Saha equation with three comments: 1. At low density, plasma may not be in thermal equilibrium and may be optically thin. The ionization level is controlled by competition between collisional ionization and three-body and radiative recombination rates, which depend upon the entire plasma configuration, such as the flux of electrons and photons. 2. The model fails at high density, where interatomic energies become dominant, giving multiple solutions: one is highly ionized, the other nearly neutral. The interatomic interactions result in electrons hopping from ion to ion, causing rapid fluctuations. 3. The free-electron partition function (4.17) could be replaced with an expression obtained from Fermi-Dirac statistics. We might think this would include degeneracy effects, but without additional modifications, the Saha equation would be unchanged and still predict excessive recombination at high densities. Thus, the free electrons do not become degenerate.

4.2 Thomas-Fermi

135

The Saha equation is a consequence of statistical mechanics where the ionization (and possibly excitation) energies are important; these are single-particle energies. As plasma becomes denser or colder, energies associated with particle interactions become important. Quantum effects change the partition functions significantly through degeneracy and through ion–ion correlations. Despite the limitations to the Saha model, it remains a powerful tool for calculating ionization levels, providing we keep in mind the temperature-density region for which it is valid. Next, we address a model that overcomes some of the objections of the Saha model, particularly the issue of high density.

4.2 Thomas-Fermi The preceding discussion assumed that free electrons obey Boltzmann statistics, which is true for high temperature or low density. Strictly speaking, an electron gas is described by Fermi-Dirac quantum statistics. The transition from Fermi-Dirac to Boltzmann statistics occurs if the temperature of the electron gas is much greater than the Fermi temperature, as discussed in Section 3.4. Another way to picture the structure of the atom is to consider the electrons, both bound and free, to be a continuous “fluid.” There are no ions in the sense that the Saha model invokes; only electrons and nuclei exist. The model postulates that electrons are distributed uniformly in phase space with two electrons in every h3 of volume. For each volume element of coordinate space, a sphere of momentum space can be filled up to the Fermi momentum pF . Equating the number of electrons in coordinate space to that in phase space gives nðrÞ ¼ 2

4π 1 3 p ðrÞ: 3 h3 F

(4.39)

The Thomas-Fermi statistical model concentrates on the spatial electron density nðrÞ (Feynman, Leighton, & Sands, 1949). The model atom assumes the boundary is that of the ion sphere R0 . Poisson’s equation r2 ΦðrÞ ¼ 4πe½nðrÞ  ZδðrÞ

(4.40)

gives the electrostatic potential ΦðrÞ within the ion sphere. The boundary condition of lim ΦðrÞ ¼ Ze=r is just the Coulomb potential of the nucleus with charge Z. r!0 Charge neutrality of the sphere requires dΦ=dr ¼ 0 at r ¼ R0 , and the zero of electron energy is chosen such that ΦðR0 Þ ¼ 0. Electrons inside R0 have E < 0, while those outside R0 have E > 0; this is the same convention as used in the Saha model. The electron density distribution is determined from the finite-temperature semiclassical electron gas

136

Ionization

2 nðrÞ ¼ 3 h

ð

h exp

1 kB T



1

p2 2me

i d3 p,  μ  eΦðr Þ þ 1

(4.41)

where p is the electron’s momentum and μ is the chemical potential determined by the requirement that the cell be charge-neutral ð∞ nðrÞd 3 r ¼ Z,

(4.42)

0

which is equivalent to dΦ=dr ¼ 0 at r ¼ R0 . The integrand in (4.41) is the FermiDirac distribution function presented in Section 3.4. Equation (4.41) is more commonly written as
 4 1 μ þ eΦðrÞ nðr Þ ¼ pffiffiffi 3 F 1=2 , (4.43) π λde kBT where F j ðxÞ is the Fermi-Dirac integral of order j. The potential ΦðrÞ must satisfy both (4.40) and (4.41); the combined equations are then
 pffiffiffi 1 d2 16 π e μ þ eΦðr Þ ½rΦðr Þ ¼ F 1=2 : (4.44) r dr 2 kB T λ3de Through the potential ΦðrÞ, electrons in one part of the ion sphere interact with the remaining electrons. The distribution of electrons (4.43) allows for thermal excitation and ionization, although the electron states are not quantized as in the Saha model, but simply represented by their classical phase-space density. The Thomas-Fermi theory includes a number of plasma density effects: 1. Ion–ion correlations enter through the ion-sphere model if the ion-coupling parameter Γii is large. 2. Free electrons may be degenerate and also experience the electrostatic potentials of neighboring ions. 3. The free electrons are not spatially uniform as their density is calculated selfconsistently. 4. At high density, the free electrons are forced into the ion core and the potential ΦðrÞ includes the resulting screening of bound electrons by free electrons. This last point is depicted in Figure 4.5, where in the case of the isolated neutral atom, the potential may be taken to be zero when all charges are at infinite separation; the chemical potential is then zero. In the case of the compressed atom, the electron density at the cell boundary (ion-sphere radius) is finite. The core electrons are “pressure-ionized” at high densities; this process is a continuous function of density.

137

n (r )

4.2 Thomas-Fermi

R02

R01 r

Figure 4.5. Electron density distribution function as a function of radius extends to infinity for the free neutral atom (extended line). But for a compressed atom, the density is finite at the ion-sphere radius R01 , and at a still higher density with radius R02 .

In the limit of low temperature, F 1=2 ðxÞ  23 x =2 and (4.43) gives the ground-state electron density of a semiclassical electron gas 3


3= 3= 8π 2me 2 nðrÞ ¼ ½μ þ eΦðrÞ 2 , 2 3 h pffiffi and in the limit of high temperature, F 1=2 ðxÞ  2π ex , and (4.43) becomes   2 μ þ eΦðrÞ nðr Þ ¼ 3 exp : kBT λde

(4.45)

(4.46)

There remains the issue of what is the ionization level Z  . By definition, free electrons have positive energy relative to the ion-sphere boundary potential. Thus, classical mechanics allows these electrons to leave the ion sphere. The first possibility for the ionization level may be determined by ð  (4.47) Z 1 ¼ nðrÞd3 r,

138

Ionization

subject to p2 2me eΦðrÞ. A second choice for the ionization level is determined by the electron density at the ion-sphere boundary: Z 2 ¼

4π 3 R nðR0 Þ: 3 0

(4.48)

This second expression neglects the polarization of free electrons produced by their attraction toward the nucleus. For low density and high temperature, Z 1 and Z 2 agree. A third choice for determining Z  is to write two potentials:

and

r2 Φ1 ðr Þ ¼ 4πen0 ΦðrÞ,

(4.49)

 r2 ΦðrÞ ¼ 4πe nb Φðr Þ þ nf ½r; Φ1 ðr Þ  Z  ni ½r; Φ1 ðrÞ :

(4.50)

The potential Φ1 ðr Þ is created only by the bound electrons, and ΦðrÞ is created by all the sources: bound and free electrons and the ions. The bound-electron density nb is given by (4.41), with the exception that the upper limit on the integral is the 1= cutoff momentum pm ¼ f2me e½Φðr Þ  ΦðR0 Þ g 2 . This definition ensures that the bound-electron density vanishes at the ion boundary and that the maximum energy of the bound electrons is the same everywhere within the ion. The number density of the free electrons and the ions involve electron–ion pair and ion–ion pair correlations, respectively. In the first approximation, the pair correlation functions are zero, which means the free electrons and neighboring ions are both assumed to be uniformly distributed and provide an electrically neutral background only. In this approximation, Poisson’s equation becomes r2 Φðr Þ ¼ 4πenb ðrÞ pðm 8π ¼ 4πe 3 h






p2



dp,

(4.51)

dΦðR0 Þ Z e Ze ¼  2 , Φðr Þ ¼ as r ! 0: dr r R0

(4.52)

0

2

1 p exp  α  eΦðrÞ kB T 2me

þ1

with boundary conditions ΦðR0 Þ ¼

Z e , R0

The parameter α in (4.51) is not the chemical potential and is to be determined. Solution of (4.51) is done numerically. The choice of parameters must be driven by the minimization of the total free energy consisting of the contributions from the bound and free electrons; the contribution to the free energy from the ion is independent of the parameters Z  , R0 , and α, and thus is irrelevant to the minimization process.

4.2 Thomas-Fermi

139

We turn now to solving equations (4.40) and (4.41) in the case of zero temperature. Using (4.45) in (4.40), and introducing a new potential ψ ¼ Φðr Þ þ μ=e, we obtain (Zel’dovich & Raizer, 1966) pffiffiffi 5 3 1 d2 64 2π 2 e =2 me=2 3=2 ðrψ Þ¼ ψ : (4.53) r dr 2 3 h3 This can be expressed as d 2 χ χ =2 ¼ , dx2 x1=2 3

(4.54)

where the dimensionless variables are χ¼

r r ψ and x ¼ , Ze a

(4.55)

with x being the distance from the nucleus measured in units of
a¼

9π 2 128

1=3

a0 Z =3 1

:

Here, a0 is the Bohr radius. The boundary conditions become

dχ

χ ð0Þ ¼ 1 and χ ðx0 Þ  x0 ¼ 0, dx x0

(4.56)

(4.57)

where x0 ¼ R0 =a is evaluated at the ion-sphere boundary. It is easy to see that a solution to (4.54) is χ ðxÞ ¼ 144=x3 . Unfortunately, this solution does not satisfy the boundary conditions. However, a satisfactory solution

dχ is found if the boundary condition dx x0 is replaced by lim χ ðxÞ ¼ 0. Then we x!∞ obtain    a2 =2 x a1 χ ðxÞ  η0 1 þ , (4.58) 1 144 =3 pffiffiffiffiffi pffiffiffiffiffi   where a1 ¼ 7  73 =2 and a2 ¼ 7 þ 73 =2. Continued evaluation of (4.55) as x ! ∞ gives a proportionality constant for (4.58) as η0  1:9958. This solution is appropriate only near the boundary x0 . It is not possible to obtain a more complete expression for χ. The solution for the potential from (4.44), by numerical integration, is a power series in x (Feynman et al., 1949) 3=

χ ðxÞ ¼ 1 þ b2 x þ b3 x 2 þ b4 x2 þ    ,

(4.59)

140

Ionization

where b2 is the slope of χ at x ¼ 0, and b3 ¼ 4=3, b4 ¼ 0, b5 ¼ 2b2 =5, b6 ¼ 1=3, and so on. Once a value is selected for the initial slope b2 , the remaining coefficients are determined. The density distribution with respect to radius can be written, using (4.55), (4.56), and (4.45) ! 1= 3 rZ , (4.60) nðr Þ ¼ Z 2 f 0:88534a0 3=

where the function f is proportional to ðχ=xÞ 2 . Once ΦðrÞ is known, the radial electron density, and thus the ionization level as well as all other quantities of interest, may be found. For zero temperature, an expression fitted to the numerical solution is Z ¼ Z

y pffiffiffiffiffiffiffiffiffiffiffiffiffi , 1 þ y þ 1 þ 2y

(4.61)

where y ¼ αðρ=ZAÞβ , with α ¼ 14:3139 and β ¼ 0:6624; ρ is the mass density, Z is the atomic number, and A is the atomic weight of the material. The effect of nonzero temperature is to alter the charge distribution of electrons in the atom; this is expressed by (4.41), which leads to Poisson’s equation (4.44). This equation has to be solved numerically along the same lines as that done for the previous zero temperature case. We do not develop the solution here, but note that the integration must be done from the outside of the atom inward, since the value of the Fermi-Dirac function (of the transformed variable) at the origin has a singular behavior. In a manner similar to that for zero temperature, the ionization level is obtained by an expression fitted to the numerical solution of (4.61). Table 4.1 gives the complete set of equations for determining Z  (More, 1982). While fairly easy to use, the basic Thomas-Fermi ionization model has limitations: 1. It is strictly correct only in the limit of infinite nuclear charge. 2. Aside from semiclassical phase-space quantization and Fermi-Dirac statistics, it omits quantum effects such as the shell structure of the atom. 3. The electron density diverges as r ! ∞, whereas the nonrelativistic quantum result remains finite. 4. It treats the outer boundary of the free atom inaccurately; this is not a major concern for plasma, since the outermost electrons are strongly perturbed by density and/or temperature effects. 5. Relativistic effects are not considered.

4.3 Pressure Ionization and Continuum Lowering

141

Table 4.1 Procedure for Determining the Ionization Level in the Thomas-Fermi Model at Zero Temperature R¼

ρðg-cm3 Þ ZA

A ¼ a1 T a02 þ a3 T a04

T0 ¼

Z

Tf ¼

4= 3

B ¼  exp b0 þ b1 T f þ b2 T 2f  1=C Q ¼ RC þ QC1

Q1 ¼ ARB

Z ¼ Z a1 ¼ 0:003323

T ðeVÞ

C ¼ c1 T f þ c2

y ¼ αQβ

y pffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ y þ 1 þ 2y

a2 ¼ 0:971832

b0 ¼ 1:7630



T0 1 þ T0

a3 ¼ 9:26148  105

b1 ¼ 1:43175

c1 ¼ 0:366667 α ¼ 14:3139

a4 ¼ 3:10165

b2 ¼ 0:315463

c2 ¼ 0:983333 β ¼ 0:6624

When considering the Thomas-Fermi ionization model, we find that the interaction of free electrons with ions never becomes large. The measure of this interaction is the electron–ion coupling parameter Γei (4.36). As shown in Figure 4.6, the coupling parameter is never much above one for nondegenerate plasma as solid density is approached. But as the density increases, further degenerate electrons appear and the coupling parameter should be redefined using the Fermi energy rather than k B T as a measure of the electron kinetic energy; this modification reduces Γei again to a small value at high pressures. If the attraction of ions for electrons strongly exceeds the electron kinetic energy, some electrons will recombine and the ionization level will decrease, thus reducing the coupling. The Thomas-Fermi model treats degenerate electrons self-consistently as well as continuum lowering/pressure ionization. The question remains of how to include pressure ionization in the Saha model.

4.3 Pressure Ionization and Continuum Lowering The Saha ionization model is based on an isolated atom model. It becomes necessary to truncate the sum in the expression for the statistical weights (4.18) to avoid a mathematical divergence. In the real world, the atoms are not isolated but are influenced by neighboring atoms. As the density of the matter increases,

142

Ionization 101 0.001 0.01 100 Temperature (keV)

0.1

1.0

10–1

10.0 10–2 100.0

10–3 10–4

10–3

10–2

10–1

100

101

102

103

Density (g-cm–3)

Figure 4.6. Contours of Γei for iron using the ionization levels from the Thomas-Fermi model.

the atoms become more closely packed until they form a crystal-like structure at higher densities. Thus, the electronic structure of the atom/ion does not extend to infinity but is terminated at some finite value of the principal quantum number. As discussed earlier, one may assign a sphere of finite volume to an atom, the radius being the ion-sphere radius R0 . Each of these spheres is in contact with its neighbors and a pressure is exerted on that sphere by all of the surrounding ones. As the density increases, the ion sphere’s volume decreases. This compression of the atom/ion is termed “pressure ionization” and results in the liberation of the outermost bound eigenstates of the ion. Those electrons become itinerant and wander among the adjacent ions. In turn, the binding energy of the remaining electrons is reduced, and the energy gap between free and bound electrons decreases, which in turn leads to a reduction of the ionization potentials. The lowering of the continuum level automatically truncates the series in (4.18). The term “pressure ionization” is a bit of a misnomer; it really should be thought of as “density ionization.” Many proposals have been put forth over the years as to the proper way to calculate this lowering of the continuum, but there is no satisfactory model for all conditions. As the density increases, the ions become more neighborly so that

4.3 Pressure Ionization and Continuum Lowering

143

the correlation between the ions becomes stronger, and thus the interatomic potential becomes stronger. There are two primary points of view for this process: the creation of an electrostatic polarization configuration or a lattice-like configuration. Depending upon the point of view, we use a statistical approach or a thermodynamic approach. Figure 4.7 sketches how the lowered continuum excludes the higher levels and reduces the ionization energy by an amount equal to the binding energy at the cutoff boundary. If ~ n is the principal quantum number of the cutoff level, the binding energy at that level is equal to its Coulomb energy at a distance of the order of the orbital dimension r~n . Using the expression for the energy of the revolving electron, the ionization energy decrement is ΔI ¼ E ~n ¼ Ze2 =2r ~n . There remains the issue of what should be the cutoff dimension ~ n. As seen earlier, there are two measures for the size of the ion: the ion Debye length λDi and the ion-sphere radius R0 . We demonstrated in Section 3.5 that for

Continuum

E4 E3 E4 E2

E3

E2

E1

E1

Figure 4.7. An isolated atom has an infinite number of bound (Rydberg) states, but atoms in dense matter are pushed together, forcing the outermost bound states to be pressure-ionized into the continuum.

144

Ionization

high density, the Debye length is inappropriate because there is a very small number of electrons in the Debye sphere; that is, when λD  R0 . Thus, the ion-sphere radius appears to be the “proper” measure for the size of the ion. Equating the cutoff radius r~n (from Section 3.11) to an “ion-sphere-like” radius gives
 3 q~n 3 ni ¼ , (4.62) 4π a0 n~2 where the effective charge seen by an electron in the cutoff state is q~n  Z  , the ion charge state. Figure 4.8 shows the cutoff in the principal quantum number for aluminum at a temperature of 1 keV. The upper curve is obtained by setting the cutoff radius to the Debye length. States above the upper line are in the continuum for either model, while states below the lower line are considered discrete bound states. The two criteria disagree for states with R0 < r~n < λD . The physical justification for the truncation is that high-lying states have large orbital radii and consequently suffer perturbations in the plasma 200

Principal quantum number

100

10

5 1017

1018

1019

1020

1021

1022

lon density (cm–3)

Figure 4.8. Cutoff of the principal quantum number as a function of ion density for aluminum at a temperature of 1 keV. The solid line is based on the Debye length and the dashed line comes from the ion-sphere radius.

4.3 Pressure Ionization and Continuum Lowering

145

environment. The orbital velocity of these outer bound electrons is slow and their position may be easily influenced by the motion of neighboring ions, which introduce fluctuations in the microfield. At higher density, there is a higher degree of thermal excitations lending to enhanced fluctuations of the bound states with r~n near R0 . For plasma at a constant temperature, compression decreases R0 and successively deeper core states are liberated into the continuum. In contrast, if the temperature is raised at constant density, Z  increases and r~n decreases so that the number of bound states will increase. The terms “pressure ionization” and “continuum lowering” are used to identify essentially the same phenomenon, but viewed from different perspectives. Continuum lowering may be characterized by the concept that as the density increases, the spatial average of the electrostatic potential becomes increasingly negative until it ultimately exceeds the isolated atom’s binding energy for any given electron. Pressure ionization occurs when ion cores are forced together with the outermost bound states hybridizing and becoming propagating waves. At lower densities, pressure destroys excited states, which are only occasionally occupied. This generates fluctuations of the continuum environment among adjacent ions. As pressure ionization proceeds by increasing the density, the effect is to lower the continuum successively through various one-electron bound states. The energy of these bound states becomes scattering resonances. From the partial-wave potential, we can calculate the population of the resonance state, which effectively adds to the total number of electrons in the continuum; we do not address this resonance process. Our discussion has assumed thermodynamic equilibrium, and as such, continuum lowering is a static configuration. In contrast, there is a dynamic, nonequilibrium picture that is appropriate when analyzing spectroscopic data to infer the continuum lowering value. The adiabatic approximation of the equilibrium model is not correct when considering, say, photo-ionization. The time for a photo-ionization event to take place is short compared to the time required for redistribution of thermal and kinetic energy of the ion to form a new state of equilibrium. We do not address nonequilibrium continuum lowering. A note of caution: Pressure ionization is often explained in terms of the virial theorem. This theorem is applicable only in very limited regimes such as an isolated atom or a periodic structure, but not in dense plasma where “cell” boundaries are ill defined. There are four relatively simple models for the lowering of the continuum, which are discussed in the following sections.

146

Ionization

4.3.1 Debye-Hückel The Debye-Hückel method (Landau and Lifshitz, 1958; Griem, 1964) may be used to calculate the Coulomb corrections for the case of “weakly” imperfect gases; “weakly” means eΦ=k B T  1. The model assumes a nonuniform, spherically symmetric cloud of electrons about an ion. The electric field produced by the nonuniform distribution of electrons weakens the bond between any electron and the rest of the atomic electrons and the ion core. For an ion in the mth stage, the electron cloud has a Boltzmann distribution corresponding to the electrostatic potential of the central charge Z m e. From Section 3.5, the solution to Poisson’s equation is Φm ðrÞ ¼ ΦDH ¼

Z m e r=λD , e r

(4.63)

where, for equal electron and ion temperatures, the length characterizing the dimensions of the cloud is the Debye radius (length): ! X e2 2 2 λD ¼ 4π (4.64) nm Z m : ne þ kBT m Near the center, for r  λD ,
 1 1 : Φm ð r Þ ¼ Z m e  r λD

(4.65)

The first term on the right-hand side of (4.65) is the potential of the central ion, and the second is the potential due to all of the other surrounding charges at the position of the ion. The Coulomb energy is then ! X ne þ nm Z m Φðr Þd 3 r Δℰc ¼ e

Ð


¼ e3

π kB T

from which the free energy Δℰc =T 2 ¼ ∂=∂T ðΔF c =T Þ to give

1=2

m

ne þ

X

!3=2 nm Z 2m

m

correction

is

found

!3=2
1=2 X 2 3 π 2 ne þ nm Z m : ΔF c ¼  e 3 kBT m Minimizing the variation gives

(4.66)

, by

integrating

(4.67)

4.3 Pressure Ionization and Continuum Lowering

ΔI mþ1

∂ΔF c ∂ΔF c ∂ΔF c þ  ∂nmþ1 ∂ne ∂nm !1=2
1=2 X π ¼ 2ðZ m þ 1Þe3 ne þ nj Z 2j , kBT j

147

¼

(4.68)

since Z mþ1 ¼ Z m þ 1. The correction to the ionization potential for the mth ionization stage is ΔI DH ðmÞ ¼ 

Z m e2 : λD

(4.69)

There remains the question of the value for the maximum principal quantum number ~ n that must be considered. If we use the Debye-Hückel potential (4.63) in the semiclassical Bohr-Sommerfeld model (see Section 3.11), we obtain n~DH

1 ¼ π

ð∞


2ϕðrÞ ea0

1=2


dr ¼

0

4Z  λD πa0

1=2 ,

(4.70)

provided n~DH 3. The Debye-Hückel model is physically meaningful only when the coupling parameter Γ  1. Then the energy given by (4.68) is much less than the average thermal energy kB T, so the Coulomb forces are weak perturbations. This is equivalent to the statement that many ions are needed to screen a test charge. This model for continuum lowering used with the Saha ionization model is moderately successful for lower plasma densities, but fails at higher densities where interatomic energies become dominant. The second model, which is more appropriate for high-energy-density plasma, is the ion-sphere model.

4.3.2 Ion-Sphere The ion-sphere model considers the ions to be strongly correlated (Armstrong et al., 1961). Consequently, they tend to repel each other, and this forces them to be packed closely together so that the spheres overlap one another to fill all of space. For each sphere with radius R0 , numerical simulations show that the average nearest neighbor distance is approximately 1:7R0 . The first step is to calculate the partition function, including the fluctuations among the particles that produce electrostatic contributions. The correlations between the charged components are between pairs of particles, which can be expressed as potentials. We then average over these potentials to produce

148

Ionization

the partition function and hence the free energy, which leads to a value for the lowering of the ionization potential. In a single sphere, the nuclear charge and the bound electrons are located at the center, and the free electrons uniformly fill the sphere. Neighboring spheres are electrically neutral. Solution of Poisson’s equation gives the potential for the mth ion as
 Zme Z me r2 (4.71) Φm ðr Þ ¼ Φis ¼ 3 2 :  r 2Rm Rm 1=

The “ion-sphere” radius for this particular ion is Rm ¼ ð3Z m =4πne Þ 3 . The potential is defined to be zero for radii greater than this radius. The last term on the right-hand side of (4.71) is well known from elementary electrostatic theory (Jackson, 1999). We proceed with calculating the electrostatic energy for one sphere. The potential energy of the electrons in the field of the ion core is ð ℰen ¼ e ne ðr ÞΦðr Þd3 r: (4.72) For a uniform distribution of electrons within the sphere, then using the first term on the right-hand side of (4.71), the potential due to the central ion, we have for the mth stage Rðm

ℰen ðmÞ ¼

4πZ 2m e2 ne

rdr ¼ 

3 Z 2m e2 : 2 Rm

(4.73)

0

The potential energy from the interaction among the electrons is ð ðð e2 ne ðrÞne ðr0 Þ 3 3 0 e d rd r ¼  ne ðrÞΦðrÞd 3 r, ℰee ¼  2 2 jr  r0 j

(4.74)

where the factor of one-half prevents double counting. Using the second part on the right-hand side of (4.71), together with a uniform electron density, then for the mth ionization stage 1 Z m e2 ℰee ðmÞ ¼ ne 4 Rm

R ðj


3

0

 r2 3 Z 2m e2 : 4πr2 dr ¼ 2 5 Rm Rm

(4.75)

A proper, but messy, approach for calculating both energies would be to use (4.45) or (4.46) for the electron density in (4.72) and (4.74). The combined potential energies ℰen þ ℰee are known as the Madelung energy.

149

4.3 Pressure Ionization and Continuum Lowering

Combining (4.73) and (4.75), and noting that Rm ¼ ðZ m =Z  Þ1=3 R0 , gives the Coulomb correction to the free energy density ΔF m ¼ 

9 2 e2 9 e2  1=3 5=3 ¼ ðZ Þ Z m , Zm 10 Rm 10 R0

(4.76)

and thus the continuum lowering potential is 3  1=3 Z m=3 e2 ΔI is ðmÞ ¼  ðZ Þ ; 2 R0 2

(4.77)

terms of Oðe2 =Rm Þ would be added when doing the proper calculation. As an aside, we note that the factor of 0:9 in (4.76) is appropriate for any fluid-like system with no discrete symmetry. In contrast, a “crystal” with “fcc” or “hcp” symmetry has a value 0:99025, and “bcc” symmetry has a value 1:01875. The effects of pressure ionization in iron are seen in Figure 4.9. Ionization contours in the density-temperature plane show the successive ionization of deeper atomic shells. For example, for a temperature of 0.01 keV, at low density the shell with n ¼ 3 and l ¼ 2 is being ionized, but as the density increases the n ¼ 3, l ¼ 1 shell, then the n ¼ 3, l ¼ 0 shell, and so on, are being ionized. 101 24 22

Temperature (keV)

100

1s 16 2s 2p

14

10–1 3s 3p

Z*=8 3d

10–2

10–3 10–4

10–3

10–2

10–1

100

101

102

103

104

Density (g-cm–3)

Figure 4.9. Equilibrium ionization contours of iron as a function of density and temperature. The effects of pressure ionization are clearly seen. In the lower-left corner, the shell with n ¼ 3 and subshell l ¼ 2 is being ionized. At higher densities and/or temperatures, deeper shells are removed.

150

Ionization

Again, the question is asked as to the value for the maximum principal quantum number ~ n that must be considered. In a fashion similar to that for (4.70), but using the Coulomb interaction, approximately represented by the ion-sphere potential given by (4.71), we obtain
n~is ¼

Z  R0 3a0

1=2 :

(4.79)

The Debye-Hückel model is appropriate for the low-density regime, while the ion-sphere model is for high density. For the intermediate region, we turn to the model of Stewart and Pyatt.

4.3.3 Stewart-Pyatt The Stewart and Pyatt (1966) model is founded on the finite-temperature ThomasFermi model for the electrons (Feynman et al., 1949). The model is extended to include ions in the vicinity of a given nucleus, thus relaxing the “frozen” nuclear positions of the ion-sphere model. The potential due to the free electrons and neighboring ions is isolated from the core ion, which includes the bound electrons. One limitation of the model is its neglect of fluctuations; only time-averaged, spherically symmetric potential, and charge distributions are considered. Consider a fixed ion immersed in a sea of electrons. The electrons are described by Fermi-Dirac statistics, while the ions are characterized by Maxwell-Boltzmann statistics. Both distributions of particles are assumed to be at the same temperature. As was done previously for the Thomas-Fermi model, the electrostatic potential is given by the spherically symmetric Poisson’s equation " # X 1 d2 2 r Φð r Þ ¼ ðrΦÞ ¼ 4πe ne ðrÞ  Z m nm ðr Þ , (4.80) r dr2 m where nm is the local number density of ions of charge eZ m . The radial distribution of electrons is h i ðr Þ F 1=2 μþeΦ kB T , ne ðr Þ ¼ ne ð∞Þ (4.81) F 1=2 kBμT and for each ion species

  eΦðrÞ nm ðrÞ ¼ nm ð∞Þ exp Z m : kBT

(4.82)

The boundary conditions on the potential are Φðr ! ∞Þ ! 0 and lim rΦðrÞ ¼ Ze. r!0

4.3 Pressure Ionization and Continuum Lowering

151

As before, the degeneracy parameter (chemical potential) μ is found from the free-electron density far from the ion (see Section 3.4.1)
 4 1 μ ne ð∞Þ ¼ pffiffiffi 3 F 1=2 : (4.83) π λde kB T We consider only the case where thepelectrons are nondegenerate, so μ=k B T is ffiffi π x large and negative; then F 1=2 ðxÞ  2 e  1. Close to the nucleus, where ðμ þ eΦÞ=kB T > 0, there is a region where the bound electrons are degenerate; this is the ion core. Charge neutrality is expressed by P Z m nm ð∞Þ ¼ ne ð∞Þ. m

Let y ¼ eΦ=k B T and x ¼ r=λDi , where λDi is the ion Debye length. Then (4.80), including (4.81) and (4.82), takes the form 2 hμþeΦðrÞi 3 2 Zy F 1= kB T 1d 1 hZe i5 2  , (4.84) ðxyÞ ¼ 4 Z x dx2 Z p F1 μ =2 kB T

where the bracketed quantity denotes an average weighted with nm ð∞Þ over species, and  P 2 Z j n j ð ∞Þ Z2 , ¼P Zp ¼ Z j nj ð∞Þ hZ i 

(4.85)

which is never less than unity; Z p is effectively Z  (see footnote #4 of Chapter 3). The second term on the right-hand side of (4.84) may be simplified since it is comparable to the first only for a fairly small y; in this region: hZeZy i hZ ð1  Zy þ   Þi  ¼ 1  Z p y þ     eZ p y : hZ i hZ i The basic equation (4.84) for the potential becomes

2 3 μ 2 F y þ 1= kB T 1d 1 2  eZ p y 5, ðxyÞ ¼ 4 2 μ x dx Zp F1

(4.86)

(4.87)

=2 k B T

with boundary conditions yð∞Þ ¼ 0 and mth ion.

lim xy ¼ Z m e2 =λDi kB T for the x!0

Assuming nondegenerate electrons, we use (Keller & Fenwick, 1953)

152

Ionization


 μ   F 1=2 y þ 1 yþk μT k T B y y
  e 1  3= e B ð1  e Þ þ    μ 22 F 1=2 kB T 2 3
3=2 μ 4 μ π2 6 7  pffiffiffi ekB T y þ 41 þ

2 þ   5: 3 π kB T 8 y þ kBμT

(4.88)

Equation (4.87) may be broken down into four regions of interest: (1) For R0  λD , where y  1=Z p , (4.87) is 1 d2 ðxyÞ ¼ y, x dx2

(4.89)

and has the solution y¼

C x e , x

(4.90)

which is the form of the Debye-Hückel potential. In this region, the charge densities of the ions nearly cancel those of the electrons, and most of the electrons are free. (2) Proceeding inward, if Z p  1, there is a region where 1=Z p  y  1, and we have 1 d2 1 ðxyÞ ¼ , 2 x dx Zp

(4.91)

with solution y¼

A x2 , þBþ x 6Z p

(4.92)

where A and B are determined by boundary conditions. Here, the density of the ions and bound electrons are small compared to that of the free electrons, which is the form of the ion-sphere model. (3) For 1  y  μ=kB T, the bound electrons outnumber the free, but the occupation of the bound states is well below their maximum. (4) For y  μ=k B T, most of the electrons are in fully occupied states. This is the ion-core region, and (4.87) is approximately the zero-temperature ThomasFermi equation. Regions three and four are not of concern here.

4.3 Pressure Ionization and Continuum Lowering

153

The complete solution to (4.87) must be done by numerical techniques. An Approximate solution to (4.87) can be obtained if only regions one and two are considered. This is equivalent to having a potential that places all of the bound electrons at the origin with the free-electron density being approximately uniform. Thus, (4.87) becomes  1 d2 1  ðxyÞ ¼ 1  eZ p y : 2 x dx Zp

(4.93)

For large Z p , the right-hand side can be approximated by y when Z p y < 1 (region 1) and by 1=Z p when Z p y > 1 (region 2). Let the transition point between the two regions be χ and require the potential and its spatial derivative be continuous there. Using the solutions for the two regions, (4.90) and (4.92), the potential and its derivative at χ are Ceχ ¼ A þ Bχ þ

χ3 χ2 and  Ceχ ¼ B þ , 6Z p 2Z p

(4.94)

which gives the result i 1 h ðχ þ 1Þ3  1 , 3Z p i 1 h B¼ ðχ þ 1Þ2  1  A, 2Z p  χ  2 χ þ 1 eχ : C¼ Zp A¼

(4.95)

Then, eliminating χ between A and B of (4.95), we have i 2= 1 h B¼ 3Z p A þ 1 3  1 : (4.96) 2Z p  2= For small Z p A, B ¼ A, and for large Z p A, B ¼  3Z p A 3 =2Z p . In region four, that of the ion core, we know xyðxÞ ! Z m e2 =λDi k B T as x ! 0. From (4.92), y ! A=x also as x ! 0, so A ¼ Z m e2 =λDi kB T. This is just the DebyeHückel continuum lowering in units of k B T for large Z m . Then 3Z p A is the ratio of the “ion-sphere” volume to the Debye-sphere volume ðRm =λDi Þ3 . Using these facts in (4.96) yields 8 9
2 >

> > 2 > > R > > 0 < = λD n~SP ¼ n~is  : (4.99) 2=3 3=2 > > > > > > R > þ 1  1> : λD0 ; A note of caution: The formulation of the Stewart and Pyatt model used here exhibits thermodynamic inconsistency (Sweeney, 1978). However, a formulation

4.3 Pressure Ionization and Continuum Lowering

155

based on the change in the ionization potential corresponds to a change in the partition function; then using this partition function in the entropy equation removes the inconsistency. Further, the model is not applicable to degenerate plasma.

4.3.4 Ecker-Kröll This model is developed along a line of thinking similar to the model of Stewart and Pyatt. In the discussion of the Debye-Hückel model in Section 4.3.1, expressions (4.63) and (4.65) give the potentials for large and small radii, respectively. The Ecker and Kröll (1963) model divides the radial dimension into three regions. The dividing point between the innermost and central regions is designated Rc , which is the largest distance from the core below which the potential distribution can be approximated by the Coulomb potential. The dividing point between the central and outermost regions is the ion-sphere radius R0 , which separates the environment of the core particle into “individual” and “collective” regions. For r  Rc , (4.65) can be written as Φm ðrÞ ¼

Zme þ a, r

(4.100)

where a is found from the boundary conditions. For r R0 , (4.63) becomes b Φm ðrÞ ¼ er=λD : r

(4.101)

The potential in the central region, where Rc  r  R0 , is not precisely known, but we can assume (4.100) and (4.101) are limiting expressions. Let ~r ¼ Rc or R0 , depending on which limiting function is used. Requiring the field to be continuous at ~r gives b ¼ Z me

λD ~r=λD e , ~r þ λD

and continuity of the potentials gives
 Z me λD Zme a¼ 1 ¼ ~r ~r þ λD ~r þ λD :

(4.102)

(4.103)

Subtracting Z m e=r from (4.100), we see that a is identical with the potential we wish to calculate. This result restricts the density region to that below a critical density defined by

156

Ionization


 3 kB T 3 nc ¼ : 4π e2

(4.104)

We note that for plasma with just electrons and singly charged ions, the average electrostatic energy is identical to the average thermal energy at the critical density. The critical density corresponds to the number of particles in a Debye-sphere where the Debye-Hückel theory breaks down: about one particle. The boundary between the innermost and central regions, Rc , is assumed to be smaller than but of the order of magnitude of R0 , and so Rc ¼ γR0 . To arrive at a potential for the central region γR0  r  R0 , we consider two limiting cases. In the range above the critical density, the screening will be definitely stronger than that given by (4.100) at the critical density. The other limit for the potential distribution is (4.101), with the Debye length evaluated at the critical density λDc . The continuities of the potential and field at γR0 requires a quantity d, corresponding to a of (4.100), such that 

Zme Zme d : γR0 γR0 þ λDc

(4.105)

Now λDc ¼ OðRc Þ and the potential is Φm ¼ C

Z me , R0

(4.106)

where C is determined from the continuity of the potential across the critical density; thus
2 1=2 e 1= nc6 : (4.107) C  2:2 kB T With the values of Φm determined, the electrostatic contribution to the chemical potential is found from ð μel ¼ Z m eΦm ðrÞdr: (4.108) The electrostatic part of the chemical potential for the region below the critical density is ðZ m eÞ2 , μ< ¼  2λD

(4.109)

while for the region above it is μ> ¼ C

ðZ m eÞ2 : 2R0

(4.110)

4.4 Collisional-Radiative

157

The lowering of the potential is given by (4.68) with ∂F c =∂n replaced by μ. Then for n  nc ΔI mþ1 ¼  Hence

 e2  2 Z mþ1  Z 2m þ 1 : 2λD

8 e2 > > <  Z m for n  nc λD ΔI EK ðmÞ ¼ : e2 > > : C Z m for n > nc R0

(4.111)

(4.112)

We include a brief discussion of the Inglis-Teller model for merging of the energy levels as the continuum is lowered. This model (Inglis & Teller, 1939) approximately takes into account the microfields, which explain the Stark effect. In the presence of an electric field, spectral lines are split. For some energy levels n, the splitting is equal to the difference between the adjoining level; beyond this energy level, the spectral lines merge. That is, as one proceeds to higher states of a given series (for a principal quantum number), the spacing between the levels decreases (see Figure 4.7) and the quasistatic broadening of the lines (which are proportional to n2 ) increases, thus the lines overlap and merge. The full theory is quite complex, and we quote the results for the cutoff quantum number log ne ¼ 23:26  7:5 log ~n m þ 4:5 log Z,

(4.113)

where Z  1 is the net charge of the atom. The use of the Inglis-Teller limit to determine the cutoff quantum number is of questionable relevance to high-energy-density plasma, for a number of reasons. For example, when considering partition functions, this limit should not be considered for truncating the summation, because bound states of the atom can exist above this limit. Although the adaptation of continuum lowering to the Saha model for moderately dense plasma is successful, the Saha approach still fails at high densities where inter-atomic energies become dominant. There exist more sophisticated descriptions of static screening effects, such as finite-temperature density functional models, but this topic is beyond the scope of this work.

4.4 Collisional-Radiative It has long been known that local thermodynamic equilibrium is more the exception than the rule in both astrophysical and laboratory plasmas. Plasma can be driven to a state of nonlocal thermodynamic equilibrium (non-LTE) by several

158

Ionization

Table 4.2 Ground-State Configurations for the Carbon Atom Ionization Stages m 0 1 2 3 4 5 6

Configuration 2

2

2

1s 2s 2p 1s22s22p1 1s22s2 1s22s1 1s2 1s1 Bare

“Looks like” Carbon Boron Beryllium Lithium Helium Hydrogen

mechanisms: (1) sharp density gradients, (2) a rapidly changing environment, (3) finite dimensions, and (4) an ambient radiation field. LTE is based on the principle of minimum entropy. Thus, any nonadiabatic change of conditions creates non-LTE situations. Plasma having strong time or space gradients are in non-LTE, because equilibration does not have time to be established. Non-LTE can also exist, even in “steady state” if its spatial extent is such that photons can escape or the radiation field is not Planckian or is of high intensity (radiation-dominated plasma). The ionization models discussed previously are based on thermodynamic arguments as evidenced by the inclusion of the Boltzmann factor. We saw in Section 4.1 that the models produce a distribution of charge states about some mean value (see (4.32)). This spectrum of charge states can be seen from the more detailed rate equation approach, where the details of the rate coefficients are not dependent on thermodynamics. As noted at the beginning of this chapter, the ionization process proceeds primarily by a collision of a free electron with an atom/ion, or by the absorption of a photon, providing it has sufficient energy. The ionization process is “balanced” by recombination or inverse collisional and radiative reactions. Consider a volume element of matter composed of a single type of atom that can exist in various charge states m. All of the atoms are assumed to be in the same charge state, but we will relax this requirement in later sections of this chapter. For example, Table 4.2 lists the ground-state configurations of the ions possible in a pure carbon plasma. The number density of these heavy particles is nm , which is just the density of particles with m electrons removed; m ¼ 0 signifies the fully recombined (chargeneutral) atom. For simplicity, we ignore excitation/de-excitation reactions as well as autoionization and its inverse, dielectronic recombination.2 2

If excitation/de-excitation terms are included in (4.114), those terms cancel exactly, reflecting the fact that these processes induce transitions only within the same charge state and cannot affect the change in the charge-state density.

4.4 Collisional-Radiative

159

The rate of change of the number of ions in charge state m is found by combining (4.1), (4.2), (4.7), and (4.8) involving three adjacent charge states: h i h i h i dnm ðeÞ ðrÞ ¼  ne αðmeÞ þ αðmrÞ nm  ne βðmeÞ þ βðmrÞ ne nm þ ne αm1 þ αm1 nm1 dt h i ðeÞ ðr Þ þ ne βmþ1 þ βmþ1 ne nmþ1 : (4.114) The rate coefficients are indicated with an “(e)” or “(r)” superscript for collisional and radiative processes, respectively. For material with atomic number Z, there are possibly Z þ 1 similar equations in addition to one for the free electrons, which is Z P mnm . The coefficients (related to cross needed to conserve particle number, ne ¼ m¼0

sections) αðeÞ , βðeÞ , αðrÞ , and βðrÞ are discussed in detail in Chapter 9. Of these four, three are electron temperature–dependent; only αðrÞ is not. The existence of the photoionization process implies a radiation field is present. Thus, we need to solve the population equations (4.114) in concert with the radiation transfer equation, a really messy situation. Our discussion here is based on the radiation distribution being Planckian in equilibrium with the electron fluid. Processes due to atom–atom, atom–ion, and ion–ion collisions are not considered, since relevant reaction rates are much smaller than those for electron collisions, at least for plasma having ionization levels exceeding a few percent. The steady state is of our primary interest, so the left-hand side of (4.114) is set to zero, which puts the system into statistical equilibrium and yields the collisionalradiative equilibrium (CRE) model. This steady-state condition can also be achieved when the thermalization processes are fast enough to adjust the population densities rapidly to any change in the total ion density or temperature. Thus, (4.114) is reduced to a simple recursive expression: nmþ1 αðeÞ ¼ ðrÞ m ðeÞ : nm βmþ1 þ ne βmþ1

(4.115)

The explicit appearance of the electron density in (4.115) divides the density dependence of the results into two asymptotic regions: a high-density region where ðeÞ ðr Þ ðrÞ ðeÞ ne βmþ1  βmþ1 , and the opposite case βmþ1  ne βmþ1 . The first case is the most important for high density, where electron collisional ðeÞ processes dominate. In this limit, ne βmþ1 nmþ1 ¼ αðmeÞ nm . We know the distribution between charge states is governed by the Saha equation (4.19), (if bound excited states were being considered, their distribution would be given by Maxwell-Boltzmann statistics,) and the free electrons are distributed according to a Maxwellian or a FermiDirac distribution. If we know the collisional ionization coefficient αðeÞ , then the Saha equation gives βðeÞ . This is just the principle of detailed balance.

160

Ionization

Earlier in this chapter we identified the basic requirements for plasma to be in complete LTE. The necessity for the electron distribution to be an equilibrium distribution states that the electron–electron collision time τ ee be much shorter than the heating or containment times. It is necessary that for free electrons τ ee be shorter than the rate of bremsstrahlung emission, which requires T e ≲ 5  104 logΛ keV, which is always the case for high-energy-density plasma. Even though excitation and de-excitation are not being considered, there is also the requirement for LTE that a particular excited level be in equilibrium with all higher levels. This is equivalent to the total radiative decay rate into the level being much smaller than the total excitation rate from the level. For a hydrogenic system with principal quantum number n, an estimate of this requirement is (Griem, 1964)
1= 7 kB T e 2 18 Z ne 7  10 17= , (4.116) n 2 Z 2 EH where Z 2 EH ¼ Em1 ð∞Þ is the ionization energy of the hydrogenic ion with charge Z  1. For complete LTE, Griem considers that the collisional rate of the ground state be ten times greater than the radiative rate via the resonance transition from the first excited state; thus
ne ≳ 9  10 Z

17 7

kB T e Z 2 EH

1=2   E m1 ð2Þ 3 : Z 2 EH

(4.117)

The quantity E m1 ð2Þ is the energy of the first excited state with respect to the ground state (E m1 ð1Þ ¼ 0). Even if this condition is met, the presence of a significant radiation field can cause the violation of conditions for LTE; the absorption of photons disrupts the collisional equilibrium, so some other equilibrium state will be established. ðr Þ ðeÞ The second case, where βmþ1  ne βmþ1 , is the high-temperature low-density, optically thin region where the coronal equilibrium (CE) model is applicable. Thus, ðrÞ βmþ1 nmþ1 ¼ αðmeÞ nm , and the requirement for the electron density is
ne ≲ 1:4  10 Z

14 7

kB T e Z 2 EH

4 

1=2 Z 2 EH : Em1 ð∞Þ

(4.118)

Figure 4.11 shows charge state distributions for argon plasma near these two limits. Figure 4.11a is in the LTE range with ne ¼ 1  1024 cm3, and Figure 4.11b is in the CE range with ne ¼ 1  1018 cm3. These results were obtained using the generalized population kinetics code, FLYCHK (Chung et al., 2005). Both limiting regions have been amply verified by more elaborate calculations and by numerous experiments. One should realize that in extreme circumstances

4.5 Screened Hydrogenic Average-Atom (b) 1.0

0.6

18 8

0 3

0.5 Charge state fraction

Charge state fraction

0.4

0.1 0.2

0.3

0.2

12 0.6 9 0.4 6 0.2

0.1

0.0

15

0.8

1.0

0.5

6

8

10

12 Charge state

14

16

18

16

0.0 0.001

lonization level

(a)

161

3

0.01

0 0.1

Temperature (keV)

Figure 4.11 (a) Distribution of charge states, in the LTE limit, for selected electron temperatures, as indicated. (b) Distribution of charge states in the CE limit, as a function of electron temperature; each curve is for a different value of m, and the ionization level is shown by the dashed line.a The filling of the quantum shells has been regular up through argon, but for potassium and calcium, the 4s shell is filled first and then the 3d shell begins filling for higher atomic numbers. This is because most of the orbit of the 4s electrons lies inside the elliptical 3d orbit. a

the rate coefficients may vary considerably as a function of charge state m, so that some charge states would be in steady state while others might not be. The rate equations (even in steady state) should be solved simultaneously with the radiative transfer equation, particularly at lower densities where photon mean free paths can be greater than the spatial scale of plasma. The long mean free paths can lead to nonlocal spatial coupling between two elements of plasma. Photons emitted from a plasma element at position r1 are absorbed in a different element at r2 , altering the distribution of charge states there, which in turn affects the distribution back at r1 because of the absorption of photons arriving from r2 . Clearly, doing this calculation is a nearly insurmountable task because of the complexity of the set of integrodifferential equations; we then resort to numerical calculations, but they still require approximations. This topic is addressed in Chapter 8.

4.5 Screened Hydrogenic Average-Atom High-energy-density plasma is composed, in general, of atoms with complex structures. To gain insight into the physical processes found in the plasma requires an adequate understanding of the nature of these atomic structures. A natural approach would be to use the self-consistent field method. One such method is to treat a single atomic nucleus surrounded uniformly by a positive charge

162

Ionization

background. The electron energies and wave functions are found from the solution to the one-electron Schrödinger equation. The exact problem to be solved depends upon the choice of electrostatic potential, always more complicated than the simple Coulomb potential. In addition to the eigenvalues from the solution, we need the distribution of the electrons. For LTE plasma, each electron state is independently occupied according to Fermi-Dirac statistics. Then the average radial distribution of electrons is found by summing over the quantum states of the product of the particle distribution with the square of the wave function. This distribution must be consistent with the potential used in the Schrödinger equation. Upon examination of the eigenstates, we find fractional occupation numbers. These one-electron eigenvalues represent the spectrum of the so-called averageatom. Electron eigenvalues are averaged over ionization and excitation states of the central ion, as well as the entire local plasma environment. The average-electron eigenvalues depend upon density and temperature. We have to distinguish the atomic properties of the bound states from the states of the background. A useful way to do this, as done previously, is to define the zero of energy to be the last bound state, so that the bound states have negative energy, and the background electrons have positive energy. The distribution of these “outside” electrons (the free electrons) decreases exponentially with radius. Using these ideas, the average-atom model was first introduced by Strömgren in the 1920s and developed in detail by many authors (Mayer, 1948; Lokke & Grasberger, 1977). Much has been written about this model; we summarize the important attributes here. The following are the essential components of the model: 1. A screening theory, based upon a Wentzel-Kramers-Brillouin (WKB) calculation, which gives the electron energy levels as a function of the shell populations 2. A Hartree-Fock approximation, which replaces the shell populations by their averages 3. LTE or (non-LTE) equations to determine the average shell populations This latter point distinguishes the average-atom model from the Saha model in that the focus is on shell populations as the fundamental set of variables. The use of shell populations rather than number densities of atoms having a given set of populations is important for many reasons, especially the use of screening corrections and opacity calculations. Consider an elementary volume of gas at uniform density and assume that hydrodynamic changes in the gas occur more slowly than the time needed to establish equilibrium distributions of either the ions or free electrons. Thus, the ions and electrons may be characterized by a single kinetic temperature T. These assumptions

4.5 Screened Hydrogenic Average-Atom

163

permit the use of the steady-state Boltzmann-Saha equations, which are functions only of the elemental composition, mass density, and temperature. The average-atom model does not attempt to calculate the level (shell) populations of many different atomic configurations. Rather, only one sequence of levels is considered, which has a few discrete energy levels, and the occupation of these levels represents an average over the many possible configurations. The averageatom is not a real entity with an integer number of bound electrons, but a fictitious atomic system with a noninteger number of electrons that are distributed among the levels in a way that approximates the average over the configurations. Again the emphasis on this model is on calculating the shell populations as the fundamental set of variables. The ionization level of the average-atom is known once the shell (level) populations and corresponding energies have been found. This information can be used further to find the pressure; energy density; and, with additional models, the x-ray absorption coefficients. The underlying theory for the average-atom model is based on the work of Sommerfeld’s (1934) semiclassical WKB theory (see Section 3.11.2). The radial wave function of the atom is related to the integral of the wave number of the orbiting electron in an electrostatic potential ΦðrÞ. The wave number is defined by More (1982), " #1=2 E þ eΦðr Þ ðl þ 1=2Þ2 pðrÞ ¼  , r2 h2 =8π 2 me

(4.119)

which vanishes at the two turning points r1 and r 2 . The integral over the radius is quantized according to the number of radial nodes nr ¼ n  k ¼ n  l  1, where n is the principal quantum number rð2




 1 pðr Þdr ¼ π nr þ : 2

(4.120)

r1

The radial wave function is then 0 1 rð2 Cnl π Rnl ðr Þ ¼ pffiffiffiffiffiffiffiffiffi sin @ þ pðr0 Þdr 0A, 4 pðrÞ

(4.121)

r1

where C nl is the normalization constant. For a hydrogen-like atom with Coulomb potential ΦðrÞ ¼ Z ðrÞe=r, we can find an effective charge qðrÞ, which is specified by the electric field at radius r:

164

Ionization

E ðrÞ ¼ 

dΦðrÞ qðrÞe ¼ : dr r

(4.122)

The relation between Z ðrÞ and qðrÞ is dZ ðr Þ : (4.123) dr qðr Þ can be interpreted as the nuclear charge minus inner screening associated with electrons at a radius less than or equal to r. The last term in (4.123) is associated with outer screening because a shell of charge located at a radius greater than r adds a constant to the potential Φðr Þ and makes a linear contribution to Z ðrÞ. For an electron shell with principal quantum number n, the charge Z ðrÞ is expanded about an average orbit radius rn to give   dZ ðrÞ Z ne Φðr Þ ¼ e þ þ , (4.124) dr n r qðrÞ ¼ Z ðrÞ  r

where Z n ¼ qðr n Þ. Thus, outer screening produces a constant potential e½dZ ðrÞ=dr n , while the second term is the Coulomb potential, including only inner screening. We can represent the energy level in terms of a wave vector k E nl ¼ E ðn0Þ 

h2 2 k , 8πme

(4.125)

where Eðn0Þ ¼ e2 ½dZ ðr Þ=dr n is the energy associated with screening by outer electrons (the electrostatic potential of those electrons). We approximate the wave number (4.119) by " #1= 2 2 2Z ð l þ 1=2 Þ n pð0Þ ðrÞ ¼ k 2 þ , (4.126)  a0 r r2 where Z n is the nuclear charge reduced by screening from electrons in inner orbits, ð0Þ and a0 is the Bohr radius. Expression (4.126) is zero at the turning points r1 and ð0Þ r2 , which are close together for circular orbits (where n ¼ l  1), and one can use the average radius of the orbit of the nth level as given by the Bohr theory: rn ¼ 

Ze2 n2 ¼ a0 : 2E Zn

(4.127)

Applying the quantization condition (4.120) defines the quantum defect Δnl ¼

1 π

rð2

r1

ð0Þ

pðr0 Þdr 0 

1 π

ð

r2

ð0Þ

r1

pð0Þ ðr Þdr:

(4.128)

4.5 Screened Hydrogenic Average-Atom

165

This quantity is produced by interactions with electrons interior to rn . Then from (4.125), the WKB approximation to the energy levels is E nl ¼ E ðn0Þ 

ðZ n e2 Þ 2a0 ðn  Δnl Þ2

:

(4.129)

The quantum defect is zero if the potential is exactly hydrogenic. Applying the WKB approach to the hydrogen-like atom, with Δnl ¼ 0, the screened nuclear charge depends upon the electron populations of the shells inside radius r n . A simple approximation is Zn ¼ Z 

n1 X

1 σ ðn; mÞPm  σ ðn; nÞPn , 2 m¼1

(4.130)

where Pn ¼ 0, 1, 2,    , 2n2 are the populations of the nth level. Outer screening is approximated by E ðn0Þ

nmax X 1 e2 e2 ¼ σ ðn; nÞPn þ σ ðm; nÞPm : 2 rn r m¼nþ1 m

(4.131)

The quantity ePm =rm is just the potential inside a spherical shell of charge ePm having radius r m . Expression (4.130) fails to predict the correct energy levels for truly hydrogenic ions. We see that a single electron in level n screens itself via the factor σ ðn; nÞ. This self-screening is of little consequence in levels with several electrons, but can be of significance when the number is small. We note, however, that self-screening can be justified in the context of the average-atom model, because Pn ¼ 1 actually stands for the average over a distribution of all possible populations, some of which require screening between different electrons in level n. The constant coefficients σ ðn; mÞ describe the screening of the nth shell by the mth shell; that is, they describe the spatial extent of the charge distribution (e.g., the n ¼ 3 shell is not entirely inside the n ¼ 4 shell). Standard values for the screening coefficients are listed in Table 4.3 (More, 1982); they were obtained by fitting to a database of eight hundred ionization potentials for thirty elements. This raises a second difficulty, that using these coefficients for calculating excitation energies is problematic. The total ion energy is not simply the sum of the one-electron eigenvalues, because that sum counts electron–electron interactions twice. Instead, we use X E ion ¼ E n Pn  Eee , (4.132) n

166

Ionization

Table 4.3 Screening Constants σ ðn; mÞ 1

2

3

4

5

6

7

8

9

10

0.3125 0.2345 0.1093 0.0622 0.0399 0.0277 0.0204 0.0156 0.0123 0.0100

0.9380 0.6900 0.4018 0.2430 0.1597 0.1098 0.0808 0.0624 0.0493 0.0400

0.9840 0.9040 0.7200 0.5150 0.3527 0.2455 0.1811 0.1392 0.1102 0.0900

0.9954 0.9722 0.9155 0.7329 0.5888 0.4267 0.3184 0.2457 0.1948 0.1584

0.9970 0.9879 0.9796 0.9200 0.7469 0.5764 0.4592 0.3711 0.2994 0.2450

0.9970 0.9880 0.9820 0.9600 0.8300 0.7350 0.6098 0.5062 0.4222 0.3492

0.9990 0.9900 0.9860 0.9750 0.9000 0.8300 0.7450 0.6355 0.5444 0.4655

0.9999 0.9990 0.9900 0.9830 0.9500 0.9000 0.8300 0.7500 0.6558 0.5760

0.9999 0.9999 0.9920 0.9860 0.9700 0.9500 0.9000 0.8300 0.7600 0.6723

0.9999 0.9999 0.9999 0.9900 0.9800 0.9700 0.9500 0.9000 0.8300 0.7650

where the electron-electron interaction energy is found from the WKB calculation   1X Ze2 Eee ¼  (4.133) Pn Ψn jeΦðrÞ  jΨn : 2 n r The expectation value is evaluated by performing a radial integral using the methods of contour integration (More, 1982). The result is Eion ¼ 

X ðZ n eÞ2 n

2a0 n2

Pn ,

(4.134)

with the consistency condition ∂Eion ðZ n eÞ2 ¼ þ E ðn0Þ ¼ En : ∂Pn 2a0 n2

(4.135)

The Hartree-Fock approximation for the average shell populations may be calculated for any well-defined plasma state. The average-atom shell energies En and the ion energy E ion are calculated by substituting the average populations into (4.130) through (4.134). Because the formulas for E n and Eion are not linear in the populations, the result of substituting Pn into the equations is not strictly equivalent to the averages hEn i or hE ion i. The equilibrium ionization level of the average-atom is given by X Z ¼ Z  Pn , (4.136) n

where the sum is over all levels. The equilibrium level populations Pn are given by the Fermi-Dirac distribution function Pn ¼

Dn , 1 þ eðEn μÞ=kB T

(4.137)

4.5 Screened Hydrogenic Average-Atom

167

with Dn being the level degeneracy (for the isolated atom Dn ¼ 2n2 ), μ is the chemical potential, and En is the level’s energy. Most often, the Pns are nonintegral populations, but they are confined to the range 0  Pn  2n2 . Expression (4.137) would be rigorously correct if the electrons were noninteracting. Expression (4.137) has the same form as the occupation of free-electron states; the distinction between bound and free states is now merely a matter of labeling part of the range of one-electron quantum numbers. Average-atom eigenvalues depend strongly on temperature. As the temperature increases, bound electrons are thermally ionized and the screening charges Zn increase. As a result, the energy levels also increase with temperature, so that the spectral lines indicative of bound-bound transitions are shifted toward higher energies, and the angular momentum splitting of levels decreases as the hydrogenic limit is approached. The density dependence of the average-atom eigenvalues is much weaker. The allowed average-atom energy levels are distinguished by the principal quantum numbers and the energies of the levels given by analytic formulas. These formulas include spin and relativistic effects, with the bare nuclear charge replaced by a screened charge that depends upon the populations of the various levels. When the relativistic change in mass of the electron is taken into account, the velocity changes in different parts of the Coulomb potential, which leads to a small splitting of the levels with angular momentum. The shell energy is given by the Dirac formula (Born, 1962) for isolated hydrogenic ions, allowing for spin and relativity 82 9 12 31=2 0 > > > > < = αZ n ð0Þ C7 B 2 6 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiA 5  1 , E n, k ¼ me c 41 þ @ (4.138) > > 2 2 2 > > n  k þ k  α Zn : ; where k is Bohr’s azimuthal quantum number (which is equal to l þ 1 in the wave mechanics model), and α ¼ 2πe2 =hc is the fine-structure constant. A simpler approximation, due to Pauli, for low atomic number, is found by expanding (4.138) in powers of ðαZ n Þ2 " # 2
ð αZ Þ 1 3 ð0Þ n E n, k ¼ me c2 1 þ  : (4.139) k 4n n Expressions (4.138) and (4.139) are modified to include dense plasma effects according to ð0Þ

ð0Þ

En, k ¼ E n, k  ΔEcl ,

(4.140)

168

Ionization

where ΔE cl is the continuum lowering energy taken from one of the aforementioned models, most often that of Stewart and Pyatt or Ecker and Kröll. The level ionization energy ignores the fine-structure splitting and is taken as the average of the

angular ð0Þ ð0Þ ð0Þ 1 momentum subshell extremes for a particular shell E n ¼ 2 E n, 1 þ E n, n . The screened level charge is now, in contrast to (4.130),
 1 1 Zn ¼ Z  σ ðn; mÞPm  1  2 σ ðn; nÞPn : 2 2n m¼1, m6¼n mmax X

(4.141)

This formula includes screening by both inner and outer electrons, and has the following property: If one electron is removed (ionized) from a given shell, and then a second electron is removed (ionized) from a (generally) different shell, the total energy for ionizing the two electrons is dependent upon the order in which they are removed. To rectify this problem, we consider a different definition for the ionization potential: ! mmax ∂ X ð0Þ In ¼ (4.142) E Pm : ∂Pn m¼1 n, m ð0Þ

The shell energy E n, m is computed with (4.138) or (4.139), except that the sum over m is restricted to m  n. This would imply that only inner shell screening is ð0Þ included, but by taking derivatives of E n, m with respect to Pn , with m n þ 1, terms involving σ ðn; mÞ with m > n will appear. Thus, the total energy required to ionize two or more electrons is independent of the order of ionization. Pressure ionization/continuum lowering is incorporated into the average-atom model by forcing the degeneracy of the nth bound shell to decrease to zero at high densities where rn ¼ R0 is satisfied. This is accomplished with a simple expression for the degeneracy of the nth level, Dn ¼

2n2 b , 1 þ a Rrn0

(4.143)

where the constants a and b are adjusted to provide approximate agreement with the Thomas-Fermi theory at high density. A comparison of the Saha, Thomas-Fermi, and average-atom models is shown in Figure 4.12, for moderate and high-atomic-number elements. The Saha theory of Section 4.1 is often compared to the average-atom theory of ionization. For hydrogen, we have exact analytic solutions for each, and they can be used to understand the difference of the two approaches. For a completely ionized plasma, the two theories give essentially equivalent chemical potentials

169

4.6 Time-Dependent Non-LTE Average-Atom (a)

(b)

Figure 4.12. Comparison of the ionization levels as a function of temperature at a density of 10–3 g-cm3 for (a) aluminum and (b) gold. The Saha model (dashed lines) and average-atom model (solid line) exhibit a “plateau-like” shell structure that the Thomas-Fermi model (dot-dashed line) does not.

and bound-state populations. On the other hand, for the nearly neutral case, where Z   1, we find Z Saha ¼

eE1 =kB T ni λ3de

!1=2 and Z AA ¼

2 E1 =kB T e : ni λ3de

(4.144)

 2 Hence, Z AA ¼ 2 Z Saha  Z Saha . This prediction is caused by an exaggerated binding energy for negative ions in the average-atom theory. Of course, the Saha theory altogether ignored the possibility of negative ions. As pointed out earlier, we rarely encounter true situations where thermodynamic equilibrium exists. Extending the discussion of Section 4.4 to encompass the average-atom model gives us a simple non-LTE model. 4.6 Time-Dependent Non-LTE Average-Atom The distribution of ions within a particular ionization stage (as well as those between stages) cannot be described by an equilibrium distribution function such as the Saha-Boltzmann equation or the LTE average-atom model of the previous section. In these plasmas, populations of energy levels and ionization balance are determined by balancing rates of excitation and ionization against the countervailing rates of de-excitation and electron capture. Further, in general, the populations of the atomic levels are not in a steady state, and thus the distribution depends upon the history of the distribution.

170

Ionization

We introduced in Section 4.4 that there are a number of atomic processes in play: electron collisional, radiative (spontaneous and stimulated), autoionization (and dielectronic recombination), energetic charged particle collisions, and so on. Clearly, depending on the processes being considered, together with their relative importance, and with the complexity of the atomic system being investigated, there can be a substantial number of equations to solve. For situations of high temperatures and atomic numbers sufficiently low that open K-shell ions dominate, the number of equations required to find the shell populations is somewhat manageable. However, with lower temperatures and/or atoms with higher atomic numbers, the electronic structure of the ions involves open L-, M-, . . . shells, the number of equations makes the problem nearly insolvable. This topic is addressed briefly in Section 4.7. The extension of the screened hydrogenic LTE average-atom model of the previous section allows the time evolution of the principal quantum shell populations to be followed. As with the LTE average-atom model, the shell occupancies are nonintegral, and the number of equations is reduced to a “handful.” The model makes the assumptions that there is no correlation between electrons occupying different shells and that the ambient radiation field is weak. Furthermore, the collisions of the free electrons may not be strong enough to establish a detailed balance of the electron transitions between bound-bound states as well as those between bound states and the continuum.

4.6.1 Population Rate Equations The pioneering work of the time-dependent non-LTE average-atom model was by Lokke and Grasberger (1977) and was subsequently extended and improved by others. The fundamental processes being considered are collisional electron transitions and radiative transitions; both types of interactions result in excitation and/or ionization, as well as the inverse processes of de-excitation and recombination. This is a collisional-radiative model. Later in this section, we consider an additional process that refines the model to be more relevant at high temperature and low density. The time development of the population of the nth level (thinking of the atom represented by the Bohr model) is governed by dPn ¼ An Qn  Bn Pn : dt

(4.145)

The first term on the right-hand side represents processes that populate the level, and the second term represents processes that depopulate the level. The population terms have a factor Qn , which represents the availability of the level to accept an

171

4.6 Time-Dependent Non-LTE Average-Atom TcÆn

Continum

TnÆc

nmax TmÆn

TnÆm

n

TmÆn

2

TnÆm

Populating

Depopulating

1

Figure 4.13. Schematic of transitions into and out of a particular energy level.

electron; it is the fraction of the shell that is, on average, empty. Referring to Figure 4.13, populating transitions into the nth level can come from levels with smaller quantum numbers as well as larger quantum numbers, in addition to transitions from the continuum; hence, An ¼

n1 X

nmax X

Pm T m!n þ

Pm T m!n þ Z  T c!n ,

(4.146)

Qm T n!m þ T n!c :

(4.147)

m¼nþ1

m¼1

and the depopulating transitions are Bn ¼

n1 X m¼1

Qm T n!m þ

nmax X m¼nþ1

The last term in these two equations accounts for transitions from and to the continuum. These two terms can be cast in the same form as for the discrete transitions if we set the availability of the continuum to one, and the “population” of the free electrons is just the average ionization level. (Note that our definition of Pn for the average-atom model differs from the interpretation used in standard kinetics models.) The transition rates are denoted by T and are defined to be the rate per electron in the initial shell, assuming the final shell is empty. This allows the use of simple scaled hydrogenic-rate formulas but introduces the complication that for an excitation from level m to level n, the contribution to dPn =dt has the form
 Pn Pm 1  (4.148) T m!n , Dn

172

Ionization

where the first term Pm multiplies the one-electron rate by the number of electrons in the initial shell. Because the rate assumes that the final shell is empty, it is multiplied by the factor in parentheses. This factor is just the availability Qn . It arises physically from the Pauli principle and plainly has the correct limiting form for an empty (Pn ¼ 0) and full shell (Pn ¼ Dn ); Dn is the shell degeneracy given by (4.143). Because the assumption is made that there is no correlation between the occupancies of the initial and final levels, we can add terms of this type together to form the simple rate equations (4.146) and (4.147). The rates of excitation and de-excitation by a single process are related by the principle of detailed balance. This is written as P0m Q0n T m!n ¼ P0n Q0m T n!m ,

(4.149)

for discrete bound-bound transitions, and ðZ  Þ0 Q0n T c!n ¼ P0n T n!c ,

(4.150)

for transitions from and to the continuum. The superscript “0” represents the equilibrium values discussed in the previous section. The equilibrium populations P0n are given by (4.137), and the equilibrium ionization value is ðZ  Þ0 ¼ Z 

nmax X

P0n ,

(4.151)

n¼1

where Z is the nuclear charge. The de-excitation and excitation rates are related by
 P0m Q0n Dm En  Em T n!m ¼ 0 0 T m!n ¼ exp (4.152) T m!n : Dn kB T e Pn Qm The shell level energy E n ¼ I n is calculated using (4.142). The transition rates are composed of both collisional and radiative processes. They are not important for the present discussion, but as mentioned in Section 4.4, will be developed in detail in Chapter 9. Solution of the set of rate equations (4.145), together with (4.136), is done numerically. Obtaining a solution is a challenge since the matrix of transition rates is very “stiff”; this means that the coefficients span many orders of magnitude. Figure 4.14 shows the time dependence of the level populations and energies for normal-density iron at temperature 1 keV. The levels for n ¼ 4 and 5 are initially empty but immediately populated by Boltzmann statistics. As time progresses, level 3 begins to strip down, followed slightly by level 2, with those electrons being removed appearing in levels 4 and 5. Equilibrium is well established by one picosecond. The LTE values for the populations corresponding to Figure 4.14 are

173

4.6 Time-Dependent Non-LTE Average-Atom (a)

(b) 25 1

10

101

2

1 100

20

1

100 2

15

10–2 10 10–3

5

4

10–2

10–16

10–15

10–14

10–13

5

10–4

5

10–17

3

10–3

4

10–4

10–5 10–18

10–1 –En (keV)

Pn

10

lonization level

3 –1

10–5 10–18

0 10–12

10–17

10–16

Time (s)

10–15

10–14

10–13

10–12

Time (s)

Figure 4.14. (a) Level populations for iron at density 7.85 g-cm3 and temperature 1.0 keV. The ionization level is shown by the dashed line. (b) The corresponding level energies, which are expressed as negative quantities because the zero of energy is defined at the bottom of the continuum. The populations P1 , P2 , and P3 were initialized for a fully recombined atom.

(b)

15

101 2

3

–3

10

9 4

6 5

3 12

1

10–1

9

4

Pn

Pn

10–1 10–2

2 100

12

1

Ionization level

100

15

101

10

–2

10

–3

5–7

3

10–4

3

10–4

0 10–5 10–18 10–17 10–16 10–15 10–14 10–13 10–12

6

Ionization level

(a)

0 10–5 10–18 10–17 10–16 10–15 10–14 10–13 10–12

Time (s)

Time (s) 3

Figure 4.15. (a) Level populations for iron at 7.85 g-cm and temperature 0.1 keV. (b) Without pressure ionization/continuum lowering. The initial conditions are the same as for Figure 4.14.

P01 ¼ 1:960, P02 ¼ 0:404, P03 ¼ 0:249, P04 ¼ 7:12  104 , P05 ¼ 6:27  105 , and Z  ¼ 23:39. One sees that as level 3 and then 2 strip down, there is an increase in the nuclear charge seen by the higher levels and therefore their binding energy increases. The importance of pressure ionization/continuum lowering is demonstrated in Figure 4.15. The reader should refer to Section 4.3 for the detailed discussion of this topic.

174

Ionization

4.6.2 Steady-State Non-LTE The rate equations will eventually converge to steady-state values for the level energies, populations, and screened charges, as seen in the preceding figures. The time required to achieve equilibrium is determined by the size of the rate coefficients. As shown in Figures 4.14 and 4.15, some levels approach equilibrium values rapidly while others seem to take “forever.” We can examine the limiting case by setting the left-hand side of (4.145) to zero, as was done in Section 4.4. For simplicity, consider an “average” ion to have only two active energy levels and that they are outside (have a higher principal quantum number) the inert core. The lower level has label “1” and the upper level (less tightly bound) has label “2.” Writing the rate equations from (4.145) through (4.147) gives for each level dP1 ¼ 0 ¼ ðP2 T 2!1 þ Z  T c!1 ÞQ1  ðQ2 T 1!2 þ T 1!c ÞP1 , dt

(4.153)

dP2 ¼ 0 ¼ ðP1 T 1!2 þ Z  T c!2 ÞQ2  ðQ1 T 2!1 þ T 2!c ÞP2 : dt

(4.154)

and

Upon summing these two expressions, the discrete (bound-bound) terms cancel, and thus

 P1 P2   P1 T 1!c  Z 1  (4.155) T c!1 ¼ P2 T 2!c þ Z 1  T c!2 : D1 D2 Assuming the occupation of the higher level is small compared to that of the lower level, we have the CRE (see Section 4.4) expression P1 ¼

Z  ðT c!1 þ T c!2 Þ :  T 1!c þ DZ 1 T c!1

(4.156)

When collisional ionization and radiative recombination dominate, T n!c e ne and T c!n e ne , P1 becomes independent of ne . This is the familiar CE, where the ionization state freezes out as the plasma density decreases. For the two-level system, Z   Z  P1 . Excitation acts to determine the upper-level population. Using (4.156) in (4.153) gives the result P2 ¼

P1 T 1!2 þ Z  T c!2

: 1  DP11 T 2!1

(4.157)

As before, the excitation and de-excitation processes become collisional and radiative, respectively, and since P1 is independent of density, we see, from the

4.6 Time-Dependent Non-LTE Average-Atom

175

ratio of the bound-bound to free-bound rates, that the upper-level population becomes proportional to the density. Thus, while the lower-level population remains constant with decreasing density, the upper population decreases.

4.6.3 Dielectronic Recombination It is well known that the basic non-LTE average-atom model fails for low-density plasma. This is due to the neglect of two-electron transitions. In particular, the rate of electron capture is substantially enhanced by dielectronic recombination. Two-electron transitions act to increase the emissivity (thermal cooling) of plasma so that the relaxation of the system proceeds much faster than that of the CRE system. Figure 4.16 shows that for high temperature, the number of doubly excited states contributes a substantial increase to the recombination coefficient. Dielectronic recombination (also known as resonant capture) occurs when the electron and ion can form a doubly excited quasibound state, that is, an autoionizing state (also known as Auger decay). Referring to Figure 4.17, an electron capture (recombination) occurs when a free electron encounters an ion and forms a new ion in a stable state n (ordinarily an excited state). The final stabilized state n

Figure 4.16. Comparison of the dielectronic recombination coefficient (solid line) with that for the radiative recombination (dashed line) for Heþ þ e .

176

Ionization

c

m, m⬘

n

gs

Figure 4.17. Schematic illustration of electron capture from the continuum c to a doubly excited state m, m0 (bold arrow), followed by a stabilizing radiative transition to a singly excited state n (thin arrow). Direct capture by radiative recombination is shown by the dashed arrow.

can arise either by direct radiative capture, in which a single electron changes from the continuum to a bound orbital, or through formation of a doubly excited state m, m0 , followed by a stabilizing radiative decay to the singly excited bound state n. This is the dielectronic recombination process and is a resonance process because of the discrete nature of the two bound electron states. The highly excited state may then autoionize, thereby inverting the dielectronic recombination, and the process may be treated as a simple scattering event. Resonances for the captured electron, which encompass high-lying Rydberg states near the series limit, can be substantially affected by electric fields (the Stark effect), which tend to enhance the dielectronic recombination rates. For dielectronic recombination to occur, the new ion must have at least two electrons; thus dielectronic recombination is not possible in hydrogenic systems. Looking at the process another way, consider the simple two-level system as presented earlier, with the state n labeled “1” and the states m and m0 labeled “2” (in general, state “2” is two distinctly separate states). The inverse process to dielectronic recombination is autoionization, whereby an electron falls from level 2 to level 1, while another electron in level 2 receives the transition energy of the falling electron and jumps into the continuum. If the (integer) population of level 1 is initially N 1 and that of level 2 is N 2 , then after the autoionizing event the configuration is N 1 þ 1 and N 2  2. Of course, there must be a vacancy in level 1 to take the falling electron.

4.6 Time-Dependent Non-LTE Average-Atom

177

For dielectronic recombination to occur, energy conservation requires the energy of the free electron to be E ¼ Em þ E m0  E n 0:

(4.158)

Recall, by convention, that the free electron has positive kinetic energy while the bound states have negative eigenvalues. A first guess for the rate of change of the population of level m is dPm ¼ Z  Pn Qm Qm0 Rn!mm0  Pm Pm0 Qn Amm0 !n : (4.159) dt In this equation, Rn!mm0 is the dielectronic recombination rate coefficient and Amm0 !n is the autoionization rate coefficient. But all is not well here. At low densities, the excited-state populations Pm and Pm0 are small and proportional to ne , the free-electron number density. Therefore, the availability factors for the two states are essentially one at low-density coronal conditions. However, the right-hand side of (4.159) predicts an autoionization rate proportional to Pm Pm0 e n2e , which is extremely small in the low-density limit. The difficulty lies in the basic assumption of our average-atom model, that there are no correlations among the electrons. Clearly in this two-electron process, the electrons of state m and m0 know something about one another; that is, there is a correlation between them. We expect the average population of Pm Pm0 e ne , which is much larger than n2e . At low densities, autoionization occurs mainly from doubly excited states created by dielectronic recombination and not from very rare ions that have been doubly excited by coincident impact excitation. Therefore, we should replace (4.159) with dPm ¼ Z  Pn hQm Qm0 iRn!mm0  hPm Pm0 iQn Amm0 !n , (4.160) dt where the terms in brackets represent averages. In a many-level model, the levels are coupled via all “triples” denoted by n, m, and m0 . For an arbitrary level n, the two-electron model population rates are
 nmax n1 X X dPn δnm  ¼Z hQn Qm iPm0 Rn!mm0 ð1 þ δnm Þ 1  dt Dm m0 ¼1 m¼m0 þ1
 nmax n1 X X δnm  Qm0 hPn Pm iAmm0 !n ð1 þ δnm Þ 1  Dm m0 ¼1 m¼m0 þ1
 nmax X nmax X δmm0 þQn hPm Pm0 iAmm0 !n 1  Dm0 m0 ¼nþ1 m¼m0
 n n max max X X δmm0 0  Pn : (4.161) hQm Qm0 iRn!mm 1  Dm0 m0 ¼nþ1 m¼m0

178

Ionization

In this equation, the final factor in each line arises from combinatoric considerations. Detailed balance requires (4.160) be zero in the LTE limit, where the two rate coefficients are related by Rn!mm0 Q0 hPm Pm0 i0 ¼  n0 ; Amm0 !n Z Pn hQm Qm0 i0

(4.162)

the superscript zero denotes LTE values. To obtain a practical formula, we may neglect correlations in the detailed-balance expression only, and calculate the autoionization rate Amm0 !n from quantum mechanics and then find the dielectronic recombination rate from λ3de ðEn Em Em0 Þ=kB T e , (4.163) e 2 where λde is the de Broglie wavelength of the thermal electron and the E i are the one-electron eigenvalues for the ith principal quantum number, given by (4.142). Equation (4.163) follows from (4.162) if the equilibrium population correlations are factored and the Fermi-Dirac distribution is used for equilibrium populations.3 The two averages in (4.162) can be determined assuming there are no electron correlations at equilibrium (LTE); then we can factor hPm Pm0 i0 ¼ hPm i0 hPm0 i0 ¼ P0m P0m0 . There still remains one additional equation to be considered, that is, the rate of change of the correlated non-LTE population hPm Pm0 i Rn!mm0 ¼ Amm0 !n

d hPm Pm0 i dPm (4.164) ¼ ð1 þ Pm þ Pm0 Þ  hPm Pm0 iQn Rm!n , dt dt where Rm!n is the one-electron radiative decay rate. Since hPm Pm0 i is essentially the number of doubly excited ions at low density, this term relaxes toward the steady-state value by radiative decay. The low-density, steady-state solution to (4.164), for Pm and Pm0 very small, is Pn Rn!mm0 hPm Pm0 i , ¼ Z Qn Rm!n þ Amm0 !n hQm Qm0 i

(4.165)

and the net rate of dielectronic recombination becomes dPm Rm!n : ¼ Z  Pn hQm Qm0 iRn!mm0 dt Rm!n þ Amm0 !n

3

(4.166)

  The quantity λ3de =2 in (4.163) is for nondegenerate electrons. One could replace it using Z  ene ¼ 2=λ3de eμ=kB T for degenerate electrons, but this would be unimportant for degenerate plasma because the high Rydberg states would be either fully occupied or destroyed by pressure ionization.

4.6 Time-Dependent Non-LTE Average-Atom

179

In this form, (4.166) is the dielectronic recombination rate multiplied by a branching ratio for radiative stabilization. The key equations are (4.162) or (4.163) and (4.166); they provide expressions that properly describe both LTE and coronal plasma limits. In the density-temperature (and radiation) regimes of many high-energy-density plasmas, the two-electron transition rates are as large as the one-electron boundbound transition rates, while the one-electron ionization and recombination rates are smaller. Hence, both one-electron and two-electron processes must be treated together; thus, (4.166) is to be combined with (4.145) through (4.147). Returning to the two-level atom, the one-electron bound-bound transitions conserve the total number of electrons in the two levels, P1 þ P2 , while the twoelectron transitions conserve 2P1 þ P2 . Using (4.161), with Z  ¼ 1 and Dn  1, and assuming no correlation between the electrons in level 2, and using (4.145) through (4.147) excluding one-electron transitions to and from the continuum, the two-level equations are dP1 ¼ P1 Q22 R1!22 þ Q1 P22 A22!1  P1 Q2 T 1!2 þ Q1 P2 T 2!1 , dt

(4.167)

and dP2 ¼ 2P1 Q22 R1!22  2Q1 P22 A22!1 þ P1 Q2 T 1!2  Q1 P2 T 2!1 : (4.168) dt The one-electron transition rates T i!j include both collisional and radiative contributions. In steady state, the one-electron and two-electron processes become separated and each are in balance: dP1 dP2 þ ¼ 0 ¼ Q22 P1 R1!22  Q1 P22 A22!1 , dt dt

(4.169)

and dP1 dP2 þ ¼ 0 ¼ Q1 P2 T 2!1  Q2 P1 T 1!2 : dt dt Together, these two expressions yield 2

P1 ¼

D1


 , A22!1 T 1!2 2 1 þ D1 R1!22 T 2!1

(4.170)

(4.171)

and P2 ¼

D2 : A22!1 T 1!2 1 þ D2 R1!22 T 2!1

(4.172)

180

Ionization

Using the detailed balance expressions (4.152) and (4.163) in (4.171) or (4.172) confirms that the populations are in LTE Pn ¼

Dn , 1 þ eðEn μÞ=kB T e

(4.173)

in agreement with (4.137). Conditions where two-electron equilibrium (2eE) (in contrast to CRE) exists can be identified by looking at the ratio of the bound-bound excitation and deexcitation rates in (4.171) and (4.172). At lower density, and in the absence of a strong radiation field, collisional excitation and radiative de-excitation are dominant, so that T 1!2 =T 2!1 e C 1!2 =R2!1 e ne . Also, A22!1 =R1!22 e 1=ne . These two observations show that the smaller, upper population P2 becomes independent of density, while the larger, lower population P1 increases with decreasing density. From (4.171) and (4.172), the two-electron equilibrium values are ð2eE Þ

P1

D1 D2 ð2eE Þ e 1 þ Oðne Þ and P2 e 1 þ Oðconst Þ :

(4.174)

In contrast, from (4.156) and (4.157), the corresponding CRE values are ðCRE Þ

ðCREÞ

(4.175) e D1 OðconstÞ and P2 e D2 Oðne Þ: ð2eE Þ ðCRE Þ > P1 . Since P1  P2 in this two-level system, Thus we expect to find P1 ð 2eE Þ    ðCREÞ Z  Z  P1 , and so ðZ Þ < ðZ Þ . Figure 4.18 shows a comparison of the ionization levels for the 2eE and CRE models as a function of electron density. The material is gold and the temperature is 0.6 keV (Rose, 1997). This value was chosen so that the N- and O-shells correspond roughly to levels one and two of the two-level model. The CRE curve is at the coronal equilibrium value for densities less than about 1021 cm3. Increased cooling (lower ionization level) is achieved at the lower densities because of recombination enhanced by two-electron processes. It is well known that the Burgess-Merts formula (Burgess, 1965) used in the 2eE calculation overestimates the dielectronic recombination contribution in high-atomic-number ions. An improved description for dielectronic recombination is available in the FLYCHK code (Chung, 2005); those results are also shown in the figure. P1

4.6.4 Reconstruction of Ionic States Solution of the non-LTE average-atom equations yields the average populations for the principal quantum levels of the atom. These noninteger values represent averages over ionic distributions. Experiments, however, measure (indirectly) the ion-stage distributions by a careful analysis of X-ray spectra. The issue is to relate the ionic populations to the shell populations of the calculation. Since we assume

181

4.6 Time-Dependent Non-LTE Average-Atom

Figure 4.18. Ionization level for gold plasma at temperature 0.6 keV. The CRE results are indicated by the filled circles, the 2eE results by the open triangles, and the FLYCHK calculations by the open circles.

that the electrons are not statistically correlated, the binomial theorem will give the probability of a particular configuration αfk 1 ; k 2 ; . . . ; knmax g, where kn is the integer population of the nth principal quantum shell. The probability is then
k n
 2 nmax Y 2n2 ! Pn Pn 2n kn Pfαg ¼ 1 2 , (4.176) k !ð2n2  k n Þ! 2n2 2n n¼1 n where 2n2 is used in place of Dn for the shell’s degeneracy. In general, the configuration α encompasses many ionic states. For example, the beryllium-like configuration {2,2,0,. . .} (i.e., two electrons in the K-shell and two in the L-shell) comprises, in LS-coupling, the ionic states 1

1s2 2s2 1s2 2s2p 1s2 2p2

S0 P1 , 3 P2, 1, 0 1 S0 , 1 D2 , 3 P2, 1, 0 1

The total degeneracy for a configuration is gfαg ¼

nmax Y

2n2 ! , k !ð2n2  kn Þ! n¼1 n

(4.177)

182

Ionization

Figure 4.19. Fractional ionization distribution for a gold plasma at a temperature 2.2 keV and ne ¼ 6  1020 cm3. The reconstruction is shown by solid points, the superconfigurational calculation by open points, and the experimental values by plus signs (+). The superconfigurational calculation without two-electron processes is also shown by open squares.

which, for this example, is 32, in agreement with adding the degeneracies of each individual ionic state. Equation (4.176) does not provide the fraction of ions in, for example, the ionic state 1s2 2s2p1 P1 . Figure 4.19 shows an example of this reconstruction. The results based upon the preceding are compared to a superconfiguration-based collisional-radiative code and the experimental data of Foord et al. (2000). The experimental value for the ionization level is Z  ¼ 49:3, the average-atom value Z  ¼ 48:6, and the superconfigurational values Z  ¼ 49:1 and 53.3, with and without two-electron processes, respectively.

4.7 Other Models There is a large collection of alternate ionization models described in the literature. Some of them are designed for particular regions of density-temperature space, while others are quite detailed and somewhat unwieldly for quotidian use. A detailed calculation requires thousands to hundreds of thousands of levels along with perhaps millions of transitions.

4.7 Other Models

183

Beyond the time-dependent non-LTE average-atom model, the next most detailed approach is that of detailed-configuration accounting (DCA). DCA models also come in a number of flavors. A direct extension of the time-dependent nonLTE average-atom model of Section 4.6 is to treat all ionization stages in a hydrogenic approximation (Lee, 1987). Referring to Figure 4.1, the assumption is made that each energy level Em, n couples with every other level of the same mth ionization stage, but only the ground state to the ground states of the adjacent ionization stages labeled ðm  1; 1Þ and ðm þ 1; 1Þ. The ground- and excited-state energy levels within each ionization stage are generated from screening constants, in a fashion essentially that described in Section 4.5. The rate equations are quite similar to (4.114), except now there is an additional index (or two) for the excited states. For the mth ionization stage, the population of the lth level is given by X dN m, l T m, l, k N m, k ¼ T m, l N m, l þ T mþ1, l N mþ1, 1  dt k X þ ðT m1, k N m1, k  T m, k N m, 1 Þδl, 1 ,

(4.178)

k

where the index k refers to another excited state. The first term on the right-hand side is the rate from level l; the correspondence to (4.114) is  total ionization  T m, l $ ne αðmeÞ þ αðmrÞ . The second term is the total recombination rate. The term P T m1, k N m1, k represents the increase of the ground-state population caused by k P ionization, and the term T m, k N m, 1 represents the decrease of the ground-state k

population caused by recombination. These two terms exactly cancel when plasma is in steady-state equilibrium, where the total ionization rate of a given ionization stage is equal to the total recombination rate of the next-higher stage. The third term on the right-hand side of (4.178) couples the excited levels together according to ðeÞ

T m, l, k ¼ ne C k, l þ Bk, l Eν for l > k, ðd Þ

T m, l, k ¼ ne C k, l þ Ak, l þ Bk, l E ν for l < k,

X

X ðd Þ ðeÞ T m, l, l ¼  ne Cl, k þ Al, k þ Bl, k E ν  ne Cl, k þ Bl, k Eν : kl ðeÞ

ðd Þ

In this expression, Ck, l and C k, l are the collisional excitation and de-excitation rates for transitions between the levels l and k. The symbol Ak, l is the Einstein coefficient for spontaneous emission, and Bk, l is for stimulated emission for a transition induced by radiation of energy density E ν dðhνÞ in the photon frequency range ν to ν þ dν. The Einstein coefficients will be discussed in Chapter 9.

184

Ionization

While this model includes ionization from excited states and recombination from ground states, it does neglect recombination from excited states. This is justified for ne < 1022 cm3 because excited-state populations are usually much less than ground-state populations and the recombination rate from a ground state is approximately equal to that from an excited state. Simulating the spectral signatures of plasma requires models with much more detail, resulting in the use of many more energy levels and a phenomenal increase in possible bound-bound transitions. A screened DCA model, due to Scott and Hansen (2010), is the next step in complexity to providing this detailed information. Still other, more sophisticated, synthetic spectral codes are available and produce results comparable to the quality of the experimental data. These are not generalized codes, as they are constructed for specific situations. There is one other topic that we haven’t addressed, but is important when high intensity laser beams are being used as an energy source. If the intensity is greater than about 1017 watts-cm2, the electric field of the electromagnetic wave is comparable to the electric field of a single proton at a distance of one Bohr radius. Hence an electron can be torn from the atom solely by the electric potential. There are several phenomena by which this type of ionization can take place; they are collectively known as high-field ionization, with the Keldysh theory being the most notable. This theory has two regimes: the multiphoton and the tunnel ionization. We will see the importance of these effects in Section 12.9.

5 Entropy and the Equation of State

One of the more important aspects of the physics of high-energy-density material is its motion in response to internal and external forces. In Chapter 3, we developed expressions for describing the motion in terms of conservation equations. We argued there that it is neither proper nor appropriate to describe the motion of the individual atoms, ions, or electrons. Because of the “smoothness” of highenergy-density material over distances that are large compared to the size of atoms or the interatomic spacing, the fluid description is the proper picture. The conservation equations of momentum and energy require information about the state of the material. The pressure enters directly in the momentum equation, while the specific heat is required for the energy equation. These quantities are material-specific and are expressed as the equation of state (EOS). The EOS relates thermodynamic properties, such as pressure and energy, to intensive variables, such as density and temperature – quantities that are readily measureable. As we shall see in the following two chapters, an EOS provides closure to the fluid conservation equations. Calculation of a proper EOS is one of great complexity. A simple perfect gas EOS is not very useful in most instances, and we must turn to models that contain a mixture of different assumptions about the macroscopic, as well as microscopic, description of matter. Iron is an excellent example of a material with much complexity. The EOS of iron has long been a topic of considerable interest, and there is a vast body of literature devoted to it. This fact is hardly surprising in view of the importance of iron and its alloys as structural materials. The study of iron’s EOS is also motivated by interest in modeling the Earth’s core, where it is necessary to understand the properties at high temperature and pressure. Four solid phases of iron have been observed at modest pressures. These are diagramed in Figure 5.1. The alpha phase, which is stable at ambient pressure and temperature, has a body-centered cubic (bcc) structure and exhibits well-known 185

186

Entropy and the Equation of State

Figure 5.1. Phase diagram for iron. Additional phases exist at higher pressures.

magnetic properties up to the Curie temperature of 1,042 K. Iron transforms to the gamma phase, which has a face-centered cubic (fcc) structure at temperatures above 1,184 K (and zero pressure). It transforms back to the bcc structure (sometimes called the delta phase) at 1,665 K before melting at 1,809 K. The epsilon phase, which was first discovered in shock wave experiments, has a hexagonal close packed (hcp) structure and is produced at pressures above about 11 GPa. The boiling point of iron is 3,135 K. Our interest in iron as an example, from the highenergy-density physics perspective, is not so much with the different solid phases, but the solid-liquid interface. In this chapter, we present models that work for different portions of the densitytemperature space that are important in high-energy-density physics. We also address the basics of more sophisticated EOS models involving ideas from condensed-matter physics. At this time, we do not consider pressures and energies arising from additional physics processes such as radiation and magnetic fields; these will be discussed in subsequent chapters. Referring to Figure 2.5, the region of density-temperature space for Γii < 1 is that of a “perfect” gas, while for Γii > 100 the material has a crystal-like structure; this region is often modeled as a one-component plasma (OCP). It is the intermediate region, where 1 < Γii < 100, is of primary interest. For convenience, in this chapter, Γ is understood to be Γii , the ion–ion coupling parameter.

5.1 Two-Temperature Thermodynamics

187

In Section 3.3, we developed a consistent approach to finding thermodynamic quantities. Explicit in that discussion is the entropy that assumes a central role in describing the thermodynamic properties of material. The entropy in classical statistical mechanics is determined by the logarithm of the volume in phase space accessible to the ensemble of particles. We established earlier that the change in entropy is an exact differential, so that once the ensemble of particles is specified in terms of the particle distribution in phase space, the entropy is known. The entropy of the ensemble has its maximum when the system is in equilibrium, and changes in the closed system necessitate that the change in entropy is nonnegative. Thus, entropy differs fundamentally from internal energy – it is not conserved simply by isolating the system from its surroundings. Thermodynamically consistent quantities are based on first finding the Helm~ Here Z~ is the total partition function of holtz free energy F ðρ; T Þ ¼ kB T log Z. the canonical ensemble for the system of particles. The entropy, pressure, and energy are found from


 ∂F ∂F F 2 ∂ S¼ , P¼ , and ℰ ¼ T ¼ F þ TS: (5.1) ∂T V ∂V T ∂T T V This expression for the internal energy ℰ emphasizes that it has contributions from both the free energy as well as the entropy.

5.1 Two-Temperature Thermodynamics High-energy-density material is strongly characterized as a two-temperature plasma. Separate electron and ion temperatures can result from a variety of physical processes such as laser-energy deposition, viscous heating due to the passage of a shock front, and so on. The nonequilibrium states arise because the electron–ion energy transfer is impeded by the large difference between the electron and ion masses. The Coulomb interactions encountered in high-energy-density physics cause substantial deviations from the EOS of a perfect gas. At low densities, we can successfully use separate EOSs for the two fluids and assume uniform intermixing. At high densities, the two EOSs are not additive because the electron–ion interaction energy cannot be separated, nor can the pressure; we cannot separately compress ions and electrons. However, the entropies are additive. Consider a system of two types of particles, one with N 1 particles and the other with N 2 particles. The phase space (thermodynamic probability) of the combined particles W is just the product space of the phase spaces of the individual particles, W 1 and W 2 . Then the thermodynamic entropy is

188

Entropy and the Equation of State

S ¼ k B logW ¼ kB logΔW 1 þ kB logΔW 2 ¼ S1 þ S2 :

(5.2)

In deriving the thermodynamic quantities in a canonical ensemble (see Section 3.3), we considered only a single type of particle. However, with two types of interacting particles, the canonical ensemble has to be modified to include two types of electron–ion interactions: rapid Coulomb interactions associated with Debye screening, and slower collisional interactions that transfer entropy between electrons and ions (More, 1982). The ion-pair interaction, represented by the Debye potential, discussed in Section 3.5, yields a free energy for the ions that depends upon both electron and ion temperatures. An instantaneous change in the electron temperature is felt by the ions through both the ion-pair potential and the slower collisional energy transfer. The electron thermal velocity is large compared to that of the ions, so we can calculate the thermodynamic properties of the electrons assuming fixed ions; this is the treatment of Section 3.3. The electron free energy   is a function of the ion positions and the electron temperature, F e ¼ F e rj ; T e . With the ions free to move, F e serves as a classical potential energy function for ion thermal motion, at least for the most important class of ion motions. It is appropriate to use the free energy rather than the electron energy ℰe (found from F e ) because the electrons are isothermal for the class of ion motions that dominate the equation of state. Skipping the justification, the effective potential energy, in the low-density limit is Fe  Fo þ

1X ℰi, j , 2 i, j

(5.3)

where F o depends upon the mass density and electron temperature but not upon the ion coordinates. The second term in (5.3) is a sum of screened Debye-like potentials. Using methods of statistical mechanics, we find the complete partition function Z~ . Because the ion kinetic energy is classical, the partition function can be written as the product of the partition function of a perfect gas (see the next section) and a configurational partition function. We then define the complete free-energy function ~ ðT e ; T i ; V Þ, from which the entropies for the electron and ion fluids F ¼ k B T i log Z are found. The pressure consists of the ion kinetic pressure and the average (over ion coordinates) of the electron pressure given by Pe ¼ ð∂F e =∂V ÞT e . Hence, dF ¼ PdV  Se dT e  Si dT i ;

(5.4)

it is straightforward to generate a complete set of thermodynamic quantities. The total energy is defined as the average ion kinetic energy plus the average of the

5.2 Perfect Gas

189

electron energy given by ℰe ¼ F e  T e ð∂F e =∂T e ÞV , from which it immediately follows that dℰ ¼ T e dSe þ T i dSi  PdV:

(5.5)

The heat capacities C at constant volume are defined according to dQe ¼ T e dSe ¼ Cee dT e þ C ei dT i dQi ¼ T i dSi ¼ C ie dT e þ C ii dT i :

(5.6)

Upon taking second derivatives of (5.5), a particular Maxwell relation ð∂Se =∂T i ÞT e , V ¼ ð∂Si =∂T e ÞT i , V is obtained and used to show Cie =T i ¼ C ei =T e . The off-diagonal heat capacities are negative quantities. Inverting (5.6) yields the more common expression dT e ¼ Aee dQe þ Aei dQi dT i ¼ Aie dQe þ Aii dQi ,

(5.7)

and Aei =T e ¼ Aie =T i . Both the electron and ion fluids have their temperatures incremented/decremented simultaneously for the addition of an amount of heat dQe added instantaneously to the electrons (with dQi ¼ dV ¼ 0). The change in ion temperature occurs along a constant-volume adiabat because the ion energy will have changed through the alteration in the electronic screening. A generalized condition of thermodynamic consistency of pressure and energy is


 ∂ℰ ∂P ∂P ¼ P þ T e þ Ti : (5.8) ∂V T e , T i ∂T e T i , V ∂T i T e , V From these basic ideas about two-temperature thermodynamics, we can address specific models for EOSs. High-energy-density material is considered to be in a state of thermal equilibrium (independently for each type of particle) and completely dissociated into individual atoms. Furthermore, we consider the material to be composed of only one atomic element. 5.2 Perfect Gas In the simplest picture, the atoms are treated as noninteracting and have no internal structure, so they can be treated as “point-like” particles. In Section 3.2.2, we developed the translational partition function Z~ trans . For N identical particles in a volume V !N
3N =2 N V 2πmk T 1 V B Z~ trans ¼ ¼ , (5.9) N! λ3d N! h2

190

Entropy and the Equation of State

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where λd ¼ h= 2πmkB T is the de Broglie wavelength. Using this in the expression for the free energy, and then taking derivatives, we find the EOS for an ideal gas of noninteracting particles ( "
) 3= # V 2πmk B T 2 3 þ 1 þ Nk B , S ¼ NkB log 2 N 2 h P¼

NkB T 3 , and ℰ ¼ Nk B T : V 2

(5.10)

The expression for the entropy is the Sackur-Tetrode formula. We recognize the three-halves factor in the energy equation to be just the three degrees of translational freedom, each with one-half kB T of energy. It is more convenient to rewrite the pressure and specific energy as P ¼ nk B T and ε ¼

P , ρðγ  1Þ

(5.11)

where n is the particle number density and ε is the specific energy. The polytropic index is γ, which is defined as the ratio of the specific heats γ ¼ cP =cV (see Section 3.3.1). For a perfect monatomic gas characterized by only the translational part of the partition function, γ ¼ 5=3. Some simple molecular structures can be characterized by a different value for the polytropic index. The partition function for a molecule with translational, rotational, and vibrational degrees of freedom is given by Z~ ¼ Z~ trans Z~ rot Z~ vib . A simple rotating diatomic molecule has two additional degrees of freedom, so γ ¼ 7=5, while a rotating and vibrating diatomic molecule has γ ¼ 9=7. Values for the polytropic index can be calculated for more complex molecules (Zel’dovich & Raizer, 1966). A note of caution: these remarks imply a constant γ and therefore constant specific heats; in general, they are not constant. Now consider an adiabatic change for a perfect gas. Using the results from Section 3.3, we can write cV d ð log T Þ ¼ ðcP  cV Þd ð log ρÞ, which yields the polytropic law
γ
γ1 ρ ρ P ¼ P0 or T ¼ T 0 , ρ0 ρ0

(5.12)

(5.13)

where the 0 subscript refers to the initial state. Expression (5.13) reveals why γ is also called the adiabatic exponent. Having made no distinction as to the type of fluid, the perfect gas EOS can be used independently for the electrons and ions. This model is relevant only for relatively low density, where the two fluids may be approximated as independent.

5.3 Realistic Gas

191

The preceding discussion has some relevance for high-energy-density situations, but mentions nothing about the effects of ionization and/or electronic excitation. 5.3 Realistic Gas In ionized plasma there is no longer a single type of particle; the charge-neutral atom and both free electrons and ions are present. As discussed previously, it is highly likely that the species will not be in thermodynamic equilibrium and thus will have separate temperatures. More importantly, we must account for the energy required to singly or multiply ionize the neutral atom. Chapter 4 was devoted to the discussion of the ionization process where bound electrons are successively removed from the atom. Each ionization event requires some amount of energy to remove an electron from an unexcited bound level to the edge of the continuum, that is, moving from state m to state m þ 1. The energy required to remove the m electrons is the sum of the ionization potentials Qm ¼ I 1 þ I 2 þ    þ I m . The m electrons removed leaves the atom in ionization level Z  . Continuing to treat both types of particles as perfect gases (with polytropic index γ ¼ 5=3), for N atoms in volume V, the energy is X 3 3 ℰ ¼ NkB T i þ Z  NkB T e þ N αm Qm , 2 2 m

(5.14)

where αm is the concentration of m-type ions. Assuming there is only one charge state for all N atoms, and using the simple Bohr ionization energies for I m , (5.14) becomes 3 Z  ð1 þ Z  Þð1 þ 2Z  Þ ℰ ¼ Nk B ðT i þ Z  T e Þ þ Nk B I H , 2 6

(5.15)

where I H is the ionization potential of the hydrogen atom. The pressure is simply P ¼ ni kB T i þ ne kB T e . So far, no allowance has been made for the effects of the Coulomb charge of the particles on one another. In high-density plasma, the atoms are in close proximity to one another, and their collisions restrict the size of the atom. This confinement of the bound orbitals manifests itself in several ways, such as continuum lowering or pressure ionization. In the high-temperature limit, where plasma is highly ionized, the electrons behave as a perfect gas within the ion sphere. The free energy of the electrons is 

Fe ¼ Z kBT e








Z  ni λ3de 9 ðZ  eÞ2 log , 1  10 R0 2

(5.16)

192

Entropy and the Equation of State

where the first term on the right-hand side comes from (5.10) and last term is the Madelung energy, ℰen þ ℰee , of Z  free electrons uniformly distributed over the ion-sphere volume, as discussed in Section 4.3.2. Note that this EOS is valid only for nondegenerate plasma, and as such the argument of the logarithm is less than one. The electron pressure is found from Pe ¼ ð∂F e =∂V ÞT "

1=3  2 # 3 4 Z e 1=3 3 Z  e2  , (5.17) Pe ¼ Z ni kB T e 1  n ¼ ne k B T e 1  π 10 3 10 kB T e R0 kBT e i and the electron-specific energy is "
1=3  2 #
 3 Z kBT e 3 4 Z e 1=3 3 ne kB T e 3 Z  e2 π 1 εe ¼ 1 n ¼ , 2 Am0 5 3 2 ρ 5 k B T e R0 kBT e i

(5.18)

where ρ is the mass density. We readily recognize the last expressions in (5.17) and (5.18) contain a correction term to the perfect gas values. The thermodynamic quantities for the ionic component are given by (5.11). The entropies for each, which are additive, are easily found from (5.10) and (5.16). 5.4 Debye-Hückel The Debye-Hückel model of plasma is founded on equilibrium statistical mechanics. At the heart of the theory is the configurational partition function. In the absence of a magnetic field, for a system of N particles of the same type in volume V, the thermal equilibrium configurational partition function (canonical ensemble) for the particles is " # ð ð X ðZ  eÞ2 1 1 d3 r1    d3 rN , (5.19) Z~ ¼ N    exp  2 i, j rij kB T V   where r ij ¼ ri  rj  is the separation of two particles labeled i and j, and each particle has charge Z  . (More complicated partition functions can be written, such as for three-particle clusters.) The probability of any specific configuration is " # 1 1 X ðZ  eÞ2 : (5.20) P ¼ exp  2 i, j r ij kB T Z~ Expression (5.19) can be written as " ð ð Z~ ¼

R3N 0



# ΓX 1   d3 x1    d3 xN : exp  2 i, j xi  xj 

(5.21)

5.4 Debye-Hückel

193

Here, R0 is the ion-sphere radius, and Γ is the ion–ion coupling parameter defined, in general, by qα qβ e2 Γαβ ¼ , (5.22) Rαβ kB T αβ   1= with a more general “ion-sphere radius” Rαβ ¼ 3=4π nα þ nβ 3 and T αβ ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   T α þ T β =2 (if one species is degenerate electrons T α ! T 2e þ T 2F ). The ion–ion coupling parameter describes the strength of ion correlations and is used to classify different theoretical approaches. Small Γ implies the kinetic energy is large compared to the electrostatic energy, while large Γ indicates the particles are closely packed, resulting in a large electrostatic energy. Alternately, Γ can be interpreted as the ratio of the distance of closest approach, ðZ  eÞ2 =kB T, to the average spacing between particles R0 . The classical ion-pair correlation function (radial distribution function) is given by " # ð ð N ð N  1Þ 1 X ðZ  eÞ2 3    exp  d r1    d3 rN : gðrÞ ¼ (5.23) 2 i, j rij kB T N 2 Z~ This function is defined such that gðr Þ is the ratio of the correlated ion density nðr Þ to the average density of ions. Alternately, ni gðr Þ is the average density of ions a distance r away from one given ion. With this definition, gð0Þ ¼ 0 and gðrÞ ! 1 as r ! ∞. This is a classic N-body problem. In general, averages over ion configurations can be evaluated by Monte Carlo or molecular dynamics methods. The simple Debye-Hückel shielding model, developed in Section 3.5, is a combination of Poisson’s equation r2 ΦðrÞ ¼ 4πZ  e½nðrÞ  ne ,

(5.24)

where ne is the mean electron density, the Maxwell-Boltzmann expression for the number density nðrÞ

Z  eΦðr Þ nðrÞ ¼ ne exp  : (5.25) kB T Using the boundary condition Φ ! 0 as r ! ∞, the solution to Poisson’s equation yields the radially symmetric Debye-Hückel potential Φð r Þ ¼

Z  e r=λD : e r

(5.26)

This derivation assumes the electrostatic energy Z  eΦðrÞ is small compared to the kinetic energy k B T. Inserting (5.26) into (5.25) gives

194

Entropy and the Equation of State

"

ðZ  eÞ2 1 r=λD gDH ðr Þ ¼ exp  e kB T r

#



 pffiffiffiffiffiffi 

Γ ! gDH ðxÞ ¼ exp  exp  3Γx , x

(5.27) pffiffiffiffiffiffi where x ¼ r=R0 . The Debye length is λD ¼ R0 = 3Γ. Expression (5.27) is nonlinear and therefore difficult to work with. For small Γ, (5.27) can be linearized to give gDH ðrÞ ¼ 1 

ðZ  eÞ2 1 r=λD : e kB T r

(5.28)

This has the undesirable property that gðr Þ, a probability, is negative for small r. We might suppose that it becomes negative when r ¼ R0 , the interparticle spacing. Then (5.28) becomes gDH ðxÞ  1 

ðZ  eÞ2 1 , R0 kB T x

(5.29)

and so gDH ðxÞ ¼ 0 for x ¼ ðZ  eÞ2 =R0 k B T ¼ Γ. Equation (5.27) displays the proper behavior as r ! 0. As demonstrated in Figure 3.7, the Debye-Hückel model is valid only for Γ  1. Figure 5.2 compares the distribution function for the linear and nonlinear models. The difference between the two is less for smaller values of the coupling parameter; the linear model is generally acceptable for Γ ≲ 1=3, where there is at least one particle in a Debye sphere.

5.4.1 Thermodynamic Properties For the N particles in volume V, the average value of the total potential energy is ð ne ΦC ðrÞ½gðrÞ  1d 3 r, ℰ¼ (5.30) 2 where ΦC ðrÞ is the Coulomb potential. The “minus one” provides the contribution to the thermodynamic properties due to the background. For small Γ, we use (5.28), and thus for N particles pffiffiffi 3 3=2 N ðZ  eÞ2 Γ Nk B T; ℰC ¼  ¼ (5.31) 2 2 λD the one-half factor prevents double counting. The total energy ℰDH is the kinetic energy ℰk ¼ 3NkB T=2 plus (5.31). Thus, correlations reduce the total energy. Physically, this originates from interactions with the opposite-signed background, which surrounds each particle as the mobile particles are locally repelled. The total energy has a zero value when Γ  1:44.

5.4 Debye-Hückel

195

Figure 5.2. Radial distribution functions for the linear (dashed-line) and nonlinear (solid-line) Debye-Hückel models for Γ ¼ 1.

In general, ℰ is a function of both temperature and density. However, for the Coulomb potential we see that a change in temperature at fixed density is equivalent to a certain change in density at fixed temperature. That is, the properties of the system do not depend on temperature and density separately, but only on a certain combination of them, which we have taken as the parameter Γ. In the same vein, the pressure is found from ð n2e dΦC ðr Þ ½gðr Þ  1d3 r: r P ¼ ne k B T  (5.32) dr 6π By the virial theorem,1 the pressure is
PDH ¼ ne k B T

1

pffiffiffi  3 3=2 Γ ; 1 6

The virial theorem of Clausius applied to a system of particles in a Coulomb field states: 2ℰk ¼ 

(5.33)

N P

r i  Fi

i¼1

where ℰk is the kinetic energy, ri is the radius vector of the ith particle, and Fi is the force acting on it. For the present case, it yields the relation between pressure and both kinetic and potential energy: PV ¼ ð2ℰk þ ℰC Þ=3.

196

Entropy and the Equation of State

the Coulomb interactions cause a decrease in the pressure from that of a perfect gas, also seen in (5.17). From (5.33), the pressure vanishes when Γ  2:29. For plasma to be stable, the total pressure must be positive. The standard formula for the heat capacity at constant volume is "
2
2 # 3 ℰ ℰ h i : (5.34) C V ¼ Nk B þ N 2 k B  2 NkB T NkB T The Helmholtz free energy is 1 3= F DH ¼ F 0  pffiffiffi Γ 2 NkB T, 3

(5.35)

where F 0 ¼ ℰ0  TS0 is the free energy of a perfect gas. The quantities ℰ0 and S0 are specified in (5.10). 5.4.2 Nonequilibrium Debye-Hückel In Section 5.1, we discussed some aspects of a two-temperature EOS. For the lowdensity regime, the Debye-Hückel model can be modified to include electron–ion pair correlations. Salpeter (1963) noted that the typical electron moves much faster than the ions and cannot polarize them, so there is an indirect screening effect because an electron is more likely to be found in the neighborhood of an ion than elsewhere. The probability of finding the ith electron in a volume element dV surrounding ðeÞ ri is simply dV=V. The joint probability of simultaneously finding the ith electron ðiÞ in dV and the jth ion in dV centered at rj depends upon the separation ðeÞ ðiÞ r ¼ ri  rj . This probability is   1

ðdV Þ2 ðeÞ ðiÞ hei ri  rj þ , Pij ¼ (5.36) V V where hei ðrÞ is the electron–ion pair correlation function. ðiÞ Consider first the case of one ion of charge Z  e situated at rj , then, on the average,   ðeÞ ðeÞ ðiÞ the particle density of electrons in the vicinity of ri is ne hei ri  rj þ ne . If the ion is now moved to the origin, the excess charge density at r is ne ehei ðrÞ ¼ ne e2 ΦðrÞ=k B T, where ΦðrÞ is given by (5.26), and so hei ðrÞ ¼

Z  e2 1 r=λD : e kB T e r ðeÞ

(5.37) ðiÞ

Conversely, located at ri , the particle density of ions near rj  for one electron  ðeÞ ðiÞ  is ðne =Z Þhei ri  rj þ ne =Z  . The equation corresponding to (5.37) would be

5.4 Debye-Hückel

hei ðrÞ ¼

197

Z  e2 1 r=λD , e kBT i r

(5.38)

ðZ  eÞ2 1 r=λD e : kB T i r

(5.39)

but this is incorrect. For a pair of ions the expression is hii ðrÞ ¼ 

Comparing (5.39) with (5.28), we see the pair distribution function gii ðrÞ ¼ hii ðrÞ þ 1. If the ions are uniformly distributed, then there are no correlations among them (hii ¼ 0), and the probability of finding an ion at some distance is 100 percent. For a pair of electrons hee ðrÞ ¼ 

e2 1 r=λD e , kB T e r

(5.40)

but this too is incorrect. The reason (5.40) is incorrect is that the electron correlation has two parts: (1) a part in which the ions play no role at all and thus the Coulomb forces cause no correlation between the ions and electrons, and (2) a part in which the correlation between a pair of electrons due to the correlation each of the two electrons has with the ion charge distribution (Salpeter, 1963). The ion-independent part yields hðeeeÞ ðrÞ ¼ 

e2 1 r=λD , e kB T e r

(5.41)

and the contribution due to the correlation of each of the two electrons with the ion is hðeeiÞ ðrÞ ¼

 T i e2 1  r=λDe  er=λD : e T e kB T e r

(5.42)

Adding (5.41) and (5.42) gives the total electron–electron correlation function
 e2 1 T i r=λD T e  T i r=λDe e þ e : (5.43) hee ðrÞ ¼  kBT e r T e Te For reference, the Debye lengths (from Section 3.5) are
 1 4πe2 ne Z  1 1 4πe2 ne ¼ þ ¼ : and kB Ti Te kBT e λ2D λ2De

(5.44)

The correlation functions can be interpreted as follows: a. If an ion is situated at the origin, then from (5.39) and (5.37), the nearby ion and electron densities are

198

Entropy and the Equation of State

ni ðrÞ ¼

ne ½1 þ hii ðr Þ and ne ðrÞ ¼ ne ½1 þ hei ðr Þ; Z

(5.45)

b. If an electron is situated at the origin, then (5.37) and (5.43) gives ni ðr Þ ¼

ne ½1 þ hei ðrÞ and ne ðrÞ ¼ ne ½1 þ hee ðrÞ: Z

(5.46)

In this two-temperature Debye-Hückel model, the thermodynamic quantities are found in a fashion similar to that in the previous section. The Coulomb energy is

where

ℰC ¼ 2πN e e4 ne ϕ,

(5.47)

h  i λ Ti Te  Ti D  2  þ λD þ 2 λDe : ϕ ¼ ðZ Þ þ 2Z k B T i kB T 2e kB T 2e

(5.48)

If the two species have equal temperatures, ϕ ¼ ðZ  þ 1Þ2 λD =kB T, and the Coulomb energy reduces to the usual form ℰC ¼ N e ðZ  þ 1Þe2 =2λD . The Coulomb free energy F C is found from ∂F C ∂F C  Ti , ∂T e ∂T i

(5.49)

2 1 F C ¼ ℰC ¼  N e 4πe4 ne ϕ: 3 3

(5.50)

ℰC ¼ F C  T e so that

The Coulomb contribution to the pressure is easily calculated because both λD and 1 λDe are proportional to V =2 . Hence,


∂F C PC ¼  ∂V

 ¼ Te, Ti

4πe4 n2e ϕ; 6

(5.51)

for equal temperatures, this reduces to PC ¼  16 ðZ  þ 1Þe2 ne =λD . The total pressure is the perfect gas pressure plus (5.51). The heat capacities can be calculated in a straightforward manner, but the results are neither simple nor beautiful. The off-diagonal coefficients C ei and Cie are negative and the corresponding inverse coefficients Aei and Aie in (5.7) are positive.

5.4 Debye-Hückel

199

5.4.3 Density Fluctuations The discussion about statistical mechanics in Section 3.2 considered the canonical ensemble in developing distribution functions. Statistical mechanics makes no attempt to describe a given macroscopic system precisely. Instead, a probability is assigned that it is in a given state. The uniformity of macroscopic phenomena is then attributed to the “sharpness” of this probability distribution. The average behavior of the members of the ensemble corresponds to the thermodynamic properties such as internal energy and pressure. For the canonical ensemble, mentioned at the end of Section 3.1, Gibbs’ H prescription gives the increase in entropy of an ensemble over time: the ensemble has a tendency to become distributed over many possible states in a definite manner, independent of the initial distribution. It is this definite, final distribution of the ensemble members that corresponds to the thermodynamic properties of physical systems in thermal equilibrium. In many systems, the equilibrium probability distribution is exceedingly sharp-peaked. Fluctuations from the mean behavior of the ensemble are often too small to be observed directly. However, in plasma, fluctuations can be significant and easily observed in scattering experiments. Other thermodynamic properties, an example being the heat capacity at constant volume, are related to the mean-square deviations of the distribution function. The full discussion of fluctuations may be found in many books on statistical mechanics. Considering the nonequilibrium electron–ion plasma, density fluctuations of the electron distribution are more readily seen than those of the ion distribution due to their considerable mass difference. Following Salpeter (1963), the development of electron and ion fluctuations begins with the spatial Fourier transform at time t of the electric charge densities. For the point electrons alone, it is ðeÞ

ρk ðtÞ ¼ e

Ne X

ðeÞ

eik  ri ,

(5.52)

i¼1

while that due to the ions is ðiÞ ρk ð t Þ



¼Z e

 NX e =Z

ðiÞ

eik  ri ,

(5.53)

i¼1 ðeÞ

ðiÞ

where the positions of the ith electron and ion are ri ðtÞ and ri ðt Þ, respectively. ðt Þ

The spatial transform of the total charge density ρk ðtÞ is the sum of (5.52) and (5.53). The time averages of the fluctuations are found by squaring the sums of the preceding

200

Entropy and the Equation of State

   h D  E i  ðiÞ 2 ðiÞ ðiÞ , ρk  ¼ N e e2 Z  þ N e cos k  ri  rj

(5.54)

   h D  E i  ðeÞ 2 ðeÞ ðeÞ , ρk  ¼ N e e2 1 þ N e cos k  ri  rj

(5.55)

ii

ee

and the total          D  E  ðtÞ 2  ðeÞ 2  ðiÞ 2 ðeÞ ðiÞ : ρk  ¼ ρk  þ ρk   2N 2e e2 cos k  ri  rj ei

(5.56)

If there is no secular change with time of any bulk properties, the time averages are equivalent to statistical ensemble averages, and the right-hand sides of (5.54) through (5.56) are related to the pair correlation functions. The cosine averages in these equations, for nonzero k, are the Fourier transforms of the correlation functions. For example, ð (5.57) h cos k  riii ¼ hii ðrÞ cos k  rd 3 r, with similar expressions for the other two averages. We shortly will need the fact that ð λ2 ΦC ðrÞeik  r d3 r ¼ 4πZ  e 2 2 D : (5.58) k λD þ 1 Evaluating the average of the cosine quantity in (5.54) Ne 4πne Z  e2 λ2D 1 cos k  r ¼  , ¼ α2i h i ii  2 2 Z kB T i k λD þ 1 1 þ α2e þ α2i

(5.59)

where (5.57) and (5.39) have been used, and αi ¼ ðkλDi Þ1 and αe ¼ ðkλDe Þ1 . In a similar fashion, using (5.40) N e h cos k  riee ¼ α2e

1 , 1 þ α2e þ α2i

(5.60)

and from (5.37) Ne 1 2 ,  h cos k  riei ¼ αe Z 1 þ α2e þ α2i

(5.61)

1 : 1 þ α2e þ α2i

(5.62)

and then from (5.38) N e h cos k  riei ¼ α2i

Using (5.59) through (5.61) in (5.54) through (5.56) yields

5.4 Debye-Hückel

201

      ðiÞ 2 ρk  ¼ N e Z  e2 1 þ α2e

(5.63)

      ðeÞ 2 ρk  ¼ N e e2 1 þ α2i

1 , 1 þ α2e þ α2i

1 , 1 þ α2e þ α2i

(5.64)

1 : 1 þ α2e þ α2i

(5.65)

and     ðtÞ 2 ρk  ¼ N e ðZ  þ 1Þe2

Since we have asserted that both (5.38) and (5.40) are incorrect, then so are (5.60) and (5.62), along with (5.64) and (5.65). The same arguments used with correcting (5.40) apply here. By putting  Z ¼ ðkλDi Þ1 ¼ 0 in (5.60), gives us N e h cos k  riðeeeÞ ¼ α2e

1 , 1 þ α2e

(5.66)

yielding   ðeÞ  ðeÞ 2 ¼ N e e2 ρk 

1 : 1 þ α2e

(5.67)

The second part, the correlation between a pair of electrons due to the correlation each of the two electrons has with the ion charge distribution, is D  EðiÞ ðeÞ ðeÞ cos k  ri  rj ðð
ee 2 2 h  i 1 ne e ðeÞ ðeÞ ¼ Φðr ÞΦðr Þexp ik  r  r d3 ri d3 rj i j i j kB T e ðN e eÞ2 1 1 ðiÞ ¼ α4 jρ j2 : (5.68) 2 e 2 k 2 ðN e eÞ ð1 þ αe Þ Taking the thermal average of the ion distribution, using (5.63), the second part becomes   ðiÞ 1 1  ðeÞ 2  : ¼ N e Z  e2 α4e  (5.69) ρk  1 þ α2e 1 þ α2e þ α2i Summing (5.67) and (5.69) gives the correct expression for N e h cos k  riee . The revised (5.64) is now

202

Entropy and the Equation of State

    ðeÞ 2 ρk  ¼ N e e2

1 þ α2i Ti  Te α2e α2i    , þ Te 1 þ α2e þ α2i 1 þ α2e 1 þ α2e þ α2i

where T i =T e ¼ Z  α2e =α2i . The replacement for (5.65) is    1 þ Z Te  Ti α2i  ðtÞ 2    : þ ρk  ¼ N e e2 Te 1 þ α2e þ α2i 1 þ α2e 1 þ α2e þ α2i

(5.70)

(5.71)

For the case of no electron screening, (5.63) becomes    k 2 λ2Di  ðiÞ 2 , (5.72) ρk  ¼ N e Z  e2 1 þ k2 λ2Di     ðiÞ 2 and for a perfect gas without screening, ρk  ¼ N e Z  e2 . Coulomb interactions tend to inhibit the long-wavelength density fluctuations relative to those that would exist in an uncharged perfect gas. This suppression of fluctuations is the wave number-space image of the Debye screening itself.

5.5 Strongly Coupled Plasma So far, in this chapter, we have considered “dilute” plasma, where the coupling parameter Γ is small compared to one. Γ  1 is a statement that the long-range Coulomb forces tend to arrange the particles in a specific configuration that minimizes the potential energy, but the kinetic energy is much larger than the Coulomb energy. This results in material behavior similar to that of a perfect gas. When the density is high and the temperature “low,” plasma has a high degree of ion correlation, signified by Γ  1. As the temperature is reduced, the electrostatic forces become stronger compared to the thermal forces and start competing with the random motion of the gas. The system of particles no longer resembles a classical gas. Correlations associated with the Coulomb force begin to dominate, and the ions, seeking to minimize the potential energy, start to form a regular structure. At this point, the plasma is in a liquid state, exhibiting both gas and solid properties. For even lower temperatures, the material transitions to the solid state. Thus, we draw upon liquid theory (Hansen & McDonald, 1976) and solid-state theory (Kittel, 2005). The standard model for extreme density and temperature is the OCP (Brush, Sahlin, & Teller, 1966; Hansen, 1973). This environment is typical of “hot” white-dwarf stars where the temperatures are in the tens of keV, the densities are ∽106 g-cm‒3, and ionization is complete. At these conditions, the ratio of the thermal de Broglie wavelength to the ion-sphere radius is λdi =R0  0:07, and the ratio of the temperature to the Fermi temperature is 0:05. The first number ensures that quantum effects are

5.5 Strongly Coupled Plasma

203

small, so that classical statistical mechanics can be used, while the second ratio ensures that the electrons are highly degenerate and can be considered as forming an inert background in which the ions move. In this model, the ions are treated as classical point charges without ion core potentials. The ionization level is fixed, and the free electrons are assumed to be uniformly distributed, thus polarization effects are neglected. The picture is further complicated for high density and low temperatures as the electrons become partially degenerate; the Thomas-Fermi model will be discussed later in this chapter. The OCP model is usually applied to plasma with Γ > 100, but our interests are more in the range 1 < Γ < 100. The previous sections have provided some detail for Γ < 1, in particular the radial density distribution function gðrÞ. Figure 5.3 shows the ion pair distribution function for several values of Γ. We see that gðr Þ is small out to r ¼ 1:68R0 , then has a peak corresponding to a ring of neighboring atoms, and then oscillates with atoms farther away toward an asymptotic limit of unity for large radii. The pair distribution function agrees with the Debye-Hückel value up to Γ  0:1. For Γ 2, gðr Þ is no longer monotonic and begins to show oscillations characteristic of lattice-like structures. The solid begins a transition to the fluid phase at about Γ  125. In the high-density limit, where Γ  1, the OCP model approximates the ion-sphere model, and we find

Figure 5.3. Density distribution function for various Γ. The dashed line shows the ion-sphere approximation.

204

Entropy and the Equation of State

 gðrÞ ¼

0 r < R0 for : r R0 1

(5.73)

That is, no ion penetrates closer than a distance R0 ; this is shown in Figure 5.3 by the dashed line. As discussed previously, gðrÞ is a spherically symmetric average and does not retain any information about fluctuating anisotropies available in detailed numerical calculations.2

5.5.1 Static Properties Some aspects of a two-component plasma were discussed in the previous section, where the electrons are treated at the same level as the ions. To keep the discussion simple, we now consider a one-component system of ions, where the electrons are fixed and form a neutralizing background. This model is accurate for conditions where the electrons are highly degenerate as their wave functions are smeared out over several lattice sites. The model also is appropriate for the high-temperature case where the electrons are quite mobile and effectively act as a spatially uniform background. Considering the heavy particles to be fixed, then the many-body system is defined by correlation functions that relate the probability of finding groups of particles at some defined location in space. The particle correlation function is a probability density that a pair of particles has a separation r and is defined by * + N X   V hðrÞ ¼ δ r  ri  rj , (5.74) N ðN  1Þ i6¼j¼1 where N is the number of particles in volume V; the normalization is ð 1 hðrÞdr ¼ 1: V

(5.75)

The microscopic density distribution is defined by ρðrÞ ¼

N X

δðr  ri Þ:

(5.76)

i¼1

2

The radial distribution functions are interesting for their own sake, because their variation as a function of interionic spacing gives an indication of the short-range order of the liquid. The small-r behavior of gðr Þ results in the enhancement of nuclear reaction rates in stellar cores, due to the reduction of the Coulomb barrier relative to the two-body case by many-body conditions (Salpeter & Van Horn, 1969).

5.5 Strongly Coupled Plasma

205

Note that a slightly different definition for density is being used now. The quantity h  i in (5.74) is an average over a number of different but statistically equivalent configurations of ion pairs with positions ri and rj . The pair correlation function is related to the pair distribution function by gðrÞ ¼ hðrÞ þ 1. The density-density correlation function for translational invariant systems is given by pðrÞ ¼ hρðrÞρð0Þi ¼ hρðr þ r0 Þρðr0 Þi:

(5.77)

Since hðrÞ is independent of r0 , its form can be simplified by integration over r0 * + ð  N   1 1 X 0 0 0 : (5.78) pðrÞ ¼ δ r  ri  rj ρðr þ r Þρðr Þdr ¼ V V i, j¼1 The relation between pðrÞ and hðrÞ is pðrÞ ¼

N N ð N  1Þ hðrÞ: δðrÞ þ V V2

(5.79)

If the force between particles has a finite range, then at some large separation the densities become uncorrelated; this means limr!∞ pðrÞ ¼ hρðrÞρð0Þi ¼

N2 limr!∞ δðrÞ; V

(5.80)

also, from (5.79), limr!∞ hðrÞ ¼ 0. It is more useful to adopt the solid-state theory approach of reciprocal space to the structure of strongly coupled plasma. Expressing the quantity of interest (5.76) in terms of its wave vector k instead of its point position r is very useful, for instance, when interpreting scattering experiments. The Fourier transform of (5.76) is ρk ðt Þ ¼

N X

eik  ri ðtÞ :

(5.81)

i¼1

Clearly, the density expressed by (5.76) commutes with (5.81), since they are defined at the same time. In the Born approximation, the amplitude of the signal corresponding to the scattered vector k is proportional to ρk and only the intensity is detectable, that is, 2

I ðkÞ∽jρðkÞj ¼

N X i¼1

e

ik  ri

N X j¼1

ik  rj

e

¼

N X

eik  ðri rj Þ :

(5.82)

i, j¼1

Taking the time average (or ensemble average, which is equivalent) of (5.82) gives

206

Entropy and the Equation of State

D E I ðkÞ∽ jρðkÞj2 ¼ NSðkÞ,

(5.83)

thus defining the static structure factor (also known as the static form factor) * + N 1 X (5.84) eik  ðri rj Þ : SðkÞ ¼ N i, j¼1 This is equivalent to SðkÞ ¼

1 1 hρk ðt Þρk ðt Þi ¼ hρk ð0Þρk ð0Þi: N N

(5.85)

In contrast with crystalline solids, liquids have no long-range order, so the structure factor does not exhibit sharp peaks. They do, however, show a certain degree of short-range order, depending on the density and on the strength of the interaction between particles. Liquids are isotropic, so that, after averaging, the structure factor depends only on the absolute magnitude of the scattering vector k. Separating out the diagonal terms i ¼ j in the double sum of (5.84), whose phase is identically zero, we find that each contributes a unit constant. Then * + N 1 X (5.86) eik  ðri rj Þ : SðkÞ ¼ 1 þ N i6¼j¼1 In the limiting case of no interaction, that of an ideal gas, the sum is zero and Sðk Þ ¼ 1 because there is no correlation between ri and rj . Now taking the Fourier transform of (5.74) V NðN  1Þ

+ N ð *X

eik  r δ r  ðri  rj Þ d3 r i6¼j¼1

V ¼ NðN  1Þ

*X N

+ ik  ðri rj Þ

e

:

(5.87)

i6¼j¼1

Comparing (5.86) with (5.87) gives ð ð N  1 ik  r N  1 ik  r 3 e e SðkÞ ¼ 1 þ hðrÞd r ¼ 1 þ ½gðrÞ  1d3 r, V V

(5.88)

since N is large. Then taking the inverse Fourier transform of (5.88) gives ð V 1 1 X ik  r gðrÞ  1 ¼ e ½SðkÞ  1 eik  r ½SðkÞ  1d 3 k ¼ 3 N  1 ð2π Þ N1 k (5.89)

5.5 Strongly Coupled Plasma

207

The static structure factor SðkÞ can be determined by scattering experiments (in principle) and by molecular dynamics or Monte Carlo simulations.

5.5.2 Scattering Experiments The structure factor can be measured directly through elastic scattering experiments, in particular by the scattering of X-rays. Typical X-rays have wavelengths of order 1 Å, corresponding to a photon energy of about 10 keV. This energy is large compared to the particle energies, which are typically of order 0.01–0.1 keV. Thus, collisions of the photons with the particles will leave the photon energies almost unaltered and the scattering can be treated, to a good approximation, as being elastic. Consider a plane wave of the form a0 eik0  r , incident on a material containing N heavy particles with charge Z  . The incident intensity is I 0 ¼ ja0 j2 . We are interested in the radiation scattered in a direction θ relative to the incident beam, ðnÞ as illustrated in Figure 5.4. Let ri be the position of the ith particle and ri be the position of the nth electron associated with the particle. The amplitude at a distance r of the wave scattered by an electron at the origin is A0 ðθÞ ¼ aðθÞ

eijk1 jr : r

(5.90)

The factor aðθÞ depends upon the polarization of the incident beam through Thomson’s formula: aðθÞ ¼ a0 ðe2 =me c2 Þ cos θ if the polarization vector is in the scattering plane, or set cos θ ¼ 1 if it is orthogonal. ðnÞ The electron located at ri scatters a photon in the direction θ (wave vector k1 ) with a phase difference ðnÞ

ðnÞ

k0  ri  k1  ri

ðnÞ

¼ k  ri ,

(5.91)

where k ¼ k0  k1 is the scattering vector. The amplitude of the scattered wave from the single electron is therefore ðnÞ

Ai ðθÞ ¼ aðθÞ

eijk1 jr ik  rðnÞ e i : r

(5.92)

As the scattering is elastic, jk0 j ¼ jk1 j and the modulus of the scattering vector is k ¼ jkj ¼ 2jk0 j sin

θ : 2

(5.93)

Then, for a given wavelength of the incident X-rays, the modulus of the scattering vector determines the scattering angle and vice versa. The amplitude of the wave

208

Entropy and the Equation of State Observer

k1

k

X-ray

k0 Electron

Figure 5.4. Wave vector geometry of a scattering event.

scattered by the ith particle is just the sum of the amplitudes scattered by each of the Z  electrons Z Z ðnÞ ðnÞ eijk1 jr X eijk1 jr ik  ri X Ai ðθÞ ¼ aðθÞ eik  ri ¼ aðθÞ eik  ðri ri Þ : e r n¼1 r n¼1 



(5.94)

ðnÞ

The quantity ri  ri is the position of the nth electron of the ith heavy particle relative to its own particle. Summing these amplitudes from each of the N particles gives the amplitude of the scattered wave. The intensity of the scattered X-rays in the direction θ is jAðθÞj2 after taking the statistical average over the positions of the particles in the material D E a2 ðθÞ 2 I ðθÞ ¼ jAðθÞj ¼ NSðkÞF ðkÞ, (5.95) r2 where  2  a ðθÞ ¼ a20




e2 me c2

2

1 þ cos 2 θ 2

(5.96)

is an average over the two polarization states. F ðkÞ is the so-called form factor, and is the same for each particle 2 + *   X Z ðnÞ   (5.97) eik  ðri ri Þ  , F ðk Þ ¼    i¼1 and SðkÞ is the structural factor from (5.85)

5.5 Strongly Coupled Plasma

1 SðkÞ ¼ N

2 + *  X N   eik  ri  :    i¼1

The total intensity can be written as I ðθÞ ¼ I ðkÞ ¼ NI 1 ðθÞSðk Þ, where
2 2 e 1 þ cos 2 θ I 1 ðθ Þ ¼ I 0 F ðk Þ 2r2 me c2

209

(5.98)

(5.99)

is the intensity scattered by one heavy particle; NI 1 ðθÞ is the intensity scattered by N independent particles. The structure factor, therefore, is a measure of the correlation between the positions of the atoms, which is consistent with its definition as the Fourier transform of the pair correlation function. In practice, difficulties arise in the calculation of gðr Þ from inversion of data on SðkÞ because measurements of SðkÞ necessarily introduce a cutoff at large values of k. 5.5.3 Dynamic Properties The system of particles is not static, but undergoes vibrations, so the time dependence of the static structure factor needs to be included. The dynamic structure factor is defined by þ∞ ð 1 Sðk; ωÞ ¼ (5.100) hρk ðt þ t 0 Þρk ðt 0 Þieiωt dt: 2πN ∞

The h  i quantity is the density correlation between two different times. Since the system is stable, time invariance is specified by t ! t þ t 0 . The spectral-density function is defined in terms of the space-time Fourier transform of the microscopic density distribution as expressed by (5.81), which is þ∞ ð 1 ρðk; ωÞ ¼ ρk ðtÞeiωt dt: (5.101) 2π ∞

Using this in the correlation component of (5.100) yields the result hρðk, ωÞρðk,  ω0 Þi ð þ∞ ð 1 0 0 hρk ðt þ t 0 Þρk ðt 0 Þieiωt eiωðtþt Þ dt 0 dðt þ t 0 Þ ¼ 2 ð2πÞ ∞ þ∞

ð

¼

1 ð2πÞ

2

ð

þ∞

eiðωω0Þt0 dt 0

hρk ðtÞρk ð0Þieiωt dt ∞

¼ NSðk, ωÞδðω  ω0 Þ :

∞

(5.102)

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Entropy and the Equation of State

The dynamic structure factor is closely related to the screening of the electrical charges in plasma, as we shall see momentarily. This makes explicit calculation of Sðk; ωÞ a significant challenge. The analysis can be simplified by considering a system of noninteracting particles whose density has Fourier components, from (5.81), which is ð0Þ

ρk ðt Þ ¼

N X

ð0Þ

eik  ri

ðtÞ

,

(5.103)

i¼1

and the average over volume yields D

E

ð0Þ ρk ðt Þ

N ð ð0Þ 1X ¼ eik  ri ðtÞ d 3 r ¼ Nδ0k : V i¼1 ð0Þ

ð0Þ

(5.104)

ð0Þ

In the absence of any perturbations, ri ðt Þ ¼ ri ð0Þ þ vi t, assuming the positions and velocities are statistically independent. The time-Fourier transform of the density (5.103) becomes ρð0Þ ðk; ωÞ ¼

N X

  ð0Þ ð0Þ eik  ri δ ω  k  vi :

(5.105)

i¼1

Thus, the correlation is hρð0Þ ðk, ωÞρð0Þ ðk,  ω0 Þi   N X ð0Þ ð0Þ     ik  ri rj ð0Þ ð0Þ 0 ¼ e δ ω  k  vi δ ω  k  vj , i, j¼1

(5.106)

referenced to time zero. If k ¼ 0, (5.106) is easily evaluated since δðωÞδðω0 Þ ¼ δðωÞδðω  ω0 Þ; if k 6¼ 0. Then using (5.104), only those terms with i ¼ j are nonzero and so ( 2 0 D E k¼0 D  N δðωÞδðω EωÞ ð0Þ ð0Þ 0 for : ρ ðk; ωÞρ ðk; ω Þ ¼ ð0Þ N δ ω  k  vi δðω  ω0 Þ k 6¼ 0 (5.107) Now using (5.102), the dynamic structure factor for noninterating particles is    ð0Þ ð0Þ S ðk, ωÞ ¼ NδðωÞδ0k þ δ ω  k  vi ð1  δ0k Þ Ð ¼ NδðωÞδ0k þ ð1  δ0k Þ f MB ðvÞδðω  k  vÞd 3 v , (5.108) where f MB ðvÞ is the Maxwell-Boltzmann distribution function. The average behavior of the system occurs for k ¼ 0, and is not interesting.

5.5 Strongly Coupled Plasma

211

In reality, the effective charge associated with each particle is reduced by polarization screening of the surrounding electrons. This polarization is represented by the dielectric response function (permittivity) ϵ ðk; ωÞ ¼

ρð0Þ ðk; ωÞ : ρðk; wÞ

(5.109)

Since ρðr; t Þ and ρð0Þ ðr; t Þ are real, and ϵ is in general complex, it follows that ϵ ðk; ωÞ ¼ ϵ  ðk; ωÞ. Then for k 6¼ 0 ð Sð0Þ ðk; ωÞ 1 f ðvÞδðω  k  vÞd3 v : Sðk; ωÞ ¼ ¼ (5.110) 2 2 jϵ ðk; ωÞj jϵ ðk; ωÞj The function f ðvÞ is usually taken to be the Maxwell-Boltzmann distribution. Expression (5.110) is known as the fluctuation-dissipation theorem. It relates the spectrum of the density fluctuations with the dielectric function. For collisionless plasma, the dielectric function can be found from the Vlasov equation.3 Then 1 ðkλD Þ2 ℑm½ϵ ðk; ωÞ: πω

(5.111)



1 1 2 Sðk; ωÞ ¼  ðkλD Þ ℑm : πω ϵ ðk; ωÞ

(5.112)

Sð0Þ ðk; ωÞ ¼ and finally

This is the generalized form of the Nyquist fluctuation-dissipation theorem. 3

The development of the permittivity (including collisions), from Section 3.13, can be reworked using an oscillatory field E ¼ E0 eiωt exp ½ik  ðr0 þ δrÞ  E0 eiωt exp ½iðk  r0  ω0 t Þ, where r0 is in the frame moving with the unperturbed electron, and ω0 ¼ ω  k  v is the Doppler-shifted frequency felt by the particle moving with velocity v in the absence of perturbations. Solving the equation of motion gives δr. This displacement for the electron leaves an unneutralized ion behind, thus producing a dipole moment. Then summing these moments yields the dielectric polarization vector P. The dielectric function is customarily defined by ð 4πe2 ∂f MB =∂v d 3 v: k  ϵ ðk; ωÞ ¼ 1  k  v  ω  iν me k 2 The integral is evaluated by contour methods, yielding a principal integral, and the pole integral is the usual semicircle integral ð ð f ðx0 Þ f ðx0 Þ 0 dx0 ¼ P 0 dx þ iπf ðxÞ: limγ!0 0 x  ðx þ iγÞ x x pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi The velocity v is in the direction of the wave vector; then, if ω  2kB T=me , a small ω=k expansion in the principal value integral is permitted. Ignoring the collision frequency, we find
3=2 ð

3=2
rffiffiffiffiffiffiffiffiffiffiffi

pffiffiffi ω2pe 2 ω2pe me me v2 ω0 me me ω0 dv þ O : exp  þ i2 π 2 ϵðk, ω Þ ¼ 1  pffiffiffi 2 π k 2k B T 2kB T k 2k B T 2ðkTÞ k k The imaginary term is also O


qffiffiffiffiffiffiffiffi ω0 me k 2ðkT Þ , then to lowest order ϵ ðk; ω Þ agrees with that of Section 3.13.

212

Entropy and the Equation of State

The derivation of (5.112) assumed collisionless plasma, but it does have utility into the weakly collisional regime. Expression (5.110) required that the polarization cloud surrounding each particle be well defined at all times. Typically, the time required for screening to establish this is τ sc ∽1=ωpe , where ωpe is the electron plasma frequency. Further, a measure of collisionality is the average time between electron–ion collisions τ ei . The range of validity for (5.110) is then τ sc  τ ei , or for weak collisionality, νei  ωpe . Inserting the dielectric function in (5.112) gives Sðk; ωÞ, and then the static structure factor ð SðkÞ ¼ Sðk; ωÞdω ¼

k2 λ2Di ¼ SDH ðkÞ: 1 þ k 2 λ2Di

(5.113)

We then find the Debye-Hückel radial distribution function from (5.89) gDH ðrÞ

¼ 1þ ¼ 1

ð∞ðπ

1 3

ð2πÞ ni

2π sin θk 2 eikr cos θ ½SDH ðkÞ  1dθdk


 pffiffiffiffiffiffi r ðZ eÞ 1 r=λD R0 e ¼ 1  Γ exp  3Γ , R0 kBT r r 

00

2

(5.114)

in agreement with (5.28). Recall from Section 5.4 that for sufficiently large values of Γ, the pair correlation function becomes negative, which is clearly unphysical; the correct expression is the nonlinear form (5.27).

5.5.4 Thermodynamic Properties Determination of the thermodynamic quantities of strongly coupled plasma begins with the Hamiltonian of the system of N particles H¼

N N X p2i 1X ðZ  eÞ2  , þ 2mi 2 i6¼j¼1 ri  rj  i¼1

(5.115)

where pi ¼ mi vi is the momentum of the ith particle. Rewriting this in terms of the density yields H¼

N X p2i 1 X þ Φk ðρk ρk  N Þ, 2mi 2V k6¼0 i¼1

(5.116)

where Φk ¼ 4π ðZ  eÞ2 =k2 is the Fourier transform of the Coulomb potential ΦC ; the term with k ¼ 0 has to be omitted due to the neutralizing background of

5.5 Strongly Coupled Plasma

213

positive charges. The second term is the interaction among the electrons. Thus, the Coulomb energy of the N particles is ℰC ¼

N X Φk jSðkÞ  1j: 2V k6¼0

(5.117)

Once the static structure factor is known, the total internal energy is known, and by the virial theorem, the pressure is known. Using Φk with (5.113) in (5.117) gives pffiffiffi ð∞ 3 3=2 1 2 ℰC ¼ 2 k Φk jSðk Þ  1jdk ¼  Γ Nk B T, 2 4π

(5.118)

0

which is identical to (5.31). For Γ ¼ 0:1, this reduction amounts to about 3 percent of the perfect gas value 3NkB T=2. In real plasma, we also need to account for the electron contribution to the energy and pressure. The electronic terms are, in general, not simple, as they must reflect partial degeneracy and quantum mechanical corrections. 5.5.5 Ornstein-Zernike The properties of a strongly coupled fluid are more conveniently described using two alternate functions: the direct correlation function cðrÞ and the bridge function bðrÞ; the bridge function accounts for higher-order terms, and will not be considered here. The purpose of the correlation function is to provide a simpler interpretation of the pair correlation functions gðrÞ or hðr Þ. We begin by assuming cðrÞ represents the direct correlation between any two particles located at r1 and r2 . Then to first order, hðr1  r2 Þ  cðr1  r2 Þ: (5.119) If now a third particle is involved, second-order indirect interactions must be considered, whereby particle one interacts with particle three, which simultaneously interacts with particle two. Then ð (5.120) hðr1  r2 Þ  cðr1  r2 Þ þ ρðr3 Þcðr1  r3 Þcðr3  r2 Þdr3 : This can be extended to even higher orders of indirect interactions ð hðr1  r2 Þ  cðr1  r2 Þ þ ρðr3 Þcðr1  r3 Þcðr3  r2 Þdr3 ðð þ ρðr3 Þρðr4 Þcðr1  r3 Þcðr3  r4 Þcðr4  r2 Þdr3 dr4 þ    : In more compact form, (5.121) is written as

(5.121)

214

Entropy and the Equation of State

ð

hðrÞ ¼ cðrÞ þ ρ cðjr  r0 jÞhðr0 Þdr 0 ¼ gðrÞ  1,

(5.122)

which goes by the name Ornstein-Zernike relation (Hansen & McDonald, 1976). Since cðrÞ describes only the direct interaction between a pair of particles, it should have a much simpler form than hðrÞ and gðr Þ, which both account for all of the interactions. This implies that cðrÞ should scale with the pair potential. Clearly, cðr Þ is shorter-ranged than hðrÞ. Taking the Fourier transform of both hðrÞ and cðrÞ, denoted by ^ h ðkÞ and ^c ðkÞ, and using the convolution theorem, we have ^ h ðkÞ ¼ ^c ðkÞ þ ρ^h ðkÞ^c ðkÞ, which yields ^ h ðkÞ ¼ Then by (5.88)

^c ðkÞ : 1  ρ^c ðkÞ

SðkÞ ¼ 1 þ ρ^ h ðk Þ ¼

1 : 1  ρ^c ðkÞ

(5.123)

(5.124)

(5.125)

This is an exact result. Expression (5.125) contains one unknown, and thus some sort of closure relation is needed. The simplest form of the direct correlation function is just the Coulomb potential divided by the kinetic energy cðrÞ ¼ 

ðZ  eÞ2 1 Φk and cðkÞ ¼  , kB T r kB T

(5.126)

and thus (5.125) gives the Debye-Hückel structure factor (5.113). Another commonly used closure relation is the Percus-Yevick approximation; it is used for hard-core potentials but is not useful for Coulombic liquids where longer-range potentials occur. 5.5.6 Hypernetted-Chain The closure relation of importance to high-energy-density plasma is the hypernetted-chain (HNC). In order to obtain a short-ranged function to approximate cðr Þ, we try cðrÞ ¼ gðrÞ  d ðrÞ. Taking the total radial distribution to be

ΦðrÞ gðr Þ ¼ exp  , kBT

(5.127)

where Φðr Þ is the mean force potential, and for the indirect distribution part

5.5 Strongly Coupled Plasma

 1 d ðrÞ ¼ exp  ½ΦðrÞ  uðr Þ , kB T

215



where uðr Þ is the direct interaction between pairs. Then



ΦðrÞ ΦðrÞ  uðrÞ cðr Þ ¼ exp   exp  : kB T kB T

(5.128)

(5.129)

Expanding the second term and introducing yðrÞ ¼ euðrÞ=kB T gðrÞ, (5.129) is approximately

ΦðrÞ ΦðrÞ  uðrÞ cðr Þ ¼ exp  1þ ¼ gðrÞ  1  log yðr Þ: (5.130) kBT kB T This is the essence of the HNC closure equation. Equivalently, hðrÞ  cðr Þ ¼ gðr Þ  1  cðrÞ ¼ log yðrÞ: Using this in the Ornstein-Zernike equation (5.122) ð hðr1  r2 Þ  cðr1  r2 Þ ¼ ρ cðr1  r3 Þhðr2  r3 Þdr3 ,

(5.131)

(5.132)

gives the HNC equation uðr1  r2 Þ log½yðr1  r2 Þ ¼ log½gðr1  r2 Þ þ kB T

ð uðr1  r3 Þ ¼ ρ hðr1  r3 Þ  loggðr1  r3 Þ  hðr2  r3 Þdr3 : kB T (5.133) The preceding few lines are not very transparent as to what the HNC equation is all about. A more easily visualized approach is to write the pair distribution function in terms of an effective potential

Φeff ðrÞ , (5.134) ghnc ðr Þ ¼ exp  kB T in which each ion interacts with another ion with a pair potential (the direct correlation function) plus the indirect interactions. We can then write 

Φeff ðrÞ ðZ  eÞ2 1 þ hðrÞ  cðrÞ: ¼ kB T kBT r

(5.135)

Expressions (5.134) and (5.135) are solved together with the Ornstein-Zernike relation. The HNC equation is known to be quite accurate for Coulomb systems.

216

Entropy and the Equation of State

Figure 5.5. Radial distribution function gðxÞ and direct correlation function cðxÞ for Γ ¼ 4. The dashed line shows the Debye-Hückel values cDH ðxÞ.

Figure 5.5 plots the radial distribution function and the direct correlation function. The Fourier transform of the Ornstein-Zernike direct correlation function is obtained from (5.125), and since SðkÞ  k2 R20 =3Γ for k ! 0 (see (5.113)), ^c ðk Þ  4πR0 Γ=k 2 for small k. The direct correlation function cðr Þ is obtained by Fourier inversion, and is given by (5.126) in the Debye-Hückel limit. Only the short-range behavior of cðr Þ differs considerably from cDH ðr Þ ¼ ΓR0 =r. This illustrates the Ornstein-Zernike theory in which much of the important information on pair correlations is concentrated in the short-range behavior of cðrÞ, which approaches the Debye-Hückel limit much faster than gðrÞ. As an aside, if we assume that the potential correction term hðr Þ  cðrÞ is small, then we have the Percus-Yevick approximation " # ðZ  eÞ2 1 ½1 þ hðrÞ  cðr Þ: (5.136) gPY ðrÞ ¼ exp  kBT r 5.5.7 One-Component Plasma (OCP) The complexity of the radial distribution function and the structure factor with increasing density forces us to calculate these quantities numerically. For dilute plasma, the ions are sufficiently far apart that they may be considered isolated, and only the nearest two ions need be considered. However, as the volume is compressed, additional ions begin to contribute to gðrÞ, with their influence becoming dominant as the liquid undergoes a phase change to a bcc solid. Numerical

5.5 Strongly Coupled Plasma

217

calculations suggest this occurs for Γ  172; we do not consider the phase change nor the solid-state-like structure here. The region of very dense plasma (Γ ≳ 100) is usually characterized as a onecomponent plasma (OCP), though this applies to smaller Γ also; an OCP is defined as a dense system of charged particles of one species imbedded in a uniform background of opposite charge, which ensures overall electrical neutrality. These high-Γ systems are studied in detail by molecular dynamics methods and Monte Carlo techniques. Molecular dynamic simulations for hard-sphere systems were pioneered by Alder and Wainwright (1959) and since extended by many others. The Monte Carlo method, introduced by Metropolis et al. (1953), was applied to dense plasma by Brush et al. (1966). Their original simulations are crude by modern standards, but they did identify many important aspects of classical liquids. Perhaps the most important issue they addressed is how to account for the long-range tail of the Coulomb force in a small computational volume; this required taking into account all of the periodic images of the particles contained in the initial cell. With the increase in computational power, their work has been extended by others; we refer to the early enhancements as reported by Hansen (1973). Brush et al. (1966) derive the “Ewald potential,” which is embodied in the total potential energy (per kB T) of the system for a given configuration X   V0 V Φ ri  rj þ ¼Γ , kBT kBT i t 1 .

or since cs ¼ cs0 in the undisturbed gas, we have for an isothermal rarefaction   ρ x þ cs0 t u x þ cs0 t : (6.54) ¼ exp  ¼ and ρ0 cs0 t cs0 cs0 t The properties of this rarefaction, described by (6.54), are shown in Figure 6.3 for two separate times. We see in Figure 6.3b there is a constant flux of matter across the interface at x ¼ 0 equal to ρ0 cs0 =e, where e = 2.178. These results contain an apparent paradox in that the density curves for t > 0 extend out to positive infinity, so that matter at large distances is moving exceedingly fast. However, the density at these distances is very small. In fact, the ratio of matter included in the expansion 2cÐs0 t from cs0 t to þ2cs0 t to that in the original matter is ð1=ρ0 cs0 tÞ ρdx  0:95. In cs0 t

reality, the leading edge does not move with infinite speed. By conservation of energy, all of the initial internal energy is converted to kinetic energy, and this sets the maximum speed. 6.3.2 Adiabatic Expansion The second case considers an adiabatic expansion in the same geometry. By the same reasoning given in the previous section, the cross-characteristic lines are straight lines through the origin to the right of the limiting line, denoted by “B” in Figure 6.2; there is no material to the right of “C.”

6.3 Method of Characteristics

265

For a polytropic EOS, P ¼ P0 ðρ=ρ0 Þγ , then the sound speed is  cs ¼

∂P ∂ρ

1=2 s

 ðγ1Þ =2 ρ ¼ cs0 : ρ0

(6.55)

Let y ¼ ρ=ρ0 , and using (6.55) in (6.45) gives ρ=ρ0  2 ðγ1Þ = 2 y ω ¼ cs0 y =2 dy ¼ cs0 γ1 1 1 " ðγ1Þ #   =2 2 ρ 2 cs ¼ 1 ¼ 1 : cs0 cs0 γ1 ρ0 γ1 cs0 ð ρ=ρ0





ðγ3Þ

(6.56)

Rearranging this equation to find cs =cs0 and using dx=dt ¼ u  cs ðωÞ results in   γ1 ω x ¼ ut  cs0 þ 1 t: (6.57) 2 cs0 Along a characteristic C þ , ω ¼ u, therefore   γþ1 ωt  cs0 t, x¼ 2

(6.58)

and then from (6.58) and (6.56) ðx þ cs0 t Þ 2 ¼ ω ¼  γþ1 cs0 γ  1 t 2

" ðγ1Þ # =2 ρ 1 : ρ0

(6.59)

Thus, we arrive at an expression for the density    2=ðγ1Þ ρ γ  1 x þ cs0 t : ¼ 1 ρ0 γþ1 cs0 t

(6.60)

Note that ρ ¼ 0 when x þ cs0 t ¼

γþ1 2 cs0 t or x ¼ cs0 t : γ1 γ1

(6.61)

Therefore, the front of the wave has a finite velocity equal to 2cs0 =ðγ  1Þ, whereas in the isothermal case, the velocity approached infinity at large x. The curves of density and velocity are given in Figure 6.4. We note that for γ ¼ 2 the matter front moves twice as fast forward as the rarefaction backward.  proceeds A N A As γ ! 1, we approach the isothermal case since lim 1 þ N ¼ e , or N!∞

266

Hydrodyamics

Figure 6.4. Profiles of density, velocity, and sound speed for an adiabatic expansion with γ ¼ 5=3.

      ρ x þ cs0 t ρ lim ¼ exp  : ¼ γ!1 ρ0 adia cs0 t ρ0 iso The velocity of matter flow for the adiabatic case, using u ¼ ω, is   u 2 x þ cs0 t ¼ , cs0 γ þ 1 cs0 t and the sound speed, from (6.63) together with u ¼ x=t þ cs , is    cs γ1 2 x  ¼ : γ þ 1 γ  1 cs0 t cs0

(6.62)

(6.63)

(6.64)

Equations (6.60), (6.63), and (6.64) show that for t > 0, the velocity increases linearly from zero at the head of the rarefaction, at x ¼ cs0 t, to a value of u ¼ 2cs0 =ðγ þ 1Þ at x ¼ 0, and then to a maximum value at the tail, whose location is given by (6.61), at which point u ¼ 2cs0 =ðγ  1Þ, the same as x=t. The sound speed decreases linearly from cs0 to zero over the same interval. 6.4 Acoustic Disturbances Acoustic waves are small-amplitude disturbances that propagate in a compressible medium through the interplay between the fluid’s inertia and the restoring force of pressure. These sound waves are really density waves. As long as the wavelength of the disturbance is large, the variation in density is very gradual, and thus

6.4 Acoustic Disturbances

267

acoustic waves cost very little energy; they do not change the properties of the fluid nor affect the fluid’s flow. We assume the medium is homogeneous, isotropic and of infinite extent, with no external forces imposed. The ambient medium is considered at rest with constant density ρ0 and pressure P0 . A small disturbance is imposed that perturbs these quantities locally to ρ ¼ ρ0 þ δρ and P ¼ P0 þ δP, where jδρ=ρ0 j  1 and jδP=P0 j  1. The fluid is given a small velocity u, such that juj  cs , where cs is the local sound speed. Velocity gradients are assumed to be small so that viscous effects may be neglected, and the time scale for thermal conduction is large compared to a characteristic fluctuation time for collisional energy exchange. Hence, there are no dissipative processes, and changes in the fluid properties are adiabatic, that is, the flow is isentropic. The inviscid conservation equations of mass (6.13) and momentum (6.21) may be linearized to give ∂ðδρÞ ¼ ρ0 ru ∂t

(6.65)

∂u ¼ rðδPÞ: ∂t

(6.66)

and ρ0

The conservation of energy equation is not needed here since the process is assumed to be adiabatic. This implies that variations in material properties can be related through derivatives taken at constant entropy. Thus, δP ¼ ð∂P=∂ρÞS δρ. Then taking the time derivative of (6.65) and subtracting the divergence of (6.66), we obtain the wave equation for the density perturbations ∂2 ðδρÞ  c2s r2 ðδρÞ ¼ 0: ∂t 2

(6.67)

Similar equations exist for δP and u. If the pressure is not provided as an explicit function of density and entropy, then ð∂P=∂ρÞs must be found implicitly. Consider the EOS P ¼ Pðρ; T Þ. Then     ∂P ∂P dP ¼ dρ þ dT: (6.68) ∂ρ T ∂T ρ Using the composite function theorem, the adiabatic sound speed is found from        ∂P ∂P ∂T ∂P 2 ¼ þ , (6.69) cs ¼ ∂ρ s ∂T ρ ∂ρ s ∂ρ T and then incorporating a Maxwell relation (see Section 3.3.4), we obtain      1 ∂P ∂P ∂P 2 þ : cs ¼ 2 ρ ∂s ρ ∂T ρ ∂ρ T

(6.70)

268

Hydrodyamics

The combined first and second law of thermodynamics gives Tds ¼ dε  ðP=ρ2 Þdρ, from which we obtain     ∂P ∂P ds ¼ Tds ¼ αTds, (6.71) ∂s ρ ∂ε ρ where α¼

ð∂P=∂T Þρ ð∂ε=∂T Þρ

:

(6.72)

Then c2s

    αT ∂P ∂P ¼ 2 þ : ρ ∂T ρ ∂ρ T

(6.73)

The two thermodynamic derivatives defining α are found from the EOS. Forpffiffiffiffiffiffiffiffiffi a polytropic EOS, P ¼ P0 ðρ=ρ0 Þγ , and the adiabatic sound speed is ffi cs ¼ γP=ρ. We may also define an isothermal sound speed c2T ¼ ð∂P=∂ρÞT ¼ c2s =γ that is useful when discussing the damping of acoustic waves by viscous and/or thermal processes. Taking the curl of (6.66) gives ∂ ðr  uÞ ¼ 0; ∂t

(6.74)

hence, the vorticity ω ¼ r  u of the disturbed fluid remains constant in time. As the fluid was initially at rest, it follows that ω ¼ 0 for all times, and the velocity u may be described as the gradient of a potential Φ. The only nonzero component of the velocity lies in the direction of propagation, hence an acoustic wave is longitudinal in nature. For one-dimensional planar geometry, (6.67), and similar equations for δP and u, may be written as 2 ∂2 ψ 2∂ ψ  c ¼ 0, s ∂t 2 ∂x2

(6.75)

ψ ¼ f 1 ðx  cs t Þ þ f 2 ðx þ cs t Þ,

(6.76)

which has the general solution

where f 1 and f 2 are arbitrary functions. This solution represents a superposition of two traveling plane waves, one moving in the negative direction and the other in the positive direction, each with speed cs .

6.4 Acoustic Disturbances

269

Letting the gradient of the potential be the velocity, u ¼ ∂ψ=∂x, and differentiating (6.76) gives ∂ 1 ∂ ½uðx∓cs tÞ ¼ ∓ ½uðx∓cs tÞ, ∂x cs ∂t

(6.77)

and using this in the continuity equation (6.65) shows the perturbation amplitudes are related ðδρÞ ¼ 

u ρ and ðδPÞ ¼ c2s ðδρÞ ¼ cs ρ0 u: cs 0

(6.78)

We see that the particle velocity is in the direction of propagation where the fluid is being compressed, and in the opposite direction where the fluid is expanding. Planar acoustic waves are not readily found in nature but can be created quite easily in the laboratory. In our everyday experience, we encounter spherical sound waves. In the absence of absorption via viscosity and/or heat conduction, the amplitude of a plane wave decreases with time; we will address the topic of dissipation later in this chapter. However, in the spherically symmetric case, the acoustic wave is attenuated. From (6.65) the density perturbation behaves as ∂ðδρÞ ρ ∂ 2 ¼  20 r u : ∂t r ∂r

(6.79)

The equation of motion is the same as before, (6.66). We can write a wave equation for ðδρÞ, whose solution describes a wave spreading from the origin ðδρÞ ¼

f ðr  cs t Þ : r

(6.80)

For short acoustic pulses whose lengths are much shorter than the radius of the disturbance, the form of the pulse given by r  cs t does not change and the amplitude of the wave decreases by 1=r. We can understand this by realizing that as the pulse propagates outward, the mass of fluid that has been set in motion is approximately 4πr2 ðδrÞ. The energy of a sound wave per unit volume is proportional to ðδρÞ2 . Since the total energy is conserved, ðδρÞ2 r2 is constant and the amplitude decreases as ðδρÞ e 1=r. A spherical wave differs from a plane wave in another respect. Using (6.79) in the equation of motion (6.66) gives   ∂u c2s 1 ∂ f ðr  cs t Þ : (6.81) ½ f ðr  cs t Þ  ¼ ∂t r2 ρ0 r ∂r

270

Hydrodyamics

Integrating this equation gives the solution for the velocity   ð ð cs f ðr  cs t Þ 1 rcs t cs 1 rcs t f ðξ Þdξ ¼ f ðξ Þdξ u¼  2 ρ0 ρ0 r 2 r r

(6.82)

This solution differs from the plane wave solution u ¼ ðcs =ρ0 ÞðδρÞ by the presence of an additional term. For a plane wave, the fluid can be compressed only in the region of disturbance, but is impossible in the spherical case, where a compression region must be followed by an expansion region. Outside the disturbance, both δρ and u become zero. In the plane wave case, this is automatically satisfied since u e δρ, but in the spherical wave case this is possible only when φðr  cs t Þ ¼ 0 outside the region of disturbance. That is, ð ð (6.83) φðr  cs t Þ ¼ f ðξ Þdξ ¼ ðδρÞrdr ¼ 0: We see that δρ changes sign in a spherical wave, and that the compression region is followed by an expansion region. The additional fluid included in the wave is equal to Ð ðδρÞ4πr2 dr, and the additional mass in the compression increases as it spreads from the origin. An increase in the amount of compressed fluid during propagation causes the appearance of the higher-density wave followed by the lower-density wave. The theory developed here applies only to small-amplitude disturbances, which propagate adiabatically and are slowly damped by dissipative processes. For increasing wave amplitude, this simple picture is no longer appropriate, since nonlinear terms in the hydrodynamic equations become important. The most significant aspect is that the regions of compression tend to overtake the rarefaction that precedes it, and the leading part of the profile progressively steepens. Eventually this profile becomes a “discontinuity,” which we identify as a shock front.

6.5 Shocks A shock front, sometimes referred to as a shock wave, represents a disturbance of large amplitude for which the linear treatment of small-amplitude (acoustic) waves no longer applies. Shock fronts are such violent disturbances across which the thermodynamic functions as well as the velocity of the fluid change so rapidly as to be discontinuous. The shock front can be represented as a “geometrical surface” passing through the fluid. The shock wave is not a true wave since periodic motion does not occur. The discontinuity in the quantities at the shock implies a thickness of the front to be of the order of a few collsional mean free paths with respect to the undisturbed medium ahead of the shock front. In the case of a very strong shock, the use of the term “mean free path” becomes ambiguous because of the dissociation, excitation,

6.5 Shocks

271

and ionization of the atoms/molecules composing the medium. This region of discontinuity is not in a state of thermodynamic equilibrium. The subject of shock propagation is very rich. However, space limitations require us to limit the discussion to “normal” shocks only. Oblique shocks, Mach stems, bow shocks, and such are fascinating in themselves, and the reader is encouraged to explore them in the literature. Consider the long cylindrical gas-filled tube from before. Now let there be a piston moving inward from the left end. As it accelerates, the motion of the piston may be thought of as divided into a large number of small, successive moments, each of which produces a pressure pulse that travels through the fluid at the speed of sound. At any given instant, the fluid to the right of the disturbance remains unchanged and at rest, while the fluid between the piston and disturbance is adiabatically compressed. A short while later, the velocity of the piston has increased by a small amount; consequently a second compression is produced and is propagated along the tube behind the first pulse. A continued increase in the piston velocity to some constant value produces the shock front. The material behind the shock front, being compressed and heated, has higher entropy than the undisturbed material in front of it. This rise in entropy across the shock front implies that the energy in the “wave” has been dissipated irreversibly, thus causing the wave to be damped and eventually destroying the shock. Courant and Friedrichs (1948) develop the theory of shock fronts in considerable detail. The excellent standard reference on hydrodynamic waves is the work by Whitham (1974).

6.5.1 Nonlinear Acoustic Waves The physics of acoustic disturbances is based upon the fluid velocity being small compared to the sound speed. The density, pressure, and velocity are completely determined by the continuity and momentum equations, together with an EOS, and are functions of a single argument, x  cs t. We can obtain a general solution of the nonlinear equations for a traveling wave for which all physical properties are a function of x  cs t, but now the sound speed is a function of the fluid velocity at each point in the disturbance. Assuming ρ ¼ ρðuÞ, the continuity equation (6.13) may be written   ∂u du ∂u þ uþρ ¼ 0, (6.84) ∂t dρ ∂x and the momentum equation (6.21) is     ∂u 1 ∂P dρ ∂u þ uþ ¼ 0: ∂t ρ ∂ρ s du ∂x

(6.85)

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Hydrodyamics

Combining (6.84) and (6.85) shows  1= du 1 ∂P 2 cs ¼ , ¼ dρ ρ ∂ρ s ρ

(6.86)

from which the general relation between the fluid velocity and the density (or pressure) in the wave is obtained ðρ u¼ ρ0

cs dρ ¼  ρ

ðP P0

1 dP, cs ρ

(6.87)

where ρ0 and P0 are the values in the undisturbed fluid. Using (6.86) in (6.84) gives ∂u ∂u þ ðu  cs Þ ¼ 0: ∂t ∂x

(6.88)

We may also invert the function ρðuÞ and write the continuity equation as   ∂ρ du ∂ρ þ ρ þu ¼ 0, (6.89) ∂t dx ∂x and upon using (6.86), implies ∂ρ ∂ρ þ ðu  cs Þ ¼ 0: ∂t ∂x

(6.90)

Equations (6.88) and (6.90) yield solutions of the form of (6.76) u ¼ f 1 ½x  ðu  cs t Þ and ρ ¼ f 2 ½x  ðu  cs t Þ:

(6.91)

These two functions define the range of fluid velocities and densities at the initial time. For a particular value of velocity (or density), it propagates through the medium with phase velocity vph ðuÞ ¼ u  cs ðuÞ,

(6.92)

where cs ðuÞ is given by (6.86) and (6.87). All physical quantities propagate in the same manner, since ρ ¼ ρðuÞ and P ¼ P½ρðuÞ. Consider a simple wave in a polytropic gas, then c2s e ργ1 and thus ðγ  1Þ

dρ dcs : ¼2 ρ cs

(6.93)

Then using this in (6.87) and rearranging, we have 1 cs ¼ cs0  ðγ  1Þu, 2

(6.94)

6.5 Shocks

which implies a density of

273

 2 ρ 1 u γ1 ¼ 1  ðγ  1Þ , ρ0 2 cs0

(6.95)

  2γ P 1 u γ1 ¼ 1  ðγ  1Þ : P0 2 cs0

(6.96)

and a pressure

Thus, for a perfect gas, the temperature is   T 1 u 2 ¼ 1  ðγ  1Þ : T0 2 cs0

(6.97)

Consider now a piston moving from the left, which applies a time-dependent pressure Pðt Þ. At t ¼ 0, the pressure is P0 . As the piston’s speed increases incrementally, the pressure, density, and gas velocity are incremented according to (6.86), with the piston-gas interface moving with u. A further increase in the piston’s speed creates an additional disturbance, which in the laboratory frame, travels with velocity u þ cs . Each of these successive disturbances travels independently until a time τ when all arrive coincidentally at x ¼ 0, the motion of each given by d ½ðu þ cs Þðτ  t Þ ¼ u: dt

(6.98)

The velocity and sound speed may be expressed in terms of the driving pressure Pðt Þ by integrating across all of the former disturbances. Expression (6.93) can be modified and integrated to give  γ1 P 2γ : cs ¼ cs0 P0

(6.99)

Eliminating ρ in (6.86) and integrating gives u¼

2 ðcs  cs0 Þ: γ1

(6.100)

Then from (6.98) the condition for coincidence of all the waves is d 2 ½cs ðτ  tÞ ¼  cs , dt γþ1

(6.101)

which has the solution for sound waves in the vicinity of the piston γ1  t γþ1 cs ¼ cs0 1  : τ

(6.102)

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Hydrodyamics

Figure 6.5. Nonlinear steepening of a simple acoustic wave into a shock discontinuity.

Then using (6.99) gives the pressure of the piston 2γ  t γþ1 : Pðt Þ ¼ P0 1  τ

The pressure, of course, becomes infinite at t ¼ τ. The position of the piston is determined from (6.98)      2 γ1  t γþ1 2 t xðt Þ ¼ cs0 τ  1 1 : γþ1 τ γ1 τ

(6.103)

(6.104)

Since the wave initially travels with speed cs0 , the initial position of the piston is cs0 τ ¼ x0 , which defines the constant τ. We can visualize this steeping by beginning with an initial sinusoidal shape moving to the right, as sketched in Figure 6.5. In the small-amplitude limit equations, (6.78) is recovered from (6.95) and from (6.96); the wave propagates with velocity cs0 . In the finite-amplitude regime, (6.92) through (6.97) show that the compressed parts of the pulse have a large fluid velocity, are hotter, and have a higher sound speed given by (6.94). The front of this pulse is moving toward the right with a higher velocity than the less compressed portions, and the pulse continues to grow in amplitude, resulting in an ever steepening of the front. Eventually the crest of the pulse overtakes the front and the solution becomes multiple valued, clearly an unphysical result. In reality, as the front of the pulse steepens, the variables undergo a sharp change in value over a thin layer, within which viscosity and thermal conductivity can no longer be ignored. We see in the figure that the initial sinusoidal shape is transformed into a triangular shape, and the velocity amplitude of the shock front decreases. Once the shock front forms, the wave continuously dissipates energy and thus is damped. At any instant, we may consider the motion to be steady because ur terms are large compared to ∂=∂t terms, in addition to the shock thickness being only of the order of a few particle mean free paths, whereas the distance over which the

275

6.5 Shocks

properties of the upstream material can change significantly is some characteristic structural length. Consequently, the ratio of the time required for material to cross the shock front to the time needed for upstream conditions to change appreciably is very small, roughly equal to the Knudsen number. Such shocks are fixed in space and have upstream and downstream properties that are independent of time. The task now is to relate the physical variables on the upstream side to those on the downstream side. (We note that in the following discussion about shock fronts, many authors work with the reciprocal of the density [specific volume] rather than density.)

6.5.2 Rankine-Hugoniot Equations Consider a volume of gas initially at rest with constant density ρ0 and pressure P0 , and assume the column of gas has unit cross section. Let a piston move from the left end into the gas at constant velocity U. A discontinuity in the gas will develop, moving with velocity us . We designate the variables in the undisturbed material ahead of the discontinuity (the upstream side) with the subscript 0 and those behind the front (the downstream side) with subscript 1. This configuration is sketched in Figure 6.6. At the end of a short period of time t, an amount of gas equal to ρ0 us t is set in motion. This mass of gas has been compressed to a density ρ1 . For zero shock thickness, conservation of mass requires ρ1 ðus  U Þ ¼ ρ0 us :

(6.105)

This mass of gas acquires momentum ρ0 us Ut that is equal to the impulse due to the pressure forces. Conservation of momentum requires ρ0 us U ¼ P1  P0 ,

(6.106)

Shock front Downstream

Upstream

r1 T1 P1 e1 u1 = U – u s

u0 = –us r0 T0 P0 e0

x

Figure 6.6. A one-dimensional steady-state shock in a reference frame moving with the shock front whose speed is us in the laboratory frame. In this reference frame, the fluid flows from right to left.

276

Hydrodyamics

where P1 is the downstream pressure. The third conservation law, that of energy, states that the increase in the sum of the kinetic and internal energies of the compressed gas is equal to the work done by the external force P1 Ut, thus   U2 (6.107) ρ0 us ε1  ε0 þ ¼ P1 U: 2 We assume the specific internal energies ε1 and ε0 may be obtained from a thermodynamic relationship εðP; ρÞ, the EOS. Let us transform to a coordinate system in which the observer is moving with the shock front. We need to find the three unknowns us , ρ1 , and P1 . Since us is the propagation velocity of the discontinuity through the undisturbed gas, the velocity at which the gas flows into the discontinuity (on which the observer is riding) is u0 ¼ us . Likewise, the propagation velocity of the discontinuity with respect to the gas moving behind it is us  U, thus u1 ¼ ðus  U Þ is the velocity of the gas flowing out of the discontinuity. Then from (6.105), the conservation of mass flux becomes ρ1 u1 ¼ ρ0 u0 ;

(6.108)

from (6.106), conservation of momentum flux becomes ρ1 u21 þ P1 ¼ ρ0 u20 þ P0 ,

(6.109)

and from (6.107), the conservation of energy becomes ε1 þ

u21 P1 u2 P0 þ ¼ ε0 þ 0 þ : 2 ρ1 2 ρ0

(6.110)

Equations (6.108) through (6.110) constitute the Rankine-Hugoniot equations in one dimension. Alternately, they may be derived from (6.13), (6.21), and (6.26). Equations (6.105) through (6.107) are not particularly useful for finding us , ρ1 , and P1 , the quantities that are most important to us. Eliminating U between (6.105) and (6.106) gives u2s ¼

ρ1 ðP1  P0 Þ : ρ0 ðρ1  ρ0 Þ

(6.111)

Alternately, eliminating us from (6.105) and (6.106) results in jU j2 ¼ ðu1  u0 Þ2 ¼

ðρ1  ρ0 Þ ðP1  P0 Þ: ρ0 ρ1

(6.112)

Eliminating u1 in the mass and momentum equations (6.108) and (6.109) gives u20 ¼

ρ1 ðP1  P0 Þ , ρ0 ðρ1  ρ0 Þ

(6.113)

6.5 Shocks

277

or eliminating u0 yields u21 ¼

ρ0 ðP1  P0 Þ : ρ1 ðρ1  ρ0 Þ

(6.114)

From (6.110) and using (6.113) and (6.114), the difference in the specific energies across the shock front is ε1  ε0 ¼

1 ðρ1  ρ0 Þ ðP0 þ P1 Þ, 2 ρ0 ρ1

(6.115)

and is analogous to the adiabatic case Δε ¼ PΔð1=ρÞ. Equation (6.115) is sometimes referred to as the Hugoniot equation, but we prefer (6.120), in the following section. Another useful quantity is the specific enthalpy, which is defined by h ¼ ε þ P=ρ. Then using (6.115), we obtain h1  h0 ¼

1 ðρ0 þ ρ1 Þ ðP1  P0 Þ: 2 ρ0 ρ1

(6.116)

The equations (6.113), (6.114), and (6.115) permit two types of solutions: (1) those that have P1 > P0 , ρ1 > ρ0 , and u1 < u0 or (2) those in which the inequalities are all reversed. The former are compression shocks, and the latter are rarefaction discontinuities. As we shall see shortly, only compression shocks can exist physically. Rarefactions, when they occur, are always continuous. A key assumption for the validity of the Rankine-Hugoniot equations is that the shock is steady in time so that the rise time is short compared to the characteristic time for which the downstream quantities of pressure, density, and specific energy are constant. The Rankine-Hugoniot equations we have discussed are the most simple. If we include electric and magnetic fields in the momentum and energy equations, we can then explore additional effects. We will discuss the inclusion of magnetic fields in the equations of hydrodynamics in Chapter 11. We note, further, that a deeper exploration of the Rankine-Hugoniot equations leads to a number of different effects, including contact discontinuities, oblique shocks, and bow shocks.

6.5.3 Jump Relations in a Polytropic Gas The EOS for a polytropic gas is P ¼ ðγ  1Þρε. In general, the polytropic index γ is not the ratio of the specific heats, but should be obtained from an empirical Hugoniot relation. In high-energy-density physics, the polytropic index may differ significantly across a shock and must be taken into account in the jump relations.

278

Hydrodyamics

Using the polytropic EOS in (6.115) produces the compression ratio across the shock front   ρ1 P1 ðγ1 þ 1Þ þ P0 ðγ1  1Þ γ0  1 ¼ : (6.117) ρ0 P0 ðγ0 þ 1Þ þ P1 ðγ0  1Þ γ1  1 This equation may be rearranged to find P1 =P0 in terms of ρ1 =ρ0   P1 ρ1 ðγ0 þ 1Þ  ρ0 ðγ0  1Þ γ1  1 ¼ : P0 ρ0 ðγ1 þ 1Þ  ρ1 ðγ1  1Þ γ0  1

(6.118)

Consider, for the moment, γ1 ¼ γ0 ¼ γ; then the density ratio is ρ1 P10 ðγ þ 1Þ þ ðγ  1Þ , ¼ ρ0 ðγ þ 1Þ þ PP1 ðγ  1Þ 0

(6.119)

ρ1 P1 ρ0 ðγ þ 1Þ  ðγ  1Þ : ¼ P0 ðγ þ 1Þ  ρρ1 ðγ  1Þ

(6.120)

P

and the pressure ratio is

0

Equation (6.120) is commonly known as the Hugoniot. For the initial upstream conditions ρ0 , P0 , and u0 , then (6.115) defines a unique curve in the density-pressure plane. This is the Hugoniot curve, passing through the point ðρ0 ; P0 Þ and having the general shape sketched in Figure 6.7. Equation (6.115), combined with an EOS state, determines ρ1 and P1 , and thus u1 . The Hugoniot is the locus of points describing the possible final thermodynamic states for the passage of a single shock through the fluid. It is therefore a set of equilibrium states and does not specifically represent the path through which a material undergoes transformation. The figure includes the Rayleigh line that intersects the Hugoniot curve at the points labeled “0” and “1.” The Rayleigh line is the locus of all permissible ðP; ρÞ states consistent with a particular shock velocity us , given by (6.111)   ρ0 2 P1 ¼ P0 þ ρ0 us 1  : (6.121) p1 Also shown is the isentrope (adiabat) defined by ðP1 =P0 Þ ¼ ðρ1 =ρ0 Þγ and the isotherm defined by P=ρ ¼ const. Let us examine the initial slope of the Hugoniot curve at the point “0.” The derivative dP1 =dρ1 comes from (6.120) dP1 P0 ¼ h dρ1 ρ0

4γ ðγ þ 1Þ 

ρ1 ρ0

ðγ  1Þ

i2 :

(6.122)

279

6.5 Shocks

Figure 6.7. Compression curves. For a shock, the thermodynamic path is a straight line (the Rayleigh line) between the initial and final points on the Hugoniot (solid line), which is the locus of shocked states. Also shown is the thermodynamic path for the isentrope (short-dashed line) and the isotherm (long-dashed line).

Evaluating this at the point “0” gives ½dP1 =dð1=ρ1 Þ0 ¼ γρ0 P0 , as it should be. This quantity is simply the slope of the isentrope; the Hugoniot curve is tangent to the isentrope at the point “0.” The vertical asymptote is 1=ρ0 ¼ ðγ  1Þ=ðγ þ 1Þð1=ρ1 Þ, and the horizontal asymptote is P1 =P0 ¼ ðγ  1Þ=ðγ þ 1Þ. The curve crosses the 1=ρ axis at ρ0 =ρ1 ¼ ðγ þ 1Þ=ðγ  1Þ. Of course, values of 1=ρ1 that are larger than 1=ρ0 represent expansion, which cannot occur in a shock. Similarly, using (6.112) and (6.117), we could find the velocity jump for different polytropic indices on each side of the shock, but the expression is not very useful here. Rather, we use (6.112) and (6.119) to find 2 31=2  2 P1  1 P0 62P0 7 u1 ¼ u0 þ 4 (6.123) 5 : P 1 ρ0 P ðγ þ 1Þ þ ðγ  1Þ 0 A perfect gas EOS has P e ρT, so the temperature ratio is " # " # ρ0 P T 1 ρ0 ðγ þ 1Þ  ρ1 ðγ  1Þ P1 ðγ þ 1Þ þ P10 ðγ  1Þ ¼ : ¼ T 0 ρ1 ρρ0 ðγ þ 1Þ  ðγ  1Þ P0 PP1 ðγ þ 1Þ þ ðγ  1Þ 1

0

(6.124)

280

Hydrodyamics

For strong shocks, P1 =P0 1, and we have the downstream quantities ρ1 ¼

γþ1 2 γ1 ρ , P1 ¼ ρ u2 and u1 ¼ u0 ; γ1 0 γþ1 0 0 γþ1

(6.125)

the first of these equations represents the maximum density jump possible across a shock front.1 The Mach number is defined as the particle velocity divided by the local sound speed. The upstream Mach number is found from (6.113)  2 P 1 u0 P ðγ þ 1Þ þ ðγ  1Þ 2 , (6.126) M0 ¼ ¼ 0 cs0 2γ and the downstream flow is from (6.114)  2 P 0 ðγ þ 1Þ þ ðγ  1Þ u1 2 M1 ¼ ¼ P1 : cs1 2γ

(6.127)

In the limiting case of a weak shock, P1  P0 , and thus u1  cs1  cs0  u0 . However, u0 is the velocity of the discontinuity moving through the undisturbed gas, and hence the speed that the shock front moves through the gas is very nearly the speed of sound, the same as for an acoustic compression wave. For a strong shock, the limiting downstream Mach number is M 21 ¼ ðγ  1Þ=2γ. Since P1 =P0 > 1, M 0 > 1 and M 1 < 1, the flows are supersonic and subsonic, respectively. Upstream and downstream flow properties can also be related in terms of the upstream Mach number M 0 . Equation (6.119) may be expressed in terms of M 0 ρ1 M 2 ðγ þ 1Þ ¼ 2 0 , ρ0 M 0 ðγ  1Þ þ 2

(6.128)

P1 2γM 20  ðγ  1Þ : ¼ ðγ þ 1Þ P0

(6.129)

and the Hugoniot (6.120) as

Then using this equation in (6.128), the downstream Mach number becomes M 21 ¼

1

ðγ  1ÞM 20 þ 2 : 2γM 20  ðγ  1Þ

(6.130)

The more degrees of freedom the individual particles of the gas have, the more ways there are for energy to be portioned, and the closer the two specific heats are to being equal; that is, γ ! 1, and thus the ratio ρ1 =ρ0 diverges.

6.5 Shocks

281

pffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi The sound speed in the downstream flow is cs1p¼ffiffiffiffiffiffiffiffi γP1 =ρ1 ¼ γðγ  1Þ=2U, where (6.125) has been used. For γ ¼ 5=3, cs1 ¼ 5=9U. Thus, in the laboratory frame, the flow is supersonic, as the sound speed is less than the particle speed. However, the pressure source that sustains the shock is moving with speed U, and the distance between the piston and the shock increases as u1 ¼ ðγ  1ÞU=2. For γ ¼ 5=3, u1 ¼ U=3 and the gas there is subsonic. In the frame of reference moving with the shock front, the downstream flow moves subsonically. The piston velocity U is also referred to as the downstream particle velocity up . The specific energy and enthalpy are ε1 ¼

u20 4γ u20 ¼ and h : 1 ðγ þ 1Þ2 2 ðγ þ 1Þ2 2 4

(6.131)

We see that a large fraction of the kinetic energy entering the shock is turned into heat. For a γ ¼ 5=3 gas, the postshock internal energy is nine-sixteenths of the incoming kinetic energy, and the enthalpy is fifteen-sixteenths of the incoming kinetic energy. The second law of thermodynamics states that the entropy of a substance cannot be decreased by internal processes alone. Thus, the downstream entropy in a shock must equal or exceed its upstream value, that is, s1 ≥ s0 . The specific entropy is s ¼ cV log ðPÞ  cP log ðρÞ ¼ cV log ðPργ Þ, so the jump in entropy across the front is   γ1  P1 ρ 0 γ0 γ1 : (6.132) s1  s0 ¼ cV log ρ0 P0 ρ 1 Then substitute (6.117) into (6.132). For γ0 ¼ γ1 ¼ γ ( " #γ ) P P1 ðγ þ 1Þ þ P10 ðγ  1Þ , s1  s0 ¼ cV log P0 PP1 ðγ þ 1Þ þ ðγ  1Þ 0

(6.133)

which is positive for all shocks. This equation implies an irreversible dissipation of energy, independent of the existence of a dissipation mechanism. At first sight, this seems paradoxical, but is easily resolved by studying the shock structure for a real gas. Now suppose we have a small pressure differential across the shock front such  that P1 ¼ P0 ð1 þ ϵ Þ, ϵ being small. Then from (6.119), ρ1  ρ0 1 þ ϵγ . Recall, for an adiabatic compression, ρ1 ¼ ρ0 ðP1 =P0 Þ =γ ¼ ρ0 ð1 þ ϵ Þ =γ  1 þ ϵγ þ   . 1

1

Therefore, in the limit of weak shocks we have adiabatic compression. We compare the density ratios for small overpressures for a γ ¼ 5=3 gas in Figure 6.8. The adiabatic approximation to compression is fairly accurate up to overpressures of a factor of two.

282

Hydrodyamics

Figure 6.8. Comparison of shock (solid line) and isentropic (dashed line) compressions as a function of overpressure for a gas with γ ¼ 5=3.

We can extend the Hugoniot (6.120) to include pressures lower than the initial pressure and assume that discontinuities exist in which an expansion of the gas occurs such that ρ1 < ρ0 and P1 < P0 . The conservation equations, for these conditions, do not preclude the possibility of the existence of discontinuities. We see, from (6.126) and (6.127), u0 < cs0 and u1 > cs1 . Further, (6.133) shows that the entropy jump across the discontinuity decreases, that is, s1 < s0 . This is a rarefaction shock wave. The second law of thermodynamics states that the entropy of a substance cannot be decreased by internal processes alone, without the transfer of heat to an external medium. Therefore, the rarefaction shock violates this law. The impossibility of the existence of a rarefaction shock can be understood by the following. Such a wave would propagate through the undisturbed gas with a subsonic velocity. But that means that any disturbance induced by the density and pressure jumps will begin to travel back into the more dense substance where the sound speed is cs0 , and will outrun the “shock wave.” After a certain time, the rarefaction region will include the gas in front of the “discontinuity,” and the discontinuity will simply disappear; that is, a rarefaction shock is mechanically unstable.

6.5 Shocks

283

It is sometimes convenient to introduce the critical velocity uc , where the flow speed equals the local sound speed. Then from conservation of energy, (6.110) using u2c ¼ γP0 =ρ0 ¼ γP1 =ρ1       u21 γ P1 u20 γ P0 u2c γ þ 1 ¼ þ ¼ , (6.134) þ γ  1 ρ1 γ  1 ρ0 2 2 2 γ1 and  u0 þ

   2γ P0 γ þ 1 u2c ¼ , γ  1 ρ0 u0 γ  1 u0

(6.135)

and 

   2γ P1 γ þ 1 u2c ¼ : u1 þ γ  1 ρ1 u1 γ  1 u1

(6.136)

Subtracting (6.136) from (6.135) and then dividing (6.108) into (6.109) and using the result, we obtain the Prandtl relation u0 u1 ¼ u2c . We can find a relation between the shock front velocity us and the piston velocity U by combining (6.105) and (6.106) to get P1 ¼ P0 þ

ρ0 ρ1 U2, ρ1  ρ0

(6.137)

and then using (6.120) in (6.137), along with (6.105), gives 1 P0 u2s  ðγ þ 1ÞUus  γ ¼ 0, 2 ρ0

(6.138)

and taking the positive root results in sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 P0 ðγ þ 1Þ2 U 2 þ γ : (6.139) 16 ρ0 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi Recall that the sound speed in the ambient gas is cs0 ¼ γP0 =ρ0 : The pressure behind the discontinuity is found from using (6.139) in (6.106) rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 1 2 ðγ þ 1Þ2 U 2 þ c2s0 , (6.140) P1 ¼ P0 þ ðγ þ 1Þρ0 U þ ρ0 U 4 16 1 us ¼ ðγ þ 1ÞU þ 4

and the density ratio across the shock comes from using (6.138) in (6.105) ρ0 U qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : ¼1 ρ1 1 ðγ þ 1ÞU þ 1 ðγ þ 1Þ2 U 2 þ c2 4

16

s0

(6.141)

284

Hydrodyamics

And if cs0 =U  1, the strong shock limit is recovered. Finally, for a strong shock, a simple relationship between the piston velocity U and the shock velocity us is us ¼

γþ1 γ1 U ¼ U  u1 ¼ U  u0 : 2 γþ1

(6.142)

For a gas with γ ¼ 5=3, the postshock fluid velocity is 75 percent of the shock speed. We can easily show that the postshock fluid velocity becomes closer to the shock speed as the density jump increases. The postshock pressure in the strongshock limit is P1 ¼ ðγ  1Þ

ρ1 U 2 : 2

(6.143)

For a polytropic gas, the postshocked fluid has equal kinetic and internal energies.

6.5.4 The Shock Tube Up to this point, we have mentioned the shock tube as a way to visualize the idealized expansion/compression of a fluid under different conditions. The shock tube of the laboratory exhibits a wealth of phenomena, which we examine in some detail. Shock fronts are often studied using a “gas gun” or similar devices driven by high explosives or electrical discharges. With the introduction of high-power lasers in the 1970s, a new source of intense energy was available for driving shocks to much higher velocities. For our purposes, we simplify the geometry of a shock tube into two regions: (1) a portion of the tube under high pressure separated from the rest of the tube by a membrane of negligible mass and (2) the remainder of the tube at low pressure. It may be that the two regions separated by the membrane contain different gases. For our purposes, we limit the discussion to only one type of gas. Initially, the gas is everywhere at rest. The shock is initiated by breaking the membrane and the high-pressure gas starts expanding into the lower-density gas. The interface between the two regions propagates with a velocity characteristic of the sound speed in the unshocked matter. However, the resulting shock travels at a velocity characteristic of the shock velocity also in the unshocked gas, which is greater than the velocity of the interface. The pressure profile extends further into the gas than the interface position. Simultaneously, the pressure behind the interface falls adiabatically (since there is no shock progressing backward), starting a rarefaction fan moving back into the high-pressure matter with a velocity characteristic of the sound speed there. The development of the adiabatic portion of the

285

6.5 Shocks

Figure 6.9. Configuration of the shock tube at a time after rupture of the membrane. Region 1 is the undisturbed high pressure, region 2 is the adiabatic rarefaction, region 3 is the downstream side of the shock, and region 4 is the undisturbed (upstream) gas.

expansion was addressed in Section 6.3.2, while the nature of the penetrating shock front was presented in the previous section. Figure 6.9 sketches the geometry and characteristics at a time after rupture of the membrane (Sod, 1978). The contact discontinuity is the location of the interface between the two material regions; it has to satisfy the jump relations the same as the shock front does. There is no flow of matter through this discontinuity, so the fluid velocity is the same on both sides. The pressure jump condition is continuous also, since there is no flow across the interface. The densities and sound speed will be different, however. First, the velocity distribution of the rarefaction in region two is described by the Riemann invariant J þ given by (6.48). Hence, u þ 2cs =ðγ  1Þ ¼ const for x ¼ ðu  cs Þt. Combining these two gives 2  x 2 uþ (6.144) u ¼ const ¼ u1 þ cs1 , γ1 t γ1 which has the solution   2 x γ1 uðx; t Þ ¼ þ u1 þ cs1 , γþ1 t 2

(6.145)

286

Hydrodyamics

which is effectively (6.63). Thus, u2 ¼ u1 þ

2 ðcs1  cs2 Þ: γ1

(6.146)

The velocity is continuous across the contact discontinuity, thus " #  γ1 2cs1 P2 2γ : 1 u2 ¼ u3 ¼ u1 þ γ1 P1

(6.147)

Second, the jump relations across the shock front are found from (6.123) 2



2

31=2

2 1 62c 7 u3 ¼ u4 þ 4 s4 P3 5 , γ P ðγ þ 1Þ þ ðγ  1Þ P3 P4

(6.148)

4

where P3 =P4 is to be replaced by ðP1 =P4 Þ=ðP1 =P2 Þ. The ratio P1 =P4 is one of the initial conditions of the problem. Equating (6.147) with (6.148) gives an equation that is to be solved by an iterative technique. We can allow dissimilar fluids on each side of the membrane by using different polytropic indices in (6.147) from those in (6.148). In the laboratory frame, the shock velocity, from (6.111), is us ¼ u3 ρ3 =ðρ3  ρ4 Þ. Then, applying (6.119), we find  1=2 cs4 P3 ðγ þ 1Þ þ ðγ  1Þ : us ¼ u4 þ pffiffiffiffiffi 2γ P4

(6.149)

The pressure distribution across the shock front is known from the Hugoniot, and the pressure across the contact discontinuity is P2 ¼ P3 . It remains to find the shape of the rarefaction wave. The velocity is given by (6.145), and the sound speed within the rarefaction is x 2 γ  1 x cs ðx; tÞ ¼ uðx; tÞ  ¼ cs1 þ u1  , (6.150) t γþ1 γþ1 t which is effectively (6.64). The pressure distribution is found from using (6.150) in (6.55), together with Pðx; tÞ ¼ P1 ½ρðx; tÞ=ρ1 γ . The density and specific energy (temperature) as a function of space and time can be readily found. Figure 6.10 sketches the profiles of these quantities for γ ¼ 5=3, an initial pressure ratio of 105 , and a density ratio of 103. The membrane was initially at x ¼ 0, with u1 ¼ u4 ¼ 0. The solution to the equations gives P2 =P1 ¼ 0:007748 and P3 =P4 ¼ 774:8. The density ratios are ρ2 =ρ1 ¼ 0:05414, ρ2 =ρ3 ¼ 13:60, and ρ3 =ρ4 ¼ 3:981 (ideally, it should be 4). The Mach number of the shock, from (6.128), is 24.97.

287

6.5 Shocks (a)

(b) 2.5 10

0

0

1 2.0

10–1

Pressure

10–1

Density

3 2

1.5

10–2

10–3

1.0

10–4

0.5

Velocity

10

10–2 4

10–3 –2.0

–1.0

0.0

2.0

1.0

3.0

10–5 –2.0

4.0

–1.0

0.0

x/t

1.0

2.0

3.0

0.0 4.0

x/t

(c) 2.0

Temperature

1.5

1.0

0.5

0.0 –2.0

–1.0

0.0

1.0

2.0

3.0

4.0

x/t

Figure 6.10. Profiles for the density, pressure, velocity (dashed line), and temperature in a shock tube. The membrane was initially at x=t ¼ 0.

The positions of the four features are x1 =t ¼ cs1 , x2 =t ¼ u2  cs2 , x3 =t ¼ u2 , and x4 =t ¼ us . The increase of entropy across the shock front requires the temperature of the shocked gas to be greater than the temperature of either the high- or low-pressure gases that were originally in equilibrium. The gas ahead of the interface will be higher because of compressive effects, while the gas behind will be lower because of expansion. At the point where the rarefaction has “eaten” into the undisturbed gas, the temperature will rise to the equilibrium temperature. The shocked gas is much hotter than the rarefied former high-pressure gas, even though the latter began one hundred times hotter than the low-pressure gas.

6.5.5 Shock Reflection A practical application of a shock being reflected from a surface (or an interface) may be found in both the terrestrial laboratory and in various astrophysical

288

Hydrodyamics

settings. An example, in the astrophysical case, occurs when a shock front in a supernova remnant encounters a molecular cloud. In the laboratory, reflected shocks are used to compress and heat material to a higher state than can be achieved by a single shock. The nature of a reflected (and possibly transmitted) shock depends upon the densities of the materials with which the shock is interacting. The problem of shock reflections in the case of normal incidence can be solved analytically. For shocks arriving obliquely at a material interface, the picture is more complicated since the incident and reflected shocks may interact, forming complex structures such as Mach stems. Understanding of such complex structures is critical in engineering, as can be visualized by the flow of air around an airplane or rocket. Even though shock reflection was discovered more than a century ago, research in this area is still very active. For our purposes, we are content to discuss the elementary theory of normal shocks. Let us consider two distinctly different materials as sketched in Figure 6.11a; the reference frame is the laboratory. The shock is approaching the interface between the two fluids from the left. As before, the unshocked state of the left-hand fluid is designated with the subscript 0, and the shocked state by 1. The second fluid, on the right, is designated by 4. For a reflection to take place, we require the initial densities to be such that ρ0 and ρ1 ρ4 ; both fluids are initially at rest. For simplicity, we work in the strong-shock limit. The density profile following the entry of the shock into the higher-density material is shown in Figure 6.11b. The state of the reflected shock is designated 2 and the transmitted shock by 3. We allow for the two media to have different (but constant) polytropic indices γ0 and γ4 . The density and gas velocity downstream of (a)

(b) Interface

Interface

r4 , P4 u1

us

u2

Density

Density

u3 r3, P3

ST

r2, P2 SR u1

r1, P1 u0 = 0

x

u4 = 0

r4, P4

r1, P1

u4 = 0 x

Figure 6.11. Density profile of the fluids (a) prior to the shock reaching the interface and (b) after the shock has entered the higher-density fluid.

6.5 Shocks

289

the shock before reaching the interface are the usual shock values given by (6.125) and from (6.123) ρ1 ¼

γ0 þ 1 2 P1 ρ0 and u21 ¼ : γ0  1 γ0 þ 1 ρ0

(6.151)

The maximum pressure that can be achieved with velocity u1 is P1 ¼ ðγ0  1Þρ1 u21 =2. At a time after interaction of the shock with the interface, the reflected shock SR is at the boundary between regions 1 and 2 (region 0 has been consumed by the passage of the shock), and the transmitted shock ST is at the boundary between regions 3 and 4. The interface between the two materials is the boundary between regions 2 and 3. Three frames of reference are involved here: there is, of course, the laboratory frame, the reflected shock frame, and the transmitted shock frame. If u2 0 is the postshock fluid velocity in the reflected shock frame, and u3 0 is the postshock fluid velocity in the transmitted shock frame, then the reflected and transmitted shock velocities in the laboratory frame are SR ¼ u2  u2 0 and ST ¼ u3 þ u3 0 :

(6.152)

In the reflected shock frame, the upstream velocity is u1 0 , so SR ¼ u1  u1 0 also. Continuity across the reflected shock front requires u2 0 ¼ u1 0 ρ1 =ρ2 , and we find u1 0 ¼

ρ2 ðu1  u2 Þ: ρ2  ρ1

(6.153)

The reflected and transmitted shock speeds are SR ¼

ρ2 u2  ρ1 u1 ρ2  ρ1

and ST ¼

ρ3 u3 , ρ3  ρ4

(6.154)

since u3 0 ¼ ST ρ4 =ρ3 . Across the material interface between regions 2 and 3, we have u2 ¼ u3 and P2 ¼ P3 . The equations governing the properties of this structure are the RankineHugoniot jump conditions (6.119) and (6.120) across each shock front; the two shock fronts are connected by the preceding expressions. First, we consider an approximate solution by writing the velocity and density jumps across each shock front using (6.123) and (6.120). Thus, for the velocities  2 P2  1 P1 2P1 , (6.155) ðu2  u1 Þ2 ¼ ρ1 PP2 ðγ0 þ 1Þ þ ðγ0  1Þ 1 and if we simplify the problem by assuming P4 is zero, u23 ¼

2 P3 : γ4 þ 1 ρ4

(6.156)

290

Hydrodyamics

The density jumps, using (6.119), are P ρ2 P21 ðγ0 þ 1Þ þ ðγ0  1Þ , ¼ ρ1 ðγ0 þ 1Þ þ PP2 ðγ0  1Þ 1

(6.157)

and in the strong-shock limit ρ3 =ρ4 ¼ ðγ4 þ 1Þ=ðγ4  1Þ. For the simple case of a rigid wall (ρ4 ¼ ∞), there is no transmitted shock. Expressions (6.153), (6.155), and (6.157) yield  2 P2 γ0  1 ¼ 1, (6.158)  1 P2 P1 P1 ðγ0 þ 1Þ þ ðγ0  1Þ which has the solution P2 3γ0  1 ¼ : γ0  1 P1

(6.159)

The density and temperature jumps across the shock are ρ2 =ρ1 ¼ γ0 =ðγ0  1Þ and T 2 =T 1 ¼ ð3γ0  1Þ=γ0 . The reflected shock speed, according to (6.154), is sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 P1 2 P1 SR ¼ u1 þ γ0 ¼ ðγ0 þ 1Þ , (6.160) γ0  1 ρ1 γ0 þ 1 ρ0 by virtue of (6.151). The effect of two shocks is shown in Figure 6.12. In a laboratory experiment using a reflected shock, the first (principal) Hugoniot takes the material to the state marked A. The reflected shock then moves the material from state A to state B. In principle, both the final density and pressure are increased above what can be achieved with just a single shock. In practice, the most significant increase is in final state density. Returning to the case for finite ρ4 , the preceding expressions yield a transcendental equation 8 #1=2 92  " < = P2 γ4 þ 1 ρ4 P2 γ0  1 ¼ 1  1 P2 , ; P1 γ0 þ 1 ρ0 : P1 P ðγ0 þ 1Þ þ ðγ0  1Þ

(6.161)

1

which is solved numerically. Knowing P2 =P1 determines ρ2 =ρ1 and thus u2 . The remaining quantities are easily found. The transmitted shock has a speed ST ¼

γ4 þ 1 u2 : 2

(6.162)

6.5 Shocks

291

Figure 6.12. Two shocks allow a higher final state to be reached.

Let us examine (6.161) a bit further. If ρ4 =ρ0 ¼ 1, then (1) if γ4 ¼ γ0 , then P2 ¼ P1 , as it should (this serves as a useful check); (2) if γ4 > γ0 , then P2 > P1 ; and (3) if γ4 < γ0 , then (6.161) has a nonphysical root for 0 < P2 =P1 < 1. In this case, a rarefaction occurs in the left-hand material, preceded by a shock in the right-hand material. A more proper solution to the problem can be found by noting that the Mach numbers at each shock front are accurately defined by M 12 ¼ u1 0 =cs1 and M 34 ¼ ST =cs4 , respectively. Then, using the jump relations (6.127) and (6.128) for both shocks, we have ρ2 M 212 ðγ0  1Þ þ 2 P2 ðγ0 þ 1Þ ¼ ¼ and , 2 ρ1 P1 2γ0 M 212  ðγ0  1Þ M 12 ðγ0 þ 1Þ

(6.163)

ρ3 M 2 ð γ þ 1Þ P3 2γ4 M 234  ðγ4  1Þ and ¼ 2 34 4 ¼ : ρ4 M 34 ðγ4  1Þ þ 2 P4 ðγ4 þ 1Þ

(6.164)

and

These four equations plus the relations among the velocities can be solved with some effort and yield a messy set of equations; the reader is referred to Drake (2006).

292

Hydrodyamics

6.5.6 Multiple Shock Reflections Having discussed the phenomenon of a shock reflecting from a rigid wall, we now turn to the problem of continued shock reflections. We use the geometry presented in Section 6.5.2, but with the tube terminated at the right end by a rigid wall. In the absence of damping mechanisms, the shock will first reflect from the fixed boundary and travel back to the piston, where it will reflect again. The density and pressure of the compressing gas will continue to increase with each reflection. Let the velocity of the first reflected shock be labeled us1 , and the gas velocity downstream from this new shock is that of the rigid wall, which is zero. The shock us1 then returns to the piston, and a new shock us2 is formed, whose direction is reversed back toward the rigid wall. The process continues until the piston “meets” the wall. We wish to find the values of the pressure and density behind the nth shock. For odd n, the shock is traveling from the piston toward the fixed boundary, and for even n the shock is traveling from the fixed boundary toward the piston. Then usn ¼ ðU n  U n1 Þ

ρn , ρn  ρn1

(6.165)

and Pn ¼ Pn1 þ ρn1 usn ðU n  U n1 Þ:

(6.166)

Equations (6.165) and (6.166) are generalized versions of (6.105) and (6.106), respectively. The quantity U n  U n1 alternates from positive the piston velocity to negative the piston velocity U. For a shock process, Pn > Pn1 , so the shock velocity changes sign upon each reflection. Eliminating usn from (6.165) and (6.166), we obtain ρn1 ρn1 ¼1 U2, ρn Pn  Pn1

(6.167)

and so 1 Pn ¼ Pnþ1 þ ðγ þ 1Þρn1 U 2 þ ρn1 U 4

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 ðγ þ 1Þ2 U 2 þ c2sn1 , 16

(6.168)

where c2sn1 ¼ γPn1 =ρn1 . The solution is complete, since all quantities with index n can be obtained for those with index n  1. Expression (6.168) is essentially (6.140). In principle, we might hope to attain arbitrarily high pressures and densities. In reality, this will not happen for a variety of reasons, of which viscosity and thermal conductivity are most important. More on the effects of viscosity and thermal conductivity will be discussed later in this chapter.

6.5 Shocks

293

6.5.7 Shocks Moving from Heavy into Light Media Let us now consider the case of a dense gas expanding into a less dense (lighter) gas using the method of characteristics. The initial configuration is shown in Figure 6.13a, assuming two distinct materials. We identify the quantities to be nearly the same as in Section 6.5.4. Initially, the shock front is moving with velocity us through the undisturbed material with subscripts 0. At the moment of shock contact with the interface, the quiescent upstream material (region 4) develops a shock front that moves with velocity u3 ¼ ui þ u1 , ui being the interface velocity. The compressed material from region 4 now forms region 3, and the transmitted shock front is moving with velocity usT ¼ ðγ4 þ 1Þðui þ u1 Þ=2. At the same time, a rarefaction forms region 2, and the inflection point moves “backward” into region 1. Since the fluid in region 1 is moving with velocity u1 , the inflection point actually moves to the right with velocity u1  cs1 . The configuration now is shown in Figure 6.13b.

(a) r

Interface

A

B

r1 P1 u1 r0 P0 u0 = 0

r 4 P4 u4 = 0

x (b) r

A

Interface

B

r1 P1 u1 r2 u2

r3 u3

r 4 P4 u4 = 0 x

Figure 6.13. (a) Density profile of a shock tube with the driving fluid having a density greater than that of the material being impacted. (b) A rarefaction begins when the shock front impacts the interface, with a shock driven into the lowdensity medium.

294

Hydrodyamics

The response of the material in regions 1 and 2 is quite similar to the adiabatic expansion discussed in Section 6.3.2, and we draw upon that material. For a simple wave, the Riemann invariant J þ ¼ ω þ u is constant. Following this characteristic back into the left-hand medium at time zero, we find ω ¼ 0, but u ¼ u1 or u ¼ u1  ω, or if we specifically solve for the matter velocity in the rarefaction, u2 ¼ u1  ω. Using (6.59) together with P ¼ P1 ðρ=ρ1 Þγ0 gives "  γ02γ1 # 0 2 P , (6.169) u2  u1 ¼ ω ¼ cs1 1  γ0  1 P1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where the sound speed in region 1 is cs1 ¼ ðγ0 P1 =ρ1 Þ. There are in effect two shock fronts in this problem: the driving one that propagates onward into the second material and a reverse shock propagating backward. From (6.151), for a strong shock  u1 ¼

2 P1 γ0 þ 1 ρ0

1=2

 and u3 ¼

2 P3 γ 4 þ 1 ρ4

1=2 :

(6.170)

Forming u2 =u1 from (6.169) and forming u3 =u1 and equating these at the interface, subject to the continuity requirement u2 ¼ u3 and P2 ¼ P3 , we arrive at the transcendental equation for y 

2γ0 1þ γ0  1

1=2 h  1= γ0 1 i γ0 þ 1 2 1=2 1=2 2γ0 1y x y , ¼ γ4 þ 1

(6.171)

where xp ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ρ0 =ρ4 andffi y ¼ P3 =P1 , and we have used the fact that cs1 =u1 ¼ γ0 ðγ0  1Þ=2. Expression (6.171) contains a wealth of information. However, not all solutions have a physical reality due to the fact that the density ratio, pressure ratio, and polytropic indices play against one another. We examine a few of these dependencies and the limitations that must be imposed. For γ0 ¼ 2, (6.171) has the solution

1=

y4¼

  1=2 1=2 1  1 þ 3 γ 3xþ1 4



1=2

:

(6.172)

3x γ4 þ1

Now 0 < y < 1, therefore the positive sign ahead of the square brackets is required. This is plotted in Figure 6.14 for γ4 ¼ 2. Also shown in the figure are solutions to (6.171) for γ4 ¼ 4=3 and 5=3, both with γ0 ¼ 5=3. We see that the stiffest material (greatest γ), and the smallest density mismatch gives rise to the

6.5 Shocks

295

Figure 6.14. Pressure ratio for allowed values of the density mismatch. The solid line is for γ0 ¼ γ4 ¼ 2, while the long-dashed line is for γ4 ¼ 4=3, and the shortdashed line is for γ4 ¼ 5=3. Values for ρ0 =ρ4 less than those shown produce nonrealistic solutions.

greatest shock pressure ratio, or allows the shock to progress with minimum pressure attenuation. The dependency of y on γ is, however, much smaller than its dependence on x, the density ratio. Reflection from the interface occurs if x > 1; we refer the reader to the discussion in Section 6.5.5. Reflection can also occur if x > 1 when P3 =P1 > 1, in which case the interface is moving back into the driving material. In this case, the limiting value of u1 at the transition from a rarefaction to a reflected shock is found from P1 ¼ ðγ4  1Þρ4 u21 =2. The pressure in region 1 is given by P1 ¼ ðγ0  1Þρ1 u21 =2, and thus the transition occurs when ρ4 =ρ0 ¼ ðγ0 þ 1Þ=ðγ4 þ 1Þ. If γ4 > γ1 , a reflection will occur even though there is still a density drop.

6.5.8 Sedov-Taylor Blast Wave So far in this chapter, we have discussed several types of compressive and expansive flows. In particular, the isothermal and adiabatic rarefactions employed the method of characteristics based on Riemann invariants (Sedov, 1959). These solutions to the conservation equations are self-similar with similarity variable x=ðcs0 t Þ. A defining characteristic of self-similar motion is that the solutions have

296

Hydrodyamics

shapes in space that are independent of time, with a spatial scale Rðt Þ, that is a function of time. A most useful parameter is the ratio ξ ¼ r=R. The conservation equations (6.13), (6.21), and (6.29) (without body forces) for one-dimensional flow may be written ∂ρ ∂ρ ∂u u þ u þ ρ þ νρ ¼ 0, ∂t ∂r ∂r r

(6.173)

∂u ∂u 1 ∂P þu þ ¼ 0, ∂t ∂r ρ ∂r

(6.174)

  ∂P ∂P ∂ρ 2 ∂ρ þ u  cs þu ¼ 0, ∂t ∂r ∂t ∂r

(6.175)

and

where ν ¼ 0, 1, or 2 for planar, cylindrical, or spherical geometry. In planar geometry, r is the linear dimension, and for cylindrical or spherical geometry, r is the radius. A polytropic EOS is needed to relate the sound speed to density and pressure. Equation (6.175) becomes     ∂ P ∂ P log γ þ u log γ ¼ 0: (6.176) ∂t ρ ∂r ρ The task is to transform these conservation equations into ordinary differential equations involving dimensionless functions of ξ that represent the shape of the fluid variables. Equations (6.173) through (6.175) have variables u, ρ, and P that need to be expressed as the product of the dimensionless function ξ and other necessary parameters. Hence _ ðξ Þ, u ¼ RU

ρ ¼ ρ0 ðr; tÞΩðξ Þ, and P ¼ ρ0 ðr; t ÞR_ 2 Πðξ Þ,

(6.177)

where U, Ω, and Π are new dimensionless functions of the similarity variable ξ. Quantities with the overdot represent a time derivative, and the initial density ρ0 is, in general, a function of both space and time. The scales R, ρ0 , and R_ are time dependent in some manner yet to be determined. Using ξ ¼ r=R , we find that to transform the derivatives we need, for example, ∂ρ dρ0 r R_ ¼ Ω  ρ0 Ω0 2 R_ ¼ ρ0̇ Ω  ρ0 Ω0 ξ , ∂t dt R R

(6.178)

∂ρ ρ0 0 ¼ Ω: ∂r R

(6.179)

and

6.5 Shocks

297

The primes indicate differentiation with respect to the similarity variable. Using (6.177) through (6.179) in (6.173), we obtain   ρ0̇ R_ Ω0 U 0 þ ¼ 0, (6.180) U þ ðU  ξ Þ þ ν ρ0 R ξ Ω and similarly for (6.174) RR€ Π0 0 U þ ð U  ξ ÞU þ ¼ 0: Ω R_ 2 Finally, for (6.176), with some effort, we arrive at  2    Rd d Π R_ log log þ ðU  ξ Þ ¼ 0: γ1 dξ Ωγ ρ0 R_ dt

(6.181)

(6.182)

We assume that the velocity, density, and pressure have no remaining dependence on time in the transformation from variables ðr; t Þ to ðξ; t 0 Þ. This imposes a restriction on the number of constraints permitted by any boundary or initial conditions. There are now three ordinary differential equations with explicit dependencies on time and space. These dependencies must cancel out if the evolution is to be self-similar. That is, we must separate the time and similarity variable in (6.180) € R_ 2 to be constant, and then use R ¼ At α , where through (6.182). In (6.181), set RR= _ A and α are constants. Next, in (6.180) set ρ0̇ =ρ0 ¼ const R=R, which leads to β ρ0 ¼ Bt , where B and β are also constants. The first term in (6.182) is therefore constant, and all the scales in the self-similar motion have a power-law dependence on time. Hence, ξ¼

r r ¼ : Rðt Þ At α

Expression (6.180) can be written in terms of the constants α and β   Ω0 U 0 β þ α ðU  ξ Þ þ ν þ U ¼ 0, ξ Ω

(6.183)

(6.184)

and (6.182) becomes  0  2ðα  1Þ β Π Ω0  ð γ  1Þ þ ð U  ξ Þ γ ¼ 0: α α Π Ω

(6.185)

Note that the polytropic index γ appears explicitly in this last equation as a parameter. Thus, the solution is not independent of the EOS. Self-similar methods are powerful in many instances; Zel’dovich and Raizer (1966) present numerous interesting examples. Because of space limitations, we

298

Hydrodyamics

cannot explore the breadth of this work. We demonstrate the power of this technique as applied to point explosions. Our discussion concerning the nature of a shock front, such as that created by a gas gun, assumed that the downstream driving pressure P1 is constant in time. If we wait long enough until the shock front is far removed from the piston, the pressure of the shocked material will eventually decrease and the shock will disappear due to viscous and/or thermal effects. A shock can decay also by a rarefaction wave propagating toward the shock front from the downstream side. This will happen if the driving piston is suddenly brought to a halt. At early time, the shock structure has an essentially steady-state profile. At times late compared to the removal of the driving force, a rarefaction wave develops and propagates toward the shock front, soon overtaking it. This is referred to as a planar blast wave. Spherical blast waves are rather common in nature, particularly in astrophysics settings such as stellar explosions. Supernova remnants develop a blast wave structure that persists for much of their evolution. Cylindrical blast waves are formed during a lightning strike where the energy is deposited rapidly in the ionized gas channel, and the outward-moving shock front cannot be maintained indefinitely, since the source of energy is finite. A blast wave represents a store of energy that allows more and more mass to be swept up, but at a continued reduction of velocity. Consider an explosion from a “point” in a uniform-density gas; an amount of energy E0 is instantaneously deposited in a minute amount of gas. This thermal energy is transformed to kinetic energy as the outward-going shock sweeps up mass. At any time, the mass within the spherical blast wave is M ðt Þ ¼ 4πρR3 =3. Then by conservation of energy dM _ 2 € (6.186) R ¼ 2M R_ R, dt _ and thus RR= € R_ 2 ¼ 3=2. where dM=dt ¼ 4πρR2 R, We seek a solution to the point-explosion problem, assuming the blast wave is sufficiently far from the energy source, yet sufficiently early that the shock front has not moved too far away from the source. That is, its strength is still large enough that it is possible to neglect the initial gas pressure. The only information we have are the initial gas density ρ0 and deposited energy E 0 . At some time t, the three quantities ρ0 , E 0 , and t can be used to construct a quantity with the dimensions of the radius 

E0 2 Rðt Þ ¼ A t ρ0

1=5 :

(6.187)

6.5 Shocks

299

From (6.183) (and the discussion preceding), we see the constant α ¼ 2=5. Further, for uniform initial density β ¼ 0. From (6.177) the velocity, density, and pressure are  2 2R 2R U ðξ Þ, ρ ¼ ρ0 Ω and P ¼ u¼ ρ0 Π: (6.188) 5t 5t Using (6.187) and (6.188), the equations (6.184), (6.181), and (6.185) for spherical geometry become   2 Ω0 U 0 (6.189) ðU  ξ Þ þ 2 þ U ¼ 0, 5 ξ Ω 3 0  UΩ þ ðU  ξ ÞΩU 0 þ Π ¼ 0, 2

(6.190)

 0  Π Ω0 3 þ ðU  ξ Þ γ ¼ 0: Π Ω

(6.191)

and

The boundary conditions are determined at the shock front where ξ ¼ 1. Thus, U ð1Þ ¼

2 γþ1 2 , Ωð1Þ ¼ and Πð1Þ ¼ : γþ1 γ1 γþ1

(6.192)

There remains the task of determining the constant A of the scaled radius in (6.187). Since the energy E 0 was deposited “instantaneously,” conservation of energy allows A to be calculated  ðR  u2 3 E0 ¼ ρ ε þ d r, 2

(6.193)

0

from which, using ρε ¼ P=ðγ  1Þ, and thus 5

A

16π ¼ 25

 ð1  1 Π 2 ΩU þ ξ 2 dξ: 2 γ1

(6.194)

0

Equations (6.189) through (6.194) are evaluated numerically. Distributions of the velocity, density, and pressure for a γ ¼ 5=3 gas are shown in Figure 6.15. There is the interesting feature of the velocity being almost linear in r, which is the ballistic relation v ¼ r=t. The pressure drops steeply behind the shock, but only by a factor of about 2.5, and is nearly constant over most of the sphere. The density has a very large dynamic range. The inner portion of the sphere is nearly void of

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Hydrodyamics

Figure 6.15. Dimensionless structure of a spherical blast wave for a gas with γ ¼ 5=3. The scaled velocity is the solid line, the scaled pressure is the shortdashed line, and the scaled density is the long-dashed line. The abscissa is scaled to the position of the shock front ξ 0 , which has the value 1:152.

material, with that material being moved to a thin shell just behind the shock front. The entropy (and thus the temperature) is much larger near the center than behind the shock. That is because each element of the material remembers how strong the shock was when the shock passed through it. The shock is stronger early in the propagation and thus the entropy is larger there.

6.6 Viscous and Heat Conducting Fluids Up to this point, we have assumed that the individual particles comprising the elemental volume all move at the same speed and direction as the volume element itself. In reality, because there is a distribution of velocities among the individual particles, some have velocities slower than the mean fluid velocity and some have higher velocities. In addition, the particles undergo collisions that alter their individual velocities, and we have no reason to believe the local velocity distribution is isotropic. This nonuniformity in phase space results in fluctuations related to gradients in the velocity and temperature. For example, for the velocity component of the fluid element uy that is increasing in the x direction, a particle that arrives in the fluid element traveling in the þx direction will have a uy value that is

6.6 Viscous and Heat-Conducting Fluids

301

negatively biased compared with the mean value within the volume element. The amount of bias is of the order of the mean free path times the x-gradient of uy . Alternatively, if the temperature is increasing in the x direction, then a particle that arrives in the element traveling toward the þx direction will have a small negative bias in its kinetic energy. These biases are determined either by laboratory measurement or found by a kinetic model (see Section 3.1 and Chapter 7). Thus, “corrections” are to be applied to the pressure and the heat flux in the momentum and internal energy equations. These processes create an additional, nonhydrodynamic transfer of momentum and energy that result in nonadiabatic flow and the irreversible transformation of mechanical energy into thermal energy. These effects determine the dissipation of thermal energy in an acoustic wave or a shock front and their subsequent damping. Since these nonadiabatic processes occur over a few mean free paths, we can determine the thickness of the shock front. Viscosity results from generalizing the isotropic pressure to a stress tensor T, which is a symmetric tensor. We can determine the properties of T using the physical consideration that the internal frictional forces exist only when one element of the fluid is moving relative to another, and that the stress tensor relaxes to its hydrostatic form when the fluid is at rest or moving with a uniform velocity. The viscous terms must depend upon the gradients of the velocity field. The total stress tensor is T ¼ PI þ σ,

(6.195)

where I is the unit tensor and σ is the viscous stress tensor, also known as the deviatoric stress tensor. The usual notation is followed here where stress is counted positive in tension and negative in compression, which is just the opposite for pressure. For small velocity gradients, the viscous forces depend only linearly on the spatial derivatives of the velocity; that is, the fluid is a Newtonian fluid. (A Stokesian fluid has the viscous stress tensor depending quadratically on the velocity gradients.) We expect viscous forces to be zero within a fluid element that is undergoing rigid rotation, and thus there are no contributions to σ from the vorticity (this is also necessary from the requirement that the stress tensor be symmetric). The existence of a velocity gradient is a statement that the fluid is undergoing a deformation. Deformation is the result of the separation of adjacent points in the fluid changing with time. Consider a fluid element in Cartesian coordinates moving 0 with velocity components ui , and an adjacent fluid  element moving with ui . The 0 connection between them is ui ¼ ui þ ∂ui =∂xj dxj , which can be decomposed into symmetric and antisymmetric tensors     1 ∂ui ∂uj 1 ∂ui ∂uj 0 ui ¼ ui þ þ  (6.196) dxj þ dxj : 2 ∂xj ∂xi 2 ∂xj ∂xi

302

Hydrodyamics

The second term on the right-hand side defines the strain-rate tensor, which describes how the fluid element is deformed by the flow. The diagonal elements represent the rates of stretching or contraction along the principal axes, while the off-diagonal elements are one-half the rate of shear deformation. The last term on the right-hand side is the antisymmetric part and is related to the vorticity; we require this to be zero, as we shall show momentarily. The strain-rate tensor can be written as     1 ∂ui ∂uj 1 ∂ui ∂uj 2 1 ¼ ϵ_ ij ¼ þ þ  ruδij þ ruδij : (6.197) 2 ∂xj ∂xi 2 ∂xj ∂xi 3 3 Any deformation can be represented as the sum of a pure shear and a hydrostatic compression. The first term on the right-hand side of (6.197) is pure shear, since the sum of the diagonal terms is zero (δii ¼ 3), while the last term is the hydrostatic compression. Returning now to the stress tensor, the most general symmetrical tensor of rank two is   ∂ui ∂uj 2 σ ij ¼ 2μϵ_ ij þ λðruÞδij ¼ μ þ  ruδij þ ζ ruδij , (6.198) ∂xj ∂xi 3 where the coefficient of bulk viscosity is defined as ζ ¼ λ þ 2μ=3. The term in parentheses on the right-hand side has the mathematical property of being traceless, that is, it sums to zero when we contract on i and j. It also has the property of vanishing identically for a fluid that dilates symmetrically, that is, such that  ð∂u1 =∂x1 Þ ¼ ð∂u2 =∂x2 Þ ¼ ð∂u3 =∂x3 Þ and ∂ui =∂xj ¼ 0 for i 6¼ j. If we assume that the fluid is isotropic, then λ and μ must be scalars. The symbols μ and λ are the coefficients of shear viscosity and dilatation viscosity (the second viscosity coefficient), respectively.2 Hence, the viscous stress tensor and strain-rate tensor are related by     ∂ui ∂uj 2 σ ij ¼ μ þ (6.199) þ ζ  μ ðruÞδij : 3 ∂xj ∂xi The coefficient of bulk viscosity for a perfect monatomic gas is predicted to be zero on the basis of kinetic theory. Experimental measurements of ζ generally 2

Some authors write the viscous stress tensor in terms of a strain tensor rather than the strain-rate tensor. The strain tensor is defined as ϵ ij ¼

  1 ∂yi ∂yj þ , 2 ∂xj ∂xi

where y is the displacement vector. Many authors use u for the displacement vector, but we use y to avoid confusion with the velocity vector u. The viscosity coefficients are replaced by moduli.

6.6 Viscous and Heat-Conducting Fluids

303

gives a zero value because its effect vanishes for incompressible flow, that is, when ru ¼ 0. Also, incompressible flows generally are high-velocity flows for which the Reynolds number is large and thus viscous effects are unimportant.3 Using (6.195) and (6.196), we see that the mean of the principal stresses is T ii =3 ¼ P þ ζ ðruÞ. For incompressible flow, the hydrostatic pressure is P ¼ T ii =3, and for a compressible fluid P is the thermodynamic pressure given by an EOS. The Cauchy equation of motion (6.21) is extremely general because it makes no assumptions about the form of the stress tensor. For a Newtonian fluid, the NavierStokes equation for the conservation of momentum is   ∂ 1 2 ðρuÞ þ rðρuuÞ ¼ f  rP þ μr u þ ζ þ μ rðruÞ: (6.200) ∂t 3 Combining the last two terms of this equation, for one-dimensional flow, allows us to define a fictitious viscous pressure   4 3 ∂ux μþ ζ Q¼ : (6.201) 3 4 ∂x Clearly there are no viscous effects in the equation of continuity (6.13), since the fluid velocity is defined as the mean mass flux divided by the density. The effect of viscosity on the total energy equation (6.26) comes from the rate of doing work by viscous forces being subtracted from the internal energy equation (6.29). The term PðruÞ in (6.29) is modified by subtracting a dissipation function defined by     1 ∂ui ∂uj 2 2 Φ¼ μ þ þ ζ  μ ðruÞ2 , (6.202) 2 ∂xj ∂xi 3 which is always nonnegative. The internal energy equation becomes ∂ ðρεÞ þ rðρuεÞ þ PðruÞ ¼ Φ: ∂t

(6.203)

An additional term for heat conduction comes from Fourier’s law q ¼ κrT and is to be added to (6.203). The thermal conductivity κ is determined by experiment or theoretical considerations; this will be addressed in more detail in Chapter 7. The magnitude of the thermal conductivity is approximately the mass density times the specific heat at constant volume times the atomic mean free path. 3

The Reynolds number is the ratio of the inertial forces to the viscous forces. It is expressed by Re ¼ uL=ν, where L is a characteristic linear dimension, and ν is the kinematic viscosity. Low Reynolds numbers are characteristic of laminar flow, where viscous forces are dominant; turbulent flow occurs at high Reynolds numbers and are dominated by inertial forces, which tend to produce chaotic eddies, vortices, and other flow instabilities.

304

Hydrodyamics

Thus, for gases, the thermal conductivity and viscosity are closely related. The effect of heat conduction on the energy equation is to add a volume source term rq. Then (6.203) becomes ∂ ðρεÞ þ r  ðρuεÞ þ Pðr  uÞ ¼ Φ þ r  ðκrTÞ: ∂t

(6.204)

We might ask about the effects of surface tension. This is not necessary, in general, for high-energy-density material, because the molecular forces creating the surface tension are absent; that is, no molecular structure is present.

6.6.1 Damping of an Acoustic Wave In our previous discussion about acoustic waves, we assumed the waves propagate adiabatically. We now address, in an elementary fashion, the behavior of the wave when dissipative processes are present. For planar geometry, the linearized continuity equation (6.65) is unaffected, while the linearized momentum equation (6.66) becomes   2 ∂u ∂ 4 ∂u ρ0 ¼  ðδPÞ þ , (6.205) μþζ ∂t ∂x 3 ∂x2 and the linearized energy equation is ρ0

∂ ∂u ∂2 ðδεÞ ¼ P0 þ κ 2 ðδT Þ: ∂t ∂x ∂x

(6.206)

No viscous term appears in (6.206) because the dissipation function ð4μ=3 þ ζ Þð∂u=∂xÞ2 is a second-order quantity. For a perfect gas, P ¼ ρRT ¼ ðγ  1Þρε, so the temperature and specific energy perturbations are δT ¼

ðδPÞ  Pρ 0 ðδρÞ 0

(6.207)

Rρ0

and δε ¼

ðδPÞ  Pρ 0 ðδρÞ 0

ðγ  1Þρ0

:

(6.208)

Using (6.207) and (6.208) and eliminating ∂u=∂x via the linearized continuity equation, (6.206) is rewritten as  γ  1 ∂2  ∂ κ 2 ðδPÞ  c2s ðδρÞ , ðδPÞ  c2s ðδρÞ ¼ ∂t Rρ0 ∂x

(6.209)

6.6 Viscous and Heat-Conducting Fluids

305

which is of course a wave equation. Taking plane wave solutions δP ¼ PeiðωtkxÞ , δρ ¼ ReiðωtkxÞ and δϕ ¼ ΦeiðωtkxÞ ,

(6.210)

where δϕ is the perturbed velocity potential. Substituting the equations (6.210) into the continuity equation, (6.205) and (6.206) give iωR  ρ0 k 2 Φ ¼ 0,     4 ikP þ ρ0 kω  i μ þ ζ k3 Φ ¼ 0, 3

(6.211) (6.212)

and  iω þ

   γ1 2 γ1 2 κk P  c2s iω þ κk R ¼ 0: Rρ0 γRρ0

(6.213)

A nontrivial solution to (6.211) through (6.213) exists only if the determinate of the coefficients vanishes. This yields the dispersion relation 0 1 4 2 γ1 k 2 1  i κ μ þ ζ k ω γRρ ω 0 Aþi 3 ω2 ¼ c2s k2 @ : (6.214) γ1 k2 ρ0 1i κ Rρ0

ω

If both 4μ=3 þ ζ and ðγ  1Þ=γRρ0 are small, we can set k ¼ k0 þ δk ¼ ω=cs þ δk, and expand to first order in small quantities to arrive at " #  ω2 4 ðγ  1Þ2 μþζ þ κ : (6.215) δk ¼ i 3 3 2cs ρ0 γR Then the pressure perturbation is δP ¼ Peiðωtk0 xÞ ex=L ,

(6.216)

where L ¼ i=δk. Similar equations can be found for δρ, δT, and δϕ. Equation (6.216) shows that the wave propagates with the sound speed cs , but its amplitude decreases with a characteristic length L. Thus, small amounts of viscosity and/or conductivity damp acoustic waves by the irreversible process of converting wave energy into entropy. Equation (6.215) shows that high-frequency waves are attenuated more than low-frequency waves. This is because, for a given amplitude, the high frequencies will have steeper velocity and temperature gradients over a smaller physical distance, a wavelength. This analysis can be extended to the more realistic problem that, depending on specific conditions, can result in a propagating thermal wave; a discussion on this topic may be found in Mihalas and Mihalas (1984).

306

Hydrodyamics

6.6.2 Structure of the Shock Front Our development of the behavior of the propagation of a shock acknowledged that a dissipative mechanism is taking place within the shock front. This irreversible process is demonstrated by the jump of entropy across the front (see (6.133)). We now investigate how the dissipative processes of viscosity and thermal conduction determine the structure and thickness of a shock front. We expect these processes to play a role within the front because the gradients of velocity and temperature are very steep there. Working in the frame of reference moving with the shock front, as shown in Figure 16.6, the conservation relations of mass flux and momentum flux, as given by (6.108) and (6.109), are now ρu ¼ ρ0 u0 ,

(6.217)

and, using one component from the stress tensor (6.199),   4 du 2 μþζ ¼ ρ0 u20 þ P0 : ρu þ P  3 dx

(6.218)

The energy flux, from (6.110) and using (6.204), becomes       1 2 4 du dT 1 2 ρu h þ u  μþζ u κ ¼ ρ0 u0 h0 þ u0 : 2 3 dx dx 2

(6.219)

u u0

u1 x d

Figure 6.16. Shock structure in a reference frame moving with the shock front. The “discontinuity” occurs over a width δ.

6.6 Viscous and Heat-Conducting Fluids

307

Quantities with a 0 subscript refer to the upstream material, where far enough from the shock front the viscosity and thermal conductivity are not operative. We also have the entropy generation equation from (6.204) (in the shock’s frame) together with the first law of thermodynamics   2   ds 4 du d dT μþζ κ ρuT ¼ þ , (6.220) dx 3 dx dx dx where s is the specific entropy. We will investigate the effects of the two dissipative processes individually. An important case is that of a perfect gas with zero thermal conductivity. The energy equation (6.219) is       1 γ P 4 du 1 γ P0 : (6.221) ρu u2 þ  μ þ ζ u ¼ ρ0 u0 u20 þ 2 γ1ρ 3 dx 2 γ  1 ρ0 Multiplying (6.218) by uγ=ðγ  1Þ and subtracting (6.221) yields     u 4 du 1 1 2 2  μþζ ¼ cs0 ðu  u0 Þ þ u0 γu  ðγ  1Þu0  ðγ þ 1Þu0 u2 : ρ 3 dx 2 2 (6.222) (The quantity ν ¼ ð4μ=3 þ ζ Þ=ρ is known as the kinematic viscosity.) Let the velocity change across the shock front be w ¼ u0  u, then (6.222) can be written     1 4 dw 1 w 2 2 μþζ ¼ u0  cs0  ðγ þ 1Þu0 w : (6.223) ρ 3 dx 2 u0  w Far downstream, the velocity drop has a maximum  2 u20  c2s0 wmax ¼ u0  u1 ¼ , u0 γþ1

(6.224)

which follows from the Prandtl relation (see Section 6.5.3), while far upstream w ¼ 0. Equations (6.223) and (6.224) show that ðdw=dxÞ≥0 with ðdw=dxÞ ¼ 0 at w ¼ 0 and w ¼ wmax . Thus, wðxÞ increases monotonically, while uðxÞ decreases monotonically going from downstream to upstream. In addition, differentiating (6.223) with respect to x gives   2    1 4 d w 1 u0 2 2 ¼ u0  cs0  ðγ þ 1Þw u0  w , (6.225) μþζ ρ 3 dx2 2 ðu0  wÞ2   shows that d 2 w=dx2 > 0 at w ¼ 0, d 2 w=dx2 < 0 at w ¼ wmax , and which pffiffiffiffiffiffiffiffiffi d 2 w=dx2 ¼ 0 at w ¼ u0  u0 u1 .

308

Hydrodyamics

Using (6.223) in (6.218) gives   1 w 2 : P ¼ P0 þ ρ0 cs0 þ ðγ  1Þu0 w 2 u0  w

(6.226)

Combining this equation with the continuity equation yields the temperature change across the shock front. Equation (6.226) shows that PðxÞ, and therefore T ðxÞ increases monotonically, as does the density. Further, (6.220) also shows the entropy sðxÞ increases monotonically through the transition layer. As sketched in Figure 16.6, we can estimate the width of the shock from δ ¼ w=ðdw=dxÞ, evaluated at the point where w ¼ wmax =2, the center of the shock front. From (6.224) and (6.223) we obtain  2 2 u0  c2s0 dw 1 ρ  , (6.227) ¼  dx 2 43 μ þ ζ γu20 þ c2s0 which results in  2   γu0 þ c2s0 2 4  : δ¼ μþζ ρ 3 ðγ þ 1Þu0 u20  c2s0

(6.228)

There is a close connection between viscosity and the collisional mean free path. A fluid with a large viscosity resists motion because its molecular makeup gives a lot of internal friction. The average speed of the particles increases with temperature, so the amount of time they spend “in contact” with their nearest neighbors decreases. If the particles are moving with roughly the sound speed, the viscosity is proportional to the product of that speed and the mean free path λ. For a weak shock u0  cs0 , then  2 43 μ þ ζ 4γ P0 e δ¼ 2 2 λ: (6.229) ρ cs0 M 0  1 γ þ 1 P1  P0 Thus, the shock thickness is of the order of a particle mean free path adjusted for the pressure jump across the shock front. For a strong shock (M 0 1), the shock thickness, according to (6.228), is δe2

γ λ , γ þ 1 M0

(6.230)

which suggests that δ  λ for M 0 1. This is incompatible with one of the basic assumptions for a fluid description. If we assume that the viscous dissipation occurs mainly in the hot material at the back edge of the transition layer, and that

6.6 Viscous and Heat-Conducting Fluids

309

ð4μ=3 þ ζ Þ=ρ e uλ e M 0 cs0 λ, then the thickness of the layer remains of the order of the mean free path. A second important case is that of heat conduction being present with the absence of viscosity. We mentioned in Section 6.5.3 that the thermodynamic path between points “0” and “1” of Figure 6.7 is a straight line (the Rayleigh line). We also showed the isentrope beginning at point “0.” Other isentropes are possible, depending on the strength of the shock. One of these isentropes will be tangent to the Rayleigh line between the initial and final points of the shock, and represents the maximum entropy. At the point of tangency, the particle velocity will exactly equal the local sound speed; at point “0” u0 > cs0 , and at point “1” u1 < cs1 . Equating the slope of the Rayleigh line (from (6.121))     ∂P 1 ∂2 P P  P0  ðV  V 0 Þ þ ðV 1  V 0 ÞðV  V 0 Þ, (6.231) ∂V s0 2 ∂V 2 s0 (V ¼ ρ1 is the specific volume) to that of the isentrope       ∂P 1 ∂2 P ∂P 2 ðV  V 0 Þ þ ðV  V 0 Þ þ ðs  s0 Þ, P  P0  2 ∂V s0 2 ∂V s0 ∂s V0 (6.232) shows that the tangency point is exactly halfway between points “0” and “1,” that is, V  V 0 ¼ ðV 1  V 0 Þ=2. We then find an expression for the pressure at that point, and then the entropy 2 ∂ P  2 1 ∂V 2 1 ΔV : (6.233) s  s0 ¼ ∂P s0 ðV 1  V 0 Þ2 , ! γðγ þ 1ÞcV 8 ∂S V0 8 V For a perfect gas the maximum entropy change within the shock front is a secondorder quantity with respect to the density jump. In contrast, the total entropy change in going from point “0” to point “1” is a third-order quantity. The presence of an entropy maximum within the front indicates that the temperature profile has an inflection point (a maximum) at the point of maximum entropy. Assuming a constant thermal conductivity, the entropy generation equation (6.220) is ρuTds=dx ¼ κd2 T=dx2 . The existence of the entropy maximum is attributable to the fact that heat transfer takes place from a high-temperature region to a low-temperature region. Therefore, the gas flowing into the shock is first heated by conduction and is then cooled, with the final state having entropy greater than its initial upstream value.

310

Hydrodyamics

The thickness of the shock front, considering only thermal conductivity, can be evaluated from the entropy generation equation 0 1 þ∞ ðT ð 2 1d T dT 1 B 1 dT C þ ρ0 u0 ðs  s0 Þ ¼ κ dx ¼ κ@ dT A: (6.234) 2 T dx T dx dx T 2 ∞

T0

Applying this expression to the final state “1,” where dT=dx ¼ 0, eliminates the first term within the parenthesis. The effective thickness of the shock front δ is defined by ðT 1  T 0 Þ=δ ¼ jdT=dxjmax , which gives ρ0 u0 ðs1  s0 Þe κ

1 ðT 1  T 0 Þ2 : δ T 20

(6.235)

Expressing the temperature jump in terms of the pressure jump gives the result   ∂T V0 ðP1  P0 Þ ¼ ðP1  P0 Þ: (6.236) T1  T0 ¼ ∂P s cP  Using (6.233) for the entropy jump, and approximately ∂2 V=∂P2 s e V 0 =P20 , κ e ρ0 cP λcs0 , with u0  cs0 , we have the shock front thickness P0 λ: (6.237) δ P1  P0 When compared to the estimate of the shock thickness for the viscous case (6.229), we see they are the same to within a constant of order unity; a more accurate calculation using (6.133) with (6.129) gives a numerical factor of 16ðγ  1Þ=3ðγ þ 1Þ. For moderate strength shocks, the kinematic viscosity ν and the thermal diffusivity χ ¼ κ=ρcP are similar in size, since they are determined by the same mean free path. The situation for strong shocks is quite different. Only the temperature varies continuously through the dissipation zone, while the other variables have discontinuous jumps. Using (6.128) and (6.129) for a perfect gas, the temperature ratio is     T ρ0 ρ0 2 ¼ γM 0 1  þ1 : (6.238) T0 ρ ρ This expression has a maximum at   ρ0 γM 20 þ 1 ¼ : ρ max 2γM 20

(6.239)

311

6.6 Viscous and Heat-Conducting Fluids

For large Mach number ðρ0 =ρÞmax ! 1=2, and ρ0 =ρ1 1=4. Combining (6.239) and (6.128) gives the critical Mach number of the transition from continuous to discontinuous variables  2 3γ  1 , M 0 crit ¼ γð3  γÞ

(6.240)

and thus from (6.129) and (6.239)     P1 γþ1 ρ0 γþ1 ¼ ¼ and : P0 crit 3  γ ρ1 crit 3γ  1

(6.241)

For ρ0 =ρ1 < ðρ0 =ρÞmax , the heat flux, from (6.219), using (6.218) and (6.128), becomes   ρ0 ρ0 ρ0 ð γ þ 1 Þ 1   ρ ρ ρ1 dT ¼ ρ0 u30 q ¼ κ

0, (6.242) dx 2ðγ  1Þ for ρ0 =ρ1 ρ0 =ρ 1. Then

 ρ0 d ρ dT dT ¼  ≥ 0, ρ dx d 0 dx ρ

(6.243)

throughout the shock front. Now d ðρ0 =ρÞ=dx is always negative (or zero), so only those portions of (6.238) for which dT=d ðρ0 =ρÞ 0 are allowed. Hence, the temperature rises continuously from T 0 to a point where ρ0 =ρ > ðρ0 =ρÞmax , and then remains constant while the relative volume collapses discontinuously to ρ0 =ρ1 . This is depicted in Figure 6.17. Because the density discontinuity occurs

Downstream

T, r

Upstream

T1 r1 T0

r0 x

Figure 6.17. Temperature variation across the shock is continuous, while the density is discontinuous in an isothermal shock.

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Hydrodyamics

at a single temperature, thisffi type of solution is called an isothermal shock. For a gas pffiffiffiffiffiffiffi with γ ¼ 5=3, M 0 ≥ 9=5, or P1 =P0 ≥ 2, or ρ0 =ρ1 2=3. For strong shocks, the thickness may be less than one mean free path. If the flow variables change rapidly over a distance of one mean free path, the hydrodynamic treatment of viscosity and heat conduction is no longer applicable. Our analysis shows that heat conduction has an indirect effect, since it serves only to transfer energy of random motion of the particles from one point to another; it does not affect the directed motion of the matter passing through the front. Viscous effects convert directed kinetic energy of the particles into random motion by dissipation of momentum. The important case of the presence of radiation on the structure of a shock front will be discussed in Chapter 10. 6.6.3 The Relaxation Layer Our discussion about the nature of the shock front assumed that the fluid remains instantaneously in local thermodynamic equilibrium as it flows through the dissipation zone. In reality, the material probably does not remain in a state of equilibrium because the time required for the material to pass through is short compared to times necessary for equilibrium to be established. Thus, there is a second zone adjacent and downstream of the dissipation zone. This “relaxation zone” is large compared to the size of the dissipation zone. In fact, this relaxation zone may be several parallel zones, one for each separate relaxation process. In general, the problem of relaxation is a complicated one, because the relaxation rates depend upon the thermodynamic state of the material and thus the flow dynamics, but also in the opposite sense of the relaxation rates determining the flow properties. An example of this latter situation is the energy loss incurred during the ionization process, or the energy gain by recombination. There are many processes that should be considered, but we select a relatively simple one for illustration. Consider a gas sufficiently hot that it is fully ionized plasma, thus containing only free electrons and heavy ions. For our purposes, we assume thermal conduction can be neglected and the only relaxation process active is thermal equilibration between the two species. Initially, the electron and ion fluids in the upstream material are in equilibrium T e ¼ T i ¼ T 0 . As the material passes through the dissipation zone, viscous forces dissipate a large fraction of the ion’s kinetic energy into thermal energy, thus increasing the ion temperature by an amount ΔT i e mi u20 =kB . This takes place in a layer of thickness δ e u1 τ ii , where the ion self-collision time τ ii is the time necessary for a velocity distribution to approach a Maxwellian; it is a special case of the deflection time τ D , as discussed in Section 3.9.2. The electron fluid also has some of its initial directed energy

6.6 Viscous and Heat-Conducting Fluids

313

converted into thermal energy, but it is a small fraction of ΔT i due to the large mass difference of the electrons and the ions; ΔT e e me u20 =kB ¼ ðme =mi ÞΔT i . We can therefore assume the electron temperature rise is negligible. Furthermore,p the electron– ffiffiffiffiffiffiffiffiffiffiffiffi ffi ion equilibration time τ eq is long compared to τ ee and τ ii [τ eq  mi =me τ ii ¼ ðmi =me Þτ ee ], and there is no energy transferred within the dissipation zone. We also know that the strong coupling of the electrons and ions, due to Coulomb forces present, prevents any charge separation over distances greater than the Debye length. Therefore, the ions are compressed in the shocks, as are the electrons. Because the electrons cannot exchange energy with the ions in the time available, the compression of the electrons is essentially adiabatic. The resulting electron temperature just downstream from the dissipation zone is now T e 0  ðρ1 =ρ0 Þγ1 T 0 . This temperature rise is much smaller than experienced by the ions, particularly for high Mach number flows. A simple quantitative relation between T e 0 and T i 0 in the relaxation zone of the downstream flow can be derived. Assuming a perfect gas, the pressure behind the dissipation zone is nearly constant P1 ¼ ne k B T e 0 þ ni kB T i 0 ¼ ni k B ðZ  T e 0 þ T i 0 Þ ¼ ni kB ðZ  þ 1ÞT 1 :

(6.244)

Using (6.128) in (6.124) gives the ratio of the temperatures across the shock front

  2γM 20  ðγ  1Þ ðγ  1ÞM 20 þ 2 T1 ¼ : (6.245) T0 ðγ þ 1Þ2 M 20 For γ ¼ 5=3, T 1 =T 0  5M 20 =16, and we have "  2=3 # 2 5 4M 0 T0 T i0  ðZ  þ 1ÞM 20  Z  16 M 20 þ 3  2=3 4M 20 0 Te ¼ T 0: M 20 þ 3 0

(6.246)

0

We see that T i decreases and T e increases, both toward T 1 . Figure 6.18 sketches the temperature profiles. A rough estimate of the thickness of the relaxation zone is 

mi Δ e u1 τ eq e me

1=2 

T e0 T i0

3=2



mi u1 τ ii e me

1=2 

T e0 T i0

3=2



mi λe me

1=2 δ,

(6.247)

if we assume T e 0  T i 0 averaged over the zone. There is a limiting upstream Mach number (Imshenik, 1962)  2 γ2 þ ð3Z  þ 1Þγ  Z  , M 0 crit ¼ γ½ð3Z  þ 1Þ  γðZ   1Þ

(6.248)

314

Hydrodyamics

for which the shocks are fully dispersed; that is, for Mach numbers less than the critical value, all flow variables are continuous. The corresponding critical pressure jump is   P1 ðZ  þ 1Þðγ þ 1Þ , (6.249) ¼  P0 crit ðZ þ 1Þðγ þ 1Þ  2Z  ðγ  1Þ and the density jump is   ρ0 ðZ  þ 1Þγðγ þ 1Þ  Z  ðγ2  1Þ ¼  : ρ1 crit ðZ þ 1Þγðγ þ 1Þ  Z  ðγ  1Þ2

(6.250)

For large Z  , (6.248), (6.249), and (6.250) reduce to (6.240) and (6.241). The preceding discussion has omitted the effects of thermal conduction. This omission is appropriate for the ions, because the characteristic length scale of the temperature gradient in the relaxation zone greatly exceeds the mean free path of the ion, λi . The situation is much different for the electron fluid, where the mean free path in plasma is independentpofffiffiffiffiffiffiffiffiffiffiffiffi the ffiparticle’s mass. The thermal diffusivity for electrons is χ e e ve λe is χ e ¼ mi =me χ i . The characteristic length scale for electron conduction is  1=2  1=2 χe mi χi mi Le e e λi , e u1 me u1 me

(6.251)

which is about the same order of thickness as the relaxation zone (6.247). Therefore, energy transport by electrons is very efficient and will strongly heat the electrons immediately behind the dissipation zone, in addition to promoting a more rapid equilibration of the ions. More importantly, the higher-velocity electrons behind the shock front are moving more rapidly than the downstream mass flow, and electrons can overtake the shock front and conduct heat into the upstream material before the shock front arrives. This is known as electron thermal preheat, and the penetration of the electrons serves also to heat the upstream ions. (We will address this important effect again in Chapter 10.) Figure 6.18 shows the major modification to the upstream electron temperature distribution.

6.7 Elastic-Plastic Behavior of Solids In contrast to fluids, the mechanics of solid bodies are much more complicated since shear forces must be accounted for. Not only are the conservation equations more complicated, the thermal properties of the material are also affected. The presence of a small shear stress component can have a large effect on the manner in which a pressure wave, an acoustic wave, or a shock wave attenuates.

315

6.7 Elastic-Plastic Behavior of Solids T

Downstream

Upstream Ti

T1

Te

T0 x

Figure 6.18. Spatial distribution of the electron and ion temperatures in the relaxation zone. The dissipation zone is a narrow region at the discontinuity. The dashed line depicts the modification to the electron temperature when thermal conduction is included.

Furthermore, it has been observed experimentally that the shear strength of some solids increases with increasing pressure. These two facts have led to the development of elastic-plastic models instead of a fluid model to describe the behavior of solids at high pressure. We are interested primarily in the elastic-plastic behavior of isotropic metals. The formalism can be extended to anisotropic materials such as crystals, but that involves many complications that do not lend insight into the basics of solid mechanics. Beginning with a body in static equilibrium and applying a compressive force in one direction only causes the material to undergo a deformation. If the deformation is small enough, releasing the applied force will cause the body to resume its original shape; this is elastic behavior. If however, the applied force is greater than some limiting value, the elastic limit, the body deforms plastically, and upon release of the force, it will not return to its original shape. In the plastic case, the unloading path is different from the loading path. Thus, application of a compressive force is a dynamic phenomenon, resulting in the propagation of waves throughout the body. The dynamic response is governed by the conservation equations in a fashion similar to that for fluids. Figure 6.19 sketches the propagation of the waves. The elastic precursor is basically a sound wave whose speed of propagation is determined by the EOS, while the plastic wave is a shock wave whose speed can be found from (6.113), where the reference density ρ0 is now the density at the elastic limit ρA . Plastic deformation results from the motion of lattice dislocations along the slip planes. In contrast, viscous fluid flow is the result of forces between individual particles, and the flow is not constrained to any particular planes.

316

Hydrodyamics

Figure 6.19. Spatial profile of a propagating elastic-plastic stress wave. The wave is moving to the right.

For one-dimensional strain under consideration, the total stress is given by T ¼ P þ σ. At some level of stress, the body no longer behaves elastically, and the deviatoric stress has a maximum value σ ¼ 2Y 0 =3, where Y 0 is the yield strength in simple tension. This is the von Mises yield condition that will be discussed later in this chapter. A further increase in T places the body in the plastic deformation region, and the deviatoric stress remains constant; the increase in the total stress is due to increasing pressure. Thus, the sound speed increases with pressure, and a shock front can form. The Hugoniot equation still applies, but now the pressure is replaced by the total stress T. Figure 6.7 can now be redrawn as Figure 6.20, where A indicates the point where the distortion stress has reached its maximum value; this is referred to as the Hugoniot elastic limit. The discontinuous decrease in slope at A will cause the stress wave to break into two steps for stress levels that are between A and B. When the stress exceeds the value at B, the plastic wave will have a speed greater than the elastic precursor speed, and the stress will propagate as a single shock. This follows from noting that the slope of the Rayleigh line is greater than the slope ðT A  T 0 Þ=ðV 0  V A Þ for stress points above B. Deformations of solids are three-dimensional by nature because of the lattice arrangement of constituent atoms. This makes the mathematical description of the physics somewhat complicated. We can use a vector-tensor notation, but we find it more instructive to use indicial notation. Further, we limit the discussion to Cartesian coordinates, although other coordinate systems may be appropriate for specific applications. We might rightly ask the question: “How does this topic relate to high-energydensity physics?” As we indicated in the early part of this book, the traditional definition of high-energy-density material is far too limiting, especially when it comes to the topic of warm dense matter. Thus, this section serves as an

6.7 Elastic-Plastic Behavior of Solids

317

Figure 6.20. One-dimensional stress-strain for an elastic-plastic material. Also shown is the unloading stress path (dashed line).

introduction to warm dense matter where structural effects of the material must be accounted for. This is the extent of the topic of warm dense matter that we address.

6.7.1 Hooke’s Law In Section 6.6, we developed the relation between strain rate and stress. Rewriting that relation in terms of strain and of moduli, rather than viscosities, the stress tensor becomes, in indicial notation,   1 σ ij ¼ Kϵ kk δij þ 2μ ϵ ij  ϵ kk δij , (6.252) 3 where ϵ ij is the strain tensor and K and μ are the bulk modulus (modulus of hydrostatic compression) and shear modulus (modulus of rigidity), respectively. As pointed out in the beginning of Section 6.6, the stress tensor is required to be symmetric, that is, σ ij ¼ σ ji . The converse formula, which expresses ϵ ij in terms of σ ij , is found by first finding the sum of the diagonal terms of the stress tensor. Since this sum is zero for

318

Hydrodyamics

the second term in (6.252), we have ϵ ii ¼ σ ii =3K. Using this expression in (6.252) leads to   1 1 1 (6.253) ϵ ij ¼ σ kk δij þ σ ij  σ kk δij : 9K 2μ 3 This expression shows the strain tensor to be a linear function of the stress tensor. That is, the deformation is proportional to the applied forces. Both (6.252) and (6.253) are known as Hooke’s law and they are valid for small deformations. 6.7.2 Homogeneous Deformations Consider two simple examples of deformations for which the strain tensor is constant throughout the material of interest.4 In the first case, the symmetric cylindrical body is placed under tension along its axis of symmetry, the z-direction. Since the strain is constant everywhere, the stress is also constant. There is no external force on the cylindrical surface, and thus the only nonzero component of σ ij is σ zz . At the ends of the cylinder, σ zz ¼ p, the applied force per unit area (not the hydrostatic pressure P). From (6.253), we see that all components of ϵ ij with i 6¼ j are zero. The remaining components are     1 1 1 1 1 1 ϵ xx ¼ ϵ yy ¼   p and ϵ zz ¼ þ p: (6.254) 3 2μ 3K 3 3K μ This last expression is often written as ϵ zz ¼ p=E, where Young’s modulus is E¼

9Kμ : 3K þ μ

(6.255)

The components ϵ xx and ϵ yy give the relative compression of the rod in the transverse directions. The ratio of the transverse compression to the longitudinal extension is ϵ xx ¼ σ P ϵ zz , where Poisson’s ratio is5 σP ¼

4 5

1 3K  2μ : 2 3K þ μ

(6.256)

The six components of the strain tensor are not entirely independent, since they are expressed in terms of the derivatives of only three independent functions, the components of the displacement vector. When reading the literature on the elastic behavior of materials, one will find some authors write the equations using the shear modulus μ, the second Lamé coefficient λ, and the bulk modulus K. They are related to E and σ P by λ¼

Eσ P E E , μ¼ , and K ¼ : 2ð1 þ σ P Þ 3ð1  2σ P Þ ð1 þ σ P Þð1  2σ P Þ

6.7 Elastic-Plastic Behavior of Solids

319

Since both K and μ are always positive, 1 σ P 1=2. In practice, there are no known substances with σ P < 0, as this would indicate an expanding transverse dimension for a longitudinal stretching. The stress tensor for this first case is   E σP (6.257) σ ij ¼ ϵ ij þ ϵ kk δij , 1 þ σP 1  2σ P and its converse expression is  1 ð1 þ σ P Þσ ij  σ P σ kk δij : E

ϵ ij ¼

(6.258)

The second case is where the same cylindrical body undergoes a compression along its z-axis. The cylindrical surface is fixed in a way that cannot move. The only component of the strain that is not zero is ϵ zz . Then, from (6.257) E ϵ zz , ð1 þ σ P Þð1  2σ P Þ E ð1  σ P Þ ϵ zz : σ zz ¼ ð1 þ σ P Þð1  2σ P Þ

σ xx ¼ σ yy ¼

(6.259)

For the compressing force (per unit area) p, the strain is ϵ zz ¼

ð1 þ σ P Þð1  2σ P Þ p, E ð1  σ P Þ

(6.260)

and the transverse stresses are σ xx ¼ σ yy ¼

σP p: 1  σP

(6.261)

6.7.3 Thermal Deformations Up to now, we have paid little attention to thermal effects. Deformations can occur because of a temperature change of the body due to the deformation process itself, or from external causes. The Helmholtz free energy of the body is F ¼ ℰ  TS. Since dℰ ¼ TdS  PdV ¼ TdS þ σ ij dϵ ij , we see dF ¼ SdT þ σ ij dϵ ij :

(6.262)

We know that the undeformed state of a body is the state in the absence of external forces, thus the strain tensor  is zero, as is the stress tensor. The components of the stress tensor are σ ij ¼ ∂F=∂ϵ ij T , and it follows that there is no linear term in the expansion of F in powers of ϵ ij . Since the free energy is a scalar, each term in the expansion of F

320

Hydrodyamics

must be a scalar also. Two independent scalars of the second degree can be formed from the symmetrical tensor ϵ ij ; they are the squared sum of the diagonal components and the sum of the squares of all the components. Then to second order 1 F ¼ F 0 þ λϵ 2kk þ μϵ 2ij , 2

(6.263)

where F 0 is the free energy of the undeformed  body and, since it is of no further interest, can be set to zero. Using the identity ϵ ij ¼ ϵ ij  13 ϵ kk δij þ 13 ϵ kk δij , (6.263) becomes  2 1 1 F ¼ μ ϵ ij  ϵ kk δij þ Kϵ 2kk : (6.264) 3 2 The first term on the right-hand side is a pure shear, since the sum of the diagonal terms is zero (δij ¼ 0). The second term is a hydrostatic compression. Let the temperature of the undeformed state of the body be T 0 . If the body is at a temperature different from T 0 , then even in the absence of external forces there will be a deformation on account of thermal expansion, and now there will be linear terms of the strain tensor in the expression for the free energy  2 1 1 F ðT Þ ¼ F 0 ðT Þ  KαðT  T 0 Þϵ kk þ μ ϵ ij  ϵ kk δij þ Kϵ 2kk : (6.265) 3 2 Differentiating with respect to ϵ ij gives

  1 σ ij ¼ KαðT  T 0 Þδij þ Kεkk δij þ 2μ ϵ ij  ϵ kk δij : 3

(6.266)

In free thermal expansion (absence of external forces), there can be no internal stresses. Setting σ ij ¼ 0 gives ϵ kk ¼ αðT  T 0 Þ. But ϵ kk is just the relative change in volume caused by the deformation, thus α is just the thermal expansion coefficient. In Section 6.3, we talked about isothermal and adiabatic expansions. In an isothermal deformation, the temperature does not change, and T ¼ T 0 in (6.265) and (6.266), thus returning to the usual formulas; the coefficients K and μ may be called isothermal moduli. An adiabatic deformation exists if there is no heat exchanged between various parts of the body (or with the surrounding environment). The entropy is constant and is found from the derivative of the free energy (6.265) with respect to temperature SðT Þ ¼ S0 ðT Þ þ Kαϵ kk :

(6.267)

We see a change in temperature is proportional to ϵ kk , thus Kαϵ kk ¼ C V

T  T0 : T0

(6.268)

6.7 Elastic-Plastic Behavior of Solids

321

Using this in (6.266) gives   1 σ ij ¼ K adia εkk δij þ 2μ ϵ ij  ϵ kk δij , 3

(6.269)

where K adia is the adiabatic modulus of compression, but the modulus of rigidity is the same as before. The relation between K and K adia can be found from the thermodynamic expression     ∂V ∂V T ð∂V=∂T Þ2P ¼ þ : ∂P S ∂P T CP

(6.270)

The derivatives give the relative volume changes in heating and compression       ∂V 1 ∂V 1 ∂V ¼ , ¼  , and ¼ α, (6.271) ∂P S K adia ∂P T K ∂T P leading to the relation 1 K adia

¼

1 α2 T:  K CP

(6.272)

Expressions for the adiabatic versions of Young’s modulus and Poisson’s ratio are easily found.  In isothermal deformations, the stress tensor is σ ij ¼ ∂F=∂ϵ ij T , and for adiabatic deformations it is σ ij ¼ ∂ℰ=∂ϵ ij S . The analogous expression to (6.264) for adiabatic expansions is  2 1 1 ℰ ¼ μ ϵ ij  ϵ kk δij þ K adia ϵ 2kk : 3 2

(6.273)

6.7.4 Elastic Deformations If motion occurs in a deformed body, its temperature is not in general constant but varies in both time and space. This complicates the exact equations of motion. However, if we assume that heat is transferred by thermal conduction only, matters are greatly simplified in that heat exchanged during a time of order of the oscillatory motions of the body is negligible. That is, the body is thermally isolated and thus the motion is adiabatic. The usual stress-strain formulas are valid, but now the isothermal values for K, E, and σ P must be replaced by their adiabatic values, given in the preceding section.

322

Hydrodyamics

The equation of motion for an isotropic elastic medium is founded on (6.19), and in analogy with (6.200), is ρ€ y¼

E E r2 y þ rðr  yÞ, 2ð1 þ σ P Þ 2ð1 þ σ P Þð1  2σ P Þ

(6.274)

where y is the displacement vector. We consider a plane elastic wave, where the displacement is a function only of one coordinate, the z-direction, and of the time. Then, in analogy with the development of acoustic waves in Section 6.4, the components of the vector y are ∂2 yz 1 ∂2 yz ∂2 yx 1 ∂2 yx  ¼ 0 and  ¼ 0; ∂z2 c2l ∂t 2 ∂z2 c2t ∂t2

(6.275)

the equation for yy is the same as that for yx . We now have two velocities of propagation (sound speeds), one for the longitudinal motion and one for the transverse motion c2l ¼

1 E ð1  σ P Þ λ þ 2μ 1 E μ and c2t ¼ ¼ ¼ , ρ ð1 þ σ P Þð1  2σ P Þ ρ 2ρ ð1 þ σ P Þ ρ

(6.276)

using (6.254). These propagation velocities, being material dependent, vary widely. For example, air has cl ¼ 3:4  104 cm s1 (with ct ¼ 0 since it is a gas), and aluminum has cl ¼ 6:5  105 cm s1 and ct ¼ 3:1  105 cm s1. Expressions (6.275) are ordinary wave equations, and the two waves propagate independently. From (6.276), the velocity of theffi longitudinal wave is always pffiffiffiffiffiffiffi greater than that of the transverse p wave, c > 4=3 ct , since σ P ≥  1; but in reality l ffiffiffi σ P ≥ 0 and we always have cl > 2ct . The volume change in a deformation is given by the sum of the diagonal terms in the strain tensor, that is, ϵ kk ¼ ry. In the transverse wave, there is no component yz , and since the other components do not depend on yx or yy , ry ¼ 0. Thus, transverse waves do not involve any change in volume of the body. However, since ry 6¼ 0, longitudinal waves do involve compression and expansion of the body. The following discussion about elastic-plastic behavior is guided by material in the book by Wilkins (1999). The six components of the stress tensor can become a bit confusing, especially for one-dimensional flow. It is always possible to choose a coordinate system in which the shear stress at a given point is zero, that is, T ij ¼ 0 for i 6¼ j. The new coordinate axes are called principal axes for the point being considered. The stress on surfaces normal to these axes are the principal stresses denoted by T i , where i ¼ 1, 2, 3.

6.7 Elastic-Plastic Behavior of Solids

323

A perfectly elastic material is characterized by a linear correspondence between stress and strain, described by Hooke’s law. It is customary to use μ and λ; the bulk modulus is given by K ¼ λ þ 2μ=3, thus (6.252) becomes σ_ i ¼ λ

V_ þ 2μϵ_ i , V

(6.277)

where the equation of continuity (6.12) allows the strain rate to be expressed in terms of the volume element V, that is, ϵ_ 1 þ ϵ_ 2 þ ϵ_ 3 ¼ V_ =V. Hooke’s law written as (6.277) gives the natural strain, which means that the strain of a volume element is referred to the current configuration rather than the original configuration. For elastic behavior, we want to limit the stress contributions that are due to shear distortions. Therefore, we decompose each of the principal stresses into a hydrostatic component and a stress deviator component σ. The hydrostatic part is P, which is the mean of the three stresses, that is, 1 P ¼  ðT 1 þ T 2 þ T 3 Þ: 3 The stresses in the principal directions are now written T i ¼ P þ σ i . With these definitions, Hooke’s law (6.277) can be written   V_ σ_ i ¼ 2μ ϵ_ i  : 3V

(6.278)

(6.279)

It follows that σ_ 1 þ σ_ 2 þ σ_ 3 ¼ 0 and also σ 1 þ σ 2 þ σ 3 ¼ 0, which says that the distortion components of the stresses do not contribute to the average pressure. At some point, the motion of the body will result in the volume element exceeding a certain combination of the stresses. This limiting point marks the onset of plastic flow. 6.7.5 Plastic Flow In plasticity theory, it is usual to assume that plastic behavior is independent of pressure. Therefore, the condition for plastic flow is written in terms of the stress deviators, f (σ 1 , σ 2 , σ 3 ) ¼ 0. This expression states that in principal stress space there is a boundary condition on the magnitude of the stresses. After this value has been attained, plastic flow begins. Prior to this condition f ðσ 1 ; σ 2 ; σ 3 Þ < 0, and the material is in the elastic region. This boundary condition is referred to as the yield condition and must be independent of the coordinate system. At the yield condition and beyond, we assume the flow is perfectly plastic; that is, f ¼ 0 retains its value throughout the whole of the plastic flow. This simplification means that there can be no strain hardening and that the material flows plastically under a constant yield

324

Hydrodyamics

stress. A further simplification to the yield condition is that it is independent of a change in the sign of the stress; that is, the material behaves similarly in tension and in compression. In addition to the yield condition, a stress-strain relation for the plastic region is needed. Plasticity theories assume that during plastic flow, the rate of plastic strain is proportional to the instantaneous stress deviator ðpÞ

ϵ_ i

_ i, ¼ λσ

(6.280)

where λ_ is a scalar plastic flow rate parameter. This parameter is different for different positions and different for the same point at different times. The stress deviators σ i are not rate dependent, but are proportional to a rate-dependent constant. This is in contrast to elasticity theory, which states that the stress is proportional to the strain so that the stress and the strain determine each other. Here, the stress is proportional to the plastic strain rate, so a state of plastic strain does not correspond to a unique state of stress. Experimentally, it is observed that there is no permanent change in volume due ðpÞ ðpÞ ðpÞ to plastic strain, that is, ϵ 1 þ ϵ 2 þ ϵ 3 ¼ 0. The total strain is the sum of the ðt Þ ðpÞ ðeÞ plastic and elastic strains ϵ i ¼ ϵ i þ ϵ i . The elastic portion of the strain is recoverable, but the plastic portion is assumed to be permanent. The yield condition limits the magnitude of the stresses. An often-used boundary that sets the limit on elastic behavior is that developed by von Mises (1913). This theory states that the principal plastic strain-rate vector associated with the principal stress vector is directed outward along the normal to the yield surface at the point ðT 1 ; T 2 ; T 3 Þ. Thus, if f ðT 1 ; T; T 3 Þ ¼ 0 denotes the yield condition, then ðpÞ

ϵ_ i

∂f ¼ λ_ : ∂T i

(6.281)

In principal stress space, this expression corresponds to the gradient of a scalar point function.

6.7.6 Yield Strength The von Mises yield condition in principal stress space is  2 f ¼ ðT 1  T 2 Þ2 þ ðT 2  T 3 Þ2 þ ðT 3  T 1 Þ2  2 Y 0 ¼ 0,

(6.282)

where Y 0 is a material-dependent constant and is the yield strength in simple tension. This expression is independent of pressure, since the pressure is an additive constant to all stress terms T i of (6.282).

6.7 Elastic-Plastic Behavior of Solids

325

Differentiating (6.282) gives ∂f ¼ 2ðT 1  T 2 Þ  2ðT 3  T 1 Þ ¼ 6T 1  2ðT 1 þ T 2 þ T 3 Þ: ∂T 1

(6.283)

Using the definition of the mean pressure, (6.283) becomes  ∂f ¼ 6 T 1 þ P ¼ 6σ 1 : ∂T 1

(6.284)

There are similar expressions for the derivatives with respect to the other two principal stresses T 2 and T 3 . The yield condition (6.282) can be written in terms of the stress deviators  2 ðσ 1  σ 2 Þ2 þ ðσ 2  σ 3 Þ2 þ ðσ 3  σ 1 Þ2 ¼ 2 Y 0 , (6.285) and since σ 1 þ σ 2 þ σ 3 ¼ 0, we can write σ 21 þ σ 22 þ σ 23 ¼

2  0 2 Y : 3

(6.286)

The left-hand side of (6.285) can be shown to be proportional to the energy required to change shape, as opposed to the energy that causes a change in volume. Thus, plastic flow begins when the elastic distortion energy reaches the limiting 0 2 value Y =6μ, and this energy remains constant during the plastic flow. Thus the term “elastic-plastic” means the state whereby the distortion component of the strained material has been loaded, followed by Hooke’s law, up to a state where the material can no longer store energy. For one-dimensional strain, expressions T i ¼ P þ σ i , together with (6.279) and (6.286), can be graphically presented in Figure 6.21. In Figure 6.21a, point A is the elastic limit. For strains between points O and A, the loading and unloading paths are the same. For strains beyond point A, the unloading path is from point B to point C. The total stress T 1 is shown in Figure 6.21b with the offset 2Y 0 =3 beyond point A. The introductory discussion about elastic-plastic behavior can be extended to include models of dislocation theory, work hardening, pressure, and temperature effects on material strength. For example, for a work-hardened material, the deviatoric stress will increase monotonically with strain for strains beyond point A of Figure 6.21a, instead of remaining constant, as for the perfectly plastic material shown. Work-hardening can be introduced by letting the yield condition Y 0 be a function of the strain energy. When sufficient work has been done so that the material begins to melt, the value of Y 0 is allowed to approach zero, at which point the only remaining stress will be the pressure.

326

Hydrodyamics

(a)

(b) –T1 or P

–s1

Compression

A

B –T1 2 0 3Y

2 0 3Y O 2 0 3Y

Tension

P

A

e1 C

O

e1

Figure 6.21. One-dimensional stresses as a function of strain for an elastic perfectly plastic material. In (a), ϵ 2 ¼ ϵ 3 ¼ 0 and σ 2 ¼ σ 3 ¼ σ 1 =2; the initial slope of σ 1 is 4μ=3. In (b), the slope of the total stress T 1 between points O and A is λ þ 2μ, and the slope of the pressure P is K ¼ λ þ 2μ=3.

The value of Y 0 can be made a slow function of pressure, as suggested by von Mises, to correspond to the observed increase in shear strength with pressure for some materials. Finally, time-dependent yielding can be introduced by way of the parameter Y 0 . If a departure is made from the elastic perfectly plastic model where Y 0 is no longer a constant, then attention should be given to the function dependence of the shear modulus μ. While a great number of constitutive (yield) models have been developed for specific materials and conditions, a few more general ones find utility in highenergy-density physics. One class of such models is for high strains: Johnson and Cook, (1983) and Steinberg, Cochran, and Guinan (1980). A second class is for strain rate–dependent materials and conditions: Hoge and Murkherjee (1977), Preston, Tonks, and Wallace (2003), Steinberg and Lund (1989), and Zerilli and Armstrong (1987).

6.8 Transitioning from Planar to Elliptical Flow It is well known from laboratory experiments on free-standing planar targets irradiated by a laser beam that a quasispherical plume appears at late time. The one-dimensional description of the planar expansion is often quite accurate at early times, while at late times the evolution of the plume is described by elementary hydrodynamics and should be sensitive only to integral quantities, such as source

327

6.8 Transitioning from Planar to Elliptical Flow

Figure 6.22. Temporal evolution of a finite diameter planar plasma into an ellipsoidal plume. The dashed lines are the marginal rays of the incident laser beam.

energy deposited, mass removal, and so on. We address here a special solution of the hydrodynamic equations to the initial conditions of planar flow. During the course of a laser irradiation, a thin layer of the surface of the planar target is rapidly heated and begins to expand into vacuum. It is assumed that the diameter of the laser spot is large compared to the thickness of the heated layer. The normal component of the pressure gradient is by far the largest, and this drives the acceleration of the material from the surface. As time progresses, the plume changes shape and becomes more elongated in the normal direction, as depicted in Figure 6.22. This evolution is nearly adiabatic, and the temperature and pressure decrease with the expansion. Once the pressure is significantly reduced, the expansion becomes inertial (i.e., the internal energy is mostly converted into kinetic energy), and the plume’s shape is largely preserved. At the beginning of Section 6.5.8, the simple theory of self-similar hydrodynamic motion was developed. Those equations can be extended for a threedimensional geometry, with a set of spatial relations for each direction. We develop the approach for two dimensions assuming the initial laser illumination pattern is circular, resulting in axial symmetry. There are two directions: the normal to the target’s surface, expressed as z, and the orthogonal direction parallel to the target’s surface, expressed as r. The motion of a particle is given by r ¼ r0

Rðt Þ r dR , ur ¼ , Rð0Þ RðtÞ dt

z ¼ z0

Z ðt Þ z dZ , uz ¼ : Z ð0Þ Z ðt Þ dt

(6.287)

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Hydrodyamics

Parameters with subscript 0 are the initial coordinates. The self-similar evolution of the plume’s density, pressure, and temperature parameters is given by gρ ½Rð0Þ; Z ð0Þ; γ  2 fρ ξ ;γ , R2 Z g ½Rð0Þ; Z ð0Þ; γ  2 Pðr; z; t Þ ¼ E P  2 γ fP ξ ;γ , RZ E gT ½Rð0Þ; Z ð0Þ; γ; A  2 fT ξ ;γ , T ðr; z; t Þ ¼  2 γ1 M RZ ρðr; z; t Þ ¼ M

(6.288)

where 

r ξ ¼ Rðt Þ 2

2



 z 2 þ : Z ðt Þ

(6.289)

The quantities E and M are the plume’s energy and mass, respectively, γ is the polytropic index, and A is the atomic weight. The specific expressions for the functions g and f depend upon the type of self-similar solution. An isentropic-type solution has the functions f expressing a parabolic profile, while an isothermaltype solution has f T constant and the other two are Gaussian profiles. The solution to (6.288) for the plume’s expansion has the parameters constant on the ellipsoid surfaces ξ ðr; zÞ ¼ const. Transformation with time is governed by d2 R dU d2 Z dU and , ¼  ¼ 2 2 dt dR dt dZ

(6.290)

where   E Rð0ÞZ ð0Þ γ1 U ðR; Z Þ ¼ βðγÞ : M RZ

(6.291)

The constant βðγÞ depends on the type of self-similar solution and is usually of order unity. The two equations of (6.290) are equivalent to an equation of motion for a particle moving in a repulsive potential U ðR; Z Þ, where ðR; Z Þ plays the role of the particle’s coordinates. If we assume an initial velocity of zero (all of the energy is initially thermal), then the particle’s velocity increases the most along the axis with the smallest initial dimension, and vice versa. That is, the plume expands the fastest in the direction normal to the surface of the target. When the particle has moved sufficiently far from its initial position, the potential is very small and the movement becomes inertial Rðt Þ  vr ð∞Þt and Z ðtÞ  vz ð∞Þt, where vr ð∞Þ and vz ð∞Þ are components of the particle’s velocity far from the initial position. Thus,

329

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the final stage of the ellipsoid’s axes increases at a fixed rate. That is, as the plume expands the shape is preserved and determined by the relative magnitude of the components of the final velocity. For the plume to be elongated in the normal direction, we must have vz ð∞Þ vr ð∞Þ. This occurs when Z ð0Þ  Rð0Þ. According to (6.287), the velocity components of the particles must be linear functions of their respective coordinates. The three-dimensional version of the theory also predicts, for a noncircular irradiation pattern, the plume will “rotate.” The smaller the initial dimension, the faster the final expansion. Therefore, if the initial illumination pattern is elongated in the x-direction, the final stage of the expansion will be elongated in the ydirection. The solution of (6.290) is found by writing that expression in the dimensionless form  0 γ1 2 0 2 0 Z ð0Þ 0d R 0d Z R 02 ¼ Z 02 ¼ , (6.292) R0 Z 0 dt dt where R0 ðt Þ ¼

Rðt Þ Z ðt Þ and Z0 ðtÞ ¼ , Rð0Þ Z ð0Þ

(6.293)

with t0 ¼

t Rð0Þ and t 0 ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi , t0 βðγÞE=M

(6.294)

where Rð0Þ is the radius of the circular laser spot on the target’s surface and Z ð0Þ is the depth of the instantaneously heated material. The time t 0 is a characteristic time of the expansion. The ratio k ð0Þ ¼ Z ð0Þ=Rð0Þ determines the initial shape of the ellipsoid. At late times, where t t 0 , the plume expansion is inertial, and the ellipsoid grows linearly with time. Numerical solution of (6.292), (6.293), and (6.294) gives the final shape of the ellipsoid kð∞Þ. Table 6.1 gives the results for different initial shapes kð0Þ for three values of the polytropic index. Table 6.1 Final Ellipsoid Shape kð∞Þ for Selected Initial Ratios kð0Þ and γ k ð 0Þ γ ¼5/3 6/4 7/5

5  105 160.22 42.993 20.256

5  104 50.683 19.831 11.563

5  103 16.018 8.9954 6.3025

5  102 5.0316 3.861 3.213

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Hydrodyamics

6.9 Fluid Instabilities This chapter has largely focused on the behavior of a fluid in one dimension. But our terrestrial experiences, as well as our observation of astrophysical phenomena, tells us that the reality of hydrodynamic motion is anything but one-dimensional. We see multidimensional phenomena such as waves, vortices, and turbulence. In fact, we have to look quite hard to find true one-dimensional hydrodynamic situations. The basic equations of hydrodynamics are essentially complete, but their solution to realistic situations is impossible except in isolated cases. The difficulty arises because of the nonlinear nature of the conservation equations. More specifically, the continuity equation (6.13) contains the divergence of the product of the density and fluid velocity, which makes it a nonlinear equation. Solutions can be obtained if we assume one of the two is constant, or by linearizing both quantities. Such approaches work well in selected cases, and even then only to a limited extent. Even with the advent of sophisticated mathematical techniques and the construction of massive computing machines, progress in the field of complex hydrodynamic motion is painfully slow. Despite these limitations, a large body of knowledge has been assembled, particularly on the topics of instabilities, which are the precursors to fullfledged vortex and turbulent motion. The best we can do here (because of space limitations) is to provide an introduction to three types of hydrodynamic instabilities of interest in highenergy-density physics (there are other instabilities also, but we omit them here). The subject of hydrodynamic instabilities is fascinating, so the reader is referred to the literature for the real substance of this topic; an excellent summary is given by Drake (2006). The first two instabilities are closely related, at least through the fundamental mathematics. The description of the Rayleigh-Taylor instability and the KelvinHelmholtz instability is founded on the acceleration equation (6.35). If we set the vorticity (r  u) to zero, (6.35) becomes (6.38). The quantity within the parentheses of (6.38) is constant in all space, though it may vary with time. Further, we take the fluid to be incompressible (ru ¼ 0). Now consider two adjacent, planar, incompressible fluids of possibly different densities, oriented in a fashion that the normal to their interface is in the direction of an acceleration (e.g., gravity), and that there may be relative motion between the two fluids along the interface. Most likely, the fluid interface is slightly perturbed, which may arise from thermal fluctuations, acoustic disturbances, and such. Let the unperturbed interface between the two regions be defined by z ¼ 0, with the density of the region with z > 0 designated as ρ0 , and for z < 0 the density is ρ1 .

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331

Then on each side of the interface there are the x-directed velocities U 0 and U 1 . A small interface perturbation zs is specified by zs e exp ðikr  iωt Þ,

(6.295)

where k has components only in the x- and y-directions. Our task is to determine the dispersion relation that gives ω in terms of k. A real ω signifies surface waves, while a complex ω with a positive imaginary part signifies an unstable interface. In the unperturbed state, the pressure is uniform across the interface, which we may define as the zero of pressure. When the interface surface is perturbed, a flow will be established on both sides of the interface. If U 0 is different from U 1 , a vortex sheet will exist at z ¼ 0 in the unperturbed state, and even if they are equal, vorticity may develop on the perturbed surface. Our discussion about Bernoulli’s equation (Section 6.2.4) introduced the concept of potential flow in (6.39). We need solutions for the potential flow in each region, from which the normal component of the velocity and the pressure across the interface are matched. Determination of the two potentials is straightforward, but we omit the details here, and refer the reader to Castor (2004). Two velocity potentials, one for each region, can be found, each of which satisfies Laplace’s equation. Since we have chosen the interface perturbation according to (6.295), the appropriate harmonic functions are exp ðkz þ ikrÞ, where k is the magnitude of k; k 2 ¼ k 2x þ k2y . Far from the interface, the fluctuations vanish, so the negative sign is chosen for z > 0 and the positive sign for z < 0. The pressure approaching the interface from each side is found from the strong form of Bernoulli’s law (6.39)   ∂Φ0, 1 1 2 1 2 (6.296) P0, 1 ¼ ρ gz þ þ u  U 0, 1 , 2 2 ∂t in each region separately, as indicated by the subscripts. The acceleration is g, and the additive constant has been chosen to ensure the pressure matches the unperturbed value ρgz far from the interface. Substituting the velocity potential for each region into (6.296) and equating the two at the interface gives the dispersion equation ðρ1  ρ0 Þg  ρ0

ðω  k x U 0 Þ2 ðω  k x U 1 Þ2  ρ1 ¼ 0: k k

(6.297)

Solving this equation for the angular frequency gives " #1= 2 2 ρ0 U 0 þ ρ1 U 1 ρ1  ρ0 ρ ρ ð U  U Þ 0 1  kg  k 2x 0 1 : ω ¼ kx 2 ρ0 þ ρ1 ρ0 þ ρ1 ðρ0 þ ρ1 Þ

(6.298)

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Hydrodyamics

Examination of (6.298) shows that if ρ0 > ρ1 and U 0 ¼ U 1 ¼ 0, the values of ω are imaginary, and in particular one root is  1=2 ρ0  ρ1 ω¼i kg : ρ0 þ ρ1

(6.299)

This root leads to exponential growth in time with the growth rate given by the square-root factor. This is the “classical” Rayleigh-Taylor instability. The growth rate contains the Atwood number α ¼ ðρ0  ρ1 Þ=ðρ0 þ ρ1 Þ, which is a positive number less than one. A second instability arises with ρ0 ¼ ρ1 and U 0 6¼ U 1 .6 The roots of (6.298) are imaginary, and the growth rate is kx jΔU j=2. This is the Kelvin-Helmholtz instability. This simplified analysis has neglected viscosity, surface tension, thermal conductivity, and so on, so dissipative effects have been ignored. For high wave numbers, dissipation is increasingly important, as those processes will stabilize the instabilities. We review both of these instabilities in a bit more detail now, but confine the discussion to two-dimensional representations. This provides us insight and establishes a foundation for extension to the full three-dimensional versions. We note that the three-dimensional instabilities tend to be specific to a particular geometry and boundary conditions.

6.9.1 Rayleigh-Taylor The introduction to instabilities just presented omitted several important pieces of physics. In particular, real fluids have viscosity and are compressible. For problems of interest in high-energy-density physics, surface tension plays no role, as molecular forces are absent. We will comment later on other physics that may be important in certain situations. The Rayleigh-Taylor instability causes the interpenetration of fluid regions having different densities. Our everyday experience suggests that the instability occurs when a less dense fluid supports a more dense fluid against gravity. More appropriately, the instability occurs whenever fluid regions that differ in density experience a pressure gradient that opposes the density gradient. Gravity is 6

There is a third instability also, that of deepwater waves. If ρ0 is negligible (air) and both U 0 and U 1 arep zero, ffiffiffiffiffiffiffiffi pffiffiffiffiffi the dispersion relation gives ω ¼  kg. These are dispersive waves, and the phase velocity ω=k h ¼ g=k pffiffiffiffiffiffiffi ffi increases with wavelength. The group velocity is dω=dk ¼ g=k=2, which is one-half the phase velocity. So for a packet of water waves, the wave crests appear at the rear of the packet, ride up over the top, and disappear at the front.

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333

not necessarily the driving force in high-energy-density physics. Rather, it is the low-density fluid pushing against the higher-density fluid causing the acceleration (or deceleration) of the higher-density fluid. It is the inertia of any fluid that will cause it to resist being accelerated; the denser fluid has more inertia and the interpenetration of the two fluids allows some of it to lag behind, thus creating the instability. The original work of Lord Rayleigh (1883) considered the higherdensity fluid to lose potential energy as it “fell” into the lower-density fluid under the effects of gravity. An important insight to this phenomenon contributed by Taylor (1950) was that this situation is equivalent to the lower-density fluid being accelerated into the higher-density fluid, such as might be encountered underwater by an expanding bubble. One astrophysical example is that of the Crab nebula, in which the expanding pulsar wind is sweeping up ejected material from the supernova explosion of one thousand years ago. The momentum equation, including viscosity and density gradients, is given by (6.200). Linearizing this expression and assuming the fluid is incompressible (ru ¼ 0), together with the continuity equation, gives a set of five equations. Then applying a growing modulation of the interface that has amplitudes with an  unknown variation in z but proportional to exp ik x þ ik y þ γt , we arrive at a new set of five equations. These can be reduced to one equation that describes the fluid’s velocity uz in terms of known parameters   2     ∂ ∂uz ∂ ∂ðρνÞ ∂2 2 ∂uz 2 ργ þ ρν þ  k þ k uz ∂z ∂z ∂z ∂z2 ∂z ∂z2  2      ∂ ∂ðρνÞ ∂uz g ∂ρ 2 2 (6.300) 2 ¼ k ργuz þ ρν  k uz þ , þ uz ∂z γ ∂z ∂z2 ∂z where γ is the growth rate and ν ¼ μ=ρ is the kinematic viscosity, also known as the momentum diffusivity; μ is the shear viscosity. Applying (6.300) to each side of the interface between the two “materials,” and noting that uz and ∂uz =∂z are continuous across the interface, we have the boundary condition   ∂uz1 2 2g ui k ðρ0  ρ1 Þ þ k 2ðρ0 ν0  ρ1 ν1 Þ γ ∂z   2    2  ∂ ∂uz0 ∂ ∂uz1 2 2  ρ1 γ þ ν1 , k k ¼ ρ0 γ þ ν0 (6.301) 2 2 ∂z ∂z ∂z ∂z where ui designates the common value of uz at the interface. We have allowed for the two materials to have different densities and viscosities. We note that for a hydrodynamic instability, the pressure is not necessarily continuous across the interface. However, the x–z and y–z components of the viscous stress tensor are continuous.

334

Hydrodyamics

The simplest case is that of two inviscid, uniform fluids. Then ∂ρ=∂z ¼ 0 and (6.300) become d 2 uz ¼ k 2 uz , dz2

(6.302)

which has the solutions uz ¼ ui exp ðkzÞ. The appropriate boundary condition, from (6.301), is g γ ui ðρ0  ρ1 Þ ¼ 2 ðρ0 þ ρ1 Þkui , (6.303) γ k pffiffiffiffiffiffiffi which gives the growth rate γ ¼ αkg, as before. The observant reader will have noted by now that if the Atwood number is negative, the instability does not form, but the initial disturbance oscillates. The density and pressure gradients must be antiparallel for the instability to grow. The next level of complexity is to include viscous effects. Assuming the viscosities and densities of each fluid are different but constant, (6.300) gives   2  2   2   ∂ ∂ uz ∂ 2 2 γ þ ν k ¼ γuz þ ν  k uz k 2 : (6.304) ∂z2 ∂z2 ∂z2 Drake (2006) provides a dispersion relation derived from (6.304) and then specializes itqto the case ffi ν0 ¼ ν1 ¼ ν, which results in a fifth-order polynomial for ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi s ¼ k 1 þ γ= k 2 ν . Analysis of the polynomial equation is rather complex, and we refer the reader to Drake (2006). However, we summarize his results whereby he notes the growth rate for small viscosity approaches the zero-viscosity case, and for large viscosity, those effects are dominant, even though the solution that joins these two extremes

may not be accurate between them. At high viscosity, an expansion gives s  k 1 þ γ= 2k2 ν , which yields a much simpler dispersion relation qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi γ ¼ αkg þ k4 ν2  k2 ν: (6.305) We see that the largest growth occurs at wave numbers slightly smaller than those for which viscosity begins to reduce the growth substantially. The wave number for maximum growth is k¼

1 αg 1=3 , 2 ν2

(6.306)

and the magnitude of the growth rate for this wave number is " #1=2 4= 2= 3 ðαgÞ 3 ðαgÞ 3 γ¼  : 4 ν2=3 4ν1=3

(6.307)

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335

Thus, the effect of viscosity is to reduce the growth rate, albeit viscosity cannot completely eliminate the instability. This is reasonable, since viscosity transforms some of the kinetic energy into thermal energy, but the system will be in a state of minimum potential energy. The growth-rate dependency on wave number saturates when the viscous force becomes comparable to the “gravitational” force. An effect not dissimilar to viscosity is the microscopic effects of binary collisions at the interface. Such collisions lead to mass diffusion across the interface, which also serves to reduce the growth rate. Another important stabilizing mechanism is provided by density gradients. Not only does it become more difficult to identify an interface between two materials, an extended density gradient exists that opposes the pressure gradient. For the case of zero viscosity, (6.300) becomes     g ∂ρ ∂ ∂uz 2 k γρ  uz ¼ γρ : (6.308) γ ∂z ∂z ∂z This equation has the solution uz ¼ ui exp ðkzÞ for an arbitrary density profile. pffiffiffiffiffi This mode, known as the global mode, has a growth rate γgm ¼ kg, which is the largest Rayleigh-Taylor mode possible. However, this mode does not always exist, because it may not satisfy the boundary conditions. The global mode, for it to exist, requires a high-pressure region of negligible density over some size L, and then that layer will be unstable to modes with kL 1. If the density is nonnegligible on both sides of the interface, the boundary conditions do not allow the global mode. Applying the boundary condition (6.301) with ρ0 ¼ ρ1 ¼ ρ causes the acceleration to vanish, and the growth rate is  γ ¼ γgm

4kL 1 þ 4k 2 L2

1=2 :

(6.309)

We see that the growth rate reaches the global mode for kL ¼ 1=2, corresponding to a wavelength of about ten times the density scale length L. In the limit of small pffiffiffiffiffiffi kL, γ  2k gL, which says more mass (L) has to move pffiffiffiffiffiffiffiffifarther as the wavelength increases. In the opposite limit of large kL, γ  g=L, and the growth rate is independent of the wave number. Figure 6.23 shows the reduction in growth rates (compared to the classical value pffiffiffiffiffiffiffi αkg) for the viscous stabilized (6.305) and the density-gradient stabilized (6.309) 1=3 models. The abscissa quantities are k ðν2 =αgÞ and kL, respectively. The linear-growth regime we have considered is valid only for a single mode and only until the amplitude of the perturbation grows to about 10 percent of the wavelength. From this point on, a nonlinear regime is established, in which a number of additional phenomena become important. For example, if the initial

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Hydrodyamics

Figure 6.23. Growth-rate reduction by viscous stabilization (solid line) and density-gradient stabilization (dashed line). The abscissa quantities are different for the two. The Atwood number is 1.

perturbation is sinusoidal, higher harmonics will begin to appear and the shape of the perturbation becomes distorted. Multiple modes interact with one another, with some modes being enhanced as others are degraded. As the Rayleigh-Taylor instability continues to grow, the distortion takes the shape of spikes and bubbles as shown in Figure 6.25. Spikes are fingers of high-density fluid penetrating into the lower-density fluid, while bubbles of the latter form due to the rising of fluid into the high-density material; this is classic buoyancy. A typical rule of thumb is that the linear theory is applicable, provided d 2 z=dt 2  g, that is, αk  1. There are additional processes that can alter the growth rate of the RayleighTaylor instability. Nearly all tend to reduce the growth rate, at least for some modes. These include compressibility effects, thermal conduction (fire polishing), magnetic fields, geometrical convergence/divergence, acceleration history, and so on. There is an important feature of the Rayleigh-Taylor instability when a single mode has grown to sufficiently large amplitude that the nonlinear nature becomes apparent. The incompressible fluid requirement leads to ik x ux þ ik y uy ¼ 

∂uz , ∂z

(6.310)

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337

Figure 6.24. Shear flow (arrows) at the interface (solid line) is introduced by the Rayleigh-Taylor instability. A vortex sheet is created at the interface that will eventually result in the formation of spikes and bubbles. The heavier fluid is on top. The dashed line representing an approaching shock front will be considered shortly.

and with the solution to (6.302) yields iu1 ¼ ui and u0 ¼ u1 . The first implies that ux and uz are spatially out of phase, while the second implies the existence of a shear flow at the interface. Figure 6.24 sketches the shear flow across the interface resulting from the fluids moving in opposite directions; this results in the formation of vorticies. The vorticity lies in the interface itself in the form of a vortex sheet. Initially, the amount of vorticity is negligible, but as the shear increases, alternate regions of positive and negative circulation develop; however, the net vorticity along the interface remains approximately zero. The magnitude of the vorticity exponentiates with time along with the magnitude of the interface perturbation. As the instability continues to grow and shear effects begin to dominate, the Kelvin-Helmholtz instability begins to develop. We see the late-time effects of this instability at the top of the spikes and along their edges. The tips of the spikes are eroded and broadened by drag forces, and that deflected material is pushed back along the spike toward the high-density region. Figure 6.25 shows the development of the nonlinear phase of the Rayleigh-Taylor instability for a fluid with a high Reynolds number, a situation most common in high-energy-density physics.

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Hydrodyamics

Figure 6.25. Cross section of the time development of spikes and bubbles in the Rayleigh-Taylor instability. The Atwood number is 0.5, and the Reynolds number is several thousand.

This discussion has not addressed the effects of fluids being compressible. Unfortunately, the mathematics required to analyze the compressible version is much too complicated to get analytic solutions for, and we must turn to numerical simulations to gain some understanding. Fortunately, the incompressible theory works quite well for compressible situations, especially in the linear-growth regime. One possible explanation for this success is that unstable fluctuations that do not compress the fluid often grow faster than those that do, because the incompressible fluctuations do not spend energy on compression.

6.9.2 Kelvin-Helmholtz As we demonstrated at the beginning of this section, the mathematics of the Kelvin-Helmholtz instability is essentially the same as for the Rayleigh-Taylor instability, but with different aspects of the underlying equations being emphasized. This instability, first analyzed by Lord Kelvin (1871) and von Helmholtz (1868), can occur when there is a velocity shear in a single continuous fluid or where there is a velocity difference across the interface between two fluids. An example is that of wind blowing across water, creating surface waves. More generally, clouds, the ocean, Jupiter’s Red Spot, and the Sun’s corona show this instability. The linearized analysis proceeds along the same lines as for the Rayleigh-Taylor instability, but now there are six equations, since all three directions are involved, unlike the simplified Rayleigh-Taylor instability. In addition, the three components of the gradient of the velocity field and wave vector matter independently. Following Drake (2006), the amplitudes of the linearized equations are

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339

  proportional to exp i kx x þ ky y þ γt , which differs from the Rayleigh-Taylor case. Thus, a growing instability will be one with negative imaginary γ. After several algebraic steps, we find the equations for uz in terms of known parameters   ∂ ∂uz ∂U 2  k x uz k ðγ þ k x U Þuz þ ðγ þ k x U Þ ∂z ∂z ∂z    2 1 ∂ρ uz k ∂uz ∂U g  ðγ þ k x U Þ  k x uz ¼ : (6.311) ρ ∂z ∂z ðγ þ k x U Þ ∂z In this equation, U is the initial flow parallel to the x-axis. The right-hand side of this equation can be ignored so long as the density-scale length L is large compared to the perturbation wavelength of interest, and unless the gravitational acceleration is very large (>k2x U 2 L). Again, we need the boundary condition at the interface. One requirement is that it be the same when approached from either side; thus uz0 uz1 ¼ : γ þ kx U 0 γ þ kx U 1

(6.312)

The rest of the boundary condition is found in the same fashion as for (6.301):   uz ∂uz0 2 gk ðρ  ρ1 Þ ¼ ρ0 ðγ þ k x U 0 Þ γ þ kx U i 0 ∂z   ∂uz1 ∂U 0 ∂U 1 ρ1 ðγ þ k x U 1 Þ þ kx ρ0 uz0 þ ρ1 uz1 : (6.313) ∂z ∂z ∂z The subscripts “0” and “1” indicate the two sides of the interface, as before, and the subscript “i” indicates the continuous quantity is to be evaluated at the interface. Let us consider the simple case of the two fluids having uniform densities and uniform initial flow velocity U. For negligible gravitational effects, (6.311) becomes ∂ 2 uz ¼ k2 uz , ∂z2

(6.314)

which is identical to (6.302), and the solutions are uz ¼ A0 ekz for z > 0 and uz ¼ A1 eþkz for z < 0. Using (6.278) implies A0 ¼ A1

γ þ kx U 0 : γ þ kx U 1

(6.315)

It is convenient to work in a frame of reference corresponding to the average velocity of the two regions, because the velocity difference is what drives the

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Hydrodyamics

instability and because we often know the velocity difference in real applications; thus, ΔU ¼ 2U 0 ¼ 2U 1 . The boundary condition (6.313) becomes 0 ¼ ρ0 ðγ þ k x U 0 Þ2 þ ρ1 ðγ þ kx U 1 Þ2 ,

(6.316)

which gives the instability growth rate γ¼

pffiffiffiffiffiffiffiffiffi ρ0 ρ1 αkx : ΔU  ikx ΔU 2 ρ0 þ ρ1

(6.317)

The real part of this is finite if α 6¼ 0, and the wave propagates along the surface in the frame of reference. The negative imaginary part of γ describes the exponential growth. For equal densities, α ¼ 0 and we find γ ¼ kx ΔU=2. There are several features of (6.317) that must be noted. First, there is no minimum wave number for this process; perturbations at all wavelengths are unstable. Second, shorter-wavelength perturbations grow more rapidly. If the initial fluctuations present at a sharp interface corresponding to broadband noise, we would expect small-scale structures to appear first, followed by the large-scale features. However, we do not normally see this, which might be the result of the small-scale fluctuations being absent at the outset, or that the large-scale fluctuations are more prevalent, or that it is a poor approximation to consider the shear layer to be infinitely sharp. The third point is that the component of k along U determines the growth rate, but there is no limitation on the y-component of k. Thus, fluctuations whose wave vector makes some angle with U grow freely, albeit more slowly than those fluctuations for which k is parallel to U. If gravity is operative, (6.314) and (6.315) remain correct, but (6.313) becomes gk ðρ0  ρ1 Þ ¼ ρ0 ðγ þ k x U 0 Þ2 þ ρ1 ðγ þ kx U 1 Þ2 ,

(6.318)

with solutions h γ¼

αkx ΔU  i 2

 i1=2 k2x ðΔU Þ2 ρ0 ρ1 þ gk ρ20  ρ21 ρ0 þ ρ1

:

(6.319)

If the argument of the square-root factor is positive, there is an unstable root. The “upper” region, which we designated with the subscript 0, is defined relative to the gravitational acceleration, and thus the instability is always present if ρ0 > ρ1 . This configuration allows both the Rayleigh-Taylor and Kelvin-Helmholtz instabilities to work together and produce larger growth rates. In the opposite case, where the upper density is lower, the condition for an instability to exist requires  g ρ21  ρ20 , (6.320) k> ðΔU Þ2 ρ0 ρ1 cos 2 θ

6.9 Fluid Instabilities

341

where θ ¼ cos 1 ðkx =k Þ. Thus, the Rayleigh-Taylor dynamics for a particular wave number opposes the instability due to the Kelvin-Helmholtz instability. For a sufficiently large wave number, Kelvin-Helmholtz wins and the growth is positive. When gravity becomes large enough and ρ1 > ρ0 , the argument of the square-root factor in (6.319) becomes negative, and any modulations of the interface oscillate but do not grow. Drake (2006) continues on with additional analysis of the Kelvin-Helmholtz instability, and the reader is encouraged to pursue those. The continued growth of both the Rayleigh-Taylor and Kelvin-Helmholtz instabilities will lead to a turbulent state. Turbulence is characterized by chaotic motion of which vorticity is one particular motion. The fundamental equation of these two instabilities is (6.38), which is just (6.35) with the vorticity removed. Hence, our simple approach to these instabilities cannot treat turbulence. We then have to appeal to direct numerical simulations to understand such complex motion. 6.9.3 Richtmyer-Meshkov Both the Rayleigh-Taylor and Kelvin-Helmholtz instabilities require time to grow to destructive proportions. Rayleigh-Taylor requires the presence of acceleration, while Kelvin-Helmholtz depends upon sustained shear. But we know high-energydensity systems often involve some combination of shock waves, rarefactions, and interfaces. So we expect that these hydrodynamic “structures” will have an influence on the stability properties of the fluid, perhaps in concert with RayleighTaylor and Kelvin-Helmholtz. The discussion about both of these instabilities largely considered them to take place in incompressible material. But now we must take into account compressibility, which greatly complicates the analysis. This shock-driven “instability” was mathematically developed by Richtmyer (1960), expanding on the work of Taylor (1950), and the first experimental measurements were made by Meshkov (1969). We put the term “instability” in quotation marks, for the Richtmyer-Meshkov phenomenon is not truly an instability. Instabilities require some sort of a feedback mechanism to create the increase in magnitude of some physical quantity with time. This can be seen in the equation df =dt ¼ γf , where γ is the (exponential) growth rate of the quantity f . In the Rayleigh-Taylor instability, the feedback mechanism is the increase in the net buoyant force resulting from increased interpenetration of the two fluids. But in the case of the Richtmyer-Meshkov instability, there is no identifiable feedback process. Even so, we will continue to call this phenomenon an instability. A second misconception is that the Richtmyer-Meshkov instability is the impulsive limit of the Rayleigh-Taylor instability. Some authors argue that the variation of the acceleration in time approaches a delta function, which corresponds to a shock.

342

Hydrodyamics

In fact, the evolution of the Richtmyer-Meshkov structure occurs after the shock passes and thus is not the limiting case of growth that occurs during acceleration. At the heart of this topic is the interaction of a shock front with a rippled interface. A proper analysis should be based upon the behavior of an oblique shock interacting with a planar surface, but we elected to forgo that discussion earlier and consider only normal shocks. Rather, we present a qualitative discussion of the process. In Section 6.5, we discussed a number of situations of a planar shock front interacting with an interface. Not only is there a transmitted shock from one material to the other, but there is most likely a reflected shock created at the interface. For the case of an increase in density across the interface (see Section 6.5.5), the shock will reflect from the interface and form a rarefaction wave. This is because the reflected shock is traveling with higher velocity than the incoming shock front. If we consider a planar shock front approaching a rippled interface, as sketched in Figure 6.24, the result upon reflection is that the phase of this wave is the same as that of the interface, but with a larger amplitude than that of the rippled interface as shown in Figure 6.26a. This can be visualized by looking at the ratio of the reflected-shock velocity to the interface velocity; the postshock velocity of the interface and the transmitted shock velocity are each smaller than that of the initial shock front. The transmitted shock has the same phase as the interface, but is reduced because of the ratio of the postshock interface velocity to the incoming shock front velocity. The transmitted shock velocity will be slightly greater than that of the interface. Consequently, the initial modulations of the transmitted shock will be somewhat larger than that of the interface, but these will be rapidly damped. The rippled interface may be present initially as a static structure with no fluid motion prior to the shock front arriving, or it may appear as the result of some previous disturbance. Figure 6.24 shows the alternating lateral velocities at an interface introduced by the Rayleigh-Taylor instability. These form the seed for vortex tubes with alternating directions of rotation. The shock front deposits vorticity on both sides of the interface. For the case of the shock passing from low to high density, as shown in Figure 6.26a, any structure of the interface will grow in time. The rotations near the reflected shock front deflect the flow away from the trough of the interface, while the rotation near the transmitted shock deflects material toward the trough. The combined effect of the two shock fronts is to move material in such a way as to deepen the troughs and raise the peaks of the interface. For the case of the shock front moving from higher-density material into lowerdensity material, the situation is somewhat different; this situation was addressed, for planar shocks, in Section 6.5.7. Now the reflected rarefaction wave moves faster than the shock front’s velocity, but not by very much as it moves at the sound speed of the initially shocked matter. As a result, the ripple of the reflected wave is in phase with that of the rippled initial interface, but a series of damped

343

6.9 Fluid Instabilities (a)

(b) upstream inflow

transmitted shock high density

upstream inflow

low density

reflected shock incoming shock

transmitted shock

low density

high density

reflected shock incoming shock

Figure 6.26. Schematic of the behavior of a shock front interacting with a rippled interface (dashed line). The density jump goes (a) from low to high and (b) from high to low. The frame of reference is moving with the shock front, thus the incoming flow is from the top down. The lateral postshock flow on each side of the interface is shown by the interior arrows.

acoustic waves are present in the rarefaction fan and propagate toward the interface. In contrast, the transmitted shock moves faster than the incident shock, but the ripple of this shock is typically inverted in phase relative to the initial interface phase. However, the postshock behavior of the interface depends upon whether its postshock velocity is larger or smaller than the incident shock velocity. If the postshock velocity is large, which will occur for strong shocks if the density ratio is large enough, the phase of the interface is inverted. As the density ratio becomes smaller, eventually the velocity will be smaller and the phase of the interface will not be inverted. This behavior, for large postshock velocity, is shown in Figure 6.26b. The transmitted shock continues to deposit vorticity, but the reflected rarefaction does not deposit additional vorticity. If the density decrease is small, then the interface will first retain the phase of the initial ripple. The flow of material then causes the interface to invert before the ripples begin to grow larger. We end this section and chapter by noting that fluid instabilities eventually lead to turbulent behavior of some sort. Upon reading the literature, we find several definitions of turbulence. We suffice it to say that the Rayleigh-Taylor and KelvinHelmholtz instabilities introduce thermal energy into the fluids, which produce small-scale fluctuations that eventually grow and redistribute the energy by viscous diffusion and heating. The result is the common swirling patterns of fluid motion characteristic of turbulent systems, in which there is a wide spectrum of spatial scales of vorticity. The reader is left to ponder turbulence independently.

7 Thermal Energy Transport

When different parts of a body are at different temperatures, heat flows from the hotter parts to the cooler parts. There are three distinct methods by which this transference of heat takes place: (1) conduction, in which heat passes through the substance of the body itself; (2) convection, in which heat is transferred by relative motion of portions of the heated body; and (3) radiation, in which thermal energy is transferred directly between distant portions of the body by electromagnetic radiation. We will not discuss convection in this work, but radiation transfer will be discussed in Chapter 8. Conduction of heat through the body is driven primarily by a temperature gradient. In general, a temperature gradient is accompanied by a pressure gradient as can be seen in an equation of state, such as that of a perfect gas. In many cases, hydrodynamic energy transport dominates over that associated with heat conduction. Thermal heat conduction transports energy comparatively slowly through a medium, while a small pressure difference causes disturbances to be propagated with the speed of sound, leaving redistributions of density, and the pressure equalizes more rapidly than the temperature. In high-energy-density matter, we must consider both modes of energy transport. The basic conservation equations of momentum and energy were introduced in Section 3.6. As discussed in that section, there are really two sets of equations: one for the electron fluid and one for the ion fluid. The reason for this is the significant mass difference between an electron and a heavy particle. The kinetic equation allows for collisions between these two species. The Boltzmann transport equation for a single species, developed in Section 3.1, is ∂f F þ v  rf þ  rv f ¼ J ð f Þ, ∂t m

(7.1)

where the velocity distribution function is f ðx; v; t Þ, rv is the gradient operator in velocity space, and there is a body force Fðx; tÞ acting on a mass m. The right-hand 344

Thermal Energy Transport

345

side is the collision term, often written as ð∂f =∂t Þcoll . In the absence of collisions, J ð f Þ ¼ 0, and (7.1) becomes the Vlasov equation. Taking the first velocity moment of (7.1) together with a Maxwellian distribution function gives the momentum equation

 ∂ 1 2 ρ þ u  r u ¼ rP þ μr u þ ζ þ μ rðr  uÞ ∂t 3
 q 1 þ ρ Eþ uB þR : (7.2) m c A note of caution: in these equations q is used both for charge and for heat flux. The third term on the right-hand side of (7.2) was discussed in Section 6.6, with μ and ζ being the shear and bulk viscosities, respectively. The next part uses the Lorentz force, and the last term on the right-hand side is the frictional force due to collisions ð R ¼ m vJ ð f Þdv: (7.3) The second velocity moment gives the total energy equation  

 ∂ u2 1 1 εþ ρ ¼ r  ρuε þ ρuu2 þ Pu  μ u  ru þ r u2 ∂t 2 2 2
  2 q  ζ  μ ðr  uÞu þ q þ ρE  u þ R  u þ Q : (7.4) 3 m The last term on the right-hand side is the energy exchanged between particles due to collisions ð m vvJ ð f Þdv: (7.5) Q¼ 2 Subtracting the kinetic energy from (7.4) gives the internal energy ∂ ðρεÞ ¼ r  ðρuεÞ  r  ðPuÞ þ Φ  r  q þ Q, ∂t

(7.6)

where the dissipation function is given by

 1 ∂ui ∂uj 2 2 Φ¼ μ þ þ ζ  μ ðr  uÞ2 , 2 ∂xj ∂xi 3

(7.7)

which is found from the stress tensor (see Section 6.6) and is always nonnegative. The quantity in parenthesis in the first term is twice the strain-rate tensor. The heat flux is given by Fourier’s law q ¼ κrT, where κ is the thermal conductivity coefficient.

346

Thermal Energy Transport

We wish to examine the nature of the internal energy equation. The easiest way to do this is to write (7.6) in one-dimensional, planar geometry ∂ ∂ ∂u ∂q ðρεÞ þ ðρuεÞ þ P þ ¼ 0: ∂t ∂x ∂x ∂x

(7.8)

We have omitted the Lorentz and collision terms for the present time, and we assume an inviscid fluid, that is, μ ¼ ζ ¼ 0. Our interest, for the moment, is focused on the last term in (7.8), that of thermal conduction, in particular the conductivity coefficient κ. 7.1 Linear Heat Conduction For planar geometry, we consider the case where there is no fluid motion. Then (7.8) can be written

 ∂ε ∂T ∂ ∂T ¼ κ ρ , (7.9) ∂T V ∂t ∂x ∂x where ð∂ε=∂T ÞV ¼ cV is the specific heat at constant volume. If we assume the thermal conductivity and specific heat change very little over a small range of temperatures, we may consider them constant. The assumption of a constant thermal conductivity is reasonable for low temperatures, meaning temperatures less than that required for “vaporization” or perhaps ionization. For one-dimensional geometry of infinite extent, (7.9) can be written
 ∂T ∂ ∂T ∂2 T (7.10) ¼ χ ¼χ 2, ∂t ∂x ∂x ∂x where the diffusivity is given by χ ¼ κ=ρcV , a constant. Let a delta-function energy source, with respect to time and space, be placed at the center of the slab, thus T ðx; 0Þ ¼ Qδx. The thermal energy is found from þ∞ ð

Q¼

Tdx:

(7.11)

1 x2 =4χt e , t 1=2

(7.12)

∞

A particular solution to (7.10) is T ðx; t Þ ¼

and has the properties T ! 0 as t ! 0 for x 6¼ 0, T ! ∞ as t ! 0 at x ¼ 0, and þ∞ Ð pffiffiffiffiffi Tdx ¼ 2 πχ for all t > 0. This last property corresponds to the instantaneous

∞

7.1 Linear Heat Conduction

347

pffiffiffiffiffi release of a quantity of heat Q ¼ 2ρcV πχ per unit area at x ¼ 0 and t ¼ 0. Expression (7.12) can be written T ðx; t Þ ¼

1

0 2

ð4πχt Þ

1= 2

eðxx Þ =4χt :

(7.13)

Since this equation is linear, the sum of any number of particular integrals is also an integral, thus T¼

þ∞ ð

1 1= 2

ð4πχt Þ

0 2

f ðx0 Þeðxx Þ =4χt dx0 :

(7.14)

∞

Suppose that at t ¼ 0 there is a region about the center of the slab a < x < a for which the temperature is T 0 and everywhere else it is zero. Thus, (7.14) becomes 80 9 ða ð < = 1 0 ðxx0 Þ2 =4χt 0 0 ðxx0 Þ2 =4χt 0 T¼ T e dx þ T e dx : (7.15) 1= ; ð4πχt Þ 2 : a

0

The general solution to (7.10) for the infinite slab is 

 T0 ax aþx : erf pffiffiffiffiffiffiffi þ erf pffiffiffiffiffiffiffi T ðx; tÞ ¼ 2 4χt 4χt

(7.16)

To lowest order, (7.16) is T ðx; t Þ ¼

Q ð4πχt Þ

ex =4χt , 2

1= 2

(7.17)

where Q ¼ 2aT 0 . While the heat is concentrated at the point of initial energy release and as time moves forward, the heat instantaneously propagates throughout all of space with the temperature tending asymptotically toward zero at x ! ∞. The majority of the energy is concentrated in a region whose dimension is pffiffiffiffiffiffiffi 1 approximately x ∽ 4χt and increases with time proportionally to t =2 . Therefore, 1 the temperature must decrease by t =2 so that the energy remains constant. A series of temperature profiles is shown in Figure 7.1. The law of heat propagation can be easily obtained by estimating the order of magnitude of the characteristic dimension of the heated region or from dimensional considerations. For constant thermal conductivity, (7.10) contains only a single parameter χ. The other dimensional parameter is the energy Q. If x is the width of the region where most of the heat is concentrated, then from pffiffiffiffiffiffi ffi dimensional considerpffiffiffiffi ations x ∽ χt, and the propagation rate is dx=dt ∽ χ=t ∽ x=t. The heated region pffiffiffiffi has a temperature T ∽ Q=x ∽ Q= χt. We may also replace the derivatives

348

Thermal Energy Transport

Figure 7.1. Temperature distribution spreads to ∞ instantly for the linear thermal conduction model. The initial temperature distribution is a delta function at x ¼ 0. Curves are for relative times of 0.1, 0.3, and 1.0.

in (7.10) by finite differences and obtain results. Evaluating (7.11), we Ð the same 1 1 see that the energy remains constant: Tdx ∽ Tx ∽ 1=t =2 t =2 ¼ 1. The asymptotic character of the temperature decrease at infinity and the instantaneous propagation of energy to infinite distance is explained by the fact that the conductivity remains finite at zero temperature. The actual behavior at large distances is governed by the gas particles arriving there without going through the diffusion process. The temperature is governed by the exponential relation T ∽ ex=λ , where λ is the mean free path. No matter the size of the pre-exponential factor, for a given time, the simple exponential will eventually become greater than the Gaussian exponential. However, this region at large distance contains negligible heat, so that consideration of it is not important.

7.2 Nonlinear Heat Conduction In contrast to linear heat conduction is the case where the conductivity is a function of temperature, such that it has zero value at zero temperature. The heat front will now propagate at a finite velocity because there is a sharp boundary between the heated region and the initially cold region; that is, the heat propagates as a “wave.”

7.2 Nonlinear Heat Conduction

349

At the position of the heat front, continuity requires that the flux goes to zero. In contrast, for the constant conductivity case, the vanishing of the heat flux is attributed only to the vanishing of the temperature gradient, while in the nonlinear case, the heat flux vanishes because the temperature vanishes. We can assume the temperature  distribution near the heat front is in the form of a standing wave T ¼ T x  vf t , where vf is the front’s velocity. Using this in (7.10), we see
 ∂T ∂ ∂T vf ¼ χ : (7.18) ∂x ∂x ∂x If the diffusivity is χ ¼ aT n , with n positive, then upon integrating (7.18) twice along with the boundary condition T ¼ 0 at the front x ¼ xf , we obtain the temperature distribution nv

1=n f x  xf , T ðxÞ ¼ (7.19) a

11 and the temperature gradient there is dT=dx ∽ x  xf n . We can develop an exact expression for the nonlinear heat propagation using the methods of self-similarity (Zel’dovich & Raizer, 1966). Rewriting (7.9) as ∂T ∂ ∂T ¼ a Tn , ∂t ∂x ∂x

(7.20)

we see there is but the single parameter a. The other dimensional parameter is the quantity Q. Combining these into an independent quantity containing only the length and time, we have aQn . This law gives the position of the heat front as a function of the total energy and the time 1

1

xf ∽ ðaQn Þnþ2 t nþ2 ,

(7.21)

and the “speed” of the heat front 1 nþ1 dxf aQn ∽ ðaQn Þnþ2 t nþ2 ∽ nþ1 : dt xf

(7.22)

For large n, the thermal wave is slowed very rapidly. The average temperature in the thermal wave is of the order T ¼ Q=xf and the average diffusivity is χ ∽ aQn =xnf , so the propagation of the heat front is of the form similar to that for pffiffiffiffi the linear theory xf ∽ χt . In contrast to the linear theory, the diffusivity is a function of time according to χ∽

2 aQn n ∽ ðaQn Þnþ2 t nþ2 : n xf

(7.23)

350

Thermal Energy Transport

From (7.21), it is evident that the only dimensionless combination of space, time, and the parameters a and Q is the similarity variable x : (7.24) ξ¼ 1 1 n nþ2 nþ2 ðaQ Þ t The quantity




Q 1

1

ðaQn Þnþ2 t nþ2

¼

Q2 at

1 nþ2

(7.25)

has the units of temperature, and thus we should seek a solution of the form 1
2 nþ2 Q T ðx; tÞ ¼ f ðξ Þ, (7.26) at where f ðξ Þ is an unknown function. Substituting (7.26) into (7.20) and transforming to the similarity variable yields two equations ∂f 1 df ¼ , (7.27) 1 1 ∂x ðaQn Þnþ2 t nþ2 dξ and ∂f 1 ξ df ¼ , ∂t n þ 2 t dξ

(7.28)

from which an ordinary differential equation for the unknown function is obtained
 d df n df ðn þ 2Þ f þ ξ þ f ¼ 0: (7.29) dξ dξ dξ The boundary conditions are, by symmetry, ∂T=∂x ¼ 0 at x ¼ 0 and T ¼ 0 at x ¼ ∞. Thus, df =dξ ¼ 0 at ξ ¼ 0, and f ðξ Þ ¼ 0 at ξ ¼ ∞. The solution to the unknown function is found in Zel’dovich and Kompaneets (1959) h i1=n 2 2 n f ðξÞ ¼ 2ðnþ2Þ ðξ 0  ξ Þ for ξ < ξ 0 , f ðξÞ

¼ 0

for

ξ > ξ0 ,

(7.30)

where ξ 0 is a constant of integration, which is found from (7.11). Performing the calculation gives 1 "  n " #nþ2 #nþ2 Γ 12 þ 1n ðn þ 2Þ1þn 21n  ξ0 ¼ , nπ n=2 Γ 1n

(7.31)

351

7.2 Nonlinear Heat Conduction

Figure 7.2. Temperature distribution spreads to a finite distance for the nonlinear thermal conduction model with n ¼ 5=2. The initial temperature distribution is a delta function at x ¼ 0. Curves are for relative times of 0.1, 0.3, and 1.0.

where Γ is the gamma function. Equation (7.24) gives the heat front location as 1

1

xf ¼ ξ 0 ðaQn Þnþ2 t nþ2 :

(7.32)

From (7.26), the temperature profile at any time is T ðxÞ ¼ T c

x2 1 2 xf

!1=n ,

(7.33)

where the central temperature is T c ¼ T =J. The average temperature is T ¼ Q=2xf and ð1

 1=n J ¼ 1  z2 dz ¼ 0

 pffiffiffi π Γ 1n  : n þ 2 Γ 12 þ 1n

(7.34)

 1= The temperature near the heat front is approximately T ∽ xf  x n . Figure 7.2 gives the temperature profiles (7.33) for several different times.

352

Thermal Energy Transport

Figure 7.3. Distribution of flux (solid line) and its divergence (dashed line) for the nonlinear thermal conduction model with n ¼ 5=2.

The heat flux and its divergence in planar geometry are 2  x 2 3 n n 1 1þ2 n xf ∂T T x ∂q 2 T c 6 7 qðxÞ ∽  T n ¼ 2 c 2 T ðxÞ and ∽ 4 2 5T ðxÞ: (7.35) 2 ∂x ∂x n xf n xf 1  xxf Figure 7.3 plots the heat flux and its spatial derivative. The flux increases linearly from x ¼ 0 to the heat front and then drops rapidly to zero. In contrast, the divergence of the heat flux is nearly constant over the entire region where the medium is cooling, but just inside the heat front the medium is being heated by energy removed from the matter behind the front. The linear heat conduction equation is recovered in the limit of the exponent pffiffiffi n ! 0. Thus, ξ 0 ! 2= n and so "
1= # Q Q nx2 n Q 2 ½ f ðξ Þn!0 ¼ T¼ 1 ¼ ex =4at , (7.36) 1= 1= 1= 2 2 2 4at ðat Þ ð4πat Þ ð4πat Þ n!0

which is just (7.17) with a ¼ χ. The important case of nonlinear heat conduction from a point source can be found by redoing the preceding analysis in spherical geometry; the results are quite

7.3 The Heat Flux

353

similar to that for the planar source. For example, the speed of heat front propagation, compared to (7.22), is 1

1

drf ðaQn Þ3nþ2 aQn T nþ3 ∽ ∽a , ∽ 3nþ1 dt r3nþ1 Q1=3 t3nþ2 f

(7.37)

1=

speed is greater than the sound since rf ∽ðQ=T Þ 3 . This heat front propagation pffiffiffiffi speed for sufficiently high n, since cs ∽ T . For a perfect gas, the pressure is P∽ρT and the pressure distribution closely follows the temperature. As the temperature decreases rapidly at the heat front, the pressure gradient there causes the gas to be accelerated and there is an “explosive” accumulation of the gas at the heat front. Because the disturbance travels at the speed of sound, the heated gas downstream of the heat front has minimal motion. The propagating thermal front rapidly decelerates. When its speed is about that of sound, the fluid is set in motion and a shock front forms, which then moves ahead of the thermal front. The problem then has the characteristics of a Sedov-Taylor blast wave discussed in Section 6.5.8. 7.3 The Heat Flux The energy transport equation (7.6) has a term for the divergence of the heat flux r  q. The heat flux q ¼ κrT is the kinetic energy of a particle times the flux of particles. In terms of the distribution function, the heat flux may be written ð
2 mv q¼ n (7.38) vf ðvÞd3 v, 2 where n is the particle number density. If we take the velocity distribution function to be a Maxwellian
f MB ðvÞ ¼

m 2πk B T

3=2

 exp mv2 =2k B T ,

(7.39)

we see in combining (7.38) with (7.39) that v2 is an even function, while v is an odd function, and thus the integral vanishes identically, that is, q ¼ 0! The true distribution function is often close to that of a Maxwellian but is not symmetric. This occurs where there is, for example, a slight temperature difference in the medium. The distribution function, being temperature dependent, is such that a slightly higher temperature results in an increase in the number of particles with energies above k B T, while the number of particles with lower energy decreases. The result is that there is a movement of particles through a point in space between the two different temperature regions such that the warmer region will include a surplus of hot particles and a deficit of cold ones. Thus, for an initial equilibrium

354

Thermal Energy Transport

distribution, there will develop a non-Maxwellian distribution that is asymmetric in velocity and thus able to carry heat. We now specify the type of particle to be electrons and can determine the nonMaxwellian distribution by looking at the collision term in the Boltzmann equation (7.1). From Section 3.1, for electron–ion collisions, ðð J ð f Þ ¼ ½ f ðv0 Þ f i ðvi 0 Þ  f ðvÞ f i ðvi ÞvR dσ ðv ! v0 Þd3 vi : (7.40) Here, vR is the relative velocity of the two species and f ðvÞ is the distribution function of the electrons before collision with the ions, with distribution function f i ðvi Þ. After the collision, the velocities take on the primed values. Because of the large disparity of masses between the two species of particles, the electron’s velocity changes only in direction but not in magnitude. We may also assume the velocity of the ion is not changed significantly, and vR  v. Thus ð J ð f Þ ¼ ni ½ f ðv0 Þ  f ðvÞvdσ ðv ! v0 Þ, (7.41) where the ion number density is defined by ð ni ¼ f i ðvi Þd3 vi :

(7.42)

We shall assume that the mean free path for collisions is small compared to macroscopic dimensions, and thus the distribution function is approximately Maxwellian. This is saying that the electron distribution function is nearly spherically symmetric in velocity space. Expanding the distribution function in terms of Legendre polynomials and keeping the leading terms gives f ðx; v; t Þ  f ð0Þ ðx; v; tÞ þ

v ð1Þ  f ðx; v; tÞ: v

(7.43)

The spherically symmetric component f ð0Þ (the Maxwellian distribution) and the next-order term f ð1Þ are functions of the magnitude of the velocity only. Using (7.43) in the Boltzmann equation (7.1), we find for each term ∂f ∂f ð0Þ v ∂f ð1Þ ¼ þ  , ∂t v ∂t ∂t v v  rf ¼ v  rf ð0Þ þ r  f ð1Þ , 3
 F F 1 ∂ v2 F ð1Þ ð0Þ  rv f ¼  rv f þ 2 f : me me v ∂v 3 me

(7.44)

7.3 The Heat Flux

355

We have used the fact that for a vector A that is independent of the direction of v, the quantity vv  A is replaced by the value averaged over all directions of v, that is, vv  A ¼ v2 A=3. Then (7.1) becomes "
# ∂f ð0Þ v 1 ∂ v2 F ð1Þ ð1Þ f þ rf þ 2 3 v ∂v 3 me ∂t ! ð0Þ v ∂f ð1Þ F ∂f ¼ Jð f Þ : (7.45) þ  þ vrf ð0Þ þ v me ∂v ∂t Returning to (7.41), since v0 ¼ v, it follows that f ð0Þ ðx; v; t Þ ¼ f ð0Þ ðx; v0 ; t Þ and f ðx; v; t Þ ¼ f ð1Þ ðx; v0 ; tÞ, and the f ð0Þ term drops out. Thus, (7.41) becomes ð J ð f Þ ¼ ni f ð1Þ ðx; v; tÞ  ðv0  vÞdσ ðv ! v0 Þ: (7.46) ð1Þ

The differential scattering cross section is dσ ¼ σ 0 ðθs Þ sin θs dθs dϕs ,

(7.47)

where θs is the scattering angle between the vectors v and v0 , and ϕs is the azimuthal angle. Define the angles θ and θ0 by f ð1Þ  v0 ¼ f ð1Þ v cos θ0 and f ð1Þ  v ¼ f ð1Þ v cos θ:

(7.48)

The law of cosines gives cos θ0 ¼ cos θ cos θs þ sin θ sin θs cos ϕs , and then integrating (7.46) over ϕs gives 2π 3 ð J ð f Þ ¼ 2πni 4 ð cos θs  1Þσ 0 ðθs Þ sin θs dθs 5f ð1Þ v cos θ:

(7.49)

(7.50)

0

Defining the collision time to be ðπ 1 ¼ 2πni v ð1  cos θs Þσ 0 ðθs Þ sin θs dθs , τ

(7.51)

0

and since v ¼ v cos θ, we arrive at v 1 J ð f Þ ¼   f ð1Þ : v τ

(7.52)

This expression for J ð f Þ is then used in (7.45). Since this equation must hold identically for all v, we may equate to zero separately the coefficient of v

356

Thermal Energy Transport

and the term independent of ðv=vÞ. This yields two coupled equations that determine f ð0Þ and f ð1Þ
 ∂f ð0Þ v 1 ∂ v2 F ð1Þ ð1Þ ¼ 0, (7.53) f þ rf þ 2 3 v ∂v 3 me ∂t and ∂f ð1Þ F ∂f ð0Þ 1 þ vrf ð0Þ þ ¼  f ð1Þ : me ∂v τ ∂t Using (7.43) in the definition for the particle density, we have ð ð ne ðx; t Þ ¼ fdv ¼ f ð0Þ d 3 v,

(7.54)

(7.55)

and similarly for the particle flux

ð ð 1 ne v ¼ vfdv ¼ vf ð1Þ d 3 v: 3

(7.56)

We have assumed that the collision time is small compared to macroscopic periods, and ∂f ð1Þ =∂t can be neglected compared to ð1=τ Þf ð1Þ in (7.54). Then ! F ∂f ð0Þ ð1Þ ð0Þ : (7.57) f ¼ τ vrf þ me ∂v Substituting this into (7.56) gives the matter flux "
ð #  ð 1 F ∂f ð0Þ 3 2 ð0Þ 3 ne v ¼  r τv f d v þ τv d v : 3 me ∂v

(7.58)

Assuming the particle velocity to be small compared to the thermal speed, that pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi is, v  2kB T e =me , then ð 3nk B T e v2 f ð0Þ d3 v ¼ (7.59) me and ð v

∂f ð0Þ 3 d v ¼ 3ne : ∂v

(7.60)

Expression (7.58) becomes ne v ¼ 

ne k B kB T e F τrT e  τrne þ ne τ : me me me

(7.61)

7.4 Thermal and Electrical Conductivities

357

The flux of particles is driven by a temperature gradient, a density gradient, and/or an external force. If the scattering cross section is constant, then (7.51) may be integrated 1 ¼ 4πσ 0 ni v: τ

(7.62)

Then for F ¼ 0, (7.57) can be written f ð1Þ ¼ 

1 rf ð0Þ ¼ λrf ð0Þ , 4πσ 0 ni

(7.63)

where λ is the mean free path. The expression for the flux, equation (7.58), becomes ð 1 λhvi ne v ¼  r vf ð0Þ d3 v ¼  (7.64) rne , 3 3 Ð since ne hvi ¼ vf ð0Þ d 3 v. The continuity equation then yields the diffusion equation ∂ne λhvi 2 r ne : ¼ 3 ∂t

(7.65)

A characteristic diffusion time τ d is obtained by setting ∂=∂t ¼ 1=τ d and r2 ∽1=L2 , where L is a dimension over which diffusion occurs. Then 1 λhvi  2: τd 3L

(7.66)

As we shall see later in this chapter, the diffusion equation breaks down when the temperature scale length exceeds about thirty times the electron mean free path. When this condition is greatly violated, the limit on the heat flux is determined by the particle energy transported at some characteristic thermal velocity; we can express this free-streaming heat flux as qfs ¼ f e ne k B T e vth , where f e is a flux reduction multiplier.

7.4 Thermal and Electrical Conductivities Having just developed the basic description of energy transport, we need to determine the “conductivities.” Since there is both heat and particle transfer (and therefore charge transfer), the conductivities must be treated simultaneously. The traditional study of thermodynamics is concerned with reversible processes, for which the expression for the change in heat energy, dQ ¼ TdS, is valid. Yet there are a number of empirical or phenomenological relations governing

358

Thermal Energy Transport

irreversible processes. Among such relations are Ohm’s law (J ¼ σE), Fourier’s law (q ¼ κrT), and Fick’s law for diffusion for particles. The electrical and thermal conductivities, σ and κ, respectively, are often based on a simple calculation of electron–ion collisions. However, the two conductivities are not independent. The presence of a temperature gradient warps the velocity distribution, and a net flow of electrons appears. Similarly, an electric field produces a flow of heat. The combined effects result in an irreversible thermodynamic process, which we now review.

7.4.1 Onsager Relations Consider an electrical conductor of finite length and cross section with an applied temperature difference ΔT and a potential difference ΔΦ between its end points. Both an electric current I and a thermal current W are present. Each “force” results in a current, and thus the entropy production rate is dS ΔΦ ΔT ¼I þW : dt T T

(7.67)

It is an interesting fact of experimental physics that, in the absence of a potential difference, a heat current depends only on the temperature difference. But when there is a potential difference as well, the heat current depends upon both the temperature difference and the potential difference. Similarly, when both temperature and potential differences exist, the electrical current also depends upon both of these differences. Both the heat flow and the electricity flow are irreversible coupled flows, which exist by virtue of departure from equilibrium conditions in the wire. If the departure from equilibrium is not too great, it may be assumed that both I and W are linear functions of the temperature and potential difference. Hence, we may write I ¼ l11

ΔΦ ΔT þ l12 T T

(7.68)

ΔΦ ΔT þ l22 : T T

(7.69)

and W ¼ l21

These are the famous Onsager equations, which express the linearity between the currents and the generalized forces ΔΦ=T and ΔT=T. The ljk are coefficients connected with the electrical resistance, thermal conductivity, and the thermoelectric properties of the wire. Only three of the four coefficients are independent, for it can be proved rigorously by means of statistical mechanics

7.4 Thermal and Electrical Conductivities

359

that if the departure from equilibrium is small, l12 ¼ l21 , which is known as Onsager’s reciprocal relation (Onsager, 1931). Consider two heat reservoirs connected by the conductor. The first reservoir is at fixed temperature T and zero potential, the second reservoir is at temperature T þ ΔT and potential ΔΦ. Assuming electrons carry the energy, let N denote the number transferred between the first and second reservoirs. The energy transfer is Δℰ ¼ Δℰ1 ¼ Δℰ2 . The combination of the first and second laws of thermodynamics is dℰ ¼ TdS  PdV þ μN:

(7.70)

Neglecting the expansion term, the differential entropy in reservoir one is ΔS1 ¼ 

ΔU μ þ N, T T

(7.71)

where μ, a function of temperature, is the chemical potential for Φ ¼ 0. Similarly, for reservoir two
 Δℰ μ½T þ ΔT  þ eΔΦ N ΔS2 ¼ : (7.72) T þ ΔT T þ ΔT Summing the two gives the differential entropy for the system

 1 1 μ½T þ ΔT þ eΔΦ μ  N  ΔS ¼ ΔS1 þ ΔS2 ¼ Δℰ T þ ΔT T T þ ΔT T
  

ΔT ΔT ∂ μ ΔΦ  Δℰ  2  eN þ : (7.73) e ∂T T T T The entropy production rate is
   dðΔSÞ dðΔℰÞ ΔT dN ΔT ∂ μ ΔΦ ¼  2 e þ ¼ J 1X1 þ J 2X2 , dt dt dt e ∂T T T T where

d ðΔℰÞ ¼ thermal energy current, dt ΔT X 1 ¼  2 ¼ generalized energy current force, T dN ¼ electric current, and J2 ¼ e dt ΔT ∂ μ ΔΦ X2 ¼   ¼ generalized electric force: e ∂T T T

(7.74)

J1 ¼

(7.75)

Using this definition of currents and forces, we can write the currents as P J j ¼ Ljk X k , where the Ljk satisfy the Onsager relation L12 ¼ L21 . Then the two currents (7.68) and (7.69) may be recast as

360

Thermal Energy Transport


I ¼ L11


  ΔT ∂ μ ΔΦ ΔT  þ L12  2  e ∂T T T T

(7.76)


  ΔT ∂ μ ΔΦ ΔT   þ L22  2 : e ∂T T T T

(7.77)

and
W ¼ L21

It remains for us to determine the Onsager coefficients Ljk , which require the use of the kinetic theory developed in Section 3.1.

7.4.2 Transport Coefficients Accurate conductivity coefficients are crucial to understanding the heat flow in dense plasma. The Chapman-Enskog theory (Chapman & Cowling, 1953) is primarily concerned with the properties of gases composed predominantly of neutral atoms. This theory, however, is not well suited to handle inverse-square forces between charged particles. The proper approach to this subject was pioneered in the work of Chandrasekhar (1943) on stellar dynamics. The basis of this work is that for inverse-square forces, the velocity distribution function is affected primarily by the many small deflections produced by relatively distant encounters. There will be many such encounters during the time a particle travels over its mean free path. The assumption is made that the large-angle deflections produced by the relatively close encounters may be neglected. This was explored by Cohen, Spitzer, and Routly (1950). The theory for transport coefficients was improved by Spitzer and Härm (1953). Their theory included the electron–electron encounters neglected by Cohen et al. They solved the Fokker-Planck equation numerically for a completely ionized hydrogen gas, taking into account all mutual interactions among the ions and electrons. (The Fokker-Planck equation is just a form of the collisional Boltzmann equation.) This produced an accurate nonequilibrium distribution function. The problem is to calculate the electron distribution function resulting from many collisions between charged particles in an inverse-square law force. Consider a single electron as it moves through the gas. The random electrical fields encountered by the electron produce deflections and changes in velocity. These two-body encounters are characterized by the impact parameter b, as discussed in Section 3.8. The situation is characterized by four quantities: d, the mean distance from the test electron to its nearest neighbor; b90 , the value of the collision parameter for 90 deflections by a stationary positive ion; λD , the Debye shielding length discussed in Section 3.5; and λ90 , the mean free path for a net deflection of 90 . For virtually all situations of interest, b90  d  λD  λ90 .

7.4 Thermal and Electrical Conductivities

361

Encounters for which b  d can be described in terms of successive two-body encounters, since usually an encounter with one particle will be effectively over before another particle approaches to a distance less than d. These successive encounters may be divided into two classes: those with b b90 produce large deflections and are labeled “close” encounters, and those with b90 < b < d produce relatively small deflections and are called “distant” encounters. The cumulative effect of many distant encounters outweighs the effect of the less frequent close encounters. Encounters for which d b < λD cannot be regarded as independent, since several such encounters will be taking place at the same time. In this case, the encounters must be attributed to statistical fluctuations of the electron density within a sphere of radius b. However, the mean-square change of electron velocity produced by such fluctuations is correctly given if the formulas derived for successive two-body encounters are applied for b > d. Lastly, particles passing at a distance large compared to the Debye length produce a negligible effect. We first take the simple approach of an electron gas in the Lorentz approximation, which assumes the ions are fully ionized and stationary and that the electron mean free path is governed solely by electron–ion collisions; electron–electron interactions are neglected. The one-dimensional, steady-state Boltzmann transport equation from Section 3.1, in the relaxation time approximation (Krook term) for the collision term, can be written
 ∂f e ∂f ð0Þ f ¼ f  τc v  E , (7.78) ∂x me ∂v where τ c is a relaxation time; we have omitted the magnetic field of the Lorentz force to keep the problem one-dimensional. If we assume a weak electric field and a small temperature gradient so that the departure of the distribution function f

ð0Þ ð0Þ ð0Þ

from its equilibrium value f is small, that is, f  f =f  1, then the occurrences of the distribution function within the parentheses of (7.78) may be replaced by the equilibrium values. In general, f ð0Þ is a function of energy ε, chemical potential μ, and electron temperature T e ; the energy is a function of the velocity.1 The two derivatives in (7.78), for an equilibrium distribution, are ∂f ð0Þ ∂f ð0Þ dμ ∂f ð0Þ dT e þ ¼ ∂x ∂μ dx ∂T e dx

(7.79)

and 1

Within this subsection, we represent the single-particle energy as ε, rather than E as used elsewhere; we do this to avoid confusion with the electric field E.

362

Thermal Energy Transport

∂f ð0Þ ∂f ð0Þ dε ∂f ð0Þ ¼ me v ¼ , ∂v ∂ε dv ∂ε

(7.80)

with ε ¼ me v2 =2. The electrical conductivity is defined under the conditions dT e =dx ¼ 0 and dne =dx ¼ 0. Since there is a relation between electron density and chemical potential, ∂f ð0Þ =dx ¼ 0, (7.78) becomes f ¼ f ð0Þ þ eEvτ c

∂f ð0Þ : ∂ε

The electric current density is defined by ð ð ∂f ð0Þ 3 3 2 J e ¼ ne e vf d v ¼ ne e Eτ c v2 d v: ∂ε

(7.81)

(7.82)

By taking τ c outside the integral, we are assuming that the relaxation time is independent of the velocity. For a Maxwellian distribution, f ð0Þ ¼ f MB ðvÞ, we find ∂f ð0Þ 1 ð0Þ ¼ f , kB T e ∂ε

(7.83)

ð ne e2 E τ c v2 f ð0Þ d3 v: Je ¼ kB T e

(7.84)

so that (7.82) is

The integral in (7.84) evaluates to kB T e =me . Thus, J e ¼ σE, where the DC electrical conductivity is σ dc ¼

ne e2 τc: me

(7.85)

The choice of a Fermi-Dirac distribution, for μ=kB T  1, yields the same result for the conductivity as for a Maxwellian distribution, except that now we do not have to assume τ c is independent of velocity; it is only the value of τ at the Fermi surface that enters the conductivity. Let us rewrite (7.76) and (7.77) as current densities J e ¼ L 11 E þ L 12

d ðk B T e Þ dx

(7.86)

J q ¼ L 21 E þ L 22

d ðk B T e Þ , dx

(7.87)

and

7.4 Thermal and Electrical Conductivities

363

and now determine the coefficients L jk . We already have L 11 ¼ σ dc . Using the Maxwellian distribution in (7.79), we have
 ∂f 0 ε 3 ð0Þ 1 d ðk B T e Þ  f , (7.88) ¼ kBT e 2 kB T e dx ∂x so that (7.78) becomes f ¼f

ð0Þ


 ∂f ð0Þ 3 1 d ðkB T e Þ :  vτ c ε  kB T e f ð0Þ þ eEvτ c 2 ∂ε ðkB T e Þ2 dx

Consequently, from (7.82) and (7.86)  ð
ne e 3 2 L 12 ¼ τ c v ε  kB T e f ð0Þ d3 v: 2 ðkB T e Þ2 The first integral evaluates to ð ð 2πme 6 ð0Þ 5 ðkB T e Þ2 2 ð0Þ 3 v f dv ¼ v εf d v ¼ , 2 me 3

(7.89)

(7.90)

(7.91)

and the second integral is 3ðk B T e Þ2 =2me , so that L 12 ¼

ne e τc: me

(7.92)

In a similar fashion, the thermal current density (energy flux) is given by ð 5 ne ek B T e 5ne kB T e dðkB T e Þ , (7.93) Eτ c  τc J q ¼ ne v ε f d 3 v ¼  2 me dx me where (7.89) has been used and the coefficients in (7.87) are L 21 ¼ 

5 ne ek B T e 5ne kB T e τ c and L 22 ¼  τc: 2 me me

(7.94)

We can verify the validity of the Onsager relation L12 ¼ L21 in (7.76) and (7.77). A direct comparison with (7.86) and (7.87), keeping in mind that E ¼ ∂Φ=∂x, gives (apart from constant factors relating to length and cross section of the wire) L11 , kB T e
 L11 1 ∂ μ L12 ¼  ,  e k2B ∂T e T e ðk B T e Þ2 L21 ¼ , kB T e
 L21 1 ∂ μ L22 ¼  :  2 e k B ∂T e T e ðkB T e Þ2

L 11 ¼ L 12 L 21 L 22

(7.95)

364

Thermal Energy Transport

Now if L12 ¼ L21 , we must have L 12


 kB T e 1 ∂ μ L 21 L 11 2 ¼ :  e kBT e k B ∂T e T e

Upon substituting the L jk from (7.85), (7.92), and (7.94), together with
 ∂ μ 3 , ¼ ∂T e T e 2k B T e

(7.96)

(7.97)

for a nondegenerate classical gas, into (7.96) gives the result 

ne e 3ne e 5ne e τc ¼ τc  τc, me 2me 2me

(7.98)

which is indeed true. Therefore, the Onsager relation is satisfied by our solution. The thermal conductivity κ is not simply given by L 22 , because the thermal conductivity is usually measured with J e ¼ 0 rather than with E ¼ 0. The requirement that the electric current vanish means that E¼

L 12 d ðk B T e Þ , L 11 dx

(7.99)

so that
J q ¼ L 22

 L 12 L 21 dðkB T e Þ d ðk B T e Þ ¼ κeff : 1 dx dx L 11 L 22

(7.100)

The thermoelectric effects represented by (7.93) and (7.100) act to reduce the effective thermal conductivity. In a steady state, no current can flow in the direction of the temperature gradient, as a current divergence would result, and electric fields would rise rapidly without limit. It is the secondary electric field produced such that the current generated by the temperature gradient is canceled. This secondary electric field, in turn, acts to reduce the flow of heat, and the effective conductivity coefficient is then κeff ¼

5ne k B T e τc: 2me

(7.101)

The determination of the relaxation time τ c takes a bit more effort, in that it requires careful calculation of the collisions among the particles. The original paper of Cohen et al. (1950) assumed a completely ionized gas in a weak electric field. They stressed the need to include both close and distant encounters. To illustrate this, we may compute the cumulative squared value of the deflection

7.4 Thermal and Electrical Conductivities

365

Table 7.1 Cumulative Mean-Square Deflection Produced by Encounters with b < bmax bmax =b90 P

sin 2 χ



0

1

2

4

0.00

0.19

0.81

1.89

102

10 3.63

8.21

104

108

17.4

35.8

angle χ produced during the interval Δt by all encounters for which the impact parameter b is less than some upper limit bmax 2 3 ! * + 2 X b 1 5,  sin 2 χ ¼ 4πv0 ni b290 Δt 4 log 1 þ max (7.102) b2 b290 1 þ b290 b T i ð0Þ and T e ð0Þ < T i ð0Þ, respectively. The curves are parametric in the initial temperature ratio γ ¼ T i ð0Þ=T e ð0Þ for ξ positive, and γ ¼ T e ð0Þ=T i ð0Þ for ξ negative.

For me  mi , integration of (7.136) gives o 3= 3= t 2n ½1 þ ξð0Þ 2  ½1 þ ξðtÞ 2 ¼ τ eq 3   n o 1= 1= ξð0Þ 2 2 þ2 ½1 þ ξð0Þ  ½1 þ ξðtÞ : þ log ξðtÞ

(7.138)

The temperature deviation ratio ξ is shown in Figure 7.6 as a function of t=τ eq .

7.6 Electron Degeneracy Effects Our discussion so far in this chapter has considered the electrons to be nondegenerate. But we know that high-energy-density plasma is quite often at least partially degenerate. The work of both Cohen et al. and Spitzer and Härm is rigorously valid only for fully ionized nondegenerate plasma. Applying the Spitzer-Härm formula for the electrical conductivity at solid density and low temperatures gives results that are incorrect by about two orders of magnitude. Ignoring electron–electron encounters, we redo the preceding analysis with the equilibrium Fermi-Dirac distribution function

7.6 Electron Degeneracy Effects

375

m 3 1 e : ¼2 h eðεμÞ=kB T e þ 1

(7.139)

f

ð0Þ

Expression (7.78) becomes f ¼f

ð0Þ

  ∂f ð0Þ ε  μ d ðk B T e Þ  τc : v eE þ k B T e dx ∂ε

(7.140)

We use (7.140) in the expressions for the current densities, (7.82) and (7.93), to obtain expressions for the electric and thermal conductivities (as well as the other thermoelectric coefficients). Following Lee and More (1984), we have σ ¼ e2 K 0

(7.141)


 1 K 21 K2  , κe ¼ Te K0

(7.142)

ð 2 me 3 ∂f ð0Þ 3 Kn ¼  τ c v2 εn d v: 3 h ∂ε

(7.143)

and

where

The transport coefficients become
 ne e2 μ α σ¼ τeA kBT e me

(7.144)

 
 ne kB ðk B T e Þ μ τ e Aβ , me kB T e

(7.145)

and κe ¼

with the average electron relaxation time 1


μ  3me=2 ðkB T e Þ3=2 μ=kB T e F 1=2 τ e ¼ pffiffiffi 1þe : kB T e 2 2π ðZ Þ2 e4 ni logΛei

(7.146)

The functions Aj ðμ=kB T e Þ are expressed in terms of Fermi-Dirac integrals


μ Fj kBT



ð∞ ¼ 0

tj dt, 1 þ exp ðt  μ=kB T Þ

(7.147)

376

Thermal Energy Transport

such that Aα




μ kBT e

 ¼

4 F3 3 ð1 þ eμ=kB T e ÞF 12=

(7.148)

2

and β

A




  μ 20 F 4 1  16F 23 =15F 4 F 2 ¼ : 2 kB T e 9 ð1 þ eμ=kB T e ÞF 1=

(7.149)

2

In the nondegenerate limit ðμ=kB T ! ∞Þ, the Fermi-Dirac integrals are F j  eμ=kB T Γð j þ 1Þ, where Γ is the gamma function. Then Aα ¼

32 128 and Aβ ¼ , 3π 3π

(7.150)

and the transport coefficients for a Lorentz gas, (7.105) and (7.108), are recovered. In the complete degenerate limit, (7.148) becomes Aα ¼ 1:0, and (7.149) becomes Aβ ¼ π 2 =3, with the relaxation time (7.146) τe ¼

3h3 1 : 32π 2 me Z e4 logΛei

(7.151)

The two conductivities are σ¼

3 h3 ni 32π 2 m2e e2 logΛei

(7.152)

κe ¼

1 h3 ni k B ðk B T e Þ : 32 m2e e4 logΛei

(7.153)

and

For the region between completely degenerate and nondegenerate plasma, Aα , Aβ , and τ e must be evaluated numerically, but this is cumbersome. These coefficients are expressed by simple analytic functions, with the new variable z ¼ log ½1 þ exp ðμ=kB T Þ, Ai ¼

a1 þ a2 z þ a3 z2 , 1 þ b1 z þ b2 z 2

(7.154)

where i ¼ α or β. The fitting coefficients are listed in Table 7.3. The degeneracy correction to screening in the Coulomb logarithm (Lampe, 1968) is to multiply the electronic component (the first term on the right-hand side of (7.120)) by the logarithmic derivative of the Fermi integral F 1=2 0 =F 1=2 . It is

377

7.6 Electron Degeneracy Effects

Table 7.3 Coefficients for Determining the Values of Aα and Aβ . α a1 a2 a3 b1 b2

3.39 0.347 0.129 0.511 0.124

β 13.5 0.976 0.437 0.510 0.126

Figure 7.7. Degeneracy and ionization level of copper as a function of temperature. The dashed lines are for density 0:1 solid, and the solid lines are for 10 solid.

easier to replace T e by

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi T 2e þ T 2F , where T F is the Fermi temperature.q Numerical ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

calculation of the F 1=2 0 =F 1=2 factor gives a result to within 5 percent of

T 2e þ T 2F .

This should be used in both the expression for the Debye length in (7.120) and in (7.123). Since electron degeneracy plays a central role in the transport coefficients, we show them in Figure 7.7 for copper at two densities. At the higher density and low temperatures, degeneracy is significant. The effects of pressure ionization are noticeable in the values of the ionization level Z . The electrical conductivity (7.144) for copper is displayed in Figure 7.8, and the thermal conductivity (7.145) is shown in Figure 7.9. For reference, the measured

378

Thermal Energy Transport 1018

Electrical conductivity (s–1)

1017 Degenerate 1016

1015

Classical

1014

1013 10–3

10–2

10–1

100

Temperature (keV)

Figure 7.8. Electrical conductivity of copper as a function of temperature. The dashed lines are for density 0:1 solid, and the solid lines are for 10 solid.

Thermal conductivity (erg–1 s–1 keV–1)

1019

1018

Degenerate

1017

Classical

1016

1015 10–3

10–2

10–1

100

Temperature (keV)

Figure 7.9. Thermal conductivity of copper as a function of temperature. The dashed lines are for density 0:1 solid, and the solid lines are for 10 solid.

7.7 Inhibited Thermal Transport

379

conductivities at room temperature are σ ¼ 6:29  1017 s‒1 and κe ¼ 4:65  1014 erg-cm‒1-s‒1-keV‒1. A recent measurement of the electrical conductivity at 1 eV gives σ  4:5  1015 s‒1, which is in good agreement with Figure 7.8. Degeneracy corrections to electron–ion energy exchange are found in much the same way as for electron thermal conduction. We use the Fermi-Dirac distribution in Section 7.5 (or the equivalent Chandrasekhar/Spitzer development) to arrive at an expression for the equilibration time (Brysk, 1974) τ eq ¼



3 h 3 mi μ=k B T e 1 þ e : 64π 2 m2e Z e4 logΛei

For small degeneracy, 1 þ eμ=kB T e  eμ=kB T e , and from Section 3.4.1
3 μ ne λde ¼ log , kBT e 2

(7.155)

(7.156)

and (7.137) is recovered, provided T e =me  T i =mi . In the opposite limit of complete degeneracy, eμ=kB T e ! 0, and thus τ eq ¼

3 h 3 mi : 64π 2 m2e Z e4 logΛei

(7.157)

Complete degeneracy implies T e ¼ 0, thus (7.134) is inappropriate. Since we assume the ions are never degenerate, we can think of the energy exchange as the ions giving up their energy to a Fermi distribution of electrons dEi mi v2i : ¼ dt 2τ eq

(7.158)

The assumption is made in the derivation of (7.134) that the root-mean-square electron velocity vrms is large compared with the ion velocity. The proper statement for the completely degenerate case is that the Fermi velocity be greater than the ion thermal velocity.

7.7 Inhibited Thermal Transport The failure of classical heat transport theory to explain laboratory experiments of high-energy-density plasma is well known. The theory overestimates the heat transport into cold, dense material in the presence of a steep temperature gradient. An ad hoc normalization of the classical theory in computer simulation codes is required to replicate laser plasma experiments. In particular, the technique employed is to reduce the heat flux to a limiting value, which is a fraction of the free-streaming value according to

380

Thermal Energy Transport

Figure 7.10. Density (solid line) and temperature (dashed line) profiles encountered in a typical laser irradiation experiment from a hydrodynamic simulation. The simulation is for a laser wavelength of 0.5 μm incident on an aluminum slab with beam intensity of 1  1015 watts-cm‒2. The simulation time is 300 ps after the onset of the irradiation. The laser beam is incident from the right.

qfl ¼

q 1 þ f jqqj

,

(7.159)

e fs

where f e is the phenomenological flux limit parameter, which is often  0:05, and qfs  0:6ne me v3th is the free-streaming flux. Many phenomena have been suggested for explaining this reduced heat flow; we do not attempt to provide a comprehensive discussion of the possibilities. Instead, we focus on just one. It is well known that the Spitzer-Härm formulation discussed previously is inadequate when large temperature (and density) gradients are encountered, such as in high-power laser experiments. Figure 7.10 depicts this situation. The effect of inhibited transport into the denser portions of the target material is to increase the ablation pressure, which in turn yields a higher ablation velocity and thus steeper density and temperature profiles. In most cases, electrons do not transport energy efficiently through high-energydensity plasma. This is due to the electron’s small mass and the fact that the plasma is highly collisional; the electrons are thus “bound” to the ions (this is what allows us to argue that quasineutrality exists). An exception to this argument is how electrons transport energy from the laser deposition region into the dense material. The absorption of electromagnetic radiation by the inverse bremsstrahlung process will be discussed in Chapter 12. The energy is most efficiently absorbed in a region

7.7 Inhibited Thermal Transport

381

where the electron density is slightly less than the critical electron density for the particular laser wavelength. (For a wavelength of 0.5 μm, the critical density is nc  4  1021 cm‒3.) We see in Figure 7.10 that this is about two orders of magnitude less than the electron density in the high-energy-density region of the target. The heating of the plasma creates an ablation pressure that drives the outer layer of the target to expand into the “vacuum.” After a sufficient period of time, some of the plasma has attained a density much less than critical, and there is little absorption there. The absorption takes place in the region from about ne  0:3nc up to nc . At densities greater than critical, the energy is transported inward toward higher density, down the temperature gradient, to the region where ablation takes place. The thermal energy transfer in this transport region is crucial, and this region typically extends for only a few tens of microns. Over this distance, the temperature drops precipitously by more than two orders of magnitude. It is this region where the collisional mean free path may become a significant fraction of the temperature scale length L ¼ T e =jrT e j. The mean free path for momentumexchanging collisions is given by λmfp ¼ λs y4 , where
1= 2 1 ðk B T e Þ2 4 2πk B T e 2 λs ¼ jhvijτ e ¼ ¼ τe, 3 π ðZ Þ2 e4 ni lnΛei 3π me

(7.160)

and y2 ¼ v2 =v2th . Near the ablation front (located at about 12 μm in the figure), the temperature scale length is less than 1 μm. At about the same position, ne  1:1  1024 cm‒3, and the temperature is about 30 eV, so the mean free path is nearly the same size as the scale length. The Spitzer-Härm prescription for the classical heat flux is known to be incorrect if λmfp =L ≳ 102 , for which we have demonstrated a case. The theory fails to treat the higher-energy electrons adequately in the assumed Maxwell-Boltzmann distribution. Electrons having energies 7  kB T e are the ones that carry the majority of the heat, and these can have mean free paths λmfp > L. Although electron collision times are short compared with the time scales of interest, their mean free paths are comparable to the scale lengths. Collisions with ions drive the electron distribution function toward isotropy over a fairly short distance, while thermalization takes place over a significantly longer distance. The issue is made more complicated since the absorption of high-intensity laser radiation in the region of interest establishes a non-Maxwellian distribution function. Shortly after the laser irradiation begins, a quasisteady state is achieved when the laser absorption is balanced by heat transport. Let us examine the nature of the electron distribution function in more detail. For plasma in steady state, small temperature gradients, and weak electric fields,

382

Thermal Energy Transport

the distribution function, can be expanded in Legendre polynomials. Keeping only the leading term in angle, we have f ¼ f ð0Þ þ μ f ð1Þ , where μ ¼ cos θ and f ð0Þ is the equilibrium (Maxwell-Boltzmann) distribution function f

ð0Þ


¼

me 2πkB T e

3=2




 E þ eΦ exp  , kBT e

(7.161)

where Φ is the electrostatic potential. The Boltzmann equation without the electron–electron scattering term becomes   ∂f ∂Φ ∂f ∂f ∂Φ ð1  μ2 Þ ∂f v ∂ 2 ∂f e þ μv þ e ¼ ð1  μ Þ : (7.162) ∂t ∂t ∂E ∂x ∂x me v ∂μ 2λs ∂μ ∂μ Solving this equation in steady state, subject to quasineutrality, the leading correction term to the distribution function is pffiffiffi pffiffiffiffiffiffi  me 2 π ð1Þ y4 y2  4 f ð0Þ : (7.163) q f ¼ 3= 32 ne ðk B T e Þ 2 We see that for f ð1Þ to be nonnegative, y ≥ 2. Also, the heat flux must remain nonnegative; otherwise, the heat will flow up the temperature gradient. The heat flux carried by electrons of energy less than ϵ is q ¼

4π 3

ϵð

v me v2 ð1Þ f dϵ: v me 2

eΦ

(7.164)

In the limit that ϵ ! ∞, we obtain the classical result q ¼ κðeLÞ rT e , where κðeLÞ is the Lorentz value of the thermal

conductivity, which is just (7.108) with δT ¼ 1. Let us require f ð1Þ ðϵ Þ=3 < f ð0Þ ðϵ Þ, then q ðϵ Þ ¼ 0:9q as required by the self-consistent heat transport theory, which must obey v∂f ð0Þ =∂x ¼ ðv=λs Þ f ð1Þ . The inequality expresses the need to have a finite flux in the interval dϵ . For the classical theory ϵ ¼ 11:4k B T e , then pffiffiffiffiffiffi me 1

ð1Þ

f ð0Þ ðϵ Þ < f ð0Þ ðϵ Þ, (7.165) f ðϵ Þ ¼ 20jqj 3= 2 3 ne ðk B T e Þ if we set jqj

pffiffiffiffiffiffi me 3= 2

ne ðkB T e Þ

¼

jqj < 0:05: qfs

(7.166)

Now using the expression for the thermal conductivity of a Lorentz gas (7.108) and δt ¼ 1, we find λmfp =L ≲ 0:01. Returning to the conditions for the simulation of

7.7 Inhibited Thermal Transport

383

Figure 7.10, the laser flux is I L ¼ 1  1015 Watts-cm‒2 ¼ 1  1022 erg-cm‒2-s‒1, the density is nc ¼ 4  1021 cm‒3, and a temperature of about 1 keV gives

ð1critical

f Þ ðϵ Þ=3  24f ð0Þ ðϵ Þ. Clearly, this violates (7.165)! Returning to (7.162) in steady state, but relaxing the quasineutrality requirement, we find f ¼f

ð0Þ




   eE0 5 1 ∂T e 2  y μ , 2 T e ∂x kBT e

 1  λmfp

(7.167)

where the pressure gradient term is included in E 0 ¼ E þ ð1=ene Þ∂Pe =∂x. Equation (7.167) is just a rewrite of (7.114). The form of (7.167) reveals the limitations of the Spitzer solution. As long as the gradients and electric fields are small, f goes negative (unphysically) at velocities beyond the range of interest. For gradients such that the scale length L is smaller than λmfp over a significant portion of the distribution function, the approximation fails completely. The method also fails for electric fields strong enough that an electron gains energy of order k B T e over one scattering length. Many attempts have been made to extend the Enskog and Chapman theory (Chapman and Cowling, 1953) to higher order, with varying degrees of success. Most of these are focused on extending the approximations beyond that of diffusion to include the long mean free path particles that are freely streaming. This situation is not unlike that encountered in radiation transport. An approach based on radiation transport theory gives a solution to the Boltzmann equation that includes particles streaming at energies for which the mean free path exceeds the temperature gradient scale length or the distance over which an electron gains an energy of k B T e from the electric field (Campbell, 1984). The electric fields are assumed smaller than the critical value for electron runaway. The resulting distribution function is positive definite for all particle energies and scale lengths. Figure 7.11 shows the distribution function in a temperature gradient, where λs =L ¼ 0:1 in the direction of the electric field. For small velocities, the distribution is diffusive and exhibits the familiar properties of the Spitzer solution. The distribution function is slightly skewed in relation to the equilibrium distribution f ð0Þ by the temperature gradient and field to produce a flow of heat with zero net particle flow. For large positive velocities, the distribution shows the presence of hot, long mean free path electrons streaming through the region from the hot side. The distribution for large negative velocities is depopulated with respect to f ð0Þ , since it represents long mean free path electrons originating from the cold side of the gradient region. Once the distribution function is known, the current and heat flux can be calculated, as in (7.103) and (7.104), to give the current density and heat flow

384

Thermal Energy Transport

Figure 7.11. Velocity profile for p theffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi distribution function with μ ¼ 0:99 and λs =L ¼ 0:1. The abscissa is y ¼ me v2 =2k B T e . The local Maxwellian f ð0Þ is shown as the dashed line.

ð∞ J ¼ e


1= 4 e2 2k B T e 2 vI 0 ω ΓDv dv ¼ pffiffiffi ne λs π kBT e me 2

2

0

ð ∞





  Z DE 0 kB Z DT ∂T e 3 x2 Γω E 2 x e dx A e B ∂x 2

(7.168)

0

and

ð∞ me q¼ 2


1= 2 eme 2k B T e 2 v I 0 ω ΓDv dv ¼  pffiffiffi ne λs π kBT e me 3

2

2

0

ð∞ 

Γω2 0



  Z DE 0 kB Z DT ∂T e 5 x2 E 2 x e dx, A e B ∂x

(7.169)

7.7 Inhibited Thermal Transport

385

where Γ contains the distribution function, and ω ¼ I=I 0 is the reduced distribution Ðx 2 function normalized by that of the Maxwellian; recall, I n ðxÞ ¼ yn DðyÞey dy. 0

The factors ðZ DE =AÞ and ðZ DT =BÞ as functions of y are tabulated (Spitzer & Härm, 1953). Expressions (7.168) and (7.169) can be put in simpler form using E0 ¼ E þ ð1=ene Þ∂Pe =∂x J ¼ σEϕE þ

σ ∂Pe ∂T e ϕE þ α ϕ ∂x T ene ∂x

(7.170)

β ∂Pe ∂T e ψ E  κe ψ , ene ∂x ∂x T

(7.171)

and q ¼ βEψ E 

where σ, α, β, and κe are the Lorentz transport coefficients given by (7.105) through (7.108). The correction factors are

ð∞ ϕE

¼


Γω

¼

 Z DE 3 y2 y e dy, A

ð∞

0

ϕT

2


 2 Z DT 3 y2 Γω2 2 y e dy 3 B

(7.172)

0

and

ð∞

ψE


 1 2 2 Z DE ¼ Γω y5 ey dy, 4 A 0

ψT

¼

ð∞


 1 Z DT 5 y2 2 Γω 2 y e dy: 10 B

(7.173)

0

In the limit of small gradients and fields, ϕE ! γE , ϕT ! γT , ψ E ! δE , and ψ T ! δT . The problem of thermal energy transport under the condition of quasineutrality is of special interest in light of the anomalous results from laser plasma experiments. The requirement of quasineutrality requires the current (7.168) to be zero, and this determines the electric field
 eE0 3 ϕT 1 ∂T e 3 1 ∂T e ¼ δ , (7.174) ¼ 2 ϕE T e ∂x 2 T e ∂x kBT e

386

Thermal Energy Transport

Figure 7.12. Calculated values of the heat flux relative to the classical (SpitzerHärm) values are shown as solid points. Values from a Legendre polynomial solution are represented by the open points. Values for fractions of the freestreaming flux, qfs , are also shown.

where δ is a measure of the deviation of the field from the classical Lorentz value. Under quasineutrality, D can be expressed in terms of the temperature gradient scale length 


 λs 3 Z DE Z DT D¼ δ 2 : (7.175) 2 L A B The heat flux becomes




 ϕT ðLÞ q ¼  ψ T  0:6ψ E κ rT e : ϕE e

(7.176)

Note the similarity between (7.176) and (7.109). The correction factors, equations (7.172) and (7.173), can be evaluated numerically for different values of Z and λs =L. Figure 7.12 shows calculated values along with a numerical solution of the Fokker-Planck equation (Campbell, 1984). These results for the heat flux are better than those of the classical model, but still do not explain the limiting value needed to completely rectify theory with laser plasma experiments. Many other mechanisms for thermal inhibition have been

7.8 Nonlocal Heat Transport

387

suggested over the years. We do not develop any discussion of those here, but are satisfied by listing five of them: 1. It is well known that the heating of the electrons by inverse bremsstrahlung creates an electron distribution that is not Maxwellian. Since the absorption region is close to the transport region, it is not surprising that the distribution function in the transport region is distorted. 2. The effects of plasma turbulence, in particular the ion acoustic instability, may play a role. 3. It is well known, experimentally, that magnetic fields exist in many extreme plasmas. The thermal conductivity is greatly reduced when there is a component of the heat flow perpendicular to the magnetic field lines. 4. Other two-dimensional effects may play a role, such as the temperature gradient having different values in different directions, that is, lateral heat flow versus normally directed heat flow. 5. Very energetic (fast) electrons may be created by plasma instabilities (e.g., wake field oscillations). In all probability, there are several mechanisms operating simultaneously to reduce the heat flow. This makes for a rich field of study. 7.8 Nonlocal Heat Transport As we have seen, the classical heat flow model is inadequate in the presence of a sharp temperature gradient, which manifests itself in an excessive heat flux. An ad hoc fix to this shortcoming is the introduction of flux limiting. A second deficiency of the model is the lack of “preheat” ahead of the heat front due to long mean free path electrons. Energetic electrons with v  2  3  vth are collisionless and stream into the cold material forming a hot tail in the local distribution function. This deformation leads to larger gradients for electrons contributing to the heat flux and hence reduced thermal fluxes, as well as the presence of a small foot ahead of the heat front. The fix for these deficiencies in the classical model is to develop a more sophisticated, nonlocal transport model such as a full Fokker-Planck calculation or convolution formulas, which are much simpler. The convolution approach is to model the local (classical) heat flux with an appropriate kernel in configuration space to obtain an approximation to the more accurate Fokker-Planck heat flow. Convolution formulas should have the desirable properties of (1) reproducing flux inhibition in sharp temperature gradients, (2) predicting preheat, and (3) yielding the classical result in the collisional limit.

388

Thermal Energy Transport

The general formula for the nonlocal heat flow, in one-dimensional planar geometry, is 1 qðxÞ ¼ βðxÞ

ð∞ qcl ðyÞGðx; yÞdy,

(7.177)

∞

where the classical (Spitzer-Härm) flux Ð is qcl , and the convolution kernel is Gðx; yÞ. The normalization factor is βðxÞ ¼ Gðx; yÞdy. One choice of kernel (Luciani, Mora, & Virmont, 1983) is   1 jx  yj Gðx; yÞ ¼ exp  , (7.178) 2λðyÞ λðyÞ where the delocalization parameter λ is an effective mean free path of the electrons dominating the heat flow. Physically, this equation states that the heat flux at position x is determined by the classical fluxes from other points y up to a distance λ away. When the electron density is nonuniform, the exponent in (7.178) is replaced jx  yj jξ ðxÞ  ξ ðyÞj ! , (7.179) λðyÞ T 2 ðyÞ  Ðx with ξ ðxÞ ¼ dy=λ ðyÞ and λ ¼ T 2e =ne F ¼ T 2e λ, where F is some function of the ∞

ionization level and the appropriate Coulomb logarithm. The evolution of the electron temperature is given by ∂T e ∂q (7.180) ¼ : ∂x ∂t Prasad and Kershaw (1989) have shown that this set of equations is intrinsically unstable in the realm of very steep gradients, which can lead to heat being transported from cold to hot regions, thus producing negative temperatures. One of the difficulties is that (7.177) is defined for an infinite slab. For finite geometry, the kernel (7.178) must guarantee that the flux vanishes at both edges of the slab. To find the modified kernel, we note that G∞ ðx; yÞ for constant λ is in fact the Green’s function of a one-dimensional Helmholtz operator
2  ∂ 1 1  2 G∞ ðx; yÞ ¼  2 δðx  yÞ, (7.181) 2 ∂x λ λ ne ðxÞcV

where δðx  yÞ is the Dirac delta function. We see that as λ ! 0, the kernel goes to δðx  yÞ, and the nonlocal flux goes to the classical value. Therefore, the kernel becomes the appropriate Green’s function solution for (7.181) for finite geometry Gs ðx; yÞ ¼

sinh ðx< =λÞ sinh ½ðs  x> Þ=λ : λ sinh ðs=λÞ

(7.182)

389

7.8 Nonlocal Heat Transport

Figure 7.13. Simulated temperature profiles using the nonlocal model with α ¼ 81 (solid points) and flux-limited diffusion (crosses), with a flux limiter of 0.05. The laser is incident from the right.

Here, the thickness of the slab is s, and x< ¼ min ðx; yÞ and x> ¼ max ðx; yÞ. The limits of the integral in (7.177) are now zero and s. For nonuniform electron density, we use (7.179). Figure 7.13 compares the electron temperature profiles for a simulation of a laser incident on a planar layer; both flux-limited diffusion and nonlocal transport results are shown for identical times. The significant difference is that the lowdensity corona is cooler and not isothermal at early times, and the peak temperature at the heat front is higher in the nonlocal model, indicating a more inhibited transport. The effective mean free path includes both electron–ion collisions and electron–electron collisions pffiffiffiffiffiffiffiffiffiffi λ ¼ α λei λee ¼

αT 2e 1=

4πne e4 ðZ logΛei logΛee Þ 2

:

(7.183)

α is an adjustable parameter whose value is in the range of 16 ! 81; these values come from the fact that the velocity of electrons carrying the nonlocal heat are in the range v  2  3  vth . The limit of α ! 0 corresponds to classical flux-limited diffusion, whereas the limit of α ! ∞ would correspond to extreme nonlocality. The model of Luciani et al. (1983) uses α ¼ 32.

390

Thermal Energy Transport

We note that other models for the kernel have been developed by many authors, but most suffer from instabilities similar to those just discussed for the model of Luciani et al. (1983). No single nonlocal model appears to be satisfactory for every situation; that is, the correct kernel is problem dependent. We must check the suitability of a particular kernel by comparing it to the results from a more accurate Fokker-Planck calculation.

8 Radiation and Radiative Transfer

In preceding chapters, we developed the equations governing the behavior of nonradiating fluids. We now address the subject of energy transport by thermal radiation. The term “thermal radiation” means electromagnetic radiation of atomic origin, as opposed to nuclear origin. Thermal radiation is generally emitted by matter in a state of thermal excitation, thus accounting for the designation of the radiation as thermal. The physical mechanisms for generating this thermal radiation will be discussed in Chapter 9. A brief discussion about the effects of radiation on matter was presented in Chapter 2. The photons that make up the radiation field transport momentum and energy just as particles with mass do. The importance of thermal radiation increases as the temperature of the matter is raised, because the radiation energy density associated with a Planck distribution varies as the fourth power of the temperature. At moderate temperatures, on the order of a few electronvolts, the role of radiation is primarily one of transporting energy by radiative processes, such as photon emission and absorption. At higher temperatures, tens to hundreds of electronvolts (and even higher), the momentum and energy densities of the radiation field become comparable to or even dominate the corresponding fluid quantities. In this case, the radiation field significantly affects the dynamics of the fluid; this is the subject of Chapter 10. The methods for describing thermal energy transport by particles, as opposed to radiation, were presented in Chapter 3. A kinetic description of the motion of charged particles undergoing Coulomb collisions resulted in the Boltzmann equation and a Fokker-Planck prescription for describing the collisions. The Boltzmann equation is too hard to work with in most instances, so different approximation methods yielded simpler equations, such as the thermal diffusion equation of Chapter 7. The situation with radiation transport is similar. The transport equation is nearly the same as for thermal energy transport, and the approximation methods are almost the same. The difference is that the subject of radiation transport was 391

392

Radiation and Radiative Transfer

largely developed within the astrophysical community, while kinetic theory was primarily the province of the plasma physicists. Each community developed their own notation, and we have that difference with us today. We will adopt the traditional language of radiation and radiative transfer in the following pages. Many astrophysics books treat the radiation field and its transport adequately. We recommend Mihalas’ Stellar Atmospheres (Mihalas, 1978), also Mihalas and Mihalas’ Foundations of Radiation Hydrodynamics (Mihalas & Mihalas, 1984). The classic book is Chandrasekhar’s Radiative Transfer (Chandrasekhar, 1960). A comprehensive treatise on the subject is Pomraning’s Radiation Hydrodynamics (Pomraning, 1973). We also recommend the more recent book by Castor, Radiation Hydrodynamics (Castor, 2004). One topic that we will not delve into is that of spectral line transfer. While it is a fascinating subject, it is also quite complex; it must be considered in problems where non-LTE effects are important. For the LTE situations of interest to us now, the collisional rates are assumed to dominate the kinetics and the radiation field does not alter the atomic level populations by very much. In non-LTE, the radiation field changes the populations significantly, and transitions between the bound states become important. The transport of the radiation created by the discrete transitions is sensitive to the surrounding environment. The shape of the line, as a function of frequency, must be taken into account in the transport equations. The literature is full of discussions about spectral line transport, and a good starting point is the book by Castor. 8.1 The Radiation Field The transport of radiant energy by photons suggests that we might develop a theory from Maxwell’s equations, leading to a wave-equation-like description for transport. We will take this approach in dealing with monochromatic photons and their interaction with matter in Chapter 12. This electromagnetic wave approach is awkward for discussing thermal radiation for a number of reasons that will become apparent as we proceed. Rather, we approach the subject treating photons as pointlike particles. According to quantum theory, an individual photon is characterized as a wave packet. The phase-space coordinates of the wave packet are specified by the variables r, ν, Ω, where r is the spatial position, ν is the frequency, and Ω is the direction of travel. If the density of photon wave packets is not too large and they are randomly distributed (with no preferred direction), the packets will not overlap and show interference effects. The contents of the wave packet are described by an amplitude (as well as a frequency), while in radiation transport theory, the fundamental quantity is the intensity. Since there is no information in the intensity about

8.1 The Radiation Field

393

the wave nature of photons, the theory cannot deal with refractive, diffractive, or dispersive effects. Another important point to consider is the quantum nature of the photon, which is not mentioned in discussions about electromagnetic wave propagation. Indeed, the explanation for the shape of the Planck distribution function depends upon the inclusion of the quantum nature of the photon; this was discussed in Section 3.10. 8.1.1 Specific Intensity The basic premise is that energy in the radiation field is carried by point, massless particles – the photons. With each photon we associate a frequency ν, such that the energy carried by the photon is E ¼ hν. It is known that a massless particle has momentum p ¼ E=c ¼ hν=c. Between collisions with matter, a photon is presumed to travel in a straight line with the speed of light. We will not explicitly define the polarization, but assume such information is carried along. The fundamental quantity in radiation transport is the frequency-dependent, specific intensity I ν . It plays the same role as the phase-space distribution function does for material particles. The intensity is a function of three spatial coordinates: two angle coordinates and one each for the photon frequency and time. The sevendimension photon distribution function is defined by f ¼ f ðr; ν; n; tÞ:

(8.1)

The number of photons at time t and spatial position r in a differential volume element dr, with frequency ν in a frequency interval dν, and traveling in a direction n in a solid angle element dΩ, is dN ¼ fdrdνdΩ:

(8.2)

Since each photon carries energy hν and moves with the speed of light, the specific intensity of radiation is defined by I ν ðr; n; t Þ ¼ chν f ðr; ν; n; t Þ:

(8.3)

Equation (8.3) appears to address a single “ray,” but a single ray makes little sense. So we present the intensity as a bundle of rays with some cross section. The intensity has units of power, but is also expressed in unit area, unit solid angle, and unit frequency. To better understand how these quantities enter in, refer to Figure 8.1. We visualize a spectrum of photons streaming from the “oven,” on the left, through a hole of area Ah . The exit hole is large compared to the wavelength of the radiation, so the emitted photons travel in straight lines; that is, we do not need to consider the wave properties of light. At a distance R away from the hole is a “viewing screen” of area As , in line with the normal through the

394

Radiation and Radiative Transfer As

Ah

R

Figure 8.1. Schematic representation of thermal radiation emitted by a radiation enclosure, or hohlraum. The specific intensity I ν is defined by the geometry of the emitted radiation.

exit hole. The size of As is large compared to Ah . The definition of the spectral intensity I ν , for a specific frequency, is given by ΔE ν ¼ I ν Ah

As ΔνΔt: R2

(8.4)

The quantity ΔE ν is the energy passing through Ah during a time Δt in a frequency band Δν. The quantity As =R2 is just the solid angle dΩ subtended by As at the aperture. That is, the intensity is the energy crossing a unit area at a given point (the exit hole) per unit time, per unit frequency, per unit solid angle. In a small increment of time Δt, the photons leaving the exit hole will travel a distance of cΔt. Let the area of the exit hole become infinitesimal, that is, let Ah ! dA. Then, at any given time, the photons occupy a cylindrical volume of cdAΔt. Dividing ΔE ν by this volume gives us the contribution to the radiation energy per unit volume of ðI ν =cÞdΩdν from the solid angle dΩ ¼ As =R2 in the frequency band Δν. Further, we must specify a significantly narrow bandwidth Δν so that the pulse of radiation will be at least 1=Δν in duration, since otherwise the light pulse would have sidebands outside the bandwidth. It can easily happen that 1=Δν is longer than the Δt of interest. The definition we ascribe to the intensity requires the system to be much larger than the wavelength, and the times of interest to be much longer than the wave period, as specified by the Fourier relations between localization in space and spreading in Fourier space, and between location in time and spreading in frequency. Radiation transfer requires the geometrical optics concept, which does not contain any information about wave optics.

8.1 The Radiation Field

395

Dividing (8.4) by the energy of a single photon gives the number of photons. Therefore, I ν =hν is the number of photons crossing a unit area per unit frequency per unit solid angle in the specified direction per unit time.

8.1.2 Radiation Energy Density and Mean Intensity Equation (8.4) is the expression for the energy density of the radiation. Integrating this over all directions, we have the spectral energy density, which is the total energy density per unit bandwidth ð 1 Eν ¼ I ν dΩ: (8.5) c 4π

Integrating this equation over frequency gives the total radiation energy density. The mean intensity J ν is defined as the average of the specific intensity over all solid angles ð 1 Jν ¼ I ν dΩ, (8.6) 4π 4π

which is just the zeroth moment of the radiation field over angles. Combining (8.5) and (8.6) shows E ν ¼ ð4π=cÞJ ν .

8.1.3 Radiation Flux and Momentum Density The first moment of the radiation field integrated over angles defines the radiative flux, which is the rate of energy flow per unit area across a surface. Hence ð Fν ¼ nI ν dΩ: (8.7) 4π

If we have an area element represented by the normal vector A and form Fν  A, we see that it is the sum of contributions like n  AI ν dΩ, which gives the sign of flow of energy across A; it is positive if going in the direction of A and negative otherwise. Notice that the flux moment does not contain a factor c, as does the energy density. The ratio of Fν to E ν is the “fluid velocity” if we think of the radiation as a fluid, the magnitude of which can be no greater than c. We can visualize the particle picture of radiation by introducing the quantum of the photon. The quantity I ν =hνc is the number density of photons per unit solid angle per unit frequency (see (8.3)). Since the momentum of a photon is hν=c, then the momentum space volume element is

396

Radiation and Radiative Transfer

 3 h d p¼ ν2 dνdΩ: c 3

(8.8)

Then, dividing the number density of photons by h3 ν2 =c3 gives the phase-space density dN Iν ¼ 4 3 2: 3 dVd p h ν =c

(8.9)

The net rate of momentum transport across an area element dA is Fν  dA=c. Since the photons are moving at the speed of light, we obtain the radiation momentum density Gν ¼

1 Fν : c2

(8.10)

8.1.4 Radiation Pressure Tensor The second moment of the radiation field is defined by ð 1 Pν ¼ nnI ν dΩ: c

(8.11)

4π

We recognize P ν as the frequency-dependent pressure tensor. The factor 1=c that was present in the expression for the energy density and absent from the flux expression has reappeared in the definition of the pressure tensor. In the absence of momentum being exchanged between the radiation and matter, the momentum density (8.10) decreases according to ∂Gν 1 ∂Fν ¼ 2 ¼ r  P ν : c ∂t ∂t

(8.12)

The pressure tensor P ν is represented by a symmetric 3  3 matrix. For a onedimensional medium, along the z-direction, the specific intensity is independent of angle, and evaluation of (8.11) yields 0 1 0 1 Pν 0 0 0 0 3Pν  Eν 1 P ν ¼ @ 0 Pν 0 A  @ 0 3Pν  E ν 0 A: (8.13) 2 0 0 Pν 0 0 0 We can define another tensor Kν ¼

1 4π

ð nnI ν dΩ, 4π

and the scalar Pν is defined as

(8.14)

8.1 The Radiation Field

Pν ¼

4π K ν, c

397

(8.15)

where the scalar K ν comes from (8.14) 1 Kν ¼ 2

ð1 I ν ðz; μ; tÞμ2 dμ,

(8.16)

1

with μ ¼ cos θ. We see that in the case of a one-dimensional medium, only the scalars Pν and Eν are needed to specify the full tensor P ν . It is common that the quantity Pν is referred to as the radiation pressure, but it is important to bear in mind that because the second term in (8.13) is not necessarily zero, P ν is not, in general, isotropic, and therefore does not reduce to a simple hydrostatic pressure. The anisotropy of P ν reflects the anisotropy of I ν , which is induced by efficient photon exchange between regions with significantly different physical properties, particularly in the presence of strong gradients and/ or open boundaries. The trace of the pressure tensor matrix (8.11) is defined as the sum of the diagonal elements. The effect of this is accomplished by replacing nn with n  n inside the integral. But n  n is just the identity matrix I, and therefore  Tr P ν Þ ¼ E ν , (8.17) which is also evident from (8.13). If we are deep inside opaque material and away from an open boundary, all directions look the same. Thus, the radiation field tends to be isotropic, and that makes the  pressure tensor look like a scalar; that is, P ν ¼ Pν I. Thus, (8.17) becomes Tr P ν Þ ¼ 3Pν , and so 1 P ν ¼ Eν I: 3

(8.18)

It is useful to define a dimensionless ratio fν ¼

K ν Pν ðr; t Þ , ¼ J ν E ν ðr; t Þ

(8.19)

which is known as the Eddington factor. It is a measure of the anisotropy of the radiation field. We shall use this factor later in this chapter when we discuss approximate solutions to the transfer equation. All frequency-dependent quantities in this section may be integrated over frequency. For example, (8.15) and (8.16) become

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Radiation and Radiative Transfer

ð∞

ð∞ 4π Pr ¼ Pν ðr; tÞdν ¼ K ν ðr; t Þdν: c 0

(8.20)

0

We have assumed, so far, that the radiation field does not interact with the matter field. 8.2 Interaction of the Radiation Field with Matter Matter and radiation are closely entwined. Radiation can be absorbed, emitted, or scattered by material particles. The atomic physics of these processes will be discussed in Chapter 9.

8.2.1 Absorption As a photon travels though matter, there is a certain probability that it will interact with the matter and disappear. To describe this process, we introduce a macroscopic absorption coefficient k ðνaÞ ¼ k ðνaÞ ðx; tÞ, which depends upon the frequency of the photon as well as its spatial and temporal coordinates. It does not have any angular dependence. The coefficient is defined such that the probability of a photon being absorbed in traveling an infinitesimal distance ds is k ðνaÞ ds. The empirical relation of attenuation of radiation passing through a slab of material with thickness Δs shows that a small amount of the radiation intensity is removed from the incident “beam.” The intensity reduction is ΔI ν ¼ k ðνaÞ ΔsI ν ,

(8.21)

provided that none of the absorbed energy is replaced by emission. Since the photon bundle will cross the slab of material in time Δt, there is a probability 1  kðνaÞ cΔt of no interaction, and a probability of k ðνaÞ cΔt of having an interaction. The energy absorbed from the radiation field must appear elsewhere within the element of material. This energy may be used for ionizing an atom (with some of the energy going into the kinetic energy of the freed electron), or excitation of an atom, which is subsequently de-excited by a collision with another particle, and the energy is dispersed into kinetic energy of the collision’s reactants. The inverses of the processes produce “thermal emission,” in which energy is extracted from the thermal energy of hot material and converted into radiation.

8.2.2 Emission Matter at any temperature emits radiation. At room temperature, most of the radiation is emitted in the infrared and is not detected by our eyes. At elevated

8.2 Interaction of the Radiation Field with Matter

399

temperatures of a few thousands of Kelvin, the emission peaks in the visible region, while at temperatures representative of high-energy-density matter, the emission is in the soft X-ray range. The emissivity is a measure of the amount of energy added to the beam by the thin slab. Normally, this emission does not depend upon the intensity of the incident beam, but does depend upon the thickness of the slab; we will find that this statement needs modifying. The intensity added to the beam is ΔI ν ¼ þjν Δs,

(8.22)

where the emission coefficient is jν . The emission coefficient and the absorption coefficient do not have the same units; the absorptivity is measured in inverse length, while emissivity is measured by the intensity divided by the length. Another way to characterize the emissivity is that it has units of energy per unit volume per unit bandwidth per unit solid angle per unit time. Our brief discussion about absorption and emission assumes simple atomic processes that do not depend upon the intensity of the radiation field. We will discuss one process that does depend upon the intensity (stimulated emission), and we recognize that there are other important processes that are nonlinear in the intensity, such as multiphoton absorption, for which the energy loss is proportional to two or more factors of the intensity for different frequencies and directions. Just as the absorption coefficient depends upon the properties of the particular material, so does the emission coefficient. Under certain conditions, the two coefficients are related to one another.

8.2.3 Kirchhoff’s Law An important relation between the emissivity and absorptivity exists in the case of thermal equilibrium. For strict thermodynamic equilibrium (TE), the spectral content of the intensity is given by the Planck function BðνPÞ ðT Þ. The discussion about the Planck function and its derivation can be found in Section 3.10. If a material element is in TE, the radiation field is isotropic and in steady state, characterized by a particular temperature. The emission of radiation must be exactly balanced by the absorption of that radiation; that is, from (8.21) and (8.22), we have jν ¼ k ðνaÞ BðνPÞ ðT Þ:

(8.23)

A note of caution: Kirchhoff’s law is valid only in conditions of complete thermodynamic equilibrium. We often assume the law is also valid when small gradients of the physical properties are present. Such conditions are defined as

400

Radiation and Radiative Transfer

being in local thermodynamic equilibrium (LTE) and are reasonably satisfactory in the diffusion limit of the transport equation (to be discussed shortly). We must also use caution when free transport of radiation occurs, since the radiation field then acquires a nonlocal and/or nonequilibrium character that tends to drive the state of the material away from LTE.

8.2.4 Scattering In addition to emission and absorption of radiation in a material element, there is the process of scattering of the radiation. In just a few words, scattering is the change in direction (and possibly frequency) of a photon interacting with the atoms of the material. In fact, the concept of scattering is complex and can be studied from different points of view; this could easily result in a very lengthy discussion, but space constraints here require us to limit the discussion to only the simplest aspects of this topic. Scattering is normally a linear process. By this we mean that scattering adds something to the absorption coefficient, k ðνaÞ , and those photons that were removed from the beam reappear at other angles and/or frequencies. These scattered photons are treated as modifications to the emission coefficient jν . Since these scattered photons are dispersed in angle and frequency, integrals over those two quantities must be performed in order for the scattering to be included in the emission coefficient. Scattering is treated in much the same way as absorption and emission. A scattering coefficient may be defined such that the probability of scattering is kðsÞ Δs. As with absorption, the probability of scattering is independent of direction. The scattering process serves to change the photon’s characteristics (frequency and angle) to a new set of characteristics. This leads to the definition of the differential scattering coefficient k ðsÞ ðr; ν0 ! ν; n0  n; t Þ; the characteristics before the scattering event are denoted by primes. The probability of a photon being scattered from frequency ν0 to ν in the interval dν, and from solid angle n0 to n contained in dΩ is kðsÞ ðν0 ! ν; n0  nÞdνdΩds. We could have written n0  n as n0 ! n, but we would then have to carry along two solid angles, where we can do just as well with the single quantity of the dot product of the incoming and outgoing solid angles. Thus, we have n0  n ¼ μ ¼ cos θ. Integrating the differential scattering coefficient over the final frequency and angle gives the scattering coefficient

ð∞ ð ðsÞ

k ðsÞ ðν0 ! ν, n0  nÞdΩdν

k ν0 ¼

0 4π ∞ 1

ðð

¼ 2π

k ðsÞ ðν0 ! ν, μÞdμdν: 0 1

(8.24)

8.2 Interaction of the Radiation Field with Matter

401

If the scattering process results in no change of photon frequency and only a change in direction, the scattering is said to be “coherent.” The scattering coefficient has the same flavor as the absorption coefficient: it is not dependent on the specific intensity. In contrast to absorption and emission, little of the photon’s energy goes into (or comes from) thermal energy of the matter. One example of this is a photon exciting an atom from one state to another of higher energy, and then the atom decaying radiatively back to the first state. A second example is the photon colliding with a free electron (Thomson or Compton scattering) or with an atom or molecule in which a resonance is excited (Rayleigh or Raman scattering).

8.2.5 Stimulated Emission Our discussion about emission assumed that the only mechanism for creating radiation is spontaneous emission. Einstein predicted that there is also a mechanism that depends upon the presence of a radiation field itself: stimulated emission. While Einstein was primarily concerned with transitions between two energy levels of the atom, for which the transitions result in line radiation, the argument is easily extended to continuum radiation. The expression for emission (8.22) thus has two terms, one for spontaneous emission and one for stimulated emission. This latter term depends upon the specific intensity in a fashion similar to that for absorption (8.21). Hence, we can take the stimulated emission term and combine it with the absorption term; that is, we introduce a “negative absorption.” The net absorption may be positive or negative, depending on whether “true absorption” or stimulated emission dominates. The reader will recognize this is the physical basis for laser action, in which a “pump” photon can cause the ejection of an additional photon. A property of stimulated emission that is not clear from discussions is that it takes place into precisely the same direction and frequency (in fact, into the same photon state) as the pump photon. The stimulated photon is precisely coherent with the pump photon. While Einstein originated the theory of stimulated, or induced, emission for discrete transitions, we do not need to appeal to quantum theory to obtain the result. Since photons are bosons, both the processes of emission and scattering are enhanced by the number of photons already in the final state following an interaction. Quantitatively, the enhancement is simply stated: If P represents the basic probability of a photon event (emission or scattering), then, due to induced effects, the actual probability P0 is given by (Feynman, Leighton, & Sands, 1966) P0 ¼ Pð1 þ N Þ:

(8.25)

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Radiation and Radiative Transfer

We shall return to this topic in Chapter 9, in the context of the Einstein coefficients. In (8.25), N is the number of photons in the final state of the transition, as given by integrating (8.2). In the present context, the final state corresponds to the unit cell of phase space. Integrating (8.9) yields ðð c2 1 N ¼ 4 3 I ν ðr; n; t Þdpdr, (8.26) h ν where the integrals are over the phase-space cell. (Recall that this cell is ΔpΔr ¼ h3 =2, where the factor of two comes from the two states of polarization.) Using this fact in (8.26), together with dp ¼ p2 dpdΩ ¼ ðh=cÞ3 ν2 dνdΩ, and then from (8.25) we have the stimulated probability   c2 0 Iν : P ¼P 1þ (8.27) 2hν3 This equation implies that the emission and scattering coefficients should be increased by this factor. In this chapter so far, we have developed the concept of the radiation field, characterized by its specific intensity I ν , and three coefficients characterizing the interaction of the radiation with real matter. The matter influences the composition of the radiation field, and vice versa.

8.3 The Equation of Transfer The transfer equation is quite simple: the specific intensity does not change as the radiation moves from one point in spacetime to another point, in the absence of real matter. If we view the bundle of radiation coming from the enclosure of Figure 8.1 at time t, as a cylindrical beam containing a total energy of I ν Ah ΔνΔtdΩ, then a short while later t þ Δt the bundle is at some other location, but with the same bandwidth Δν, and fills the same solid angle element dΩ. The vector characterizing the direction n is defined by two angles, such as θ for the colatitude (polar angle) and ϕ for the longitude (azimuthal angle). The element of solid angle is dΩ ¼ sin θdθdϕ. Since the specific intensity of the bundle does not change during time Δt, we can write I ν ðr þ ncΔt; t þ Δt Þ ¼ I ν ðr; n; t Þ:

(8.28)

Expanding (8.28) in a Taylor series about the point r and t, keeping terms up through OðΔt Þ, subtracting I ν ðr; n; t Þ and dividing by cΔt, gives the result

8.3 The Equation of Transfer

1 ∂I ν þ n  rI ν ¼ 0: c ∂t

403

(8.29)

This equation is reminiscent of the Boltzmann transport equation of Section 3.1, but this is to be expected. The fundamental process of transport does not distinguish between types of particles. The Boltzmann equation appears to be more complicated because of body forces that must be included (e.g., electromagnetic fields), but since no body forces operate on photons, we do not need those terms. The “collision” terms of the Boltzmann equation act like sources or sinks, which in this chapter we call emission and absorption. Before we assemble the complete transfer equation, we need to examine the scattering equation (8.24) a bit more. Consider a volume element ΔV, into which photons are being scattered from a neighboring volume element, and photons are being scattered out of ΔV. The time rate of out-scattering is c ΔV hν

ð∞ ð

k ðsÞ ðr; ν ! ν0 ; n  n0 ; tÞI ν ðr; n; t ÞdΩ0 dν0 ,

(8.30)

kðsÞ ðr; ν0 ! ν; n0  n; t ÞI ν0 ðr; n0 ; t ÞdΩ0 dν0 :

(8.31)

0 4π

and for the in-scattering c ΔV hν0

ð∞ ð 0 4π

In light of (8.27), equations (8.22), (8.30), and (8.31) should be increased by 1 þ c2 I ν =2hν3 . Combining (8.29), (8.21), (8.22), (8.30), and (8.31) with the appropriate signs and remembering that the specific intensity is a function of solid angle, we arrive at the complete radiative transfer equation   1 ∂I ν ðnÞ c2 þ n  rI ν ðnÞ ¼ jν 1 þ I ðnÞ  kνðaÞ I ν ðnÞ ν c ∂t 2hν3 ð∞ ð   ν ðsÞ 0 c2 0 0 0 þ k ðν ! ν, n  nÞI ðn Þ 1 þ I ðnÞ dΩ0 dν0 ν ν ν0 2hν3 0 4π

ð∞ ð

  c2 0 k ðν ! ν , n  n ÞI ν ðnÞ 1 þ I ν0 ðn Þ dΩ0 dν0 : 2hν03 ðsÞ



0

0

(8.32)

0 4π

We see that (8.32) is severely complicated by the induced scattering terms in that it leads to nonlinear (quadratic) terms in the specific intensity. Note that in the

404

Radiation and Radiative Transfer

limiting case of no frequency change upon scattering, ν0 ¼ ν, the induced inscattering and out-scattering terms identically cancel one another. Then the integrals in the out-scattering term may be carried out to yield   1 ∂I ν c2 I ν  kðtÞ þ n  rI ν ¼ jν 1 þ ν Iν c ∂t 2hν3

ð∞ ð

þ

ν ðsÞ 0 k ðν ! ν, n0  nÞI ν0 dΩ0 dν0 , ν0

(8.33)

0 4π

k ðνtÞ

where is the sum of the absorption and scattering coefficients, kðνaÞ þ kðνsÞ . Unfortunately, (8.33) does not give the correct distribution function when considering scattering in a thermodynamic equilibrium environment. Consider a free-electron gas described by a Maxwellian distribution. The scattering kernel kðsÞ ðν0 ! ν; n0  nÞ must also satisfy the detailed balance conditions. This condition states that in complete TE, the number of photons that scatter from dν0 dΩ0 about ðν0 ; n0 Þ to dνdΩ to ðν; nÞ must equal the number scattered from dνdΩ to dν0 dΩ0 . That is,   1 ðsÞ 0 c2 ðPÞ ðPÞ 0 k ðν ! ν, n  nÞBν0 1 þ B 3 ν hν0  2hν 2  1 ðsÞ c ðPÞ 0 0 ðPÞ (8.34) B0 , ¼ k ðν ! ν , n  n ÞBν 1 þ hν 2hν03 ν where BðνPÞ is the Planck distribution function. Inserting the Planck function in (8.34) yields 1 ðsÞ 0 1 k ðν ! ν; n0  nÞW ðν0 Þ ¼ k ðsÞ ðν ! ν0 ; n  n0 ÞW ðνÞ, hν0 hν

(8.35)

where W ðνÞ e ν3 ehν=kT is the Wien approximation to the Planck function. This result can be interpreted as the detailed balance condition in the absence of induced scattering and shows that the neglect of the induced terms leads to Wien’s law, rather than Planck’s, as the equilibrium distribution of the scattering operator. Equation (8.32) is not restricted to LTE situations but describes a more general class of problems. To see the effect of the LTE assumption, we can replace jν and 0 kðνaÞ with Bν and k ðνaÞ defined by jν ¼

0 kðνaÞ Bν

and

k ðνaÞ

¼

0 k ðνaÞ



 c2 1þ Bν : 2hν3

(8.36)

It is important to note that Bν is not the Planck function BðνPÞ ; it is just a new variable defined in terms of kðνaÞ and jν . The transfer equation (8.32) becomes

8.3 The Equation of Transfer

405

0 1 ∂I ν ðnÞ þ n  rI ν ðnÞ ¼ k ðaÞ ν ½Bν  I ν ðnÞ c ∂t ð∞ ð   ν ðsÞ 0 c2 0 0 þ k ðν ! ν, n  nÞI ν0 ðn Þ 1 þ I ν ðnÞ dΩ0 dν0 ν0 2hν3

0 4π

ð∞ ð

  c2 0 k ðν ! ν , n  n ÞI ν ðnÞ 1 þ I ν0 ðn Þ dΩ0 dν0 : 2hν03 ðsÞ



0

0

(8.37)

0 4π

If we exclude scattering in (8.37), then in complete TE the radiation field is independent of space and time, and (8.37) becomes 0

kνðaÞ ½Bν  I ν ðnÞ ¼ 0:

(8.38)

In this situation, the transfer equation must give the Planck function for the specific intensity; hence, according to (8.38), Bν ¼ BðνPÞ . The LTE assumption states that the radiation field does not affect the properties of the matter. Putting the Planck function into the second part of (8.36) shows that   0 kðνaÞ ¼ kðνaÞ 1  ehν=kT : (8.39) The form of (8.37) is preferred for the radiative transfer equation even if the LTE assumption is not imposed. This assumption is generally made in radiation hydrodynamic work because of the simplification it affords; namely, thermodynamics can be used to describe the matter. Equation (8.32) is an integro-differential equation, which can be daunting if not almost impossible to work with. To have practical applications, we need to find simpler representations, which we shall do momentarily. Quite often the scattering process results in negligible frequency shift between the incoming and outgoing photons, and if the angular distribution of the radiation is essentially isotropic, (8.37) can be written in a much more userfriendly form 0 1 ∂I ν þ n  rI ν ¼ k ðνaÞ Bν  k ðνtÞ I ν þ k ðνsÞ J ν , c ∂t

(8.40)

where the mean intensity J ν is given by (8.6). 8.3.1 Boundary Conditions on the Transfer Equation Our description of radiation transfer is incomplete without specifying the boundary conditions. Since (8.32) (without the stimulated emission term) is a first-order

406

Radiation and Radiative Transfer

differential equation in space and time, a solution cannot be obtained without knowledge of the boundary conditions of both variables. We assume that the medium of interest is bounded by a surface that is nonreentrant. By nonreentrant, we mean that if any photon leaves the volume, it will not reenter through another part of the surface of that volume. If the body is reentrant, we enclose it with a nonreentrant surface. The new system is then partly vacuum and partly real matter. For a nonreentrant volume, we must specify the specific intensity at all points on the bounding surface. That is, for a particular point on the surface ^  n < 0, I ν ðrs ; n; t Þ ¼ Γν ðrs ; n; t Þ and n

(8.41)

^ is the outward normal unit vector at the point and Γ is the specified where n boundary function. For a “vacuum” surface, Γ ¼ 0, which states that no photons enter the system through its surface. The initial condition is specified by I ν ðr; n; 0Þ ¼ Λν ðr; nÞ:

(8.42)

8.3.2 Equation of Transfer in Various Coordinate Systems The equation of transfer is a little opaque (pun intended) when it comes time to use it in a calculation. In a specific coordinate system, the n  rI ν term in (8.33) needs some interpretation. Let us write n  rI ν ¼ ∂I ν =∂s, where s is a length in the direction n. In particular, for planar geometry with only one dimension, that of z, the intensity depends only on the spatial coordinate and a single angular coordinate θ, the angle between n and the z-axis. The geometry is depicted in Figure 8.2. With μ ¼ cos θ, we have

Figure 8.2. Geometry used to construct the planar transfer equation.

8.3 The Equation of Transfer

∂I ν ∂I ν dz ∂I ν dμ þ : ¼ ∂s ∂z ds ∂μ ds

407

(8.43)

From the figure, we see that dz=ds ¼ cos θ ¼ μ and dμ=ds ¼ 0. Equation (8.33) then becomes   1 ∂I ν ðμÞ ∂I ν ðμÞ c2 þμ ¼ jν 1 þ I ν ðμÞ  kðtÞ ν I ν ðμÞ c ∂t ∂z 2hν3

ð∞ ð

ν ðsÞ 0 k ðν ! ν, n0  nÞI ν0 ðμ0 ÞdΩ0 dν0 : ν0

þ

(8.44)

0 4π

The scattering term is easily simplified in planar geometry by expanding the angular dependence of the differential scattering coefficient in Legendre polynomials k ðsÞ ðν0 ! ν; n0  nÞ ¼

∞ X 2n þ 1

4π

n¼0

k ðnsÞ ðν0 ! νÞPn ðn0  nÞ:

(8.45)

The expansion coefficients follow from the orthogonality property of the Legendre function ð1 ðsÞ 0 k ðν ! νÞ ¼ 2π kðsÞ ðν0 ! ν; μ0 Þdμ0 : (8.46) 1

We also need the addition formula for the Legendre polynomials Pn ðn0  nÞ ¼ Pn ðμÞPn ð μ0 Þ þ 2

n X ðn  mÞ!

ðn þ mÞ! m¼1

m 0 0 Pm n ð μÞPn ð μ Þ cos mðϕ  ϕ Þ,

(8.47)

where θ is the polar angle and ϕ is the azimuthal angle in the direction of n, and similarily μ0 and ϕ0 define n0 ; the Pm n ð μÞ are the associated Legendre polynomials. The angular portion of the scattering term of (8.44) becomes

ð

ðð

2π 1

kðsÞ ðν0 ! ν, n0  nÞI ν0 ðμ0 Þdμ0 dϕ0

k ðsÞ ðν0 ! ν, n0  nÞI ν0 ð μ0 ÞdΩ0 ¼ 4π

¼

∞ X 2n þ 1 n¼0

þ2

4π

ð1 0 k ðsÞ n ðν

n X ðn  mÞ! m¼1

ðn þ mÞ!

0 1 2π 0

ð

! νÞ I ν0 ð μ Þ 1

Pn ðμÞPn ð μ0 Þ

0

m 0 Pm n ðμÞPn ðμ Þ cos mðϕ

  ϕ Þ dϕ0 dμ0 : 0

(8.48)

408

Radiation and Radiative Transfer

Because of the periodicity of the cosine function, m, (8.48) simplifies to

2Ðπ

cos mðϕ  ϕ0 Þdϕ0 ¼ 0 for all

0

ð k ðsÞ ðν0 ! ν, n0  nÞI ν0 ðμ0 ÞdΩ0 4π

¼

∞ X 2n þ 1 n¼0

2

ð1 0 0 0 0 k ðsÞ n ðν ! νÞPn ð μÞ Pn ð μ ÞI ν0 ð μ Þdμ :

(8.49)

1

Using this equation in (8.44) gives the equation of transfer in planar geometry   1 ∂I ν ðμÞ ∂I ν ðμÞ c2 I ν ðμÞ  kðtÞ þμ ¼ jν 1 þ ν I ν ðμÞ c ∂t ∂z 2hν3 ∞ X 2n þ 1

þ

n¼0

2

ð∞

ð1

ν ðν ! ν0 Þ Pn ð μ0 ÞI ν0 ðμ0 Þdμ0 dν0 : Pn ðμÞ 0 kðsÞ ν n 0

(8.50)

1

In practice, the sum can be truncated after only a few terms. This is true for two reasons: The differential scattering coefficients of interest are generally well represented by a few terms in the Legendre expansion, and the solution to the transfer equation is relatively insensitive to the higher terms of the expansion. A similar approach can be used for cylindrical geometry and spherical geometry but are slightly more tedious; the mathematical development may be found in Pomraning (1973). For those readers who might contemplate doing actual calculations, we present results for three-dimensional Cartesian, cylindrical, and spherical geometries. In Cartesian coordinates, (8.33) is easily written as   1 ∂I ν ∂I ν ∂I ν ∂I ν c2 þ nx þ ny þ nz ¼ jν 1 þ I ν  kðtÞ ν Iν c ∂t ∂x ∂y ∂z 2hν3

ð∞ ð

þ

ν ðsÞ 0 k ðν ! ν, n0  nÞI ν0 ðn0 ÞdΩ0 dν0 , ν0

(8.51)

0 4π

where nx , ny , and nz are the direction cosines with respect to the coordinate axes. Any two of these three cosines can be taken as the two variables needed to define n; it thus follows n2x þ n2y þ n2z ¼ 1. In cylindrical coordinates, we have (Reed & Lathrop, 1970)

8.4 Moments of the Transfer Equation

409

  1 ∂I ν ∂I ν ξ ∂I ν ∂I ν ξ ∂I ν c2 I ν  kðtÞ þη þ þμ  ¼ jν 1 þ ν Iν c ∂t ∂r r ∂Θ ∂z r ∂ϕ 2hν3

ð∞ ð

þ

ν ðsÞ 0 k ðν ! ν, n0  nÞI ν0 ðθ0 , ϕ0 ÞdΩ0 dν0 , ν0

(8.52)

0 4π

where I ν ¼ I ν ðr; Θ; z; θ; ϕ; tÞ and μ ¼ cos θ, η ¼ sin θ cos ϕ, ξ ¼ sin θ sin ϕ,

(8.53)

and μ2 þ η2 þ ξ 2 ¼ 1. In spherical coordinates, we have (Reed & Lathrop, 1970) 1 ∂I ν ∂I ν η ∂I ν ξ ∂I ν ð1  μ2 Þ ∂I ν ξ cot Θ ∂I ν þμ þ þ þ  c ∂t r r ∂r r ∂Θ r sin Θ ∂Φ ∂μ ∂ϕ ∞ð ð   c2 ν ðsÞ 0 ¼ jν 1 þ I ν  kðtÞ k ðν ! ν, n0  nÞI ν0 ð μ0 , ϕ0 ÞdΩ0 dν0 , ν Iν þ 3 ν0 2hν 0 4π

(8.54) where I ν ¼ I ν ðr; Θ; Φ; μ; ϕ; tÞ. The angle cosines are given in (8.53). 8.4 Moments of the Transfer Equation In Section 8.1, we discussed the first three angle moments of the intensity; they are the energy density, flux, and pressure tensor, respectively. We can do something similar to the transfer equation to yield a hierarchy of terms. Unfortunately, when we do this, each moment of the transfer equation introduces the next higher moment of the intensity, so the set of moment equations up through a given order is always one equation shy of being closed. We often get around this difficulty by prescribing an ad hoc relation that gives the highest moment, and that expression may contain elements of the lower moments. If we limit our discussion to the first two moments, we see that in the kinetic theory of gases, these are precisely the moments that lead to Euler’s equations after the pressure tensor is approximated. The transport equation (8.40) is written in coordinate-free notation. While appearing simple, it is actually deceptive. If a curvilinear system of spatial coordinates is used, then it is not correct to think of the angle vector as being constant during spatial transport. Instead, we have to imagine that the unit vector n is moved along, remaining parallel to itself in a physical sense, which means that its components along the three coordinate directions are changing. To avoid this

410

Radiation and Radiative Transfer

problem, we refer to a Cartesian coordinate system, and then the components of n are truly constant. A second point to consider is that emission and absorption are not necessarily isotropic. This can occur in a medium that has a nonisotropic structure such as a crystal or in highly magnetized plasma, where the normal atomic physics processes are modified by the Zeeman effect and depend on the photon direction with respect to the magnetic field. Anisotropy of the emissivity is more common, since scattering is in most cases different for different angles between the incoming and outgoing photons; since the intensity itself may be quite anisotropic, the scattered distribution is anisotropic also. Of course, there are the Doppler and aberration effects that will be discussed later. In our Cartesian coordinate system, the solid angle is constant and it can be taken inside the divergence term of (8.37). Setting aside the scattering terms, we have 0 1 ∂I ν þ r  ðnI ν Þ ¼ k νðaÞ ðBν  I ν Þ: c ∂t

(8.55)

The next two moments are found by successively multiplying by n and integrating over angles; thus 0 ∂E ν þ r  Fν ¼ kνðaÞ ð4πBν  cEν Þ ∂t

(8.56)

0 1 ∂Fν þ cr  P ν ¼ kνðaÞ Fν : c ∂t

(8.57)

and

Since E ν , Fν , and P ν are a proper scalar, vector, and tensor, respectively (not in spacetime, but just in space), we can use curvilinear coordinates for these moment equations, which are written in coordinate-free form. Integrating (8.56) and (8.57) over frequency gives expressions for “mass” conservation and momentum conservation: ð 0 ∂E r (8.58) þ r  Fr ¼ kðνaÞ ð4πBν  cEν Þdν ∂t and

ð 0 1 ∂Fr þ cr  P r ¼  kνðaÞ Fν dν: c ∂t

(8.59)

The “r” subscript on the energy density, flux vector, and pressure tensor denote radiation, as opposed to matter.

8.4 Moments of the Transfer Equation

411

Since photons have no mass, (8.58) is really a statement about conservation of relative mass, that is, energy. The terms on the left-hand side of (8.58) express the familiar energy density and energy flux, while the integral on the right-hand side represents the net rate of gain and loss of the radiation energy per unit volume. In contrast to the case where real matter is present, the emission term corresponds to a loss of material energy, and the absorption term would be a gain for the matter. Dividing (8.59) by the speed of light, we have ð 1 ∂Fr ðaÞ0 Fν þ r  P ¼ c k dν, (8.60) ν r c2 ∂t c2 where the radiation momentum density is Fν =c2 and the radiation momentum flux (pressure) is P r . The divergence terms in (8.58) and (8.60) account for the flow of radiant energy and radiant momentum, respectively, across the boundary surface of the volume element. The integral on the right-hand side of (8.60) is the momentum lost per unit time by the radiation and transferred to the matter. This provides the avenue for connecting the radiative transfer equation to the material conservation equations of momentum and energy. We will pursue this in Chapter 10. The moment equations provide powerful tools for solving transfer problems because they eliminate angle variables and thus reduce the dimensionality, but as pointed out earlier, they introduce the problem of “closure.” It is useful to compare the radiation moment equations to their hydrodynamic counterparts. We developed Euler’s equations from the Boltzmann equation in Sections 3.6, 6.2, and 6.6. The energy density, heat flux, and stress in the fluid were found by taking suitable averages over the distribution function. We assumed that the distribution function is isotropic, so that the system of fluid equations closes exactly, then the heat flux and the stresses vanish identically. The same is true for radiation, if we assume I ν is perfectly isotropic. We know from (8.13) that the radiation stress tensor becomes diagonal and isotropic with Pν ¼ E ν =3 and F ν ¼ 0, and no further closure is necessary. On the other hand, if, as in the case of the material conservation equations, the particle mean free path is much less than structural scale lengths, we are able to achieve closure by deriving explicit expressions for the fluid heat flux and viscous stress tensor. We can do an analogous process for the radiation in the limit that the photon mean free path is small compared to the density scale length; this will be presented shortly. In one-dimensional calculations, the easy solution is to close the system with variable Eddington factors f ν , represented by (8.19). The moment equations are solved assuming f ν is known, and then we subsequently determine f ν from a separate angle-by-angle solution to the full transfer equation, assuming that the radiation energy density Eν is known. This is an iterative process until the two steps converge. Some authors have developed computationally easier approaches

412

Radiation and Radiative Transfer

to determining f ν based on analytic formulas for specific problems. In multidimensional calculations, the problem becomes significantly more complex. The difficulty arises near the boundary surfaces where the mean free path (for both particle and photon) exceeds any structural length. There we must find other methods for evaluating the averages that appear as the energy flux or as nonisotropic contributions to the stress tensor. For ordinary particle flow, the issue is not important except for extremely rarefied flows. But this is not the case for radiation, where there might be regions where the particle mean free path is short compared to the radiation mean free path.

8.5 Optical Depth The concept of optical thickness (depth) is central to the discussion of radiation transfer. It is defined by Δ ðs

τν ¼

kðνtÞ dl,

(8.61)

0

where the integral is over the path through the material. The total absorption coefficient is a function of the local properties of the material (density, temperature, and composition), so kðνtÞ will most likely vary along the path. If the medium is moving, the optical depth can depend both on the separation between the two points and on the direction in which the integration is performed. Since λν ¼ 1=kðνtÞ is the mean free path of a photon of frequency ν, we see that τ ν is equal to the number of mean free paths along Δs. In terms of the attenuation of the specific intensity, τ ν ¼ 1 corresponds to the reduction by an amount e1 . For a static planar medium, optical depth is customarily measured from the boundary inward. Of course, if not integrating in the direction antinormal to the surface, the integral must take into account the angle of the trajectory with respect to the normal direction. In the case of a solar atmosphere, integrating inward in the radial direction, the photosphere is defined as the surface where the optical depth is 2/3. We shall return to this shortly.

8.6 Approximate Descriptions of the Radiative Transfer Equation The radiative transfer equation (8.37) is much too complicated for everyday use. Even the reduction of this equation to simple geometries does not yield a “simple” equation. The problem, as pointed out earlier, is that the transfer equation has independent variables of photon frequency and angle. So, we are forced into using approximations

8.6 Descriptions of Radiative Transfer Equation

413

that give tractable versions of the transfer equation. Solutions to the equation of radiative transfer form an enormous body of work. The majority of the differences are due to the various forms people assign to emission and absorption coefficients. In opaque material, such as the interior of a star, photons are trapped and the radiation field is nearly isotropic and approaches thermal equilibrium; the photon mean free path is much smaller than the characteristic structural length of the body. In this limit, the energy transport is well described by the diffusion process; the photons undergo random walk paths through the material. This is true until the bounding surface of a radiating medium is approached. There, the material becomes transparent, and photon mean free paths become vastly greater than the scale lengths. That is, the transport is nonlocal and we must treat the photon transport with a kinetic-like theory. The problem is further complicated by photon scattering. After being thermally emitted, a photon may scatter many times, changing only its direction of travel before being absorbed by the material. In doing so, the collection of photons forming the radiation field is no longer uniquely determined by local conditions. The transport is determined by conditions within a large interaction volume whose size is set by the extent of the photon’s travel from birth to death. Thus, the specific intensity is not a local variable. The transfer equation is (superficially) linear in the radiation field, and it is thus possible to solve it for very general physical situations by numerical techniques. But as pointed out in the discussion about induced processes, inclusion of those terms makes the transfer equation nonlinear. In addition, the emission and absorption coefficients depend on the internal state of the material, such as ionization level. In general, the radiation field and the internal state of the matter must be determined simultaneously and self-consistently. Many prescriptions have been put forth for solving the transfer equation (8.37) for specific situations. We turn now to a few of the more simple approximations – those that have found utility in modeling high-energy-density physics. 8.6.1 Free-Streaming Approximation Let us recalculate the moment equations. Integrating (8.40) over angles gives 0 ∂E ν þ r  Fν ¼ kðνaÞ ð4πBν  cEν Þ, ∂t

(8.62)

which is identical to (8.56) since the integral over J ν is zero. The flux equation becomes 1 ∂Fν 1 ∂Fν þ cr  P ν ¼ þ cf ν rEν ¼ k ðνtÞ Fν , (8.63) c ∂t c ∂t where f ν is the Eddington factor from (8.19).

414

Radiation and Radiative Transfer

In the limit of very small optical depth, which is the case in the absence of matter, there is no emission or absorption. Thus, (8.62) becomes ∂Eν þ r  Fν ¼ 0, ∂t

(8.64)

1 ∂Fν þ cf ν rE ν ¼ 0: c ∂t

(8.65)

and (8.63) becomes

For planar geometry, eliminating Fν between these two equations results in the wave equation 2 ∂2 Eν 2 ∂ Eν  c ¼ 0: ∂t2 ∂z2

(8.66)

This equation is familiar, since it can also be derived from Maxwell’s equations in the absence of matter. We note that in this approximation, the Eddington factor is one.

8.6.2 Diffusion Approximation In the opposite limit of great optical depth, the time derivative in (8.63) may be neglected relative to k ðνtÞ Fν . We then have 1 r  P ν ¼ f ν rE ν ¼  kðνtÞ Fν : c

(8.67)

From a previous discussion in Section 8.1.4, we know for these conditions that the pressure is one-third that of the energy density; that is, the Eddington factor is 1=3. Equation (8.67) becomes an expression for the radiative flux Fν ¼ 

c 1 rEν , 3 k ðνtÞ

(8.68)

where the “diffusion” coefficient is defined by Dν ¼ c=3k ðνtÞ . Using this in (8.62) produces the diffusion equation 0 ∂Eν  r  ðDν rEν Þ ¼ kνðaÞ ð4πBν  cEν Þ: ∂t

(8.69)

By neglecting the time derivative in (8.63), we have lost the finite propagation speed as well as the radiation momentum density. Any momentum imparted to the matter by the radiation is not compensated for in the radiation momentum.

8.6 Descriptions of Radiative Transfer Equation

415

Equation (8.69) is a parabolic differential equation. If the right-hand side is dropped temporarily, then the equation describes a radiation wave that spreads qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi according to z e ct=3k ðνtÞ . Even so, the propagation of radiation from a pulse at

time zero can be faster than the speed of light at early time when kðνtÞ ct < 1, and the transport is not diffusive. This is related to the fact that there is no limit to the flux given by (8.68). We will return to this topic momentarily. Some comments are in order about the diffusion coefficient Dν . In some respects, it is analogous to the diffusion coefficient for material particles. For the case of a material particle, it does not vanish upon a collision, but only changes its direction of motion; the mean free path is the collisional mean free path, as presented in Chapter 3. In contrast, the photon, after traversing an average distance λν ¼ 1=k ðνtÞ , is “absorbed” by the medium. And if the fluid is in a state of thermodynamic equilibrium, the energy of the photon is distributed among the particles according to the laws of statistical equilibrium. At the place of absorption, new photons are emitted with different frequencies and in random directions. In considering the “diffusion” of photons of a given frequency, among all newly born photons we look only at those at the same frequency. This process repeats itself, with the newly born photon traveling some distance, being absorbed, and then being born again. Another view of the diffusion equation is found from (8.37). Since τ ν  1, the matter is in LTE, and we expect Bν  I ν . Setting aside the two scattering terms for the moment, we start with I ν ðnÞ ¼ Bν 

1 ∂I ν ðnÞ n  rI ν ðnÞ : þ 0 0 ck νðaÞ ∂t k ðνaÞ

(8.70)

Now Bν is by definition isotropic, and this suggests we can expand the specific intensity in a series I ν ðr; n; tÞ  I ðν0Þ ðr; t Þ þ I ðν1Þ ðr; t Þ þ    :

(8.71)

We must retain at least two terms in the expansion because it is the small asymmetry produced by the gradient of the specific intensity that yields a nonzero flux. Now I ðν0Þ ¼ Bν , and using this in the right-hand side of (8.70), we have   1 1 ∂Bν ð0Þ ð1Þ ð0Þ I ν þ I ν  I ν  ðaÞ0 (8.72) þ n  rBν : c ∂t kν Thus, I ðν1Þ is a correction term that depends upon the isotropic component of the specific intensity and the absorption coefficient.

416

Radiation and Radiative Transfer

We could include more terms to attempt an even better approximation, but to do so may be counterproductive. We have assumed that the absorption coefficient is large and the next term of the expansion would be too small for us to care about. On the other hand, if the absorption coefficient is not large, then we anticipate that including more terms will make the answer worse rather than better. The infinite series is really an asymptotic expansion, and including one term after another will make the answer better for a while, then it will start to diverge. Returning to (8.68), we see the photon flux is determined by the gradient of the energy density. As with the diffusion of thermal energy (see Chapter 7), the diffusion approximation is applicable only to small gradients of the radiation density. That is, the radiation energy density should change little over a distance of the order of the radiation mean free path λν . The radiation field for small gradients is almost isotropic, and it was this condition that allowed us to derive (8.68). In fact, the photons arriving at a given point originate primarily in a region with dimensions of the order of the mean free path. If the radiation density in this region is almost constant, then the photons arrive at that point uniformly from all directions, and the radiation field is considered isotropic. It is obvious that at a vacuum boundary, the radiation changes very rapidly over a distance of order λν , and the angular distribution of the photons is very anisotropic. The photons leave the medium in the direction of the vacuum, and none arrive from the vacuum. Hence, the diffusion approximation can lead to appreciable errors under these circumstances. In contrast, for an optically thick medium, the radiation density gradients are small, and we expect the diffusion approximation to be valid, and we can approximate (8.68) by jF ν j e λν cEν =z. The greater the optical depth τ ν e λν =z, the smaller the flux is compared with cEν , and the diffusion approximation is more exact. If the body is optically thin (τ ν  1), then jF ν j e cEν , and in the extreme of τ ν < 1, the flux becomes greater than cEν . This is physically impossible and thus indicates the diffusion approximation is inapplicable to optically thin bodies. The flux can never be greater than cEν since F ν ¼ cEν corresponds to the case where all of the photons travel in exactly the same direction, the case of maximum anisotropy. Hence, the ratio of F ν =cEν is a measure of the anisotropy of the radiation field. An optical depth τ ν ¼ 1 corresponds to a reduction in the specific intensity of 1 e , and we might expect a distance corresponding to τ ν  2  3 would be sufficient to define the thickness of the boundary layer. In many cases, this is a good estimate. However, scattering clouds the issue (pun intended). Consider a photon being emitted in an infinite, homogeneous medium. It travels a displacement r1 before being scattered, and then a displacement r2 in a new direction before being scattered again, and so on. The net displacement after N scatterings is

8.6 Descriptions of Radiative Transfer Equation

R ¼ r1 þ r2 þ    þ rN :

417

(8.73)

The distance jRj traveled can be estimated by taking the square of (8.73) and averaging over all paths. Thus, 2 2 2



(8.74) R ¼ r1 þ r2 þ    þ r2N þ 2hr1  r2 i þ 2hr1  r3 i þ    : Each term involving the square of a displacement averages to the mean square of the free path λν . The cross terms in (8.74) average to zero. Therefore 2

(8.75) R ¼ Nλ2ν : Using this, we can estimate the mean number of scatterings in a finite medium. Assuming for the moment that the photon is not absorbed, it will scatter until it escapes completely. For a medium with a large optical depth, the number of pffiffiffiffiffiffiffiffi scatterings required for escape can be determined by setting hR2 i equal to the structural size, L. Thus, N  L2 =λ2ν . Now L=λν is approximately the optical thickness τ ν , so N  τ 2ν . In contrast, for a medium of small optical depth, the number of scatterings is of order 1  eτ ν  τ ν , and so N  τ ν . For an order-of-magnitude estimate, we can use N  τ 2ν þ τ ν for any optical thickness. Turning absorption back on, we can define the single-scatter albedo as the probability that a photon will undergo a single scattering event ϖν ¼

k ðνsÞ

kðνsÞ þ kðνaÞ

:

(8.76)

In an infinite medium, a photon is created, undergoes a number of scatterings in a random-walk fashion and is then destroyed. Since the random walk can be terminated with probability 1  ϖν at the end of each free path, the mean number of paths is N ¼ 1=ð1  ϖν Þ. Thus, from (8.75) 2

Rν ¼

λ2ν : 1  ϖν

 1 Now λν ¼ kðνsÞ þ k ðνaÞ and we find qffiffiffiffiffiffiffiffiffiffi 2

1 Rν  qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  : ðaÞ ðsÞ ðaÞ kν kν þ kν

(8.77)

(8.78)

This length represents the net displacement of a photon between the points of creation and destruction. This distance is defined as the effective mean path or diffusion length. In a finite medium, the random walk behavior depends strongly on the size of L. Let us define the effective optical depth

418

Radiation and Radiative Transfer

L τ ν ¼ q ffiffiffiffiffiffiffiffiffiffi ¼ L R2ν

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  kðνaÞ k ðνsÞ þ kðνaÞ :

(8.79)

When the effective free path is large compared to L, we have τ ν 1, and the medium is said to be effectively thin or translucent. Most photons will randomly walk their way to freedom before being truly absorbed. In the other extreme, when the effective free path is small compared to L, τ ν  1 and the medium is effectively thick. An albedo approaching unity says that the photon will undergo many scatterings before being absorbed. In this case, the thickness of the boundary layer becomes the distance the photon can move in a random walk with that many steps, which can be many mean free paths. In the limit of an albedo of one, the whole of the medium is boundary layer. In most of the cases we encounter in high-energydensity physics, the albedo is not very close to unity, and the boundary layer is at most a few mean free paths thick. The reader is probably more familiar with the definition of albedo (Latin for whiteness) as a measure of the diffuse reflectivity of an object. It is defined as the ratio of the reflected radiation from a surface to the incident radiation upon it. In addition to being frequency dependent, the albedo also depends upon the directional distribution of the incident radiation. An exception to this last dependency is a Lambertian surface that scatters radiation in all directions according to a cosine function, and so the albedo does not depend on the incident distribution.

8.6.3 Telegrapher’s Equation Before we leave the discussion about simple radiation transfer models in the two extremes, let us consider a combination of the free-streaming and diffusion regimes. Equations (8.66) and (8.69) were obtained from the moment equations when the flux F ν was eliminated under special circumstances. When the absorption coefficient is a function of frequency only, the flux can always be eliminated from the moment equations. From (8.62), assuming the total absorption coefficient is a function of frequency only, we find 1 ∂ 1 ∂Eν c ∂t kðtÞ ∂t ν

! þ

0 ∂E ν  kðaÞ ν ð4πBν  cE ν Þ ∂t

! " 0 # 1 ∂ k ðaÞ 1 ν ð4πBν  cEν Þ þ cf ν r  ðtÞ rE ν , ¼ c ∂t k ðtÞ kν ν

(8.80)

8.6 Descriptions of Radiative Transfer Equation

419

which is a form of the telegrapher’s equation (Landau & Lifshitz, 1960). Although the time and space derivatives involving k ðνtÞ are zero, we place k ðνtÞ within the derivatives so that (8.80) will have the same form as the diffusion equation (8.69). In general, we have no clear indication of the proper value for f ν , but we expect it to have values 13 f ν 1.

8.6.4 Eddington’s Approximation The diffusion equation is the result of applying the lowest-order approximation to the radiative transfer equation. This approximation is often termed Eddington’s approximation (Eddington, 1926), but Eddington’s approximation is a special case of the diffusion equation. In a manner similar to the discussion about forming the diffusion equation, Eddington’s approximation begins with the angular dependence of the specific radiation intensity being approximated by a spherical harmonic expansion. That is, I ν ðr; n; t Þ ¼

1 ð0Þ 3 I ν ðr; tÞ þ n  Iðν1Þ ðr; t Þ þ    , 4π 4π

(8.81)

where the expansion coefficients I ðν0Þ , Iðν1Þ , and so on are to be determined. The higher-order terms involving Iðν1Þ , and so on represent anisotropy corrections to the (presumed) dominant I ðν0Þ term. Integrating (8.81) over all solid angles establishes the zeroth coefficient I ðν0Þ ðr; tÞ

ð ¼

I ν ðr; n; t ÞdΩ,

(8.82)

4π

which is to within a factor of the speed of light, just the energy density per unit frequency, E ν . Multiplying (8.81) by n and then integrating gives Iðν1Þ ðr; t Þ ¼

ð nI ν ðr; n; t ÞdΩ,

(8.83)

4π

which is the radiative flux per unit frequency, Fν . Similar operations on the higher-order terms of (8.81) can be performed; in fact, the next moment is related to the pressure tensor. Keeping only the first two terms of (8.81) and using this in (8.37), we find

420

Radiation and Radiative Transfer

1 ∂I ð0Þ ðaÞ0 ð0Þ ðsÞ ð0Þ ν þ r  Ið1Þ ν ¼ k ν ½4πBν  I ν   k ν I ν c ∂t

ð∞

þ

ν ðsÞ 0 ð0Þ k ðν ! νÞI ν0 dν0 ν0 0

0

c2 ð0Þ I þ 8πh ν

ð∞  0

3c2 ð1Þ I þ 8πh ν

ð∞ 

 1 ðsÞ 0 1 ðsÞ ð0Þ 0 k 0 ðν ! νÞ  03 k0 ðν ! ν Þ I ν0 dν0 2 0 νν ν  1 ðsÞ 0 1 ðsÞ ð1Þ 0 k ðν ! νÞ  03 k1 ðν ! ν Þ Iν0 dν0 : ν2 ν0 1 ν

(8.84)

0

Similarly, multiplying (8.37) by n and then integrating yields 0 1 ∂Ið1Þ 1 ðaÞ ð1Þ ν þ k ðsÞ þ rI ð0Þ ν ¼ ½k ν ν Iν c ∂t 3

ð∞

þ

ν ðsÞ 0 ð1Þ k 1 ðν ! νÞIν0 dν0 0 ν

0

c2 ð0Þ I þ 8πh ν

ð∞ 

 1 ðsÞ 0 1 ðsÞ ð1Þ 0 k ðν ! νÞ  03 k1 ðν ! ν Þ Iν0 dν0 ν2 ν0 1 ν

0

þ

c2 ð1Þ I 8πh ν

ð∞ 

 1 ðsÞ 0 1 ðsÞ ð0Þ 0 k ðν ! νÞ  k ðν ! ν Þ I ν0 dν0 : ν2 ν0 0 ν03 0

(8.85)

0

In (8.84) and (8.85), we have defined

kðnsÞ ðν0

ð1 ! νÞ ¼ 2π

k ðsÞ ðν0 ! ν; μ0 ÞPn ðμ0 Þdμ0 ,

(8.86)

1

where Pn ðμ0 Þ is the Legendre polynomial of degree n, and the total scattering coefficient is kðνsÞ

ð∞

ðsÞ

¼ k0 ðν ! ν0 Þdν0 : 0

(8.87)

8.6 Descriptions of Radiative Transfer Equation

421

Equations (8.84) and (8.85) form a closed set of equations for I ðν0Þ ðr; t Þ and providing initial and boundary conditions are specified. For the initial conditions, using (8.42), we have

Iðν1Þ ðr; t Þ

ð I ð0Þ ν ðr, 0Þ

¼

Λν ðr, nÞdΩ,

ð

4π

Ið1Þ ν ðr, 0Þ ¼

nΛν ðr, nÞdΩ:

(8.88)

4π

The boundary conditions on (8.84) and (8.85) are not as straightforward to write down. At each point on the boundary, we need a single relation between I ðν0Þ ðr; t Þ and Iðν1Þ ðr; tÞ. Because of its simple angular dependence, the Eddington representation of the specific intensity (8.81) cannot satisfy the boundary condition (8.41) for an arbitrary incoming distribution Γν ðrs ; n; t Þ. The best we can do is to satisfy (8.41) in an integral sense. That is, we use (8.81) in (8.41), multiply the result by a weight function wðnÞ, and integrate over all incoming directions. This gives ð ^  n > 1 > > for μ < 0 exp ½ðτ  τ 0 Þ=μSðτ 0 Þdτ 0 > > jμj < τ I ðτ; μÞ ¼ ðτ > > 1 > > I s ð μÞ exp ðτ=μÞ þ exp ½ðτ  τ 0 Þ=μSðτ 0 Þdτ 0 for μ > 0 , > > : μ 0

(8.132) where I s ð μÞ is the incoming intensity at the boundary point xs .

8.8 Steady-State Transfer

435

Unless the source function is directly known, we need to conjure up J, which is to be found from (8.131) or by other means. The integrals over μ in (8.132) can be written in terms of the exponential integral functions, defined by ð∞ Ein ðxÞ ¼

exy dy: yn

(8.133)

1

The properties of these functions and their recursion relations can be found in Abramowitz and Stegun (1964). For our purposes, we rewrite (8.133) as ð1

Ein ðxÞ ¼ t n2 ex=t dt:

(8.134)

0

Then from (8.6), we find ð∞ c 1 1 Ei1 ðjτ 0  τ jÞSðτ 0 Þdτ 0 : J ðτ Þ ¼ E ðτ Þ ¼ I s Ei2 ðτ Þ þ 4π 2 2

(8.135)

0

We also find the next two moments of the intensity according to (8.7) and (8.15) with (8.16), which are the flux ð∞ F ðτ Þ 1 1 ¼ H ðτ Þ ¼  I s Ei3 ðτ Þ þ sgnðτ 0  τ ÞEi2 ðjτ 0  τ jÞSðτ 0 Þdτ 0 , 4π 2 2

(8.136)

0

and the pressure ð∞ c 1 1 Ei3 ðjτ 0  τ jÞSðτ 0 Þdτ 0 : K ðτ Þ ¼ Pðτ Þ ¼ I s Ei4 ðτ Þ þ 4π 2 2

(8.137)

0

These expressions were first obtained by Milne (1966). All of the exponential integrals behave as exp ðxÞ=x for large x, so for optical depths greater than a few, the lower boundary of the integral in (8.135) can be extended to ∞ with little effect on the answer. The exponential integral functions then evaluate to ð∞ ∞

2 Ein ðjxjÞdx ¼ : n

(8.138)

If the source function Sðτ Þ varies slowly in the interior of the slab, then Sðτ Þ can be taken outside the integrals and evaluated at τ. Then from (8.19) the Eddington factor becomes

436

Radiation and Radiative Transfer

f ðτ Þ ¼

K ðτ Þ 12 I s þ 23 Sðτ Þ ¼ : I s þ 2Sðτ Þ J ðτ Þ

(8.139)

If there is no radiation incident on the boundary, I s ¼ 0, and we find f ¼ 1=3, as expected for τ  1. Even in the case where τ is not large, we can still move Sðτ Þ outside the integral in (8.135), resulting in the escape probability approximation   1 1 J ðτ Þ  I s Ei2 ðτ Þ þ 1  Ei2 ðτ Þ S: (8.140) 2 2 The quantity 12 Ei2 ðτ Þ in (8.140) is called the escape probability. Equation (8.140) can be interpreted as an average of the one-sided escape probability going toward larger τ, namely zero, and the one-sided escape probability going toward smaller τ, which is Ei2 ðτ Þ. The function Ei2 has the value of one at τ ¼ 0 and zero as τ ! ∞. There are some similarities among (8.135), (8.136), and (8.137). To see how these fit together, we use the recursion relations among the exponential integral functions 1 Ein ðxÞ ¼ ½ex  xEin1 ðxÞ (8.141) n1 and dEin ðxÞ ¼ Ein1 ðxÞ: dx

(8.142)

Using these in the preceding equations, we find expressions relating the moment equations dH ¼JS dτ

(8.143)

dK ¼ H: dτ

(8.144)

and

In the diffusion limit, we can expand the source function according to Sðτ Þ  BðPÞ ðτ Þ þ

∂BðPÞ ðτ Þ 0 1 ∂2 BðPÞ ðτ Þ 0 2 ðτ  τ Þ þ ðτ  τ Þ þ    : 2 ∂τ 2 ∂τ

(8.145)

Applying (8.145) to (8.132) gives I ðτ; μÞ ¼ BðPÞ ðτ Þ þ μ

∂BðPÞ ðτ Þ ∂2 BðPÞ ðτ Þ þ , þ μ2 ∂τ ∂τ 2

for the outgoing radiation ( μ < 0), and

(8.146)

8.8 Steady-State Transfer

I ðτ; μÞ ¼ I s ð μÞ þ BðPÞ ðτ Þ þ μ

∂BðPÞ ðτ Þ ∂2 BðPÞ ðτ Þ þ , þ μ2 ∂τ ∂τ 2

437

(8.147)

for μ > 0. Omitting the source term I s in (8.147) reveals that (8.146) and (8.147) are identical, since only terms of O eτ=μ would make a difference. We then find for the moment equations J ðτ Þ ¼ BðPÞ þ

1 ∂2 BðPÞ þ , 3 ∂τ 2

(8.148)

1 ∂BðPÞ 1 ∂3 BðPÞ þ , þ 3 ∂τ 5 ∂τ 3

(8.149)

1 1 ∂2 BðPÞ K ðτ Þ ¼ BðPÞ þ þ : 3 5 ∂τ 2

(8.150)

H ðτ Þ ¼ and

The series in (8.148)  2 2 through (8.150) converge rapidly, since successive terms are 2 Oð1=τ Þ ¼ O λ =L , and we retain only the first term in each series. In contrast, we must retain one additional term in (8.147) because it is the small asymmetry in the specific intensity, which gives rise to the gradient that yields the nonzero flux. From (8.148) and (8.150), we recover the usual Eddington factor. From (8.149), we have the flux H ðτ Þ ¼

1 ∂BðPÞ 1 ∂BðPÞ ∂T 1 ¼ : 3 ∂τ 3 ∂T ∂z k ðaÞ

(8.151)

The equations in this subsection involving I, S, B, J, E, F, K, and P are really frequency dependent, but we omitted that notation to save on ink. These same equations can be integrated over frequency to give results in the gray approximation.

8.8.3 Milne’s Equation Suppose we have a semi-infinite medium with no external radiation impinging on the boundary; that is, I s ¼ 0 at τ ¼ 0, and we wish to know the character of the radiation leaving the surface into the vacuum. We can use the diffusion approximation to obtain a simple estimate of the flux of radiant energy flowing outward, but we know this is incorrect since the diffusion approximation doesn’t apply there. Using the diffusion result relating the pressure to the energy density Pr ¼ Er =3, integrating (8.144) over frequency, gives the gray approximation H¼

F r dK c dPr 1 c dEr ¼ ¼ ¼ : dτ 4π dτ 4π 3 dτ 4π

(8.152)

438

Radiation and Radiative Transfer

We know from Section 3.10 that the energy density is E r ¼ 4σT 4 =c, where σ is the Stefan-Boltzmann constant. At the surface, the flux is just the one-sided flux being emitted at the surface temperature, that is, F r ¼ cEr =2 ¼ 2σT 4 ð0Þ. Using these facts in (8.152) yields   3 4 4 T ðτ Þ ¼ T ð0Þ 1 þ τ : (8.153) 2 We see that for the Sun where the boundary between the photosphere and chromosphere is defined, where τ ¼ 2=3; the temperature there is about 20 percent higher than the surface temperature. This result is known as the Milne problem, and is the most simple of a whole class of solutions. The next best approximation is the homogeneous problem arising from (8.135). Since we are in an equilibrium situation, J ¼ S, and (8.143) tells us the flux is a constant. A simple integration of (8.144) gives K ¼ H ðτ þ constÞ:

(8.154)

Eddington’s approximation, J ¼ 3K, is appropriate for large τ, and the solution for (8.154) is J ¼ 3H ½τ þ qðτ Þ: (8.155) The function qðτ Þ is named the Hopf function, and it tends to a constant for τ  1. We can use the flux leaving the surface to define an effective temperature of the medium. The effective temperature is also known as the brightness temperature. The Planck function is given by BðPÞ ¼ σT 4 =π, and thus (8.155) gives the temperature distribution in the gray atmosphere 3 T 4 ðτ Þ ¼ T 4eff ½τ þ qðτ Þ: 4

(8.156)

In Eddington’s approximation, the Hopf function is replaced by a constant where the value is found at the surface τ ¼ 0. Then from (8.155)

Ð1

μI ð0; μÞdμ H ð0Þ 1 ¼ h μi: ¼ ¼ Ð01 J ð0Þ 3qð0Þ I ð 0; μ Þdμ 0

(8.157)

The value for h μi depends upon the angular distribution for the intensity at the surface. Various estimates have been suggested, each with its corresponding value for the Hopf function. If the intensity is approximately constant over the hemisphere, then h μi ¼ 1=2 and qðτ Þ ¼ 2=3. If we use a two-point Gauss-Legendre pffiffiffi quadraturep(from the discussion of the S-N approximation), μ ¼ 1= 3 and h i ffiffiffi qðτ Þ ¼ 1= 3. The exact value for the Hopf function at large optical depth is pffiffiffi qð∞Þ  0:7105; the exact value of qð0Þ ¼ 1= 3. If we choose the constant

439

8.8 Steady-State Transfer

intensity approximation with qðτ Þ ¼ 2=3, we arrive at an expression similar to (8.153), namely   1 4 3 4 T ðτ Þ ¼ T eff 1 þ τ : (8.158) 2 2 Expression (8.156) is shown in Figure 8.3. The difference between (8.156) and (8.158) is for τ ≲ 0:3, and for τ ¼ 0 the error is less than 4 percent. The effective temperature is always less than the blackbody temperature for depths τ > 2=3. Returning momentarily to the discussion of the Eddington factor, f is not a constant except in Eddington’s approximation. If f is known, then the mean of hμi ¼ f H is probably known also, and using this in (8.157) gives the answer to the homogeneous Milne problem   H f ð0Þ f ¼ : (8.159) τþ J fH We then find an expression for the Eddington factor as a function of optical depth f ðτ Þ ¼

1 τ þ qð∞Þ : 3 τ þ qðτ Þ

(8.160)

Figure 8.3. Temperature distribution (solid line) as a function of optical depth in the diffusion approximation. The viewer sees the temperature that exists at the surface of the photosphere, which is two-thirds of a mean free path inward from the surface where τ ¼ 0. Also shown is the Hopf function (long-dashed line) and the Eddington factor (short-dashed line).

440

Radiation and Radiative Transfer

This expression is shown in Figure 8.3. We can gain more understanding of the nature of the effective temperature by applying (8.136) at the surface of the semiinfinite slab. For sufficient optical depth, Sν ¼ BðνPÞ and (8.136) can be written BðνPÞ ðT eff Þ

ð∞

¼ 2 BðνPÞ ½T ðτ 0 ÞEi2 ðτ 0 Þdτ 0 :

(8.161)

0

We have set the left-hand side of (8.161) to represent an effective surface flux based on F ν ð0Þ ¼ πBðνPÞ ðT eff Þ. Using the expression for the Planck function, we have ð∞ 1 1 ¼ 2 hν=k T ðτ 0 Þ Ei2 ðτ 0 Þdτ 0 , (8.162) hν=k T e B eff  1 e B 1 0

which shows that the effective temperature is a function of frequency. Only in the case of a perfect blackbody are the two the same at all frequencies and equal to the material’s temperature. Upon examining (8.136) and (8.162) at τ ¼ 0, we see that the flux at the surface is frequency dependent and is determined by the integral of the source functions from all depths. Since the exponential integral function decreases rapidly with τ ν for self-absorption, the main contribution to the integral comes from a depth of the order of one mean free path inward from the surface. Photons reaching the surface at frequencies for which the absorption is strong are emitted in the layers closer to the surface, and vice versa. Thus, if the temperature of the material is cooler at the surface (as is often the case), the effective temperature for the strongly absorbing frequencies is less than for the weakly absorbing frequencies. The overall effect is that photons of higher frequency are emitted from the deeper layers. Photons of each frequency are emitted from a depth of about one mean free path, and thus the depth is a function of the frequency. Clearly, the emitted spectrum from a nonuniformly heated body is different from that of the Planck function. As we mentioned previously, there are a number of Milne problems based on different physical scenarios; the reader is referred to the literature for these more elaborate problems.

8.8.4 Eddington-Barbier Relation Equation (8.156) gives the apparent temperature of the medium, but it is also necessary to know the distribution of the intensity. The solution requires solving the general Milne relation for the source function and then doing an additional integration to get the emergent intensity. A useful semiquantitative formula can be

8.9 The Comoving Frame Representation

441

found, assuming that the Planck function varies slowly with optical depth; this is the Eddington-Barbier relation. Neglecting scattering, the angular distribution of the intensity at the surface is found from (8.132) ð∞ I r ð0; μÞ ¼ 0

1 τ=μ ðPÞ e B ðτ Þdτ: μ

(8.163)

Using a linear approximation for the Planck function BðPÞ ¼ a þ b τ, we see I r ð0; μÞ  a þ bμ ¼ BðPÞ ð μÞ:

(8.164)

The intensity the observer sees is that given by the Planck function at one mean free path into the medium measured along the ray. We can also find the value of the flux leaving the surface. Multiplying the intensity by 2πμ and integrating over angles gives     2 ðPÞ 2 : (8.165) F r ð0Þ  π a þ b ¼ πB 3 3 We can use (8.155) with qðτ Þ ¼ 2=3 in (8.163) to arrive at I r ðτ ¼ 0; μÞ ¼ H ð2 þ 3μÞ,

(8.166)

which shows that the emitted radiation is peaked in the direction of outward flow; indeed, I ð0; 1Þ=I ð0; 0Þ ¼ 2:5. We then use (8.166) to calculate J and K at τ ¼ 0, obtaining J ð0Þ ¼ 7H=4 and K ð0Þ ¼ 17H=24, and then f ð0Þ ¼ qð∞Þ=3qð0Þ ¼ 0:410. These results show that using a rough estimate of f ¼ 1=3 to solve the transfer equation yields a source function that gives a reasonably accurate angular distribution of intensity; then we can compute a much better estimate of f throughout the entire material. The Eddington-Barbier relation is useful for qualitatively understanding the nature of the emergent spectrum from a body with a complicated temperature structure when the opacity is very different at different frequencies.

8.9 The Comoving Frame Representation So far, this chapter has been somewhat lackadaisical in specifying the coordinate system for the equations. Mostly, a fixed (laboratory) system is assumed and the motion of the material is measured relative to these coordinates. However, this coordinate system introduces some complications in the radiative transfer equation, namely Doppler and aberration effects.

442

Radiation and Radiative Transfer

The velocity field in a flow is, in general, a function of both position and time. Thus, the comoving frame is a noninertial frame. Photon trajectories in the comoving frame are not straight lines in the Euclidian sense, but are geodesics whose shapes are determined by the metric of the curved spacetime through which the photons move. It is clear that photon frequencies are not constant in spacetime either. Consequently, the transfer equation in the comoving frame is more complicated than the laboratory frame representation. The comoving frame equations contain derivatives with respect to angle and frequency in addition to spacetime coordinates. In a coordinate system moving with an elemental volume of the material, the comoving frame, the Doppler and aberration effects are nonexistent, and it is in this frame that the emissivity and absorptivity have the values specified by the atomic physics. The emission process (recombination) generates photons with an isotropic distribution. Similarly, the absorption process (photoionization) will have the shell edges at the same frequency for every angle. Since the emission and absorption processes are isotropic in the comoving frame, the equilibrium intensity I 0ν will be isotropic, and therefore the flux will vanish. Recall that in the fixed laboratory frame, the flux does not vanish no matter how opaque the medium may be. We can anticipate that the equation of radiative transfer will contain velocitydependent terms. This is readily seen by considering two fluid elements adjacent to one another but moving with different velocities. First, consider the advection effect whereby one fluid element will “sweep up” (“leave behind”) photons traveling against (along) the relative velocity vector, thus increasing (decreasing) the radiation energy density. Second, the Doppler effect shifts the spectral distribution of the radiation. Since we assume the two fluid elements are moving with respect to one another, then in addition to the change in the photon number density produced by the advection, the photon will be blue (red) shifted and have a higher (lower) energy when the two elements approach (recede from) one another. Both of these effects are Oðu=cÞ, and in certain cases of interest, terms that are formally only of Oðu=cÞ actually dominate over all others in the equations. The derivation of the radiative transfer equation in the comoving frame and the transformation of the radiation quantities between the fixed and comoving frames require the use of some concepts from relativistic kinematics. Speaking a little more precisely, the transformation properties come from the photon’s fourmomentum. 8.9.1 Doppler and Aberration Transformations Before we derive the transformation equations between the fixed and comoving frames, we need to briefly review the Lorentz transformation. For those readers

8.9 The Comoving Frame Representation

443

who are a bit rusty on this subject, the material is well presented in Mihalas and Mihalas (1984) and Rybicki and Lightman (1979). We rely heavily on Mihalas and Mihalas (1984) and Castor (2004) for the following material. The photon’s four-momentum is ðpμ Þ ¼

hν ð1, ncÞ, c2

(8.167)

where the superscript index μ ¼ 0 indicates the time component, μ ¼ 1, 2, 3 designate the space components, and the photon is traveling in the direction n. The transformation of the four-momentum between the two coordinate systems involves a 4  4 matrix designated Aμλ . Then x μ ¼ Aμλ x0λ :

(8.168)

The four-momentum transforms similarly. The “zero” subscript designates the comoving frame. For reference purposes, we reproduce the matrix without substantiating it   γ γuT =c2 μ ðAλ Þ ¼ : (8.169) γu 1 þ ðγ  1ÞuuT =u2 The velocity u is represented by a column vector, and p its ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi transpose is the row T vector u . The scalar γ is the usual relativistic factor 1= 1  u2 =c2 . The 3  3 matrix in the lower-right corner is arranged to leave unchanged a vector it multiplies that is perpendicular to u, and to multiply by γ one that is parallel to u. In the comoving frame, a photon has a frequency ν0 and is moving in direction n0 . An observer in the inertial (laboratory) frame sees the comoving frame moving with velocity u, and the photon has frequency ν moving in direction n. The transformation of the frequencies and directions are obtained from (8.168) and (8.169). The Doppler shift is given by  ν u (8.170) ¼ γ 1 þ n0  , ν0 c and the aberration by n¼

γ uc þ n0 þ ðγ  1Þðn0  uÞ uu2  γ 1 þ n0  uc

  γn0  uc ν0  u n0 þ γ ¼ 1þ : c ν γþ1

(8.171)

The classical formulas are obtained by setting γ ¼ 1 and retaining terms only through Oðu=cÞ.

444

Radiation and Radiative Transfer

For the special case of motion in the z-direction, (8.170) and (8.171) become  u ν ¼ ν 0 γ 1 þ μ0 , c    u u μ ¼ μ0 þ = 1 þ μ0 , c c   1=2  1=2 u (8.172) 1  μ2 ¼ 1  μ20 =γ 1 þ μ0 , c where μ and μ0 are the cosines of the polar angle in the two coordinate frames. The azimuthal angle is unchanged, that is, ϕ ¼ ϕ0 . We see that isotropic emission in the comoving frame does not appear isotropic in the laboratory frame. As γ becomes large, the radiation is observed to lie near the direction of u in the laboratory frame.

8.9.2 Transforming the Specific Intensity, Emissivity, and Absorptivity To find the transformation properties of the specific intensity, we note that the momentum space volume element d 3 p is not Lorentz invariant, but d3 p=E is; E is the relativistic energy. For photons d 3 p e ν2 dνdΩ and E e v. From (8.172), dν ¼ ðν=ν0 Þdν0 , dμ ¼ ðν0 =νÞ2 dμ0 , and dΩ ¼ dμdϕ, thus νdνdΩ ¼ ν0 dν0 dΩ0

(8.173)

is also Lorentz invariant. The spatial volume element d3 r is also not Lorentz invariant, but Ed3 r is; Thus, the phase-space volume element d3 rd 3 p is an invariant. Equation (8.9) gives the phase-space density of photons and we can define the invariant number of modes to be the number per volume h3 in phase-space ℓν ¼

Iν , 2hν3 =c2

(8.174)

which is Lorentz invariant; the factor of one-half comes from the two polarizations. The transformation rule for the specific intensity is then  3 ν I ν ðnÞ ¼ I 0ν ðn0 Þ: (8.175) ν0 The rules for transforming the emissivity and absorptivity are found by considering two times separated by dt, with matching three-volumes d 3 r on each, in addition to a momentum volume d3 p. The number of photons added to d3 r that lie in d 3 p in a time increment dt is

8.9 The Comoving Frame Representation

ΔN ¼

jν 3 j d rdtdνdΩ ¼ 4 3ν 3 d 3 rdtd 3 p: hν h ν =c

445

(8.176)

The product d 3 rdt is the invariant four-volume element, but the product d3 rdtd3 p is not invariant, but in light of the preceding discussion, d 3 rdtd 3 p=E is, and (8.176) becomes ΔN ¼

jν d 3 rdtd 3 p : hν h3 ν2 =c3

(8.177)

Since the number of photons added should be invariant, the quantity eν ¼

c jν 2 ν2

(8.178)

is Lorentz invariant. Similar reasoning applies to kν I ν , and we find another invariant h aν ¼ νkν : c

(8.179)

We then have the two transformations  2 ν j0 jν ¼ ν0 ν

(8.180)

and kν ¼

ν0 0 k : ν ν

(8.181)

We use the “zero” superscript for the specific intensity I, the emissivity j, and the absorptivity k to indicate the comoving frame; for example, I 0ν ¼ I ðn0 ; ν0 Þ. We will also use it for the energy density E, the flux F, and the pressure tensor P. Note: In this section, we use E for the energy density to avoid confusion with the material velocity u.

8.9.3 Transforming the Transfer Equation The equation of transfer (8.32), without induced emission and scattering, is found using (8.170), (8.180), and (8.181). For γ ¼ 1, (8.181) expanded to Oðu=cÞ yields   u 0 ∂k0ν 0 kν ¼ kν  n  k þν , (8.182) c ν ∂ν and (8.180) yields

446

Radiation and Radiative Transfer

jν ¼

j 0ν

  u ∂j 0ν 0 2j ν  ν þ n : c ∂ν

Note that in (8.182) and (8.183), ν is the laboratory-frame frequency. The transfer equation becomes     1 ∂I ν u ∂j 0ν ∂k 0ν 0 0 0 0 2j ν  ν þ n  rI ν ¼ j ν  kν I ν þ n  þ kν þ ν Iν : c ∂t c ∂ν ∂ν

(8.183)

(8.184)

Integrating (8.184) over solid angle gives the monochromatic radiation energy equation   ∂Eν ∂k0ν u 0 0 0  Fν , þ r  Fν ¼ 4πj ν  ck ν E ν þ k ν þ ν (8.185) ∂t ∂ν c and then integrating (8.184) against ndΩ gives the monochromatic radiation momentum equation     1 ∂Fν 4π u ∂j 0ν ∂k0ν 0 0 0 2j ν  ν þ cr  P ν ¼ k ν Fν þ þ kν þ ν u  P ν : (8.186) c ∂t 3 c ∂ν ∂ν These last three equations use a combination of coordinate systems and are referred to as mixed-frame equations.

8.9.4 Transforming the Moment Equations Transforming the moments of the radiation field requires a bit more effort because of their tensor nature; we reproduce here much of the work from Castor (2004). The moments of the radiation field are just parts of the stress-energy tensor defined by ð λ μ p p 3 λμ T ¼ d pℓ: (8.187) Er The integral is evaluated using the previously derived relations to yield T

λμ

ð∞ ð  ¼c 0 4π

ð∞



ð

¼

 ν2 I ν ν2 dΩdν 1  1 nT c nc c ν3 ν

Iν

1=c n

 nT dΩdν, nnT c

(8.188)

0 4π

where incidental factors of h and c have been discarded. Then using (8.5), (8.7), and (8.11), we have

8.9 The Comoving Frame Representation

T

λμ

 ¼

Er Fr

 FTr : c2 P r

447

(8.189)

The time–time part is the energy density, the time–space part is the flux vector, and the space–space part is the pressure tensor. Since T λμ is a good tensor, we can apply the Lorentz transformation matrix Aμλ , from (8.169), to obtain its components in the fixed frame from those in the comoving frame. The rule is to multiply the T matrix by one factor of A from the left and by one factor of A from the right. Then, ignoring terms Oðu2 =c2 Þ and higher, we have 0  T 1 u 0 0 0 0 0 Er þ 2 2  Fr F r þ uE r þ u  P r C B c  (8.190) T λμ ¼ @   0 T A: 0 0 0 T 0 2 0 F r þ uE r þ u  P r c Pr þ F r u þ u F r  T Note that F0r ¼ F0r because F0r is symmetric. The quantities cEr , Fr , and cP r are potentially all the same order of magnitude. In the diffusion regime, the flux will be smaller than the other two. The velocity corrections to the moments are formally Oðu=cÞ, but in the diffusion regime the relative magnitude of the correction to the flux may be much larger than that, while the relative corrections to the energy density and pressure are even smaller than u=c. The point here is that the correction to the flux is most important. Expression (8.188) has been integrated over frequency. Transformation of the monochromatic radiation moments between the two frames cannot be performed. The reason is that the monochromatic moments are not spacetime tensors, and therefore the Lorentz transformation rules do not apply. A second reason is that an angle moment at a constant value of the fixed-frame frequency is an integral over a different slice through photon momentum space than an angle moment at any constant fluid-frame frequency.

8.9.5 Comoving-Frame Transfer Equation The radiative transfer equation in the comoving frame is relatively straightforward to obtain. Consider (8.32) without the scattering (including induced) terms. Then using (8.170), (8.171), (8.175), (8.180), and (8.181) in (8.32), expanding and discarding terms of Oðu2 =c2 Þ and higher, we have (Buchler, 1983)    u 1 ∂I 0ν u 0 þ  rI ν þ n0  rI 0ν 1 þ n0  c c ∂t c

448

Radiation and Radiative Transfer

 ν 0 a þ n0 ru  rν0 n0 I 0ν c c  3 a þ n0  þ n0  ru  n0 I 0ν ¼ j 0ν  k 0ν I 0ν : c c 

(8.191)

For simplicity here and below, the subscripts indicating that all the frequencydependent quantities in the comoving frame are evaluated at ν0 have been omitted. Again, the vector u is the velocity of the comoving frame as viewed from the fixed frame. The term involving the acceleration a should be ∂u=∂t þ u  ru, but the convective part is dropped since it is Oðu2 =c2 Þ. Further, we argue that the acceleration terms as well as the n0  u=c part of the coefficient of the first term should be discarded. The reasoning is that the time derivative and the flow derivative are generally of the same order of magnitude, so that if the acceleration terms are ordered in this way, then all of them are Oðu2 =c2 Þ compared to the dominant terms. The same argument is true of the n0  u=c term at the beginning of (8.191). These simplifications lead to   1 ∂I 0ν ν0 0 þ u  rI ν þ n0  rI 0ν  n0  ru  rν0 n0 I 0ν c ∂t c 3 þ n0  ru  n0 I 0ν ¼ j 0ν  k0ν I 0ν : c

(8.192)

The term containing the gradient with respect to the momentum needs some explanation: A photon with a fixed momentum travels along its ray, and as the local fluid velocity changes, so do its momentum components when referred to the local comoving frame. Thus, the transport operator must account for this change with a gradient term multiplied by the rate of change of the comoving momentum along the ray. We can separate the vector n0  ru into its radial and angular components in momentum space  n0  ru ¼ ðn0  ru  n0 Þn0 þ n0  ru  I  n0 n0 Þ, (8.193) where I is the unit tensor. When dotted with the momentum-space gradient, the first term picks up the radial derivative, that is, ∂=∂ν, and the second one picks up 1=ν times the angle gradient. Using this in (8.192), we find   1 ∂I 0ν ν0 ∂I 0 0 þ u  rI ν þ n0  rI 0ν  n0  ru  n0 ν c ∂t c ∂ν0  1 3  n0  ru  I  n0 n0  rn0 I 0ν þ n0  ru  n0 I 0ν ¼ j 0ν  k 0ν I 0ν : c c

(8.194)

The projection-operator term here looks more complicated than it really is. The frequency-derivative term is the Doppler correction, and the angle-derivative term is the aberration correction.

8.9 The Comoving Frame Representation

449

The complete transfer equation (8.32), including the scattering terms, can be developed using the scattering kernel transformation k ðsÞ ðν ! ν0 ; n ! n0 Þ ¼

1  n  uc ðsÞ k ðν0 ! ν0 0 ; n0 ! n0 0 Þ, 1  n0  uc 0

(8.195)

where the kernel depends only upon the scattering angle.

8.9.6 Comoving-Frame Moment Equations The monochromatic moment equations are based on (8.191). The energy density in the comoving frame is    0 1 ∂   ∂E 0ν 0 0 u  F ν þ u  r u  F ν þ r  F 0ν þ r  uE ν þ 2 c ∂t ∂t  ∂  0 i þ P 0ν  ν0 P ν Þ : ru ∂ν0   1 ∂  ν0 F 0ν  a ¼ 4πj 0ν  ck 0ν E 0ν , (8.196) þ 2 F 0ν  c ∂ν0 and the flux is

   i  0 ∂F0ν ∂ u  P 0ν Þ þ u  r u  P 0ν Þ þ c2 r  P 0ν þ r  uF ν þ ∂t ∂t  ∂ ∂  0 0 0 ν0 Q 0ν Þ : ru  ν0 P ν Þ  a ¼ ck 0ν F 0ν : (8.197) þaE ν þ F ν  ru  ∂ν0 ∂ν0

The colon operator (:) in (8.196) and (8.197) indicates summing the product of the tensor on the left with the tensor on the right, Ð namely, R : S ¼ Rij Sij . The object Q 0ν is the symmetric third-rank tensor Q ν ¼ 4π nnnI ν dΩ in the laboratory frame. The corresponding frequency-integrated moment equations are    1 ∂  ∂E 0r u  F 0r þ u  r u  F 0r þ r  F 0r þ r  uE 0r þ 2 c ∂t ∂t ð  a 0 0 þP r : ru þ 2  F r ¼ 4πj 0ν  ck 0ν E 0ν dν0 c

(8.198)

and    i  0 ∂F 0r ∂ u  P 0r Þ þ u  r u  P 0r Þ þ c2 r  P 0r þ r  uF r þ ∂t ∂t ð þaE0r þ F0r  ru ¼ c k0ν F 0ν dν0 :

(8.199)

450

Radiation and Radiative Transfer

Expressions (8.198) and (8.199) can be rewritten in terms of u=c, and the question is raised as to the importance of the various terms. The acceleration terms a ¼ du=dt are Oðu=cÞ compared to the divergence terms, hence Oðu2 =c2 Þ overall and can be dropped. However, if the velocity evolves on a radiation flow time scale, the acceleration terms should be retained. For most applications, we can neglect the terms with 1=c2 in (8.198). The corresponding momentum-density equation becomes ð  ∂F0r (8.200) þ r  uF0r þ c2 r  P 0r ¼ c k0ν F0ν dν0 : ∂t Similarly, (8.198) becomes ð  0  ∂E 0r 0 0 þ r  uE r þ r  Fr þ P r : ru ¼ 4πj 0ν  ck 0ν E0ν dν0 : ∂t

(8.201)

8.9.7 Transforming the Radiation-Matter Coupling Terms The right-hand sides of (8.200) and (8.201) are source/sink terms for the radiation field; that is, they take energy away from or supply energy to the matter. In the rest frame, these two equations are given by (8.58) and (8.59). The transformation between the two frames of reference can be found by taking the divergence of the tensor T λμ (8.187). This gives a four-vector gμ , which has components ð∞ ð g ¼

ð jν  k ν I ν ÞdΩdν

0

0 4π

ð∞ ð g¼g ¼c

nð jν  kν I ν ÞdΩdν:

i

(8.202)

0 4π

These two quantities are the right-hand sides of (8.58) and (8.59), to within a factor of the speed of light. Since gμ is a four-vector, it can be evaluated in the comoving frame where the atomic properties are known, and then transformed to the fixed frame. To Oðu=cÞ we then have u (8.203) g0 ¼ g00 þ  g0 c and u g ¼ g0 þ g00 : c

(8.204)

8.9 The Comoving Frame Representation

451

The last term in (8.203) is the rate of doing work by the force exerted on the radiation by the matter, while the last term in (8.204) comes about because the addition of energy increases the relativistic mass density and therefore the relativistic energy density. As mentioned earlier, the absorptivities and emissivities are easily found from the atomic properties of the matter in the comoving frame, without the need for Doppler and aberration corrections. Furthermore, we can almost always assume isotropy of the absorptivity and emissivity in the fluid frame. We then obtain ð  g00 ¼ 4πj 0ν  ck 0ν E 0ν dν0 (8.205) and

ð g0 ¼ c k0ν F0ν dν0 :

(8.206)

The important difference is that the fixed-frame expressions are incorrect if there is an appreciable velocity, while the fluid-frame expressions are correct as long as the velocity is nonrelativistic. 8.9.8 Diffusion in the Comoving Frame The diffusion approximation in the comoving frame should yield zero flux when the radiation mean free path is small compared to the characteristic structural length. We can verify this by taking the comoving-frame transport equation (8.191) and rearranging it in a fashion similar to (8.70). Then, identifying I 0ν with the thermodynamic equilibrium Planck function, we have    1  u 1 ∂BðνPÞ u 0 ðPÞ ðPÞ þ  rBν I ν ¼ Bν ðT Þ  0 1 þ n0  þ n0  rBðνPÞ c c ∂t c kν    ν0 a 3 a ðPÞ ðPÞ þ n0  ru  rν0 n0 Bν þ n0  þ n0  ru  n0 Bν :  (8.207) c c c c The Planck function BðνPÞ ¼ j 0ν =k0ν is isotropic but does have spatial and temporal gradients. Its momentum-space gradient is rν0 n0 BðνPÞ ¼ n0 ∂BðνPÞ =∂ν0 . Using this fact in (8.207) gives us (see (8.213) for the Lagrangean derivative)    1 dBðνPÞ 1 DT u DT 0 ðPÞ þ n0  rT þ n0  2 I ν ¼ Bν ðT Þ  0 c Dt c Dt k ν dt   ∂BðPÞ 3   ν0  a a ðPÞ ν n0  þ n0  ru  n0 þ n0  þ n0  ru  n0 Bν : (8.208)  c c c c ∂ν Integrating this expression over angles leads to the diffusion equation in the comoving frame

452

Radiation and Radiative Transfer

E 0ν

  4π ðPÞ 4π dBðνPÞ DT T ¼ Bν  2 0 þ ru , c Dt 3 c k ν dT

(8.209)

where we have used the Planck function explicitly, and thus T

dBðνPÞ ∂BðPÞ ¼ 3BðνPÞ  ν ν : dT ∂ν

Integrating (8.210) over frequency and noting 4πBðPÞ =c ¼ aT 4 yields    4 1 DT 1 0 4 þ ru , E r ¼ aT 1  ðRÞ T Dt 3 ck

(8.210)

(8.211)

where kðRÞ is the Rosseland mean absorption coefficient (see (8.120)). The correction term in (8.211) is of the order λðRÞ =½c  min fτ; L=ug, where λðRÞ ¼ 1=kðRÞ and τ is the characteristic time scale. If the flow time L=u is longer than the characteristic time, hthe isi O λðRÞ =cτ , while if the flow time is shorter,  correction  the correction is O λðRÞ =L ðu=cÞ . If the characteristic time is so  shortthat it is comparable to the light transit time c=L, then the correction is O λðRÞ =L . The monochromatic flux derived from (8.208) is   4π dBðνPÞ u DT a 0 þ T : rT þ 2 Fν ¼  0 (8.212) c Dt c2 3k ν dT Comparing this to (8.68), we find correction terms with both velocity and acceleration. The velocity term is explained by realizing we are using the fixed-frame coordinates, and the transformation to the comoving frame involves two Lorentz transformations. The time derivative transforms according to the Lagrangean derivative ∂ D ∂ ! ¼ þ u  r, ∂t Dt ∂t

(8.213)

and the spatial derivative changes from one at time t to one at time t0 according to r ! r0 ¼ r þ

u ∂ : c2 ∂t

(8.214)

Hence, the velocity term represents a relativistic effect on the diffusion flux. The acceleration term has a physical interpretation as well. The mean free path 1=k0ν corresponds to a flight time 1=ck 0ν , during which the local fluid velocity has increased by an amount a=ck 0ν . The change to the flux is given by a=ck 0ν times the sum of the energy density and pressure (which are related by the Eddington factor of 1=3). We can visualize this if we pick the flux to be zero at the time when the photon flight begins. Then by the end of the flight the fluid has accelerated

8.9 The Comoving Frame Representation

453

away from the rest frame of that photon, which produces a flux in the new fluid rest frame in the direction opposite to the acceleration that is proportional to the acceleration times the flight time. The frequency-integrated flux is F0r ¼ 

4π dBðPÞ  0 a  r T þ T , c2 3k ðRÞ dT

(8.215)

where the gradient operator is the Lorentz-corrected one from (8.214). The monochromatic pressure tensor, derived from (8.208), requires tensor analysis manipulations, the details of which may be found in Castor (2004). We are satisfied quoting the result ! ðPÞ 1 4π 4π dB DT P 0ν ¼ BðPÞ  2 0 ν 3 c ν c k ν dT Dt 

 4π dBðνPÞ  T I ru þ ðruÞT þ r  u , 0 2 dT 15c k ν

(8.216)

and the frequency-integrated expression   1 4 1 DT P 0r ¼ aT 4 1  ðRÞ 3 ck T Dt 

4 15ck

ðRÞ

  aT 4 I ru þ ðruÞT þ r  u :

(8.217)

This form of the pressure tensor implies that radiation in the diffusion approximation contributes viscosity to the bulk fluid. Both Castor (2004) and Mihalas and Mihalas (1984) include discussions about this subject. The reader has no doubt suffered some mental fatigue in following this discussion about the comoving-frame picture. It is easy to forget that some of the issues are more significant than others. We summarize the more important points here for three regimes of interest: a. In the streaming limit, ðλ=LÞ≥1, E r  Pr , and F r  cEr . Here L is the characteristic length and λ the mean free path. b. In the static diffusion limit, Er ¼ 3Pr and ðu=cÞ ðλ=LÞ. Then from (8.190), the first term on the right-hand side of Fr ¼ F0r þ uE 0r þ u  P0r dominates, hence F r =cEr is Oðλ=LÞ. The net emission-absorption term is O cλ=L2 E r . c. In the dynamic diffusion limit, Er ¼ 3Pr and ðu=cÞ≳ðλ=LÞ and the last two terms dominate. Now F r =cEr is Oðu=cÞ. The net emission-absorption term is Oðu=LÞE r .

454

Radiation and Radiative Transfer

Let us examine the inertial-frame equations for radiative transfer, energy, and momentum for the three regimes: 1. First, consider the transfer equation (8.184). In the streaming regime, dimensional analysis suggests that on a fluid-flow time scale the terms scale as ðu=cÞ : 1 : ðL=λÞ : ðL=λÞ : ðu=cÞðL=λÞ. Here, the radiation field is quasistatic, so the time-derivative term and the velocity-dependent term on the right-hand side may be ignored, retaining only the spatial operator and the emissionabsorption terms. In the diffusion regime, where the time-derivative term may be omitted, and treating j0ν  k0ν I ν as one term, then for static diffusion the terms scale as ðu=cÞ : 1 : ðL=λÞ : ðu=cÞðL=λÞ. For dynamic diffusion, they scale as ðu=cÞ : 1 : ðu=cÞ : ðu=cÞðL=λÞ; and the velocity-dependent term may dominate all others. Even for static diffusion, it will dominate the net mission-absorption term if ðu=cÞ≥ðλ=LÞ2 . Of course, the velocity-dependent term must be retained as well. 2. Next, consider the radiation energy equation (8.185) integrated over frequency to give an expression for gray material   ∂Er u þ r  Fr ¼ k ðRÞ 4πBðPÞ  cEr þ k ðRÞ  Fr ¼ g0 : (8.218) c ∂t In the streaming limit, dimensional analysis suggests that on a fluid-flow time scale, the terms scale as ðu=cÞ : 1 : ðL=λÞ : ðL=λÞ : ðu=cÞðL=λÞ. Thus, we need only retain the r  Fr and the emission-absorption terms. If the situation arises that the medium is nearly in radiative equilibrium, the emission-absorption terms may cancel almost exactly, and the time-derivative term and the velocity-dependent terms can then determine the energy balance, and all terms must be retained. In the limit of static diffusion, the terms scale as ðu=cÞðL=λÞ : 1 : 1 : ðu=cÞðL=λÞ, where the net emission-absorption terms are grouped together. Now we can drop both the time-derivative term and the velocity-dependent terms because ðu=cÞ ðλ=LÞ. But when ðu=cÞ ! ðλ=LÞ, all terms are of the same order and must be retained. In the dynamic diffusion limit, the terms scale as 1 : 1 : 1 : ðu=cÞðL=λÞ, and the velocity-dependent term may dominate all others. 3. Now consider the radiation momentum equation (8.186), also written for gray material   1 ∂Fr u ðPÞ g 0 ðRÞ 0 (8.219) þ cr  P r ¼ k Fr þ 4π B þ u  P r ¼ : c ∂t c c Again, in the streaming limit, the terms scale as ðu=cÞ : 1 : ðL=λÞ : ðu=cÞðL=λÞ : ðu=cÞðL=λÞ. We need retain only r  P 0r and the term with Fr , all

8.9 The Comoving Frame Representation

455

other terms being at most Oðu=cÞ. In the static diffusion limit, the scaling is ðu=cÞðλ=LÞ : 1 : 1 : ðu=cÞðL=λÞ : ðu=cÞðL=λÞ, and we can drop the timederivative and velocity-dependent terms. But as ðu=cÞ ! ðλ=LÞ, the velocitydependent terms become of the same order as r  P 0r and must be retained, while the time-derivative term is only Oðu2 =c2 Þ. In the dynamic diffusion limit, the terms scale as ðu2 =c2 Þ : 1 : ðu=cÞðL=λÞ : ðu=cÞðL=λÞ : ðu=cÞðL=λÞ, and we can drop the time-derivative term but keep r  P 0r . In addition, all three terms on the right-hand side must be retained, as they are of the same size and may actually dominate the solution. In all three situations, if we need to follow the radiation flow, the time-derivative terms need to be retained. There are a few other points to take note of; we copy these directly from Castor (2004): • Ignoring the Doppler effect entirely and using the fixed-frame transport equation with the absorptivity and emissivity appropriate for matter at rest gives the wrong answer for the radiation when the velocity is supersonic and spectral lines dominate the opacity. It also makes a significant error in the energy coupling rate of radiation and matter. • The coupling terms are correctly given by the usual relations (8.202), but those values are in the fluid frame. The transformations (8.203) and (8.204) must be used to get the correct coupling terms for the fixed-frame equations. • The coupling term in the material internal energy equation is indeed the fluidframe energy term g00 from (8.205), but in the material total energy equation it is combined with the work done by the radiation force, which turns it into a fixedframe energy term equation. • The diffusion limit gives a Fick’s law form for the flux in the fluid frame, not the fixed frame. In the fixed frame, the convective flux of radiation enthalpy is added on. • Confusion of the correct frames for the coupling terms and the Fick’s law flux is more serious than the other moderate corrections that the u=c terms give. • The velocity terms in the comoving-frame energy equation (8.201) matter, especially the Doppler-shift frequency-derivative term; the ones in the comoving-frame momentum equation are less important. The aberration (angle-derivative) term in the transport equation does not survive in the energy moment and does not matter in the flux moment, so this can be dropped with little consequence. This section has barely touched the subject of radiative transfer in matter moving with a constant velocity where special relativity is appropriate. For the more

456

Radiation and Radiative Transfer

general case of the comoving-frame transfer equation being transformed to a noninertial Lagrangean frame, we must appeal to general relativity. The reader is directed to the literature for the full wealth of material. 8.10 View Factors In many laboratory experiments, an intense energy source is used to heat a finite body to a temperature sufficiently high that thermal radiation is produced from the surface. This surface becomes the source of energy for irradiating some other body in close proximity, that is, the subject of the experiment. One example of such an arrangement is a cavity being heated by a laser beam. If the cavity’s walls are made of high-atomic-number material (e.g., gold), it will produce thermal radiation that can be directed to a test sample either within the heated cavity or adjacent to it. A classic example is the hohlraum-with-capsule experiments being evaluated for possible energy production by thermonuclear reactions (radiatively driven inertial confinement fusion). These enclosures are not normally of “simple” geometry, so calculations must be performed to understand the distribution of the radiation field within the cavity and the concomitant heating of the interior components. Clearly, this can be a tedious task since the “surfaces” may be at various angles with respect to one another. The technique for calculating the radiative interchange between surfaces involves quantities known as view factors (Őzişik, 1973). We consider diffuse view factors, as opposed to specular view factors, for surfaces that are diffuse reflectors and diffuse emitters. Consider two such surfaces A1 and A2 , as depicted in Figure 8.4. The space between the two surfaces is composed of nonparticipating material, that is, a material that does not emit, absorb, or scatter radiation. A vacuum meets all of these requirements. The two surfaces are at temperatures T 1 and T 2 , respectively. Let dA1 and dA2 be two elemental surfaces on A1 and A2 , respectively, and r12 be the vector joining dA1 and ^ 1 and n ^ 2 , drawn normal to dA1 and dA2 , make angles θ1 and dA2 . The unit vectors n θ2 , respectively, with the line joining these two elemental surfaces. The view factor between dA1 and dA2 is the ratio of the radiative energy leaving surface element dA1 that strikes surface dA2 directly to the radiative energy leaving surface element dA1 into the entire hemispherical space. Let I 1 be the specific radiation intensity leaving surface element dA1 , and dΩ12 be the solid angle under which an observer at dA1 sees dA2 . The amount of radiation energy leaving dA1 that strikes dA2 per unit time may be found from (8.4) dE 1 ¼ dA1 I 1 cos θ1 dΩ12 , where the solid angle is given by

(8.220)

457

8.10 View Factors

A2

dA2 q2

T2

r12

q1 A1

T1 dA1

Figure 8.4. Coordinates for determining the diffuse view factor.

dΩ12 ¼

dA2 cos θ2 : r2

(8.221)

Here r is the length of the vector r12 joining dA1 and dA2 . The radiant energy leaving dA1 in all directions in the hemispherical space is found by integrating (8.220) over angles =2 2ðπ πð

E 1 ¼ dA1

I 1 cos θ1 sin θ1 dθ1 dϕ ¼ πI 1 dA1 ,

(8.222)

0 0

since the intensity is independent of direction for a diffusely emitting/reflecting surface. The view factor between dA1 and dA2 is obtained using (8.220), (8.221), and (8.222) dF dA1 dA2 ¼

dE1 cos θ1 cos θ2 ¼ dA2 : E1 πr 2

(8.223)

This represents the fraction of the energy leaving dA1 in all directions that strikes surface dA2 directly. Obviously, the view factor between dA2 and dA1 is

458

Radiation and Radiative Transfer

dF dA2 dA1 ¼

dE 2 cos θ1 cos θ2 ¼ dA1 : E2 πr2

(8.224)

Then from (8.223) and (8.224), we have the reciprocity relation dA1 dF dA1 dA2 ¼ dA2 dF dA2 dA1 . Integrating over the surface A2 gives the view factor for the energy leaving dA1 and striking A2 ð ð cos θ1 cos θ2 F dA1 A2 ¼ dF dA1 dA2 ¼ dA2 : (8.225) πr 2 A2

A2

Conversely, the fraction of energy leaving surface A2 in all directions that strikes surface dA1 directly is Ð  cos θ1 cos θ2 I2 dA1 dA2 πr2  F A2 dA1 ¼ Ð Ð A2Ð π= : (8.226) 2π 2 I cos θ sin θ dθ dϕ dA 2 2 2 2 2 0 0 A2

Since I 2 is independent of direction and position over A2 , (8.226) simplifies to ð dA1 cos θ1 cos θ2 F A2 dA1 ¼ dA2 : (8.227) A2 πr 2 A2

We have the reciprocity relation, from (8.225) and dA1 F dA1 A2 ¼ A2 F A2 dA1 . We can then integrate over dA1 to get the view factor from A1 to A2 ð ð 1 cos θ1 cos θ2 F A1 A2 ¼ dA2 dA1 , A1 πr2

(8.227)

(8.228)

A1 A2

while the view factor from A2 to A1 is ð ð 1 cos θ1 cos θ2 F A2 A1 ¼ dA2 dA1 , A2 πr2

(8.229)

A1 A2

which gives the reciprocity relation A1 F A1 A2 ¼ A2 F A2 A1 . In general, an enclosure consists of a number of small surface elements, so there are many view factors between the N elements. The reciprocity relation between two such elements i and j is Ai F ij ¼ Aj F ji . The view factors for an enclosure N P F ik ¼ 1. The physical significance of this obey the summation relationship k¼1

relation is obvious from the definition of the view factor. If a surface is flat or

8.10 View Factors

459

convex, none of the radiation leaving that surface will strike itself directly. If the surface is concave, part of the radiation leaving the surface will strike itself directly. Thus, F ii ¼ 0 for a plane or convex surface, and F ii 6¼ 0 for a concave surface. Using view factors in practice can be painful, especially for complicated geometries. Fortunately, a great number of numerical tools have been developed, so we don’t have to start with the basics to get usable results. This chapter has addressed only the basic issues of radiative transfer. Things get much more complicated when considering the transport of spectral lines and the effects of nonconservative scattering, let alone relativistic effects. We leave these topics for the reader to pursue in the literature. We turn now to the subject of the emission and absorption coefficients, which, as we have seen, appear throughout the equations of radiative transfer.

9 Transition Rates and Optical Coefficients

Portions of Chapters 4 and 8 have referred to transition rates and absorption, emission, and scattering coefficients. This chapter addresses the physics of determining these quantities and how they play an important role in establishing the ionization balance and transport of radiant energy in high-energy-density-plasma. The full development of the physics behind the concepts is beyond the scope of the book, so we make many simplifications to bring out the essence of the physics. One such simplification is that only principal quantum numbers are considered. A second simplification is that the model atom has only two levels, defined by states n and m, with the latter state being located closer to the continuum. A third simplification is the neglect of dielectronic recombination, as it is not usually important in highenergy-density physics; this topic was addressed briefly in Section 4.6.3. Because much of the development in this chapter is relevant to the light elements at astrophysical and terrestrial laboratory conditions (densities of order unity and ∽keV temperatures), additional simplifications allow us to (mostly) assume thermodynamic equilibrium, use hydrogenic wave functions, and treat the individual ions as noninteracting. To set the stage for the following sections of this chapter, we begin by repeating the population rate equations from Section 4.6.1. Processes that populate the atomic level n are given by An ¼

n1 X m¼1

Pm T m!n þ

nmax X

Pm T m!n þ Z  T c!n ,

(9.1)

m¼nþ1

where the first term on the right-hand side accounts for transitions from lowerlying levels m, the second term is for levels lying higher than n, and the last term is for transitions from the continuum. Here, nmax allows us to consider the atom to have more than two levels. The companion equation representing processes that depopulate level n is given by 460

9.1 Radiative Transitions

Bn ¼

n1 X m¼1

Qm T n!m þ

nmax X

Qm T n!m þ T n!c :

461

(9.2)

m¼nþ1

In these two expressions, Pm is the population (number of electrons) of the level and Qm is the availability of the level. In statistical equilibrium, An Qn ¼ Bn Pn . The transition rates indicated by T are composed of radiative and collisional processes, for example, T m!n ¼ Rm!n þ C m!n . For the first term in (9.1), the two processes are radiative and collisional excitation, and for the second term, the two are the inverse processes of radiative and collisional de-excitation. Radiation plays an important role in high-energy-density physics, and so we devote the majority of this chapter to developing the physics behind transition rates and optical coefficients. 9.1 Radiative Transitions The interaction of radiation with matter has a basis in Planck’s hypothesis that the exchange of energy and momentum between material particles and radiation occurs by the creation and annihilation of photons. Letting E and p be the energy and momentum, respectively, of a matter particle before the collision with a photon of energy hν and momentum ℏk, and E 0 , p0 , hν0 , and ℏk0 after the collision, the conservation laws become hν þ E ¼ hν0 þ E 0 and ℏk þ p ¼ ℏk0 þ p0 :

(9.3)

If ν0 ¼ 0, absorption of the photon takes place, and if ν ¼ 0, emission of a photon occurs. If both ν and ν0 are zero, the equations define the scattering of radiation. It is important to note that these simple laws are in conflict with both the wave and corpuscular concepts of radiation and cannot be interpreted within the framework of classical physics. According to the wave theory, the energy of the field is found  2electromagnetic  2 from Poynting’s vector and is given by S ¼ E þ B =8π, which makes no mention of frequency. There is no general relation between the wave amplitude and the oscillation frequency. The assumption that a photon is a particle located somewhere in space is also invalid. A photon, by definition, is associated with a monochromatic plane wave. Such a wave is a purely periodic process, infinite in both space and time. The assumption that the photon is localized is in contradiction with the periodicity of the wave. In the classical limit, the position and momentum of an electron can be specified without any uncertainty at each point of space and at each moment. Once the external fields are specified, all the radiation processes can, in principle, be found from Newton’s second law and Maxwell’s equations. In the quantum limit,

462

Transition Rates and Optical Coefficients

Newton’s second law is modified according to the rules of wave mechanics. Additionally, the radiation field should be quantized, but fortunately for our purposes, we have no need to do this. Thus, we can use the semiclassical treatment in which the motion of the electron is treated quantum mechanically, while the radiation field is treated classically, except that the wave-particle nature of photons is taken into account. Photons interact with an atom or ion in several ways. Of interest to us, at the moment, are changes in the structure of the atom resulting from absorption or emission of a photon. The first process we consider is the absorption of a photon of energy hν, such that an electron undergoes a transition from level n to a higherlying level m; the energy difference of the two levels is just the energy of the photon. These photoexcitations are labeled “upward bound-bound” transitions, and the inverse process of an electron moving from level m down to level n is labeled a de-excitation, or “downward” transition. The second process is the absorption of a photon, such that the electron transitions from a bound level into the continuum. This is photoionization, also known as bound-free absorption. The inverse process has four titles: radiative recombination, two-body recombination, radiative capture, or free-bound emission. Both types of processes involve the promotion of an electron, or its demotion, and so play a role in determining the ionization level Z ∗ of the atom as well as its structure; these processes also play a role in the coupling of the radiation field to the matter. The first of these, for the most part, is represented by transition rates, while the latter describes the energy exchange; the radiation field is coupled exclusively to the electron fluid, and we ignore any coupling to the ion fluid, as its effect is small because of the disparate mass difference. The transition rates are related to the absorption/emission rates. Another important process is the interaction of the radiation field with free electrons. Again, both absorption and emission are possible, but in addition, there is a similar process: that of Compton scattering and its inverse. We address only conservative (Thomson) scattering here, with a few words about nonconservative Compton scattering; we leave the details of nonconservative scattering to the literature. Having determined the transition rates, we can calculate the absorption and emission coefficients. We will find rates for bound-bound, bound-free, and freefree processes. The principal contributions to the opacity come from bound-free and free-free transitions and, at very high temperatures and very low densities, from Thomson scattering. The development of the topic of absorption and emission of radiation dates back to the early work of Kramers (1923), Sommerfeld (1934), and others. We augment those works with those of Heitler (1954) and Bethe and Salpeter (1977).

9.1 Radiative Transitions

463

9.1.1 Einstein Relations Consider the radiative bound-bound processes of excitation and the inverse process of de-excitation. There are three types of transitions between levels n and m, which are described in terms of the Einstein rate coefficients. First, the direct absorption of radiation proceeds at a rate proportional to the specific intensity of the radiation field at the proper frequency I ν and the Einstein coefficient Bnm . In general, there is a range of frequencies between ν and ν þ dν involved, so the transitionÐ is not sharp but can be described by an absorption profile ϕν , normalized such that ϕν dν ¼ 1.1 This spreading of frequencies arises primarily from the perturbations exerted by

1

The line absorption profile can arise from several processes. We consider the “natural” profile produced by an oscillating dipole. If the driving electromagnetic wave is given by E0 eiωt , the equation of motion for the electron is that of a damped harmonic oscillator   _ me €xþω20 x ¼ eE0 eiωt  me γ0 x, which has the steady-state solution " x ¼ ℜe 

# E0 eiωt  : ω2  ω20 þ iγ0 ω e me

The power absorbed from the wave is Pðt Þ ¼

2e2 2 x€ , 3c3

and upon averaging over one period gives hPðωÞi ¼

e 2 ω4 2 e4 ω4 E20 x0 ¼ 2 3  :  3 3c 3me c ω2  ω2 2 þ γ2 ω2 0

0

The classical damping constant, due to electron collisions with the radiating atom, is 2e2 ω20 , 3me c3 which is the reciprocal of the time during which a harmonic oscillator will reduce its energy by e. The correspondence between the macroscopic and electromagnetic descriptions of radiation is given by I 0 ¼ cE20 =8π. Thus ð cE2 hPðωÞi ¼ σ ðωÞ IdΩ ¼ 0 σ ðωÞ: 8π γ0 ¼

Comparing the two expressions shows the cross section to be σ ðωÞ ¼

8πe4 ω4 1 :   3m2e c4 ω2  ω2 2 þ γ2 ω2 0

0

Now  2 γ0 2  ω and the cross section is a sharply peaked function around ω0 . To a good approximation, ω  ω0  2ω0 ðω  ω0 Þ and thus #  2 " 2πe γ0 =2 σ ðωÞ ¼ : me c ðω  ω0 Þ2 þ ðγ0 =2Þ2 The term in square brackets is the normalized Lorentz line profile ϕν . The line shape resulting from the Stark effect and Doppler shifts (motion of the ions) is commonly used in conjunction with the Lorentz profile (Griem, 1960). The natural emission profile ψ ν is the same as ϕν .

464

Transition Rates and Optical Coefficients

nearby atoms and ions (the Stark effect) as well as the finite lifetime of the upper level. In the comoving (fluid) frame, the number of upward transitions from level n to level m per unit time per unit volume is

Ð

nn Rn!m ¼ nn Bnm ϕν J ν dν ¼ nn Bnm J nm ð σ nm σ nm ðvÞ ¼ 4πnn J nm ¼ 4πnn J ν dν, hν hνnm

(9.4)

where nn is the number density of ions in level n, Bnm is the absorption probability, J ν is the mean intensity, and σ nm is the absorption cross section. In the laboratory frame (see Section 8.9) ðð h  1 ui J nm ¼ ϕ ν 1  n I ðn; νÞdΩdν: (9.5) 4π c We see the line shape is different when viewed in the laboratory frame. The second and third cases are for a transition from level m to level n (a “downward” transition). Of these two, the first is the spontaneous emission of a photon, where the number of transitions per unit time per unit volume is ð 0 nm Rm!n ¼ nm Amn ψ ν dν; (9.6) Amn is the spontaneous emission probability; it is simply related to the mean life of the energy state E m against radiative decay. The second process is the transition induced by the radiation field (stimulated emission) ð 0 (9.7) nm Rm!n ¼ nm Bmn ψ ν J ν dν ¼ nm Bmn J nm with Bmn being the stimulated emission probability. Be aware that some authors use energy density Eν instead of the intensity to define the B-coefficients. The units of the Einstein coefficients are Amn , s1, Bnm , and Bmn , cm2-erg1-s1. The line-shape function ψ ν is for de-excitations; it is not necessarily the same as ϕν used for excitations. Note that spontaneous emission takes place isotropically, but induced emission has the same angular dependence as absorption. Induced emission is sometimes considered to be “negative absorption,” though this is not quite correct. In general, ψ ν will not be identical to ϕν . Combining (9.6) and (9.7) gives the total rate of downward transitions per unit volume     gn hν3nm 0 nm Rm!n ¼ nm Amn þ Bmn J nm ¼ nm Bnm 2 2 þ J nm , (9.8) gm c

9.1 Radiative Transitions

465

where (9.14) has been used. A more useful form of this expression is n0n Rm!n n0m   ð n0 σ nm hν3 ¼ 4πnm 0n 2 2 þ J ν ehν=kB T dν, nm hν c

nm R0m!n ¼ nm

(9.9)

where σ nm is the macroscopic absorption cross section, uncorrected for stimulated emission. The last part of (9.9) was obtained using Maxwell-Boltzmann statistics n0n g ¼ n ehνnm =kB T , 0 nm gm

(9.10)

where the zero superscripts are quantities at thermodynamic equilibrium, and the g’s are the statistical weights of the levels. The coefficients Amn , Bmn , and Bnm are simply related. In strict thermodynamic equilibrium, the radiation field is isotropic and I ν ! I 0ν ¼ BðνPÞ . Assuming ψ ν ¼ ϕν , the upward transitions must be exactly balanced by the downward transitions, thus n0n Bnm I 0ν ¼ n0m Amn þ n0m Bmn I 0ν : Solving for I 0ν yields BðνPÞ

n0 Amn Amn ¼ 0 m 0 ¼ nn Bnm  nm Bmn Bmn

1  gn Bnm hνnm =kB T 1 : e gm Bmn

(9.11)



(9.12)

The Planck distribution function, from Section 3.10, is BðνPÞ

1 hν3  hν=kB T ¼2 2 e 1 , c

(9.13)

and we conclude that Amn ¼ 2

hν3nm Bmn and gn Bnm ¼ gm Bmn : c2

(9.14)

The Einstein relations of (9.14) were derived from thermodynamic equilibrium arguments, but the Einstein coefficients are really properties of the atom only, and they must be independent of the nature of the radiation field. These relations are an expression of detailed balance that connect any microscopic process and its inverse process. They are the extensions of Kirchhoff’s law to include the nonthermal emission that occurs when matter is not in thermodynamic equilibrium. Einstein was led to include stimulated emission in his formulation because he could not otherwise obtain Planck’s law, but only Wien’s law, which was known to be incorrect. Recall that Wien’s law is the portion of Planck’s law for hν  k B T.

466

Transition Rates and Optical Coefficients

In this regime, nm  nn and thus stimulated emission is unimportant compared to absorption, since they are proportional to nm and nn , as shown in (9.4) and (9.7), respectively. This development implicitly assumed the atomic levels to have integer statistical weights. These formulas carry over directly to the non-LTE average-atom model of Section 4.6. Replacing gn with Dn , and so on, where Dn is the level degeneracy, and nn with Pn , the level populations, (9.10) becomes P0n Dn ðEn Em Þ=kB T ¼ e , 0 Pm Dm

(9.15)

and the level degeneracies are related to the Einstein coefficients according to Dn Bnm ¼ Dm Bmn . We have assumed, of course, that for a bound-bound excitation to take place, there must be at least one electron in level n and a “hole” in level m, and vice versa, for de-excitation.

9.1.2 Bound-Bound Optical Coefficients We are yet to determine the amount of energy absorbed by photoexcitation or emitted by the inverse process of de-excitation. The amount of energy emitted by a transition from level m to level n from a volume element dV, into a solid angle dΩ, in the frequency interval dν in time dt, is by definition jν dVdΩdνdt. In the comoving frame, each atom contributes an energy hνnm , distributed uniformly over 4π. This is expressed as dW e ¼ hνnm nm Amn ψ ν , dVdνdt

(9.16)

so that the spontaneous emission coefficient is jðνbbÞ ¼

1 dW e hνnm ¼ nm Amn ψ ν : 4π dVdνdt 4π

(9.17)

The energy absorbed out of a beam of photons is dW a ¼ hνnm nn Bnm ϕν I ν , dVdνdt

(9.18)

where I ν is the specific intensity. The absorption coefficient (uncorrected for stimulated emission) is k ðνbbÞ ¼

hνnm nn Bnm ϕν : 4π

(9.19)

9.1 Radiative Transitions

467

Since the absorption and stimulated emission processes both depend upon the intensity of the radiation, and stimulated emission can be viewed as negative absorption, we can combine the two to define a line absorption coefficient (in the comoving frame) hνnm ðnn Bnm ϕν  nm Bmn ψ ν Þ 4π     nm Bmn ψ ν nm gn ψ ν ¼ hνnm nn Bnm ϕν 1  ¼ hνnm nn Bnm ϕν 1  : nn Bnm ϕν nn gm ϕν

kðνbbÞ ¼

(9.20)

We can also define a line-source function, which is the ratio of emissivity to absorptivity, as  1 nm Amn ψ ν hν3 nn gm ϕν Sν ¼ ¼ 2 nm  1 : (9.21) nn Bnm ϕν  nm Bmn ψ ν c2 nm gn ψ ν In the case of complete redistribution, ψ ν ¼ ϕν , and in the case where there is thermodynamic equilibrium, (9.10) is applicable and we have h i k ðνbbÞ0 ¼ hνnm nn Bnm ϕν 1  ehνnm =kB T : (9.22) Similarly, the source function becomes   i1 hν3nm h hνnm =kB T 0 e 1 ¼ BðνPÞ : Sν ¼ 2 2 c

(9.23)

We can now write the equation of transfer in terms of the Einstein coefficients. Thus, from Section 8.3, but omitting the scattering terms, we have 1 ∂I ν þ n  rI ν ¼ hνnm nm Amn ψ ν  hνnm ðnn Bnm ϕν  nm Bmn ψ ν ÞI ν : c ∂t

(9.24)

Once Bnm is known, the other two Einstein coefficients may be found using (9.14). 9.1.3 Quantum Mechanics of Radiative Processes Calculation of Bnm requires a quantum mechanical treatment of the electron moving in an external electromagnetic field. Beginning with the nonrelativisitc interaction Hamiltonian, including the rest mass energy of the electron, H ¼ me c2 þ

p2 e e2 ðp  A þ A  pÞ þ  A2  eΦ, 2me 2me c 2me c2

(9.25)

where p is the electron’s momentum, A is the vector potential of the electromagnetic field, and Φ is its scalar potential.

468

Transition Rates and Optical Coefficients

The nonrelativistic wave equation is found from ðH  me c2 ÞΨ ¼ EΨ. Replacing the conjugate momentum and energy with p ! iℏr and E ! iℏ∂=∂t, respectively, gives the result 

ℏ2 2 eℏ e2 ∂Ψ r Ψþi A2 Ψ  eΦΨ ¼ iℏ : ðA  r þ r  AÞΨ þ 2 2me c ∂t 2me 2me c

(9.26)

This expression is valid for a single electron with no spin; for our purposes, we do not need to include spin in the Hamiltonian because it is of second order. We can decompose the energy operator into an unperturbed part and a small perturbing part: H ¼ H ð0Þ þ H ð1Þ . The unperturbed part has the terms of (9.25) containing first-order terms in the vector potential. We assume that H ð1Þ does not contain the time explicitly. Expanding the wave function in terms of eigenfunctions uk X Ψð t Þ ¼ ak uk eiωk t , (9.27) k

where ωk are the eigenvalues. Multiplying (9.27) by uk and integrating gives only one surviving expansion coefficient ð (9.28) a_ k ¼ eiωk t uk Ψd 3 x, because of the orthonormal properties of the uk . The Schrödinger equation becomes ð X Ψ H ð0Þ Ψd 3 x ¼ jak ðtÞj2 E k :

(9.29)

k

The left-hand side of (9.29) is the average value of H ð0Þ , whereas the right-hand side is the sum of terms, each of which is a possible value of H ð0Þ multiplied by a term that must be the probability of that given value of H ð0Þ . Thus, jak ðtÞj2 is the probability that the system is in the state k at time t. If the system was initially in a state described by ui , then the transition probability from state i to state k is jak ðt Þj2 . The task is then to determine ak ðt Þ from the wave equation with the initial condition ak ð0Þ ¼ δki . We can now rewrite Schrödinger’s equation as  X X  ak E k þ H ð1Þ uk eiωk t ¼ ðiℏa_ k þ ak E k Þeiωk t : (9.30) k

Multiplying both sides by

k

uf eiωk t

and integrating over the volume, we obtain X ð1Þ iℏa_ f ¼ ak H fk eiωfk t , (9.31) k

9.1 Radiative Transitions

469

  where the subscript f indicates a final state, ωfk ¼ ωf  ωk ¼ E f  Ek =ℏ, and ð ð1Þ (9.32) H fk ¼ uf H ð1Þ uk d3 x; it is assumed H ð1Þ commutes with eiωk t . To solve (9.31), we write ak ðt Þ as the sum of two parts, and we have  X  ð0Þ ð0Þ ð1Þ ð1Þ ð1Þ iℏa_ f þ iℏa_ f ¼ ak þ ak H fk eiωfk t :

(9.33)

k ð0Þ

All terms on the right-hand side of (9.33) are first order, so a_ f ð0Þ condition is ak ¼ δki . We then find ð1Þ

iℏa_ f

ð1Þ

¼ H fi eiωfi t ,

¼ 0 and the initial (9.34)

which has the solution ðτ i iωfi t0 h ð1Þ 0 i 0 ¼ e H fi ðt Þ dt : ℏ

ð1Þ af

(9.35)

0

We assume the perturbation is nonzero for 0 < t < τ, and thus the lower limit of ð1Þ the integral can be replaced by ∞ and the upper limit by þ∞. For t > τ, af is independent of htime, since i the energy is an integral of the motion. The Fourier ð1Þ components of H fi ðt 0 Þ are given by  ð1Þ  H fi ωfi

1 ¼ 2π

Hence, ð1Þ

af

¼ i

þ∞ ð

h i 0 ð1Þ H fi ðt 0 Þ eiωfi t dt0 :

(9.36)

∞

2π h ð1Þ  i H fi ωfi , ℏ

(9.37)

resulting in the transition probability from state i to state f being 4π 2

ð1Þ  

2 (9.38) Pfi ¼ 2 H fi ωfi : ℏ The transition probability exhibits a resonance behavior; that is, it is nonzero only for perturbations that contain frequencies equal to the eigenfrequencies of the system. For example, in the case of absorption of radiation by an atom, only photons having a frequency ωfi will be absorbed. We may assume the perturbation Hamiltonian is constant during the characteristic interaction time τ, which is much greater than the characteristic time scale of the system given by ω1 fi . Performing the integral in (9.35) yields

470

Transition Rates and Optical Coefficients ð1Þ

af

¼

 ð1Þ 1  iωfi τ e  1 H fi : ℏωfi

(9.39)

Then

2 ω τ  1 4  ð1Þ 2

ð1Þ

fi sin 2 ,

af ¼ 2 H fi 2 ω2fi ℏ

(9.40)

and the transition probability per unit time is

wfi ¼

2

ð1Þ

af

τ

¼

 2π  ð1Þ 2  H fi δ Ef  Ei , ℏ

(9.41)

since sin 2 ðαt Þ=παt ! δðαÞ as t ! ∞. We need to ask the question about what conditions are required for the firstorder perturbation approach for determining transition rates to be valid. Taking the ratio of the first- and second-order terms in the vector potential of (9.25), we   have ðepA=me cÞ= e2 A2 =2me c2 ¼ ð2pc=eAÞ, and the first-order perturbation theory is valid provided ð2pc=eAÞ  1. From the uncertainty principle, p ∽ h=b, where b is the impact parameter of the incident electron, and its kinetic energy is W ∽ e2 =b. If all of the kinetic energy is carried away by one photon, qffiffiffiffiffiffiffiffiffiffiffiffi W ¼ hν, and thus p=e ∽ h=νb3 . The electric field of the electromagnetic wave is related to the vector potential by A ∽ ðc=νÞE, and the wave’s energy density is ∽ E2 =4π. This, in turn, is related to the average number of photons nph according qffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 to E =4π ∽ hνnph , and so A=c ∽ 4πhnph =ν. Then, ð2pc=eAÞ ∽ 1= nph b3 , which gives the perturbation validity requirement nph b3  1. That is, the average number of photons in the interaction volume must be much less than one. If the reverse were true, there would be many photons present in the emitting region. Thus, we say that keeping only terms linear in the vector potential corresponds to considering only cases where a single photon is emitted or absorbed by the system. To put this in perspective, if the impact parameter is taken to be the Bohr radius, nph ∽ 1025 cm3; as a reference point, nph ∽ 1012 cm3 at the Sun’s surface, so we are well justified in ignoring the quadratic terms. For the emission or absorption of two or more photons, the quadratic term in the vector potential must be used. The bound-bound absorption coefficient is found using the perturbed part of the Hamiltonian. From (9.26), we have H ð1Þ Ψ ¼ i

eℏ ðA  r þ r  AÞΨ: 2me c

(9.42)

9.1 Radiative Transitions

471

In the Coulomb gauge,2 r  A ¼ Φ ¼ 0, and the factor ðr  AÞΨ ¼ ðA  rÞΨþ Ψðr  AÞ ! A  rΨ. Expression (9.38) applies, and the problem becomes one of calculating ð    eℏ ð1Þ  A  eik  r r fi eiωfi t dt, H fi ωfi ¼ i (9.43) 2πme c where we have separated out the time dependence of the vector potential as Aðr; t Þ ¼ Aðt Þeik  r , and ð  ik  r  e r fi ¼ uf eik  r rui d 3 x, (9.44) which is independent of time. Since r ! ip=ℏ, (9.43) becomes    e  ik  r  ð1Þ  e p fi : H fi ωfi ¼ A ωfi me c

(9.45)

Then (9.38) is written  Pfi ¼

2π ℏ

2 

e me c

2



2   2

 ik  r

e I  p fi A ωfi ,

(9.46)

where I is a unit vector specifying the polarization of the electromagnetic wave. We expand the vector potential in plane waves according to ð  1  A0 ðωÞeiðk  rωtÞ þ A0 ðωÞeiðk  rωtÞ dω : (9.47) A¼ 2 The Fourier transform is   1   A ωfi ¼ A0 ωfi eikfi  r , 2

(9.48)

assuming ωfi > 0, corresponding to absorption. Using this in (9.46), we arrive at the absorption probability   

2   2 πe 2

 ik  r Pfi ¼ I  p fi A0 ωfi : (9.49)

e me ℏc The “” symbol in the exponential signifies absorption for the positive sign and emission for the negative sign. We wish to express the vector potential in terms of the time average of the intensity I of the radiation wave; the connection will be developed in Chapter 12, 2

The need for gauge transformations arises when trying to calculate the scalar and vector potentials from the observables E and B of the electromagnetic field. In general, there is no unique way to do this. To make them uniquely calculable, we require another condition, for example, a differential equation. For a complete specification of a vector field, it is necessary to state both its curl and its divergence. For more information on gauge transformations, see Jackson (1999).

472

Transition Rates and Optical Coefficients

but for now we note I ðωÞ ¼ ðc=τ ÞjEðωÞj2 , while E ¼ ð1=cÞ∂A=∂t, and thus EðωÞ ¼ iðω=cÞAðωÞ ¼ ωA0 ðωÞeik  r . Hence,  c 2 1 cτ 2 (9.50) jA0 ðωÞj ¼ jEðωÞj2 ¼ 2 I ðωÞ, 4 ω ω and the transition rate for absorption becomes   

2 Pfi 4π 2 e2 I ωfi

 þik  r wfi ¼ e I  p : (9.51) ¼ 2 2

fi

τ me ℏ c ω2fi The transition rate for induced emission is the same as (9.51), except that ωfi is replaced by ωif , and the minus sign in the exponential is used. Hence, the transition rates for the two processes of absorption and induced emission are the same; this is an example of the principle of detailed balance. The expressions for the transition probabilities contain a term of the form ð uf eik  r I  rui d 3 x: (9.52) A simplification is made by noting k  r ∽ ðν=cÞðh=pÞ ∽ ðE=hcÞðh=pÞ ∽ v=c, and for nonrelativistic problems the exponential can be replaced by unity to first order in v=c. This is just the quantum mechanical analog of the classical dipole approximation. Then (9.52) becomes ð i (9.53) uf I  rui d3 x ¼ ðI  pÞfi : ℏ We evaluate the matrix element using the commutation relation rp2  p2 r ¼ 2iℏp, and it follows that r commutes with the unperturbed Hamiltonian H ð0Þ according to  i 1  ð0Þ ð0Þ rH  H r ¼ p: (9.54) ℏ ℏ2 Using this to replace ip=ℏ in (9.53) yields ð  i me   ð0Þ ðI  pÞfi ¼ 2 uf I  rH  H ð0Þ r ui d 3 x ℏ ℏ ð  me  ¼ 2 E i  E f uf I  rui d3 x: ℏ

(9.55)

The transition rate (9.51) becomes wfi ¼

2   4π 2



ð I  d Þ

fi I ωfi , 2 ℏc

where d ¼ er is the electric dipole operator.

(9.56)

9.1 Radiative Transitions

473

As we are concerned only with unpolarized radiation from atoms with random

2

  2



orientations, ðI  dÞfi ¼ dfi =3, since cos 2 θ ¼ 1=3. Then the average transition rate is 

 4π 2 2   wfi ¼ 2 d fi I ωfi : 3ℏ c

(9.57)

We can relate this to the Einstein coefficient for absorption Bnm in our two-level atom. From (9.4), we have hwnm i ¼ Bnm J nm ¼ Bnm I ðνnm Þ=4π ¼ Bnm I ðωnm Þ=2, since the intensity is unidirectional. For nondegenerate levels, we arrive at Bnm ¼

32π 4 64π 4 ν3mn 2 d and A ¼ j j jd mn j2 : nm mn 3hc3 3h2 c

(9.58)

However, since the levels are degenerate, the transition rates are found by averaging over the initial states and summing over the final states. Then we have for absorption Bnm ¼

4π 2 e2 f , me hcνnm nm

(9.59)

where the absorption oscillator strength is f nm ¼

X 2me 1 ð E  E Þ jdnm j2 : m n 3ℏ2 e2 gn

(9.60)

The reason for naming it such is that the B coefficient associated with a classical oscillator can be defined in terms of the total energy extracted from a beam of radiation, for which f nm ¼ 1 in (9.59), and thus the oscillator strength is a correction factor. The emission oscillator strength is f mn , and since gn Bnm ¼ gm Bmn , together with νmn ¼ νnm it gives gn f nm ¼ gm f mn . Thus, emission oscillator strengths are negative. We can now write the Einstein spontaneous emission coefficient Amn ¼ 

8π 2 e2 2 8π 2 e2 gn 2 ν f ¼ ν f : me c3 mn mn me c3 gm mn nm

(9.61)

The usefulness of oscillator strengths stems from the simplest oscillator strength sum rule. The Thomas-Reiche-Kuhn rule states that the algebraic sum of all oscillator strengths, emission and absorption, beginning from a given state, equals the number of electrons in the atom. Other types of sum rules can also be defined for different assumptions about the nature of the atomic states.

474

Transition Rates and Optical Coefficients

We can perform a calculation for determining the oscillator strength for the simple case of a transition from the ground state of hydrogen. Beginning with (9.60), but rewriting it as X 2me gn f nm ¼ 2 ðE m  En Þ jrmn j2 , (9.62) 3ℏ since d ¼ er, and we have used the Hermitian property, jrmn j ¼ jrnm j. The square of the matrix element is defined by

2

ð

 2 3

(9.63) jrmn j ¼ ψ m rψ n d r : The initial wave function is the ground state of hydrogen, with quantum numbers ðn; l; mÞ ¼ ð1; 0; 0Þ. From Section 3.11, this wave function is 1 1 ψ 100 ¼ pffiffiffi 3= er=a0 , π a0 2

(9.64)

where a0 is the Bohr radius. For the two-level atom, the final state wave function may be any of the three states ðn; l; mÞ ¼ ð2; 1; 1Þ, ð2; 1; 0Þ, ð2; 1; 1Þ with corresponding wave functions 1 1 1 ψ 211 ¼ R21 Y 11 , ψ 210 ¼ R21 Y 10 , and ψ 211 ¼ R21 Y 11 , r r r

(9.65)

where R21 ¼

1 1 1 2 r=2a0 pffiffiffi r e , 3 5 2 =2 a0=2 3

(9.66)

and Y lm are the spherical harmonics. It is easily shown that the operator jrmn j2 may be written as

2 1

2 1

(9.67) jrmn j2 ¼ ðx þ iyÞmn þ ðx  iyÞmn þ jzmn j2 , 2 2 where thepffiffiffiffiffiffiffiffiffiffi operators are expressible pffiffiffiffiffiffiffiffiffiffi in terms of spherical harmonics: x  iy ¼ r 8π=3Y 11 and z ¼ r 4π=3Y 10 . We then find (9.67) to be jr21 j2 ¼

1 1 2 2 ℛ jAj , 18 a80

(9.68)

where  5 2 dr ¼ 4!a50 ℛ¼ r e 3

2 ð

2 ð

2

ð









 2







jAj ¼ Y 1m Y 11 dΩ þ Y 1m Y 10 dΩ þ Y 1m Y 11 dΩ

: ð

4 3r=2a0

(9.69)

475

9.1 Radiative Transitions

Now, by the orthogonality relations of the spherical harmonics, only one term in jAj2 contributes for each magnetic quantum number m, and this contribution is unity. Performing the sum over m in (9.62) and multiplying by two for the two possible spin states of the initial electron, we obtain g1 f 12 ¼

217 a20 me 217 ℏ3 ¼ ω12 : ω 12 310 ℏ 310 me e4

(9.70)

Now the transition frequency between levels m ¼ 2 and n ¼ 1 is   e2 1 3 me e4 , ω12 ¼ ¼ 1 4 8 ℏ3 2a0 ℏ

(9.71)

and we obtain the result g1 f 12 ¼ 214 =39  0:8324. A convenient form for hydrogen-like oscillator strengths is given by (Kramers, 1923)    32 1 1 1 3 ðK Þ p ffiffi ffi f nm ¼  , (9.72) n 2 m2 3π 3 n5 m3 for large n and m; for n, m < 5, the values are given in Table 9.1. It is customary to express the exact f value in terms of Kramer’s approximation as bbÞ ðK Þ f nm ¼ gðnm f nm ,

(9.73)

bbÞ where gðnm are the bound-bound Gaunt factors, and are of order unity. From the table we see that the principal lines are at the heads of the series. The higher transitions become weaker and closer together, eventually blending into the bound-free continuum edge as shown in Figure 9.1. The effect of the large number of weak, closely spaced lines is to extend the photoionization edge to lower frequencies. Note that the absorption coefficient varies over many orders of magnitude. The spontaneous emission transition probability, found from (9.17) and using the Einstein A coefficient from (9.58), is

Pfi ¼

64π 4 e2 3



2 ν rfi : 3 hc3 fi

(9.74)

Table 9.1 Hydrogenic Oscillator Strengths for Small n, m n, m 1 2 3 4 5

1

2

3

4

5

– –0.1041 –0.0088 –0.0018 –0.0006

0.4161 – –0.2848 –0.0298 –0.0072

0.0791 0.6408 – –0.4736 –0.0542

0.0290 0.1193 0.8420 – –0.6637

0.0139 0.0447 0.1506 1.037 –

476

Transition Rates and Optical Coefficients

Figure 9.1. Schematic of bound-bound, bound-free, and free-free absorption processes. The bound-bound contributions (dashed lines) are much narrower than shown here.

Summing over all final states with energy less than that of the initial state, we find the total probability that the initial state is emptied. Hence, the mean lifetime of the  1 P initial state is τ i ¼ Pfi . Using arguments similar to those following (9.41) f 0, we have the interesting result that electron scattering of a completely unpolarized incident wave produces a scattered wave with some degree of polarization, the degree depending on the viewing angle with respect to the incident direction. If we look along the incident direction (θ ¼ 0), we see no net polarization, since, by symmetry, all directions in the plane are equivalent.

492

Transition Rates and Optical Coefficients

However, if we look perpendicular to the incident wave (θ ¼ π=2), we see 100 percent polarization, since the electron’s motion is confined to a plane normal to the incident direction. The scattering is coherent for hν=me c2  1, since the recoil of the scattering electron is negligible. In this limit, the degeneracy of the electron gas has no effect on the cross section since in the absence of recoil the electron occupies the same phase-space cell after the scattering event as before. For hν=me c2  1, the recoil of the electron is no longer negligible and the scattering is noncoherent, and degeneracy of the electron gas will reduce the cross section by reducing the number of phase-space cells available to the electron after the scattering event. The simple picture of scattering needs two refinements. Kinematic effects must be taken into account since a photon possesses momentum as well as energy. Thus, the scattering is not really elastic (conservative) because the electron will recoil during the scattering event. The collision process is completely determined by conservation of energy and momentum. Let the incident photon have frequency νi and νf following the collision. Conservation of momentum in the direction of the incident photon is hνi hνf ¼ cos θ þ me v cos ϕ, c c

(9.145)

where θ is the angle of the scattered photon and ϕ is that for the electron, measured from the photon’s initial direction. Momentum conservation in the orthogonal direction is 0¼

hνf sin θ  me v sin ϕ; c

(9.146)

energy conservation is expressed by hνi þ me c2 ¼ hνf þ E,

(9.147)

where, relativistically, the square of the total electron energy is 2 2 E2 ¼ ðme c2 Þ þ ðme v2 Þ . Solving for the final photon energy gives the well-known result hνf ¼

hνi hνi ð1  cos θÞ 1þ me c2

:

(9.148)

We may rewrite this in terms of wavelengths λf  λi ¼ λc ð1  cos θÞ,

(9.149)

493

9.1 Radiative Transitions

where λc ¼ h=me c is the Compton wavelength of the electron. Thus, the change in wavelength (or frequency) of a photon is independent of the initial wavelength and depends only upon the scattering angle of the photon. For long wavelengths λ  λc , the scattering is mainly elastic and there is essentially no change in photon energy in the rest frame of the electron. The second consideration is quantum effects on the differential scattering cross section. The derivation is beyond the scope of this work; it can be found in Heitler (1954). Briefly, the scattered radiation is not as sharply defined in wavelength as the incident radiation, because of the Doppler shift due to the motion of the electron. For unpolarized incident radiation, the differential cross section is given by the Klein-Nishina formula   dσ ðνsÞ 1 2 ν2f νi νf 2 ¼ r0 2 þ  sin θ : (9.150) 2 νi νf νi dΩ Using (9.148) in (9.150) gives dσ ðνsÞ 1 2 ð1 þ cos 2 θÞ ¼ r0 2 ½1 þ xð1  cos θÞ2 dΩ

(

) x2 ð1  cos θÞ2 , 1þ ð1 þ cos 2 θÞ½1 þ xð1  cos θÞ (9.151)

where x ¼ hν=me c2 . The behavior of (9.151) is shown in Figure 9.3.

Figure 9.3. Differential cross section for Compton scattering for different values of x.

494

Transition Rates and Optical Coefficients

Integration over solid angles gives ( )  σ ðνsÞ 3 1 þ x 2xð1 þ xÞ log ð1 þ 2xÞ 1 þ 3x  log ð1 þ 2xÞ þ  : ¼ ðsÞ 4 x3 1 þ 2x 2x ð1 þ 2xÞ2 σ T

(9.152) In the nonrelativistic regime, (9.152) expands to   26 2 ðsÞ ðsÞ σ ν ¼ σ T 1  2x þ x þ    : 5

(9.153)

ðsÞ

For all values of x, σ ðνsÞ < σ T , eventually going to zero as x ! ∞. Free electrons are never really at rest. If the moving electron has sufficient kinetic energy compared to the photon’s energy, net energy may be transferred from the electron to the photon, in contrast to the situation expressed by (9.148). This process is referred to as inverse Compton scattering. The development of this is rather complex and lengthy, so we will avoid it; it is sufficient to say that the low-energy incident photons are converted to high-energy photons by a factor of order x2 . An in-depth discussion about both Compton and inverse Compton scattering may be found in the book by Castor (2004). 9.1.8 Maximum Opacity Theorem The basic parameter of radiation transport theory is the Rosseland mean absorption coefficient, discussed in Section 8.7. This occurs in the expression for the frequency-integrated flux c Fr ¼  ðRÞ rPr , (9.154) K ρ where K ðRÞ is the Rosseland opacity and ρ is the mass density. The Rosseland mean is a weighted integral over frequency of the total absorption coefficient, including induced emission ð∞  1 ∂BðPÞ 0 ν ð∞ dν k νðaÞ þ kðνsÞ 1 ρ 1 4 e2u ∂T 0 ¼ ¼ ρ u du, (9.155) ¼ ρ ð ∞ ðPÞ ∂Bν K ðRÞ kðRÞ kðνaÞ ðeu  1Þ3 dν 0 0 ∂T where u ¼ hν=kB T. In the last term of (9.155), the scattering term k ðνsÞ has been 0 omitted and induced scattering has been factored out. We note that kνðaÞ obeys a conventional dipole sum rule, while k ðνaÞ , in general, does not. The cross section for a single electron to absorb a photon is, in the dipole approximation,

9.1 Radiative Transitions

σ ðνÞ ¼ with

πe2 df ðνÞ , me c dν



2 

me X 

δ Ef  Ei  hν ðI  dÞfi

df ðνÞ ¼ 8π 2 2 he f 2

495

(9.156)

(9.157)

where the subscripts i and f designate initial and final states of the transition, ð∞ df ðνÞ respectively. The dipole sum rule is dν ¼ 1. dν 0

Summing over all electrons gives the frequency-dependent opacity πe2 φðνÞ, (9.158) me c Ð where φðνÞdν ¼ Z  =Am0 , where A is the atomic weight and m0 is the atomic 0 mass. K ðRÞ ðuÞ includes induced emission, which can be factored out according to 0 K ðRÞ ðuÞ ¼ K ðRÞ ðuÞð1  eu Þ. Then, by definition 2∞ 31 ð 1 du5 , K ðRÞ ¼ 4 W 0 ðuÞ ðRÞ0 (9.159) K ðuÞ 0

K ðRÞ ðuÞ ¼

0

with 15 u4 eu : 4π 4 ð1  eu Þ2 Figure 9.4 plots this weighting function. Using Schwarz’s inequality, we have 2∞ 32 ∞ 3 2∞ 32 ð ð ð qffiffiffiffiffiffiffiffiffiffiffiffi 0 1 4 W 0 ðuÞ du54 K ðRÞ ðuÞdu5≥4 W 0 ðuÞdu5 : ðRÞ0 K ðuÞ W 0 ðuÞ ¼

0

0

(9.160)

(9.161)

0

The maximum value is attained under the conditions that all of the bound-bound pffiffiffiffiffiffiffiffiffiffiffiffi 0 transitions are weak, thus the equality exists only when K ðRÞ ðuÞ ¼ const W 0 ðuÞ. Then the Rosseland mean opacity is such that K

ðRÞ

ð∞ h 1 πe2 h Z  1 ðRÞ0

K ð u Þdν ¼ , k B T s2 me c kB T Am0 s2

(9.162)

0

where pffiffiffiffiffi ð∞ 2 u=2 15 u e du  3:30: s¼ 2π 2 eu  1 0

(9.163)

496

Transition Rates and Optical Coefficients

Figure 9.4. Rosseland mean weighting function W 0 ðuÞ. Dashed line shows the weighting function for the Planck mean, B ¼ ð15=π 4 Þu3 =ðeu  1Þ.

This gives the Bernstein-Dyson opacity limit (Bernstein & Dyson, 1959)    R∞ ðRÞ 5Z K 4:43 10 , (9.164) A kBT where R∞ is the Rydberg. In general, the Rosseland mean opacity is less than the value computed from (9.164). However, this upper bound provides a useful check on approximation methods. In closing this section, we remark about the use of the Planck and Rosseland means. If the spectral distribution is very complex, with a great many strong lines and photoionization edges, the dynamic range may be very large. For example, the opacity at the center of a strong line might have a value of 1010 cm2-g1, while in the windows between lines, and below absorption edges, the opacity may become as low as the scattering part, which is around 0:2 cm2-g1. In this circumstance, the Planck mean, which is a linear mean, may be orders of magnitude larger than the Rosseland mean, which is a harmonic (reciprocal) mean. In very dilute plasma, the scattering opacity is often much larger than the absorption opacity, with the result that the Rosseland mean, which includes scattering, can be much larger than the Planck mean, which does not contain scattering. Thus, we must consider the appropriateness of the particular mean to a specific problem.

9.2 Collisional Transitions

497

9.2 Collisional Transitions The assumption that high-energy-density material is in thermodynamic equilibrium is not always valid, since electron impact processes can determine the competition between “upward” transitions and “downward” transitions, as summarized at the beginning of this chapter. In certain situations, such as the solar corona and chromosphere, collisional excitation and ionization (and their inverse processes) are dominant over radiative processes in determining the populations of the atomic levels. Our discussions about Coulomb collisions in Section 3.8.3 gave the energy transferred during a single collision by an electron as ΔW ðbÞ 

ðΔpÞ2 2e4  , 2me me v2 b2

and the momentum impulse is given by ð e2 2b Δp  Fdt  2 , b v

(9.165)

(9.166)

where F is the Coulomb force. As the electron passes through a medium, it will encounter target atoms at various impact parameters b. The flux of electrons, all with the same velocity incident on a single ion, is ne v, thus the energy loss rate will be ð dW 4πe4 ne ¼ ne v ΔW ðbÞ2πbdb  logΛ: (9.167) dt me v The classical treatment for energy transfer gives the cross section for the hydrogen ion as σ ∽ πb2 

πe4 IH IH ¼ 4πa20 , WΔW W ΔW

(9.168)

where W is the total energy transferred from the electron during the collision; this is a reasonable estimate.

9.2.1 Excitation and De-excitation In the semiclassical theory, the incident free electron is treated as a classical charged particle that produces a variable field in the neighborhood of the heavy particle, and quantum perturbation theory is used to calculate the probability that the field induces a transition to an excited state. If Pnm ðbÞ is the probability that an

498

Transition Rates and Optical Coefficients

electron with impact parameter b will cause a transition from a bound level n to another bound level m, with m > n, the cross section is given by ð∞ σ nm ¼ Pnm ðbÞ2πbdb:

(9.169)

0

The quantum theory of atomic structure p (see Section 3.11.3) introduced the ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi quantization of the angular momentum as ℏ lðl þ 1Þ. Classically, this is me vb, and we have the correspondence   ℏ 2 2πbdb ! π ð2l þ 1Þ Δl: (9.170) me v Selection rules for allowed transitions require Δl ¼ 1, and we replace the integral in (9.169) by a sum to obtain σ nm ¼ πa20

IH X ð2l þ 1ÞPnm ðlÞ: W

(9.171)

Here, a0 is the Bohr radius and I H is the hydrogen ionization potential. The probability can be written in terms of a collision strength Ωnm Ωnm ðlÞ : 2l þ 1

(9.172)

IH 1 Ωnm , W gn

(9.173)

Pnm ðlÞ ¼ Averaging over initial states gives σ nm ¼ πa20

where Ωnm is the sum of Ωnm ðlÞ over angular momentum states, and gn is the statistical P weight of level n. Since Pnm is a probability, that is, Pnm 1, we have m6¼n

X1 Ωnm ðlÞ 2l þ 1, g m6¼n n

(9.174)

which is analogous to the oscillator strength sum rules discussed earlier. Many authors have evaluated the collision strength. The general results, when compared to the rough estimate (9.168), show that Ωnm IH , ∽4 gn ΔW

(9.175)

where the transition energy is ΔW ¼ E m  En ¼ Enm . A better estimate can be obtained from the Born approximation, which is expressed in terms of the

9.2 Collisional Transitions

499

dipole-allowed matrix element connecting the two states and is valid at high energies (Seaton, 1962; Van Regemorter, 1962) Ωnm 8π I H ∽ pffiffiffi f nm gnm , gn 3 E nm

(9.176)

where f nm is the same oscillator strength (also known as the dipole length matrix element) discussed in Section 9.1.2, and gnm is an effective Gaunt factor (also known as the dipole acceleration matrix element). This approximation, also known as the Bethe approximation, is fairly accurate for hydrogenic ions. The standard expression for the excitation cross section (Seaton, 1962) is 8π I2 1 σ nm ðE Þ ¼ pffiffiffi πa20 H f nm gnm : Enm E 3

(9.177)

The rate coefficient is the number of collisions producing an excitation from state n to state m, with velocity-dependent cross section σ nm ðvÞ ð∞ C n!m ¼ ne

vσ nm ðvÞf MB ðvÞdv,

(9.178)

vmin

where vmin is the speed corresponding to the threshold energy of the transition and is set by Enm ¼ me v2min =2. For a Maxwellian distribution of electrons, we have C n!m

 1= 2πa20 2k B T 2 2I H ¼ ne Ω nm , πme gn kB T

(9.179)

where ð∞ Ω nm ¼

Ωnm ðuÞeu du,

(9.180)

unm

with unm ¼ Enm =k B T. Using (9.176) we find Cn!m

 1=2 ð∞ 4π 2 4 2 f nm ¼ pffiffiffi e ne gnm ðuÞeu du, πme k B T E nm 3

(9.181)

unm

where f nm is the oscillator strength (9.73). The Gaunt factor requires considerable calculation (Sampson & Golden, 1971). Letting g nm represent the integral in (9.180), then a fit to the results found in the literature (Lokke & Grasberger, 1977) is

500

Transition Rates and Optical Coefficients

g nm

       mðm  nÞ 2 unm 1þ 1 unm e E 1 ðunm Þ , ¼ 0:19 1 þ 0:9 1 þ 20 Z (9.182)

where Z is the atomic number. For a collisionally induced de-excitation, from level m to level n, the rate is the same as (9.178), but with the lower limit of the integral set to zero. In thermal equilibrium, the levels n and m are described by a Boltzmann distribution, and the excitation rate C n!m and de-excitation rate Cm!n must be the same. This requires Ωnm ¼ Ωmn , and thus Cm!n ¼ C n!m

gn ðEm En Þ=kB T e : gm

(9.183)

When the free electrons are degenerate, the Maxwell-Boltzmann distribution is replaced by the Fermi-Dirac distribution function, giving the excitation rate 

Cn!m

2me ¼ 4π h2

3=2 ð∞ E nm

vσ n!m ðEÞE

1= 2



1 1 þ eðEμÞ=kB T

 1



1 1 þ eðEEnm μÞ=kB T

dE,

(9.184) where we have neglected the correction factors. The last term in square brackets accounts for the probability that the unit cell in phase space is not occupied. For non-LTE situations, we should include excitation to highly excited bound levels, with the total ionization cross section being lowered by the depressed continuum.

9.2.2 Ionization and Three-Body Recombination For our last topic of this chapter, we consider the impact ionization of an atom by an electron, and the inverse process of the collision of two free electrons in the field of an ion, referred to as three-body recombination. Calculation of the two transition rates from microscopic cross sections is problematic. Because the initial state of the recombination process involves three bodies, the concept of a collision cross section is not applicable. Thus, we must approach the two processes independently, but require that they satisfy the principle of detailed balance. Returning, for the moment, to a discussion about detailed balancing, we consider the general reaction in which two systems in states i and j approach with relative velocity v, and recede in states i0 and j0 with relative velocity v0 , viewed from the center-or-mass system. For each system, there are g states with the same energy, but we will not include the sums over those states explicitly. The total cross section is

9.2 Collisional Transitions

501

ð  v 1 02 0 2π 1 2 k dk V σ ðij ! i j Þ ¼ dv0 , jðΨ0 jAjΨÞj ℏ gi gj ð2π Þ3 dE 0 V 0 0

(9.185)

with a similar expression for the inverse process, in which the primed and unprimed quantities are interchanged. The last factor accounts for the density of states divided by the incident current for box normalization of the relative motion. In general, the two amplitudes jðΨ0 jAjΨÞj and jðΨjAjΨ0 Þj are not equal, but only the time-reversed amplitudes are the same; however, the average amplitudes are the same. Therefore  0 02 dk k σ ðij ! i0 j0 Þ gi0 gj0 v0 dE0  : ¼ (9.186) σ ði0 j0 ! ijÞ gi gj v 2 dk k dE Recognizing that ð1=vÞðdE=dkÞ ¼ ð1=v0 ÞðdE0 =dk 0 Þ ¼ 1, we arrive at the result of detailed balancing gi gj k2 σ ðij ! i0 j0 Þ ¼ gi0 gj0 k 02 σ ði0 j0 ! ijÞ:

(9.187)

We used this result for photoionization and radiative recombination earlier. Using this result to find the rate at which one of the states is changed by the two processes gives 

dnr dt





r!r 0

dnr0 þ dt

 r 0 !r

ð∞

¼ nr f ðE Þvσ ðrE ! r 0 E 0 ÞdE ð∞

I rr0

þnr0 f ðE0 Þv0 σ ðr0 E0 ! rE ÞdE0 ,

(9.188)

0

where f ðEÞ is taken to be the Maxwell-Boltzmann distribution function. The quantity I rr0 is the difference in the ionization potentials. We then have 

dnr dt 2





r!r0

dnr0 þ dt

 r 0 !r

rffiffiffiffiffiffiffiffi 8 ne ¼ πme ðkB T Þ3=2

ð∞

ð∞

3

0 6 7 4nr σ ðrE ! r 0 E 0 ÞEeE=kB T dE  nr0 σ ðr 0 E 0 ! rE ÞE 0 eE =kB T dE0 5:

I rr0

Now, using (9.187) in (9.189) leads to

0

(9.189)

502



dnr dt

Transition Rates and Optical Coefficients

 r!r0



dnr0 þ dt

 r0 !r

rffiffiffiffiffiffiffiffi 8 ne ¼ πme ðk B T Þ3=2

"  # 0 2 r σ ðr0 E 0 ! rE ÞE 0 eE =kB T dE0 eI rr0 =kB T nr  nr0 : r ð∞

0

0

(9.190) In equilibrium, this must be zero, and we have the necessary and sufficient condition  0 2 I 0 =kB T nr 0 r e r g0 ¼ ¼ r eðEr0 Er Þ=kB T , (9.191) I =k T nr r gr e r B which is the equilibrium distribution. We now extend these ideas to the two processes of interest. For the ionization process, let the unpolarized electron have momentum k0 , incident on an atom in the rth state with statistical weight gr . Following the collision, the two product electrons have momenta k1 and k2 , and the ion is in the sth state  with weight gs . ^ 1 and another in The cross section for producing an electron in the solid angle d k the momentum-space element d ðk2 Þ is     ^ 1 k2 s d k ^ 1 dðk2 Þ ^ 0r ! k σ ei k     ^ 1 2π V 2 2 dk 1 d k 1 V d ðk2 ÞV ¼ : (9.192) jðk1 k2 sjAjk0 rÞj k 1 dEðk1 Þ ð2π Þ3 ð2π Þ3 v0 gr ℏ The energy of the electron with momentum k1 is determined by conservation Eðk0 Þ  I r ¼ E ðk1 Þ þ  E ðk22 Þ  I s . Now averaging and summing over both states, 2 and noting dE=d k ¼ ℏ =2me , gives us       ^ 1k ^1 d k ^ 2 dE 2 ^ 2s d k σ ei k0 r ! k   2me 3 3 V 1 ℏ2 1 k 1 k 2 ^  ^  ¼ d k 1 d k 2 dE 2 : (9.193) jðk1 k2 sjAjk0 rÞj2 5 8 ð2π Þ gr k0 The total cross section for electrons incident with energy E 0 and the production of one electron with energy E1 and another in the energy range dE 2 is σ ei ðE 0 r ! E 1 E2 sÞdE 2   2me 3 ðð V     1 k1 k2 ℏ2 1 ^1 d k ^ 2 dE2 : (9.194) ¼ jðk1 k2 sjAjk0 r Þj2 d k 5 8 k0 gr ð2π Þ For the three-body recombination process, we find the total transition rate

9.2 Collisional Transitions



dPðE 1 E 2 s ! E 0 rÞ dt

3b

1

1 2π ¼ 2g ℏ ð4π Þ s

503

ððð jk0 r jAjk1 k2 sj2

      dk 0 ^2 d k ^1 d k ^0 : Vd k ð2π Þ3 dEðk0 Þ k20

(9.195)

In this expression, in addition to averaging over the initial degenerate states of state s and summing over the final degenerate states of state r, we have averaged over all initial electron directions and integrated over all final electron directions, as required by Fermi’s golden rule. This expression states that the transition probability per unit time for two electrons with energy E1 and E2 and an ion in state s will transition into state r with the outgoing free electron having energy E0 . This can be ^ 1 and k ^ 2 , the expression is simplified further by noting that after integrating over k ^ 0. independent of k The average of the matrix elements in both (9.194) and (9.195) are equal and we can write  3 2me 3 1 k 1 k2 ℏ2 V 1 8 k0 ð2π Þ5 gr σ ðE r !E 1 E2 sÞ    ei 0 ¼ 2me dPðE 1 E 2 s ! E0 rÞ 1 k 0 ℏ2 1 dt 3b 4 ℏ ð2π Þ3 gs 

¼

2me ℏ2

2

V 2 k1 k2 gs ℏ 2 , gr 2ð2π Þ2 k0

(9.196)

which leads to the balance equation for these two processes 8 9  > > 2me 2 2 > > > > V  < = 2 ℏ dP ð E E s ! E r Þ ℏ 1 2 0 2 gr k0 σ ei ðE0 r ! E1 E2 sÞ ¼ gs k1 k2 : 2 > > 2 dt ð 2π Þ > > 3b > > : ; (9.197) In the preceding, the three-body term has dimensions T1, while the ionization term has dimensions T2 M1. From (9.197), we note  dPðE 1 E 2 s ! E 0 rÞ E0 ∽ pffiffiffiffiffiffiffiffiffiffi σ ei ðE 0 r ! E1 E2 sÞ: (9.198) dt E1 E2 3b

504

Transition Rates and Optical Coefficients

In the limit E 1 , E 2 ! 0, which for the inverse process corresponds to approaching the threshold, since E0 ¼ E1 þ E2 þ I r  I s . For this particular ionization, the cross section remains finite at threshold, but the integrated cross section vanishes. pffiffiffiffiffiffiffiffiffiffi Thus, in the limit, (9.198) is proportional to a constant divided by E1 E2 . To calculate the rate at which the number of atoms in the rth state changes by recombination, we multiply (9.195) by the number of ions and the number of electrons in two energy intervals f MB ðE ÞdE for electrons with energy dE 1 and dE2 . Then   dnr 2 ð2π Þ2 gr k 20 ¼  ns σ ei ðE0 r ! E 1 E 2 sÞf ðE1 ÞdE1 f ðE 2 ÞdE 2 : (9.199)  dt 3b ℏ 2me 2 gs k1 k2 ℏ2 Integrating this over all electron energies gives the rate of increase in state r as 

dnr dt

 3b

ð∞ð∞ 32π gr 2 1 1 ¼  ne ns σ ei ðE 0 r ! E1 E2 sÞ  ðk B T Þ3 2 2me 2 gs 00 ℏ ℏ2 E 0 eðE1 þE2 Þ=kB T dE 2 dE 1 ,

(9.200)

where the factor of one-half accounts for the electrons’ spin states, and the corresponding loss rate from state r is 

dnr dt

 ei

rffiffiffiffiffiffiffiffi ð∞ 8 ne nr ¼ πme ðkB T Þ3=2

ð∞

σ ei ðE 0 r ! E 1 E 2 sÞE 0 eE0 =kB T dE0 dE 2 :

0 E 2 þI r I s

(9.201) We now seek to find the electron impact ionization cross section σ ei . For more than a century, numerous authors have approached the problem from different directions, depending upon the particular scenarios they had in mind. This was first done by Thomson in 1912 using basic mechanics. If an electron with kinetic energy E 0 passes near another “stationary” electron, the differential cross section for an amount of energy, ΔE 0 , transferred to the target h i electron is dσ ¼ ðπe4 =E 0 Þ dðΔE 0 Þ=ðΔW 0 Þ2 . The cross section for the transfer of energy exceeding E (E ΔE 0 E 0 ) is   πe4 1 1 σ¼ :  E0 E E0

(9.202)

We assume that ionization takes place each time the impact imparts to the bound target electron an amount of energy exceeding the binding energy. Then,

505

9.2 Collisional Transitions

if E is the binding energy of that electron, we have for the hydrogen atom in its ground state σ ei ! σ 1c ¼ 4πa20 I 2H

ðE 0  EÞ : E 20 E

(9.203)

While this gives a qualitatively correct description of the dependence of the ionization cross section as a function of incident energy for an unexcited atom, and it is of the correct order of magnitude for hydrogen, it gives a cross section (near threshold) for other atoms far in excess of experimental values. Expression (9.203) can be extended for ionization from the nth level of a hydrogen atom, where the ionization potential is E n ¼ I H =n2 σ nc ¼ 4πa20 I H n4

ðn2 E 0  I H Þ ðn2 E0 Þ2

:

(9.204)

The dependence of the cross section on incident energy is shown in Figure 9.5. It is interesting that the quantum-mechanical formula, based on the Born approximation, leads to a slightly different formula, with the n4 term replaced by n3 . The ionization rate is found using (9.178), but adapted for the collisional process

Figure 9.5. Representative collisional ionization cross section from n ¼ 1, 2.

506

Transition Rates and Optical Coefficients

 Cn!c

¼ 4πa20

8kB T πme

 ¼

4πa20

1=2 

8k B T πme

IH kB T

1=2 

 2 ð∞  1 1 u ne  e du un u un

  I H 2 eun ne  Ei1 ðun Þ Γn , kB T un

(9.205)

where un ¼ E n =kB T, and Ei1 ðxÞ is an exponential integral; we have assumed an equilibrium distribution function for the electrons in this expression. The Gaunt factor is taken from Lokke and Grasberger (1977) " # 0:622 0:0745 Γn ¼ 12:18en=ðnþ5Þ ð1 þ 0:0335nÞ 1    (9.206) Z ðZ  Þ2 ð1  eun Þjf ðyÞj, where f ðyÞ ¼ 0:23 þ 0:046y þ 0:1074y2  0:0459y3  0:01505y4 ,

(9.207)

with y ¼ log10 ðun =4Þ. In thermodynamic equilibrium, for the level n, the number of ionizations is balanced by the number of recombinations per unit volume per unit time, expressed as αðneÞ nn ne ¼ βðneÞ n2e nþ , where nþ is the number density of ions in the excited state. Letting αðneÞ nn ¼ C n!c and then using the Saha equation results in a three-body recombination coefficient,    4 a20 h3 I H 2 2 un eun ðeÞ βn ¼ ne  E 1 ðun Þ Γn ; (9.208) π m2e k B T kB T un for hydrogen-like atoms, the statistical weights are gþ ¼ 1 and gn ¼ 2n2 . From this expression, we see that captures into high-lying levels, with En  kB T, occur much more frequently than into the lower levels; the probability of capture increases very rapidly with an increase in the principle quantum number n. Physically, this is simply a result of increasing the radius and area of the orbit of the bound electron with increasing n (the orbit area is proportional to n4 ). We note in passing that for collisional excitation, the expression corresponding to (9.203) is σ nm ¼ 4πa20 I 2H

ðE 0  Enm Þ 3f nm , E 20 E nm

(9.209)

with similar adjustments made to (9.205) for C n!m . An alternate expression for the ionization cross section, by extension of (9.177), is found by replacing E nm with I n þ E, where E is the kinetic energy of the ejected electron

9.2 Collisional Transitions EI   ðn 27 2 I n 4 n3 σ ðE 0 Þ σ nc ðEÞ ¼ 2 πa0  g dE0 , I þ E0 n Z E 3

507

(9.210)

0

with σ ðE0 Þ being the photoionization cross section (see (9.114)). It is common to include the excitation to highly excited bound levels with the total ionization cross section by properly lowering the continuum (Sampson & Golden, 1971), but we do not do so here. Lastly, electron degeneracy effects are accounted for by multiplying    1 1 1 1  ðE μÞ=k T 1  ðE μÞ=k T , (9.211) B B eðE0 μÞ=kB T þ 1 e 1 e 2 þ1 þ1 within the integrals of (9.194) for ionization, and by    1 1 1 1  ðE μÞ=k T , B eðE0 μÞ=kB T þ 1 eðE1 μÞ=kB T þ 1 e 2 þ1 within the integrals of (9.195) for recombination.

(9.212)

10 Radiation Hydrodynamics

When electromagnetic radiation is present, the motion of the fluid is affected because the fluid will exchange energy with the radiation field by emission and/or absorption. This net gain or loss of energy may be sufficient to modify the pressure and therefore its motion. That is, the net momentum exchange between the matter and radiation will affect the motion. Ignoring radiation contributions to the energy and momentum equations may result in an incorrect description of the hydrodynamic response. There may be situations where substantial radiation is present but has little effect on the motion. For example, in optically thin plasma, radiation from an external source will stream through, and there is almost no momentum coupling of the radiation to the matter. However, there also may be circumstances in which the absorption of radiation is negligible, but there may be significant emission, and this will modify the behavior of plasma; this is the topic of radiative cooling, which can be significant in materials with high-atomic-number composition. In the opposite extreme where plasma is optically thick, the radiation mean free path of photons is entirely negligible compared to the size of the system or to the distances that quantities change appreciably. Such a situation is encountered deep within the Sun, where radiation takes one hundred thousand years to diffuse outward from the core, through the radiation zone and into the convection zone; the radiant flux is essentially zero. (The thermonuclear heat is transported to the outer layers of the Sun by convection.) In this case, we can consider the radiation to be in local equilibrium with the matter, and its energy, pressure, and momentum can be combined directly with those of the matter. That is, it is no longer necessary to treat the photons as being separate from the electrons and heavy particles. The need for a more sophisticated treatment of the radiation itself arises in the regime “between” the limits just mentioned. Radiation hydrodynamics addresses the problems where radiation energy and momentum coupling with the matter are important. That is, the photon mean free path is neither extremely small nor large 508

10.1 Incorporating Radiation into Euler’s Equations

509

compared with the size of the system. As we shall see, the inclusion of radiation in the hydrodynamic equations presents a challenge for obtaining “accurate” solutions. In order to make the radiation hydrodynamic equations tractable, we are forced to make approximations that may limit the region of applicability. We must do this because the photon field is intrinsically relativistic (photons travel at the speed of light), while the matter is treated using Newtonian (nonrelativistic) mechanics. There is a perfectly consistent relativistic kinetic theory, and a corresponding relativistic theory of fluid mechanics, which is suited for describing the photon gas, but it is cumbersome to use for fluids in general. As we are primarily interested in fluid flows where u
c, we need to ensure that the source/sink terms relating to the radiation field are included in the equations in a form that preserves overall conservation of energy and momentum. Thus, we require the equations to be accurate to at least Oðu=cÞ. There are two measures of the importance of radiation affecting hydrodynamic motion. The first is the ratio of the radiant flux to that of the matter flux. For a parcel of matter in thermodynamic equilibrium, the ratio is given by the inverse of the Boltzmann number 1 σT 4 ¼ ; (10.1) B0 ρcP Tu for a perfect gas, the specific heat at constant pressure is cP  ½γ=ðγ  1ÞZ  kB =Am0 . A second measure is the comparison of the energy densities (or pressures). For a perfect gas 4σ 4 T 1 ¼  c : (10.2) R ðZ þ 1Þ ni k B T ðγ  1Þ For fully ionized hydrogen gas, the energy densities are equal for T  2:8ρ =3 keV. When temperatures reach a few kiloelectronvolts, radiation dominates the energy (and pressure) even at high densities. The first measure (10.1) is routinely achieved in laboratory experiments using machines capable of creating high-energy-density conditions, but the second is not yet readily attainable. The topic of radiation hydrodynamics has been extensively studied and documented. A large body of material is readily available, but we draw primarily upon the works of Castor (2004), Drake (2006), and Mihalas and Mihalas (1984). 1

10.1 Incorporating Radiation in Euler’s Equations The radiation energy and momentum equations discussed in Chapter 8 are to be solved simultaneously with the conservation equations for the matter, as discussed in Chapters 6 and 7.

510

Radiation Hydrodynamics

To obtain the dynamical equations for a radiating fluid, we can adopt either of two equivalent physical pictures. On one hand, we can treat the radiation field as providing an additional four-vector force acting on the matter. Alternatively, we can consider the externally imposed four-vector force to act on a radiating fluid, comprising both radiation and the matter; this yields a total stress-energy tensor for which the dynamical equations follow. The second approach provides a conceptually more satisfying formulation in the diffusion regime, whereas the first is more natural in the streaming limit.

10.1.1 Fixed-Frame Equations Beginning in Section 8.9.4, the radiation stress tensor is defined and is then used to develop equations for the radiation energy and radiation momentum. A most natural way to combine radiation with matter is to form the matter stress tensor (which we did implicitly in Chapter 6) and then add it to the radiation stress tensor. The dynamical equations are then obtained from the divergence of the total stress tensor. For simplicity, we ignore viscous and thermal conduction terms in the momentum and energy equations. In Section 8.9.7, we discussed the coupling of the radiation to the matter in terms of g0 and g. In the streaming limit, the frame-dependent term ug0 =c can be Oðu=cÞ relative to the radiation force g. The momentum equation for the matter is given by Section 6.2.2, which is already correct to Oðu=cÞ. Thus, the radiatingfluid momentum equation, correct to Oðu=cÞ, is ρ

Du u ¼ F  rP  g þ 2 g0 , Dt c

(10.3)

where D=Dt is the Lagrangean derivative and F represents an externally applied body force. Using expressions for g and g0 from Section 8.9.7, and expressions for the frequency-integrated flux and pressure from Section 8.9.8, we obtain in the Eulerian frame 

∂ 1 ∂Fr u ∂E r þ r  Pr þ 2 þ r  Fr : ðρuÞ þ r  ðρuuÞ ¼ F  rP  2 ∂t ∂t c ∂t c (10.4) We distinguish the radiation quantities with subscript “r.” On the fluid-flow time scale, the term containing ∂E r =∂t is Oðu2 =c2 Þ relative to r  P r and can be omitted. The third term on the right-hand side of (10.3) and the third term in parentheses on the right-hand side of (10.4) accounts for the radiation force, expressed as the momentum absorbed by the material from the radiant flux, and as the

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511

divergence of the radiation pressure tensor. The last terms in (10.3) and (10.4) account for changes in the equivalent mass density of the material, as measured in the laboratory frame, resulting from any net gain or loss of energy by the material through its interaction with the radiation field. This term is often omitted in discussion of radiation hydrodynamics, but at a sacrifice in logical consistency. In particular, it is essential to retain this term in order to make an exact correspondence between the laboratory frame and comoving frame of the radiating fluid. In the diffusion limit, ug0 =c, is Oðλu=LcÞ or Oðu2 =c2 Þ relative to g in the static and dynamic diffusion regimes, respectively. Again, λ is the mean free path for the radiation and L is the characteristic length. Similarly, Fr =c is Oðλ=LÞ or Oðu=cÞ relative to P r in the static and dynamic limits. Hence, on a fluid-flow time scale, both terms containing Fr are at most Oðu=cÞ relative to r  P r and can be omitted. We then have the Eulerian momentum equation ∂ ðρuÞ þ r  ðρuuÞ ¼ f  rP  r  P r , ∂t

(10.5)

where f is a body force. The total energy equation, from Section 6.2.3, with the addition of the radiating fluid, can be written for the Lagrangean frame as 
D 1 2 ∂E r ρ (10.6) εþ u þ þr  ðPuÞþr  Fr ¼ u  f  g0 , Dt 2 ∂t or the Eulerian frame as 
  
∂ ρu2 u2 ρε þ þ E r þ r  ρu ε þ þ Fr þ Pu ¼ u  f: ∂t 2 2

(10.7)

Here, ε is the specific energy of the matter. Integrating (10.7) over a fixed volume element and applying the divergence theorem, we obtain the statement that the rate of change of the total energy (internal, kinetic, and radiative) equals the rate of work done on the volume element by external forces and fluid stresses, minus the flux of total energy (material plus radiative) out of that volume. The continuity equation remains unchanged since the radiation “has no mass.” We often see that r  Fr is used for the heating rate of the matter, but this is incorrect, since it fails to account for the rate of change of the radiation energy density. In fact, r  Fr is a contribution to the rate of increase to the energy density of the matter plus radiation energy. This can be seen in (10.7). The correct heating rate of matter due to the radiation is g0 . In steady state, r  Fr and g0 are equal, as shown in Section 8.4 and with the first equation in Section 8.9.7.

512

Radiation Hydrodynamics

The mechanical energy for a radiating fluid is found by taking the dot product of (10.3) with u and dropping terms of Oðu2 =c2 Þ, which gives 
D u2 ρ ¼ u  ðf  rP  gÞ (10.8) Dt 2 or 
 2

∂ u u2 1 ∂Fr ρ þ r  Pr : þ r  ρu ¼ u  f  u  rP  u  2 ∂t c ∂t 2 2

(10.9)

On the fluid-flow time scale, the ∂Fr =∂t term is at most Oðu=cÞ relative to r  P r ; hence this term is Oðu2 =c2 Þ and can be dropped. On the radiation-flow time scale, this term is of the same order as r  P r in the streaming limit. The internal energy in the laboratory frame is just the total energy minus the mechanical energy ρ

Dε þ Pr  u ¼ g0 þ u  g, Dt

(10.10)

or 
 
 Dε D 1 ∂E r 1 ∂Fr ρ þP ¼ r  Fr  þ u 2 þ r  Pr : Dt Dt ρ c ∂t ∂t

(10.11)

The Eulerian form is 
∂ 1 ∂Fr þ r  Pr : ðρε þ E r Þ þ r  ½ðρε þ PÞu þ Fr  ¼ u  rP þ 2 ∂t c ∂t

(10.12)

Dimensional analysis of (10.10) indicates u  g is of the same order as g0 in the dynamic diffusion regime and may exceed g0 in the streaming limit if the material is approximately in radiative equilibrium; hence, both terms must be retained. Further, in (10.12), c2 u  ð∂Fr =∂tÞ is Oðu2 =c2 Þ relative to r  Fr on a fluid-flow time scale, and can also be dropped. For situations of interest where the fluid-flow time scale is relevant, the fluid momentum, total energy, and mechanical energy equations are all standard Newtonian equations, which include radiative terms in exactly the way we would expect from heuristic arguments. Only the internal energy equation contains a velocity-dependent radiation term that would not be anticipated from simple Newtonian arguments. This term is often ignored in the equations of radiation hydrodynamics, but must be included to convert the net rate of radiant energy input into the matter to its value in the comoving fluid frame. In contrast, it is the radiation energy and momentum equations that require special care, since all

10.1 Incorporating Radiation into Euler’s Equations

513

velocity-dependent terms must be retained to obtain the correct radiation energy and momentum balance. For completeness, we reproduce the radiation energy and momentum equations to Oðu=cÞ from Section 8.9.3 and from the comments at the end of Section 8.9.8 ð ð  0  ∂E r u 0 þ r  Fr ¼ 4πjν  ck ν Eν dν þ  k0ν Fν dν c ∂t   u ¼ k ðRÞ 4πBðPÞ  cEr þ k ðRÞ  Fr ¼ g0 (10.13) c and

ð ð ð 1 ∂Fr u 0 0 j dν þ u  k0ν Pν dν þ cr  P r ¼  k ν Fν dν þ 4π c ∂t c ν  u ¼ k ðRÞ Fr þ 4π BðPÞ þ u  P r ¼ cg: c

(10.14)

We emphasize again the need to retain the velocity-dependent terms in (10.13) and (10.14).

10.1.2 Comoving-Frame Equations Since we are considering only nonrelativistic motion, the simplest way to obtain the comoving-frame momentum equation is to take (10.3) but use the comoving-frame radiation-matter coupling term. Then to Oðu=cÞ we have the Lagrangean equation ρ

Du ¼ f  rP  g0 : Dt

(10.15)

The last term in this equation is given in Section 8.9.7, and (10.15) becomes ð Du 1 0 0 (10.16) ¼ f  rP þ k F dν0 : ρ Dt c ν ν The velocity-dependent terms in the fixed-frame momentum equation vanish in the Lagrangean frame. We can eliminate the integral in (10.16) by using the equation for the frequencyintegrated momentum-density of Section 8.9.6 to give 
D 1 F0r 1 ρ (10.17) uþ 2 ¼ f  rP  r  P 0r  2 F0r r  u, Dt c ρ c which states that a fluid element accelerates in response to an externally applied force, a pressure gradient, and the force exerted by the radiation on the material because the radiant energy flux has inertia.

514

Radiation Hydrodynamics

On a fluid-flow time scale, we can write (10.17) as ρ

Du ¼ f  rP  r  P 0r , Dt

(10.18)

because terms containing F0r are Oðu=cÞ in the streaming limit and Oðλu=LcÞ in the diffusion limit relative to r  P 0r and thus can be dropped. In the diffusion lim0 it,r  P 0r ¼ rP  r , so0 that the fluid acceleration depends on the total pressure gradient r  P þ Pr . The internal energy equation for the matter follows from (10.10), to Oðu=cÞ,  
 ð   Dε D 1 0 þP ρ (10.19) ¼ g0 ¼  4πj0ν  ck 0ν E 0ν dν0 , Dt Dt ρ again using an expression from Section 8.9.7. This equation states that the rate of change of the material energy density plus the rate of work done by the material pressure equals the net rate of energy input from the radiation field. The radiation energy equation is specified in Section 8.9.6 as   ∂E 0r (10.20) þ r  uE 0r þ r  F0r þ P 0r : ru ¼ g00 : ∂t  The dyadic notation P 0r : ru can be replaced with P 0r  rÞ  u, and we find 
 ð   D E0r 0 ρ (10.21) þ P r  rÞ  u ¼ 4πj0ν  ck 0ν E 0ν dν0  r  F0r , Dt ρ where the equation of continuity ðDρ=Dt Þ þ ρðr  uÞ ¼ 0 has been used. This equation states that the rate of change of the radiation energy density plus the rate of work done by radiation pressure equals the net rate of energy input into the radiation field from the material minus the net rate of radiant energy flow out of the fluid element by transport. Combining (10.19) with (10.21) gives the first law of thermodynamics for the radiating fluid 

D E 0r D 1 1 1 (10.22) þP εþ þ P 0r  r  u ¼  r  F0r : ρ Dt Dt ρ ρ ρ The mechanical energy equation is found by multiplying the momentum equation (10.16) by u 
ð D u2 u (10.23) ρ ¼ u  f  u  rP þ  k 0ν F0ν dν0 : Dt 2 c Combining this with (10.22) yields the total energy equation.

10.1 Incorporating Radiation into Euler’s Equations

515

For completeness, we repeat the radiation momentum equation from Section 8.9.6 ð  0 ∂F0r 2 0 (10.24) þ r  uFr þ c r  P r ¼ c k0ν F0ν dν0 , ∂t which can be written as 
ρ D F0r 1 þ r  P 0r þ 2 F0r  ru ¼ g0 : 2 c Dt ρ c

(10.25)

10.1.3 Consistency of the Equations in Both Coordinate Frames We have presented two sets of equations for the energy and momentum of a radiating fluid. We need to verify their mutual consistency. Consider first the laboratory-frame internal energy equation (10.11). On a fluid-flow time scale, the ∂Fr =∂t term is Oðu2 =c2 Þ relative to r  Fr and  hence  can be ignored. Similarly, the terms with u=c in the transformations of E r ; P r to E 0r ; P 0r , as given in Section 8.9.4, will yield terms Oðu2 =c2 Þ,and so we need to retain Oðu=cÞ terms only to transform Fr to F0r ; then, r  Fr ¼ r  F0r þ uE0r þ u  P 0r . Using the Lagrangean derivative, it is then straightforward to show that (10.11) becomes  
  0   Dε D 1 DEr 0 0 ρ þP ¼ þ r  Fr  u  r  P r , (10.26) Dt Dt ρ Dt which is identical to the comoving-frame internal energy equation (10.22). If the velocity-dependent term on the right-hand side of (10.11) had been  omitted, we would be left with an extra term in (10.26) of the form u  r  P r , which is the rate of work done by the fluid against the radiation pressure gradient. The second consideration is the laboratory-frame momentum equation (10.4). On a fluid-flow time scale, the term containing ∂E r =∂t is Oðu2 =c2 Þ relative to r  P r , and can be dropped. Similarly, the terms containing Fr are at most Oðu=cÞ 0 relative to the terms in E r and P r . Thus, it is sufficient  to set Fr ¼ Fr , but all terms 0 0 must be retained in transforming Er ; P r to Er ; P r . Making these conversions in (10.3) gives an equation identical to the comoving-frame equation (10.18). Thus, consistency of the momentum equation between the two frames is ensured if we account for Oðu=cÞ terms in both frames. We have presented a set of equations for the momentum and energy in the comoving frame, but under what conditions should they be used? In an optically thin medium or near a radiating surface of an opaque medium, the radiation field is typically nonequilibrium, the flux is large, and the radiation pressure tensor is

516

Radiation Hydrodynamics

highly anisotropic. It is thus natural to describe the energy exchange between the material and the radiation in terms of direct gains and losses, as in (10.19), and momentum exchange in terms of radiation forces acting on the matter, as in (10.16). In contrast, in the diffusion regime, the net emission-absorption term in (10.19) vanishes and the flux becomes quite small. The radiation energy density and pressure both approach their equilibrium values, and the radiation pressure becomes isotropic. It is now natural to calculate the total energy content expressed by (10.22) and the momentum by (10.18). In nearly all real-world situations, both optically thin and thick regimes are encountered simultaneously. However, only one form of the fluid equations can be used. If the Oðu=cÞ terms are retained in the radiation energy equation (10.21), and this equation is solved simultaneously with either fluid energy equation, the choice is immaterial because exact consistency between the two is guaranteed. But if the Oðu=cÞ terms in (10.21) are omitted, and then if (10.19) is used, satisfactory results will be obtained in the optically thin regime but have serious error in the optically thick regime. This is because the right-hand side of (10.19) is nearly identically zero, because we have not accounted for the rate of change of the internal energy in the radiation or the rate of work done by radiation pressure. If, instead, we use (10.22), the difficulty is reversed. We then obtain a satisfactory solution at great optical depth but make serious errors in the optically thin regime where the radiation is decoupled from the material. Hence, it is important to retain terms of Oðu=cÞ in (10.21) to cover optically thick and thin limits. The situation for the momentum equation is different. Here DF 0r =Dt and the velocity-dependent terms multiplying F 0r in (10.17) are never larger than Oðu=cÞ and are much smaller in the diffusion limit. Dropping these terms gives consistency with (10.16) even if we drop the time-derivative and velocity-dependent terms from the radiation momentum equation. Furthermore, the mechanical energy equation (10.23) when combined with (10.22), leading to the total energy equation, all Oðu=cÞ terms in the radiation momentum equation become Oðu2 =c2 Þ and can be omitted. In summary, we do not adversely affect consistency among various forms of the energy or momentum equations by dropping all Oðu=cÞ terms from the radiation momentum equation. We have completely ignored radiative viscosity and its relevance to radiation hydrodynamics. Certain situations can arise where it is an important effect. 10.1.4 Equilibrium Diffusion The equations of radiation hydrodynamics, whether formulated in the laboratory frame or the comoving frame, result in needing to solve for seventeen variables related by nine partial differential equations. There are also two material

10.1 Incorporating Radiation into Euler’s Equations

517

constitutive relations, the pressure and caloric equations of state (as well as values for the emissivity and absorptivity). So we are short by six equations needed for solution. The six equations are in effect the closure relations relating the six nonredundant components of the pressure tensor to the energy density.1 In general, some iterative method is required to complete the solution, and these are far from trivial, especially for two- and three-dimensional flows. But for studying high-energy-density plasma, there is some relief, in that the diffusion approximation is often quite adequate. In this approximation, the closure relations between the components of the radiative pressure tensor and energy density can be specified a priori. Thus, there are only eight variables, five differential equations, and two constitutive relations. We need only specify the Eddington factor f ¼ Pr =E r in order to complete the system. The fundamental assumption of radiation diffusion theory is that the material is extremely opaque, so that ðλ=LÞ
1. If the material is moving, we must consider a second independent small parameter, namely ðu=cÞ
1. The task, then, is to solve the transfer equation through an expansion in terms of these small parameters. In the comoving frame (but not the laboratory frame), the lowest-order terms that appear in the solution are Oðλ=LÞ and terms of Oðu=cÞ are not present. We can then ignore the motion of the fluid even though it is moving.   The next higher-order terms will be of Oðλu=LcÞ, O λ2 =L2 , and Oðu2 =c2 Þ, the most important being those of Oðλu=LcÞ. Two regimes within the diffusion approximation are determined by the relative sizes of the two independent expansion parameters: one has static diffusion when ðu=cÞ
ðλ=LÞ, and dynamic diffusion whenðu=cÞ  ðλ=LÞ. In the static case, photon diffusion limits the rate of energy flow, and in the latter, the advection of energy by the moving fluid sets the effective rate of energy transport. The first-order approximation, where the radiation field is in equilibrium and is specified by the Planck function, assumes the material is optically thick. In moving material, the radiation field can achieve local thermal equilibrium (LTE) only if a photon is destroyed in essentially the same physical environment as it was created, before local conditions are modified significantly by fluid flow. That is, the mean time between absorptions must be much smaller than the fluid-flow time, hence, ðλu=LcÞ
1. We first assume that only terms of Oðλ=LÞ are present and all others are dropped. The radiation momentum equation in the comoving frame, from Section 8.9.8, in the isotropic limit and ignoring the acceleration terms, is 1

For present purposes, we consider the matter to be just one fluid; separation of the electronic and ionic components is easy to accomplish, bearing in mind that the radiation field couples only with the electronic fluid. The equations can also be cast such that the matter has a temperature different from that of the radiation, but for the present discussion we shall assume complete thermodynamic equilibrium.

518

Radiation Hydrodynamics

F0r ¼ 

c

k

r  P 0r !  ðRÞ

4π dBðPÞ c rT ¼  ðRÞ arT 4 ¼ κr rT, ðRÞ dT 3k 3k

(10.27)

where k ðRÞ is the Rosseland mean absorptivity evaluated in the comoving frame, a ¼ 4σ=c, with σ being the Stefan-Boltzmann constant, and κr is the radiative conductivity. It is clear that F 0r =c is Oðλ=LÞ relative to E 0r or P0r . Now consider the radiation energy equation (10.21). Given (10.27), dimensional analysis shows that terms on the left-hand side of (10.21) are either Oðλu=LcÞ or   O λ2 =L2 relative to E 0r , and so E0r is different from aT 4 only to second order. Therefore, we have 1 E 0r ¼ aT 4 and P 0r ¼ E0r I: 3

(10.28)

I denotes the 3 3 unit matrix. If the material is so homogeneous within an elemental volume where ðλ=LÞ ! 0, we can neglect all gradients. Thus, F 0r ! 0 and P0r ¼ E 0r =3. We emphasize that the stress-energy tensor forming E0r , F0r , and P 0r applies only in the comoving frame. The expression for Fr in the laboratory frame differs from F0r by terms that are often much larger than F0r itself. The comoving-frame momentum equation (10.17) in the first-order approximation is 
Du 1 4 ¼ f  r P þ aT : ρ (10.29) Dt 3 Thus, the radiating fluid behaves dynamically like an ideal gas whose total pressure is the sum of the material and radiation pressures, as previously stated. The energy equation in the equilibrium diffusion regime, given by (10.22), is the first law of thermodynamics 



 D T4 1 4 D 1 1 4σ 3 εþa ¼ r  4 ðRÞ T rT : þ P þ aT Dt 3 Dt ρ ρ ρ 3k

(10.30)

Expressions for the laboratory-frame radiation quantities are obtained using the transformations from Section 8.9.4. To Oðu=cÞ the main effect is to replace the Lagrangean flux by the Eulerian flux. We then have the Eulerian energy equation, correct to Oðu=cÞ, in the equilibrium diffusion regime 



  ∂ 1 4 4σ 3 4 4 ρε þ aT þ r  u ρε þ aT þ P þ aT r  u ¼ r  4 ðRÞ T rT : ∂t 3 3k (10.31)

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519

In the first-order theory, we omitted the time derivatives and velocity-dependent terms to find an expression for the radiation stress-energy tensor, with components of energy density, isotropic pressure, and an energy flux proportional to the temperature gradient. If we retain terms of Oðλu=LcÞ, the radiation stress-energy tensor appears, which contains dissipative terms corresponding to radiative viscosity. We hinted at this in Section 8.9.8, but it is beyond the scope of this book to include a discussion of this topic, although it can be quite important in certain instances (see Mihalas & Mihalas, 1984, for an excellent discussion). Fortunately, nearly all important dynamical effects of radiation are correctly described by the comoving-frame radiation momentum and energy equations, provided that all terms in those equations are retained; there is no need to resort to the second-order diffusion approximation.

10.1.5 Nonequilibrium Diffusion The basic assumption of equilibrium diffusion theory is that the matter and radiation are in thermal equilibrium. This implies that the Planck distribution function describes the distribution of the photons in terms of the material’s temperature. However, we know from previous discussions that certain physical events, such as the passing of a shock front through the matter, cause the matter to have a different temperature from that characteristic of the radiation. A second instance of nonequilibirum is when the radiation field varies too rapidly for the material to follow instantaneously. In the comoving frame where the radiation field isotropizes and P0r ! E 0r =3 as the photon mean free path becomes small, we drop terms of Oðλu=LcÞ and higher to give the nonequilibrium diffusion flux F0r ¼ 

c rE0r , 3k

(10.32)

where k is similar to the Rosseland mean kðRÞ , but defined by ð 1 rE 0ν dν0 : rE 0r ¼ k0ν k

(10.33)

Using this in (10.21), we have 
  0

  D Er 1 0D 1 c 0 ρ rEr þ c k ðPÞ aT 4  kðEÞ E0r : (10.34) ¼ r þ Er Dt ρ 3 Dt ρ 3k The Planck mean absorption coefficient is kðPÞ (see Section 8.7) and the absorption mean is defined by

520

Radiation Hydrodynamics

k

ðE Þ

ð 1 ¼ 0 E0ν k0ν dν0 : Er

(10.35)

Expressions (10.32) and (10.34) are extensions of the first-order equilibrium diffusion theory of the previous section. Defining the “radiation temperature” via E 0r ¼ aT 4r , expression (10.34) becomes 
  4

  D aT r 1 D 1 4ac 3 ρ ¼ r T r rT r þ ac k ðPÞ T 4  k ðEÞ T 4r : þ aT 4r Dt ρ 3 Dt ρ 3k (10.36) Note the use of the material temperature T and the “radiation temperature” T r . T r is only a parameter describing the total radiation energy density, and the spectral distribution need not be Planckian, though it may be. In this two-temperature description, we also need the material energy equation to solve for T and T r  
  Dε D 1 ðE Þ 4 ðPÞ 4 ρ þP ¼ ac k T r  k T : (10.37) Dt Dt ρ There is the question of what to use for the various mean absorptivities. The simplest approximation is to set, by analogy with equilibrium diffusion, k ¼ kðRÞ and k ðEÞ ¼ kðPÞ , with all quantities evaluated at the material temperature. An alternative is to assume that the spectral distribution of E0r is the Planck function for temperature T r . Then two-temperature mean absorptivities can be defined ð 4π (10.38) k 0 ðν0 ; T ÞBðPÞ ðν0 ; T r Þdν0 , kðEÞ ðT; T r Þ ¼ acT 4r ν and  ðPÞ
ð 1 π 1 dB dν0 : ¼ 4 0 k ðT; T r Þ acT r kν ðν0 ; T Þ dT T r

(10.39)

These two quantities become the standard diffusion expression when T r ¼ T. We must be careful in using nonequilibrium diffusion theory, as the equations are developed in the comoving frame. The derivation in the laboratory frame is more involved because we must discriminate between Fr and F0r , as well as how the laboratory-frame energy density relates to T r . If we naively write E r ¼ aT 4r =3, the radiation-energy and internal-energy equations contain extra terms and do not reduce correctly to their comoving-frame counterparts (although the result for the combined equations is correct). Instead, the radiation temperature must be defined by E0r ¼ aT 4r , and then Er ¼ E0r þ ð2u=c2 ÞF 0r implies

10.1 Incorporating Radiation into Euler’s Equations

E r ¼ aT 4r 

8a 3ck

ðRÞ

T 3r u  rT r :

521

(10.40)

The second term here is important only when calculating the net emission and absorption. Using (10.40) and Fr ¼ F0r þ uE0r þ uE 0r =3 in (10.13) shows that the Eulerian nonequilibrium-diffusion radiation energy equation, correct to Oðu=cÞ, is    1 ∂  4 aT r þ r  aT 4r u þ aT 4r r  u ¼ ac k ðPÞ T 4  kðEÞ T 4r ∂t 3 
4ac 3 (10.41) T r rT r , þr  3k ðRÞ which is identical to (10.36). Similarly, from (10.11) we find that the Eulerian internal-energy equation, correct to Oðu=cÞ, in the nonequilibrium diffusion limit is  ∂ ðρεÞ þ r  ðρεuÞ þ Pr  u ¼ ac kðEÞ T 4r  kðPÞ T 4 , (10.42) ∂t which is identical to (10.37). The discussion about equilibrium and nonequilibrium diffusion can be extended to frequency-dependent versions of the equations. In particular, for nonequilibrium diffusion, examination of these equations further reveals a deeper understanding of radiation transfer, but we forego a discussion here. Before we leave this topic about the radiation hydrodynamic equations in the two coordinate systems, we need to examine a statement made in the opening chapter of this book. There we suggested that radiation should be treated on an equal basis with the matter. That is, the equations of the dynamic behavior of the matter should resemble those for the radiation. Of course, we must account for the finite mass of the matter. If we rewrite the total energy equation in the laboratory frame (10.7) as 

∂ 1 2 1 2 (10.43) ρε þ ρu þ Er þ r  ρuh þ ρuu þ Fr ¼ u  f, ∂t 2 2 we see there is a term for the enthalpy flux of the matter but no term for that of the radiation. Yet we make the claim that there is no intrinsic difference between material particles and photons. So why is there this apparent discrepancy that would persist even when the absorptivity is so great that the flux vanishes? The explanation is that the flux depends upon the frame of reference from which it is viewed. The same is also true for the energy density and radiation pressure; these last two quantities are not normally affected much, and the corrections are small. Recall that the connection between the radiant flux in the laboratory frame to that in the comoving frame is Fr ¼ F0r þ uE0r þ u  P 0r to Oðu=cÞ. In the limit of

522

Radiation Hydrodynamics

small λ, F0r ! 0, not Fr . In this limit, Fr tends to the convective radiation enthalpy flux E r þ Pr ¼ ð4=3ÞaT 4r . If this is used in (10.43), the symmetry between matter and radiation is recovered. 10.1.6 Flux Limiting One of the basic assumptions of diffusion theory is that ðλ=LÞ
1. Yet we often use the diffusion approximation in all regions of plasma, even where this condition is violated. The effect is that there is too large an energy transport in optically thin material. Moreover, diffusion theory usually gives a serious overestimate of the energy deposited by a radiation front from penetrating into cold material, particularly at early times. Nonequilibrium diffusion generally gives better results than equilibrium diffusion, but both are significantly in error. In extremely transparent material, diffusion theory gives jFr j > cEr , implying that the effective speed of energy propagation is cE ¼ jFr j=Er ¼ cλjrEr j =3Er ∽cðλ=3LÞ, which exceeds the speed of light. This difficulty arises because the diffusion equation tacitly assumes that a photon always travels a distance of the order of λ, even if λ exceeds the free-flight distance cΔt. This is to be expected, since we know from Section 7.1 that the linear diffusion equation has a formally infinite signal speed. One way to overcome the problem is to introduce a flux limiter, in a fashion similar to that for electron thermal conduction. The idea is to alter the diffusiontheory formula for the flux in such a way as to yield the standard result in the extreme optically thick regime while simulating free streaming in thin regimes. One useful form is 
3 jrE r j (10.44) Fr ¼ crE r þ Er λ but numerous others have been proposed. While flux-limited diffusion has been widely used, this approach is only an ad hoc fix of generally unknown accuracy. In the two limits the results are reasonable, but the intermediate region is perhaps seriously wrong. Fundamentally, the fluxlimiting problem results from dropping the time derivate from the radiation momentum equation, which precludes recovery of the wave-equation character of the coupled radiation energy and momentum equations in the optically thin limit. Indeed, dimensional analysis suggests that on a radiation-flow time scale, c2 ð∂F r =∂t Þ is Oðλ=LÞ relative to k ðPÞ F r =c, and will dominate the solution for a radiation front in transparent material. Fortunately, computational techniques have been developed to treat this problem more realistically. One such approach, which we have not discussed, is the

10.2 Thermodynamic Relations in Presence of Radiation

523

variable Eddington method for solving the transfer equation, which yields adequate results in all optical thickness regimes. The reader is encouraged to consult the literature for the details.

10.2 Thermodynamic Relations in the Presence of Radiation The thermodynamics of matter was reviewed in Section 3.3 and that for radiation in Section 3.10.3. In the extreme optically thin regime, we don’t really need radiation thermodynamics, and in the very thick regime radiation thermodynamics dominates, but for the intermediate regime the thermodynamic quantities of a gas with radiation are not simple combinations of their individual quantities. As we are interested in conditions of thermodynamic equilibrium, this necessarily assumes optically thick matter. When material and radiation are not in equilibrium, the thermodynamical physics is poorly understood.

10.2.1 Equilibrium Radiation and a Perfect Gas Consider a “fluid” composed of thermal radiation and a gas of material particles in thermal equilibrium. For a particle number density n, the total pressure is Ptot ¼ nk B T þ aT 4 =3. If we have a perfect gas, with γ ¼ 5=3, the total specific internal energy is 
1 3 4 (10.45) εtot ¼ nkB T þ aT : ρ 2 We are being a bit cavalier here with the assumption that no free electrons are present, and thus the connection between mass density and particle density is ρ ¼ Amo ni , where A is the atomic weight and mo is the atomic mass; it is straightforward to include a free-electron gas. The specific heat at constant volume of the combined “fluids” is 

∂εtot 12ni k B aT 3 1 cV ¼ ¼ þ : (10.46) ∂T ρ ρ 3ni kB 8 The total specific enthalpy is htot ¼ εtot þ Ptot =ρ, from which we find the specific heat at constant pressure  cP ¼

∂htot ∂T


P

" 
2 # 5 ni k B 8 aT 3 32 1 aT 3 1þ : ¼ þ 2 ρ 3 ni k B 5 3 ni k B

(10.47)

524

Radiation Hydrodynamics

This expression goes to the correct limit for the perfect gas if P Pr , but diverges in the opposite case, as mentioned in Section 3.10.3. If we define α ¼ Pr =P, the polytropic index is   2 cP 5 1 þ 8α þ 32 5 α γ¼ ¼ : (10.48) cV 3 ð1 þ 8αÞ Because cP diverges for large α, (10.48) cannot yield the correct γ for pure radiation; we know the correct value is γ ¼ 4=3. It is obvious that γ ¼ cP =cV is no longer constant. Chandrasekhar (1939) introduces generalized adiabatic exponents Γ1 , Γ2 , and Γ3 , defined by 
d log Ptot Γ1 ¼ d log ρ s 
Γ2 d log Ptot ¼ Γ2  1 d log T s and  Γ3  1 ¼

d log T d log ρ


:

(10.49)

s

Only two of these are independent, thus Γ1 =ðΓ3  1Þ ¼ Γ2 =ðΓ2  1Þ. To evaluate these quantities, we return to the first law of thermodynamics. Since we know from Section 5.1 that entropies are additive, then for an incremental change in the specific entropy of the combined fluids we have  
 1 1 dεtot þ Ptot d dstot ¼ T  ρ

 1 3 4 4 dρ 3 dT ¼ : ni kB þ 4aT  ni kB T þ aT (10.50) T 2 ρ 3 ρ2 For an adiabatic process dstot ¼ 0, and we have ð1 þ 4αÞ : Γ3  1 ¼ 3 2 þ 12α

(10.51)

Since the total pressure is a function of density and temperature only, we find 5  þ 20α þ 16α2 2 : Γ1 ¼ 3 27 (10.52) 2 2 þ 2 α þ 12α Both Γ1 and Γ3 are correct in the two limits of the pressure ratio α. For sound-wave applications, Γ1 is the appropriate index, while for thermal transport applications, including radiative heat transport, Γ3 is the proper one.

10.2 Thermodynamic Relations in Presence of Radiation

525

The speed of sound in our composite fluid is not given by the simple relation c2s ¼ ð∂Ptot =∂ρÞs , as discussed in Section 6.4. Rather, we must begin anew with the relativistic dynamical equations. Referring to Mihalas and Mihalas (1984), and proceeding along the same lines as before, we have the momentum equation ∂u1 þ c2 rP1 ¼ 0, ∂t

(10.53)

∂E 1 þ ðE þ Ptot Þ0 r  u1 ¼ 0, ∂t

(10.54)

ðE þ Ptot Þ0 and the energy equation

where the total energy density is E ¼ ρ0 ðc2 þ εtot Þ. The subscripts 0 and 1 refer to the zeroth and first expansion terms of the subscripted quantitiy; for example, Ptot ¼ P0 þ P1 , with ðP1 =P0 Þ
1. Combining (10.53) with (10.54) gives a wave equation ∂2 E 1  c2 r2 P1 ¼ 0, ∂t 2 which implies that an acoustic wave propagates with a speed c where 


∂ρ0 2 2 ∂Ptot 2 ∂Ptot ¼c : cs ¼ c ∂E s ∂ρ0 s ∂E s

(10.55)

(10.56)

But 

∂E ∂ρ0


s


E ∂εtot ¼ þ ρ0 , ρ0 ∂ρ0 s

and for an adiabatic process ds ¼ 0 
∂εtot Ptot ¼ 2 , ∂ρ0 s ρ0

(10.57)

(10.58)

from which we obtain 

∂E ∂ρ0


¼ s

E þ Ptot , ρ0

(10.59)

and thus (10.56) becomes (now using Ptot ) c2s ¼ c2 Γ1

Ptot : E þ Ptot

(10.60)

526

Radiation Hydrodynamics

For a nonrelativistic gas, E ! ρ0 c2 Ptot which yields c2s ¼ Γ1 Ptot =ρ, and for an extremely pffiffiffi relativistic gas E ! ρ0 εtot ¼ 3Ptot , and that Γ1 ! 4=3, giving cs ¼ c= 3.

10.2.2 Equilibrium Radiation and an Ionizing Gas Perhaps a more interesting situation is examining the thermodynamic quantities of an ionizing gas. The presence of significant radiation indicates that the states of the atom are constantly changing in response to emission and absorption of the radiation. If those radiative processes involve ionization and/or recombination, the effective charge of the ions is changing, indicating a change in the number of free electrons, and this in turn affects the dynamical equations. Although we are focusing on radiation here, we do not intend to diminish the importance of collisional processes in determining the ionization level of the matter; we will return to this topic at the end of the chapter. Now consider a gas composed of thermal radiation and ionizing hydrogen. Denoting y as the ionization fraction (the ionization level Z  ) of the hydrogen gas, with heavy particle (ion) number density ni , the total pressure is 1 Ptot ¼ ð1 þ yÞni kB T þ aT 4 , 3 and the total internal specific energy   1 3 4 ð1 þ yÞni kB T þ ni yI H þ aT : εtot ¼ ρ 2

(10.61)

(10.62)

The middle term on the right-hand side is the ionization energy for hydrogen. While there are choices for an ionization model, we pick the Saha model for its simplicity. Section 4.1 of this book develops the Saha ionization model with some detail. There we write it in the form y2 g 2 T 3=2 I H =kB T : ¼ 1 3 eI H =kB T ¼ const e 1  y g0 ni λde ρ The specific heat at constant volume is 


∂εtot 3 ni kB ni kB 3 IH ∂y 4aT 3 þ ð1 þ yÞ þ : cV ¼ ¼ þ T ∂T V ∂T ρ 2 ρ ρ 2 kB T ρ From (10.63), we find

(10.63)

(10.64)

10.2 Thermodynamic Relations in Presence of Radiation



∂y T ∂T


V


yð1  yÞ 3 IH þ ¼ , ð2  yÞ 2 k B T

527

(10.65)

yielding the result ni k B cV ¼ ρ

"



# 3 yð1  yÞ 3 IH 2 , þ 12α ð1 þ yÞ þ þ 2 ð2  yÞ 2 kB T

(10.66)

which reduces to (10.46) when y ¼ 0. It is evident that both ionization effects and radiation pressure can make a large contribution to cV . The specific heat at constant pressure is obtained from the specific enthalpy 


∂htot 5 ni kB ni kB 5 IH ∂y cP ¼ ¼ þ ð1 þ yÞ þ T ∂T P ∂T P 2 ρ ρ 2 kB T  
 3 4 aT T ∂ρ 4 þ : (10.67) 3 ρ ρ ∂T P From (10.63), we find 

 5 ∂y 1  IH 2 ¼ y 1y þ þ 4α , T ∂T P 2 2 kB T and from (10.61)  
 
∂ρ ρ 1 5 IH þ 4α : ¼  1 þ 4α þ yð1  yÞ þ ∂T P T 2 2 kBT

(10.68)

(10.69)

Recall a ¼ Pr =P. Then, using "

2 #   ni kB 5 1 5 I H , þ 20α þ 16α2 ð1 þ yÞ þ y 1  y2 þ cP ¼ þ 4α 2 2 2 kB T ρ (10.70) which reduces to (10.47) when y ¼ 0. The generalized adiabatic exponents are found from the specific entropy in much the same way as was done in the previous section. Performing total differentiation on expressions (10.61) and (10.62), then relating them through the adiabatic relation (10.58), gives an expression in y and dy. To eliminate dy, we need the logarithmic derivative of (10.63), which is 
ð2  yÞ dy 3 I H dT dρ ¼ þ  : (10.71) ð1  yÞ y 2 kB T T ρ We then find

528

Radiation Hydrodynamics

 1 þ 4α þ 12 yð1  yÞ 52 þ kIBHT þ 4α  : Γ3  1 ¼ 2 IH 3 1 3 3 þ 12α þ y ð 1  y Þ þ þ þ 12α kB T 2 2 2 2 After considerable algebra, we arrive at   2  IH 5 2 1 5 β 2 þ 20α þ 16α þ 2 yð1  yÞ 2 þ kB T þ 4α   , Γ1 ¼ 2 IH 3 1 3 3 þ 2 þ 12α 2 þ 12α þ 2 yð1  yÞ 2 þ kB T

(10.72)

(10.73)

where β is the ratio of the gas pressure to the total pressure. Both expressions, (10.72) and (10.73), reduce to (10.51) and (10.52), respectively, when y ¼ 0. The sound speed can be found in a similar fashion as before, using Γ1 from (10.73), the pressure from (10.61), and the total energy density obtained from (10.62). The theories developed in this and the preceding subsection give an accurate description of the thermodynamic properties of a radiating fluid when both the matter and radiation are in thermal equilibrium. Even though these equations might provide some guidance, use caution when applying these results to nonequilibrium situations. Nonequilibrium conditions require a complete reformulation of the thermodynamics. While we can write down expressions for the specific energy and specific enthalpy, the entropy becomes more problematical: It is not clear how such a quantity can be defined. Further, the generalized adiabatic exponents no longer make any sense because of the presence of a nonequilibrium radiation field. This also makes the concept of a sound speed meaningless.

10.3 Marshak Waves The equations of radiation hydrodynamics, while looking rather innocuous, contain a complexity that precludes analytical solutions. We must almost always resort to numerical solutions. However, there are a few situations where the radiation hydrodynamic equations can be solved on paper. While these examples may not be particularly useful in high-energy-density physics, they are illustrative and allow us to gain insight; they also are important from a pedagogical point of view. We discussed, in some detail, the penetration of a thermal wave into matter by conduction. Sections 7.1 and 7.2 developed the equations for linear and nonlinear thermal conduction, respectively. A similar process is the penetration of radiation into cold material, and because the radiative conductivity depends strongly on temperature, the equations of Section 7.2 are relevant. Because radiative energy

10.3 Marshak Waves

529

exchange is very efficient, significant penetration and energy deposition can take place in a time much too short for the fluid to be set in motion. However, eventually the material becomes hot, pressure gradients build, and the fluid begins to move. The penetration of radiant energy into material is described by a class of problems known collectively as Marshak waves. These “waves” are not waves in the usual sense, as they do not come from a hyperbolic system of partial differential equations, the dispersion relation does not yield wave speeds, and so forth. Solution to the hyperbolic equations predicts the penetration distance to be proportional to the time, while the radiation hydrodynamic equations, having a pffi diffusive nature, give the distance ∽ t . They are waves in the sense that there is a characteristic structure that remains relatively constant in shape during propagation. Marshak (1958) first showed that the simple problem of a constant temperature source placed at the boundary of a stationary medium can be treated by self-similar methods. This nonlinear diffusion problem begins with the internal energy equation (10.12), omits the velocity-dependent terms, and ignores the rate of change of radiation energy density, to give the simple diffusion equation ρ

∂ε ∂T ¼ ρcV ¼ r  Fr : ∂t ∂t

(10.74)

We assume constant density and specific heat for the material. The heat is transferred only by radiation, and the radiative conductivity scales as some power of the temperature. For planar geometry, the first law of thermodynamics becomes 
∂T ∂ ∂T ¼ κr , (10.75) ρcV ∂t ∂x ∂x      where the radiative conductivity is κr ¼ 16σλðRÞ T 3 =3 ¼ 4acT 3 = 3k ðRÞ , which comes from (10.27), and k ðRÞ is the Rosseland mean absorption coefficient. Typical high-energy-density materials have the Rosseland mean scaling as ðRÞ kðRÞ ¼ k 0 ðT=T 0 Þn , where n  4  5. The subscript 0 denotes the value of the absorption coefficient at temperature T 0 . Hence κr varies as T 7 or T 8 and (10.75) becomes ∂T ∂2 T nþ4 , ¼ χr ∂t ∂x2

(10.76)

with diffusivity χr ¼

4ac ðRÞ

3ðn þ 4ÞT n0 k 0 ρcV

:

(10.77)

530

Radiation Hydrodynamics

The boundary condition at x ¼ ð0; t Þ is T ¼ T 0 . The absorption coefficient is for temperatures near T 0 , not the coefficient of the cold material at the start. Adopting the similarity variable x x ξ ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ A pffi t 2χ r T nþ3 0 t

(10.78)

∂f 1 ξ df ¼ ∂t 2 t dξ

(10.79)

∂ 2 f A2 d 2 f ¼ : ∂x2 t dξ 2

(10.80)

implies

and

Defining the scaled temperature τ ðx; t Þ ¼ T ðx; t Þ=T 0 as the dependent variable, (10.76) becomes dτ d 2 τ nþ4 : (10.81) ξ ¼ dξ dξ 2 The boundary conditions are τ ¼ 1 at ξ ¼ 0, and τ ¼ ðdτ nþ4 =dξ Þ ¼ 0 at some point ξ f yet to be determined. The latter implies that the flux vanishes at ξ f , since 
1=2 nþ4 ∂T T nþ5 dτ 0 ¼ ρcV χ r : F r ¼ κr ∂x 2t dξ

(10.82)

The total energy in the wave (per unit area) is ðxf ℰ ¼ ρcV where e ¼

Ðξ f

 1=2 Tdx ¼ ρcV 2χ r T nþ5 e, 0 t

(10.83)

0

τdξ, thus defining ξ f . Energy conservation implies ð∂ξ=∂t Þ ¼ F ð0; t Þ,

0

which is equivalent to

 e¼

dτ nþ4 dξ


ξ¼0

:

(10.84)

There are an infinite number of solutions that go to zero at a particular ξ f , but only one of these has a flux going to zero in the limit ξ ! ξ f . This follows from (10.81) if we integrate from a value ξ < ξ f to a value of ξ in the zero-temperature region ahead of the heat front. An integration by parts of the left-hand side of (10.81) leads to

10.3 Marshak Waves

ð∞ ξτ þ τdξ ¼  ξ

531

dτ nþ4 : dξ

(10.85)

The left-hand side clearly tends to zero as ξ ! ξ f , so the right-hand side, which is proportional to the flux, must tend to zero also. We can determine the behavior of τ near the front by approximating the lefthand side of (10.85) with ξ f τ, which leads to  1  nþ3 ðn þ 3Þ  ξ ξ ξ τ∽ : ðn þ 4Þ f f

(10.86)

This relation is the actual boundary condition at ξ ¼ ξ f , and when used in conjunction with τ ¼ 1 at ξ ¼ 0, determines a unique solution. The values of ξ f and e are listed in Table 10.1 for several values of n. Figure 10.1 shows the scaled temperature distributions from (10.86). We pick the value n ¼ 3 because it is representative of free-free and bound-free absorption (see Sections 9.1.5 and 9.1.6). The Marshak model overestimates the heat penetration as shown in Figure 10.2, where the Marshak calculation is compared to a full radiation transport calculation. Campbell and Nelson (1964) used a Monte Carlo calculation, with four thousand particles, to produce a scaled-temperature profile for three times. It is instructive to compare the solutions of other diffusion equations to the Marshak solution. Figure 10.3 shows the scaled profiles using the equilibrium diffusion model (10.31) and the nonequilbrium diffusion model (10.42). The velocity-dependent terms have been omitted from both models. Campbell and Nelson also include a solution from the telegrapher’s equation (see Section 8.6.3), which we do not include here since the results are quite similar to the full transport solution.

Table 10.1 Values of Marshak Quantities n

ξf

e

0 1 2 3 4 5 10

1.232 1.178 1.144 1.121 1.103 1.091 1.057

0.940 0.952 0.960 0.965 0.970 0.973 0.982

532

Radiation Hydrodynamics

Figure 10.1 Self-similar temperature distributions for Marshak waves for different values of n, the exponent of the temperature in the absorption coefficient.

Figure 10.2. The full transport solution evolves over time into a Marshak wave; profiles for times ct ¼ 0:1λðRÞ , λðRÞ , and 10λðRÞ . For comparison, the dashed line shows the Marshak solution for n ¼ 0.

533

10.3 Marshak Waves (b)

(a) 1.0

1.0

Marshak

0.8

0.8

T/T0

Er /aT04

0.6

0.4

0.2

Marshak

0.6

0.4

0.2

0.0 0.0

2.0

1.0

4.0

3.0

0.0 0.0

5.0

1.0

2.0

3.0

4.0

x

x

(d)

(c)

0.5

0.8

0.4

0.6

0.3

T/T0

Er /aT04

1.0

0.4

0.1

0.2

0.0 0.0

0.2

0.1

0.2

0.3

0.4

0.5 x

0.6

0.7

0.8

0.9

0.0 0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

x

Figure 10.3. (a) Comparison of temperature profiles from equilibrium diffusion (short-dashed lines), nonequilibrium diffusion (long-dashed lines), and Monte Carlo transport (solid lines) calculations to the self-similar Marshak model; and (b) the radiation energy density; the equilibrium and nonequilibrium models give essentially identical results. These calculations are for ct ¼ 10λðRÞ ; (c) and (d) are for ct ¼ λðRÞ . The Marshak solution for (c) and (d) is off to the right; ξ f ¼ 1:06.

Figure 10.1 shows that the higher the value of n, the closer the temperature profile resembles a “square” shape. This observation is important because nearly all of the material that has been penetrated by the wave is at a temperature close to T 0 ; the drop in temperature near the heat front is a consequence of the “bleaching” associated with the heat wave. The material, which is very opaque initially, becomes transparent as it is heated. The more transparent it becomes, the more the heat can flow toward the front of the wave. The radiant heat flux is found by substituting the similarity variables into the flux F r ¼ κr ð∂T=∂xÞ, so that

534

Radiation Hydrodynamics

F r ¼ F 1

dτ nþ4 , dξ

(10.87)

with the scaling factor " F1 ¼

2acT 40 ρcV T 0

#1=2

ðRÞ

3ðn þ 4Þk0 t

:

(10.88)

In the free-streaming limit at the boundary F 0 ¼ cE0 ¼ 4σT 40 ¼ acT 40 , and if the flux is scaled to this, we have " #1=2 ðRÞ Fr 2ρcV T 0 λ0 dτ nþ4 ¼ : (10.89) cE0 dξ 3ðn þ 4ÞacT 40 t    ðRÞ The ratio ρcV T 0 λ0 = acT 40 t compares the heat content of a layer one mean free path thick to the energy deposited by the radiation in a time t. If this ratio is small, the heated material is many mean free paths thick, and the deposition slows down so that the net flux at the boundary is zero. In the other extreme, the ratio being large, the front has not penetrated even one mean free path; thus, the diffusion approximation is not very good. For this model to be valid, t>

ðRÞ

2ρcV λ0 : 3ðn þ 4ÞacT 30

(10.90)

The penetration of the heat wave into the undisturbed material is initially supersonic. As the thickness of heated material increases, the speed of the heat wave decreases. Simultaneously, hydrodynamic expansion begins with the heated material expanding into the vacuum. This regime ends when the speed of the heat wave decreases to approximately the sound speed. It is then overtaken by a “shock wave,” and the subsonic phase begins. We can use the preceding equations to find the heat front speed. Rather, a simple estimate (good enough for our purposes) is obtained by noting that the temperature behind the heat front is approximately constant. Then the left-hand side of (10.75) is small away from the heat front, so the temperature profile approximately satisfies the steady-state equation 
∂ 4σ ∂T 4 ¼ 0: (10.91) ∂x 3kðRÞ ∂x The quantity in parentheses is essentially the radiative power (per unit area) reaching the heat front. The radiation supplies the internal energy needed to advance the heat front through the material and keep the heated material at a near-uniform temperature. At the heat front

10.3 Marshak Waves

535

dxf 4σ T 4  ðRÞ 0 , dt 3k xf

(10.92)

ρε0 which is easily solved to give

 xf ¼

8σ T 40 t 3kðRÞ ρε0


1=2 ,

(10.93)

from which the heat front speed is obtained. (Bear in mind that this estimate may be too large inpffiffiffiffiffiffiffiffiffiffi light of Figure 10.2). The isothermal sound speed is pffiffi cT ¼ cs = γ ¼ cs0 Z  T 0 , so the supersonic phase lasts while t
0, the wave must be damped as z ! ∞ (ζ !
∞) and at z ¼ L (ζ ¼ 0) ϵ ¼ 0, and for z > L (ζ < 0) ϵ < 0. There is, then, a standing wave for z < L 3A E ðζ Þ ¼ π

ð∞




 1 2 x
ζ x dx ¼ 3AAiðζ Þ, cos 3

(12.27)

0

where A is a constant determined by the boundary conditions, and Aiðζ Þ is the Airy function. For ζ < 0, the electric field is exponentially damped, and for ζ > 0, the field oscillates; this behavior is shown in Figure 12.2. Redefining (12.25) slightly, 1= ζ ¼ ðω2 =c2 Þjdη=dzj0 3 Δz, where the zero subscript indicates the gradient is measured at z ¼ L, and Δz is measured from this point also. The field is zero at the points given by (Ginzburg, 1970)
Δz0m ¼ βm


1=3 ω2 dη , c2 dz 0

(12.28)

where m is the number of the zero; β1 ¼ 2:338, β2 ¼ 4:088, β3 ¼ 5:521, and so on. For large ζ , the asymptotic form of the Airy function is
 3A
1=4 2 3=2 π p ffiffiffi ζ
: E ζ cos (12.29) π 3 4

634

Electromagnetic Wave-Matter Interactions

Figure 12.2. Normalized electric field intensity (solid line) and the geometrical approximation (dashed line).

If the amplitude of the incident electric field is E 0 , the solution (12.27) is a standing wave and at z ¼ 0

 2ωL π 2ωL π
i þi Eð0Þ ¼ Eþ þ E
¼ 3A exp i þ 3A exp
i 3c  4 3c 4


4ωL π (12.30) ¼ 3A 1 þ exp
i þi , 3c 2 with
1=
 2 pffiffiffi ωL 6 2ωL π A¼ E0 exp i π
i : 3 c c 4

(12.31)

As can be seen from Figure 12.2, the amplitude of the electric field 1reaches a = maximum at ζ ¼ 1, which corresponds to ðL
zÞ ¼ ðc2 L=ω2 Þ 3 ; there 1= jEmax j2  3:6ðωL=cÞ 3 . If a mirror was placed at ζ ¼ 0, we would expect a factor of four increase in E 2 because a standing wave exists. There is additional swelling due to the decrease in the group velocity of the light wave as the dielectric permittivity becomes small; that is, as ϵ becomes smaller, the wavelength becomes longer.

12.2 Propagation in Inhomogeneous Isotropic Medium

635

The magnetic field is found from (12.27) H ðζ Þ ¼
i

c ∂E ¼
3iAAi0 ðζ Þ, ω ∂z

(12.32)

where Ai0 ðζ Þ is the derivative of the Airy function with h 2 respect ito ζ . 0 At the reflection point, ζ ¼ 0, Aið0Þ ¼ 1= 3 =3 Γð2=3Þ and Ai ð0Þ ¼ h 1 i 1= 3 =3 Γð1=3Þ (Abramowitz & Stegun, 1964). The electric field and magnetic 1=

1=

induction there are jEð0Þj  1:259ðωL=cÞ 6 E0 and jH ð0Þj  0:918ðc=ωLÞ 6 E0 , respectively. Note that as E swells, H decreases. The phase shift between the reflected and incident waves, from (12.30) and (12.31), is ðL ðL 4ωL π ω z 1=2 π ω π
¼2 φ¼ 1
dz
¼ 2 ηðzÞdz
: 3c 2 c L 2 c 2 0

(12.33)

0

The factors of π=2 in (12.33) are due to reflection from the critical density surface. We now consider the case including absorption. The dielectric permittivity can be written ϵ ðzÞ ¼ ðη þ iχ Þ2 ¼ 1
z=L þ iðα þ βz=LÞ ¼ a
bz=L, where α and β are real constants. Upon changing variable
ζ ðzÞ ¼

ω2 b c2 L

1=3



2=3 aL ωL ϵ ðzÞ ¼ ξ þ iϱ,
z ¼ b cb

(12.34)

which is now a complex quantity, while ξ and ϱ are real, (12.23) is written in the form of (12.26). The solution is of the form (12.27), but now ξ ≷ 0 instead of ζ ≷ 0. The asymptotic form is


   4 3=2 π E ¼ E þ þ E
¼ E 0 1 þ exp i ζ 0
i (12.35) ¼ E0 1 þ eiΨ , 3 2 with A¼

2 pffiffiffi 1=4 iΨ=2 πζ 0 e : 3

(12.36)

The quantity ζ 0 is ζ evaluated at z ¼ 0. In the case where there is no absorption, the discontinuity in the permittivity at z ¼ 0 allows reflection. This reflection is weak if jdϵ=dzj ¼ 1=L is small. However, if the medium is absorbing, then even for small jdϵ=dzj the reflection is weak only when the parameter α is small. When α is large, nearly total reflection will take place at the boundary.

636

Electromagnetic Wave-Matter Interactions

12.2.2 Reflectivity and Phase Shift The reflection coefficient from an absorbing layer is given by R ¼ exp ½ℑmðΨÞ, where, from (12.35), ζð0 pffiffiffi π 4 3=2 π ζ dζ
Ψ ¼ ζ0
¼ 2 2 2 3 0 aL=b
1=3 ζð0 ð ωL π ω π ¼2 ðη þ iχ Þdζ
¼ 2 ðη þ iχ Þdz
, cb 2 c 2

(12.37)

0

0

1=

since ζ ¼ 0 for z ¼ aL=b and dζ ¼
ðω2 b=c2 LÞ 3 dz: The upper limit of the integral is al=b ¼ ð1
iαÞL=ð1 þ iβÞ. For the moment, we set β ¼ 0 and α is a constant, then we find ω Ψ¼2 c

ð1
iα ð ÞL

ðη þ iχ Þdz


π 2 3

2 L0 2L 3 pffiffiffi pffiffiffi ð ð ω ω 2 3=2 5 2 3=2 5 π ¼ 2 4 ηðzÞdz
α L
2i 4 χ ðzÞdz þ α L
, c c 2 3 3 0

(12.38)

0

  1= since for z ¼ L þ iy, with α ¼ 4πσ=ω being constant, gives η ¼ χ ¼ 12 α þ Ly 2 . The phase shift of the reflected wave is φ ¼ ℜeðΨÞ. Using (12.38), we have 2L 3 pffiffiffi ð ω 2 3=2 5 π φ ¼ 2 4 ηðzÞdz
(12.39) α L
, c 2 3 0

and for the reflectivity, we find 2L 3 pffiffiffi ð ω 2 3=2 5 α L : log R ¼
2 4 χ ðzÞdz
c 3

(12.40)

0

If we now set α ¼ 0 and β is a constant, then ω Ψ¼2 c

ð1
iβÞL=ð1þβ2 Þ

ð 0

π ðη þ iχ Þdz
: 2

(12.41)

12.2 Propagation in Inhomogeneous Isotropic Medium

637

In general, to terms of order α2 , β2 , and αβ, the reflectivity is
 ðL ω 2 ω α þ βz=L pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dz: αþ β L¼ log R 
2 c 3 c 1
z=L

(12.42)

0

12.2.3 Geometrical Optics Approximation While the simple linear density profile is important, we seek a solution to (12.23) for an arbitrary function of the permittivity. One approximation method, the Wentzel-Kramers-Brillouin (WKB) method, is based on the geometrical optics approximation; it is also known as the quasiclassical method. We recognize that (12.23) is the analog to the quantum mechanical Schrödinger wave equation for a particle moving in a spatially dependent potential; the WKB method was originally developed in this context. As it is an approximate method, The WKB method gives good results if the properties of the medium change sufficiently slowly with distance. The condition for geometrical optics to be applicable is that the wavelength of the electromagnetic wave should be small in comparison with the characteristic dimensions of the problem, that is, λ  L. One measure might be the density scale height L ¼ ne =ðjdne =dzjÞ. The relation between geometrical optics and wave optics is that, for λ  L, any quantity ϕ that describes the wave field is given by a formula of the type ϕ ¼ ςeiψ , where the amplitude ς is a slowly varying function of the coordinates and time, and the phase ψ is a large quantity that is “almost linear” in the coordinates and the time. The time derivative of the phase gives the frequency of the wave, and space derivatives give the wave vector k. The governing equation for the electric field of the electromagnetic wave for normal incidence in homogeneous, isotropic media is given by (12.23) with ϵ ¼ ϵ ðωÞ. The solution is
pffiffiffi  ω ϵ z : (12.43) E ¼ E 0 exp i c If the medium is inhomogeneous, but ϵ ðω; zÞ varies slowly over the length of the electromagnetic wave, the propagation of the wave is similar to that for a homogeneous medium; that is, ð1=ϵ Þðdϵ=dzÞ  ð1=λÞ. The for the electric field is Ð psolution pffiffiffi ffiffiffi similar to (12.43), but ϵ z is replaced with ϵ dz. Thus, (12.43) is a first approximate solution to (12.23).

638

Electromagnetic Wave-Matter Interactions

A better approximation is to use ω

EðzÞ ¼ E0 ðzÞ exp i ψ ðzÞ , c

(12.44)

where E0 ðzÞ and ψ ðzÞ are slowly varying functions of the coordinates. Using this in (12.23) yields E0 00 þ i

 ω ω2  ð2Ψ0 E0 0 þ ψ 00 E 0 Þ þ 2 ϵ
ψ 0 2 E0 ¼ 0, c c

(12.45)

where the primes to E0 and ψ denote differentiation with respect to z. An approximate solution to this equation may be found by equating to zero the terms of each order (in ω=c) separately. This gives   ψ 00 ϵ
ψ 0 2 E0 ¼ 0 and E0 0 þ 0 E 0 ¼ 0: 2ψ

(12.46)

The first of these gives ϵ ¼ ψ 0 2 and upon integration gives the phase of the field Ðz pffiffiffiffiffiffiffiffi ψ¼ ϵ ðzÞdz, where z0 is some constant, while the second gives z0

pffiffiffiffiffi 1= E0 ðzÞ ¼ C= ψ 0 ¼ C ½ϵ ðzÞ
4 , where C is a constant. Thus, the solution to (12.23) in the geometrical optics approximation is 0 z 1 0 1 ð pffiffiffiffiffiffiffiffi ðz pffiffiffiffiffiffiffiffi Cþ ω C ω
ϵ ðzÞdzA þ ϵ ðzÞdzA, exp @i exp @
i E ðzÞ ¼ 1= 1= 4 4 c c ½ϵ ðzÞ ½ϵ ðzÞ z0

z0

(12.47) and depends upon the two arbitrary complex constants C þ and C
. The lower limits on the integrals are also constant. We see that in this approximation, two waves propagate in opposite directions. If the equations (12.46) are satisfied, all the terms in (12.45), except the first, are zero. Hence, (12.47) is an approximate solution if the first term of (12.45) is found to be small upon substituting in (12.46). Then the inequality jE 0 00 j 

ω 00 ω jΨ E0 j∽ jΨ 0 E0 0 j c c

(12.48)

should hold. Using the relations from (12.46), we find the condition for the geometric optics approximation to be valid ψ 000 3ψ 002 ω 00
(12.49) 2ψ 0 þ 4ψ 02  c jψ j:

12.2 Propagation in Inhomogeneous Isotropic Medium

Since ψ 0 ¼

pffiffiffi ϵ ¼ η þ iχ, the inequality holds if 1  02 0  =2 η þχ2 ω  2 2 c η þχ

639

(12.50)

and 1=

½η 002 þ χ 002  2 1= 2

1= 2

ðη2 þ χ 2 Þ ½η0 2 þ χ 0 2 



ω : c

(12.51)

With no absorption, the two inequalities (12.50) and (12.51) become c jη0 j c jη 00 j  1 and  1: ω η2 ω jηη0 j

(12.52)

It is thus verified, using these inequalities, that jE0 00 j 

ω2 02 ω2 ψ E ¼ ϵE 0 : 0 c2 c2

(12.53)

Further, from the first part of (12.52), dλ=dz  2π, since λ ¼ λ0 =η, where λ0 is the vacuum wavelength. The geometrical WKB solution breaks down in regions of small ϵ; for a nonabsorbing medium, the condition ðλ=ηÞðdη=dzÞ  1 is not applicable, as η ! 0 and λ ! ∞. Expression (12.47) can be written as a standing wave with a node where ϵ ¼ 0. In the absence of absorption, we have 0 z 1 ð 2 ω E ¼ pffiffiffi sin @ ηðzÞdzA, (12.54) η c 0

as measured from the point where ϵ ¼ 0. The factor of two accounts for the amplitude of the incident wave outside the layer, which is assumed to be unity. The normalization, similar to that used with (12.27) is 4π ω dη jAj2 ¼ 9 c dz 0

!1=3 :

(12.55)

For the linear layer, the square of (12.54) is "
3=2 # 4 dη 2ω jE j2 ¼ dη 1=2 dη Δz sin 2 Δz dz 0 3c dz dz 0

4 ω ¼ pffiffiffi dη ζ c dz 0

0

!1=3


sin

2

2 3=2 ζ 3




 9 2 2 2 3=2 p ffiffiffi ¼ : ζ jAj sin 3 π ζ

(12.56)

640

Electromagnetic Wave-Matter Interactions

The first three zeros of (12.56) are now β1 ¼ 2:811, β2 ¼ 4:461, and β3 ¼ 5:847. The first zero is approximately 20 percent of that for the exact solution, and the second zero is about 10 percent of its corresponding exact value. The ratio jE=Aj2 using (12.55) is shown in Figure 12.2. Examination of (12.56) (and Figure 12.2) shows that the intensity of the electromagnetic wave increases as it propagates toward higher density. This swelling of the wave is explained by noting that the energy flux must be conserved; that is, vg jEðzÞj2 ¼ cjE ð0Þj2 . The dispersion relation relates the group velocity to pffiffiffi the permittivity according to vg =c ¼ ϵ , and thus as ϵ decreases so does vg , and EðzÞ must increase to conserve the energy flux. The WKB method just outlined is only an approximation and must be used with caution since it is based on estimating and neglecting various terms in the original expression (12.23). A small term in that equation may nevertheless considerably affect the solution, and the only completely legitimate procedure is to neglect small terms in the solution, but not in the original equation. Another, more accurate, approach is to expand the E0 ðzÞ term of (12.44) in powers of c=ω and then develop the WKB solution. We will not pursue this here, but the details may be found in Ginzburg (1970). 12.2.4 Weak Reflection In a medium with no steep gradients of η, and where η ¼ 0, there are no strong reflections. Reflection is completely absent only in homogeneous media. In the case of weak reflection and no absorption (χ ¼ 0), the solution to (12.23) may be written as the sum of an incident and reflected wave 0 z 1 0 1 ð ðz E0 ω ω E ðzÞ ¼ pffiffiffiffiffiffiffiffi exp @i ηðzÞdzA þ E 1 exp @
i ηðzÞdzA, (12.57) c c ηðzÞ z0

z0

pffiffiffi wherejE 1 j  E 0 = η . E 0 is the incident amplitude and E 1 is the reflected amplitude, which is easily found using (12.57) in (12.23) 0 1
 ðz dE1 1 dη c d 1 dη 2ω ηðzÞdzA ¼ ΧðzÞ, (12.58) E1 ¼ i E0 exp @i þ 3= 2 2η dz ωη dz 4η dz c dz z0

where the term d E 1 =dz has been neglected. Integration of this equation gives 2

2

ðz pffiffiffiffiffiffiffiffi 1 ηðzÞX ðzÞdz E1 ðzÞ ¼ pffiffiffiffiffiffiffiffi ηðzÞ z1

ðz

¼i






0

ðz

1

c E0 1 d 1 dη 2ω ηðzÞdzAdz, exp @i pffiffiffi pffiffiffi 3= 2 4ω η η dz η dz c z1

z0

(12.59)

12.2 Propagation in Inhomogeneous Isotropic Medium

641

where z0 and z1 are arbitrary constants. Consider the case where the wave is incident from the negative z side. Then E 1 ðz ! ∞Þ ¼ 0 and we require z1 ¼ ∞, and dη=dz ¼ 0 for z ¼ z1 . In front of the inhomogeneous layer, dη=dz ¼ 0 also. After working through the integration of (12.59), the reflected wave is given by 0 1 ðz ð∞ E0 1 dη 2ω exp @i ηðzÞdzAdz: E 1 ðzÞ 
pffiffiffi (12.60) c η 2η dz z

z0

The reflected amplitude is the second term of (12.57) divided by the first term 0 1∞ 0 1 ð ðz ðz 2ω 1 dη 2ω ηðzÞdzA ηðzÞdzAdz: (12.61) exp @i R 
exp @
i c 2η dz c z0

z

z0

An important case is that of reflection from a discontinuity of the derivative of dη=dz. This corresponds to a large variation in the derivative over a distance small compared to c=ω ¼ λ=2π, and all quantities in (12.59) may be considered constant except d 2 η=dz2 . If the discontinuity is at z0 ¼ 0, the reflectivity is 0 1



 ðz c 1 dη dη 2ω exp @
i
ηðzÞdzA, (12.62) R¼i 2 4ω η ð0Þ dz 2 dz 1 c 0

and thus c 1 jRj ¼ 4ω η2 ð0Þ




dη dz




 dη
, dz 1 2

(12.63)

where ðdη=dzÞ1, 2 are the values of the derivative on the two sides of the discontinuity. The lower limit of the integral in (12.62) was set to zero to fix the phase factor, since the argument of the exponential is just the variation in the phase in going from the point z to the discontinuity at z ¼ 0 and back to z. In this approximation, jRj  1. Equation (12.63) may also be obtained by using (12.57) on both sides of the discontinuity. In medium (1), the incident and reflected waves are 0 z 1 0 1 ð ðz E0 ω c ω 1 Eð1Þ ¼ pffiffiffi exp @i ηðzÞdzA þ pffiffiffi exp @
i ηðzÞdzA, (12.64) c c η η z0

z0

and in medium (2), the transmitted wave is 0 z 1 ð c2 ω ηðzÞdzA: Eð2Þ ¼ pffiffiffi exp @i c η z0

(12.65)

642

Electromagnetic Wave-Matter Interactions

Reflection from a discontinuity of the derivative dη=dz is similar to that from a discontinuity of η itself. In both cases, the discontinuity may be regarded as sharp, and its structure does not affect the reflectivity if the thickness of the discontinuity is small compared to c=ω. That is, reflection occurs at a sharp discontinuity such that the transition layer where dη=dz or η changes is small compared to the wavelength. Of course, the discontinuity in η may occur at the boundary between two homogeneous media, which we address later in this chapter. If there is a discontinuity in dη=dz, at least one of the two media on either side is inhomogeneous, and the reflection is not localized at the discontinuity alone. The phase shift between the incident and reflected waves is given by the WKB solution if we subtract π=2 to account for reflection at the critical density. The result is identical to (12.39) if α ¼ 0. As promised at the beginning of this chapter, we now get into the messy part.

12.2.5 Oblique Incidence Consider, as before, a plane-parallel medium with the interface to the vacuum at z ¼ 0, as shown in Figure 12.1. The incident wave vector k lies in the y–z plane. For TE polarization (S-polarization), the electric vector is perpendicular to the plane of incidence and the wave is pure transverse, so r  E ¼ 0, and (12.17) becomes ∂2 Ex ∂2 Ex ω2 þ 2 þ 2 ϵ ðω; zÞE x ¼ 0: ∂y2 ∂z c

(12.66)

The incident wave is described by E ¼ E0 exp ðik0  xÞ where the components of the wave vector are k0 ¼ ðω=cÞ½0; sin θ0 ; cos θ0 . Since only a transverse wave exists and propagates as E x ðzÞ ¼ F ðzÞ exp ð iky sin θÞ, the wave equation becomes (

2 )   d2 F dðkαÞ dF d 2 ðkαÞ 2 2 2 d ðkαÞ F ¼ 0, þ k 1
α F ¼ ∓2i þy y þ ∓iy dz2 dz dz dz2 dz pffiffiffiffiffiffiffiffi where αðzÞ ¼ sin θðzÞ. Recall, kðzÞ ¼ ðω=cÞ ϵ ðzÞ. Thus   ∂ 2 F ω2 2 þ ϵ ð z Þ 1
α ð z Þ F ¼ 0, ∂z2 c2

(12.67)

(12.68)

and d ðkαÞ=dz ¼ 0. Also, since the y coordinate is arbitrary, the right-hand side of pffiffiffi (12.67) mustp beffiffiffiffiffiffiffiffi zero. Hence, kα ¼ ðω=cÞ ϵ α = k0 α0 is constant. In the absence of absorption, ϵ ðzÞ ¼ ηðzÞ and ηðzÞ sin θðzÞ ¼ sin θ0 ¼ α0 , which is Snell’s law. Expression (12.68) has the same form as (12.23), with ðω2 =c2 Þϵ ðzÞ replaced by

12.2 Propagation in Inhomogeneous Isotropic Medium

  ω2 ω2 2 ϵ ð z Þ 1
α ð z Þ ¼ 2 ϵ ðzÞ cos 2 θ0 : c2 c

643

(12.69)

Now ϵ ðzÞ cos 2 θ0 ¼ ϵ ðzÞ
ϵ ð0Þα20 and the solution of (12.68) reduces to the solution of (12.23). Expression 1  (12.68) is transformed into (12.26) by using the variable    2 2 =3 2 ζ ¼ ðω =c LÞ L 1
ϵ ð0Þα0
z . The reflection point z0 is defined in (12.68) by the term ϵ ðzÞ½1
α2 ðzÞ ¼ ϵ ðzÞ
ϵ ð0Þα20 ¼ 0. In the nonabsorbing case, ϵ ðzÞ ¼ 1
ω2pe ðzÞ=ω2 , then ωpe ¼ ω cos θ0 and the electron density at the reflection point is ne ¼ ncr cos 2 θ0 , with critical density ncr ¼ me ω2 =4πe2 . Because (12.23) and (12.68) are fundamentally the same, the solution to (12.68) can be written from (12.47) 2 0z 13 ð qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p ffiffiffiffiffiffiffiffi E x0 ω ϵ ðzÞ
ϵ ð0Þα20 dz þ εð0Þα0 yA5, Ex ¼  exp 4 i @  1= 4 c ϵ ðzÞ
ϵ ð0Þα2 0

0

(12.70) p ffiffiffiffiffiffiffiffi ffi with ϵ ð0Þα20 being a constant; we assume ϵ ð0Þ ¼ ηð0Þ ¼ 1 for the vacuum. Besides depending on y, expression (12.70) differs from (12.47) for normal incidence in that pffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffi  1= ϵ ðzÞ is replaced by ϵ ðzÞ
ϵ ð0Þα20 2 ¼ ϵ ðzÞ cos θðzÞ. The wave is reflected from the point where ϵ ðz0 Þ ¼ sin 2 θ0 ¼ α20 . For the approximation of geometrical optics to be valid for oblique incidence, the first part of (12.52) should be replaced by c jη0 j c dðη cos θÞ=dz  1: (12.71) ! 2 ω η2 ω η cos 2 θ This condition is violated when cos θ ! 0 near the reflection point, and the geometrical optics approximation is not valid. In the geometrical optics approximation, we work with “rays” as opposed to “waves.” In a homogeneous, isotropic medium, the direction of the ray is the same as the normal to the wave surface. For the nonabsorbing case, the direction of the normal is found by differentiating the exponential factor in (12.70). The components of the wave vector are k ¼ ðω=cÞ½0; ηðzÞ sin θ; ηðzÞ cos θ. At any point, the unit vector along the normal to the wave front is k=k. In a quasihomogeneous isotropic medium, this vector is also tangential to the ray path. Referring to Figure 12.1, the ray path is found from dy k y sin θ α0 ¼ ¼ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2ffi : dz kz cos θ η2 ðzÞ
α0 Integration of (12.72) gives the ray path.

(12.72)

644

Electromagnetic Wave-Matter Interactions

Near the reflection point, the geometrical optics approximation is invalid and the direction of the normal to the wave surface does not coincide with the direction of motion of the “center of gravity” of the wave. For the linear layer, and for any layer far from the reflection point, there is total internal reflection. From Section 12.2.2, the phase change of the wave upon reflection is ω φ¼2 c ðz0

ðz0 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ω π η2 ðzÞ
α20 dz þ ðy2
y1 Þα0
c 2 0

π ¼ 2 kz dz þ ky ðy2
y1 Þ
, 2

(12.73)

0

where y2 and y1 are the points on the y axis (the surface of the layer) for which the phase difference is determined. The distance between the two points is found from the integration of (12.72). For TM polarization (P-polarization), the electric vector is in the plane of incidence and the two components of the electric field are ∂2 Ey ∂2 Ey ω2 ∂ þ 2 þ 2 ϵ ðω; zÞE y
r  E ¼ 0 2 ∂y ∂y ∂z c

(12.74)

∂2 E z ∂2 E z ω2 ∂ þ 2 þ 2 ϵ ðω; zÞEz
r  E ¼ 0: 2 ∂z ∂y ∂z c

(12.75)

and

The presence of the divergence of the electric field in these two equations indicates that the wave is no longer pure transverse. The components Ey and E z are not independent and depend upon the coordinates. In contrast to the TE case, the direction of the wave vector changes with increasing depth, and E must also rotate. From Figure 12.1, we see that there is a longitudinal component. The divergence term is found from Poisson’s equation
 4π (12.76) r  D
i J ¼ r  ðϵEÞ ¼ 0, ω from which E  rϵ þ ϵr  E ¼ E z and thus

dϵ þ ϵr  E ¼ 0, dz

 d ð log ϵ Þ rðr  EÞ ¼
r Ez : dz

(12.77)



(12.78)

12.2 Propagation in Inhomogeneous Isotropic Medium

645

We note that when ϵ ¼ 0, a resonance is achieved, where ωpe ¼ ω. Thus, the oscillations of the electron between layers of slightly differing density create a charge density fluctuation δne ¼ ne ðx þ xosc Þ
ne ðxÞ  xosc rne ,

(12.79)

where xosc ¼ eE=me ω2 is the amplitude of the oscillation. Hence an electron plasma wave is excited at the resonance point, which is located at the critical density; electron plasma waves were introduced in Section 3.7 and will be discussed in further detail later. The important points here are that an obliquely incident TM wave reflects at a density less than the critical density, and its electric field penetrates into the critical density, thereupon exciting the resonance. Clearly, there is little resonance action if the wave is near-normal incident, because there is essentially no Ez at the critical density, even though the reflection point and critical density point are nearly coincident. Further, there is little resonance taking place for highly oblique angles of incidence, since the turning point z0 is far from the resonance point. Expression (12.78) can be used in the wave equations (12.74) and (12.75). Considering only the second equation, (12.75), we have

 ∂2 E z ∂2 E z ω2 ∂ d ð log ϵ Þ þ þ ϵ ð ω; z ÞE þ E ¼ 0: (12.80) z z ∂z dz ∂y2 ∂z2 c2 Adopting the solution E z ¼ αðzÞF z ðzÞe ikðzÞαðzÞy ,

(12.81)

and the fact that ð2=αÞdα=dz ¼
dð log ϵ Þ=dz because dðkαÞ=dz ¼ 0, we then have (

)   1 d2 ð log ϵ Þ 1 dð log ϵ Þ 2 d2 Fz ω2 2 F z ¼ 0: (12.82) þ ϵ ðzÞ 1
α ðzÞ þ
2 dz2 4 dz dz2 c2 Expression (12.82) differs from (12.68) only by terms involving derivatives of ϵ. In the approximation of geometrical optics, these terms may be neglected and the two expressions are equivalent. Again, the geometrical optics approximation is not valid near the reflection point z0 , where the nature of the field is different for the different polarizations. For TM polarization, (12.82) is equivalent to (12.23) if an effective dielectric permittivity " # 2 2 2 2 c d ϵ=dz 3 ð dϵ=dz Þ (12.83) ϵ eff ðzÞ ¼ ϵ ðzÞ
ϵ ð0Þα20 þ 2
4 ω 2ϵ ϵ2 is used. For TE polarization, the effective permittivity is ϵ eff ðzÞ ¼ ϵ ðzÞ
ϵ ð0Þα20 .

646

Electromagnetic Wave-Matter Interactions

Expression (12.83) tends to infinity as ϵ ! 0. Near the reflection point, the difference between ϵ eff and ϵ ðzÞ
α20 is especially great for small values of α0 , therefore ϵ ðzÞ ! 0 as sin θ0 ! 0. For a linear layer, the coefficient of F z in (12.82) becomes ϵ eff ðzÞ ¼ ϵ ðzÞ
ϵ ð0Þα20


3c2 ðdϵ=dzÞ2 , 4ω2 ϵ2

(12.84)

which is zero when z < z0 . Thus, the wave begins to be damped somewhat before the point where α ¼ sin θ ¼ 1. Working with (12.74) and (12.75) is somewhat cumbersome. It is advantageous to work from (12.22), which for TM polarization becomes ∂2 H x ∂2 H x 1 dϵ ∂H x ω2 þ 2
þ 2 H x ¼ 0: ϵ dz ∂z ∂y2 ∂z c  ω pffiffiffi  Substituting H x ¼ GðzÞ exp i c ϵ y sin θ into (12.85), we obtain  d2 G 1 dϵ dG ω2 
þ 2 ϵ ðzÞ
α20 G ¼ 0: 2 dz ϵ dz dz c

(12.85)

(12.86)

Once GðzÞ is determined, H x is known, and then from Ampere’s law the electric field components are Ey ¼ i

c ∂H x c ∂H x and E z ¼
i : ωϵ ∂z ωϵ ∂y

(12.87)

If the absorption varies slowly along the density gradient, that is, νeff depends only slightly on z, then the imaginary part of the permittivity may be taken as constant and thus ω ¼ ωpe . For a linear layer, ϵ ðzÞ ¼
ðz=LÞ þ iðνeff =ωÞ, where L is the layer thickness; thus, ϵ ðzÞ < 0 for z > 0. Then (12.86) becomes i d2 G ð1=LÞ dG ω2 h z νeff 2 þ þ i
α

0 G ¼ 0: dz2 ðz=LÞ þ iðνeff =ωÞ dz L c2 ω

(12.88)

Introducing the new variables ζ ¼ ðz=LÞ þ iðνeff =ωÞ and ρ ¼ ωL=c ¼ 2πL=λ0 , with L ¼ jdϵ=dzj
1 , and where λ0 is the vacuum wavelength, (12.88) may be written   d2 G 1 dG þ ρ2
ζ
α20 G ¼ 0:
2 ζ dζ dζ

(12.89)

When absorption is taken into account, the “reflection” point ζ ¼
α20 corresponds to a complex z.

12.2 Propagation in Inhomogeneous Isotropic Medium

647

Examination of (12.89) at the point where ϵ ðzÞ ¼ 0, reveals the solution of that equation has a nonzero value for Gðζ ¼ 0Þ, and the component of the electric field in the direction of the density gradient, using the second part of (12.87), is Ez ¼

α0 GðzÞeiωα0 y=c e
iωt , ϵ ðzÞ

(12.90)

which is infinite at that point. The E y component has a logarithmic singularity. When absorption is taken into account, the maximum field at the resonance point is jE z j ¼ ðω=cÞ sin θ0 jGð0Þj, which for small collision frequency can be quite large. For oblique incidence, the behavior of the electric field is essentially the same for both polarizations up to the reflection point where ηðz0 Þ ¼ sin θ0 . For TE polarization, Ex beyond the reflection point decays exponentially. However, for TM polarization, E y and E z begin to increase near the critical point where ϵ ¼ 0 and will become infinite at z ¼ L if absorption is neglected. Clearly, at normal incidence, the singularity disappears, and for a sufficiently large angle of incidence, and if there is any absorption, the fields E y and Ez remain finite. For increasing angles of incidence, the distance between the reflection point and the critical point increases; that is, the electric field cannot penetrate to L. The spatial distribution of the squared modulus of the electric field into an inhomogeneous layer for oblique incidence is shown in Figure 12.3, for both TE and TM polarizations. If the medium is nonabsorbing and the electromagnetic wave is normally incident (θ0 ¼ 0), total reflection occurs and a standing wave is formed. The amplitude of the wave is oscillatory from the entry point to the reflection point and then decreases exponentially farther into the layer where ϵ < 0. The oscillatory part exhibits swelling, which is also seen in Figure 12.2. Faraday’s law requires H to decrease as E swells. The maximum field can be estimated, for a linear density gradient, by taking the value of the magnetic intensity at the turning point multiplied by an exponential decay from the turning point to the critical density. Since there is little difference in the standing-wave structure at the turning point for the fields between TE and TM polarization, we can use the Airy solution for TE polarization, and we find using (12.32) c 1=6   H x L cos 2 θ0  0:918 E0 , ωL

(12.91)

where E 0 is the value of the electric field in vacuum. The decay of the field beyond the turning point is estimated by 1 β¼ c

ðL L cos 2 θ0

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ω2pe
ω2 cos 2 θ0 dz,

(12.92)

648

Electromagnetic Wave-Matter Interactions

Figure 12.3. An electromagnetic wave obliquely incident on an inhomogeneous layer has a field distribution that is essentially the same in the underdense layer for both TE and TM polarizations. There is an added feature for TM polarization near the resonance point, shown by the dashed line. The TE polarization is labeled jEx j2 and the TM polarization jE z j2 .

which for the linear density profile is β ¼ ð2ωL=3cÞ sin 3 θ0 . We then obtain
 c 1=6 2ωL 3 H x ðLÞ  0:918E 0 exp
(12.93) sin θ0 : ωL 3c The electric field at the resonance point is found from Ampere’s law Ez ¼ ðH x α0 =ϵ Þ, thus E0 E z ðLÞ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi φðτ Þ, 2π ðωL=cÞ

(12.94) 1=

where the resonance function is φðτ Þ  2:3τ exp ð
2τ 3 =3Þ, with τ ¼ ðωL=cÞ 3 α0 (Kruer, 1988). This field vanishes as τ ! 0, as it should, and becomes small for large τ. Between these two limits there is an angle of incidence that yields the maximum E z ; this optimum angle is for τ  0:8. We compare this simple estimate for φðτ Þ, with the result obtained by numerically integrating the wave equation (12.88) (Denisov, 1957), as shown in Figure 12.4.

649

12.2 Propagation in Inhomogeneous Isotropic Medium

Figure 12.4. Resonance function φðτ Þ (solid line) and the approximation (dashed line).

The magnitude of E z is determined by the collision frequency and the function α0 jGð0Þj. For normal incidence, α0 jGð0Þj ¼ 0; as the angle of incidence increases, it reaches a maximum, and for still higher angles of incidence it decreases. This dependence is expressed by
 1=
 ωL
2 Lνeff α0 jGð0Þj ¼ 2π exp
φðτ Þ: c c

(12.95)

We then write (12.90) as Ez ¼

α0 jGð0Þj iωα0 y=c
iωt e , e ζ

(12.96)

with ζ ¼ ðz=LÞ þ iðνeff =ωÞ. The nature of the singularity in (12.85), when ϵ ¼ 0, is somewhat unphysical. The rapid increase in the electric field near this point makes it necessary to take account of the spatial dispersion; that is, it is not possible to use only the local properties of the permittivity of the medium. Away from this point, in the absence of collisions, spatial dispersion is negligible, since the path an electron traverses in one period is small compared with the characteristic dimension of the inhomogeneity of the field. However, near the resonance point, where the electric field is

650

Electromagnetic Wave-Matter Interactions

particularly strong, the electron acquires a large velocity and the condition for neglecting the spatial dispersion is violated. As a result of induction, DðzÞ differs from ϵEðzÞ and at the point where ϵ ¼ 0, DðzÞ is not zero, but is determined by the electric field at a point whose distance is of the order of the amplitude of the oscillations of the electron. Hence, the denominator in the second term of (12.88) is not zero when ϵ ¼ 0 and is replaced by an integral of the z-directed velocity of the electron, vz ∽ϵEz =me ω, which in a sense is the oscillatory velocity. Since this function appears in the denominator of expressions for Ez , the problem is no longer linear. This is reasonable, since the motion of the electron in the field that is harmonic in time but varies rapidly in space is not harmonic. The nonlinear effect becomes important if the integral of vz is comparable to νeff =ω when absorption is taken into account. When absorption is taken into account but spatial dispersion is neglected, the previously stated condition, that the path, 2πv=ω, traversed by the electron should be small, is insufficient. This statement is equivalent to the requirement that vη  c. There is now an additional requirement when absorption is considered, that vωχ=c  νeff (which is interpreted as the mean free path) must be small compared to the distance c=ωχ, over which the wave amplitude varies appreciably, owing to absorption. In fact this corresponds to the theory of the normal skin effect in metals. In an inhomogeneous medium, the conditions for spatial dispersion to be negligible are more stringent than for a homogeneous medium. This is because the amplitude of the field in an inhomogeneous medium is likely to vary considerably more than in a homogeneous medium even when absorption is absent. There is yet one other effect that causes the field at the resonance point, where ϵ ¼ 0, to remain finite. For TM polarization, we have already seen that a portion of the electric field is converted into a plasma wave, because at that point ω ¼ ωpe . At h i
1 the resonance point, we see from (12.88), jE z j2 ∽ ðz=LÞ2 þ ðνeff =ωÞ2 . The physical explanation for the appearance of plasma waves, when transverse electromagnetic waves are incident on the medium, is that in an inhomogeneous medium the incident wave is, in general, no longer purely transverse, since 1 Ez dϵ 6¼ 0: r  E ¼
E  rϵ ¼
ϵ ϵ dz

(12.97)

Poisson’s equation gives r  E ¼ 4πeΔne , where Δne is the departure of the electron density from the equilibrium value. Hence, when E z 6¼ 0 and dϵ=dz 6¼ 0, charges appear in the wave. At the point where ϵ ¼ 0, the density of these charges fluctuates with plasma frequency ωpe . Thus, a wave with Ez 6¼ 0 causes plasma oscillations whose amplitude increases as the resonance point is approached. These local oscillations are not independent, because any change in

12.2 Propagation in Inhomogeneous Isotropic Medium

651

electron density in one part of the medium is transmitted to a neighboring region through collisions and results in the formation of plasma waves. Thus, allowance for thermal motion removes the singularity. Ultimately, the energy of the plasma waves goes to heating the plasma. We shall discuss heating by plasma waves and magnetic field generation later in this chapter. The effect of interaction between the transverse and plasma waves is of additional interest when the problem is stated differently. It leads to the possibility of the transformation of plasma waves into electromagnetic waves in an inhomogeneous medium; an example of this is found in the solar corona.

12.2.6 Ponderomotive Force and Momentum Deposition An intense electromagnetic field incident on an inhomogeneous isotropic plasma layer can lead to a deconfining force that is directed away from regions of high particle density. This force arises from the general tendency of any nonuniform dielectric fluid to move toward regions of high dielectric permittivity in the presence of an electric field. This electromagnetic force is known as the ponderomotive force and manifests itself as a pressure. The relation between the volume force and the total stress tensor, discussed in Chapter 6, is F ¼ r  T, with T ¼
PIþσ, where P is the hydrostatic pressure and I is the unit tensor. The viscous stress tensor σ has contributions from a number of physical effects, including the presence of electromagnetic energy; other possible contributions are from viscosity (see Section 6.6) and radiation pressure. From Section 3.12, the energy density of the electromagnetic field is ur ¼

1 ðE  D þ H  BÞ: 8π

(12.98)

Setting aside for the moment the magnetic portion of (12.98), an incremental change in the energy density is
 1 2 1 1 1 1 (12.99) δur ¼ D δ þ E  δD ¼
E 2 δϵ þ E  δD: 8π ϵ 4π 8π 4π Now E  δD ¼
rΦ  δD ¼ Φδr  D ¼ 4π Φδρe , having used Poisson’s equation r  D ¼ 4πρe , where ρe is the charge density. The rate of change of the energy density is then dur ∂ρ 1 ∂ϵ ¼ Φ e
E2 : dt ∂t 8π ∂t

(12.100)

The quantities ∂ρe =∂t and ∂ϵ=∂t are determined from separate continuity equations.

652

Electromagnetic Wave-Matter Interactions

The total derivative of a particular quantity, when evaluated so that the observation point for this derivative moves with a specific volume element in a velocity field, is the substantive derivative. As discussed in Section 6.1, the total derivative of the permittivity is related to the partial derivatives and to the velocity u, moving with the dielectric, according to Dϵ ∂ϵ ¼ rϵ  u þ : Dt ∂t

(12.101)

(We must be a bit careful here as we are using ur for the energy density and u for a velocity.) The permittivity depends only on the mass density. Then Dϵ=Dt ¼ ðdϵ=dρÞðDρ=Dt Þ and the factor Dρ=Dt is evaluated by another equation of continuity. We then have
 Dϵ dϵ ∂ρ dϵ dϵ ¼ þ rϵ  u ¼ ½rρ  u
r  ð ρuÞ ¼
ρr  u: (12.102) Dt dρ ∂t dρ dρ Therefore, ∂ϵ dϵ ¼
ρr  u
rϵ  u, ∂t dρ

(12.103)

and now (12.100) can be written
 dur E2 dϵ ¼
Φr  ð ρe uÞ þ ρ r  u þ rϵ  u : dρ dt 8π

(12.104)

We need to bring this equation into the form of a dot product of an expression with the velocity by integrating by parts over a small volume. The first term on the righthand side is then
Φr  ð ρe uÞ ¼ ρe rΦ  u, and the first portion of the second term becomes

 E2 dϵ 1 2 dϵ  u: ρ ru ¼
r E ρ dρ 8π dρ 8π Expression (12.104) therefore becomes


 dur 1 1 2 2 dϵ ¼
ρe E
r E ρ þ E rϵ  u: 8π dρ 8π dt

(12.105)

(12.106)

(12.107)

The rate of change of the electromagnetic energy density is related to the force according to F  u ¼
du=dt, and this force is related to the viscous stress by

12.2 Propagation in Inhomogeneous Isotropic Medium

653

F ¼ r  σ. We then find the electric component of the stress tensor, in component form, to be

 E2 dϵ ϵ ðeÞ σ ij ¼
(12.108) δij þ Ei Ej : ϵ
ρ dρ 4π 8π In a similar fashion, the magnetic portion is ðmÞ

σ ij ¼


H2 1 δij þ H i H j , 4π 8π

(12.109)

using our assumption that μ ¼ 1 everywhere. Now the permittivity for a collisionless plasma is ϵ ¼ 1
ω2pe =ω2 ∽1
n=ncr ∽1
aρ, thus ρðdϵ=dρÞ ¼ ϵ
1. The viscous stress tensor is thus

 1 1 ϵE i E j þ H i H j
ðE k E k þ H k H k Þδij : σ ij ¼ (12.110) 4π 2 This expression can also be written as i 1 h M σ ij ¼ σ ij þ ðϵ
1ÞEi Ej , 4π where Maxwell’s stress tensor in Cartesian coordinates is ! 2 E2x
E 2y
E2z 1 ϵE x Ey þ H x H y 6 2 2 2 6 2 þH x
H y
H z 6 ! 6 E 2y
E2z
E 2x 1 6 σM ¼ 6 ϵE x E y þ H x H y 6 2 þH 2y
H 2z
H 2x 6 6 1 4 ϵE x E z þ H x H z ϵE y Ez þ H y H z 2

(12.111)

3 ϵEx E z þ H x H z ϵEy E z þ H y H z E 2z
E2x
E 2y þH 2z
H 2x
H 2y

7 7 7 7 7 7: 7 !7 7 5

(12.112) But there is a problem. The divergence of Maxwell’s stress tensor yields a force that does not vanish in the absence of charges and currents, and this violates a basic tenant of the theory of special relativity that states that no measurement can be devised which can determine the velocity or other properties of the ether. Therefore, a correction is needed. Taking the divergence of (12.110) yields
 ∂σ ij 1 ∂Dj ∂E i 1 2 ∂ϵ ∂Ej ∂H j ∂H i ∂H j þH i : ¼ þ Dj
E
Dj þ Hj
Hj Ei 4π ∂xj ∂xj ∂xj 2 ∂xi ∂xi ∂xj ∂xj ∂xi (12.113)

654

Electromagnetic Wave-Matter Interactions

By virtue of Maxwell’s equations, the first term in parentheses is 4πρe E, the second and fourth terms combine to give
D  ðr  EÞ ¼ 1c D  ð∂H=∂t Þ, the fifth term is Hr  H ¼ 0, and the sixth and seventh terms together are
H  ðr  HÞ ¼
4πc H  J
1c H  ð∂D=∂t Þ. The force equation thus reads F¼

∂σ ij 1 1 1 ∂ ðE  HÞ: ¼ ρE
E 2 rϵ
H  J
8π c 4πc ∂t ∂xj

(12.114)

The first three terms on the right-hand side are the forces acting on material bodies in an electromagnetic field, and the last term is proportional to the rate of change of Poynting’s vector (the energy-flux density) S ¼ cðE  HÞ=4π. The material force terms vanish in the absence of real material, but the last term does not. This suggests the idea of there being a force in the vacuum, which is consistent with the ether theory in which the vacuum possesses various mechanical properties that enable it to transmit elastic waves and also to sustain body forces. The general requirements of relativistic invariance have the result that the energy-flux density must be the same, apart from a factor of c2 , as the space density of the field momentum ðE  HÞ=4πc. This expression must be used in determining the forces on a dielectric in a variable electromagnetic field. Thus, the rate of change of the field momentum per unit volume must be subtracted to give the (nonlinear) ponderomotive force density F ¼ rσ


1 ∂ 1 ∂S ðE  HÞ ¼ r  σ
2 : 4πc ∂t c ∂t

Using (12.111) in (12.115), we find h i 1 1 ∂ F ¼ r  σ M
ðϵ
1ÞEE
ðE  HÞ: 4π 4πc ∂t

(12.115)

(12.116)

Consider now normal incidence on the linear layer. Expression (12.116) gives

 1 ∂ 2 1 ∂ Fz ¼
E x þ H 2y
Ex H y : (12.117) 8π ∂z 4πc ∂t Multiplying both the electric field and magnetic intensity by e
iωt and averaging over several oscillations of the wave yields

1 ∂ 2 2 Ex þ Hy : (12.118) hF z i ¼
8π ∂z Using (12.27) and (12.32) for the linear density gradient of length L gives (Lindl & Kaw, 1971)

12.2 Propagation in Inhomogeneous Isotropic Medium

655


 ω 2 ωL νeff hF z i ¼ E 0 exp
2 f½ℜeðAiÞℜeðAi0 Þ þ ℑmðAiÞℑmðAi0 Þ c c ω c 2=3 þ fℜeðAi0 Þ½ℜeðζ ÞℜeðAiÞ
ℑmðζ ÞℑmðAiÞ ωL (12.119) þℑmðAi0 Þ½ℑmðζ ÞℜeðAiÞ þ ℜeðζ ÞℑmðAiÞgg, 0

where Ai and Ai are the Airy function and its derivative with respect to the 2= argument; the argument is ζ ¼ ðωL=cÞ 3 ðz=LÞ. Figure 12.5 plots (12.119) for a collisionless plasma. The positive force acts in the direction of the incident wave and is a confining force, whereas a negative force does the opposite. The force to the right of the first zero, near ζ ¼ 1, is pushing the plasma toward higher density, while the force to the left is oscillatory in space and leads to ponderomotive bunching. This may lead to striations in the medium, and these in turn may have an influence on various instabilities. Averaging the force to the left of the first zero gives a net outward-directed force, with the largest contribution coming from between the first and second zeros, as the remainder effectively averages to zero. The combination of the inward and outward forces near the first zero leads to a modification to the density profile from its linear form; we will discuss this in more depth later in this chapter. If collisions are included, the force at the first negative 2= maximum decreases as ðωL=cÞ 3 ðνeff =ωÞ increases. We note in passing that the 0.20

F/[w/c)E02]

0.12

0.04

–0.04

–0.12

–0.20 10.0

8.0

6.0

4.0 z

2.0

0.0

–2.0

Figure 12.5. Force profile in the neighborhood of the reflection point, ζ ¼ 0, for normal incidence.

656

Electromagnetic Wave-Matter Interactions

WKB solution of Section 12.2.3 cannot give this oscillatory structure and in fact predicts an infinite (negative) force at ζ ¼ 0. The pressure is the argument of the gradient term in (12.118). Noting that pffiffiffi H ¼ cðk  EÞ=ω and so H y ¼ ϵ E x , the nonlinear ponderomotive pressure is P¼

1 1 ϵþ1 2 pffiffiffi I, ð1 þ ϵ ÞE x ¼ 8π 2c ϵ

(12.120)

pffiffiffi 2 where I ¼ ðc=4π Þ ϵ E 0 is the laser intensity, with E 0 being the free-space value of E. While we are on the subject of normal incidence, we might ask how the preceding development compares to the radiation pressure in a cavity (hohlraum) as discussed in Chapter 8. If we specialize to the case of an isotropic homogeneous radiation field (unpolarized and of equal intensity in all directions), we see that by symmetry the only nonvanishing component of (12.116) is that normal to the cavity’s wall. For the moment, we take the x-axis to be normal to the surface under consideration (to try to avoid confusion with the linear layer geometry), and the force is

  ∂ ϵ ∂E 2x ∂ Fx ¼ Ex E y þ ðE x E z Þ þ 4π ∂x ∂y ∂z

2     1 ∂H x ∂ ∂ 1 ∂ 2 þ H x H y þ ðH x H z Þ
ϵE þ H 2 : (12.121) þ 4π ∂x ∂y ∂z 8π ∂x Since the different components of the electric (and magnetic) field are uncorrelated, the time averages of all the cross terms vanish and we are left with ! 2 2 2 2 1 ∂E x ϵ ∂E ∂H x 1 ∂H : (12.122) ϵ
þ
hF x i ¼ 4π 2 ∂x ∂x 2 ∂x ∂x 2

2

2

2

Because the fields are oriented completely at random, E x ¼ E y ¼ E z ¼ E =3 and 2 2 similarly, H x ¼ H =3. Therefore, (12.122) becomes

1 ∂ 2 ∂ ur 2 ϵE þ H ¼
, (12.123) hF x i ¼
24π ∂x ∂x 12π where ur is the energy density of the radiation. Integrating over 4π gives the familiar relation of the radiation pressure being equal to one-third of the energy density. Turning now to oblique incidence, for TE polarization (S-polarization), the nonlinear force is

1 ∂ 2 2 2 Ex
H z þ Hy , (12.124) hF z iTE ¼
8π ∂z which yields the nonlinear pressure PTE ¼

1 2 ð1 þ ϵ cos 2θ0 ÞE x , 8π

(12.125)

12.2 Propagation in Inhomogeneous Isotropic Medium

657

where H y ¼ k z Ex =k0 and H z ¼
ky E x =k0 . The transverse wave numbers are pffiffiffi pffiffiffi ky ¼ k 0 ϵ sin θ0 and k z ¼ k0 ϵ cos θ0 , where k0 ¼ ω=c. The TE pressure is always less than that for normal incidence, so (12.125) reduces to (12.120) for normal incidence. For TM polarization (P-polarization), we have i  1 ∂ 1 ∂h 2 2 2 ^ z, ^ ϵE z E y n y þ ð2ϵ
1ÞE z
E y
H x n FTM ¼ (12.126) 4π ∂z 8π ∂z ^ z are unit vectors in the plane of incidence. Multiplying (12.126) by ^ y and n where n
iωt e , taking the real part and averaging over several periods, gives i 1 ∂h 2 2 2 ð2ϵ
1ÞE z
E y
H x : (12.127) hF z iTM ¼ 8π ∂z Now Ey ¼
k z H x =k0 ϵ and Ez ¼ k y H x =k0 ϵ, and the pressure is
 1 1 2 2 1 þ
2 sin θ0 H x : PTM ¼ 8π ϵ

(12.128)

The pressure can be greater than that for TE polarization and is limited only by the magnitude of the imaginary part of ϵ. The enhanced force arises because of the excitation of large longitudinal fields in the region where ϵ  0. This also explains the optimization of the force with the angle of incidence. For in the case of normal incidence, the wave stays transverse all the way to the reflection point and so there is no such enhancement. Again, if the angle of incidence is too large, the region where ϵ  0 is far from the reflection point ϵ ¼ sin 2 θ0 , and the wave amplitude decays before reaching the resonant region. 2 To lowest order, (12.127) is determined by only the E z term. Then using (12.96), we arrive at   exp
2 Lνceff 2 z ωL φ ðτ Þ h  hF z iTM ¼ (12.129)  2 i2 : 2 2 16π c L2 Lz þ νωeff pffiffiffi   This force has a maximum at z ¼ Lνeff = 3ω , and its value is   pffiffiffiffiffi 27 ω exp
2 Lνceff 2 2 (12.130) hF TM imax ¼     φ ðτ ÞE 0 : 256π 2 c ωL 2 νeff 3 c

ω

Expression (12.130) shows that for oblique incidence, the force becomes larger when νeff =ω decreases. If νeff is the electron–ion collision frequency, we see that 3 hF TM imax increases with increasing temperature, since νei ∽T
=2 . As the collision frequency becomes smaller, other dissipative processes, such as plasma waves, may become important through the permittivity.

658

Electromagnetic Wave-Matter Interactions

For inhomogeneous plasma in one dimension, the momentum transferred to the plasma (per unit area) between points z1 and z2 during a time dt is Δ ðt z2ððtÞ

p¼

hF z idzdt,

(12.131)

0 z1 ðt Þ

where the spatial integral may change with time. For collisionless conditions, at normal incidence, we obtain Δ 
  pffiffiffiffiffi2 ðt z2ððtÞ
1 ∂ ϵþ1 p0 ϵ 2 þ 1 p0 1
ϵ 2 2 pffiffiffi dzdt ¼
p¼
E0 , pffiffiffiffiffi
2 ¼
pffiffiffiffiffi ϵ2 16π ∂z ϵ2 2 2 ϵ 0

z1 ðt Þ

(12.132) where ϵ 2 is the value of the permittivity at z2 , with the integration beginning at the vacuum interface where ϵ ¼ 1. The momentum (per unit area) of the electromagnetic wave in free space is 1 p0 ¼ 8π

Δ ðt

E 20 ðtÞdt:

(12.133)

0

Examination of (12.132) shows that to get a high momentum transfer, we need low pffiffiffiffiffi values of ϵ 2 . The magnitude of the momentum is limited by the extent to which the light pressure can be neglected. The nonlinear, collisionless interaction can transfer to the plasma a momentum much larger than the magnitude of the momentum given by all of the photons, because the momentum at the front of the plasma is compensated for by that at the back of the plasma. The inclusion of collisions will increase the momentum transfer. For oblique incidence, the direction of the momentum is that of the density gradient and is independent of the direction of the incident wave. The momentum transfer for TE polarization is usually smaller than for TM polarization. We shall continue the discussion of the effects of the ponderomotive force later in this chapter. 12.2.7 Ray-Trace “Equation of Motion” As discussed earlier, the geometrical optics approximation can be used in nearly all instances where an electromagnetic wave is present. Thus, we can think of the propagation of electromagnetic energy in terms of rays moving through the medium. This is analogous to the familiar case of a massive particle moving under

12.3 Reflection at an Interface

659

the influence of an external force. We have already used these thoughts in the preceding sections, especially when drawing Figure 12.1. The starting point is again the field equation (12.17) with r  E ¼ 0, since we are considering only transverse waves. Substituting the electric field E ¼ E0 exp ½ iðk  x
ωt Þ into the field equation gives the dispersion relation k2 ¼ ðω2 =c2 Þϵ ðω; zÞ or jkj ¼ ðω=cÞðη þ iχ Þ. In the absence of absorption, the phase is propagated with velocity vph ¼ ω=jkj ¼ c=η, and the whole pulse is propagated at the group velocity vg ¼ dω=dk ¼ cη; thus vph vg ¼ c2 . Since the index of refraction is a function of the coordinates only, it may be written as a gradient η ¼ rS. Differentiating the group velocity with respect to the time, we arrive at the “equation of motion” for the ray   d 2 x dvg d ¼ ¼ c rS ¼ c vg  r rS ¼ c2 ðrS  rÞrS 2 dt dt dt
2 

c2 c 2 2 η : ¼ r jrSj ¼ r 2 2

(12.134)

For transparent materials, η2 ¼ ϵ ¼ 1
ω2pe =ω2 ¼ 1
ne =ncr , and the equation of motion becomes
2  d2 x c ne ¼r
: (12.135) 2 2 ncr dt We see that rays move as unit mass particles in a potential Φ ¼ ðc2 =2Þðne =ncr Þ, and the ray propagation is completely determined by the electron density gradient. According to (12.135), the trajectory is parabolic, which is characteristic of a constant force field (see also Figure 12.1).

12.3 Reflection at an Interface In the case of two adjacent materials with dissimilar permittivities ϵ 1 and ϵ 2 , we find the familiar Fresnel equations. This configuration of a sharp interface between two different materials is not often encountered in the physics of high-energydensity materials, but finds a utility in diagnostic tools for laboratory experiments, such as velocity interferometry. The kinematic properties are: the angle of reflection equals the angle of incidence and Snell’s law. We are interested in the dynamic properties, which are the intensities of the reflected and refracted (transmitted) waves and the phase changes and polarizations. The kinematic properties follow directly from the wave nature and the existence of boundary conditions at the interface between the two materials but do not depend on the properties of the materials. In contrast, the dynamic properties follow from the specific nature of the electromagnetic fields and the boundary conditions.

660

Electromagnetic Wave-Matter Interactions

The electric field of the incident wave is identified as E0 with angle of incidence θ0 measured from the normal to the interface. The reflected wave is labeled E1 with angle θ1 , and the refracted wave leaving the surface is identified as E2 with angle θ2 . The kinematic properties require that the wave vectors of all three waves sum to zero at the interface, thus k0  x ¼ k1  x ¼ k2  x, independent of the boundary conditions. This gives Snell’s law pffiffiffiffiffi ϵ1 sin θ2 jk 0 j ¼ (12.136) ¼ pffiffiffiffiffi : ϵ2 sin θ0 jk 2 j The dynamical properties require that the normal components of D ¼ ϵE and B ¼ H be continuous across the interface, and the tangential components of E and H are also continuous. In terms of fields, these boundary conditions at the interface are ^ ¼0 ½ϵ 1 ðE0 þ E1 Þ
ϵ 2 E2   n ^¼0 ½k0  E0 þ k1  E1
k2  E2   n ^¼0 ðE0 þ E1
E2 Þ  n ^ ¼ 0, ðk0  E0 þ k1  E1
k2  E2 Þ  n

(12.137)

^ is the unit vector normal to the surface. where n For TE polarization (S-polarization), the electric field being perpendicular to the plane of incidence, the first equation of (12.137) yields nothing. The second equation, using Snell’s law, duplicates the third. The third and fourth equations yield the relative amplitudes of the reflected wave to that of the incident wave pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffi ϵ 1 cos θ0
ϵ 2
ϵ 1 sin 2 θ0 E1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi , ¼ (12.138) ffi E 0 pffiffiffiffi ϵ 1 cos θ0 þ ϵ 2
ϵ 1 sin 2 θ0 and that of the refracted wave

pffiffiffiffiffi 2 ϵ 1 cos θ0 E2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : ¼ ffi E 0 pffiffiffiffi ϵ 1 cos θ0 þ ϵ 2
ϵ 1 sin 2 θ0

(12.139)

For TM polarization (P-polarization), the boundary conditions involved are normal D, tangential E, and tangential H. These correspond to the first, third, and fourth equations of (12.137). The amplitudes of the magnetic intensity of the reflected and refracted waves, relative to that of the incident wave, are pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi H 1 ϵ 2 cos θ0
ϵ 1 ϵ 2
ϵ 21 sin 2 θ0 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (12.140) ¼ H 0 ϵ 2 cos θ0 þ ϵ 1 ϵ 2
ϵ 21 sin 2 θ0 and H2 2ϵ 2 cos θ0 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : ¼ H 0 ϵ 2 cos θ0 þ ϵ 1 e2
ϵ 21 sin 2 θ0

(12.141)

12.3 Reflection at an Interface

661

For TE polarization, where the incident electric field is in the x-direction, Ey ¼ Ez ¼ 0, and the components for the magnetic intensity are H y ¼ ðc=ωÞk z Ex h pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffii and H z ¼
ðc=ωÞk y Ex , where k ¼ ðω=cÞ 0; sin θ0 ; ϵ 1
sin 2 θ0 . For TM polarization, where the magnetic field is in the x-direction, H y ¼ H z ¼ 0, and the components for the electric field are E y ¼
ðc=ωϵ 1 Þkz H x and Ez ¼ ðc=ωϵ 1 Þky H x . For normal incidence, (12.140) reduces to (12.138), and (12.141) reduces to (12.139) pffiffiffiffiffi pffiffiffiffiffi pffiffiffiffiffi ϵ1
ϵ2 2 ϵ1 E1 E2 ¼ pffiffiffiffiffi pffiffiffiffiffi and ¼ pffiffiffiffiffi pffiffiffiffiffi : (12.142) ϵ1 þ ϵ2 ϵ1 þ ϵ2 E0 E0 For the reflected wave, the sign convention is that for TM polarization. Thus, if ϵ 2 > ϵ 1 , there is a phase reversal for the reflected wave. The reflectivity is defined as the ratio of the time-averaged normal components of the energy flux reflected from the surface to that of the incident energy flux. Averaging Poynting’s vector gives S ¼ ðc=8π ÞℜeðE  H Þ, from which we find pffiffiffiffiffi ϵ 1 cos θ1 jE1 j2 jE1 j2 R ¼ pffiffiffiffiffi ¼ , (12.143) ϵ 1 cos θ0 jE0 j2 jE0 j2 since θ1 ¼ θ0 . For normal incidence, both modes of polarization are equivalent and the reflection coefficient is given by
pffiffiffiffiffi pffiffiffiffiffi2 ϵ1
ϵ2 R ¼ pffiffiffiffiffi pffiffiffiffiffi : (12.144) ϵ1 þ ϵ2 This expression is valid whether the reflecting medium is transparent or not. The boundary between the two media is in reality not a geometrical surface but a thin transition layer. The validity of the tangential boundary condition does not rest on any assumptions concerning the nature of this layer. In contrast, the dynamic quantities assume that the thickness of the transition layer is small compared to the wavelength. This is appropriate, since the thickness is usually comparable with the distance between the atoms. If the case exists where the layer’s thickness is large compared to the wavelength, geometrical optics is valid and the propagation of the wave can be regarded as the propagation of rays, which undergo refraction in the transition layer but are not reflected from it; the reflection coefficient is zero. If we assume both media are transparent, the proportionalities between E1 , E2 , and E0 are real. This means the phase remains unchanged or changes by π, depending on the sign of the coefficients. In particular, the phase of the refracted wave is always the same as that of the incident wave. On the other hand, the reflected wave may undergo a phase change. For normal incidence, the phase of the reflected wave is unchanged if ϵ 1 > ϵ 2 , but if ϵ 1 < ϵ 2 , the vectors E1 and E0 are in opposite directions, so there is a phase change of π.

662

Electromagnetic Wave-Matter Interactions

The reflectivities for the two polarizations, in the absence of absorption, are RTE ¼

sin 2 ðθ2
θ0 Þ tan 2 ðθ2
θ0 Þ ¼ and R : TM sin 2 ðθ2 þ θ0 Þ tan 2 ðθ2 þ θ0 Þ

(12.145)

These quantities are unaltered upon an interchange of θ0 and θ2 . An interesting case is the reflection for an incidence angle such that θ0 þ θ2 ¼ π=2, the reflected and refracted rays being perpendicular. Let this particular angle of incidencepbe θp , then sin θp ¼ sin ðπ=2
θ2 Þ ¼ cos θ2 , and ffiffiffiffiffiffiffiffiffiffiffi Snell’s law gives tan θp ¼ ϵ 2 =ϵ 1 . Then in the second part of (12.145), the denominator is infinite and RTM ¼ 0. Hence, for any direction of polarization (including unpolarized) of the incident wave at this angle, the reflected wave will have TE polarization. This particular angle θp is called the angle of total polarization, or Brewster’s angle. Note that this effect cannot be produced by refraction for any angle of incidence. The reflection and refraction of plane-polarized electromagnetic radiation always result in plane polarization, but the direction of the polarization is in general not the same as that of the incident wave. Let γ0 be the angle between the direction of E0 and the plane of incidence, and γ1 and γ2 are the corresponding angles for the reflected and refracted waves, respectively. For transparent material, using (12.138) and (12.140), we have tan γ1 ¼


cos ðθ0
θ2 Þ tan γ0 , cos ðθ0 þ θ2 Þ

(12.146)

and using (12.139) together with (12.141), tan γ2 ¼ cos ðθ0
θ2 Þ tan γ0 :

(12.147)

The angles γ0 , γ1 , and γ2 are equal for all angles of incidence only for γ0 ¼ 0, π=2, and for normal incidence θ0 ¼ 0 or θ0 ¼ π=2, which is grazing incidence for which there is no refracted wave. In all other cases, (12.146) and (12.147) give (by virtue of the inequalities 0 < θ0 , θ2 < π=2 and, as we shall assume, 0 < γ0 < π=2, 0 < γ1 , γ2 < π) the inequalities γ1 > γ0 , γ2 < γ0 . Thus, the direction of E is turned away from the plane of incidence on reflection but toward it on refraction. Comparison of the two expressions (12.145) shows that for all angles of incidence except normal or grazing, RTM < RTE . Hence, for example, when the incident radiation is of random polarization, the reflected radiation is partly polarized with the predominant direction of the electric field perpendicular to the plane of incidence. The refracted wave is partly polarized too, with its polarization predominantly in the direction of E in the plane of incidence.

663

12.3 Reflection at an Interface (b)

(a)

1.0

0.8

0.8

0.6

0.6 Total internal reflection

R

R

1.0

0.4

0.4 Brewster´s angle

0.2

0.0 0.0

0.2

15.0

30.0

45.0 q0(°)

60.0

75.0

90.0

0.0 0.0

15.0

30.0

45.0

60.0

75.0

90.0

q0(°)

Figure 12.6. Reflection coefficients for TE (solid lines) and TM polarization (dashed lines): (a) is for η1 ¼ 1: and η2 ¼ 1:5, while (b) has η1 ¼ 1:5 and η2 ¼ 1.

The quantities RTM and RTE depend differently on the angle of incidence. RTE increases monotonically with the angle, but RTM decreases toward zero at θ0 ¼ θp before monotonically increasing. Two distinct cases can occur: (1) if the reflection occurs from the two media where ϵ 2 > ϵ 1 , then RTM and RTE increase to the common value of unity at grazing incidence (θ0 ¼ π=2); and (2) if the reflection is from an “optically less dense” medium, where ϵ 2 < ϵ 1 , both coefficients become pffiffiffiffiffiffiffiffiffiffiffi equal to unity at θ0 ¼ θr , where sin θr ¼ ϵ 2 =ϵ 1 . This special angle is called the angle of total reflection, and the refracted wave is propagated along the surface separating the two media. These two cases are illustrated in Figure 12.6, for the pffiffiffi absence of absorption ( ϵ ¼ η). Reflection from this latter situation, where ϵ 2 < ϵ 1 , requires special consideration for angles with θ0 > θr . In this case, the z-directed component of the refracted wave’s wave vector is purely imaginary. This is the situation discussed in Section 12.2.1, where the electric field is damped in medium two. In the absence of true absorption, the average energy flux from medium one to medium two is zero. That is, all the energy incident on the boundary is reflected back into medium one, so that the reflection coefficients are RTE ¼ RTM ¼ 1; this phenomenon is called total internal reflection. For θ0 > θr , the proportionality coefficients between E1 and E0 become complex quantities of the form ða
ibÞ=ða þ ibÞ. The quantities RTE and RTM are given by the squared moduli of these coefficients, which are equal to unity. We also find the difference in their phases. Let ðE 1 ÞTE ¼ eiδTE ðE 0 ÞTE and ðE 1 ÞTM ¼ eiδTM ðE0 ÞTM , since ða
ibÞ=ða þ ibÞ ¼ eiδ and so tan ðδ=2Þ ¼ b=a. Then

664

δTE tan ¼ 2

Electromagnetic Wave-Matter Interactions

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ϵ 1 sin 2 θ0
ϵ 2 pffiffiffiffiffi ϵ 1 cos θ0

and

δTM tan ¼ 2

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ϵ 21 sin 2 θ0
ϵ 1 ϵ 2 : ϵ 2 cos θ0

(12.148)

Thus, total internal reflection involves a change in the wave phase, which is in general different for field components parallel and perpendicular to the plane of incidence. Hence, on reflection of a wave polarized in a plane inclined to the plane of incidence, the reflected wave will be elliptically polarized. The phase difference δ ¼ δTE
δTM is δ tan ¼ 2

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ϵ 1 sin 2 θ0
ϵ 2 cos θ0 , pffiffiffiffiffi 2 ϵ 1 sin θ0

(12.149)

which is zero only for θ0 ¼ θr or θ0 ¼ π=2. We have largely omitted a discussion about the transmissive properties of the sharp interface between two media. They do not play a role in high-energy-density physics nor as a diagnostic technique in laboratory experiments.

12.4 Density Profile Modification One of the effects of the ponderomotive force (see Section 12.2.6), especially near the critical density, is to modify the structure of the medium. This alteration of, say, a smooth linear profile can become a “source term” for other plasma processes, some of which we address later. For an electromagnetic wave normally incident on a linear density layer, in the absence of absorption, the wave will be reflected at the critical density surface, and twice its pressure is transmitted to the medium via the ponderomotive force. This momentum deposition is highly localized at the critical density, and the resulting pressure gradient is large. The pressure gradient, in turn, causes a redistribution of the medium as described by the hydrodynamic equations. The immediate effect is a local steepening of the density profile, which introduces an additional scale length, and this in turn determines both the mix and scaling of interaction processes near the critical density. For oblique incidence and TM polarization, the effect of the ponderomotive pressure is even greater as the resonant process comes into play. We can study the nature of the profile modification by first considering the simple case of a planar isothermal expansion discussed in Section 6.3. There, the method of characteristics yielded an equation for the density profile

12.4 Density Profile Modification

ρ ¼ e
x=cs t , ρ0

665

(12.150)

where ρ0 is the density at x ¼ 0, and cs is the adiabatic sound speed. Note that the density gradient length increases with time according to L ¼ ρ=ð∂ρ=∂xÞ ¼ cs t. As shown in Figure 6.3, in a coordinate frame moving with a point at constant density, the velocity u is just the sound speed. If this expanding medium is subjected to a pressure at a specific location, say the critical density, the density profile will be locally steepened. Continuing with the example of an isothermal expansion, we now include a normally incident electromagnetic wave that reflects from the critical density. Again, we assume the medium is collisionless, so there is no absorption. Rather than using the method of characteristics, we write a two-fluid description in the framework of the hydrodynamic conservation equations, also discussed in Section 6.2. The following discussion draws heavily from Kruer (1988). Neglecting the inertia of the electrons, the electron momentum equation is related to the electric field according to ne eE ¼
rPe


ne me rv2osc , 4

(12.151)

where the last term comes from the ponderomotive force; vosc ¼ ejAj=me c is the electron’s oscillatory velocity in the presence of the high-frequency electric field. This electric field transmits the electron pressure to the ions. The continuity equation for the ion fluid is ∂ni ∂ þ ðni uÞ ¼ 0, ∂t ∂z

(12.152)

where u is the one-dimensional fluid velocity. The corresponding momentum equation is
 ∂u ∂u þu n i mi (12.153) ¼ Z ni eE
rPi , ∂t ∂z where the subscript i denotes the ion fluid. The electron number density is ne ¼ Z ni . Using (12.151) in (12.153) yields ∂u ∂u c2 ∂ni Z me ∂v2osc þu ¼
s
: ∂t ∂z ni ∂z 4mi ∂z

(12.154)

Using (12.152) and (12.154) gives the variation in density and flow velocity in a coordinate system moving with the steepened surface

666

Electromagnetic Wave-Matter Interactions

∂ ðni uÞ ¼ 0 ∂z

(12.155)


 ∂ u2 ∂ Z me ∂v2osc ¼
c2s ð log ni Þ
: ∂z 2 ∂z 4mi ∂z

(12.156)

and

Normalizing the flow velocity to the sound speed and using (12.155) in (12.156), we obtain
   ∂ vosc 2 2 1 ∂ni ¼ 0, (12.157) 1
u þ ni ∂z ∂z 2vth where vth is the electron thermal velocity. Examination of (12.157) shows that the sonic point, where u ¼ 1, must be at the maximum of the standing electromagnetic wave, where ∂v2osc =∂z ¼ 0. Integrating (12.155) and (12.157) gives
2


 ns ni 1 vosc 2 1 vmax 2 þ þ 2 log ¼1þ , (12.158) 2 vth 2 vth ni ns where ns is the density at the sonic point and vmax is the value of vosc at the maximum of the standing wave, which is the sonic point. There are two solutions to (12.158), one for n1 < ns , which is supersonic and corresponds to a lowerdensity plateau, and n2 > ns , which is subsonic and corresponds to the upperdensity shelf. Figure 12.7 sketches the nearly steady-state density profile in the vicinity of the critical surface (Estabrook, Valeo, & Kruer, 1975). The standing wave pattern of the electric field creates the density striations along the lower (supersonic) shelf in accordance with Figure 12.5. The profile is steepened down to the lower-density shelf that is determined by the ponderomotive force and the local heat flow, including thermal transport inhibition. In Section 12.2.1, we found an exact solution for an electromagnetic wave penetrating a linear density gradient. We approximate the modified density profile as being locally linear from ns to ncr , with a density scale length L. Then using the Airy solution (12.27), we have "
1= # ω2 3 E ¼ 3AAi ðL
zÞ , (12.159) c2 L where A is a constant determined by the incident electromagnetic wave. This locally linear assumption is reasonable provided vosc =vth  1, but fails greatly for vosc =vth  1, since the jump in density becomes too large. Matching the peak

667

12.4 Density Profile Modification

Figure 12.7. Schematic of the steepening of the density profile by the ponderomotive force. The initial density profile is shown by the dashed line.

of the Airy function solution (see Figure 12.2) to the field at the sonic point gives v2osc ðn ¼ ncr Þ  0:44v2max . We can now write (12.158) as
2

 ns ncr vmax 2 ¼ 1 þ 0:28 þ 2 log : (12.160) ncr ns vth For a small density jump, (12.160) has the solution ns =ncr  1
0:38vmax =vth . As we shall see shortly, the solution to the equation for the evolution of the electrostatic field amplitude E es in terms of the oscillating driver field Ed 1=

gives E max  3:7ðc2 L=ω2 Þ 3 Ed , and thus the oscillating velocity of an electron in

the

driver

field

vd

is

1= 3

given

by

1=

v2max  3:7ðc2 L=ω2 Þ 3 v2d .

Since

ncr =L ¼ ðncr
ns Þðc2 L=ω2 Þ , the sonic point and the critical density point are 1=

separated by Δz  ðc2 L=ω2 Þ 3 . The sonic point is at ζ ¼ 1 and the critical density is at ζ ¼ 0, where ζ is the argument of the Airy function. Considering weak fields (vd =vth ≲0:1), we easily find the density ratio of the sonic point to that of critical to be
0:8 ns vd  1
0:77 , (12.161) ncr vth and the density ratio for the lower and upper shelves

668

Electromagnetic Wave-Matter Interactions


0:8
0:8 ns vd ns vd  1 þ 0:97 and  1
0:97 , n1 vth n2 vth

(12.162)

where ðvd =vth Þ  ð3c=2ωLÞ0:8 . Note the jump in density scales as a fractional power of the intensity. For more intense fields, numerical solutions of the transcendental equations are required. Since an expanding planar medium flows through a point of constant density at the sound speed (see Figure 6.3), the momentum deposition at the critical density resonantly perturbs the flow. Hence the profile is steepened over a significant range of densities even for a relatively weak electromagnetic wave. For an incident TM electromagnetic wave, where resonance absorption can take place at the critical density, the profile modification is generated both by the pressure of the reflecting wave and by the pressure of the resonantly generated electrostatic field near the critical density. Numerical simulations of this process, for short-wavelength, high-intensity laser light on an initially linear density profile, show that after the laser light has penetrated to its turning point, an electrostatic field is resonantly excited at the critical density and the field grows linearly in time, becoming more and more localized at the critical density. Eventually, the resonantly driven field becomes sufficiently intense and localized that electrons can be accelerated through it in one oscillation period. This increase in electron energy, assuming the phase is proper, efficiently heats the electrons. The feedback of the intense fields coupled with the localized heating on the density profile creates conditions ripe for a variety of instabilities.

12.5 Absorption of Electromagnetic Energy The fundamental expression for the energy density in electrodynamics, 1 ue ¼ 4π

ðt E
∞

∂D dt, ∂t

(12.163)

is troublesome in the presence of an absorbing, dispersive medium, because it can have a negative value, owing to the fact that the permittivity can be less than zero. This begs the question: What is the proper way to define the energy density? In fact, is the concept of an internal energy even meaningful when absorption is present? The field energy in a dispersive medium was introduced in Section 3.13. There we combined Faraday’s and Ampere’s laws that gave rise to Poynting’s theorem
 1 ∂D ∂ H 2 c E þ (12.164) þ J  E ¼
r  ðE  HÞ: 4π ∂t ∂t 8π 4π

12.5 Absorption of Electromagnetic Energy

669

For electromagnetic radiation in an isotropic medium, D ¼ ϵ ðωÞE, J ¼ σ ðωÞE, and ϵ ¼ 1 þ i4πσ=ω, where the Drude electrical conductivity is given by σ ðωÞ ¼

ω2pe 1   ðνeff þ iωÞ: 4π ω2 þ ν2eff

In the absence of dispersion, (12.164) becomes
 ∂ ϵE2 þ H 2 c þ σE 2 ¼
r  ðE  HÞ; ∂t 4π 8π

(12.165)

(12.166)

the argument of the time derivative is often interpreted as the field energy density in the medium. Including dispersion and averaging over the high frequencies, Ginzburg (1970) shows that the two terms on the left-hand side of (12.164) are    ∂ 1 ∂D 1 dðωϵ Þ ∂  (12.167) E ¼ E0  E 0 ¼ hue i 4π ∂t 16π dω ∂t ∂t and
 1 1 dσ ∂E0

∂E 0

 E0
E0  , hJ  Ei ¼ σE0  E0
i 2 4 dω ∂t ∂t

(12.168)

where the  electric field is  described by the quasimonochromatic field Eðt Þ ¼ 12 E0 ðt Þeiωt þ E 0 ðt Þe
iωt , with E0 ðt Þ and E 0 ðtÞ being slow functions of time. A similar expression for the magnetic intensity exists. The expression (12.164), averaged over the high frequencies, becomes

 1 ∂ d ðωϵ Þ 1



E0  E0 þ H0  H0 þ σE0  E 0 16π ∂t dω 2


   1 dσ ∂E0

∂E c
i  E0
E0  0 ¼
r  E0  H 0 þ E 0  H0 : (12.169) ∂t 4 dω ∂t 16π The total energy density of the electromagnetic wave is hur i ¼ hue i þ H 2 =8π. In the absence of absorption (σ ¼ 0), the entropy of an isolated system is constant even for a variable electromagnetic field. In the presence of absorption, the entropy is no longer constant but involves the left-hand side of (12.164), and thus hue i is not an internal

 energy  in the thermodynamic sense. For example, ue ¼ 1
ω2pe =ν2eff E 2 =8π has a negative value for νeff < ωpe . This result does not contradict the principle of increase of entropy since u is not the internal energy. For a closed system, in which there is no flow of energy into or out of a given volume (i.e., E  H ¼ 0), we have from (12.166)

670

Electromagnetic Wave-Matter Interactions

ð

½ur ðt 0 Þ
ur ðtÞdV ¼


ðt 0 ð σE 2 dVdt:

(12.170)

t

Now if the electromagnetic field were cut off at time t, then as t 0 ! ∞ the field is entirely damped and ur ð∞Þ ! 0, and according to the law of increase of entropy, all the energy must be converted to heat, that is ð

ð ð∞ ð  1  2 2 ϵE þ H dV ¼ ur ðtÞdV ¼ σE2 dVdt > 0: 8π

(12.171)

t

Hence, it follows that σ > 0 and ϵ > 0. Continuing with the assumption E  H ¼ 0, (12.164) yields ∂D ∂E þ 4πJ ¼ ϵ þ 4πσE ¼ 0; ∂t ∂t

(12.172)

in the second expression, we assume dispersion is absent. A solution to (12.172) is E ¼ E ð0Þ exp ð
4πσt=ϵ Þ, and for σ > 0 and ϵ < 0 the field would increase. In a quasiequilibrium system, this is impossible, and (12.172) cannot be used for plasma. The characteristic time of the variation of the field is τ∽ϵ=4πσ, and the frequencies ω are mainly 4πσ=ϵ. Thus, at such frequencies the condition ω  νeff for dispersion to be absent and the condition ϵ < 0 cannot both be satisfied. Thus, for (12.164) to be valid, certain restrictions on the properties of the medium apply. The damping of the electromagnetic wave in plasma results in the electrons gaining energy. The motion of the electrons is given by a solution to the force equation ∂ve ¼ eE
me νeff ve , (12.173) ∂t where the last term is a “frictional force.” Using the form of the electric field of the wave from the preceding, the velocity of the electron is

 e E0 eiωpe t E 0 e
iωpe t : (12.174) þ ve ¼ 2me iωpe þ νeff
iωpe þ νeff me

The kinetic energy of the electron is   1 e2 E0  E 0 2 me ve ¼ 2 4me ω2pe þ ν2eff

(12.175)

and   e2 E0  E 0 νeff 2 , heve  Ei ¼ me νeff v2e ¼ 2me ωpe þ ν2eff

(12.176)

where the quantities in angular brackets signify averages over the high frequencies.

12.5 Absorption of Electromagnetic Energy

671

The sum of the kinetic energy of all the electrons (per unit volume) and the field energy is 2 3    2 2 E 1 1 4 4πe ne

5E0  E 0 : ¼ (12.177) 1þ hue i ¼ ne me v2e þ 8π 2 16π 2 2 me ω þ ν pe

eff

In the absence of absorption, we have " #

 1 4πe2 ne 1 d ðωϵ Þ

1þ E0  E0 ¼ E0  E 0 , hue i ¼ 16π 16π dω ωpe me ω2pe

(12.178)

which is always positive, and is thus the mean energy density of the electric field in the medium. However, in the presence of absorption, there is the question of the relation between (12.177) and 2

3

 2 2 2 4πe ne ωpe
νeff 7 1 dðωϵ Þ 1 6

E0  E 0 ¼ hue i ¼ 41 þ

2 5E0  E0 , 16π dω ωpe 16π me ω2pe þ ν2eff (12.179) having used ϵ ¼ 1 þ i4πσ=ω in (12.165). Multiplying (12.173) by ne ve , we have
 ∂ me v2e ne ¼ ene ve  E
ne me νeff v2e : ∂t 2

(12.180)

The total current density, including the polarization P, is ene ve ¼ J þ

∂P 1 ∂ ¼Jþ ðD
EÞ: ∂t 4π ∂t

Using this in (12.180) gives
 ∂ me v2e E 2 1 ∂D ne
ne me νeff v2e : þ ¼ JE þ E ∂t 4π ∂t 2 8π We can now write (12.164) as
 ∂ me v2e E 2 H 2 c ne þ þ ¼
r  ðE  HÞ
ne me νeff v2e : ∂t 4π 2 8π 8π

(12.181)

(12.182)

(12.183)

This equation expresses conservation of energy, with the last term on the righthand side being the work done by the friction on the particles, and this is converted into heat.

672

Electromagnetic Wave-Matter Interactions

Averaging (12.183) over the high-frequency oscillations gives 82 9 3   < = 2 2 2 2 ∂ me ve E H 1 ∂ 4 4πe ne

5E0  E 0 þ H0  H 0 ne 1þ þ þ ¼ ; ∂t 16π ∂t : 2 8π 8π me ω2pe þ ν2eff     c r  E0  H 0 þ E 0  H0
ne me νeff v2e : ¼
16π (12.184) Using the phenomenological expression (12.169), we obtain 82 9

3 > > 2 2 2 < = 4πe n ω
ν e pe eff 7 1 ∂ 6



41 þ

2 5E0  E0 þ H0  H0 > 16π ∂t > : ; me ω2pe þ ν2eff
 e2 ne νeff e2 ne ωpe νeff ∂E0

∂E 0



E0  E0 þ i  E0
E0  þ

2 ∂t ∂t 2 2 2me ω2pe þ ν2eff 2me ωpe þ νeff   c (12.185) r  E0  H 0 þ E 0  H0 : ¼
16π Expressions (12.184) and (12.185) cannot be in contradiction, thus the frictional force is to be equated to the heat generated   ne me νeff v2e ¼

 e2 n ν e2 ne ν2eff ∂ e eff E0  E 0
E0  E 0

2 ∂t 2me ω2pe þ ν2eff 2me ω2pe þ ν2eff
 e2 ne ωpe νeff ∂E0

∂E 0  E0
E0  þi : (12.186)

2 ∂t ∂t 2me ω2pe þ ν2eff

Comparing expressions (12.169) and (12.185), we identify

 2 2 2 4πe n ω
ν e pe eff d ðωϵ Þ ¼1þ

2 , dω ωpe me ω2pe þ ν2eff
 e2 ne νeff dσ 2e2 ne ωpe νeff

, and ¼
σ¼

2 : dω ωpe 2 2 me ω2pe þ ν2eff me ωpe þ νeff

(12.187)

In the phenomenological expression (12.169), there enters as characteristics of the medium only the frequency-dependent functions ϵ ðωÞ, σ ðωÞ, and their derivatives. The usual interpretation of the term J  E ¼ σE 2 as the heat generated per unit time (per unit volume) is in general incorrect. If the electromagnetic wave is monochromatic, we have

12.5 Absorption of Electromagnetic Energy

  1 e2 n ν e eff E0  E 0 ¼ ne me νeff v2e : σE0  E 0 ¼ 2 2me ω2pe þ ν2eff

673

(12.188)

In the more general case of a quasimonochromatic wave, expression (12.188) is not to be used. That is, each term on the left-hand side of (12.169) has in general no definite significance as regards energy (change of internal energy, dissipation, etc.). It is for this reason that in the presence of absorption, the expression 1 d ðωϵ Þ E0  E 0 16π dω

(12.189)

may be negative, and this result involves no paradox, at least for our plasma model.

12.5.1 Collisional (Inverse Bremsstrahlung) The conversion of electromagnetic energy into thermal energy in underdense plasma is primarily due to collisional damping (also known as inverse bremsstrahlung absorption) and instability heating. These dominate in plasma with a shallow density gradient. On the other hand, a steep density gradient will increase the efficiency of resonant absorption because the critical surface is more accessible to the electromagnetic wave. First, we consider collisional absorption where the spatial variations of the density profile have been created by other processes such as hydrodynamic expansion, thermal transport inhibition of the energy flow, or stochastic fluctuations. The simplest situation is to consider the heavy ions to be fixed, and the electron fluid, with velocity ue , interacts with the ions according to ∂ ðne ue Þ ¼ νei ne ue , ∂t

(12.190)

where νei is a collision frequency that describes electron scattering by the ions. An expression for the electron–ion collision rate was developed in Section 7.4; its inverse is pffiffiffiffiffi 4 2π ðZ Þ2 e4 ni logΛei : (12.191) νei ¼ 3 m1=2 ðkB T e Þ3=2 e

In Section 3.13, we developed the complex permittivity, taking into account electron–ion collisions, though in that section we left the nature of the collisions vague. For electron–ion encounters, we have ϵ ðωÞ ¼ 1


ω2pe ω2pe νei  1
i ¼1
 2 : ωðω þ iνei Þ ω ω þ ν2ei

(12.192)

674

Electromagnetic Wave-Matter Interactions

Since the wave frequency ω is real, the dispersion relation k2 ¼ ðω2 =c2 Þϵ ðωÞ is now complex and the wave vector is, in general, complex. The difference in the directions kr and ki correspond to the difference in the planes of equal phase and equal amplitude. Such planes are said to be inhomogeneous. For homogeneous plane waves, the planes of equal phase and amplitude are parallel, and the wave vector may be written in terms of the indices of refraction and absorption ^ where k ^ is a unit vector. k ¼ ωðη þ iχ Þk, The electromagnetic wave propagates according to E ¼ E0 eiðkr  x
ωtÞ e
ki  x ,

(12.193)

and using this in (12.17) gives the wave vector, for νei =ω  1, 1h νei i k2 ¼ 2 ω2
ω2pe 1
i : c ω

(12.194)

Assuming ki  kr , we have the components kr ¼

1=2 ω2pe νei 1 2 and ki ¼ 2 , ω
ω2pe c 2ω vg

(12.195)

with k r ¼ ðω=c2 Þvg . The imaginary part of the wave vector is the rate at which energy decays in space, and the energy-damping length is vg =νei . The absorption coefficient α may be defined such that the intensity of the electromagnetic wave decreases by a factor 1=e in a distance Δz. Then, from the second exponential in (12.193) jEðzÞj2 jE 0 j2

¼ e
2ki Δz ,

(12.196)

  and the absorption coefficient is α ¼ 2ℑmðkÞ. Now ℑm k2 ¼ 2ℜeðkÞℑmðkÞ, so that α¼

ω2 ω ℑmðϵ Þ νei pe 2 2 pffiffiffi ¼ c ℜeð ϵ Þ c ω2 1 þ νei2 ω 41


1

31=2 :

(12.197)

ωpe 2 5 2

ν ei ω2

ω2 1þ

Further, since ϵ ¼ ðη þ iχ Þ2 , α ¼ 2ðω=cÞχ. In the limit νei =ω  1, and for ne < ncr , we find α

2 νei ωpe c ω2

1


ω2pe ω2

1=2 :

(12.198)

12.5 Absorption of Electromagnetic Energy

675

Inserting the classical collision frequency (12.191), we have D E

2 6 pffiffiffiffiffi 16π ni ðZ Þ ne e logΛei , (12.199) α ¼ 2π 3 cðme k B T e Þ3=2 ωη D E P where ni ðZ Þ2 ¼ ni j xj ðZ Þ2j , with xj being the atomic fraction of the jth species. The index of refraction (in the collisionless limit) is η¼

ω2pe 1
2 ω

!1=2


¼

ne 1
ncr

1=2 :

(12.200)

Two modifications to (12.199) are needed for dense plasma: stimulated emission and electron degeneracy effects. From the discussion for induced processes in Section 8.2, the absorption coefficient becomes

 4π 3 c2 ðPÞ 0 α¼α 1þ B ðT e Þ , (12.201) hω3 ν where BðνPÞ ðT e Þ is the frequency-dependent Planck function (here, the subscript   0
hω=2πk B T e ν ¼ ω=2π); thus α ¼ α 1
e . We then arrive at D E


2 6 5

28 π =2 e ni ðZ Þ μ 0
hω=2πk B T e 1
e F (12.202) α ¼ logΛei , 1

1= =2 k T 3 ch3 2 ω2pe 2 B e ω 1
ω2 where F 1=2 ðxÞ is a Fermi-Dirac integral. In the nondegenerate limit, pffiffiffi F 1=2 ðxÞ ! ð π =2Þex , and the classical result (12.199), with the additional stimulated emission term, is recovered. Having arrived at a local absorption coefficient, we can now calculate the overall collisional absorption coefficient in plasma. For a linear density profile, the permittivity (12.192) becomes ϵ ðzÞ ¼ 1


2

z 1 , L 1 þ i ðνei Þcr

(12.203)

ω

where ðνei Þcr ¼ ðncr =ne Þνei ¼ ω=ωpe νei is the collision frequency evaluated at the critical density. For normal incidence of the electromagnetic wave, the disperpffiffiffiffiffiffiffiffi sion relation is k ðzÞ ¼ ðω=cÞ ϵ ðzÞ. Now using the WKB theory of Section 12.2.3 and (12.196), we can write the absorption as 2L 3 ðL ð hpffiffiffiffiffiffiffiffii ω 4L 4 5 ℑm ϵ ðzÞ dz ¼ ðνei Þcr : (12.204) δ ¼ 2ℑm k ðzÞdz ¼ 2 c 3c 0

0

676

Electromagnetic Wave-Matter Interactions

Taking account of the fact that there is absorption in both directions of propagation, we can approximate the fractional energy loss from the electromagnetic wave as

 8L f ib ¼ 1
exp
ðνei Þcr : (12.205) 3c For the case of oblique incidence, things are slightly different. The z-directed component of the dispersion relation is now " #1=2 2 1=  ω ω ω pe 2 : ϵ ðzÞ
sin 2 θ0 ¼ cos 2 θ0
kz ðzÞ ¼ c c ωðω þ iνei Þ Using this expression in (12.204), we have 2 3 L cos ð 2 θ0 ω 6 7 16L δ ¼ 2 ℑm4 kz ðzÞdz5 ¼ ðνei Þcr cos 5 θ0 , c 15c

(12.206)

(12.207)

0

where the upper integral is the turning point. The fractional absorption is thus

 32L 5 (12.208) ðνei Þcr cos θ0 : f ib ¼ 1
exp
15c For normal incidence, expression (12.208) disagrees with (12.205) because the density dependence of νei has reduced the coefficient 8=3 to 32=15. Note that the absorption is a sensitive function of the angle of incidence. An obliquely incident wave reflects from a point of lower density than critical, so less plasma is traversed, and the absorption is reduced from that at normal incidence. Collisional absorption depends in detail on the nature of the density profile. For example, in an exponential profile, ne ¼ ncr exp ð
z=LÞ, for which the fractional absorption is

 8L (12.209) f ib ¼ 1
exp
ðνei Þcr cos 3 θ0 : 3c 12.5.2 Nonlinear Inverse Bremsstrahlung We have been somewhat cavalier about the details of the collisions among the particles in underdense material to the presence of electromagnetic radiation. We now take a closer look at the role of electron scattering. We know from Section 3.9 that the collision frequency depends on an average over the velocity distribution of the electrons, which is dictated by their motion in the electromagnetic field.

12.5 Absorption of Electromagnetic Energy

677

Electron–electron collisions play a significant role in determining the nature of the velocity distribution. The inverse bremsstrahlung process heats electrons with a “flat-topped” distribution that then is relaxed to a Maxwellian distribution by electron–electron collisions. The competition between the heated and relaxed distributions produces a non-Maxwellian distribution. Establishment of a Maxwellian distribution is determined by the e-folding time τ ee for electron–electron collisions. For a Maxwellian distribution, the heating time is approximately the ratio of the thermal energy present to J  E heating t  3τ e

v2th , v2osc

(12.210)

where τ e is given by the inverse of (12.191), with a small modification to logΛ. The heating time is shorter than τ ee ð τ e =Z Þ when Z v2osc =v2th ≳3, in which case a nonMaxwellian distribution is possible. Because of the presence of the ionization level Z , the nonlinear modification to the distribution function is important at lower electromagnetic field intensities (Langdon, 1980). The evolution of the electron distribution function f e is described by the Fokker-Planck equation (omitting the spatial gradient term)
 ∂f e e ∂f e ¼ C ei ðf e Þ þ C ee ðf e Þ, (12.211)
E  rv f e ¼ ∂t me ∂t coll where the two terms on the right-hand side are for electron–ion and electron– electron collisions respectively, both functions of the electron velocity distribution. Electron–electron collisions are important for determining the form of the zeroorder distribution function but otherwise are unimportant. (We have assumed that the electromagnetic field is not too strong so that magnetic induction may be ignored.) The electron–ion collision term, from Section 3.9, is   ∂ 1 ∂2  ½hΔvi if e ðve ; t Þ þ Δvi Δvj f e ðve ; t Þ , (12.212) ∂vi 2 ∂vi ∂vj   where hΔvi i represents the slowing down of the electrons and Δvi Δvj describes their diffusion in velocity space. This term has the effect of a restoring force tending to redistribute the electrons toward its isotropic equilibrium distribution function. Expression (12.212) can be written (Shkarofsky, Johnston, & Bachynski, 1966)

2  ∂ ve I
ve ve ∂f e   C ei ðf e Þ ¼ A , (12.213) ∂ve v3e ∂ve   where A ¼ 2πne Z e4 =m2e logΛei and I is the unit tensor. Cei ðf e Þ ¼


678

Electromagnetic Wave-Matter Interactions

The form of the electron–electron collision term can be written as a divergence of the particle flux


  ∂ ∂ ∂ ∂f e ∂ Cee ðf e Þ ¼ B  G 
ðH f e Þ , (12.214) ∂ve ∂ve ∂ve ∂ve ∂ve   where B ¼ e4 =4πm2e logΛee . G ðve Þ and H ðve Þ are the Rosenbluth potentials discussed in Sections 3.9 and 7.5. Setting aside the electron–electron collision term, and assuming the true distribution function is not too far removed from that of an equilibrium distribution function, valid for moderate electromagnetic field strengths, we can approximate the distribution function as the first terms of a Legendre polynomial expansion: f e  f ð0Þ þ f ð1Þ cos θ, where θ is the angle between ve and E. Then (12.213) becomes A

∂ v2e I
ve ve ∂ ð1Þ 2A   f cos θ ¼
3 f ð1Þ cos θ: 3 ∂ve ∂ve ve ve

(12.215)

Using this in (12.211), we have the linearized kinetic equation ∂f ð1Þ e ∂f ð0Þ 2A ¼
3 f ð1Þ :
E me ∂ve ve ∂t

(12.216)

The electromagnetic wave is described by E ¼ E0 e
iωt , and the solution to (12.216) is
 eE0 ∂f ð0Þ 2A
1 ð1Þ f ¼i ωþi 3 : (12.217) ve me ∂ve Ð The perturbed current density is given by Jð1Þ ¼
e f ð1Þ cos θve dve . The average rate of energy absorbed into the medium is  ð D E 1

ð1Þ ð1Þ 2 J  E ¼ ℜe
eE0 f ve cos θdve : (12.218) 2 Using (12.217) in (12.218) and integrating over angles gives ð∞ D E 4π 1 ∂f ð0Þ ð1Þ 2 dve : J  E ¼
Ame vosc  2 3 1 þ 2A=v3 ω ∂ve 0

(12.219)

e

The term in the denominator, within the integral, accounts for the changeover from the primarily reactive response of the faster electrons to resistive response of the slow, strongly scattered electrons. Since we are ignoring electron–electron collisions, we note   2A=v3e ω ∽νei ðve Þ=ω, where νei ðve Þ is a velocity-dependent collision frequency.

12.5 Absorption of Electromagnetic Energy

679

We can define the velocity vω to be where the scattering and optical frequencies are equal; that is, νei ðvω Þ=ω ¼ 1. Numerical evaluation of (12.219) shows that   absorption is reduced by a factor (within 3 percent) given by exp v2ω =2v2th , for vω ≲2vth . Assuming νei ðve Þ=ω  1 for all but a small portion of the electron h  2 i
1  1, and the integral in distribution, we can approximate 1 þ 2A=v3e ω   (12.219) is
f ð0Þ ð0Þ=ne . Balancing Jð1Þ  E against the rate of energy loss from the electromagnetic field, which is νE 2 =8π, we have the damping rate ν¼

8π ω2pe f ð0Þ ð0Þ A : 3 ω2 ne

(12.220)

The damping rate depends on the zero-order electron distribution function, which, in turn, depends on whether the electron–electron collisions can equilibrate the distribution faster than the electron–ion collisions can cause heating. If this is so (i.e., Z v2osc =v2th  1), the distribution function remains Maxwellian, and we have from (12.220) ν¼

ω2pe 1 Z ω4pe ω2pe ¼ 2 νei , ω2 3ð2π Þ3=2 ni v3th ω

(12.221)

with νei given by (12.191), and there is a slight modification to logΛ. This expression determines the electron–ion collision frequency, which describes the damping of a high-frequency wave in plasma, the result being that absorption is enhanced at lower densities. If electron–electron collisions cannot equilibrate the distribution function sufficiently rapidly (i.e., Z v2osc =v2th  1), then the distribution function is determined by collisional heating. In this limit, we can modify (12.211) to balance the change in the zero-order distribution function against the heating term (averaged over angles) * + ∂f ð0Þ eE ∂f ð1Þ eE 1 ∂ 2 ð1Þ ¼  v f ¼ : (12.222) me ∂ve me 3v2e ∂ve e ∂t h  2 i
1 ¼ 1, we find a selfSubstituting (12.217) for f ð1Þ , and setting 1 þ 2A=v3e ω similar solution

 1 1 ve 5 ð0Þ , (12.223) f ∽ 3 exp
u 5 u where
u¼

5Av2osc t 3

1=5 :

(12.224)

680

Electromagnetic Wave-Matter Interactions

Hence, the self-consistent distribution function is a super Gaussian, which changes with time. Such a distribution has fewer particles than a Maxwellian near ve ¼ 0, and thus the collisional damping rate is reduced by a factor of about two. We demonstrate the evolution of the electron distribution function in Figure 12.8. In the limit Z v2osc =v2th  1, electron–electron collisions can be neglected. And the initially monoenergetic distribution from inverse bremsstrahlung absorption diffuses and slows in balance so that no net gain in kinetic energy results. When more electrons slow to low velocities, such that ω=νei ðve Þ≲1, their loss of energy is slowed while upward diffusion of faster particles continues, so net absorption begins. By the time the electrons have gained 10 percent in energy, the distribution function is approaching equilibrium and is described by the selfsimilar solution (12.223). The absorption is only about 45 percent of what it would be if electron–electron collisions were included, which enforce a Maxwellian distribution. Electron–electron collisions alter these results only slightly for α ¼ Z v2osc =v2th ≳1. For example, with α ¼ 6 the absorption is still only 49 percent

Figure 12.8. Evolution of the electron distribution function. The initial monoenergetic distribution from inverse bremsstrahlung, shown by the vertical line, has velocity 5:8vosc . The dashed curve corresponds to a distribution function that has gained only 1 percent in energy. The dot-dashed curve is when the energy has increased by 10 percent, and the solid curve is close to the self-similar solution.

12.5 Absorption of Electromagnetic Energy

681

of its Maxwellian value, and inverse bremsstrahlung contributes equally with electron–electron collisions to diffusion of the higher-energy electrons into the “tail” of the distribution. With α ¼ 0:5, the distribution is still depressed and flattened near zero velocity, and absorption is reduced to 67 percent of normal. For α ¼ 0:05, the reduction is 12 percent. Langdon (1980) gives the absorption factor for any α as f nl ¼ 1


0:553  0:75 : 1 þ 0:27 α

(12.225)

Alteration of the distribution function from that of equilibrium impacts other parts of high-energy-density physics, such as thermal (and radiative) transport as well as atomic processes such as ionization, especially for high-atomic-number materials.

12.5.3 Resonance The classical collisional absorption model of inverse bremsstrahlung becomes inefficient in a hot medium due to the rapid decrease of the electron–ion collision frequency with increasing temperature; see (12.191). The absorption of electromagnetic energy at high intensity may then be governed by collective effects, in particular the conversion of the radiant energy into electron plasma waves, which in turn heat the electrons. This process occurs most efficiently for densities near the critical density where the electron plasma frequency equals the electromagnetic wave frequency. Collective absorption processes become important for oblique incidence with TM polarization. As discussed in Section 12.2.5, there results an angle-dependent resonance function, which is efficient over a narrow range of angles of incidence where a portion of the electromagnetic energy is converted into plasma waves. Propagation of the electrostatic wave down the density gradient, away from the resonance region, is limited by convection (among other processes). The evolution of the electrostatic field amplitude Ees is found from the equations of continuity and motion, together with Poisson’s equation (see Section 12.1). For warm electrons, the equation to be solved is

2 2 ∂ 2 2 ∂ þ ωpe ðzÞ
3vth 2 E es ðzÞ ¼
ω2pe ðzÞEd sin ωt, (12.226) ∂t2 ∂z where E d is the electric field at the resonance point driving the electrostatic wave and is given by (12.94). It is a function of the free-space wave amplitude E0 , the angle of incidence θ0 , and the density scale length L . The boundary conditions for

682

Electromagnetic Wave-Matter Interactions

the electrostatic wave are such that there is an outward-directed wave for z  L and an evanescent wave for z  L. In a locally linear density gradient, the solution is (Estabrook et al., 1975)
E es ðηÞ ¼

L2 ω2 3 v2th

1=3

  Ed ℜe I ðηÞe
iωt ,

(12.227)

where the change of variable from z is


1 ω2 η ¼ ðL
zÞ 3L v2th

1=3 ,

(12.228)

with ð∞ I ðηÞ ¼



 iξ 3
iηξ dξ: exp 3

(12.229)

0

Numerical evaluation of (12.229) shows the maximum value occurs for η  2:6 and has the value I max  1:7. Then
2=3 ðEes Þmax ω ≲1:2 L : vth Ed

(12.230)

An estimate of the amount of electromagnetic energy absorbed by the resonance process is made by noting the resonantly driven field is ∽ν
1 eff and the width of the resonance region is ∽νeff , where νeff is a generalized loss term. (Normally we think of this as being the electron–ion collision frequency, but it could be linear or nonlinear wave-particle interactions, or even the propagation of the wave out of the resonance region.) For νeff =ω  1, the absorbed energy flux is ð∞

ð∞ E2z νeff E2d ðzÞ dz: I ra ¼ νeff dz ¼ 8π 8π jϵ j2 0

(12.231)

0

The permittivity for a linear density profile is ϵ ¼ 1
ðz=LÞ þ iðνeff =ωÞðz=LÞ: For the driver field being approximately constant over the narrow width of the resonance function, (12.231) becomes I ra 

ωL 2 E : 8 d

(12.232)

By conservation of energy, we can write I ra ¼ f ra cE20 =8π, where f ra is the fraction of the incident energy absorbed by resonance absorption (assuming no inverse

12.6 Dielectric Permittivity (Revisited)

683

Figure 12.9. Measured fraction of electromagnetic energy absorbed by the resonant process as a function of the angle of incidence and density scale height for three values of the field strength a0 .

bremsstrahlung). Using (12.94) in (12.232) gives the resonance absorption fraction f ra  φ2 ðτ Þ=2. This is an overestimate when compared to a numerical solution of (12.227); Figure 12.9 shows the fractional absorption as a function of 1= τ ¼ ðωL=cÞ 3 sin θ0 for three normalized electromagnetic field strengths a0 ¼ eE0 =me cω (see the footnote of Section 12.1). A discussion about the conversion of the energy in the plasma waves into heat should continue here; however, this would lead us away from our primary purpose, and so we refer the reader to the literature (e.g., Kruer, 1988) for the details.

12.6 Dielectric Permittivity (Revisited) We step aside, for the moment, to consider in more detail some aspects of the permittivity. Our view up to now has been that of plasma, but as high-energydensity materials can include solids and liquids, we need to consider modifications to the simple Drude model. First we look at the plasma conductivity as represented by the Drude model, and then we look at modifications to the permittivity/ conductivity for near-free electron metals.

684

Electromagnetic Wave-Matter Interactions

12.6.1 Plasma Conductivity Repeating (12.165), the Drude model for the conductivity is σ ð ωÞ ¼

ω2pe 1   ðνeff þ iωÞ; 4π ω2 þ ν2eff

(12.233)

the DC conductivity is defined for ω ¼ 0, thus σ dc ¼ ω2pe =4πνeff . As before, the effective collision frequency is determined from kinetic theory. The fundamental parameters νeff and ω of the complex conductivity allow three different regimes to be distinguished: (1) the so-called Hagen-Rubens regime for ω  νeff , (2) the relaxation regime for νeff  ω  ωpe , and (3) the transparent regime for ω  ωpe . The optical properties of the Hagen-Rubens regime are determined, for the most part, by the DC conductivity. The real part of the conductivity is frequency independent and large compared to the imaginary part, while the imaginary part increases linearly with frequency ω ω ω2pe σ i ðωÞ  σ dc ¼ : νeff 4π ν2eff

(12.234)

In the relaxation regime, where the term ω=νeff cannot be neglected, the effective scattering frequency is defined where the real and imaginary parts of the conductivity are equal: σ r ¼ σ i . For ω  νeff , we find σ r  ðνeff =ωÞ2 and σ i  σ dc ðνeff =ωÞ. For high frequencies, σ r  σ i . The transparent regime occurs where the optical frequency is greater than the plasma frequency. There is no clear appearance of ωpe , and both real and imaginary components of the conductivity fall monotonically with increasing frequency; σ r ∽ω
2 and σ i ∽ω
1 . These three regimes are shown in Figure 12.10. If we define the crossing point where σ r ¼ σ i ¼ σ dc =2, we see that for low frequencies σ i ∽ω, and for high frequencies σ i ∽ω
1 . A different perspective for the complex permittivity is shown in Figure 12.11. In the low-frequency regime, the real part is negative and large; for ω  νeff ,  2 and ϵ i  ω2pe =ωνeff . In the relaxation regime, both ϵ r  1
ωpe =νeff  2 parts decrease with increasing frequency: ϵ r  1
ωpe =ω and  2  2 ϵ i  ωpe =ω ðνeff =ωÞ. And in the transparent regime ϵ r  1
ωpe =ω approaches unity in the high-frequency limit, while ϵ i ! 0 at the critical density qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where ω ¼ ωpe . The zero crossing of the real part occurs at ω ¼ ω2pe
ν2eff . For νeff  ωpe , the real part of the permittivity becomes positive for optical frequencies

greater than the plasma frequency; also, σ ! νeff ω2pe =4πω2 . As ω ! ∞, ϵ ! 1;

685

12.6 Dielectric Permittivity (Revisited) 1017 1016

wpe

neff

1015

s (w)

1014 1013 1012 1011 1010 109 108 107 108 109 1010 1011 1012 1013 1014 1015 1016 1017 w

Figure 12.10. Frequency-dependent complex electrical conductivity according to the Drude model for ωpe ¼ 2:355  1015 s
1, which corresponds to an electromagnetic wave length of 800 nm. The real part is shown by the solid line and the imaginary part by the dashed line. The collision frequency is νeff ¼ 3:165  1012 s
1.

(b)

(a)

20.0

1010 109

15.0

108 107

10.0

106

wpe

105

5.0

103

neff

e(w)

e(w)

104 0.0

102 101

–5.0

0

10

10–1 –2

10

–10.0 –15.0

10–3 10–4 7 10 108 109 1010 1011 1012 1013 1014 1015 1016 1017 w

–20.0 1013

1014

1015

1016

1017

w

Figure 12.11. Frequency-dependent permittivity for the same conditions as Figure 12.10. The real part (solid line) is negative and constant up to the collision frequency and then increases with ω
2 until changing sign at the plasma frequency, finally approaching unity. The imaginary part (dashed line) is always positive, but decreases monotonically with increasing frequency, and changes slope from ω to ω
3 at the collision frequency.

686

Electromagnetic Wave-Matter Interactions

physically, this is easily seen. When the field changes sufficiently rapidly, the polarization process responsible for the difference between the electric field E and the displacement D cannot occur; thus η ! 1 and χ ! 0. In the opposite case,

 2 where ω  νeff , ϵ ! 1
ωpe =νeff þ i ω2pe =ωνeff . Plasma and metals with free electrons (conduction electrons) have ϵ < 1, since the negatively charged electrons include a negative polarizability of the medium. In contrast, bound electrons in atoms and molecules feel a restorative force that, in general, gives a positive polarizability and ϵ > 1. A third view is that of the complex refractive index, as shown in Figure 12.12. Writing ϵ ¼ ϵ r þ iϵ i ¼ ðη þ iχ Þ2 , we have for the indices of refraction and absorption



qffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 1 2 2 2 2 (12.235) η ¼ ϵ r þ ϵ r þ ϵ i and χ ¼
ϵ i þ ϵ 2r þ ϵ 2i : 2 2 In particular, for metals inpthe frequency range where ϵ ðωÞ  i4πσ=ω is valid, ffiffiffiffiffiffiffiffiffiffiffiffiffi ffi ϵ i  ϵ r and thus η ¼ χ ¼ 2πσ=ω. Further, we can write

Figure 12.12. Real and imaginary parts of the refractive index, η (solid line) and χ (dashed line). The conditions pffiffiffiffi are the same as in Figure 12.10. In the lowfrequency regime η ¼ χ∽ ω, while in the relaxation regime χ > η with η∽ω
2 and χ∽ω
1 .

12.6 Dielectric Permittivity (Revisited)

687


 ffi pffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1 χ 2 2 ϵ ¼ η þ χ exp i tan : (12.236) η pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi Since H ¼ ðc=ωÞk  E ¼ ðω=cÞ ϵ k E, (12.236) shows H is η2 þ χ 2 times E in magnitude, and tan
1 ðχ=ηÞ from it in phase. When η ¼ χ, the phase difference is π=4. 12.6.2 Near-Free Electron Metals Near-free electron metals (also referred to as simple metals) are characterized by the kinetic energy of the “free” electrons being large compared to the potential energy of the underlying periodic lattice. Examples of simple metals include the alkaline metals as well as metals such as aluminum and copper. Consequently, the free-electron gas theory applied to these metals gives generally good agreement with experiments. In many metals, the electromagnetic wave can interact with the bound electrons, and the equation of motion for the electrons needs to include an additional restoring force, thus e E: (12.237) x€ þ Γx_ þ Ω2 x ¼ me Here, Γ is a friction coefficient that includes effects of collisions, and Ω2 is a material-specific oscillation frequency. For an electromagnetic wave described by E ¼ E0 e
iωt , the solution to this equation is x¼

eE0 : me Ω
ω2
iωΓ 

2

(12.238)

The conductivity and permittivity become σ ðωÞ ¼
i

ω2pe ω , 2 4π Ω
ω2
iωΓ

(12.239)

and ω2pe

: (12.240) Ω2
ω2
iωΓ This is known as the Lorentz model for the dielectric permittivity. The optical properties are similar to those for plasma. The exception is that when the electromagnetic frequency is near a resonance (ω  Ω), the imaginary part of the permittivity is proportional to the inverse of Γ. This parameter includes the effects of the oscillator friction Γ0 and collisions; thus Γ ¼ Γ0 þ νeff . Both the Drude and Lorentz models are widely used. However, the absorption of electromagnetic radiation in near-free electron metals may be more complex, ϵ ðωÞ ¼ 1 þ

688

Electromagnetic Wave-Matter Interactions

since the photon energy can be as high as several electronvolts, which is approximately the energy of interband transitions. Models for the permittivity should include interband as well as intraband transitions through their conductivities 4π ½σ intra þ σ inter : (12.241) ω Intraband transitions are usually described by a modified Drude model ϵ ðωÞ ¼ 1 þ i

ω2pe 1 σ intra ðωÞ ¼ i , 4πM ωð1 þ iνeff Þ

(12.242)

where the parameter M describes the strength of intraband transitions. One interpretation of M is that it represents the effective number of free electrons participating in the intraband transitions and is different from the free-electron density ne . A second interpretation is that the electron’s effective mass is different from its single-particle mass me . In near-free electron metals, M has a value slightly larger than unity; for example, for aluminum it is 1.6. Use some caution when looking in the literature to be sure it is the optical mass and not the thermal mass. The interband contribution is more complex and depends strongly on the band structure of the particular metal (Ashcroft & Sturm, 1971). A particularly simple model for parallel bands is obtained by modifying the Lorentz model (12.239). Then the conductivity becomes σ inter ðωÞ ¼
i

ω2pe ω X fn , 2 4π n Ωn
ðω þ iΓn Þ2

(12.243)

where the sum is over different transitions with frequencies Ωn and oscillator strengths f n . Combining the intraband and interband models, assuming just one interband transition (such as might be found in aluminum; aluminum actually has two parallel band transitions at 3.4 μm [0.35 eV] and 0.82 μm [1.5 eV]; Smith & Segall, 1986) results in the Drude-Lorentz model ϵ ðωÞ ¼ 1


ω2pe Aω2pe 1 þ 2 : M ωðω þ iνeff Þ Ω
½ω þ iðΓo þ νeff Þ2

(12.244)

The parameters M, A, Ω2 , and Γ0 can be determined from room-temperature data. Figure 12.13 sketches the behavior of the Lorentz permittivity ϵ L ðωÞ, the last term in (12.244). Optical effects are generally most pronounced in the solid and liquid states, where some sort of structural order exists. Thermal excitation destroys this order or symmetry on which band structure is modeled, so the effects become less

12.7 Kinetic Instabilities

689

Figure 12.13. Real (solid line) and imaginary part (dashed line) of the Lorentz permittivity. The width of the interband transitions is γ ¼ Γ0 þ νeff .

important at elevated temperatures. The preceding discussion assumes that intraband contributions are not dominant. Metals that have a complex band structure, such as copper with d-band transitions, cause the intraband contribution to be large and thus the Lorentz model is not appropriate. There is the matter of the effective collision frequency νeff . The physics of collisions in metals is quite complicated, and we omit an in-depth discussion. Real metals contain impurities as well as dislocations. These regular lattice disruptions cause electron–electron and electron–phonon collisions to become significant. In a perfect crystalline solid, the quantum mechanical picture allows electrons to propagate through the lattice without experiencing collisions. But even in a roomtemperature, perfect lattice, electrons can scatter off the lattice vibrations caused by thermal excitations – the phonons. Electron–phonon collisions become important when the kinetic energy of the electrons is comparable to the strength of the random potential energy. Electron–electron interactions are important for narrow bands, while electron–phonon interactions are expected to be strong for soft lattices.

12.7 Kinetic Instabilities Plasma, much like fluids, can experience a multitude of instabilities. Broadly speaking, instabilities in high-energy-density matter may be classified as

690

Electromagnetic Wave-Matter Interactions

Figure 12.14. Schematic of the density profile in the underdense region where many instability processes occur that affect laser-plasma coupling.

hydrodynamic or kinetic, with various subclasses. Certain hydrodynamic instabilities were addressed in Section 6.9; we now discuss a few of the kinetic instabilities that can exist in the presence of an intense electromagnetic wave. These are, for the most part, the more important ones drawn from a considerable list of possibilities. Figure 12.14 is a schematic picture of a laser-produced density profile, showing the approximate locations of the plasma instabilities. We tend to visualize the plasma as being quiescent, but as we have seen before, the plasma environment is anything but benign. There are processes that produce turbulence, fluctuations, and profile modification, to name just a few. These perturbations can interact strongly with an electromagnetic wave that scatters the wave and creates other types of waves. These “manufactured” waves are unstable and can grow to large amplitude, which may result in substantial scattering of the incident electromagnetic wave as well as the production of energetic electrons or ions. Laser-plasma instabilities, common in laboratory high-energy-density physics experiments, have a few analogies in astrophysical situations, particularly in certain types of solar bursts. Our focus is on laboratory conditions for which there has been an extensive development of theory augmented by experiments. We can easily imagine the complexity of interactions among multiple waves, so we seek to provide a somewhat simplistic understanding of a few of the interactions. The mathematical analysis of instabilities is well developed, but is far too complex for us to present here. This section is intended to provide the reader with an overview of plasma instabilities encountered in high-energy-density physics, without the detailed development. An excellent introduction to instability analysis and dispersion relations may be found in Kruer (1988).

12.7 Kinetic Instabilities

691

Of the several nonlinear wave–wave interactions of interest to us, “parametric instabilities” are historically among the earliest investigated. Parametric amplifiers are well-known devices in electrical engineering for which the theory is basically a linear one, but it is linear about an oscillating equilibrium. There is a close analogy between the coupled-oscillators problem of classical mechanics and the plasma physics version of the problem. Rather than mechanical components, we have waves. The following work closely follows that of Chen (1974).

12.7.1 Matching Conditions The equation of motion of a particle behaving as a simple harmonic oscillator (oscillator #1) is d 2 x1 þ ω21 x1 ¼ 0, dt 2

(12.245)

where x1 is the amplitude and ω1 is its resonant frequency. If it is driven by a timedependent force that is proportional to the product of the amplitude E 0 of the driver (pump) and the amplitude x2 of a second oscillator (oscillator #2), the equation of motion becomes d 2 x1 þ ω21 x1 ¼ c1 x2 E, dt 2

(12.246)

where c1 is a constant that specifies the strength of the coupling. A similar equation holds for x2 d 2 x2 þ ω22 x2 ¼ c2 x1 E: dt 2

(12.247)

Let x1 ¼ x 1 cos ωt, x2 ¼ x 2 cos ω0 t, and E ¼ E 0 cos ω0 t. Then we find  2  ω2
ω02 x 2 cos ω0 t ¼ c2 E0 x 1 cos ω0 t cos ωt 1 ¼ c2 E0 x 1 f cos ½ðω0 þ ωÞt  þ cos ½ðω0
ωÞt g: 2 (12.248) The driving terms on the right-hand side can excite oscillator #2 with frequencies ω0 ¼ ω0 ω. The nonlinear driving terms can cause a frequency shift, so ω0 does not need to be exactly ω2 , but only approximately so. Furthermore, ω0 can be complex because there can be damping or growth, so oscillator #2 has finite Q and can respond to a range of frequencies. Now let x1 ¼ x 1 cos ω00 t and x2 ¼ x 2 cos ½ðω0 ωÞt . The equation of motion for oscillator #1 becomes

692

Electromagnetic Wave-Matter Interactions

 2  ω1
ω002 x 1 cos ω00 t 1 ¼ c1 E 0 x 2 ð cos f½ω0 þ ðω0 ωÞt g þ cos f½ω0
ðω0 ωÞt gÞ 2 1 ¼ c1 E 0 x 2 f cos ½ð2ω0 ωÞt  þ cos ωt g: (12.249) 2 The driving terms can excite not only the original oscillation x1 ðωÞ, but also new 00 frequencies ω ¼ 2ω0 ω. Considering the case jω0 j  jω1 j so that x1 ð2ω0 ωÞ can be neglected, coupled equations among x1 ðωÞ, x2 ðω0
ωÞ, and x2 ðω0 þ ωÞ are derived  2  2 ω x1 ðωÞ

ω 1 h i c1 E ½x2 ðω0
ωÞ þ x2 ðω0 þ ωÞ ¼ 0 2 ω2
ðω0
ωÞ x2 ðω0
ωÞ
c2 Eðω0 Þx1 ðωÞ ¼ 0 h 2 i ω22
ðω0 þ ωÞ2 x2 ðω0 þ ωÞ
c2 Eðω0 Þx1 ðωÞ ¼ 0: (12.250) Moreover, the values of ω and k for each wave must satisfy dispersion equations ϵ 0 ðω0 ; k 0 Þ ¼ 0, ϵ 1 ðω1 ; k1 Þ ¼ 0 and ϵ 2 ðω2 ; k 2 Þ ¼ 0. The dispersion relation is given by setting the determinate of the coefficients of (12.250) to zero 2 ω
ω21 c1 E c1 E 2 2 ¼ 0: c2 E (12.251) ð ω
ω Þ
ω 0 0 2 2 cE 2 0 ð ω þ ω Þ
ω 2 0 2 A solution with ℑmðωÞ > 0 indicates an instability. For small frequency shifts and small damping or growth rates, ω and ω0 can be set approximately equal to the undisturbed frequencies ω1 and ω2 , so the frequency-matching condition can be written ω0  ω2 ω1 . When the oscillators are waves in plasma, ωt should be replaced by ωt
k  x. There is also a wavevector matching condition k0  k2 k1 , which describes spatial beating. These two conditions can be interpreted as conservations of energy and momentum. The simultaneous satisfaction of frequency and wave-vector matching conditions is possible only for certain combinations of waves. For one-dimensional problems, the required relationships can be shown on an ω-k diagram. Figure 12.15 shows the dispersion curves of ion acoustic waves (straight lines), electron plasma waves (wide parabola), electromagnetic waves (narrow parabola), the incident pump wave (ω0 Þ and the two decay waves (ω1 and ω2 ). The requirement that the three waves be eigenmodes implies that the end points of the vectors be situated on dispersion curves. The parallelogram construction ensures that the frequency and wave number matching conditions are satisfied simultaneously. Four instabilities are shown: a. Parametric decay: An incident electromagnetic wave of large phase velocity (ω0 =k0  c) excites an electron plasma wave and an ion acoustic wave moving in opposite directions. Since k0 is small, k1 
k2 .

12.7 Kinetic Instabilities (a)

(b)

(c)

(d)

693

Figure 12.15. Parallelogram constructions showing the frequency and wave vector matching conditions for four parametric instabilities. In each case ω0 is the incident electromagnetic wave, and ω1 and ω2 the decay waves.

b. Stimulated Raman backscattering: A light wave can also excite an electron plasma wave (also known as a Langmuir wave) and a backward-moving light wave. c. Two-plasmon decay: An incident light wave decays into two oppositely propagating electron plasma waves (plasmons). Frequency matching can be satisfied only if ω0  2ωpe (i.e., ne ¼ ncr =4). d. Stimulated Brillouin backscattering: A light wave excites an ion acoustic wave and another light wave moving in the opposite direction. We note that from the constructions of Figure 12.15, an electromagnetic wave cannot decay into two other electromagnetic waves, nor can an electron plasma wave decay into two other electron waves. However, an ion acoustic wave may decay into two ion waves.

694

Electromagnetic Wave-Matter Interactions

Stimulated Brillouin and stimulated Raman backscattering can lead to reduced absorption of the incident electromagnetic radiation, while the other two instabilities may lead to increased absorption. Parametric instabilities will occur at any amplitude of the driving force unless there is a damping (collisional or Landau damping), which will limit the growth rate of the instability or prevent it altogether if the pump wave is not strong enough.

12.7.2 Damping Electron–ion collisions are the simplest mechanism for removing energy from electron plasma waves. The regular motion of the electrons in the electric field of the wave is converted to random motion, and thus thermal energy, by the collisions. The thermal energy created is balanced by the loss from the energy of the electric field, with damping rate ν ν

E 20 ne me v2osc ¼ νei , 8π 2

(12.252)

where the oscillatory velocity is vosc ¼ eE0 =me ω. We see that ν ¼ ω2pe νei =ω2 , and for an electron plasma wave ω  ωpe , and so ν  νei . Collisional damping can play an important role in determining the threshold intensity for instabilities. The electron–ion collision frequency νei is a function of ionization level, electron density, and electron temperature, so it can be significant in high-atomic-number plasma. However, collisional thresholds are often greatly exceeded, particularly for very intense and/or long wavelength electromagnetic radiation. Another damping mechanism comes into play; it is a collisionless interaction. The purely mathematical discovery of wave damping without energy dissipation by collisions is one of the most astounding results of plasma physics research. This unexpected result has been verified in the laboratory and also is seen in galactic structure. The kinetic treatment of galaxy formation from a gas of stars interacting via gravitational forces shows that the growth of instabilities results in the formation of spiral arms, but the process is limited by Landau damping. Conceptually, Landau damping is straightforward, but the rigorous mathematical development is difficult, and we present only the key results. The derivation of the dispersion relation for electron plasma oscillations using the Vlasov equation gives 1¼

ω2pe k

ð∞

2
∞

1 ∂^f 0 ðvÞ dv, v
ðω=k Þ ∂v

(12.253)

where ^f 0 is a normalized one-dimensional velocity distribution function. The integral in (12.253) is not straightforward to evaluate because of the singularity at v ¼ ω=k.

12.7 Kinetic Instabilities

695

Using the methods of contour integration, we arrive at 2 þ∞ ! 3 ð ^ ^ ω2pe 1 ∂f 0 ðvÞ ∂f 0 5, 1 ¼ 2 4P dv þ iπ v
ð ω=k Þ ∂v ∂v k
∞

(12.254)

vph

where P is the Cauchy principal value. The integral evaluates to þ∞ ð


∞

1 ∂^f 0 ðvÞ dv ¼ v
ðω=kÞ ∂v

þ∞ ð


∞

1 v
vph

2 ^f 0 ðvÞdv,

(12.255)

where vph ¼ ω=k is the phase velocity of the wave. Assuming vph  v, we can expand the denominator of the right-hand side of (12.255), then performing the integral, we obtain the average !
2 D 
2 E k v2th (12.256) 1þ3 2 : v
vph ¼ ω vph Assuming the thermal correction term is small, then (12.254) becomes 2 ! 3 ^ ω2pe k 2 ∂f 0 5, 1 ¼ 2 4 2 þ iπ ω ∂v k

(12.257)

vph

and if the imaginary term is small, then 2 ω ¼ ωpe 41 þ

π ω2pe i 2 2 k

∂^f 0 ∂v

! 3 5:

(12.258)

vph

Using the one-dimensional Maxwellian distribution for ^f 0 , we arrive at the Landau damping rate !
 ω2pe ωpe 3 pffiffiffi
3=2 exp
2 2 γ ¼ ℑmðωÞ ¼
π e ωpe kvth k vth " # rffiffiffi π
3=2 ωpe 1 ¼
: (12.259) exp
e 8 2ðkλDe Þ2 ðkλDe Þ3 The negative sign indicates a damping. From this expression, we see the damping of an electron plasma wave is a strong function of its phase velocity. The damping is small for small kλDe , but becomes important whenever kλDe ≳0:4; pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi λDe ¼ kB T e =4πne e2 is the electron Debye length. For these values of kλDe , the equilibrium distribution function is distorted by the electron plasma wave.

696

Electromagnetic Wave-Matter Interactions

A simple mechanical analogue of linear Landau damping is that of a surfboard rider trying to catch an ocean wave. If the surfer is not moving, a passing wave causes the surfboard to bob up and down and the wave goes by with the surfer gaining no energy on the average. However, when the surfer sees a wave to his or her liking, he or she paddles furiously to gain a velocity close to the phase velocity of the wave. If the timing of the surfer is correct, the wave is caught and the surfboard is pushed along by the wave gaining energy at the expense of the wave, and the wave is damped. However, if the surfboard is moving slightly faster than the wave, it pushes on the wave and its energy is transferred to the wave. It is a bit different in plasma, where there are electrons moving both faster and slower than the wave. A Maxwellian distribution has more slow electrons than fast ones, so there are more particles taking energy from the wave than vice versa, and the wave is damped. Particles with velocity approximately equal to the phase velocity are trapped in the wave, resulting in the distribution function being flattened near the phase velocity. The perturbed distribution function has the same number of particles, but has gained energy at the expense of the wave. In contrast, if the distribution function contains more fast particles than slow particles, a wave can be excited. From (12.258), it is apparent that ℑmðωÞ is positive for ∂^f 0 =∂v positive at v ¼ vph . Waves with phase velocity in the region of positive slope will be unstable, taking energy from the particles. The physical picture of a surfer catching a wave is not precise, as there are actually two types of Landau damping: linear and nonlinear. The linear theory cannot adequately treat the trapping phenomenon. We can see this from the simple equation of motion me d2 x=dt 2 ¼
eEðxÞ. If we insert an exact value for x, the equation is nonlinear since E ðxÞ∽ sin kx. The linear theory uses x ¼ x0 þ v0 t, but this is no longer valid when a particle is trapped. When it encounters a potential large enough to reflect it, its position and velocity are affected by the wave and are not given by their unperturbed values. For the motion of a particle resonant in an electric field E0 sin ðkx
ωtÞ, and transferring to a frame moving with the phase velocity, the equation of motion becomes me ξ€ ¼
eE0 sin kξ, where ξ ¼ x
ðωt=k Þ. For electrons near the bottom of the potential trough, we can take sin kξ  kξ and we have pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 € ξ ¼
ωb ξ, where ωb ¼ ekE 0 =me is the bounce frequency. The linear theory describes only the early phase of this motion, for ωb t  1. Alternatively, the linear theory requires γ  ωb , which is to say, the wave damps before electrons can oscillate in the trough. In the opposite limit, where γ  ωb , electrons are trapped in the potential trough. Since the motion of the particles is resonant, the amplitude of the wave oscillates, as it first gives and then recovers energy from the particles. In other words, first there are more electrons moving slightly slower than the phase velocity of the wave and the wave gives up energy to the particles. This leads to more electrons moving slightly faster than the phase velocity, and now the wave regains

12.7 Kinetic Instabilities

697

energy from the particles. The process tends to repeat. To be a bit more precise, electrons in a sinusoidal trough actually have bounce frequencies that depend on their initial positions. Hence the periodic interchange of energy between the wave and the particles gradually phase-mixes away, as the slope of the distribution function flattens in the neighborhood of the phase velocity. The nonlinear theory of Landau damping is required when a large amplitude electron plasma wave can bring the oscillation velocity of an initially cold, mainbody particle into resonance with the field. That is, when ðeE0 =me ωÞ  ω=k, numerous particles can interact “resonantly” with the wave. A strong, nonlinear damping results as electrons are efficiently accelerated by the wave. The wave amplitude is often referred to as the amplitude at which “wave breaking” occurs in cold plasma. At the onset of wave breaking, large numbers of formerly nonresonant particles become strongly “trapped.” The wave energy is suddenly damped as these slow particles are accelerated by falling into the potential troughs of the wave. At this wave breaking point, ωb ¼ ω, and its inverse is the characteristic time for resonant particles to take energy from the field. The amplitude of the field at which particles are nonlinearly brought into resonance with the wave is reduced in warm plasma because faster electrons are more easily brought into resonance, and the sizable pressure force associated with density fluctuations of the electron plasma wave gives an additional acceleration. The topic of wave breaking has a place in the discussion of high-energy-density physics, but space limitations preclude us from exploring this further.

12.7.3 Instability Threshold Even a small amount of damping will prevent an instability from growing unless the pump wave is strong enough. Introducing Γ1 and Γ2 as the threshold damping rates for each of the two oscillators considered in the preceding, we have for oscillator #1 d 2 x1 dx1 þ ω21 x1 þ 2Γ1 ¼ c1 x2 E, 2 dt dt

(12.260)

with a similar expression for oscillator #2. The equations of motion become  2  ω1
ω2
2iωΓ1 x1 ðωÞ ¼ c1 x2 E h i ω22
ðω
ω0 Þ2
2iðω
ω0 ÞΓ2 x2 ðω
ω0 Þ ¼ c2 x1 E: (12.261)

698

Electromagnetic Wave-Matter Interactions

Consider the case where ω  ω1 and ω0
ω  ω2 , but ω0 þ ω is far enough from ω2 to be nonresonant. Expressing x1 , x2 , and E in terms of their peak amplitudes, we have i  2 h ω
ω21 þ 2iωΓ1 ðω0
ωÞ2
ω22
2iðω0
ωÞΓ2 ¼ c1 c2 E2 : (12.262) At threshold, ℑmðωÞ ¼ 0. The lowest threshold will occur at exact frequency matching, ω ¼ ω1 and ω0
ω ¼ ω2 , giving   c1 c2 E2 thresh ¼ 4ω1 ω2 Γ1 Γ2 : (12.263) The threshold goes to zero as the damping of either wave goes to zero.

12.7.4 Parametric Decay and Two-Stream Instabilities Consider Figure 12.15a, where an electromagnetic wave (ω0 , k 0 ), the pump, drives an electron plasma wave (ω2 , k2 ) and a low-frequency ion acoustic wave (ω1 , k 1 ). Since ω1 is small, ω0  ωpe . The behavior is different for ω0 < ωpe (the oscillating two-stream instability) from that for ω0 > ωpe (the parametric decay instability). Specifying a particular planar geometry with the pump wave normally incident in the z-direction, with electric field E0 cos ω0 t. Let there be a density perturbation in plasma of the form n1 cos k1 z1 , which can occur spontaneously as a component of the thermal fluctuations. The pump wave satisfies the dispersion relation ω20 ¼ ω2pe þ c2 k20 , so k0  0 for ω0  ωpe and E 0 can be taken as spatially uniform. If ω0 < ωpe , the electrons will move in the direction opposite to E 0 (ions, being heavy, do not move on this time scale). The density perturbation causes charge separation, which creates an electric field E 1 , which oscillates at frequency ω0 . For collisionless plasma, ϵ ðωÞ ¼ 1
ω2pe =ω20 . Then, from Section 12.2.6 we find the ponderomotive force due to the combined fields is E 1 ω2pe D 1 ω2pe ∂h2E 0 E 1 i 2 , F¼
r ðE 0 þ E1 Þ 
∂z 8π ω20 8π ω20

(12.264)

since E0 is spatially uniform and is much larger than E 1 . Since E1 changes sign with E 0 , this force does not average to zero. As seen in Figure 12.16a, the ponderomotive force is zero at the peaks and troughs of n1 , but is large where rn1 is large. The spatial distribution of the ponderomotive force pushes electrons from regions of low density to regions of high density (see Section 12.2.6 and Figure 12.5). The resulting electric field drags the ions, and the density perturbation grows. The threshold value for the ponderomotive force is the value just sufficient to overcome the pressure gradient, which tends to flatten the density

12.7 Kinetic Instabilities (a)

699

(b)

Figure 12.16. (a) For ω0 < ωpe , the ponderomotive force acts to increase the density fluctuations, whereas (b) for ω0 > ωpe , a reduction takes place.

profile. The density perturbation does not propagate, so ℜeðω1 Þ ¼ 0. This is labeled the oscillating two-stream instability because the time-averaged movement of the electrons creates a distribution function that is double-peaked. If ω0 > ωpe , the direction of the electron velocity E1 and F are reversed, and the ponderomotive force moves ions from dense regions to less-dense regions. The density perturbation would decay if it did not move, but could grow if it traveled at an appropriate phase velocity, so that the inertial delay between the application of the ponderomotive force and the change of ion positions causes the density maxima to move into the region into which the ponderomotive force is pushing the ions. This speed is the ion acoustic speed. The phase of F is exactly the same as the phase of the electrostatic restoring force in an ion acoustic wave, where the potential is maximum at the density maximum. Consequently, F adds to the restoring force. The electrons oscillate with large amplitude if the pump wave field is near the natural frequency of the electron plasma wave. That is, ω22 ¼ ω2pe þ 3k2 v2th , the Bohm-Gross frequency, the last part being a thermalcorrection term. The pump wave cannot have exactly the frequency ω1 because the beat between ω0 and ω2 must be at the ion acoustic frequency ω1 ¼ kcs . If this frequency is matched so that ω1 ¼ ω0
ω2 , both an ion acoustic wave and an electron plasma wave are excited at the expense of the pump wave. This is the parametric decay instability. Analysis of the instabilities is tedious, so we summarize the result. The dispersion relation for the oscillating two-stream instability (Kruer, 1988) in the weak limit
 νe 2 1 vosc 2 2 γþ þδ þ ω0 δ ¼ 0, (12.265) 4 vth 2

700

Electromagnetic Wave-Matter Interactions

where νe accounts for either collisional or Landau damping of the electron waves, and δ ¼ ω0
ω2 is the mismatch between the pump and electrostatic wave frequencies. The mismatch that corresponds to maximum growth is found by taking the derivative ∂=∂δ of (12.265) and setting ∂γ=∂δ to zero. Hence δ ¼
ðvosc =vth Þ2 ðω0 =8Þ. Using this in (12.265) yields the maximum growth rate
 1 vosc 2 νe (12.266) γ¼ ω0
: 8 vth 2 For growth to exist, the threshold value is found when γ ¼ 0
2 vosc νe ¼4 : vth thresh ω0

(12.267)

Plasma inhomogeneity affects growth rates. Typically the threshold for the onset of an instability is raised because the plasma wave convects out of the interaction region. Absolute instabilities tend to build up at the location of their generation, whereas convective instabilities move away from the interaction region. On the other hand, if the convection is in the direction perpendicular to that of the incoming electromagnetic wave, the product of the instability (the electron plasma waves) still accumulate. Clearly, these accumulations can have a profound effect on the subsequent development of the density distribution. For plasma with a density gradient, the oscillating two-stream instability is excited over a limited region of space where ω0  ωpe . There is then a loss mechanism where unstable waves can propagate out of the creating region. We can estimate the effect of a linear density variation of scale length L by noting that the excitation of the plasma wave is strongest where its parallel wave number kǁ is aligned with the electric field vector of the pump wave. However, at a lower density, the wave vector must have a component k z down the density gradient so that the increase in frequency due to the thermal correction balances the decrease due to lower density. Hence z 3k2z v2th ¼ ω2pe , L

(12.268)

where z is measured from the point where k ¼ kǁ . As kz increases, the efficiency of the coupling between the pump wave and the electron plasma wave decreases, since the plasma wave begins to propagate more and more in the direction perpendicular to the electric field vector of the pump. Estimating the size of the interaction region by the condition kz  k ǁ , l  3v2th k2ǁ L=ω2pe , then the time it takes for the wave to propagate out of this creation region is the integral of v
1 gz over the 2 region extent; vgz ¼ 3kz vth =ωpe is the z-directed component of the group velocity of the plasma wave. This gives an effective damping rate, and the right-hand side of (12.267) becomes 2=kǁ L.

12.7 Kinetic Instabilities

701

12.7.5 Stimulated Raman Scattering To those readers familiar with molecular physics, Raman scattering is a familiar topic. Raman scattering arises when light is scattered from molecular vibrations. The analogous process in plasma is scattering from electron oscillations. Because the pump wave stimulates growth of the electron plasma wave, we speak of stimulated Raman scattering. In Figure 12.15b, we see that the Raman process may lead to a backward-propagating electromagnetic wave (ω2 , k2 ). Since the minimum frequency of a light wave in plasma is the electron plasma frequency, the frequency-matching condition ω0 ¼ ω1 þ ω2 requires ω0 ≳ 2ω1 ¼ 2ωpe . Thus, the Raman instability occurs at ne ≲ ncr =4 and can scatter a large amount of the incident radiation long before it gets to electron densities where collisional absorption is efficient. The fraction of the incident energy going into the plasma wave is ω1 =ω0 , and this energy will heat the plasma as the wave damps. Because the electron plasma wave can have a very high-phase velocity (approaching that of the speed of light), very energetic electrons will be created. The physics of the Raman instability is simple. For an incident electromagnetic wave with electric field amplitude E0 , propagating in the direction of the electron density fluctuations δne associated with an electron plasma wave, the electrons are oscillating with vosc ¼ eE0 =me ω0 , which produces a transverse current δJt ¼
evosc δne . If the wave numbers and frequencies are properly matched, this transverse current generates the scattered electromagnetic wave with amplitude δE. This scattered wave can then interfere  2withthe pump wave to produce a variation in the ponderomotive force F∽r E =8π ¼ rðE 0  δE Þ=4π. Variations in the pressure result in pushing plasma to enhance the density fluctuations, as previously discussed. Thus, there exists an unstable feedback loop, where an initially small density fluctuation can grow to a large value before limiting processes set in. Drawing upon the work of Kruer (1988), the expression describing the Raman instability begins with the wave equation for the electromagnetic waves, expressed in terms of the vector potential, which is (12.6)
2  ∂ 2 2 ϵ 2
c r A ¼ 4πcJt : (12.269) ∂t As we are considering only backscatter, A  rne ¼ 0, the transverse current can be expressed as Jt ¼
ene vosc , and the equation of motion is ∂vosc e e ∂A : ¼
E¼ me me c ∂t ∂t Setting the permittivity to unity, (12.269) becomes
2  ∂ e2 ne 2 2
c r A: A ¼
4π ∂t2 me

(12.270)

(12.271)

702

Electromagnetic Wave-Matter Interactions

The scattering of the large-amplitude electromagnetic wave Al by a smallamplitude density fluctuation δne is easily determined by using A ¼ Al þ δA and ne ¼ ne0 þ δne in (12.271); ne0 is the uniform background electron density. We then find the scattered wave is produced by
2  ∂ e2 δne 2 2 2 δA ¼
4π
c r þ ω δne Al ¼
ω2pe Al , (12.272) pe ∂t2 me ne0 where the right-hand side is proportional to the transverse current. We see that the interaction of the electromagnetic wave with the density fluctuations produces an additional transverse current proportional to the right-hand side of (12.272). This expression describes wave beating; the fluctuations and pump wave beat together to drive a scattered electromagnetic wave. This process produces beat waves having both the sum and the difference of the frequencies and wave numbers of the two driving waves. Density fluctuations were discussed in Section 3.7. We redo that work to take into account both electromagnetic waves. The continuity equation is ∂ne þ r  ðne ue Þ ¼ 0, ∂t and the momentum equation, for fixed ions, is

 ∂ue ue  B 1
rPe , þ ue  rue ¼
e E þ me c ne ∂t

(12.273)

(12.274)

Pe being the electron pressure. The fluid velocity of the electrons can be separated into longitudinal and transverse components, ue ¼ ul þ vosc , and the longitudinal component of (12.274) becomes
 ∂ul e 1 eA 2 1 ¼ rϕ
r ul þ
rPe , (12.275) me 2 me c me ne ∂t where we have used E ¼
c
1 ð∂A=∂t Þ
rϕ. The second term on the right-hand side is the ponderomotive force, which involves both longitudinal and transverse components of the electric field. We now take ul ¼ δu, ϕ ¼ δϕ, and A ¼ Al þ δA to get ∂ðδne Þ þ neo r  ðδuÞ ¼ 0 ∂t

(12.276)

for the continuity equation and ∂ðδuÞ e e2 v2 ¼ rðδϕÞ
2 2 rðAl  δAÞ
3 th rðδne Þ ∂t me me c ne0

(12.277)

12.7 Kinetic Instabilities

703

for the momentum equation, where we have used the adiabatic equation of state for the electron pressure. These two expressions may be combined to give a description of how electron density fluctuations are generated by variations in the intensity of the electromagnetic wave
2  ∂ ne0 e2 2 2 2 2 δn þ ω
3v r ¼ r ðAl  δAÞ: (12.278) e pe th ∂t 2 m2e c2 Expressions (12.272) and (12.278) describe the coupling of the electrostatic and electromagnetic waves. Development of the dispersion relation for the Raman instability from these two equations is more complicated than we wish to go into, so we summarize the results. Using Al ¼ A0 cos ðk0  x
ω0 tÞ and Fourier analyzing the two expressions gives i ω2 k2 v2  2 h pe osc , ω
ω21 ðω
ω0 Þ2
ðk
k0 Þ2 c2
ω2pe ¼ 4

(12.279)

where ω1 is the frequency of the electron plasma wave given by the Bohm-Gross expression. Let ω ¼ ω1 þ δω, then we find the maximum growth occurs when the scattered electromagnetic wave is also resonant, that is ðω1
ω0 Þ2
ðk
k0 Þ2 c2
ω2pe ¼ 0:

(12.280)

The growth rate γ is given by δω ¼ iγ, and so " #1=2 ω2pe 1 : γ ¼ kvosc 4 ω1 ðω0
ω1 Þ

(12.281)

The growth rate maximizes for direct backscatter, and we see the wave number, from (12.280), is
1= ω0 ωpe 2 k ¼ k0 þ 1
2 : c ω0

(12.282)

The wave number starts from k ¼ 2k0 for ne  ncr =4 and goes to k ¼ k0 when ne ¼ ncr =4. In the preceding, we have addressed only backscatter. When A  rne 6¼ 0 for ne  ncr =4, sidescatter can occur, and the growth rate of the instability is reduced, since the electric vectors of the incident and scattered light waves are no longer aligned. In the limit δA  A0 ¼ 0, the growth rate vanishes. Sidescatter occurs preferentially out of the plane of polarization. The threshold for the instability is set by damping of the unstable waves. Add the term ν2 ½∂ðδAÞ=∂t  to (12.272), where ν2 is the energy-damping rate for the

704

Electromagnetic Wave-Matter Interactions

scattered light wave, and add vth ½∂ðδne Þ=∂t  to (12.278), then redo the instability pffiffiffiffiffiffiffiffi analysis. Kruer (1988) obtains a threshold condition γ 0  γ1 γ2 , where γ1 and γ2 are the amplitude-damping rates for the electron plasma wave and scattered light wave, respectively; γ0 is the growth rate in the absence of damping. For backscattered radiation, where ωpe =ω0  1=2, we find v 2
ω 2 ν2 osc pe ei > : (12.283) c ω0 ω0 ωpe As in the previous discussion about the oscillating two-stream instability, the threshold for stimulated Raman scattering is determined primarily by the plasma density gradient. A heuristic calculation of the threshold in plasma with a linear density gradient profile yields a backscatter threshold ðvosc =cÞ2 > 2=k0 L. Stimulated Raman scattering creates electron plasma waves, which are damped either by collisional or collisionless (Landau damping) mechanisms. Typically, Landau damping dominates over collisional damping. As noted in Section 12.7.2, very energetic electrons are created by Landau damping.

12.7.6 Two-Plasmon Decay An instability related to the stimulated Raman instability is that of two-plasmon decay, in which the incident electromagnetic wave decays into two electron plasma waves, as depicted in Figure 12.15c. Since the two plasma waves have frequencies approximately ωpe , this instability takes place at a density ne  ncr =4. Instability analysis follows along the same lines as that for stimulated Raman scattering. Assuming fixed ions, then linearizing (12.273) and (12.275) and eliminating the term ∂ðr  δul Þ=∂t between them gives

∂2 ðδne Þ 2 ∂½vosc  rðδne Þ 2 2 þ ω
3v r
ne0 r2 ½vosc  δul  ¼ 0: δne þ pe th 2 ∂t ∂t (12.284) Fourier analysis of this expression yields two coupled equations. Choosing ω  ωpe and neglecting any responses at ω þ ω0 or ω
2ω0 as being offresonant, we find equations for δne ðω; kÞ and δne ðω
ω0 ; k
k 0 Þ. We then approximate δul with a Fourier analysis of the continuity equation (12.273) to obtain the dispersion equation h i92 8 h i