Plasmas: The First State of Matter 9781107037571, 2013019974

Table of contents :
Cover
Title
Copyright
Dedication
Contents
List of Illustrations
Preface
1 The Plasma Universe
1.1 Plasma, a Matter of State
1.2 Plasma, the First State of Matter
1.3 Plasma in Superclusters of Galaxies
1.4 Intergalactic Plasma
1.5 Galactic Plasma
1.6 Interstellar Plasma
1.7 Plasmas in Stars
1.8 Sun, a Plasma Laboratory
1.9 Solar Wind
1.10 Cometary Plasma
1.11 Planetary Plasma
1.12 Terrestrial Plasma
1.13 Earth’s Magnetosphere
1.14 The Ionosphere
1.15 Plasmas in Laboratory
1.16 The Clean Energy
1.17 Thermonuclear Fusion
1.18 Magnetic Confinement Fusion
1.19 Inertial Confinement Fusion
1.20 Space Travel-Plasma Rockets
1.21 Plasma Accelerators
1.22 Plasma Materials and Methods
1.23 Fully Ionized Plasma
1.24 Partially Ionized Plasma
1.25 Ultracold Plasmas
1.26 Dusty Plasma
1.27 Quantum Plasma
1.28 Techniques for Studying Plasmas
1.29 Summary
Problems
2 Plasma Basics
2.1 Making Plasmas
2.2 Plasma Formation by Photoionization
2.3 Collisional Ionization
2.4 Thermal Ionization
2.5 Pressure Ionization
2.6 Electric Discharge through Gases
2.7 Critical Velocity Ionization
2.8 Measuring Plasmas
2.9 Plasma Specifics
2.10 Debye Screening
2.11 Plasma-Typical Time Scales
2.12 The Plasma Parameter
2.13 When is it a Plasma?
2.14 Summary
Problems
3 Plasma Confinement
3.1 Introduction
3.2 The Grip of Gravity
3.3 Hydrostatic Equilibrium
3.4 Hydrodynamic Equilibrium
3.5 Magnetic Bottles
3.6 Motion of a Charged Particle in a Magnetic Field
3.7 Plasma in a Magnetic Field
3.8 Magnetostatic Equilibrium, the Z Pinch
3.9 Magnetostatic Equilibrium, the Θ Pinch
3.10 Magnetostatic Force Free Equilibrium, the Reversed Field Pinch
3.11 Magnetic Mirrors
3.12 Confinement of Plasmas under Radiation Pressure
3.13 Inertial Confinement
3.14 Summary
Problems
4 The Waving Plasmas
4.1 Introduction
4.2 Single Fluid Description of a Plasma
4.3 Ideal Magnetohydrodynamics
4.4 Linear Waves in Ideal Magnetofluid
4.5 Transverse MHD Waves: The Linear Alfvén Wave
4.6 Polarization of the Alfvén Waves
4.7 Energy Partition in the Alfvén Waves
4.8 Nonlinear Alfvén Waves
4.9 Dissipation of the Alfvén Waves
4.10 The Longitudinal Magnetohydrodynamic Waves
4.11 Polarization of the Fast Wave
4.12 Energy Partition in the Fast Wave
4.13 Dissipation of the Fast Wave
4.14 Oblique Propagation of Magnetoacoustic Waves
4.15 Polarization of the Oblique Fast and Slow Waves
4.16 Energy Partition in the Fast and the Slow Waves
4.17 Dissipation of the Oblique Fast and the Slow Waves
4.18 Inclusion of the Displacement Current
4.19 Detection and Observation of the Magnetohydro dynamic Waves
4.20 Waves in a Two-Fluid Description of a Plasma
4.21 The Hall Wave
4.22 The Electron Plasma Waves
4.23 Polarization of the Electron Plasma Wave
4.24 Energy Partition in the Electron Plasma Wave
4.25 Dissipation of the Electron Plasma Wave
4.26 Inclusion of Thermal Pressure
4.27 Detection of the Electron Plasma Waves
4.28 Ion Acoustic Waves
4.29 The Plasma Approximation
4.30 Polarization of the Ion Acoustic Waves
4.31 Energy Partition in the Ion Acoustic Waves
4.32 Dissipation of the Ion Acoustic Wave
4.33 Detection of the Ion Acoustic Wave
4.34 Electrostatic Waves in Magnetized Fluids
4.35 The Upper Hybrid Wave
4.36 The Lower Hybrid Wave
4.37 Electrostatic Magneto-Ion Acoustic Waves in Magnetized Plasma
4.38 Electromagnetic Waves in an Unmagnetized Plasma
4.39 Electromagnetic Waves in Magnetized Plasmas
4.40 Ordinary Wave
4.41 Extraordinary Wave
4.42 Polarization of the Extraordinary Wave
4.43 Electromagnetic Waves Propagating along
4.44 Circularly Polarized Radiation
4.45 The Whistler Wave
4.46 The Faraday Rotation
4.47 Cutoff Frequencies of the Electromagnetic Waves
4.48 Resonances of the Electromagnetic Waves
4.49 Propagation Bands of the Electromagnetic Waves
4.50 Summary
Problems
5 The Radiating Plasmas
5.1 Radiation and Plasmas
5.2 A Quick Revisit of Waves
5.3 Electromagnetic Radiation
5.4 Polarization of Electromagnetic Waves
5.5 Propagation of Electromagnetic Waves in a Plasma
5.6 Absorption of Electromagnetic Waves in a Plasma
5.7 Electron–Ion Collision Frequency
5.8 Generation of the Electromagnetic Radiation
5.9 Radiation from an Oscillating Electric Dipole
5.10 Radiation from an Accelerated Single Charged Particle
5.11 Relativistic Generalization of the Larmor Formula
5.12 Radiation Spectrum
5.13 In a Plasma
5.14 Radiation from Collisions between Charged Particles
5.15 Bremsstrahlung, Radiation Generated by Coulomb Collisions
5.16 Bremsstrahlung in a Plasma
5.17 Bremsstrahlung in a Thermal Plasma
5.18 Scattering of Radiation by Plasma Particles
5.19 Thomson Scattering
5.20 Scattering in a Plasma
5.21 Summary
Problems
6 Supplementary Matter
6.1 Derivation of Eq. (3.11)
6.2 Collisional Processes
6.3 Derivation of Eq. (5.109)
6.4 Physical Constants
6.5 Electromagnetic Spectrum
6.6 Astrophysical Quantities
6.6.1 Planets
6.6.2 The Sun
6.6.3 The Milky Way
6.6.4 The Hubble Constant
6.6.5 Electron Density and Temperature of some of the Astrophysical Plasmas
6.7 Vector Identities
6.8 Differential Operations
6.8.1 Cartesian coordinates (x, y, z)
6.8.2 Cylindrical coordinates (r,θ,z)
6.8.3 Spherical coordinates (r,θ,ϕ)
Select Bibliography
Index

Citation preview

Plasmas

The First State of Matter

Vinod Krishan

Cambridge House, 4381/4 Ansari Road, Daryaganj, Delhi 110002, India Published in the United States of America by Cambridge University Press, New York 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/9781107037571 © Vinod Krishan 2014 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 2014 Printed in India A catalogue record for this publication is available from the British Library Library of Congress Cataloging-in-Publication Data Krishan, V. (Vinod), author. Plasmas : the first state of matter / Vinod Krishan.    pages cm Includes bibliographical references and index. Summary: “Develops a discussion about plasma, the first state of matter from which evolved the other three states” – Provided by publisher. Includes bibliographical references and index. ISBN 978-1-107-03757-1 (hardback) 1. Plasma (Ionized gases) I. Title. QC718.K75 2013 530.4’4–dc23 2013019974 ISBN 978-1-107-03757-1 Hardback Cambridge University Press has no responsibility for the persistence or accuracy of URLs for external or third-party internet websites referred to in this publication, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate.

To my parents Shri Om Prakash Pabbi and Shrimati Raj Dulari Pabbi who continue to adore me, educate me and inspire me.

.

Contents

List of Illustrations Preface

xi xix

1. The Plasma Universe 1.1

Plasma, a Matter of State

1

1.2

Plasma, the First State of Matter

2

1.3

Plasma in Superclusters of Galaxies

5

1.4

Intergalactic Plasma

6

1.5

Galactic Plasma

9

1.6

Interstellar Plasma

10

1.7

Plasmas in Stars

11

1.8

Sun, a Plasma Laboratory

11

1.9

Solar Wind

15

1.10

Cometary Plasma

17

1.11

Planetary Plasma

19

1.12

Terrestrial Plasma

20

1.13

Earth’s Magnetosphere

21

1.14

The Ionosphere

24

1.15

Plasmas in Laboratory

25

1.16

The Clean Energy

28

1.17

Thermonuclear Fusion

28

1.18

Magnetic Confinement Fusion

29

1.19

Inertial Confinement Fusion

30

1.20

Space Travel-Plasma Rockets

31

vi

Contents

1.21

Plasma Accelerators

32

1.22

Plasma Materials and Methods

34

1.23

Fully Ionized Plasma

35

1.24

Partially Ionized Plasma

36

1.25

Ultracold Plasmas

36

1.26

Dusty Plasma

37

1.27

Quantum Plasma

38

1.28

Techniques for Studying Plasmas

39

1.29

Summary

40

Problems

40

2. Plasma Basics 2.1

Making Plasmas

41

2.2

Plasma Formation by Photoionization

41

2.3

Collisional Ionization

42

2.4

Thermal Ionization

43

2.5

Pressure Ionization

45

2.6

Electric Discharge through Gases

45

2.7

Critical Velocity Ionization

46

2.8

Measuring Plasmas

46

2.9

Plasma Specifics

48

2.10

Debye Screening

49

2.11

Plasma-Typical Time Scales

53

2.12

The Plasma Parameter

54

2.13

When is it a Plasma?

54

2.14

Summary

55

Problems

55

3. Plasma Confinement 3.1

Introduction

56

3.2

The Grip of Gravity

56

3.3

Hydrostatic Equilibrium

59

3.4

Hydrodynamic Equilibrium

60

3.5

Magnetic Bottles

62

Contents

3.6

Motion of a Charged Particle in a Magnetic Field

vii 62

3.7

Plasma in a Magnetic Field

65

3.8

Magnetostatic Equilibrium, the Z Pinch

66

3.9

Magnetostatic Equilibrium, the Θ Pinch

71

3.10

Magnetostatic Force Free Equilibrium, the Reversed Field Pinch

71

3.11

Magnetic Mirrors

75

3.12

Confinement of Plasmas under Radiation Pressure

82

3.13

Inertial Confinement

84

3.14

Summary

86

Problems

87

4. The Waving Plasmas 4.1

Introduction

88

4.2

Single Fluid Description of a Plasma

89

4.3

Ideal Magnetohydrodynamics

95

4.4

Linear Waves in Ideal Magnetofluid

98

4.5

Transverse MHD Waves: The Linear Alfv´en Wave

99

4.6

Polarization of the Alfv´en Waves

102

4.7

Energy Partition in the Alfv´en Waves

103

4.8

Nonlinear Alfv´en Waves

104

4.9

Dissipation of the Alfv´en Waves

105

The Longitudinal Magnetohydrodynamic Waves

106

4.11

Polarization of the Fast Wave

108

4.12

Energy Partition in the Fast Wave

109

4.13

Dissipation of the Fast Wave

109

4.14

Oblique Propagation of Magnetoacoustic Waves

110

4.15

Polarization of the Oblique Fast and Slow Waves

113

4.16

Energy Partition in the Fast and the Slow Waves

114

4.17

Dissipation of the Oblique Fast and the Slow Waves

114

4.18

Inclusion of the Displacement Current

115

4.19

Detection and Observation of the

4.10

Magnetohydrodynamic Waves 4.20

Waves in a Two-Fluid Description of a Plasma

117 117

viii

Contents

4.21

The Hall Wave

119

4.22

The Electron Plasma Waves

123

4.23

Polarization of the Electron Plasma Wave

126

4.24

Energy Partition in the Electron Plasma Wave

127

4.25

Dissipation of the Electron Plasma Wave

127

4.26

Inclusion of Thermal Pressure

129

4.27

Detection of the Electron Plasma Waves

131

4.28

Ion Acoustic Waves

132

4.29

The Plasma Approximation

136

4.30

Polarization of the Ion Acoustic Waves

137

4.31

Energy Partition in the Ion Acoustic Waves

137

4.32

Dissipation of the Ion Acoustic Wave

137

4.33

Detection of the Ion Acoustic Wave

138

4.34

Electrostatic Waves in Magnetized Fluids

138

4.35

The Upper Hybrid Wave

139

4.36

The Lower Hybrid Wave

142

4.37

Electrostatic Magneto-Ion Acoustic Waves in Magnetized Plasma

144

4.38

Electromagnetic Waves in an Unmagnetized Plasma

145

4.39

147

4.40

Electromagnetic Waves in Magnetized Plasmas Ordinary Wave, k ⊥ B0 , E 1  B0

4.41

E 1 ≈ ⊥ B0 Extraordinary Wave, k ⊥ B0 , 

149

4.42

Polarization of the Extraordinary Wave

151

4.43

Electromagnetic Waves Propagating along B0

151

4.44

Circularly Polarized Radiation

152

4.45

The Whistler Wave

153

4.46

The Faraday Rotation

154

4.47

Cutoff Frequencies of the Electromagnetic Waves

157

4.48

Resonances of the Electromagnetic Waves

158

4.49

Propagation Bands of the Electromagnetic Waves

159

4.50

Summary

163

Problems

163

148

Contents

ix

5. The Radiating Plasmas 5.1

Radiation and Plasmas

165

5.2

A Quick Revisit of Waves

165

5.3

Electromagnetic Radiation

167

5.4

Polarization of Electromagnetic Waves

170

5.5

Propagation of Electromagnetic Waves in a Plasma

175

5.6

Absorption of Electromagnetic Waves in a Plasma

181

5.7

Electron–Ion Collision Frequency

183

5.8

Generation of the Electromagnetic Radiation

185

5.9

Radiation from an Oscillating Electric Dipole

186

5.10

Radiation from an Accelerated Single Charged Particle 193

5.11

Relativistic Generalization of the Larmor Formula

5.12

Radiation Spectrum

203

5.13

In a Plasma

206

5.14

Radiation from Collisions between Charged Particles

207

5.15

Bremsstrahlung, Radiation Generated by

199

Coulomb Collisions

209

5.16

Bremsstrahlung in a Plasma

209

5.17

Bremsstrahlung in a Thermal Plasma

211

5.18

Scattering of Radiation by Plasma Particles

213

5.19

Thomson Scattering

213

5.20

Scattering in a Plasma

217

5.21

Summary

220

Problems

221

6. Supplementary Matter 6.1

Derivation of Eq. (3.11)

223

6.2

Collisional Processes

225

6.3

Derivation of Eq. (5.109)

228

6.4

Physical Constants

234

6.5

Electromagnetic Spectrum

235

6.6

Astrophysical Quantities

236

6.6.1 6.6.2

236 236

Planets The Sun

x

Contents

6.6.3 6.6.4 6.6.5

The Milky Way 237 The Hubble Constant 237 Electron Density and Temperature of Some of the Astrophysical Plasmas 237

6.7

Vector Identities

238

6.8

Differential Operations

240

6.8.1 6.8.2 6.8.3

Cartesian Coordinates (x, y, z) Cylindrical Coordinates (r, θ , z) Spherical Coordinates (r, θ , ϕ )

240 241 243

Select Bibliography

245

Index

247

List of Illustrations

1.1 The little bang of a lead ion–lead ion collision as seen in NA49 CERN-EX-9600007. This is an image of an actual lead ion–lead ion collision taken from tracking detectors on the NA49 experiment. The collisions produce a very complicated array of hadrons as the heavy ions break up and create a new state of matter known as the quark–gluon plasma, credit: CERN. 3 1.2 A neutron (n) has a life time of about 10 minutes after which it decays into a proton (p), an electron (e), and a neutrino (ν ). 3 1.3 Two protons (p) fuse together to produce a deuterium nucleus D(p,n), a positron (e+), and a neutrino (ν ). 4 1.4 A deuterium nucleus D(p,n) fuses with a proton (p) to produce helium three, He(2p,n), and gamma rays γ . 4 1.5 Two helium three nuclei He(2p,n) fuse to produce one helium four nucleus, He(2p,2n), and two protons (p). 4 1.6 Production of beryllium, Be(4p,3n), nucleus.

5

1.7 Abell supercluster 1689 containing several clusters of galaxies, credit: NASA/ESA. 6 1.8 X-ray image of galaxy cluster Abell 2412. The image on the right was taken on August 20, 1999 with the Chandra X-ray Observatory’s 0.3–10.0 keV Advanced CCD Imaging Spectrometer (ACIS), and covers an area of 7.5 × 7.2 arc minutes. It shows a colossal cosmic ³weather system´ produced by the collision of two giant clusters of galaxies. For the first time, the pressure fronts in the system have been traced in detail, and show a bright, but relatively cool, 50 million degree Celsius central region (white) embedded in a large elongated cloud of 70 million degree Celsius gas, all of which are rolling in a faint ³atmosphere´ of 100 million degree Celsius gas.

xii

List of Illustrations

The bright source in the upper left is an active galaxy in the cluster, credit: http://en.wikipedia.org/wiki/File:Abell2142chandraxray.jpg. This file is in the public domain because it was solely created by NASA. 7 1.9 A Jet from Galaxy M87 Image, credit: J.A. Biretta et al., Hubble Heritage Team (STScI /AURA, NASA). 8 1.10 Chandra X-ray image of the innermost 10 light years at the center of our galaxy, credit: NASA/MIT/PSU. 9 1.11 Stars, gas, and dust in the orion nebula, credit: hubblesite, NASA.

10

1.12 Solar flares observed with various instruments onboard SOHO satellite, credit: NASA/ESA. 12 1.13 This image of coronal loops was taken by the TRACE satellite at 171 angstrom wavelength pass band, characteristic of plasma at a temperature of one million degree Kelvin, on November 6, 1999, credit: NASA. 13 1.14 This X-ray image of the Sun, captured on February 21, 2000, by the Japanese Yohkoh X-ray Observatory shows a coronal hole that sent high-speed solar wind particles toward Earth. The resulting gusts of solar wind struck Earth’s magnetic field and triggered moderate geomagnetic disturbances over the next few days, credit: NASA/ISAS. 14 1.15 Sun Storm: A Coronal Mass Ejection, credit: SOHO Consortium, ESA, NASA. 15 1.16 The sun flings out solar wind particles in much the same manner as a garden sprinkler throws out water droplets. The artist’s drawing of the solar wind flow was provided with courtesy of NASA. 16 1.17 Artist’s impression of Earth’s bow shock and magnetosphere. The solar wind shapes Earth’s magnetosphere and causes magnetic storms. Courtesy of SOHO/[instrument] consortium. SOHO is a project of international cooperation between ESA and NASA. 16 1.18 This photograph of Comet West was taken by amateur astronomer John Laborde on March 9, 1976. The picture shows the two distinct tails. The thin ion tail is made up of gases, while the broad tail is made up of tiny dust particles, credit: NASA, courtesy, John Laborde. 17

List of Illustrations

xiii

1.19 Helical structures in the plasma tail of comet Ikeya-Saki, credit: NASA, courtesy, John Laborde. 18 1.20 This image is taken by NASA’s Cassini spacecraft. Jupiter’s magnetic field has been sketched over the image. The disk of Jupiter is shown by the black circle and the approximate position of the doughnut-shaped torus, created from material spewed out by volcanoes on Io, is shown by the white circles, credit: NASA. 20 1.21 Irving Langmuir (1881–1957), wikipedia.org/wiki (public domain).

credit:

http://www.en. 21

1.22 Artist’s illustration of events on the sun changing the conditions in Near-Earth space, credit: NASA. 22 1.23 A coronal mass ejection hit Earth’s magnetic field on October 8, 2012 sparking a dramatic display of Arctic lights that is only slowly subsided three days later. The Aurora appears to be casting rays of green light through the clouds. Hugo Løhre photographed the aurorae over Lekangsund, Norway, on October 10, 2012. Text: Dr. Tony Phillips, Image courtesy of Hugo Løhre, NASA. 23 1.24 Conjugate aurorae on both the north and the south polar regions, observed with NASA’s Polar spacecraft, credit: NASA. 23 1.25 Various layers of the ionosphere and their predominant ion populations are listed at their respective heights above ground. The density in the ionosphere varies considerably, credit: http://www.commons.wikimedia.org/wiki (public domain). 24 1.26 The Maltese Cross tube, credit: http://www.crtsite.com (public domain). 25 1.27 The Crookes phosphorescent flower tube, credit: http://www.crtsite.com (public domain).

26

1.28 Lightning over the outskirts of Oradea, Romania, during the August 17, 2005 thunderstorm, credit: http://www.en.wikipedia.org/wiki (public domain). 27 1.29 Nuclear fuel deuterium and tritium fuse to produce helium and energetic neutrons. 28 1.30 Charged particles execute circular motion around the magnetic axis. Negatively charged particles (e) go anticlockwise and positively charged particles (p) go clockwise for the direction of the magnetic field B shown here. 29

xiv

List of Illustrations

1.31 ITER, the world’s largest tokamak, credit: Michel Claessens, ITER organization. 30 1.32 Upper portion of the NIF’s target chamber, credit: http://www.en.wikipedia.org/wiki (public domain).

31

1.33 A plasma rocket could significantly reduce the travel time to the Mars, credit: NASA. 32 1.34 An electron beam repels electrons of the plasma and forms a wake of positive ions behind it. The positive ions and the displaced plasma electrons create an electric field, which accelerates the electron beam. 33 1.35 The dusty plasma ring system of Saturn in visible and radio, credit: NASA. 38 2.1 The positive charge Q is surrounded by negative charges forming the Debye cloud. 48 2.2 The variation of the screened potential, Eq. (2.32) (solid line) and the Coulomb potential (dashed line) due to a charge Q with distance r. . 52 3.1 Simulation of an accretion disk, credit: Michael Owen, John Blondin (North Carolina State University). 61 3.2 Charged particles move in a helical path around the magnetic axis. Negatively charged particles go anticlockwise and positively charged particles go clockwise for the direction of the magnetic field B pointing toward the observer. 64 3.3 An extragalactic plasma jet retains its shape for distances of millions of parsecs from the center of the galaxy. The stability of the jet against its diffusion into the neighborhood is believed to be ensured by a current density flowing along the jet giving rise to an azimuthal magnetic field, which contains the plasma. Plasma Jets from Radio Galaxy Hercules A, image Credit: NASA, ESA, S. Baum and C.´o Dea (RIT), R. Perley and W. Cotton (NRAO/AUI/NSF), and the Hubble Heritage Team (STScI/AURA) 70 3.4 Variation of Bessel function J0 (x) with x.

74

3.5 Variation of Bessel function J1 (x) with x.

74

3.6 Magnetic moment μ B generated by a current density distribution J from which magnetic induction at any point P can be calculated. 76

List of Illustrations

xv

3.7 Magnetic moment of a current loop.

77

3.8 Confined plasma in a magnetic mirror.

79

3.9 A schematic representation of Van Allen inner and outer radiation belts. The inner belts contain more energetic particles than the outer belts. 81 3.10 A deuterium–tritium shell is bombarded with several lasers. The shell surface evaporates and the evaporated plasma moves outwards (ablation). The inner shell surface recoils inwards due to the back reaction (implosion) and causes compression, confinement, and heating. 85 4.1 (a) The equilibrium form of the uniform magnetic field frozen to the fluid. (b) The perturbed sinusoidal form of the fluid and the field. 100 4.2 The relative orientations of the Alfv´en wave fields and the direction of the wave propagation. 103 4.3 (a) The equilibrium form of the uniform magnetic field frozen to the fluid. (b) The compressions and the rarefactions produced in the fluid and the magnetic field by the magnetosonic wave propagating perpendicular to the ambient magnetic field B0 . 107 4.4 The relative orientations of the direction of propagation and the wave fields of the magnetosonic waves. 109 4.5 Plot of (ω 2 /k2VA2 ) vs. the angle θ for the slow, the Alfv´en and the fast wave for β = 0.5. 113 4.6 The dispersion relation of the Hall wave, Eq. (4.175) with the plus sign (solid line), is displayed as a plot of ω /ωic vs. kλH . The dashed line represents the dispersion relation of the co-propagating Alfv´en wave. 121 4.7 The dispersion relation, Eq. (4.175) for the minus sign (solid line), is displayed as a plot of ω /ωic vs. kλH . The dashed line represents the counter-propagating Alfv´en wave. 122 4.8 The dispersion relation, Eq. (4.235), for the electron plasma wave is displayed as a plot of (ω /ωep ) vs. kλeD . The dashed line marks the value of ωep . 130 4.9 The dispersion relation, Eq. (4.262), for the ion acoustic wave is displayed as a plot of ω /ωip vs. kλeD . 136

xvi

List of Illustrations

2 /ω 2 vs. the angle θ for the upper hybrid wave. 4.10 The plot of the ω+ uh . 142

4.11 Top: a plane polarized wave is split into a right circularly polarized wave and a left circularly polarized wave. Below: after a time interval T , the two waves traverse different distances and develop a phase difference. Their sum gives the rotated plane polarized wave. . 155 4.12 The ordinary wave propagates in the region where μ0 > 0. The cutoff frequency is ωep where the refractive index vanishes. 159 4.13 The extraordinary wave propagates in the region where μEx > 0. The two cutoff frequencies are at ωL and ωR . The resonance occurs at ωuh . This wave cannot propagate for ω < ωL and for ω lying between ωuh and ωR . The refractive index becomes unity at the electron plasma frequency and at frequencies much larger than ωR . . 160 4.14 The right circularly polarized wave propagates in the region where μR > 0. The frequency region between ω2ec and ωec is the propagation region of the whistlers. The cutoff frequency is at ωR . The resonance occurs at ωec . This wave cannot propagate for ω lying between ωec and ωR .The refractive index approaches unity at frequencies much larger than ωR . 161 4.15 The left circularly polarized wave propagates in the region where μL > 0. The cutoff frequency is at ωL . The resonance does not occur for this wave. The wave cannot propagate for ω < ωL . The refractive index approaches unity at frequencies much larger than ωL . 162 5.1 Orientations of the propagation vector k and the electric field  E for a plane polarized electromagnetic wave. 171 5.2 The two possible orientations of the fields  E and  B for a fixed k for a linearly polarized electromagnetic wave. 172  ) obtained by the superposition of the 5.3 A linearly polarized wave (E two linearly polarized waves, E 1 and  E 2 propagating in phase with each other. 173 5.4 A right (anticlockwise) and a left (clockwise) circularly polarized waves (E) when viewed along the z direction. 173 5.5 Deflection of an electron in the Coulomb field of an ion.

184

5.6 Electric field E at a point P due to a static charge q.

186

List of Illustrations

xvii

5.7 Orientations of the Poynting vector S = n = k, the electric dipole 192 moment P and the vector n p . 5.8 Trajectory of the accelerated charged particle e and the point P where the emitted radiation is observed. 193 5.9 Spectral distribution of radiation generated due to collisions among charged particles. 208 5.10 Thomson Scattering of radiation by an electron.

214

5.11 Thomson Scattering of unpolarized radiation.

215

5.12 Raman Scattering of radiation.

219

5.13 Nonlinear absorption of an intense radiation near the electron plasma frequency. 219 5.14 Nonlinear reflection of intense radiation at a frequency much larger than the electron plasma frequency. 220 6.1 Contours (C’ for τ > 0, C for τ < 0) in the complex ω plane.

231

* Unless otherwise mentioned, all figures have been drawn by the author.

.

Preface Plasma physicists are initiated into the field with the line that plasma is the fourth state of matter since it is produced by the three-stage process of melting a solid into a liquid, evaporating a liquid into a gas, and ionizing a gas into a plasma. Astronomers have long known that the universe originated from a very hot soup of plasma and radiation. The other three states of matter, namely gas, liquid, and solid, came into being, in that order, as the universe expanded and cooled. It is high time that we set the record straight and coronate plasma as the first state of matter. Some may ask: Does it make a difference? It just might. Plasmas are already playing a tremendous role in creating new materials. In the face of rapidly depleting conventional energy sources, plasmas emerge as the last hope for mankind to generate green energy. This paradigm shift from solid–liquid–gas–plasma to plasma–gas–liquid–solid is likely to usher in a completely novel way of dealing with the material world. The universality of plasmas has however not made it any easier to understand them. Astronomers consider plasmas, at best, an unavoidable presence and reluctantly accept the plasma often without the plasma phenomena. Here, in this book, I have attempted to introduce the subject of physics of the plasmas to graduate and undergraduate students in an accessible style in the hope of catching them young. Each chapter stands on its own for the most part. The first chapter is essentially an inventory of the first state of matter in the cosmos and on terra firma. The reader is introduced to the phenomenal variety of plasmas and their purposes. The second chapter consists of ways and means of making plasmas, followed by their defining properties. Confinement techniques of often extremely hot natural and man-made plasmas are discussed in the third chapter. ’What are the wild waves saying’? Plasmas are known by the waves they can support. The wave properties of single-fluid Magnetohydrodynamics waves and two fluid waves, electrostatic, electromagnetic, and the combination thereof, make up the stuff of Chapter 4. Radiation and plasmas, the embryonic fluid of the universe, is the subject of Chapter five. This Chapter has been made completely self-sufficient at the cost of some repetition.

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Preface

A little extra help never hurts. Detailed derivations of a few important equations are provided in Chapter 6. I have written this book after my retirement from office. It is only natural that my thoughts race back to my beginnings. I gratefully recall the family and the friends, the colleagues, and the collaborators who have nourished and nurtured me with their indulgence and instruction. Among the many colleagues and collaborators, I wish to particularly record my deep gratitude toward Drs Rufus Ritchie, Ed Harris, Professor Vainu Bappu, Ch.V. Sastry, S.M. Chitre Ram Varma, Paul Wiita, H.S. Sawant, Swadesh Mahajan, Zensho Yoshida, Padma Shukla, R.T. Gangadhara, Santoshi Masuda and Baba Varghese. I have been extremely fortunate to have found my first teacher of plasma physics and my life partner in my husband Dr Som Krishan who continues to stoically bear with my idiosyncracies. My daughter Dr Monika Krishan has always been my total support system and I feel blessed with her presence in my life. There is one person who has been there for me much longer than my immediate family. And this is my younger sister Saroj Ishwarlal who flows like a subsoil stream and nourishes my roots. I wish to place on record my deep gratitude to the Raman Research Institute for providing me with all possible support to enable me to continue my work after my retirement from the Indian Institute of Astrophysics. I shall ever remain indebted to my home institution, the Indian Institute of Astrophysics. I hope the book will serve as a primer for students who wish to have a taste of the embryonic fluid of the universe.

1

1.1

The Plasma Universe

Plasma, a Matter of State

A plasma has always been described as the fourth state of matter, the other three being the solid, the liquid, and the gas. A solid transforms into a liquid by heating (ice into water); heating transforms a liquid into a gas (water into water vapor). When a gas is heated to a rather high temperature, the outermost electrons are detached from the gas atoms and the remaining part of the atoms, called ions, acquire positive electric charge. A mixture of the positively charged ions and the negatively charged electrons is formed. This mixture of a sufficiently large number of ions and electrons is called a plasma, a term borrowed from the blood plasma in which the red blood corpuscles and other organisms float. The name plasma for this ionized gas was first suggested by Irving Langmuir in 1939 while he was studying the electric discharges of gases by passing a high electric current through a gas of Cesium atoms. But this is not how Nature did it! Nature began with the plasma. Cooling of the plasma converted it into a gas. Cooling of the gas converted it into a liquid. Cooling of the liquid converted it into a solid. Even though a plasma is not a result of a phase transition in the manner of ice–water–vapor, it is considered to be a state of matter because the characteristics of a plasma are significantly different from its parent neutral gas. A plasma, similar to a gas, has no definite shape or volume. However, unlike a gas, a plasma has a high electrical conductivity; not all its constituent particles behave alike; the plasma particles can have Maxwellian velocity distributions with different temperatures; they can have even non-Maxwellian velocity distributions and interactions among the plasma particles are more often than not collective wherein a large

2

Plasmas

number of particles get in phase and act cooperatively with other particles and waves in the system.

1.2

Plasma, the First State of Matter

Most astronomers believe that the universe began in a big bang about 15 billion years ago. There was an explosion of a not-so-well-understood state of matter under not-so-well-known extreme conditions of density and temperature. The aftermath of the explosion was the cooling and the expansion that continues till date. It was first observed by Edwin Hubble in 1930, and gloriously reconfirmed by the Hubble Space (optical) Telescope at present, that the galaxies are rushing away from each other at high speeds. The present state of the universe when extrapolated back in time, using the known laws of physics, compels us to believe that the universe was extremely hot and dense in its infancy. In the beginning there was intense radiation, the photons, that produced equal amounts of matter and antimatter (particles with equal masses and equal and opposite electric charges such as electrons and positrons). And a plasma soup of particles and antiparticles was all there was. Thus a plasma is the first state of matter out of which all the other states of matter originated. The particles and the antiparticles, could, however, recombine back into radiation and this would go on until physics would forbid further production of particles and their recombination. To our good fortune, there is a tiny difference between the amounts of matter and antimatter and this tiny difference drove the evolution of the universe to what it is today. The cause of this difference is still not well understood. The quarks and the antiquarks constituted the very first, after ten microseconds of the big bang, plasma called the quark–gluon plasma (Fig. 1.1). The gluons are the glue that binds the quarks together. The quarks and the antiquarks are the building blocks of protons and neutrons. Scientists at CERN believe that they have been able to create such a state of quark– gluon plasma through the lead ion–lead ion collisions during which the ions break up into their constitutive quarks. The electrons and the positrons were created from the decay of highenergy photons. During the period between the microsecond to the tenth of a second since the big bang, the protons, the neutrons, and other particles came into being from the quarks. At three minutes after the big bang explosion, the nuclear reactions set in. The neutrons began to decay into protons, electrons, and neutrinos (Fig. 1.2). The mass of a neutron is more than the mass of a proton. The difference in the rest

The Plasma Universe

3

masses (Δm) of the neutron and the proton is, by Einstein’s mass energy relation, converted into energy Δmc2 where c is the speed of light. Two protons fused together to form deuterium (Fig.1.3). This reaction needs energy input to overcome the Coulomb repulsion between the positively charged protons. One proton is converted into a neutron and a positron is emitted along with a neutrino.

)LJXUH  The little bang of a lead ion–lead ion collision as seen in NA49 CERNEX-9600007. This is an image of an actual lead ion–lead ion collision taken from trackingdetectors on the NA49 experiment. The collisions produce a very complicated array of hadrons as the heavy ions break up and create a new state of matter known as the quark–gluon plasma, credit: CERN.

)LJXUH  A neutron (n) has a life time of about 10 minutes after which it decays into a proton (p), an electron (e), and a neutrino (ν ).

The deuterium nucleus, D(p,n), consists of one proton and one neutron. One deuterium nucleus fuses with a proton to produce helium three nucleus He(2p,n), containing two protons and one neutron, gamma rays along with a release of 5.49 million electron volts (MeV) of (energy Fig. 1.4).

4

Plasmas

p D(p,n) e+ p n )LJXUH  Two protons (p) fuse together to produce a deuterium nucleus D(p,n), a positron (e+ ), and a neutrino (ν ).

)LJXUH  A deuterium nucleus D(p,n) fuses with a proton (p) to produce helium three, He(2p,n), and gamma rays γ .

Two helium three nuclei He(2p,n) fuse together to produce helium four nucleus, He(2p,2n), and two protons along with a release of 12.86 MeV of energy. This completes the cycle of proton–proton fusion (Fig. 1.5); remember we started with two protons. He(2p,n) He(2p,2n)

2p He(2p,n)

)LJXUH  Two helium three nuclei He(2p,n) fuse to produce one helium four nucleus, He(2p,2n), and two protons (p).

One helium four nucleus fuses with one helium three nucleus to produce a nucleus of beryllium seven, Be(4p,3n), along with a release of 1.59 MeV of energy (Fig. 1.6). Lithium seven is an unstable nucleus. Only a small amount of lithium seven is formed in the primordial nucleosynthesis.

The Plasma Universe

5

He(2p,2n) Be(4p,3n)

He(2p,n) )LJXUH 

Production of beryllium, Be(4p,3n), nucleus.

This is as far as the thermonuclear fusion, at this stage, contributes to the formation of elements. The rest of the elements of the periodic table would be manufactured in stars and their supernovae explosions. These fusion processes continued up to the fifteenth minute since the big bang. At this epoch, the temperature of the universe is about a million degrees. The universe is a fully ionized plasma consisting of photons, nuclei, and electrons. The primordial helium abundance of 25 percent by mass is a robust prediction of the big bang model and it has been sufficiently confirmed by observing the very distant objects such as quasars or objects in which stellar nucleosynthesis has not occurred. The agreement between observations and the theoretical prediction for the lithium abundance is not so good. Recently, scientists at the Brookhaven National Laboratory of the department of the US atomic energy have created the hottest temperatures ever, a whopping four trillion degrees at which the protons and the neutrons melt to produce a quark–gluon plasma, akin to the earliest, the primeval plasma from which the universe is believed to have emerged.

1.3

Plasma in Superclusters of Galaxies

The largest gravitationally bound structures in the universe are the superclusters of galaxies, each containing thousands of clusters of galaxies (Fig. 1.7). Superclusters of galaxies have typical dimensions of 100 megaparsecs (one parsec = 3 × 1018 cm) and carry signatures of the initial conditions in the early universe. The clusters of galaxies in a supercluster swim through the intercluster plasma with temperature of several million degrees Kelvin, an electron density ≈ 10−3 cm−3 and perhaps a magnetic field of the order of a microgauss. The observed emission in X-rays (Fig. 1.8) confirms the existence of the hot intercluster plasma. The electrons of this hot plasma are so energetic that they impart some of their energy to the photons of the cosmic microwave radiation

6

Plasmas

producing an observable bump in its otherwise smooth nearly perfect black body spectrum.

)LJXUH  Abell supercluster 1689 containing several clusters of galaxies, credit: NASA/ESA.

1.4

Intergalactic Plasma

The million degree intergalactic plasma was opaque to the radiation and both radiation and the ionized matter were strongly coupled and evolved together. The plasma was too hot to form atoms. As the universe expanded, it cooled. It took about a million years for the universe to cool down to a temperature of 3000 K, appropriate for the recombination of electrons and nuclei to form the first neutral matter. This is known

The Plasma Universe

7

as the recombination epoch characterized by a partially ionized plasma. The matter and radiation decoupled. The thermal radiation emitted by the matter at 3000 K pervaded the universe unchallenged by the mostly neutral matter. This is the cosmic microwave background radiation.

)LJXUH  X-ray image of galaxy cluster Abell 2412. The image on the right was taken on August 20, 1999 with the Chandra X-ray Observatory’s 0.3–10.0 keV Advanced CCD Imaging Spectrometer (ACIS), and covers an area of 7.5 × 7.2 arc minutes. It shows a colossal cosmic ‘‘weather system’’ produced by the collision of two giant clusters of galaxies. For the first time, the pressure fronts in the system have been traced in detail, and show a bright, but relatively cool, 50 million degree Celsius central region (white) embedded in a large elongated cloud of 70 million degree Celsius gas all of which is rolling in a faint ‘‘atmosphere’’ of 100 million degree Celsius gas. The bright source in the upper left is an active galaxy in the cluster, credit: http://www.en.wikipedia.org/wiki/File: Abell2142 chandra xray.jpg. This file is in the public domain because it was solely created by NASA.

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Plasmas

)LJXUH  A Jet from Galaxy M87 Image, credit: J.A. Biretta et al., Hubble Heritage Team (STScI /AURA, NASA).

The expansion and the cooling has brought the present temperature of the universe to 2.7 K as inferred from the observed spectrum of the cosmic microwave background radiation. Although the neutral matter is mostly homogeneously distributed, as the near isotropy of the cosmic microwave background radiation tells us, there does exist a very very small anisotropy in the radiation, which betrays a very small clumping of the matter. These clumps would then grow into stars and galaxies by attracting more matter toward them due to the attractive gravitational forces. The matter in between the galaxies is again hot, ionized and magnetized and this is the intergalactic plasma. The intergalactic plasma clouds are of enormous dimensions, harboring several galaxies and black

The Plasma Universe

9

holes. The plasma is observed through the detection of the synchrotron radiation, which is emitted by electrons gyrating in a magnetic field. The radiation is in the radio part of the electromagnetic spectrum and there are several big radio telescopes such as the world’s largest radio telescope at Arecibo, Puerto Rico, capable of detecting this radiation. The properties of the radiation tell us that the intergalactic plasma has a particle density of ≈ 10−5 − 10−4 per cubic centimeter, a temperature of ≈ 106 − 107 K and a magnetic field of ≈ 10−9 Gauss. Compare it with Earth’s magnetic field of about 0.3 Gauss. The picture (Fig. 1.9) shows the brightest central region where a plasma jet from the galaxy M 87 hits the intergalactic medium.

1.5

Galactic Plasma

A galaxy is made up of stars, nebulae (star-forming regions), gas, dust, magnetic field, and the medium in between the stars, the interstellar medium. Thus the temperature of a region and hence its state of ionization depends on its location in a galaxy. Around the galactic center exists a hot magnetized plasma observed through its emission in the radio part of the electromagnetic radiation. One sees plasma loops where the plasma illuminates the bipolar magnetic field configuration.

)LJXUH  Chandra X-ray image of the innermost 10 light years at the center of our galaxy, credit: NASA/MIT/PSU.

10

Plasmas

The observed radio radiation has the characteristics of the synchrotron radiation and this reveals that the typical plasma density is 0.04 per cubic centimeter, the magnetic field is of the order of a milliGauss, and the temperature is of the order of hundred million degree Kelvin. A huge envelope of plasma, a kind of galactic corona (a crown) has been observed in our own galaxy, the Milky Way, using the Far Ultraviolet Spectroscopic Explorer (FUSE) satellite. The plasma envelope may extend as far as the nearest galaxies, the magellanic clouds. Figure (1.10) shows the central region of our galaxy, which harbors the supermassive blackhole candidate Sagittarius A* (larger white dot at the very center of the image). The X-ray emission is from the plasma surrounding the black hole. The millions of degree plasma is produced by the shock waves caused by supernova explosions in the galaxy.

1.6

Interstellar Plasma

There are huge spaces among the stars in a galaxy. The spaces are so huge that the stars almost never collide. Nevertheless, the interstellar medium is sizzling and buzzing because stars are in the habit of spewing out hot winds. Just as our sun blows the hot solar wind. Stars also, not so infrequently, explode, the supernovae explosions, throwing large amounts of steaming streaming plasma in their vicinity. The interstellar medium is a potpurri of hot and cold regions containing plasma, gas, and dust (Fig. 1.11).

)LJXUH 

Stars, gas, and dust in the orion nebula, credit: hubblesite, NASA.

The Plasma Universe

11

The gas in the hot phase at a temperature of 100,000 K is necessarily ionized and christened as the interstellar plasma, a rather dilute one with an electron density of one particle or less in a cubic centimeter and a magnetic field of nearly a microgauss. Dilute it is but dull it is not! All cosmic radiations pass through the interstellar plasma. The time taken by an electromagnetic signal to traverse a certain distance in a plasma depends on the frequency of the signal and the electron density of the plasma. By determining the time delay between signals of different frequencies it has been possible to estimate the distance of pulsars. The magnetized interstellar plasma also rotates the plane of polarization of the signal. The most exciting property of the interstellar plasma is that it is turbulent and that, in one way, is a blessing. The turbulence has several universal properties irrespective of the material medium. These properties can then be attributed, with some confidence, to the otherwise inaccessible interstellar plasma.

1.7

Plasmas in Stars

Stars shine while the planets do not. It is because the stars have, in their cores, the right conditions for igniting the thermonuclear fusion reactions. The stellar cores at temperatures of tens of million degree Kelvin and densities of more than 100 grams per cubic centimeter, about ten times as dense as gold, continue to burn the plasma (for example, it is the hydrogen plasma in the case of the sun and other elements such as helium and carbon in more evolved stars) for billions of years. The solar core has been on fire for the past 4.5 billion years and would keep on cooking the elements for another 4.5 billion years, essentially converting hydrogen into helium through a chain of reactions involving protons, deuterium, and neutrons. The radiation generated in the stellar core traverses through a variety of plasma states before it emerges to illuminate the skies. A host of processes of absorption, emission, and scattering of radiation in plasmas are at play in this outward journey of the radiation. Remember for later that it is the gravitational force that keeps the core hot and dense enough to ignite the thermonuclear fusion reactions.

1.8

Sun, a Plasma Laboratory

Our sun has often been described as a plasma physics laboratory as it manifests almost each and every plasma-based process and more that one has found in a laboratory. In fact, there are concerted efforts to simulate the solar phenomenon such as a solar flare on a laboratory scale to grasp

12

Plasmas

the underlying processes. The temperature of the mostly hydrogen gas decreases to a mere 6000 K and the density to less than a millionth of a gram per cubic centimeter at the visible surface of the sun, the photosphere. At this temperature the hydrogen is only weakly ionized. There also exist a few ions of other metals such as calcium and potassium. The photosphere presents an excellent example of a weakly or partially ionized plasma, nevertheless, the plasma it is. With a typical electron density of the order of 1010 per cubic centimeter and the neutral atomic hydrogen density of 1015 per cubic centimeter, the ionization fraction is about a hundred thousandth (10−5 ). The plasma particles also collide with each other. The electron–ion, electron–neutral, and the ion–neutral collisions play a very important role in the dynamics of the magnetized plasma such as exists in the sunspots and the solar active regions. Astonishingly, the plasma temperature increases to more than 10,000 K at the next higher layer, the solar chromosphere. The ionization fraction increases accordingly. The temperature increases to more than a million degrees Kelvin at the solar corona, the uppermost layer of the sun. Why is it astonishing? Because experience tells us that the temperature should decrease away from a hot object. So the temperature of the upper layer, the chromosphere should be less than that of the photosphere and the temperature of the next upper layer, the solar corona should be less than that of the chromosphere. And the sun violates this arrangement. It is the physics of the plasmas that can account for this additional heating of

)LJXUH  Solar flares observed with various instruments onboard SOHO satellite, credit: NASA/ESA.

The Plasma Universe

13

the chromosphere and the corona. The magnetic fields have an energy of tension in the way of a stretched rubber band and plasmas have a way of releasing this energy by converting it into heat. The strong charged particle–magnetic field interaction empowers a plasma to display its diversity in myriad forms. A combination of these complex processes accounts for the inverted temperature profile of the solar atmosphere on the one hand and explosive events such as the solar flares (Fig. 1.12) on the other. Large amounts of electromagnetic radiation are emitted during a flare over a wide spectral band, from low-frequency radio radiation to extremely energetic gamma rays. Some of the microwave radiation specifically needs plasma specific mechanism, the coherent radiation mechanisms, to account for its properties. The solar corona is far from a quiet diminishing outermost layer of the solar atmosphere. It becomes visible to the naked eye only during solar eclipses. It is observed to be highly variable both in space and in time. The modern telescopes with coronagraphs (which have a mechanism to cover the blinding photospheric disk) of high-resolution capability reveal a plethora of structures. Prominent among them are the coronal loops and the coronal holes. The coronal loops are seen to trace the closed dipole magnetic field configuration, lighting up in microwave, ultraviolet, and X-ray radiations (Fig. 1.13).

)LJXUH  This image of coronal loops was taken by the TRACE satellite at 171 angstrom wavelength pass band, characteristic of plasma at a temperature of one million degree Kelvin, on November 6, 1999, credit: NASA.

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Plasmas

The coronal holes (Fig. 1.14) appear as dark regions of different shapes in X-ray images indicating that they contain plasmas colder than that found in the coronal loops.

)LJXUH  This X-ray image of the Sun, captured on February 21, 2000, by the Japanese Yohkoh X-ray Observatory shows a coronal hole that sent high-speed solar wind particles toward Earth. The resulting gusts of solar wind struck Earth’s magnetic field and triggered moderate geomagnetic disturbances over the next few days, credit: NASA/ISAS.

While the coronal loops are the sites from where the solar flares originate, the coronal holes are understood to be the sources of the faststreaming plasma called the solar wind. The solar flares often throw a large number (nearly ten billion tons or 1016 grams) of hot plasma blobs hurtling through the solar wind at a speed of nearly 2000 km per second. These are the coronal mass ejections (CMEs) (Fig. 1.15) hitting Earth’s atmosphere and producing the vagaries of space weather causing havoc for satellite operations.

The Plasma Universe

)LJXUH  ESA, NASA.

1.9

15

Sun Storm: A Coronal Mass Ejection, credit: SOHO Consortium,

Solar Wind

The Solar Wind is composed of electrons and protons at a density of 5–10 particles per cubic centimeter and a temperature of nearly 100,000 K. There are also a few metallic ions. The wind speeds vary from 300 km/s to as high as 800 km/s. While a 100 miles per hour wind on Earth can cause irreparable damage and destruction, a million miles per hour solar wind would not even lift a straw. Why? The solar wind is of extremely low mass density and therefore has hardly any momentum to share. The rotating sun throws out solar wind in the fashion of a garden sprinkler (Fig. 1.16). Nevertheless, the solar wind carries with it the solar magnetic field, which has dire consequences when the wind impinges upon Earth’s magnetosphere to produce the bow shock and the tail structure (Fig. 1.17) and CME-induced magnetic storms. The interaction of the solar wind with Earth’s atmosphere has given rise to an important area of research called the solar–terrestrial physics. The solar wind flows over all the planets and their satellites covering the entire solar system and finally merges with the interstellar medium.

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Plasmas

)LJXUH  The sun flings out solar wind particles in much the same manner as a garden sprinkler throws out water droplets. The artist’s drawing of the solar wind flow was provided with courtesy of NASA.

)LJXUH  Artist’s impression of Earth’s bow shock and magnetosphere. The solar wind shapes Earth’s magnetosphere and causes magnetic storms. Courtesy of SOHO/[instrument] consortium. SOHO is a project of international cooperation between ESA and NASA.

The Plasma Universe

1.10

17

Cometary Plasma

Comets, known as the dirty snowballs, are composed of frozen gases and volatile and non-volatile dust grains. The gases become ionized as a comet approaches the sun producing a plasma tail (Fig. 1.18) distinct from the dust tail carrying micron-sized dust grains. The plasma tail is much less massive and shows much less gravitational bending than the heavier dust tail. The plasma tail consists of highly accelerated plasma particles. Interestingly, from the existence of the cometary plasma tails, it was concluded that the solar wind must have a speed of a few hundred kilometers per second for otherwise it could not ionize the cometary gases. The solar radiation can cause ionization only when the comet is

)LJXUH  This photograph of Comet West was taken by amateur astronomer John Laborde on March 9, 1976. The picture shows the two distinct tails. The thin ion tail is made up of gases, while the broad tail is made up of tiny dust particles, credit: NASA, courtesy, John Laborde.

18

Plasmas

very close to the sun. There is a strong interaction between the solar wind and the plasma tail. The two streaming magnetized plasmas exchange mass, momentum, and energy and the plasma tail in the bargain acquires intriguing forms such as the helices (Fig. 1.19) and the sausages. The comets have also been observed to emit ultraviolet radiation as well as X-rays, both, most likely, due to interaction with the solar wind.

)LJXUH  Helical structures in the plasma tail of comet Ikeya-Saki, credit: NASA, courtesy, John Laborde.

The Plasma Universe

1.11

19

Planetary Plasma

Mercury, the closest planet to the sun, has no stable atmosphere. Although the solar wind impacting the planet tears off material from its surface, the very weak gravity of the planet has no hold and the released gases are lost in space. Venus has an extended region containing electrons and ions, the ionosphere, located at altitudes 120–300 km above the surface. The ionization is due to the solar ultraviolet radiation and exists only over the sun-facing side of the planet. The anti-sunside has practically no ionization. The ionosphere of Venus like the ionosphere of the Earth consists of distinct layers with varying degrees of ionization. The maximum electron density ≈ 3 × 105 cm−3 is reached at a height of 150 km. The ionosphere extends up to ≈ 300 km beyond which exists the magnetoplasma generated by the solar wind. The magnetoplasma has most of the characteristics of Earth’s magnetosphere, which is described next. The predominant ions are of + oxygen, O+ 2 and O . The ions of hydrogen, oxygen, and helium are also being lost due to the solar wind drag. The observation that the planet is losing twice as much hydrogen as oxygen indicates a continuous loss of water. The giant gaseous planet Jupiter has the largest magnetosphere extending about three million km on the sun-ward side and more than six hundred million km on the tail side. The large magnetosphere is due to the rather strong magnetic field of the Jupiter. It is nearly 14 times that of the Earth. The field is believed to be generated inside Jupiter’s metallic hydrogen plasma core where due to enormous pressures the electrons are squeezed out of the hydrogen atoms and are free to conduct electricity. This is the mechanism of pressure ionization. The highly volcanic satellite IO of the Jupiter (Fig. 1.20) lies within the magnetosphere of the Jupiter. The satellite spews out plasma at a rate of a thousand kilograms per second. This plasma appears as a torus around the planet and is extremely hazardous for spacecrafts. The next planet Saturn has a simpler and a lower magnetic field than that of the Jupiter and consequently a much smaller magnetosphere. The seventh planet Uranus is most peculiar as, unlike other planets, its axis of rotation lies in its orbital plane. This, combined with the inclination of its magnetic axis, makes its magnetotail acquire the shape of a corkscrew. The planet Neptune has a magnetosphere similar to that of the Uranus but with a simpler geometry of its magnetotail. Pluto’s status is under consideration!

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)LJXUH  This image is taken by NASA’s Cassini spacecraft. Jupiter’s magnetic field has been sketched over the image. The disk of Jupiter is shown by the black circle and the approximate position of the doughnut-shaped torus, created from material spewed out by volcanoes on Io, is shown by the white circles, credit: NASA.

1.12

Terrestrial Plasma

By terrestrial plasmas we mean the naturally occuring plasmas such as in the magnetosphere, the ionosphere, and the lightning plasmas in Earth’s upper atmosphere as well as the man-made laboratory plasmas. It was Nobel Laureate Irving Langmuir (Fig. 1.21) who, in 1928, first gave the name plasma to an ionized gas as the charged particles floating in the conducting gas reminded him of red blood corpuscles and other organisms swimming in a blood plasma. Some of the inventions made

The Plasma Universe

21

by Irving Langmuir are the gas-filled incandescent tungsten lamp, technique of atomic hydrogen welding, and the radio vacuum tubes. He was awarded the 1932 Nobel Prize in chemistry for his work in surface chemistry. He discovered the seeding of clouds with dry ice and iodide to produce rain or snow. The high-frequency electric oscillations of plasma electrons, the Langmuir waves, one of the most important characteristics of a plasma, are named after him. The Langmuir probe, a device with one or more electrodes, is flown on board satellites to measure in situ properties of space plasmas.

)LJXUH  Irving Langmuir (1881–1957), credit: http://www.en.wikipedia.org /wiki (public domain).

1.13

Earth’s Magnetosphere

The region of the magnetic influence around the Earth is the magnetosphere of the Earth. Its far from spherical shape, more of an oval actually, is determined by the predominant dipole magnetic field of the Earth, the magnetized swirling flows of the solar wind, and the highly variable interplanetary magnetic field. The sun-facing boundary of the magnetosphere lies where the solar wind flow is deflected by the Earth to go around it (Fig. 1.22). This happens at a distance of about 10 to 12 times

22

Plasmas

Earth’s radius RE (RE ≈ 6000 km) from the center of the Earth. In this region the solar wind pressure is equal to the magnetic pressure due to Earth’s magnetic field. The solar wind stretches the magnetosphere into a long tail that extends to nearly ≈ 200RE on the anti-sunward side. The observations by the Cluster spacecrafts of the European Space Agency show the highly dynamic (wavy) nature of the bow shock. The interrupted solar wind flow creates a bow shock structure typical of most obstructed flows. The magnetospheric plasma contains ions and electrons both from the solar wind as well as from the lower atmospheric layer, the ionosphere of the Earth. The magnetosphere protects Earth’s atmosphere from the direct assault of the solar wind and thus prevents the erosion of the life-supporting atmosphere. Nevertheless, it is in the magnetosphere that the magnetic storms (by which is meant the large variations in plasma properties of the magnetosphere) caused by the solar eruptions, the coronal mass ejections, play havoc with spacecrafts and communication systems. The magnetic storms also bring with them highly energetic electrons and protons. Streaming along Earth’s magnetic field, the energetic particles end up at the polar regions of the upper atmosphere of the Earth. Here, they collide with the atmospheric gases such as oxygen and nitrogen. The collisions produce the excited atomic states of the gases. The excited states relax back to the ground states by the emission of radiation at their characteristic colors. This is the Aurora (Fig. 1.23). A rare catch, the conjugate aurora, is shown in Fig. (1.24). NASA’s Polar spacecraft observed auroral displays simultaneously at the north (Aurora Borealis) and the south (Aurora Australis) polar regions on October 22, 2001.

)LJXUH  Artist’s illustration of events on the sun changing the conditions in Near-Earth space, credit: NASA.

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23

)LJXUH  A coronal mass ejection hit Earth’s magnetic field on October 8, 2012 sparking a dramatic display of Arctic lights that only slowly subsided three days later. The Aurora appears to be casting rays of green light through the clouds. Hugo Løhre photographed the aurorae over Lekangsund, Norway, on October 10, 2012. Text: Dr. Tony Phillips, Image courtesy of Hugo Løhre, NASA.

)LJXUH  Conjugate aurorae on both the north and the south polar regions, observed with NASA’s Polar spacecraft, credit: NASA.

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Plasmas

1.14

The Ionosphere

All those short-wave radio listeners must know how they receive their favorite tunes from across the world. It is this region of Earth’s atmosphere, the ionosphere, that makes it possible. The ionosphere lies just below the magnetosphere. It is a relatively thin region extending from 50 km to 1000 km above the surface of the Earth. In fact, the magnetosphere and the ionosphere are defined by the predominance or otherwise of the collisions between the plasma particles. The magnetosphere is collision dominated and the ionosphere is mostly collisionless. The ionosphere contains highly ionized plasma produced by the ultraviolet radiation of the sun and the cosmic rays. At night the ionosphere is ionized only by the cosmic rays and therefore has much lower electron density. There are several layers in the ionosphere distinct by their electron density (Fig. 1.25).

1000

Topside

O+ H +

600 N max

Altitude in km

H max F2 Layer F Region O+ F1 Layer

150

O 2+ NO +

E Region 90 D Region Density:

10 4

10 5 Electrons/cm 3

10 6

)LJXUH  Various layers of the ionosphere and their predominant ion populations are listed at their respective heights above ground. The density in the ionosphere varies considerably, credit: http://www.commons.wikimedia.org/wiki (public domain).

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25

The ionosphere plays a most essential role in the short wave transmission by bouncing off the radio waves incident on it. The ionosphere is also subjected to the violent outbursts of the sun which changes the bounce-off frequency of the radio radiation. The ionosphere is the most effective sunscreen, against the ultraviolet radiation of the sun, without which life on Earth, as we know, would be impossible. The lightning also influences the ionospheric electron density, most surely at night.

1.15

Plasmas in Laboratory

The laboratory studies of plasmas began with electric discharge through gases. One of the earliest discharge tubes is the Crookes tube named after the British physicist Sir William Crookes. It consists of two cold electrodes placed in a partially evacuated glass tube containing an inert gas along with a small quantity of Mercury vapor.

)LJXUH  domain).

The Maltese Cross tube, credit: http://www.crtsite.com (public

When voltage of a few to 100 kilovolts is applied between the electrodes, the gases in the tube ionize. The high voltage accelerates the electrons to high speeds and they collide with the Mercury atoms to mostly produce

26

Plasmas

ultraviolet radiation, which is invisible to the naked eye. However, if the glass tube is coated with a phosphor such as zinc sulfide, the ultraviolet radiation, through the process of fluorescence, is converted into the visible light. This is the underlying principle of all plasma-based displays. Crookes Maltese Figure (1.26) shows the Maltese Cross tube, one of the most famous Crookes tubes. It is found that the electrons go in a straight line and do not go through the metal of the cross. When the cross is on the floor of the tube, the electrons strike the glass wall and emit green light. When the cross stands upright, the shadow of the cross is seen. The coating could be shaped into any design which would then become visible due to fluorescence (Fig. 1.27).

)LJXUH  The Crookes phosphorescent flower tube, credit: http://www.crtsite. com (public domain).

Lightning is another example of a high-current electric discharge in air (Fig. 1.28).

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27

)LJXUH  Lightning over the outskirts of Oradea, Romania, during the August 17, 2005 thunderstorm, credit: http://www.en.wikipedia.org/wiki (public domain).

Huge electric fields are generated and the associated energy is released in extremely short pulses of less than a microsecond duration during lightning. The understanding of this spectacular phenomenon is yet mostly speculative.

28

Plasmas

1.16

The Clean Energy

The consumption of energy is a direct measure of the development index of a nation. As the aspirations of the peoples of the world soar so does the need for more and more energy. The fossil fuels aren’t going to last forever. Nuclear fission has its well familiar hazards. Thus the need for a clean source of energy has been long felt. Controlled thermonuclear fusion has held the hopes of the scientific community for several decades now. Once realized, the fusion energy is almost free and free of radioactive hazards. Perhaps the cleanest source of energy, it leaves no carbon foot prints and no pollutants in the air.

1.17

Thermonuclear Fusion

Controlled thermonuclear fusion, envisioned as the future energy source, is not much different from the nuclear fusion reactions that produced the primordial helium and other light elements in the early universe. It is the thermonuclear fusion that takes place in the interiors of stars and endows them with brightness. Our sun shines due to the energy generated by the proton–proton nuclear reactions. On the terra firma, the nuclear fuel, a mixture of deuterium and tritium, at appropriately high temperatures is all that is needed to generate energy. The fusion produces helium four, He(2p,2n) and neutrons (Fig. 1.29). The dream is to convert the energy of the neutrons into electricity. n

D(p,n)

(2p,3n)

T(p,2n)

He(2p,2n)

)LJXUH  Nuclear fuel deuterium and tritium fuse to produce helium and energetic neutrons.

But this has not turned out to be as easy as it appeared to be. The hot plasma needs to be maintained at the required temperature for a required amount of time for igniting the nuclear reactions. Deuterium, the raw fuel, in the form of Heavy water, HDO, is an immense gift of nature that has the potential to serve the multitudes for billenia. There are two

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29

molecules of heavy water in every 6500 molecules of water. When we succeed in tapping fusion energy from heavy water, a mere one gallon of water would furnish energy equivalent of 250 gallons of gasoline! The experts estimate that the world availability of energy in deuterium could last for six hundred billion years. Alas the universe may not last that long, the solar system, presumably, at 4.5 billion years is already undergoing a mid-age crisis. In the meantime, however, concerted efforts are on to find the right recipe to recreate a sun in the laboratory. While the sun has its gravity to hold the plasma of several million Kelvins, the earthlings depend on the magnetic containers and the inertial confinement. The former has led to the development of a class of machines called tokamaks and the latter to the laser fusion devices.

1.18

Magnetic Confinement Fusion

Charged particles such as electrons and protons feel the Lorentz force in the presence of a magnetic field. The charges execute circular motion in a plane perpendicular to the direction of the imposed magnetic field. So they kind of stick to the magnetic field lines (Fig. 1.30). B

B

p

e

)LJXUH  Charged particles execute circular motion around the magnetic axis. Negatively charged particles (e) go anticlockwise and positively charged particles (p) go clockwise for the direction of the magnetic field B shown here.

This was the origin of the optisim in magnetic confinement of a plasma. However, charges feel no force along the direction of the magnetic field and flow freely along this direction (Fig. 1.30). This has been the bane of magnetic confinement. But what if there are no free ends and the field lines curve back onto themselves bringing the charges back with them. This concept has given rise to a torus or a doughnut-shaped confining magnetic field and the machine tokamak (Fig. 1.31). The word

30

Plasmas

tokamak was formed from the Russian words ‘‘TOroidalnaya KAmera ee MAgnitnaya Katushka,’’ or Toroidal Chamber and Magnetic Coil.

)LJXUH  organization.

ITER, the world’s largest tokamak, credit: Michel Claessens, ITER

A tokamak has a doughnut-shaped vacuum vessel in which the deuterium–tritium plasma is sustained at temperatures exceeding hundred million Kelvins. The plasma is confined by strong magnetic fields of the order of tens of thousands of Gauss. The International Thermonuclear Experimental Reactor (ITER) is the world’s largest tokamak being fabricated at St Paul-lez-Durance, France by an international consortium, of which India is an equal partner.

1.19

Inertial Confinement Fusion

To achieve the fusion conditions, i.e., maintaining the fuel, a mixture of deuterium and tritium at high densities and high temperatures for a sufficiently long time, powerful lasers are used to rapidly heat a pellet of a few micrograms of the fuel. The ablation of the surface material creates a coronal plasma exploding outwards, which exerts a back reaction

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31

(rocket-like) inwards and compresses and heats the pellet to the ignition point. The fusion reaction proceeds outwards more rapidly than the outward expansion of the hot core plasma. Thus the fuel-plasma is confined by the inertia of its own mass and hence the name. Efforts are underway to make the inertial fusion a viable commercial source of energy. The National Ignition Facility (NIF) at the Lawrence Livermore National laboratory of the United States, with the world’s largest and highest-energy laser, is conducting experiments that will focus the energy of 192 giant laser beams on a hydrogen fuel target (Fig. 1.32).

)LJXUH  Upper portion of the NIF’s target chamber, credit: http://www.en.wikipedia.org/wiki (public domain).

1.20

Space Travel-Plasma Rockets

The amount of thrust for rocket propulsion depends on the mass flow rate of the fuel. In conventional liquid fuel propulsion, a large mass flows at a low speed. The liquid fuel can only reach temperatures of the order of a few thousand degrees. This makes the system heavy and inefficient. However, if the liquid fuel could be replaced by a low mass fuel, such as a million degree plasma ejecting out at rather high speeds, both the efficiency of propulsion as well as the the weight of the payload could be significantly increased. This is the expectation from the plasma rockets (Fig. 1.33), which, at present, are the subject of active research.

32

Plasmas

)LJXUH  A plasma rocket could significantly reduce the travel time to Mars, credit: NASA.

One such plasma rocket is The VAriable Specific Impulse Magnetoplasma Rocket (VASIMR). A small-scale prototype of VASIMR has been jointly fabricated and successfully tested by the NASA and the Radio Frequency and Microwave Technology Center at the Oak Ridge Centers for Manufacturing Technology. The VASIMR uses an extremely hot helium plasma shielded from all metallic parts by confinement with strong superconducting magnetic fields. The magnetic nozzle throws out the plasma exhaust at a whopping speed of 70 km per second from the back of the engine providing a huge forward thrust to the rocket. We see here principles of both the inertial confinement as well as the magnetic confinement in action!

1.21

Plasma Accelerators

Since a plasma consists of electrically charged particles, it may appear that it is easy to accelerate these particles by applying electric fields. But the plasma is also, overall, neutral. When an electric field is applied to a plasma, the positively charged particles and the negatively charged particles move in opposite directions, creating a charge separation and a net charge density, albeit in a region of specified size. The electric field

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33

associated with this charge separation is of the same order as the externally applied electric field. The charge density oscillations set in, in order to maintain the overall neutrality of the plasma; negative charges move toward region of excess positive charge and move away from region of deficient positive charge. These oscillations are called electron plasma waves. These are longitudinal waves with associated electric field and can, in principle, interact with particles and transfer their energy to the particles. However, there is a severe limit to the magnitude of the external electric field because it can cause dielectric breakdown damages. One successful way to excite intense plasma waves is to subject the plasma to a powerful laser. The laser can, by a nonlinear mechanism akin to Raman scattering, excite plasma waves. Two counterpropagating laser beams can excite plasma waves with their phase speeds close to the speed of light c. The already high-energy particles with speeds near c now remain in phase with the plasma waves for a significant duration of time and suck up tremendous amounts of energy and achieve high acceleration. This is the basic principle of beat wave plasma accelerators. There are other types of plasma accelerators, which differ by the way they generate plasma waves. The strong plasma wave can also be excited by, for example, a bunch of electrons (Fig. 1.34).

)LJXUH  An electron beam repels electrons of the plasma and forms a wake of positive ions behind it. The positive ions and the displaced plasma electrons create an electric field that accelerates the electron beam.

34

Plasmas

The table-top plasma accelerators have been demonstrated to achieve much larger acceleration at much lesser cost than the conventional linear accelerators. Particles have been accelerated to billions of electron volts in a space of a few centimeters. There are other types of plasma accelerators which use different ways of generating plasma waves. Plasma accelerators are of immense value in conducting high energy studies in medicine and many other areas.

1.22

Plasma Materials and Methods

Stars manufacture almost all the elements. Humans are made up of star dust. Perhaps a bit of the stellar inheritance inspires humans to invent new materials. The unquenchable longing for long lives of men and materials is the driving motivation. Plastic is perhaps the best example of the indestructible so much so that it has now become nearly impossible to get rid of it. Electroplating and thermal vapor deposition are some of the traditional processes of modifying the metal surfaces for various technological and aesthetic purposes. These processes are not only energy guzzlers but also have rather limited scope. Plasma surfacing in general and plasma nitriding in particular is a process much in use in diverse applications. In this process, high voltage is applied to a low-pressure discharge tube containing hydrogen, nitrogen, and other gases. The gases are ionized. The molecular nitrogen, dissociates into its atomic form. The ions are accelerated and driven on to the metal surface where they heat and clean the surface. The nitrogen atoms are absorbed on to the metal surface. What is the role of the plasma? It provides a high-temperature environment, dissociates nitrogen molecules, produces ions that can be accelerated for impacting the required surface. Plasma nitriding hardens the metal surface and prolongs its life against corrosion. The plasma nitriding is a one-step energy-efficient and eco-friendly process. It can take place at a wide range of temperatures without changing the basic properties and dimensions of the working surface. This is very important for precision components. The ability of plasmas to sustain high temperatures makes it an attractive medium for enhancing the reaction rates of some chemical reactions, opening a new area of research, the plasma chemistry, an area impregnated with the inventions of new materials. Plasma welding and etching are already well-known processes. The display panels used in plasma televisions work on the basic principle of a discharge tube with phosphor coating. In plasma display panels, many tiny gas cells exist between two glass plates. The gas in them is ionized and generates the ultraviolet radiation, which on hitting phos-

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35

phors becomes visible. Some of the advantages of these panels are that they can have huge size, light weight, high resolution, and no blurring. Plasma-based air and water purifiers are based on electrically charging the suspended particulate matter, which can then be collected at appropriately charged electrodes. The technique of plasma arc gasification has been found to be a very efficient method of handling hospital waste. In this process, the waste is decomposed into its elemental form by passing a high electric current through it. The elemental forms are more easily disposable. The scope of plasma applications is only limited by the human imagination. The plasmas are ubiquitous, be it in inorganic or in organic plane!

1.23

Fully Ionized Plasma

Since a plasma is created by the ionization of neutral gases, it consists of positively charged ions and negatively charged electrons along with some neutral atoms. The positive charge density is equal to the negative charge density. The number density of electrons or ions is determined by the processes of ionization, which would be discussed in chapter two. One can characterize a plasma by its degree of ionization. By fully ionized plasmas one does not necessarily mean that there are no neutral atoms in the plasma. But the number density of the neutral atoms is smaller than the number density of electrons or ions. How much smaller? This can be determined by considering the collisions among the three species of particles, the electrons, the ions, and the neutral atoms. The electrons and the ions feel the Coulomb force between them. The rate of their collisions is determined essentially by their number density and the range of the Coulomb force between them. The electrons also undergo collisions with neutral atoms and so do the ions. The charged particle–neutral atom collisions are governed by the induced charge displacements on the atoms. As a result of which the atoms acquire electric dipole moment and the collisions depend on the polarizability of the neutral atom. Of course, electron–neutral atom collisions are more frequent than the ion–neutral atom collisions, electrons being more agile. So we can now give a quantitative definition of a fully ionized plasma: the one in which the rate of the electron–ion collisions is much larger than the rate of electron– neutral atom collisions. The much larger is, let us say, at least ten times. Thus in a fully ionized plasma the electron–neutral atom collisions can be ignored and we deal with a simpler system with two species of particles, the electrons and the ions. A fully ionized plasma exists at a rather high temperature. For example, it takes 13.6 electron volts of energy to

36

Plasmas

separate the electron from an hydrogen atom. One electron volt corresponds to about 11, 000 Kelvin. Thus a hydrogen plasma would be fully ionized at temperatures higher than this. The solar corona at a million degree is a good example of a fully ionized plasma. At such high temperatures, the electron–ion collisions become rare. This may sound counterintuitive because one has more experience with collisions increasing with temperature. However, the rarity of electron–ion collisions at high temperature is due to the nature of the Coulomb force. Such a plasma is called collisionless. This further simplifies the study of a fully ionized plasma. But that’s a short-lived relief. Collisionless plasmas go easily unstable. Another simplification is possible in an electron–ion plasma. The ions are much heavier than the electrons and therefore are slow to respond. In many a studies, one can assume the ions to simply provide a background and ignore their active participation in the dynamics.

1.24

Partially Ionized Plasma

The presence of enough neutral atoms in a plasma makes it a partially ionized plasma. The particle collisions become very important. The dynamics becomes more involved as one has to, now, deal with at least three species of particles with their mutual interactions. The additional processes such as the ionization of the neutral atom, electron attachment to the neutral atom need to be included. This applies, for example, to the chromospheric region of the solar atmosphere. A simplification of a partially ionized plasma is a weakly ionized plasma in which the rate of collisions between electrons and ions is much smaller than the rate of collisions between electrons and neutral atoms. This is precisely due to electrons being much lesser in number than the neutral atoms. Such a plasma exists on the solar photosphere. A novel effect called ambipolar diffusion arises in a weakly ionized plasma. It is essentially due to coupling of neutrals and ions; the neutrals in the process acquire some of the properties of charged particles, begin to be affected by electric and magnetic fields and participate in electromagnetic phenomena. The ambipolar diffusion is actually responsible for the diffusion and transport of magnetic flux in many weakly ionized astrophysical plasmas.

1.25

Ultracold Plasmas

The production of a plasma necessitates heating of a neutral gas, and therefore a plasma is usually at a fairly high temperature. For example, the temperature of the solar coronal plasma is as high as several million

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37

degree Kelvin. However, extremely cold plasmas have also been produced in the laboratory. This is done by first cooling the gas atoms to a very small fraction of a degree Kelvin, in fact, just a tiny bit above the absolute zero temperature. The cooled atoms are then subjected to a laser powerful just enough to remove an electron from the cold atom. The electrons, so separated, have a temperature of the order of 1 Kelvin above the absolute zero. This is an extremely cold plasma. Of course, this plasma has a very short lifetime as the tendency for the recombination of the electrons and the ions at this low a temperature is rather strong. Such plasmas are believed to exist in the interiors of white dwarf stars and planets such as Jupiter.

1.26

Dusty Plasma

It is a dusty universe. The dust in astrophysical settings consists predominantly of silicates and graphites in micron-sized particles called dust grains. The star-forming regions, the interstellar medium, the cometary tails, and the molecular clouds in galaxies are some of the dust-rich sites. Some of these sites are also rich in their electron content, that is, they have a reasonably high degree of ionization. The electrons attach on to the dust grains which then are endowed with a negative electric charge. The dust grains are also subjected to the ultraviolet radiation of the stars and may emit electrons by the photoelectric effect and be endowed with a positive electric charge. Thus the electrically charged dust grains become the third species of particles in an electron–ion plasma. The plasma is always overall neutral. So with positively charged ions and positively and/or negatively charged dust grains the electron density is determined from the condition of overall charge neutrality. The electron density is now not equal to the ion density. It depends on the electric charge on the dust grain. The electric charging of a dust grain is a dynamic process. The quantity of charge on the dust grain is determined by the amount of electron flux impinging on the grain. Are we going in circles? Well, this is what compels us to deal with this highly coupled system in a selfconsistent manner so that all particle densities and charges are estimated correctly without violating the overall charge neutrality. It is obvious then that the electric charge of a grain is a variable quantity unlike, say, the charge on an electron or an ion. The grain charge is generally time dependent and is a dynamical parameter. This adds to the already complex nature of the multispecies plasma in no mean a manner. Charged dust is often a by-product in several industrial processes. In the laboratory, dusty plasmas are being studied for their exotic properties such as their ability to form crystal-like structures. Some experiments and

38

Plasmas

computer simulations suggest that a dusty plasma can organize itself in DNA-like double helix structures capable of storing information in their configuration. The stability of the spectacular ring system of the planet Saturn (Fig. 1.35) is proposed to be due to electrically charged dust grains and their collective levitation in the presence of gravitational and electromagnetic forces. Do the rings support double-helix structures too?

Visible Light (ISS)

Radio Signals (RSS) )LJXUH  NASA.

1.27

The dusty plasma ring system of Saturn in visible and radio, credit:

Quantum Plasma

At a temperature in the range of a few thousand to a few tens of thousand Kelvin and at a mass density of nearly one gram per cubic centimeter, hydrogen gas shows a transition to a metallic liquid state. In this state there are no identifiable individual atoms or molecules. Under conditions of extremely high pressures, the electrons are squeezed out of their parent atoms and they attain enough freedom to move around in the vicinity of more than one nucleus. The high density of the nuclei and the electrons make it a strongly interacting plasma in which the Coulomb potential energy between the particles is much larger (the interparticle distance is very small) than the kinetic energy of their thermal motion. Such a plasma is called a strongly interacting plasma. It is highly collisional. It must be studied using quantum mechanical techniques as opposed to the low-density classical plasmas. These are the circumstances of the interior of the giant gaseous planet Jupiter, for example, where pressure is nearly sixty million times the Earth’s atmospheric pressure. Fire and flames are other examples of strongly interacting plasmas. At even higher densities a proton is crushed into its building blocks, the quarks. The particles holding these quarks together are called gluons. Thus a quark–gluon plasma is perhaps the ultimate high-density quantum plasma. Such a plasma is speculated to exist within a microsecond of the big bang explosion. Our journey through the plasma universe started

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39

with the big bang and here we are at the end of our journey again at the big bang!

1.28

Techniques for Studying Plasmas

Molecules in a gas or liquid, men in a metropolis, and stars in a galaxy are in a state of incessant motion resulting from the action of various forces: internal, due to other molecules or (wo)men or stars, and external, due to the rest of the universe. It would be a frustrating task if we had to know the position and velocity of every molecule or star at every instant of time, in order to deal with the gas or the galaxy. The saving grace is that we need not know every move of every molecule or star and we will still be able to manipulate the system to our advantage. For most purposes, it suffices to know the average properties such as density, momentum, pressure, and energy, transport properties such as thermal and electrical conductivities, and mechanical and electromagnetic stresses of a large system. A system with a large number of particles such as a plasma is studied in its most simplified form using a single fluid description. In this description, the particles lose their identity. There is only one fluid made up of an average mix of positively and negatively charged particles. This is an electrically conducting fluid, much like a liquid metal such as Mercury, with only one density, one velocity, one temperature, and one pressure. It is very much like the transformation of a gas of water molecules to the liquid water. This process entails taking the continuum limit of an otherwise discrete system. In its next level of detail, a plasma is described by two fluids, one fluid of positively charged particles and the second of negatively charged particles. The two-fluid description recaptures some of the reality that there are two types of particles. The mass, momentum, and energy conservation laws are derived for each fluid. The microphysics of the many body charged particle system in its full glory is recovered with the kinetic description that summons the detailed distribution functions of the particles, somewhat like the kinetic theory of gases, only made much more complex with the presence of electromagnetic phenomena. The quantum plasmas are studied within the ambit of quantum mechanics of many body systems. These descriptions are further subjected to various simplifying assumptions to enhance the tractability of rather trying circumstances!

40

Plasmas

1.29

Summary

That plasmas are the first state of matter out of which arose the other three states of matter has been amply demonstrated. The ubiquity of plasmas in the universe needs no demonstration. The phenomenal diversity of plasmas is there all over the universe for all to see. The industrial and technological roles of plasmas are no less spectacular. Plasmas for humans is an unshakable bond. Techniques for understanding plasmas are many and varied. Happy learning!

Problems 1.1 Argue that plasmas are the first state of matter. 1.2 Collect the values of plasma densities and temperatures of all the plasmas mentioned in the chapter. Make a plot of the density vs. temperature and show the location of the various plasmas. 1.3 Revisit the kinetic theory of gases. 1.4 Calculate the most probable and the root mean speed for the Maxwellian velocity distribution of particles of mass m at a temperature T. 1.5 Estimate the ratio of electron to proton density in a plasma containing electrons, protons, and singly ionized helium ions.

2

2.1

Plasma Basics

Making Plasmas

We learnt, in chapter one, how the universe originated from a soup consisting of electrons, positrons, and radiation and how the matter came into being and became almost electric charge neutral as the universe cooled. The reionization of matter commenced with the formation of stars spewing out ultraviolet radiation. Ionization of matter by radiation is the most common process in nature. The stars in galaxies gave rise to the highly ionized state of the interstellar medium. The ultraviolet radiation of the sun created the ionosphere of the Earth. Matter can also be ionized, in addition, by heating, compression, and passing an electric current through it. Let us learn about these mechanisms in the following sections.

2.2

Plasma Formation by Photoionization

A gas can be ionized by illuminating it with electromagnetic radiation of the appropriate frequency ν . This process is known as photoionization. It is also called the photoelectric effect generally in the context of emission of electrons from metal surfaces that are subjected to radiation of the right frequency. It is essentially a quantum mechanical effect that confirmed the quantum nature of the electromagnetic radiation and for which Albert Einstein was awarded the Nobel prize in 1921. The ionization process can be described as the release of electrons from the atoms of a gas. The molecules of a gas dissociate into atoms much before the ionization process occurs. A singly charged ion results when a single electron is emitted from a neutral atom. This reaction can be expressed as A + hν = A+ + e

(2.1)

where A represents an atom or a molecule, A+ is a singly charged ion

42

Plasmas

of the atom or the molecule A, and e is the emitted electron. Here hν is the energy of a photon of the radiation and it must exceed the ionization energy I of the atom. For example, a photon of energy 13.6 electronvolts is required to ionize a hydrogen atom. The excess energy of the photon mainly goes to the kinetic energy of the emitted electron. The energy conservation is written as hν = I +

meVe2 2

(2.2)

where Ve is the velocity of the emitted electron. It is clear that it is the frequency of the radiation that has to be of the right value no matter what its intensity is. The ionization energy I defines the threshold of the photon frequency ν required for the ionization process. The degree of ionization increases rapidly at the threshold frequency and then decreases as the photon frequency increases. There is also the possibility of ionization with radiation of frequency less than the threshold frequency. This can be achieved through a twostep absorption of photons by an atom. In this process, a photon of frequency less than the threshold frequency is absorbed by the atom that is raised to a virtual or a real excited state. The second photon must then be absorbed by the atom while still in its excited state and the ionization ensues. The catch is that the second photon must be avalable to the excited atom before it decays back to its ground state. A three-photon process and higher multiphoton absorption processes leading to ionization have also been observed in the laboratory. An extremely intense photon beam is required to facilitate multiphoton absorption processes. Lasers beams provide such intense radiation fields.

2.3

Collisional Ionization

The high-energy electrons emitted via photoionization can further produce ionization by their collisions with the atoms and the ions. The reverse process of recombination also goes on, which reduces the number of electrons and ions. In addition, electrons may get attached to the atoms to produce negatively charged ions. A steady level of ionization may be obtained under the influence of all these processes. Collisions of heavy particles such as protons with the gas atoms or molecules can cause ionization in multiple ways. A proton colliding with a helium atom can release one electron from the helium atom or both the electrons of the helium atom without undergoing any change in its own charge state. Conversely, the proton could recombine with one of the emitted electrons to become a neutral hydrogen atom. There

Plasma Basics

43

are more possibilities too! These collisions are governed by the Coulomb force between the incident particle and the electrons bound to the atom. Collisions between atoms of gases of different ionization potentials can also cause ionization. This happens when the ionization potential of the atoms of the first gas is lower than the energy of the excited state of the atoms of the second gas. The energy released by the deexcitation of the atoms of the second gas is used up in ionizing the atoms of the first gas. The condition is that the life time of the excited state must be longer than the collision time. This mechanism is known as the Penning ionization.

2.4

Thermal Ionization

Plasmas are also generated by the mechanism of thermal ionization as a substance goes through the solid–liquid–gas–plasma sequence with an increase in its temperature. At sufficiently high temperatures, the collisions among atoms can often knock off their electrons, resulting in the formation of positively charged ions and the negatively charged electrons along with the leftover neutral atoms. It is known from the kinetic theory of gases and the classical statistics that at temperature T , the number density of particles (atoms, ions, electrons) nL with an energy EL are governed by the Boltzmann law:   EL nL ∝ exp − (2.3) KB T indicating clearly that the number density of high-energy particles diminishes exponentially in a system in thermodynamic equilibrium. Here KB is the Boltzmann constant. Similarly, the number density of particles nM with the energy EM are   EM (2.4) nM ∝ exp − KB T If some of the atoms or molecules of this gas are ionized at the temperature T , we have a mixture of neutral atoms as well as electrons and ions. There is one electron for one positive electric charge on the ion. If there are two positive charges on the ion, there would be two electrons. We can identify the state L with the electron–ion pair and define the ion density ni ≡ nL . The state M can be identified with the neutral atoms. Then the ratio p defined as p=

ni nM

(2.5)

44

Plasmas

describes the fractional ionization and the energy difference I given by I = EL − EM

(2.6)

represents the ionization energy of the atom under consideration. The number density ni of the ions then becomes   I ni ∝ exp − (2.7) KBT The proportionality factor is determined from quantum mechanical and statistical considerations and is a function of the electron mass, the Planck constant h, the Boltzmann constant, and the number density ne = Zni of the electrons where Ze is the electric charge on the ion with e the magnitude of the charge of an electron. Here it would suffice to give an approximate expression as   nM I (2.8) cm−3 ni ≈ 2.4 × 1015 T 3/2 exp − ne KB T where T is in Kelvin. Equation (2.8) is the celebrated Saha Ionization formula. The Indian scientist M.N. Saha derived this formula in 1920. It is applicable only under conditions of thermodynamic equilibrium. The Saha equation found tremendous utility in accounting for the atomic spectral lines from stars. These lines originate from different atomic and ionic species, which could be in various excited and ionization states. Saha’s equation with its modifications by other scientists enabled the estimation of the number densities of the ionic species of different ionization energies at a given temperature. The inverse dependence of the ion density ni on the electron density ne can be understood by realizing that the ionization process is also accompanied by the reverse process of recombination. As the electron density increases due to ionization, the probability of an electron recombining with an ion also increases. The net ionization is a sum total of these two mechanisms and is accounted for by the Saha equation. It is clear from Eq. (2.8) that there is a finite amount of ionization even at temperatures much less than the ionization temperature KB Ti = I but the degree of ionization increases tremendously near T ≈ Ti . Let us take the example of hydrogen gas. Its ionization energy is 13.6 electronvolts, which corresponds to the ionization temperature Ti ≈ 157000 K. With ni = ne , the quantity n becomes   n2i Ti 15 3/2 n= ≈ 2.4 × 10 T exp − (2.9) nM T

Plasma Basics

45

As an example, let us estimate n at three different temperatures. we find n ≈ 6 × 109 at T = 6000 K, n ≈ 3.6 × 1014 at T = 10000 K, and n ≈ 6 × 1022 at T = 157000 K. This shows that there is a substantial ionization even at temperatures much less than the ionization temperature and that the ionization is nearly total at the ionization temperature Ti . At temperatures larger than Ti , the electrons and ions become highly energetic as a result of which the collisions between them become quite rare, the plasma begins to depart from its state of thermal equilibrium and Saha’s formula no longer applies.

2.5

Pressure Ionization

At extremely high mass densities of matter when the volume per atom becomes lesser than the normal volume occupied by an atom, the atoms can no longer be identified as individual units. There is an overlap of the outer electronic orbits of the atoms, the outermost electrons become loosely bound to the atoms and behave more like a free electron gas. The substance becomes an aggregate of nuclei and electrons, a highly compressed plasma. This phenomenon is called the pressure ionization. Under such circumstances the electrons behave like a cold degenerate quantum gas provided the temperature is not too high. The energy distribution of the degenerate electrons is very much different from the Boltzmann distribution. The degenerate electrons obey the Fermi–Dirac statistics. This statistics is a consequence of the Pauli’s exclusion principle. This principle states that no more than one electron can be accomodated in each quantum state specified by the energy of the electron. In contrast, in the classical statistics, the temperature determines the occupation of an energy level through the Boltzmann factor. The degree of ionization of a degenerate gas is mainly a function of the pressure or the density and not that of the temperature. The Indian physicist D.S. Kothari (1937) was among the first ones to work out the theory of pressure ionization. From this theory it was predicted that the stellar material in the interior of white dwarf stars should be fully ionized. The theory also precluded the existence of a cold body larger in size than the Jupiter. The theory also predicted that the Jupiter and the Saturn should have metallic hydrogen in their interiors in contrast to the Earth, which has a core of molten iron.

2.6

Electric Discharge through Gases

A plasma is generated when an electrical field is applied across a gas or an electrically nonconducting fluid. The neutral gas or the fluid is

46

Plasmas

pumped into a discharge tube, which is fitted with a cathode and an anode. The electric field pulls out the bound electrons of the gas atoms. The electrons move toward the anode and the nuclei move toward the cathode. As the voltage increases, the electrical breakdown of the gas occurs and it begins to get ionized. A stage of avalanche ionization occurs wherein the collisions between electrons and neutral gas atoms create more ions and electrons. The most common application of this mechanism is seen in fluorescent lamps. For example, the everyday use compact fluorescent light tube consists of two metal electrodes across which either direct current or a low-frequency electric field is applied to breakdown the gas vapors and ionize them. There are more advanced variants of this basic process, which are used in different technological applications.

2.7

Critical Velocity Ionization

This is an ionization mechanism in which an already formed plasma ionizes a neutral gas. When the relative velocity between the neutral gas and the plasma exceeds a certain critical value, the neutral gas begins to ionize. The critical value of the relative velocity v is found from mn v2 = eVI 2

(2.10)

where mn is the mass of the atom of the neutral gas and VI is its ionization potential; e is the electron charge. If more energy is supplied to the gas– plasma system, the relative velocity remains at its critical value until the neutral gas is fully ionized. In this phenomenon, the kinetic energy of the plasma ions is transfered to the plasma electrons via the generation of lower hybrid waves (see chapter 4). The transfer of energy via the Coulomb collisions is not an efficient process. The energetic electrons can then collide with the neutral atoms and ionize them. The critical velocity ionization mechanism has been demonstrated in the laboratory though its possible role proposed to be in cometary coma and solar flares among other astrophysical environs remains to be confirmed.

2.8

Measuring Plasmas

After a gas is ionized and a plasma is formed, it is essential to measure the plasma parameters such as its density and temperature. A host of techniques are available depending on whether the in situ or the remote measurements are to be carried out.

Plasma Basics

47

The remotely located plasmas such as most of the space and astrophysical plasmas are not accessible for direct measurements. The instruments aboard satellites have sent data on near-Earth plasmas as, for example, for the Earth magnetospheric plasma. Indirect methods mostly based on the emission and absorption of radiation from the plasmas turn out to be the best bet for measuring the remote plasmas. The spectroscopic observations of the solar eclipse of August 18, 1868 from Madras (now Chennai), India, revealed a hitherto unknown yellow line near the sodium D lines. This line was later identified and christened as the helium line since it was first seen from the sun. These observations not only confirmed the presence of helium on the sun, but they also pointed to the existence of high temperatures of the emission regions of the spectral line. It is a common exercise these days to infer the density and temperature of the solar hydrogen plasma containing helium by taking the ratios of the intensities of the two selected helium lines. A temperature of the order of 20 electron volts and an electron density of the order of 109 per cubic centimeter have been inferred for the solar chromospheric region from where these spectral lines are believed to originate. The early observations of the emission spectrum of the solar corona during a solar eclipse also revealed spectral lines possibly emitted by highly ionized states of iron. It required the coronal plasma temperature to be of the order of a million degree Kelvin. The spectroscopic studies under such extreme conditions in the laboratory were taken up much later to confirm the eclipse results. The absorption profiles of the radiation passing through a plasma containing different species of gas atoms not only furnishes the plasma density and the temperature but also the total number of absorbers present in the plasma. This is how the abundances of elements present in the solar photosphere, for example, are estimated. The rotation of the plane of polarization of the radiation passing through a magnetized plasma offers a reliable method to determine the magnetic field of a remote plasma. This process is known as the Faraday rotation and its detailed discussion can be found in chapter 4. In addition to all these methods, laser-induced fluorescence also enables one to measure the plasma parameters in plasmas containing fluorescing ions. The scattering of laser radiation by the plasma electrons, Thomson scattering, (see chapter 5), is a much used technique in the laboratory. The Doppler broadening of the laser line gives the temperature and the intensity of the scattered light, the electron density. The direct measurements of the plasma parameters is achieved by inserting probes into a plasma. The most commonly used probe is the

48

Plasmas

Langmuir probe, which consists of two metal electrodes. When inserted in a plasma, the electrodes accumulate charges around them and the rest of the plasma is screened from the effects of the electrodes. A sheath region is formed around the electrodes in the nature of the Debye shielding. The current–voltage characteristics of the sheath regions are impregnated with the information on the plasma parameters.

2.9

Plasma Specifics

Are all ionized gases plasmas? Ionization is a necessary but not a sufficient attribute of a plasma. There are conditions that an ionized gas must satisfy before it can qualify to be a plasma. We know that the Coulomb potential V produced by a charged particle of charge Q at a distance r from it is given by V=

Q r

(2.11)

But if the charge Q is situated in a plasma that has a large number of charged particles, is the Coulomb law still valid? Let us find out.

Q

)LJXUH  The positive charge Q is surrounded by negative charges forming the Debye cloud.

Plasma Basics

2.10

49

Debye Screening

The Coulomb law does not apply when a charged particle is in the presence of many other charges. Since each charged particle exerts the Coulomb force on all the other charged particles, we have to determine the effective electrostatic potential φ due to the introduction of a charge Q in a plasma in a self-consistent manner. Let us say a positive charge Q is introduced in an otherwise neutral plasma in which the electron number density is equal to the ion number density. The positive charge Q will attract the electrons of the plasma toward it. These electrons would form a cloud around the positive charge Q and shield its effect from the rest of the plasma. This cloud is called the Debye cloud. What is its size? Its size is determined by the electron density and the electron and the ion temperatures. If the electron density is high, a small-sized cloud can contain enough electrons to shield the effect of the positive charge Q. Conversely, if the plasma temperature is high, the electrons would have sufficient kinetic energy to resist the Coulomb force and they could stay at a large distance from the charge Q. This would increase the size of the Debye cloud. The radius of the Debye cloud is known as the screening length, which can be determined as follows. We make use of the Poisson equation: ∇ · E = 4πσ

(2.12)

∇2 φ = −4πσ , E = −∇φ

(2.13)

Or

where E is the electric field produced by the electric charge density σ due to all the charged particles in a plasma. We know that the number of particles of a given energy ε in a gas or a plasma at a temperature T is determined by the Boltzmann law:   ε (2.14) N(ε ) = N0 exp − KBT where N0 are the number of particles at ε = 0. Now the electrostatic energy of a charged particle of charge q in a plasma of electrostatic potential φ is ε = qφ . For an electron q = −e and for a singly charged positive ion, q = e. Thus the number density of the electrons in a plasma is given by 

eφ ne = n0 exp KB Te

 (2.15)

50

Plasmas

and the number density of the ions is given by   −eφ ni = n0 exp KBTi

(2.16)

where n0 is the number density of the electrons as well as of the ions in the absence of the potential φ ; since the plasma initially is neutral and has equal number of oppositely charged particles. The temperature Te of the electrons can in general be different from the temperature Ti of the ions in a plasma. The net charge density in a plasma is found to be     −eφ eφ − en0 exp (2.17) σ = e (ni − ne ) = en0 exp KB Ti KB Te The self-consistent potential can be found by solving Eq. (2.13) after substituting for σ from Eq. (2.17). This is a nonlinear equation in φ and not so easy to solve. We can linearize the Poisson equation by expanding the exponential functions in Eq. (2.17) for their arguments to be much less than unity. Thus, for eφ  KBTe and eφ  KB Ti , the charge density becomes   Te n0 e2 1+ σ =− φ (2.18) KB Te Ti This is possible when both the electron and the ion temperatures are high enough. The Poisson equation takes the form ∇2 φ =

φ λ2

(2.19)

where

λ = 2

λe2

 −1 Te 1+ Ti

(2.20)

and 

λe =

KB Te 4π n0 e2

1/2 (2.21)

is called the electron Debye length, which is the measure of the size of the Debye cloud. Its dependence on the electron density and the temperature is as it was predicted from the physical arguments. The Debye length λ can also be expressed as 1 1 1 = 2+ 2 2 λ λe λi

(2.22)

Plasma Basics

51

where 

λi =

KBTi 4π n0 e2

1/2 (2.23)

is the ion Debye length, which is usually much less than the electron Debye length since Ti is usually smaller than Te . Equation (2.19) can be solved by writing ∇2 φ in the spherically symmetric case as   dφ 1 d d2 φ 2 dφ ∇2 φ = 2 r2 = 2 + (2.24) r dr dr dr r dr so that

φ d2 φ 2 dφ + =0 − dr2 r dr λ 2

(2.25)

The asymptotic solution of Eq. (2.25) as r → ∞ is found by ignoring the term with the first-order radial derivative. We find

Or

d2 φ φ − =0 dr2 λ 2

(2.26)

 r φ = φ0 exp − λ

(2.27)

where φ0 is a constant. Thus we see that the form of the potential at large distances from the charge is quite different from the form of the Coulomb potential. It falls exponentially with the radial distance. The fall is much faster than the r−1 fall for the Coulomb potential. The e fall distance is the screening distance λ . The solution of Eq. (2.25) for small r, in fact for r  λ , can be obtained by neglecting the term containing λ since now the term with the firstorder derivative is the dominant term. We find: d2 φ 2 dφ + =0 dr2 r dr

(2.28)

which can be easily solved by assuming a solution of the form

φ = Ar p

(2.29)

where A is a constant. Substituting in Eq. (2.28), we find p = −1

(2.30)

52

Plasmas

12 φ/φ 0 8

4

0 0

0.5

r/λ

1

1.5

2

)LJXUH  The variation of the screened potential, Eq. (2.32), (solid line) and the Coulomb potential (dashed line) due to a charge Q with distance r.

and therefore A φ= (2.31) r Now the small r solution turns out to be of the Coulomb potential form for A = Q. The complete solution of Eq. (2.25) can be written as:  r Q (2.32) φ = exp − r λ This is the potential due to a charge Q in a plasma. The potential is felt by the plasma particles predominantly within a distance λ of the charge Q; the rest of the plasma is screened from the effect of the charge Q. This also tells us that the Debye length must be much smaller than the plasma size L so that most of the plasma is neutral and free of the large electric potentials. Departures from the charge neutrality can exist on short spatial scales but not on the macroscopic system scales. It is quite likely that the ion temperature Ti may be much lesser than the electron temperature Te . Besides the ions are massive. Under the circumstances the ions do not respond to the introduction of the charge Q and remain frozen in their initial state of uniform density n0 . In this case Eq. (2.17) modifies to   eφ (2.33) σ = e (ni − ne ) = en0 − en0 exp KBTe and the Debye length λ becomes equal to the electron Debye length λe .

Plasma Basics

53

The Debye shielding is arguably the most important characteristic of a plasma. We see that electrostatic potential energy of the order of KBTe does exist at the edge of the Debye cloud and this is what gives rise to all the electromagnetic phenomena in a plasma. This also shows that a plasma is not strictly charge neutral. Electric charge density variations (Eq. 2.17) can exist in a plasma on spatial scales of the order of the Debye length. The plasmas are quasi neutral!

2.11

Plasma-Typical Time Scales

We shall see in chapter 4 that plasmas support a variety of wave modes. The frequencies and the propagation vectors of the waves are determined by the density, the temperature, and the magnetic field of a plasma. Plasma particles also undergo collisions among themselves, which are predominantly binary collisions involving two particles. Collisions cause damping of the waves. The wave modes, by contrast, are generated by the cooperative motion of a large number of plasma particles. Thus the two phenomena, the collisions and the waves, are in competition in a plasma. The plasma waves survive if there are not many collisions within a wave period. The wave periods and the collision times are the typical time scales in a plasma. An ionized gas qualifies to be a plasma if the wave period τw is shorter than the collision time τc or the wave frequency ω is larger than the collision frequency νc . We have already found a typical spatial scale, the Debye length, in a plasma. A typical velocity is the root mean velocity vT of the particles, say electrons. For a one-dimensional Maxwellian velocity distribution of the electrons, vT is found to be   KB Te 1/2 (2.34) vT = me A typical time scale τw can now be defined as

τw =

λe = vT



KBTe 4π n0 e2

1/2 

me KBTe

1/2 (2.35)

Or 

τw =

me 4π n0 e2

1/2

−1 ≡ ωep

(2.36)

where ωep , as we shall see in Chapter 4, is the frequency of the electron plasma waves. The condition for plasma behavior then becomes

54

Plasmas

τw  τc

(2.37)

ωep  νc

(2.38)

Or

This ensures that the wave does not damp within a wave period τw and the plasmas can exhibit their cooperative character in full glory!

2.12

The Plasma Parameter

We have learnt the importance of the Debye length and that it is a function of the statistical quantities, the number density and the temperature of the particles. The statistics is valid only if the system is large enough to contain a very large number of states over which averages can be taken. This condition, in a plasma, translates to a large number of particles being present in the electron Debye sphere. The number of electrons ND in the electron Debye sphere can be determined as ND =

4πλe3 n0 3

(2.39)

and the statistical argument requires that ND >>> 1

(2.40)

The triple sign emphasizes the largeness of ND !

2.13

When is it a Plasma?

An ionized gas qualifies to be a plasma if it meets the following specifications: (1) the size L of the plasma must be much larger than the Debye length λ, (2) the typical frequencies ω associated with the cooperative phenomena in a plasma, such as waves, must be much larger than the particle collision frequencies νc , (3) the number of particles, ND , in the Debye sphere must be large enough to justify the use of the statistical quantities such as the density and the temperature.

Plasma Basics

55

Check for these three criteria when asked if a certain ionized gas could be a plasma.

2.14

Summary

We have learnt, in this chapter, the different mechanisms for ionizing a neutral gas and creating a plasma. Some of these mechanisms such as photoionization, thermal and pressure ionization are more appropriate and common in cosmic objects while the others such as the electric discharge are more suitable for laboratory plasmas. A few techniques for measuring plasma parameters have been briefly mentioned. The very important effect of the Debye shielding of electric charges and fields in a plasma has been discussed with the rider that the electric potential energy of the order of the thermal energy can, nevertheless, leak out into the plasma. This is the source of all the electromagnetic phenomena in plasmas. The criteria that an ionized gas must meet in order to exhibit cooperative behavior, the hallmark of the plasmas, have been derived.

Problems 2.1 The average temperature of the solar chromosphere is 100, 000 K. Determine the ionization fractions of hydrogen and helium using the Saha equation. 2.2 Show that the critical ionization velocities of helium, carbon, and oxygen are, respectively, 34 km/s, 13.4 km/s, and 12.7 km/s. 2.3 Estimate electron Debye lengths for the solar coronal plasma and the ionospheric plasma. 2.4 Discuss the Debye shielding for a negative charge Q in a plasma. 2.5 Estimate the electron plasma frequencies for the solar coronal plasma and the ionospheric plasma. 2.6 What is the typical timescale if the typical length scale is the ion Debye length and the typical velocity is the ion thermal velocity?

3

3.1

Plasma Confinement

Introduction

A fully ionized plasma is a collection of a large number of negatively charged electrons and positively charged ions with an insignificant amount of neutral atoms. Plasmas by nature are hot. A suitable container is the first and the foremost need of a plasma-based process. Clearly no material vessel can hold, say, a million degree plasma. We have learnt in chapter one that the cosmos abounds in such hot and even hotter plasmas. The nature holds plasmas predominantly by the force of gravity and by the force of magnetism. In rare circumstances plasmas could also be trapped by the force of the electromagnetic radiation. In this chapter we shall discuss different methods of plasma confinement.

3.2

The Grip of Gravity

It is a common experience that in order to rotate a stone tied to one end of a string, one has to pull the other end of the string toward oneself for if one does not, the stone with the string would fly off. The necessary inward pull is called the centripetal force, i.e., it acts toward the center around which the object is rotating. The opposite of the centripetal force is the centrifugal force due to which a rotating object has a tendency to fly off if not pulled in sufficiently. The Earth revolves around the sun, the sun pulls it toward itself. The centripetal force, here, is the gravitational pull of the sun that keeps the Earth in its orbit. In fact, not only the Earth, but the gravitational pull of the sun also holds the entire solar system with all its planets, moons, asteroids, and comets. A plasma can also be held by gravity since gravitational force is independent of the electric

Plasma Confinement

57

charge. Electrons and ions are equally accelerated by the gravitational force as can be easily seen from Newton’s second law of motion which can be written as: me

dV e = meg dt

(3.1)

Here (me , V e ) are, respectively, the mass and the velocity of an electron and g is the acceleration due to gravity. For example, at the surface of the Earth, g = (GME /R2E ) where G is the gravitational constant, ME is the mass of the Earth and RE is its radius. One observes that the mass of the electron cancels on both sides of Eq. (3.1). This is due to the well-verified postulate that the inertial mass (the one appearing on the left-hand side of Eq. (3.1)) is equal to the gravitational mass (the one appearing on the right-hand side of Eq. (3.1)). Thus the acceleration and hence the velocity is independent of the electron mass. An electron and an ion would feel identical acceleration. So both electrons and ions would move with a common velocity and stay together in a gravitational field. However, Eq. (3.1) describes the motion of a single particle. A plasma is essentially a many-particle system. The motion of a plasma in a gravitational field can be described by multiplying both sides of Eq. (3.1) by the number density ne of electrons. This gives

ρe

dV e = ρeg dt

(3.2)

where ρe = me ne is the mass density of electrons. Note that the ion number density ni (assumed to be singly charged) is equal to the electron number density ne in a plasma although their mass densities differ greatly; ni mi = ρi  ρe . Eq. (3.2) is still good for ions as the mass density cancels on both sides. So all the electrons and the ions of a plasma move together and stay together in a gravitational field. Gravity holds plasmas. But the kinetic theory of gases tells us that a system with a large number of particles also has kinetic or thermal pressure. One could also say that a plasma like any fluid exerts a pressure and there is the corresponding force that would act on the particles and must be included in Eq. (3.2) which modifies to

ρe

dV e = ρeg − ∇Pe , dt

Pe = ne KBTe

(3.3) (3.4)

where the last term in Eq. (3.3) is the force density due to pressure of the electron fluid. The pressure is related to the density through the equation

58

Plasmas

of state. Here, we take the isothermal equation of state, Eq. (3.4); KB is the Boltzmann constant and Te is the temperature of the electron fluid. The motion of the ion fluid can be similarly described as

ρi

dV i = ρig − ∇Pi , dt

(3.5)

Pi = ni KB Ti

(3.6)

Adding equations (3.3) and (3.5) begets: dV = ρg − ∇P dt

(3.7)

ρV = ρeV e + ρiV i ,

(3.8)

ρ = ρe + ρi , P = Pe + Pi

(3.9)

ρ where

Here ρ and P are, respectively, the total mass density and the total pressure due to both the electron and the ion fluids. The velocity V so defined is a common velocity with which the entire plasma moves. In addition, the electron temperature Te is taken to be equal to the ion temperature Ti as are the number densities (ne = ni = n, ρi  ρe ). Equation (3.7), as it stands, describes a plasma as a single fluid retaining no memory that this single fluid is composed of a mixture of electrons and ions. The single fluid plasma necessarily has one temperature T , one mass density ρ , one velocity V and one pressure P = 2nKBT . The total time derivative of velocity actually is made up of two parts: (1) the time variation of the velocity at a fixed space point (the explicit or the partial derivative) and (2) its variation due to the passage of the fluid to another space point (implicit time variation) and it is expressed as

∂ V dV = + (V · ∇)V dt ∂t

(3.10)

Written in this form, it is known as the convective derivative of the fluid velocity. The force balance of a single plasma fluid is, thus, written as 

ρ

∂ V + (V · ∇)V ∂t

 = ρg − ∇P

(3.11)

The force balance equation must be supplemented with the mass conservation law stated as

Plasma Confinement

∂ρ     + ∇ · ρV = 0 ∂t

59 (3.12)

The detailed derivations of equations (3.11) and (3.12) can be found in Chapter 6. To find out whether a plasma is confined by the gravitational force, we first have to determine the equilibrium configuration of the plasma. This is done by equating the time derivatives of the velocity and the mass density to zero. Now, there are two possible configurations: (1) without flow, i.e., V = 0 and (2) with flow, i.e., V = 0.

3.3

Hydrostatic Equilibrium

The hydrostatic equilibrium is achieved when the plasma fluid is at rest, i.e., V = 0. The pressure force balances the gravitational force:

ρg = ∇P

(3.13)

One may appreciate that Eq. (3.13) describes a neutral fluid as well as a plasma. For the neutral fluid n and m may refer to the molecules or the atoms of the fluid and for a plasma to the ions which are the mass carrying species. This force balance, for example, describes the variation of pressure with height in Earth’s atmosphere. For illustration purpose let us assume that the variation is only one-dimensional and in the vertical direction z (in a direction opposite to the gravitational force, which is toward the center of the Earth). Thus g is in the negative z direction. We also need a relationship between the pressure and the mass density, the equation of state. If the total number of particles per cubic centimeter in the atmosphere is N, the total pressure is P = NKBT . The mass density is ρ = Nm, where m is the mean mass per particle. One-dimensional hydrostatic equilibrium is, therefore, given by

∂N mg =− N ∂z KB T

(3.14)

and the result is an exponentially decreasing density profile:  z N = N0 exp − H

(3.15)

where H = KB T /mg is called the scale height. At a height z = H the density falls by a factor e = 2.77 of the density N0 at the initial point z = 0,

60

Plasmas

say the surface of the Earth. For a given temperature and acceleration due to gravity, the scale height is inversely proportional to the mass m of the particle. The lighter the particle the larger the scale height. The large-scale height implies that the particle would be found that far from the surface of the Earth. This is precisely the reason for the absence of hydrogen gas in the Earth atmosphere, though it is the most abundant element in the universe. In contrast, the heavier gases such as oxygen and nitrogen are the main constituents of the terrestrial atmosphere. We can say that the hydrogen gas is less confined than the other gases. As a matter of fact, if one calculates the speed of a hydrogen atom at the atmospheric temperature of nearly 300 Kelvin, one would find that it exceeds the escape speed from the Earth (≈ 11 km/s). The hydrogen gas cannot be confined by Earth’s gravity at the atmospheric temperature. The weak gravity of the moon is the reason for the absence of a lunar atmosphere. In contrast, the gravity at the core of the sun is so strong that it can confine hydrogen plasma even at a temperature of 15 million Kelvin. √ This can be seen by comparing the thermal velocity, VT = KBT /m p of √ a hydrogen ion with the escape velocity, Ves = 2GMS /Rco at the solar core. Here m p is the mass of a hydrogen ion, Mco is the mass contained in the solar core, and Rco is the core radius. The thermal velocity is less than the escape velocity; the hydrogen plasma is confined.

3.4

Hydrodynamic Equilibrium

More interesting possibilities emerge for an equilibrium with a flow, i.e., for V = 0. The hydrodynamic equilibrium obtains from the force balance:

ρ (V · ∇)V = ρg − ∇P

(3.16)

This is a case of great relevance for astrophysical plasmas. Let us use the cylindrical polar coordinate system with coordinates (r, θ , z) and assume spherical symmetry by which is meant that all the physical quantities are functions only of the radial coordinate r. Further we now introduce the variable g = −(GM/r2)er , i.e., the acceleration due to gravity is defined at a distance r from the center of the object (say Earth or a star) responsible for the gravitational force. Here er is a unit vector pointing radially outward from the center and since the gravitational force acts radially inward toward the center, there is a minus sign. Equation (3.16) is a vector equation. One should write each component of this equation. A situation of common occurrence is where a plasma rotates around a massive object such as a star. Assuming only the azimuthal, Vθ (r), component of the velocity, the radial component of Eq. (3.16) reads

Plasma Confinement

Vθ2 GM = 2 r r

61 (3.17)

where the pressure has been assumed to be a constant. The velocity determined from Eq. (3.17) is known as the Keplerian velocity VK (since it is the very velocity with which the planets go around the sun according to Kepler’s laws) and given by  VK (r) =

GM r

1/2 (3.18)

Such a plasma with Keplerian flow settles down in the form of a disk, generally called an accretion disk (Fig. 3.1), around a compact object such as a star or even a black hole.

)LJXUH  Simulation of an accretion disk, credit: Michael Owen, John Blondin (North Carolina State University).

The Keplerian flow is another example of a confined or a gravitationally bound plasma. The plasma could also deconfine or escape from the gravitational field if its speed exceeds the escape speed. By escape one

62

Plasmas

means escape to infinity where both the potential and the kinetic energies of the plasma must vanish. The law of energy conservation according to which the initial energy (kinetic energy plus potential energy) must be equal to the final energy (which is zero in this case) gives

ρVes2 ρ GM − 2 =0 2 r

(3.19)

and the escape speed is found to be  Ves (r) =

2GM r

1/2 (3.20)

The Keplerian flow is gravitationally bound because its speed VK is less than the escape speed Ves . It should be instructive to ask what combinations of the mass M and the radius r would make the escape speed equal to the speed of light c, the maximum attainable speed. Black holes fall into this category of objects. The gravitational pull of a black hole is so strong that even light cannot escape it. The hot plasmas also radiate and could produce intense radiation. Under these circumstances, it becomes necessary to include radiation pressure along with the thermal pressure P for confinement considerations as for example in accretion disks around black holes. The confinement of a plasma with radiation pressure is discussed in section 3.11.

3.5

Magnetic Bottles

Strong magnetic fields can confine plasmas. This mechanism is the basis of the most thermonuclear fusion devices. The boundary of Earth’s magnetosphere is defined where the magnetic pressure balances the solar wind pressure. To learn how magnetic fields could confine plasmas, we first study the motion of a single charged particle in a uniform magnetic field.

3.6

Motion of a Charged Particle in a Magnetic Field

A magnetic field exerts a force on a charged particle. Under the action of this force, called the Lorentz force, the motion of a charged particle of charge q, mass m, and velocity V can be described using Newton’s second law of motion as m

V × B dV =q dt c

(3.21)

Plasma Confinement

63

where B is the magnetic induction. For illustration purposes let us take the magnetic induction to be in the z direction and write B = Bez with ez the unit vector along z. The first observation from Eq. (3.21) is that the direction of the Lorentz force is perpendicular to both the velocity and the magnetic induction. There is no force along the magnetic induction and the partcle moves with a uniform speed along z. In the perpendicular plane (x, y) the components of the particle acceleration can be written as m

Vy B dVx =q dt c

(3.22)

m

Vx B dVy = −q dt c

(3.23)

and

Differentiating Eq. (3.22) once more with respect to time and substituting from Eq. (3.23), one finds  2 qB d2Vx = − Vx dt 2 mc

(3.24)

Equation (3.24) is a standard form of the equation for a simple harmonic oscillator, and its solution is Vx (t) = V⊥ sin(ωc t + φ0 )

(3.25)

where ωc = (qB/mc) is known as the particle cyclotron frequency. The y component of the velocity can be determined by substituting for the time derivative of Vx in Eq. (3.22) to get Vy (t) = V⊥ cos(ωc t + φ0 )

(3.26)

Here V⊥ is the initial velocity, i.e., the velocity at t = 0. The phase angle φ0 is a constant of integration and fixes the relation between the x and the y components of the initial velocity: Vx0 = V⊥ sin(φ0 ) and Vy0 = V⊥ cos(φ0 ). The x component of the displacement of the particle can be found by integrating Eq. (3.25) with respect to time: X (t) =



Vx dt = x0 − Rc cos(ωc t + φ0 )

(3.27)

where Rc = V⊥ /ωc is called the cyclotron radius of the particle and X0 = x0 −Rc cos(φ0 ) is the x coordinate of the initial position of the particle. The

64

Plasmas

y component of the displacement can be found by integrating Eq. (3.26) with respect to time: Y (t) =



Vy dt = y0 + Rc sin(ωc t + φ0 )

(3.28)

Here Y0 = y0 + Rc sin(φ0 ) is the y coordinate of the initial position of the electron (at t = 0). It is now easy to see that the particle executes circular motion along a circle of radius Rc centered at (x0 , y0 ) in the (x, y) plane since (X − x0 )2 + (Y − y0 )2 = R2c

(3.29)

which is the equation of a circle. The particle motion is thus limited and confined to just going around the magnetic axis in the (x, y) plane (Fig. 3.2). However, it is free to move along the direction of the magnetic B

)LJXUH  Charged particles move in a helical path around the magnetic axis. Negatively charged particles go anticlockwise and positively charged particles go clockwise for the direction of the magnetic field B pointing toward the observer.

Plasma Confinement

65

field and may even exit the system unless the magnetic field lines curve back into the system. The motion of both the positively charged ion and the negatively charged electron can be determined by using the appropriate values of the charge and the mass. The ion, being more massive than the electron, has a lower cyclotron frequency and larger cyclotron radius for a fixed velocity V⊥ . The sense of the circular motion of a charged particle can be determined by inspecting the time variation of the phase angle: tan Φ (t) =

Y (t) − y0 = − tan (ωc t + φ0 ) , Φ (t) = − (ωc t + φ0 ) X (t) − x0

(3.30)

The time rate of change of the phase angle Φ (t) is (−ωc ) and it is positive for an electron with q = −e. This implies that the sense of circular motion of the electron is anticlockwise for the magnetic field pointing towards an observer. The time rate of change of the phase angle Φ (t) is negative for an ion with q = e. The sense of circular motion of an ion is, therefore, clockwise.

3.7

Plasma in a Magnetic Field

To include the effect of a magnetic field on a conducting fluid or a plasma, we follow the procedure outlined in section 3.2, for arriving at the single fluid Eq. (3.12) by summing the electron fluid Eq. (3.3) and the ion fluid Eq. (3.5). We multiply Eq. (3.21) by the number density n of the particles and write the corresponding equations for the electrons (with suffix e) and the ions (with suffix i) as n e me

V e × B V i ×  dV e dV i B = −ene , n i mi = eni dt c dt c

(3.31)

Equations (3.31) are the momentum equations for the electron and the ion fluids. We can also include the pressure gradient forces ∇Pe and ∇Pi to the respective fluid equations of the electrons and the ions. By adding them up we find:   ∂ V J × B    ρ + (V · ∇)V = −∇P + (3.32) ∂t c where the current density J is defined as   J = ne V i − V e

(3.33)

66

Plasmas

Other quantities have been defined in section 3.2. We have taken ne = ni = n. We have assumed singly charged ions. If the ions are not singly charged and have a charge, say, qe then the electron density and the ion density cannot be equal and are related as qeni = ene . For example, if the ion has twice the charge of the electron, q = 2, the electron density would be twice the the ion density. This ensures the overall charge neutrality of the plasma. Equation (3.32) describes the flow of a conducting fluid (a plasma) of mass density ρ and velocity V under the action of the Lorentz force (J × B) and the pressure gradient force. We have now ignored the gravitational force. One could again explore equilibria without flow, (V = 0), and with flow (V = 0).

3.8

Magnetostatic Equilibrium, the Z Pinch

The magnetostatic equilibrium obtains when the Lorentz force balances the pressure force. J × B  = ∇P c

(3.34)

An important and special case of the magnetostatic equilibrium is the scheme called the Z pinch. In this scheme the direction of the current density J is along the z axis of the cylindrical plasma column. The magnetic field generated by this current density can be determined from Ampere’s law ∇ × B = 4π J c

(3.35)

1 d (rBθ ) 4π = Jz r dr c

(3.36)

as:

The other components of the vector Eq. (3.35) should also be satisfied. This is achieved by assuming that the magnetic field has only an azimuthal component and there are no variations with respect to θ and z. The axial current density produces an azimuthal magnetic field and the Lorentz force being perpendicular to both is in the negative radial direction (ez ×eθ = −er ). The plasma is pushed radially inwards and remains confined. This confinement scheme has been found to be quite efficient.

Plasma Confinement

67

The force balance Eq. (3.34) along with Eq. (3.36) then determines the radial gradient of pressure as:

dP 1 d r2 B2θ =− dr 8π r 2 dr

(3.37)

which on integration with respect to r gives: 1 P = P0 − 8π



 r 1 d r2 B2θ (r ) r2

0

dr

dr

(3.38)

where P0 is the plasma pressure at r = 0. Since we wish the plasma to be confined to a finite radius R, the pressure must vanish at r = R. The condition that P = 0 at r = R determines P0 as 1 P0 = 8π



 R 1 d r2 B2θ (r ) 0

r2

dr

dr

(3.39)

Substituting for P0 in Eq. (3.38) provides the radial variation of the plasma pressure as: 1 P (r) = 8π



 R 1 d r2 B2θ (r ) r

r2

dr

dr

(3.40)

Note the limits of integration. One can define an average pressure inside the cylinder by integrating over the cross section of the cylinder and dividing by the cross-sectional area π R2 : < P >=

1 π R2

 2π 0

dθ 

 R

P r r dr 0

(3.41)

The integration over θ  gives a factor of 2π . The integration over R can be carried out by the method of integration by parts as  R

 R 2 r dP (r )  r2

P r r dr = P r |R0 − dr 2 dr 0 0 2

(3.42)

The first term vanishes at both the upper and the lower limits of integration. The second term is easy to evaluate when one substitutes for the pressure gradient from Eq. (3.37) as the integrand becomes a total differential:

 R     1 R 1 d r2 B2θ (r )  P r r dr = dr (3.43) 8π 0 R2 dr 0

68

Plasmas

and the average pressure gets related to magnetic induction as < P >=

B2θ (R) 8π

(3.44)

Since the quantity on the left-hand side of Eq. (3.44) is pressure, so must be the quantity on its right-hand side. The quantity B2θ (R) /8π is called the magnetic pressure. The steady state demands that the average thermal pressure of the plasma must be equal to the magnetic pressure at the surface, r = R. This is how magnetic fields contain plasmas and hence the name magnetic bottles. The force due to magnetic pressure also acts in the manner of the thermal pressure: from high magnetic pressure towards low magnetic pressure or from high B to low B. We can find out the total current carried by the plasma by integrating the current density over the area of cross section of the cylindrical plasma but we first integrate Eq. (3.36) to find Bθ (r) =

4π cr

 r 0



r Jz r dr

(3.45)

Then the total current I carried by the cylindrical plasma is found to be I=

 2π 0

dθ 

 R 0



r Jz r dr

(3.46)

The magnetic induction outside the plasma is found from Eq. (3.45) to be Bθ (r) =

2I cr

(3.47)

By using this relation in Eq. (3.44), the average thermal pressure can be expressed as a function of the total current and the radius of the plasma column as < P >=

I2 2π c2 R2

(3.48)

One can now estimate the strength of the magnetic induction B or equivalently the total current I for confining hot plasmas of a specified dimension R. At temperatures of the order of a hundred million Kelvin and densities of the order of 1015 particles per cubic centimeter, the thermal pressure P = 2nKBT is of the order of 107 dynes per square centimeter. The equality of this pressure P to the magnetic pressure requires a magnetic induction Bθ of the order of 15 kilogauss corresponding to the total current I of 2.25 × 1014R statamperes or about 105 R amperes. This is a

Plasma Confinement

69

huge current even for a radius R of one centimeter. It is interesting to check what strength of gravitational force would be required to balance the thermal pressure of 107 dynes per square centimeter. This can be done by equating P to gravitational energy density ρ gh where h is the linear dimension of the plasma. For a hydrogen plasma of particle number density of 1015 per cubic centimeter, the mass density is ρ = 1015 × 1.67 × 10−24 gram per cubic centimeter where 1.67 × 10−24 grams is the mass of a hydrogen ion or a proton. One finds that an acceleration due to gravity g = 1016 /h centimeter per square second is needed. The value of g at the solar core, the site of the thermonuclear reactions, is of the order of 106 centimeter per square second. But this would suffice since the dimensions of the plasma are also huge, about a third of the solar radius (≈ 7 × 1010 centimeter). The surface gravity of a pulsar of nearly 1.4 times the mass of the sun (≈ 2 × 1033 grams) and of a typical radius of 10 km is about 1014 centimeters per square second and it can hold the plasma of size 100 centimeters at a pressure of 107 dynes per square centimeter. However, for small meter size hot laboratory plasmas, it is the magnetic confinement that works! To determine the actual pressure profile in a Z pinch, one has to assume a model for the current density J(r). The simplest model is to take J(r) to be independent of r for r < R so that J = I/π R2 . The magnetic induction for r < R is then found from Eq. (3.45) as Bθ (r) =

2Ir cR2

(3.49)

Integration of Eq. (3.40) after substituting for Bθ (r) furnishes the radial variation of pressure as   r2 I2 (3.50) P (r) = 2 2 1 − 2 πc R R which can also be expressed in terms of the average pressure by using Eq. (3.44)   r2 P (r) = 2 < P > 1 − 2 (3.51) R The pressure on the axis, r = 0, of the cylindrical plasma column is twice the average pressure. The pressure is maximum at r = 0 and decreases toward the surface where it vanishes. The magnetic induction is zero on the axis and maximum at the surface. This shows that the two regions, one containing only plasma and the other containing only magnetic induction, can be in equilibrium with each other. Such a situation

70

Plasmas

approximates a good description of structures called flux tubes on the solar atmosphere. They are even referred to as evacuated flux tubes as their strong magnetic induction expels the plasma out of the tubes. One also defines a dimensionless parameter called the plasma β as

β=

8π P B2

(3.52)

which is used to describe the dominance or otherwise of the plasma pressure or the magnetic pressure. A value of plasma β less than unity is an indicator of a confined plasma. Laboratory thermonuclear plasmas have plasma β less than or equal to unity. On the solar atmosphere, a large range of β values exists. The photospheric plasma has β > 1 while the solar coronal plasma has β  1. The Z pinch is often invoked to model the very long and very stable extragalactic plasma jets (Fig. 3.3).

)LJXUH  An extragalactic plasma jet retains its shape for distances of millions of parsecs from the center of the galaxy. The stability of the jet against its diffusion into the neighborhood is believed to be ensured by a current density flowing along the jet giving rise to an azimuthal magnetic field, which contains the plasma. Plasma Jets from Radio Galaxy Hercules A, image Credit: NASA, ESA, S. Baum and C. o´ Dea (RIT), R. Perley and W. Cotton (NRAO/AUI/NSF), and the Hubble Heritage Team (STScI/AURA)

Plasma Confinement

3.9

71

Magnetostatic Equilibrium, the Θ Pinch

A plasma can also be confined in the form of a Θ pinch wherein an azimuthal current density Jθ produces an axial magnetic field Bz as dBz 4π = − Jθ dr c

(3.53)

All the other components of B and J are zero. The direction of the Lorentz force is again radial as eθ ×ez = er . The radial component of the force balance now reads dP Jθ Bz = dr c

(3.54)

Substitution for Jθ gives

Or

dP 1 dB2z =− dr 8π dr   B2z d =0 P+ dr 8π

(3.55) (3.56)

which clearly shows that the total pressure, a sum of the magnetic and the thermal pressure must be a constant at any point in a plasma. A region of high thermal pressure would have a low magnetic pressure and vice versa for equilibrium. For a confined plasma (dP/dr) < 0, i.e., the plasma pressure must decrease outwards so that most of the plasma remains near the axis of the cylindrical column. And this would happen if (dB2z /dr) > 0, i.e., the magnetic field must increase with r, towards the surface. Contrast this situation with the Z pinch. The radial force balance for Z pinch is B2 dP 1 dB2θ =− − θ dr 8π dr 4π r

(3.57)

The additional last term arises due to the curved nature of the azimuthal magnetic induction Bθ and represents the tension in the magnetic field lines. Equation (3.56) again demonstrates the possibility of balancing thermal pressure with magnetic pressure.

3.10

Magnetostatic Force Free Equilibrium, the Reversed Field Pinch

Another magnetostatic equilibrium is obtained for constant plasma pressure P and zero Lorentz force, which can happen when the current

72

Plasmas

density  J is parallel to the magnetic induction B. The corresponding configuration is called the force-free magnetic field since such a magnetic field does not exert any force on a plasma. The Ampere law then furnishes ∇ × B = 4π J = α B c

(3.58)

where α stands for all the constants in front of  B. The cylindrically symmetric solution of Eq. (3.58) can be found by writing its components as Br = 0,

(3.59)

dBz = −α Bθ , dr

(3.60)

1 d (rBθ ) = α Bz r dr

(3.61)

The radial component of the magnetic induction vanishes since there is no variation in the θ and z directions. To determine the other components, substitute for Bθ from Eq. (3.60) into Eq. (3.61) to get   1 d dBz r = −α 2 Bz r dr dr

(3.62)

Or d2 Bz 1 dBz + + α 2 Bz = 0 dr2 r dr

(3.63)

This is a second-order ordinary differential equation and there are standard methods to solve such equations. Let us first see if we can find an approximate solution. To do that we assume the magnetic field is not a very rapidly varying function of the radial position r. What this implies is that we can ignore all derivatives of the magnetic induction except the first derivative. This reduces the second-order equation to the first-order equation as 1 dBz + α 2 Bz = 0 r dr

(3.64)

which can be easily integrated. The integration gives: 

dBz = −α 2 Bz



rdr

(3.65)

Plasma Confinement

73

Or 

α 2 r2 Bz = B0 exp − 2

 (3.66)

and Bθ = B0

  αr α 2 r2 exp − 2 2

(3.67)

where B0 is the axial component of the magnetic induction at r = 0. Notice that Bθ = 0 at r = 0. A more simplified form of the magnetic induction can be obtained by expanding the exponential function for its small ar

gument, i.e., for α 2 r2 /2  1. We find   α 2 r2 Bz = B0 1 − 2

(3.68)

and 

α 2 r2 Bθ = B0 (α r) 1 − 2

 (3.69)

Thus, near the axis (r → 0) of the plasma column the axial field Bz is a constant and the azimuthal component Bθ is directly proportional to r. Note that this approximate behavior of the magnetic induction is valid only near the axis. What does the exact form of the force-free magnetic field look like? The answer lies in the solution of Eq. (3.63). We shall not go into the details of the method of solving Eq. (3.63). Suffice it to say here that the solution is in the form of a series in r. The series are expressed in terms of a special mathematical function called the Bessel function. The magnetic induction satisfying Eqs. (3.59)–(3.61) have the form Bz = B0 J0 (α r)

(3.70)

Bθ = B0 J1 (α r)

(3.71)

where J0 (α r) and J1 (α r) are, respectively, the Bessel functions of order zero and one. It must be understood that the small argument (r → 0) expansion of the Bessel functions would lead to the form of the solution derived from Eqs. (3.68) and (3.69) in the same limit. The graphical forms of J0 (α r) and J1 (α r) are shown as a function of the dimensionless argument α r ≡ x in Figs.(3.4) and (3.5).

74

Plasmas

1 0.8 0.6 0.4 0.2 0 2

4

6

8

-0.2

10 x

12

14

16

18

20

16

18

20

-0.4 )LJXUH 

Variation of Bessel function J0 (x) with x.

0.6

0.4

0.2 0 2

4

6

8

10 x

12

14

-0.2

)LJXUH 

Variation of Bessel function J1 (x) with x.

It is easy to see the linear rise of J1 (x) and a fall of J0 (x) for small x. What is more interesting is that both the functions vanish at some values of x and then rise and fall again. Thus the axial component of the magnetic induction reverses its sign at x ≈ 2.4. For this reason, the force free field configuration is called the reversed field pinch. This is one of the most favored schemes for confining thermonuclear plasmas. The largest reversed field experiment (RFX) is in Padua, Italy. The RFP scheme requires much lower magnetic field strength than the toroidal

Plasma Confinement

75

(doughnut shaped) confinement schemes deployed, for example, in a tokamak. The Z, the Θ, and the reversed field pinches have been described for an infinite plasma column. We have not specified the length of the cylindrical column. In practical situations such as in a tokamak, the cylindrical column is bent into a torus and the axial field Bz is then referred to as the toroidal field. In such a geometry there is no escape of the charged particles along the direction of the magnetic field. A tokamak has both the toroidal Bz and the poloidal Bθ field components. The magnetic field being stronger toward the center pushes the plasma outwards away from the center. This is contrary to the Z pinch where the magnetic field is stronger near the surface of the cylinder. The poloidal field with magnetic lines spiraling around the torus is imposed to confine the plasma accumulated near the surface. The various geometries of the magnetic fields themselves cause additional drifting motion of the charges and end up deconfining the plasma. Designing a magnetic configuration meeting all the demands of confinement is an extremely challenging task. It is gratifying to note that scientists and technologists are in good control of the confinement schemes used in a tokamak. The force free field has been invoked in several astrophysical situations. The observed solar photospheric magnetic field is modeled by force free and current free (α = 0) magnetic profiles. This configuration also corresponds to a state of minimum magnetic energy and therefore believed to be stable.

3.11

Magnetic Mirrors

As the name implies, a magnetic mirror is a magnetic configuration in which the charged particles undergo reflections. We know that a magnetic field does not exert any force on a charged particle in its own direction. A charged particle moves with a constant speed along the direction of the magnetic field. This is true if the magnetic field is stationary (does not vary with time) and uniform (does not vary with space). Motion of a charged particle is much different in a nonuniform magnetic field. One can arrange a magnetic configuration in which a charged particle decelerates as it moves along the direction of the magnetic field, comes to a halt, and reverses its direction of motion. The particle is said to be reflected back in this particular magnetic configuration. This enables the particle to remain within the system and be confined. To understand the mechanism of reflection of a charged particle, we first need to remind ourselves some important characteristics of a magnetic field. The most important characteristic is the absence of a

76

Plasmas

monopole field. The most basic magnetic configuration is a dipole configuration with a north pole and a south pole. Unlike electric charges, which can be positive or negative, a magnetic field cannot have only a north or only a south pole. Thus the simplest magnetic field is due to a magnetic dipole such as a bar magnet. Associated with it, therefore, is a magnetic dipole moment, μ B , a vector quantity. The Ampere law relates to the current distributions and the magnetic induction they produce. Therefore, the magnetic dipole moment surely must be related to the current distribution. The magnetic dipole moment μ B is defined as (Fig. 3.6): 

1 x × J x d3 x (3.72) 2c

where J x is the current density over a specified volume containing the point x .

μ B =

P

J(x )

μB x O

)LJXUH  Magnetic moment μ B generated by a current density distribution J from which magnetic induction at any point P can be calculated.

For a current I confined to a plane and flowing along any closed loop   (Fig. 3.7), one can replace Jd   so that:  3 x by I dl with a line element dl μB =

1 2c



x × Idl 

(3.73)

and its direction is perpendicular to the plane of the current loop. Now 1    = da  is the area enclosed by the line element dl   joining the x × dl 2  origin. The integral over the entire loop circumference gives the total area A. Thus the magnitude of the magnetic moment is found to be

μB =

IA c

(3.74)

Plasma Confinement

77

dl x

da

O

I

)LJXUH 

Magnetic moment of a current loop.

and it does not depend on the shape of the loop. If the current distribution is generated by particles of charge qi , mass Mi , velocities V i and positions xi so that J = ∑ qiV i δ (x −xi )

(3.75)

the corresponding magnetic moment is given by μ B =

qi  1 Li ∑ 2c Mi

(3.76)

  where the sum is over all the particles and Li = xi × MiV i can be easily identified to be the orbital angular momentum of the i particle. The quantity (qi /Mi ) = (q/M) can be factored out of the summation sign for particles with identical charge to mass ratio, and the magnetic moment becomes μ B =

q Li = q L 2Mc ∑ 2Mc

(3.77)

where L is the total orbital angular momentum. The magnetic moment and the orbital angular momentum are so related in classical mechanics and this relation is valid even on atomic scales. We are now prepared to define the magnetic moment of plasma particles in a magnetic field. The circular motion of a charged particle in the presence of a magnetic field is equivalent to a current loop of radius

78

Plasmas

equal to the cyclotron radius Rqc of a particle of charge q and mass M. The charge q traverses the circumference 2π Rqc in a time period (2π /ωqc ) where ωqc is the cyclotron frequency of the charge q. Thus the current flowing in the loop is calculated to be I=

qωqc 2π

(3.78)

and the magnitude of the magnetic moment from Eq. (3.74) is found to be

μB =

qωqc A 2π c

(3.79)

where A = π R2qc is the area of the current loop. Substituting the expressions for Rqc and ωqc from section (3.5) begets for the magnetic moment of a particle of charge q and mass M in a uniform magnetic field Bz :

μB =

MV⊥2 2Bz

(3.80)

This attribute of the magnetic moment becomes especially useful while discussing the motion of charged particles in spatially varying magnetic fields, for spatially slowly varying fields in particular. We introduce the concept of the adiabatic invariance. The adiabatic invariance implies that the quantities that are strictly invariant under certain conditions remain so even when those conditions change provided the change is very gentle. In the context under discussion here, i.e., the motion of charged particles in spatially slowly varying magnetic field, the relevant adiabatic invariant is the magnetic flux linked with the particle’s orbit. The magnetic flux ΦB is defined as the product of the magnetic induction Bz and the area enclosed by the orbit of the particle. The magnetic flux ΦB = Bz × π R2qc

(3.81)

is an invariant, i.e., it does not change with time for a constant and uniform magnetic field Bz . The adiabatic invariance grants that the flux ΦB would continue to remain a constant, an invariant, provided the variation in the field Bz is rather slow. How slow? Well the field should not change by any significant amount over a region whose dimensions are comparable to the radius of the particle orbit. The variation in the magnetic field would be observable only over a region much larger than the radius of the particle’s orbit. Accepting then the adiabatic invariance

Plasma Confinement

79

of the magnetic flux in a slowly varying field one can easily infer the adiabatic invariance of the magnetic moment of the particle since the two are related as    MV⊥2 2π Mc2 ΦB = (3.82) q2 2Bz which is found by substituting for R2qc . The presence of the magnetic moment μB is quite conspicuous. Remember that now a slow variation in Bz with z is allowed. One may further conclude that the quantity V⊥2 = Constant Bz

(3.83)

since all the other factors are constants. This relationship is the basis of a magnetic mirror. Consider a magnetic field configuration where the field Bz increases with z (the field lines are close to each other in high magnetic field regions as seen near the ends while they are far apart in weak field regions as seen near the middle) (Fig. 3.8). B0

BM

)LJXUH 

PLASMA

BM

Confined plasma in a magnetic mirror.

Recall that the cyclotron radius is inversely proportional to the field strength Bz . The radius of the particle’s orbit, therefore, decreases as the field Bz increases. A particle then begins with a large orbital radius in a region of low Bz and subsequently circulates around the field lines with a continuously diminishing orbital radius. The invariance of the magnetic moment or magnetic flux then dictates that the perpendicular velocity V⊥ must increase as Bz increases. It can be easily checked that the total energy of a particle in a magnetic field is a constant of motion. As the Lorentz force acts in a direction

80

Plasmas

perpendicular to the particle velocity, it does no work on the particle as can be seen by taking the scalar product of the velocity and the acceleration:    V e · me dV e = −eV e · V e × B = 0 dt c

(3.84)

Thus the total energy: E=

2 meVe⊥ meVez2 meVe2 = + 2 2 2

(3.85)

is a constant. Here Vez is the velocity of the particle parallel to the magnetic field Bz . This then ensures that the total energy remains the same at two different z locations. Or 2



V⊥ +Vz2 z=z = V02 = V⊥2 +Vz2 z=z 0

1

(3.86)

Here V02 is the square of the total velocity at z = z0 . Since the energy conservation is true irrespective of the charge and the mass of a particle, the subscript e has been removed. Using the subscript 0 to represent quantities at z = z0 and the subscript 1 to represent quantities at z = z1 , the adiabatic invariance of the magnetic moment gives 2 V2 V⊥0 = ⊥1 B0 B1

(3.87)

We find the parallel velocity at z = z1 to be 2 Vz12 = V02 −V⊥1

(3.88)

Or using Eq. (3.87) Vz12 = V02 −

2 V⊥0 B1 B0

(3.89)

Now it is clear from the above equation that Vz1 decreases as the field B1 increases. In other words, the particle decelerates as it moves toward region of high magnetic field and that at some point it would even vanish. Let BM be the value of Bz where Vz = 0. Then V02 =

2 V⊥0 BM B0

(3.90)

Plasma Confinement

81

And Eq. (3.89) becomes Vz12 =

2 V⊥0 V2 BM − ⊥0 B1 B0 B0

(3.91)

It is seen again that the axial velocity vanishes at B1 = BM , the particle is said to be reflected at this point. It reverses its direction of motion and begins to accelerate toward the low field region. We can now find the condition on the initial velocities, say at z = z0 so that the particle is reflected at the point where the field strength is BM . From Eq. (3.91) we find that at z = z0 Rotation axis

Magnetic axis )LJXUH  A schematic representation of Van Allen inner and outer radiation belts. The inner belts contain more energetic particles than the outer belts.

Vz02 =

2 V⊥0 V2 BM − ⊥0 B0 B0 B0

(3.92)

Or  1/2 Vz0 = BM − 1 V B0 ⊥0

(3.93)

The condition that the particle is reflected before it reaches the region of the maximum field BM is that  1/2 Vz0 < BM − 1 V B0 ⊥0

(3.94)

The particle will remain trapped in the weak field region under these circumstances assuming that the magnetic configuration is symmetric about z = z0 . The equally strong field at both the ends is produced by appropriately placing the current-carrying coils at the two ends of a cylindrical plasma column, creating bottlenecks, which prevent any leakage of the

82

Plasmas

particles. The ratio (BM /B0 ) is called the mirror ratio. Note that the trapping condition is independent of the charge and the mass of a particle. So both the electrons and the ions are trapped equally efficiently. A natural magnetic mirror exists in Earth’s magnetosphere due to the field being stronger at the two poles than at the equator. The solar wind particles are trapped in the Earth’s magnetic field and they form a torus of plasma around the Earth known as the Van Allen radiation belts (Fig. 3.9).

3.12

Confinement of Plasmas under Radiation Pressure

Cosmic sources, such as stars, emit strong electromagnetic radiation. This radiation exerts tremendous outward pressure on the plasma in the stellar atmosphere. The plasma atmosphere is held by the star against the radiation pressure by the gravitational pull of the stellar mass and the plasma atmosphere is said to be stable. However, if the brightness of the star exceeds a certain limit, the atmospheric plasma is blown away and the star is said to suffer a mass loss via the ejected stellar wind. The limiting value of the luminosity is known as the Eddington luminosity. It can be estimated by equating the gravitational force to the force due to radiation on an element of a plasma. We must find the force exerted on a plasma element due to the radiation impinging on it. Let L be the luminosity (energy radiated per second) of an object. The radiative energy flux, FE , i.e., the average energy < E > falling per second on a unit area at a distance r from the source is then given by FE =

d L = dAdt 4π r 2

(3.95)

The energy EP of a relativistic particle of rest mass m and momentum P is given by

1/2 EP = P2 c2 + m2 c4

(3.96)

Since the rest mass of a photon is zero, the energy E and the momentum P of a photon are related as: P=

E c

In the classical limit, i.e., in the limit of large intensity formed by a very large number of photons, the average radiation energy, and the average radiative momentum are also related as < P >=

c

(3.97)

Plasma Confinement

83

Therefore, the radiative momentum flux, FP , which is nothing but the rate of radiative momentum transfer per second per unit area, becomes FP =

d

L = dAdt 4π cr2

(3.98)

An electron, at a distance r from the source of radiation, then receives this radiative momentum flux. The force on the electron can be found by multiplying the radiative momentum flux by the effective area of cross section of the electron. The main mechanism underlying the radiative momentum transfer is the electron photon scattering, also called the Thomson scattering. Other scattering mechanisms could also be included and an effective scattering cross section can be defined. The Thomson cross section χT is given as (See Eq. 5.266): 

χT =

8π e2 3me c2

2 (3.99)

where (e2 /me c2 ) is the electron radius. The radial force balance of an electron fluid in the presence of the radiative and the gravitational forces then demands that

ρe

dVer d

GMS χT − ρe 2 = ne dt dAdt r

(3.100)

The radial force balance of the ion fluid in the presence of the radiative and the gravitational forces demands that

ρi

dVir dP m2 GMS χT e2 − ρi 2 = ni dt dAdt mi r

(3.101)

where MS is the mass of, say, a star. By adding these equations and imposing the static equilibrium, one finds 0=n

  m2 GMS dP χT 1 + e2 − (ρe + ρi ) 2 dAdt r mi

(3.102)

Observe that the radiative force on the ion is much smaller than on the electron while the gravitational force on the electron is much smaller than on the ion, all due to the very different masses of the two particles. Thus substituting for the radiative momentum flux and retaining the large terms gives for a plasma (ne = ni ):

84

Plasmas

LE =

4π cGMS mi χT

(3.103)

where LE is the Eddington luminosity, the maximum luminosity against which an electron–proton plasma can be held in static equilibrium by the gravitational pull of a star of mass MS . It is customary to express the Eddington luminosity in terms of the mass of the sun as (plugging in the values of all the constants): LE = 1.2 × 1038

MS erg s−1 Msun

(3.104)

This is a huge luminosity. The luminosity of the sun, for comparison, is ≈ 4 × 1033 erg s−1 . Thus the sun is a stable star, its gravitational pull being sufficiently strong to hold its plasma atmosphere. But you might recall that there is a solar wind. So be assured that it is not driven by the radiation pressure. There are processes in the solar corona responsible for its own heating and for driving the solar wind. It should also be emphasized that the Eddington luminosity would be different if the plasma composition is different. For a fully ionized helium plasma consisting of two free electrons for every nucleus of four times the mass of a hydrogen nucleus, the Eddington luminosity would be twice as large as for a hydrogen plasma. This can be understood by realizing that the radiative force would double because ne = 2ni but the gravitational force becomes four times, resulting in a net increase in the Eddington luminosity by a factor of two. Massive black holes harboring the centers of galaxies have been inferred to have masses of the order of 108 –109 times the mass of the sun. The corresponding Eddington luminosity is ≈ 1046 –1047 erg per second. This is the order of energetics appearing in extremely energetic cosmic sources such as nuclei of active galaxies and gamma ray bursts.

3.13

Inertial Confinement

Laser-assisted thermonuclear fusion is based on the inertial confinement of a hot plasma. In fact the inertial confinement mechanism not only confines the plasma but compresses and heats it too. The compression increases the density of the plasma several hundred folds. The temperature of the plasma rises to tens of million Kelvin. These are the conditions favorable for initiating the fusion reactions. A thin shell of a mixture of deuterium and tritium is subjected to the intense radiation of one or more lasers. The radiation generates pressure of the order of a few tens of megabar on the surface of the shell. The

Plasma Confinement

85

material at the shell surface ionizes and is blown off the surface. This is called the ablation of the shell surface. The rocket-like back reaction produces an inward motion of the shell generating shock waves. This is called implosion of the shell, which results in compression and heating of the remaining shell (Fig. 3.10). Laser

Laser Ablating plasma

Implosion

)LJXUH  A deuterium–tritium shell is bombarded with several lasers. The shell surface evaporates and the evaporated plasma moves outwards (ablation). The inner shell surface recoils inwards due to the back reaction (implosion) and causes compression, confinement, and heating.

Thus the plasma confines itself by its own inward motion and hence the name inertial confinement. Let PL be the pressure applied by the laser radiation on the target surface. The work WL done by the pressure on the shell in compressing it from an initial radius Ri to the final radius R f is WL =

 Rf Ri

PL 4π R2 dR

(3.105)

where 4π R2 dR is the volume element of the shell. Integrating Eq. (3.105) gives WL =

4π 3 R f − R3i PL 3

(3.106)

86

Plasmas

For high compression, R f  Ri . The energy Wplasma contained in the plasma shell after the implosion is estimated to be  Wplasma =

3 Pplasma 2



4π R3f 3

 (3.107)

where Pplasma = nKBT is the thermal pressure of the compressed plasma and 32 nKBT is the thermal energy density of the plasma (recall that 12 KB T is the energy per particle per degree of freedom). Granting that the plasma has gained this energy from the compression caused by the lasers, we can equate the magnitudes of WL (it is negative because it represents the loss of energy of the lasers) and Wplasma to find Rf ≈ Ri



2PL 3Pplasma

1/3 (3.108)

The criterion for thermonuclear fusion reactions can be met if the compressed deuterium–tritium shell has a density of nearly 300 gm cm−3 and a temperature of nearly 10 KeV. Such a plasma has a pressure of 106 megabars. The maximum direct irradiation by a laser can generate a pressure PL of the order of 102 megabars. Thus a volume compression by a factor of at least 104 is required. This corresponds to a radial compression of 104/3 . Although this is within the reach of the present-day technology, the ablating and the imploding plasma could become highly unstable as for example by expanding (due to the release of nuclear energy) at a rate which may undermine the compression and compromise the fusion reaction rates.

3.14

Summary

Plasmas are hot. With typical temperatures of more than tens of thousands of Kelvin, they cannot be stored in ordinary material containers. They can be held in place for a finite duration of time by applying external forces on them. It is the gravitational force that has a strong enough grip on hot plasmas to keep them where they are in the cosmos. For example, hydrogen plasma at tens of million Kelvin in the solar core undergoing thermonuclear reactions is held in place by the gravity of the sun’s mass. In terrestrial conditions, strong magnetic fields with exotic configurations have the capacity to confine fusion plasmas for sufficiently long time so that the nuclear reactions could take place. In inertial confinement it is the laser power that confines, compresses, and heats the plasma to the

Plasma Confinement

87

levels required for thermonuclear reactions to occur. Plasma confinement is an active area of research in physics and engineering. A lot could be learnt from the mother nature.

Problems 3.1 Compare the number density scale heights for hydrogen and nitrogen on Earth’s atmosphere. 3.2 Estimate the Keplerian velocity at the surface of a pulsar and compare it with the escape velocity. 3.3 Calculate the value of the radiation pressure exerted by sun’s radiation on the surface of the Earth. 3.4 What is the Eddington luminosity for the sun? 3.5 Get the required data on any extragalactic jet and estimate the magnitude of the azimuthal magnetic field needed for its confinement. 3.6 Get acquainted with the observed solar photospheric magnetic field and try to model it with a force free field configuration.

4

4.1

The Waving Plasmas

Introduction

The method of provocation is the easiest way of learning about a system. Provoke it and watch it reacting. The response holds a wealth of information on the nature of the system. One is quite familiar with the example of a pendulum, a simple pendulum, a string hanging vertically with its one end fixed to a beam and the other end carrying a small ball. In the steady state of the pendulum the ball is at rest. When the ball is displaced a little from its resting position and released, it sets into oscillations. It oscillates about its resting point with a period that depends on the restoring force that comes into play when the ball is displaced. The restoring force is the gravitational pull of the Earth. The period is a function of the length of the pendulum and the acceleration due to the gravity of the Earth. Plasmas are no different. A plasma can be displaced from its steady state. Some of the steady states of a plasma have been studied in Chapter 3 in some detail. A plasma has a variety of steady states and there are as many ways of disturbing them. Plasmas are set into oscillations and for each type of disturbance there is a characteristic restoring force and the corresponding period of oscillation. The restoring forces could arise from pressure gradients, Lorentz and gravitational forces, or even their combination. The periods of oscillations are functions of the plasma properties such as the density, the temperature, and the magnetic field. Therefore, the waves in a plasma have a great diagnostic value. One has to devise ways and means to observe them in a laboratory or in a remote location such as space and astrophysical environs. A plasma supports longitudinal waves (sound waves are longitudinal), transverse waves (light waves are transverse), as well as waves of

The Waving Plasmas

89

mixed polarization. The constituents of a plasma are in an incessant state of motion undergoing collisions among themselves. The collisions act contrary to the wave motion. Collisions tend to destroy the organized wave motion. The waves lose their energy to the colliding particles and end up heating the plasma. A class of waves known as the magnetohydrodynamic waves have been a favorite candidate for heating the solar coronal plasma to a whopping million degrees. In the following sections, we shall study the properties of linear plasma waves. The waves are said to be linear when their characteristics such as the period and the polarization do not depend on their amplitudes. A plasma must be displaced by a very tiny amount in order to excite linear waves. How tiny? The properties of linear waves should not depend on the magnitude of the displacement. Approximate perturbation methods are deployed, which give reasonably good results, good, since they stand confirmed by experiments. The plasma parameters such as the density, the temperature, and the magnetic field can be given tiny variations singly or jointly. For example, we may wish to know the response of a plasma to a tiny variation in its density. The change in density would effect a change in velocity, pressure, the gravitational force, and the magnetic field. Only those changes are permitted that are commensurate with the conservation laws of mass, momentum, and energy along with the laws of electrodynamics as annunciated in the Maxwell equations. The application of these laws furnishes a dispersion relation, which is a relation among the period, the wavelength, and the steady-state plasma parameters. The wave polarization can also be determined. What cannot be determined, however, is the amplitude of the linear waves. For the determination of the amplitude necessarily takes us into the domain of nonlinear plasma dynamics into which we would not venture at present. Recall that a plasma can be described as a collection of particles: electrons, ions, and neutrals or a collection of fluids: an electron fluid, an ion fluid, and a neutral fluid or a single fluid, which is neither the electron nor the ion or the neutral fluid. The nature of waves differ from each of these descriptions of a plasma. Let us get on with it!

4.2

Single Fluid Description of a Plasma

The single fluid description of a plasma, also known as the Magnetohydrodynamic description, is obtained when the mass, the momentum, and the energy conservation laws for the multifluids, here specifically the electron and the singly charged ion fluids, are combined appropriately to define a single fluid with some average properties. This two-fluid system

90

Plasmas

can be studied the way the two-particle system is studied in terms of their center of mass motion and the relative motion. The center of mass behavior of the two fluids is extracted by adding the corresponding equations for the conservation laws for each fluid. The relative behavior of the two fluids is extracted by taking the difference of the corresponding conservation laws. Let us see how it is done. The mass conservation law for the electron fluid is written as ∂ ρe     (4.1) + ∇ · ρeV e = 0 ∂t where ρe is the mass density of the electron fluid and V e is its velocity. The mass conservation law for the ion fluid is written as ∂ ρi     (4.2) + ∇ · ρiV i = 0 ∂t where ρi is the mass density of the ion fluid and V i is its velocity. The center of mass behavior of the two fluids is extracted by adding the two conservation laws to obtain the mass conservation law of the single fluid as ∂ρ     (4.3) + ∇ · ρV = 0 ∂t where

ρ = ρe + ρi

(4.4)

is the mass density of the single fluid. The velocity V of the single fluid is defined as   V = ρeV e + ρiV i ρe + ρi

(4.5)

Note that the mass conservation law of the single fluid, Eq. (4.3), does not carry any property either of the electron fluid or of the ion fluid. It represents the mass conservation law of a new fluid with mass density ρ and velocity V . The difference of the equations (4.1) and (4.2) furnishes:  ∂ (ni − ne )    + ∇ · niV i − neV e = 0 ∂t

(4.6)

where ρe = ne me and ρi = ni mi have been used. Defining the charge density σ as:

σ = e (ni − ne )

(4.7)

The Waving Plasmas

 as and the current density, J,   J = e niV i − neV e

91

(4.8)

Equation (4.6) can be recast as

∂σ   +∇·J = 0 ∂t

(4.9)

One easily recalls from electrodynamics that Eq. (4.9) is a statement of the conservation of electric charge density. If no fluctuations are allowed in the electron and the ion number densities so that ne = ni is strictly held, Eq. (4.9) becomes ∇ ·  J=0

(4.10)

This completes the treatment of the mass conservation law. The momentum conservation law of the electron fluid is written as     V e × B ∂ V e  + ρe + (V e · ∇)V e = −ne e E − ∇Pe ∂t c   + ρeg − ρe νei V e − V i + νee ∇2V e (4.11) The momentum conservation law of the singly charged ion fluid is written as:     V i × B ∂ V i ρi + (V i · ∇)V i = ni e E + − ∇Pi ∂t c   + ρig − ρi νie V i − V e + νii ∇2V i (4.12) where  E and B are the total (external plus internal) electric and the magnetic fields; Pe and Pi are, respectively, the electron fluid and the ion fluid thermal pressures and g is the acceleration due to gravity. Here νei is the collision frequency of the electron–ion collisions and this term represents the collisional force on the electron fluid due to the ion fluid; whereas νee is the electron viscosity due to the electron–electron collisions and this term represents the viscous force of the electron fluid. Here νie is the collision frequency of the ion–electron collisions and this term represents the collisional force on the ion fluid due to the electron fluid; whereas νii is the ion viscosity due to the ion–ion collisions and this term represents the viscous force of the ion fluid. All the collisional processes are discussed in Chapter 6.

92

Plasmas

The center of mass motion of the two fluids can be determined by adding equations (4.11) and (4.12) to get   ∂ V J × B ρ (4.13) + (V · ∇)V = −∇P + − ρg ∂t c where J is the total current density in the single fluid, (Eq. 4.8), and P = Pe + Pi

(4.14)

is the total thermal pressure of the single fluid. Note that Eq. (4.13), a statement of the momentum conservation law of the single fluid, does not carry any telltale signs that this single fluid has been prepared by mixing the electron and the ion fluids. Equation (4.13) describes the momentum conservation law of a magnetized conducting fluid of mass density ρ , velocity V , pressure P and a current density J in the presence of the Lorentz and the gravitational forces. The electric force ne E does not appear as it is equal and opposite for the electron fluid and the singly charged ion fluid. The electron–ion and the ion–electron collisions are elastic and therefore the rate of momentum transfer from an electron to an ion is equal and opposite to the rate of momentum transfer from an ion to an electron. This is also expressed as

ρe νei = ρi νie

(4.15)

This is the reason why the electron–ion collisional forces do not appear in the single fluid momentum conservation law. We shall ignore the viscous forces for the present. The details of the derivation of Eq. (4.13) can be found in Chapter 6. To understand the relative behavior of the two fluids, we multiply Eq. (4.11) by the mass of the ion, mi , and Eq. (4.12) by the mass of an electron, me , and take their difference. We find     ∂ V i − V e  + e meV i + miV e × B me mi = e (me + mi ) E ∂t c   − me mi (νie + νei ) V i − V e (4.16) where we have neglected the pressure gradient force, the gravitational force and the viscous force terms for the present. Note that the electron and the ion velocity differences can be expressed in terms of the current density  J. We also need to solve for V e and V i in terms of V and J using equations (4.5) and (4.8). We find  V e ≈ V − J (4.17) en

The Waving Plasmas

93

and V i ≈ V (4.18) where terms of the order of me /mi have been neglected. On substituting for V e and V i and making use of Eq. (4.15) we rewrite Eq. (4.16) as     V × B me mi ∂ J J + = e (me + mi ) E − me mi νei (4.19) e ∂t n c en By taking the limit me VA2 and the second for Cs2 < VA2 . The slow wave has a phase speed smaller than that of the sound and the Alfven speeds. We can determine the phase velocities (in fact its square) of the oblique fast and the slow waves as   2 1/2 2  2  Cs +VA2 ω±2 1  2 (4.127) ± v ph ± = 2 = Cs +VA2 − 4Cs2VA2 cos2 θ k 2 2 Clearly the phase speeds are functions of the propagation angle θ . The phase speed of the fast wave is the largest for θ = 90 degrees and the smallest for θ = 0. The converse is true for the slow wave. The ratio of the squares of the sound speed and the Alfv´en speed is an important parameter that describes the dominance or otherwise of the magnetic and the thermal pressures. It can be expressed for an isothermal plasma with Te = Ti = T as 2nKBT 4πρ nKB T Cs2  =β = = 2 2 2 nmi B0 VA B0 /8π

(4.128)

The quantity β is called the plasma β . It is the ratio of the thermal pressure and the magnetic pressure. If the thermal pressure is of the electron fluid, then it is called the plasma electron β or βe . If the thermal pressure is of the ion fluid, then it is called the plasma ion β or βi . If the plasma β is less than or equal to one, the plasma is said to be confined by the magnetic field. A larger than unity value of the plasma β indicates that the high thermal pressure fluid controls the magnetic field, which is reshaped by the motion of the fluid. The solar atmosphere has a whole range of β values. The plasma β is much larger than unity, say of the order of 20 or so, at the solar photosphere and it diminishes to a value near 0.001 at the solar corona. The dispersion relation, Eq. (4.127), can be expressed in terms of the plasma β as 1/2 ω±2 (1 + β ) 1

2 2 (1 + = β ) − 4 β cos θ ± 2 2 k2VA2

113

The Waving Plasmas

  The plot of the ω 2 /k2VA2 vs the angle θ for the slow, the Alfv´en and the fast wave is shown in Fig.(4.5) for β = 0.5. We observe that at θ = 0, the Alfv´en and the fast waves have their phase speeds equal to the Alfv´en speed whereas the phase speed of the slow wave is equal to the sound speed as discussed earlier (see problem 9). At θ = 90 degrees, the phase speeds of the Alfv´en and the slow waves vanish and the square of the phase speed of the fast wave is equal to (1 + β ) times the squares of the Alfv´en speed. 1.6 Fast Wave

1.4 1.2 1 0.8

Alfvén Wave

0.6

Slow Wave

0.4 0.2 0

0

10

20

30

40

50

60

70

80

90

  )LJXUH  Plot of ω 2 /k2VA2 vs. the angle θ for the slow, the Alfve´ n and the fast wave for β = 0.5.

4.15

Polarization of the Oblique Fast and Slow Waves

Both the fast and the slow waves have their velocity field, the magnetic field and the propagation vector in the same plane. Here, we have (V1x ,V1z ), (B1x , B1z ), and (k1x , k1z ), but the electric field E1y =

V1x B0 c

(4.129)

is in the y direction. These waves have mixed polarization, they are neither completely longitudinal nor completely transverse. The relative magnitudes of the wave amplitudes are obviously a function of the angle θ . Note from Eqs. (4.118) and (4.120) that the divergence-free nature of the perturbation B1 is ensured as it always should be.

114

4.16

Plasmas

Energy Partition in the Fast and the Slow Waves

The relative magnitudes of the wave kinetic energy density, the wave magnetic energy density, and the wave electric energy density for the oblique fast and the slow waves can be estimated from Eqs. (4.118–4.121) B21x + B21z , 8π  2  ρ0 V1x +V1z2 , WK = 2  2 2    2 2  ω± − k2Cs2 sin2 θ − k2VA2 WB k VA 1+ = WK ± ω±2 k4Cs4 sin2 θ cos2 θ

(4.131)

ω2 WE = 2 ±2 WB k c

(4.133)

WB =

(4.130)

(4.132)

and

by appropriately substituting for the frequency of the waves. The electric energy density is much less than the magnetic energy density. It is easy to check that Eq. (4.132) reduces to Eq. (4.108) for the fast wave, in the limit θ tending to 90 degrees.

4.17

Dissipation of the Oblique Fast and the Slow Waves

As we have done before, the dissipation rate of the oblique waves can be determined by including the dissipation term in the induction Eq. (4.117) as     ω B1 = − k · B0 V 1 + k · V 1 B0 − iη k2B1 (4.134) Accordingly, the components of B1 are given by B1x = −

kB0 cos θ V1x , ω + iη k2

(4.135)

B1y = −

kB0 cos θ V1y ω + iη k2

(4.136)

and B1z =

kB0 sin θ V1x ω + iη k2

(4.137)

The velocity Eq. (4.116) remains unchanged. The pair in Eq. (4.121) modifies to

The Waving Plasmas

115

 ω 2 − k2CS2 sin2 θ −

   k2VA2 V1x − k2CS2 sin θ cos θ V1z = 0 2 1 + i η k /ω    2 2  −k CS sin θ cos θ V1x + ω 2 − k2CS2 cos2 θ V1z = 0 (4.138)

The dispersion relation, Eq. (4.123) becomes   VA2 k2VA2 4 2 + k ω 4 − ω 2 k2CS2 + C cos2 θ = 0 S 1 + i η k 2 /ω 1 + i η k 2 /ω

(4.139)

We should now take ω to be a complex number and write

ω = ωR + iωI

(4.140)

Substitute in Eq. (4.139) and separate the real and imaginary parts to determine ωR and ωI . We shall assume that the dissipation term is too small to affect ωR by which we mean that we shall neglect the dependence of ωR on η . However, ωI is directly proportional to η . Therefore, ωR is determined from   ωR4 − ωR2 k2CS2 + k2VA2 + k4CS2VA2 cos2 θ = 0 (4.141) and ωI is found to be  2 2 2   k CS cos θ − ωR2 η k2 k2VA2 ωI = 2 ωR2 ωR2

(4.142)

Again the decay rate of the oblique waves depends on the angle of propagation. Remember that ωR itself is a function of θ . It is clear that the waves are damped when the condition k2CS2 cos2 θ < ωR2 is satisfied for the respective modes ωR . A positive value of ωI is meaningless as it implies that the wave amplitude is growing with time and there is no reason for this to happen. There is no source of energy in the system at the expense of which the waves can grow. The waves can grow if the circumstances of nonequilibrium exist and there is free energy in the fluid. Such a fluid then becomes unstable. We shall not study unstable systems in this book.

4.18

Inclusion of the Displacement Current

In our study of the magnetohydrodynamic waves so far, we have neglected the effect of the displacement current on the dispersion relation of the waves. We shall illustrate this effect by taking the example of the Alfv´en waves. We shall use the full Ampere law, Eq. (4.38) and rewrite Eq. (4.48) as c     1 ∂ E  J × B = ×B (4.143) ∇×B ×B− 4π 4π ∂ t

116

Plasmas

The linearized form of the Faraday law, Eq. (4.46), is k × E 1 = ω B1 c which on taking the cross product with k gives  k  ω  E 1 = − 2k × B1 − k · E 1 2 ck k

(4.144)

(4.145)

Now taking the cross product with  B0 and substituting for B1 from Eq. (4.69) we get    2  k × B 1 0        (4.146) E 1 × B0 = 2 ω B0 · B1 k + k · B0 V 1 − k · E 1 ck k2 The linearized form of Eq. (4.143) can be written as  B0 · B1 E1  1 ∂ c      0 = −c∇ B0 · ∇ B1 − J1 × B + × B0 4π 4π 4π ∂ t

(4.147)

The displacement current term, the third term of the right-hand side of Eq. (4.147), can be expressed as 1 ∂ E 1  iω  E 1 × B0 (4.148) × B0 = 4π ∂ t 4π With the inclusion of the displacement current, Eq. (4.71) modifies to   2  k × V 1 B0 ·k   ρ0 ωk × V 1 1 +VA2 /c2 = (4.149) 4πω and the dispersion relation of the Alfv´en wave becomes   B0 ·k ω =± (4.150)  1/2 (4πρ0 )1/2 1 +VA2 /c2 −

The dispersion relation of the fast wave, Eq. (4.106) is also similarly modified. In most circumstances, the Alfv´en speed is much less than the speed of light and the correction due to displacement current is rather insignificant. However, in some astrophysical environs, the plasma is so rare, i.e., ρ is so small that the Alfv´en speed may become a significant fraction of the speed of light and the displacement current would reduce the frequency of the Alfv´en wave. The special relativistic effect would have to be included so that the magnetohydrodynamic equations would need to be written in their relativistic form.

The Waving Plasmas

4.19

117

Detection and Observation of the Magnetohydrodynamic Waves

After Hannes Alfv´en reported the existence of the Alfv´en waves, S. Lundquist detected the waves in his laboratory experiments using magnetized Mercury and showed that they obeyed the dispersion relation predicted by the theory. The phase speed of the waves was found to be directly proportional to the ambient magnetic field and inversely proportional to the square root of the mass density of the plasma thus confirming the dispersion relation. The Alfv´e n waves have been proposed to exist everywhere in the universe and their existence has been mostly inferred indirectly from the observations of the cosmic objects. However, in the near reaches such as Earth’s magnetosphere, the solar wind, and the solar atmosphere, concerted attempts have been made to detect these waves more directly. In more recent attempts, the Japanese launched the Hinode spacecraft in 2006, which has been orbiting the Earth along a path that keeps it constantly in view of the sun. The spacecraft has spectrometers that observe the sun in optical, X-ray, and extreme ultraviolet wavelengths with particularly high resolution in time and space, revealing highly variable structures and magnetic fields in the solar atmosphere. The existence of the Alfv´en waves in the solar atmosphere is one of the prized results of the mission. The possible existence of the magnetosonic waves in the solar atmosphere was inferred from the solar eclipse observations and reported in the journal Solar Physics (volume 170) in 1997. These waves were shown to have enough energy flux to heat the solar corona.

4.20

Waves in a Two-Fluid Description of a Plasma

The two-fluid description of a plasma is needed when one wishes to study waves of frequencies higher than the frequencies of the Alfv´en and the magnetosonic waves. This would essentially entail the inclusion of the dynamics of the electron fluid. Electrons being much less massive than the ions have a rather fast response to any disturbance and therefore are capable of exciting short time period waves. Again there are a variety of waves that can be excited in a two-fluid description of a plasma. These are electrostatic as well as electromagnetic waves. We already have the mathematical tools to study the two-fluid dynamics. These constitute the mass, the momentum, and the energy conservation laws of the electron and the ion fluids along with the Maxwell equations. Let us collect them all here. Equations governing the electron fluid are ∂ ρe     (4.151) + ∇ · ρeV e = 0 ∂t

118

Plasmas

V e × B − ∇Pe ρe = −ne e E + c   + ρeg − ρe νei V e − V i + νee ∇2V e (4.152) The energy conservation law or the equation of state of an electron fluid is 

∂ V e + (V e · ∇)V e ∂t





Pe = ne KBTe

(4.153)

where ne is the electron number density and Te is the temperature of the electron fluid in Kelvin. For an adiabatic electron fluid, it is written as Pe = ke ρeγ

(4.154)

where the constant ke is now specific to the electron fluid. Equations governing the ion fluid are

∂ ρi     + ∇ · ρiV i = 0 ∂t   V i × B ∂ V i ρi + (V i · ∇)V i = ni e E + − ∇Pi ∂t c   +ρig − ρi νie V i − V e + νii∇2V i

(4.155)

(4.156)

The energy conservation law or the equation of state of an ion fluid is Pi = ni KB Ti

(4.157)

where ni is the ion number density and Ti is the temperature of the ion fluid in Kelvin. For an adiabatic ion fluid, it is written as γ

Pi = ki ρi

(4.158)

where the constant ki is now specific to the ion fluid. The electrodynamics of the two fluids is governed by the following set of the four Maxwell equations. The Poisson equation: ∇ · E = 4πσ ,

(4.159)

σ = e (ni − ne )

(4.160)

The divergence-free magnetic equation: ∇ · B = 0

(4.161)

The Waving Plasmas

119

The Faraday law

∂ B = −c∇ × E ∂t

(4.162)

The Ampere law, as modified by Maxwell:  ∇ × B = 4π J + 1 ∂ E , c c ∂t along with the definition of the current density:   J = e niV i − neV e

(4.163)

(4.164)

Thus equipped, we venture into the study of some of the high-frequency waves.

4.21

The Hall Wave

The partial inclusion of the electron fluid dynamics along with the ion fluid dynamics forms a kind of hybrid system with its own characteristic waves. The Hall wave is one of them. It can also be seen as a highfrequency extension of the Alfv´en wave. The restoring force is again the tension in the magnetic field but the magnetic field has now an extra degree of freedom as we shall see that it is not frozen to both the fluids. We shall ignore electron inertia by which we mean the acceleration term on the left-hand side of the electron momentum Eq. (4.152) will be ignored. In addition, we neglect all forces except the Lorentz force. Equation (4.152) then gives the Ohm law: V e × B  E=− c

(4.165)

The Faraday law, Eq. (4.162) becomes  ∂ B   = ∇ × V e × B ∂t

(4.166)

Note the difference between this equation and Eq. (4.57). Equation (4.166) tells us that the magnetic field is now frozen to the electron fluid. We can rewrite Eq. (4.166) as  ∂ B    = ∇ × V i + V e − V i × B ∂t

(4.167)

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Plasmas

Or   × B ∂ B  J = ∇ × V i × B − ∂t ene

(4.168)

where we have used ne = ni . Equation (4.168) tells us that the magnetic field is not frozen to the ions. Thus the two fluids are now revealing their differences.The second term in Eq. (4.168) is called the Hall term, a term borrowed from the Hall effect in condensed matter physics where the electric field perpendicular to both the current density and the magnetic  e) field, as it is here, is called the Hall electric field. The quantity (J/en is also called the Hall velocity. The linearized form of Eq. (4.168), on substituting the wave form space time dependence, becomes     1 ω B1 = − k · B0 V 1 + k × J1 × B0 ene

(4.169)

The linearized form of Eq. (4.48) is  J1 × B0 = −c∇

B0 · B1 4π

+

c     B0 · ∇ B1 4π

(4.170)

and the triple cross product can be written as     k × J1 × B0 = ic B0 ·k k × B1 4π

(4.171)

where the gradient operator has been replaced by ik for the plane wave variations. Substituting in Eq. (4.169) for the triple cross product and for V 1 from Eq. (4.66) or (4.70), we find  2  ω − k2VA2 B1 =

   iB0 k · B0 k × B1 4πρ0 ωωic

(4.172)

where

ωic =

eB0 mi c

(4.173)

is the ion cyclotron frequency. By writing the components of Eq. (4.172) and solving the simultaneous equations so obtained, we find the dispersion relation of the Hall wave as

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ω 2 − k2VA2 = ±

k2VA2 ω ωic

121 (4.174)

The new term arising from the Hall term in the magnetic induction equation appears on the right hand side of Eq. (4.174). We also observe that the signatures of the ion fluid are very visible in the form of the ion cyclotron frequency, which depends on the charge and the mass of an ion. The characteristic speed, the Alfv´en speed, of the single fluid is also present. Since we expect to find frequencies higher than the frequency of the Alfv´en wave, i.e., ω 2 > k2VA2 , we retain the plus sign and solve the quadratic to obtain two roots as

ω± =

k2VA2 1 ± 2ωic 2



k4VA4 + 4k2VA2 ωic2

1/2 (4.175)

It is advisable to dwell on the ratio VA2 = λH2 ωic2

(4.176)

where λH is a spatial scale characteristic of the Hall effect and is called the ion inertial scale. Thus the two terms under the square root sign can be compared in terms of the dimensionless number kλH . For kλH 1, the frequency of the co-propagating wave becomes:

ω+ ≈ (kVA ) kλH

(4.178)

These are the strongly dispersive Hall waves with their frequency increasing as the square of the wavevector. This is what we meant when we said the Hall effect allows the system to support high-frequency waves. The frequency of the counter-propagating wave, in the limit kλH >> 1, becomes

ω− ≈ −ωic Thus the counter-propagating wave attains the ion cyclotron frequency in the high wavenumber limit. The minus sign implies that the wave is propagating in the negative k direction.

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123

The dispersion relation, Eq. (4.175) for the plus sign (solid line), is shown in Fig.(4.6) where we have plotted ω /ωic vs kλH . The dashed line represents the dispersion relation of the co-propagating Alfv´en wave. It is clear that the Alfv´en wave is the low k limit of the Hall wave. The dispersion relation, Eq. (4.175) for the minus sign (solid line), is shown in Fig.(4.7) where we have plotted ω /ωic vs kλH . The frequency of this branch of the Hall wave approaches the ion cyclotron frequency at extremely large values of k. The dashed line represents the dispersion relation of the counter-propagating Alfv´en wave. We shall stop here the discussion of the Hall waves although one could go on to discuss all the other aspects of the waves such as the polarization, the dissipation, the energy partition, etc.

4.22

The Electron Plasma Waves

As the name implies, electrons are the major players in the electron plasma waves. These are one of the most important of the high-frequency waves as they can be easily excited and measured in order to infer the physical parameters of a plasma. They can be most easily studied in the two-fluid treatment of a plasma. They are also studied in the kinetic description of a plasma in which emerges a new damping mechanism of the waves. The electron plasma waves are electrostatic and longitudinal in nature. A plasma is an electric charge quasi-neutral system. It can support small departures from strict charge neutrality. If there exists an excess of charge in any small volume of a plasma, the charges of the opposite kind would be attracted to this volume due to the electrostatic attraction. However, a complete cancellation of the electric charge never happens. Let us say the electrons are pulled toward an excess of positive charges. It is highly unlikely that the exact amount of negative charge would just reach the position of the excess positive charges and stop there. The electrons have their kinetic energy and more often than not they would overshoot their target of excess positive charge. And then they would be pulled back. This goes on and on. The electrons are said to execute oscillations in their attempt to restore charge neutrality in the plasma, Fig.(2.1). These are the oscillations of the electron density christened as the electron plasma waves. What is the role of the ions? Well, the ions provide a background overall neutrality as we shall see. There would also be fluctuations in the ion density concomitant with the fluctuations in the electron density. But the response time of the two species of particles is quite different due to difference in their masses. We shall make use of the mass, momentum, and energy conservation laws of the electron and the ion fluids. In

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the first instance, we shall neglect all forces except the electrostatic force. Thus the relevant equations take the simplified form: ∂ ρe     (4.179) + ∇ · ρeV e = 0 ∂t  ∂ V e  ρe (4.180) + (V e · ∇)V e = −ne eE ∂t

∂ ρi     + ∇ · ρiV i = 0 ∂t  ∂ V i ρi + (V i · ∇)V i = ni eE ∂t

(4.181) (4.182)

∇ · E = 4πσ ,

(4.183)

σ = e (ni − ne )

(4.184)

We linearize the equations by writing ne = n0 + ne1 , V e = V e0 + V e1

(4.185)

ni = n0 + ni1, V i = V i0 + V i1

(4.186)

 E = E 0 + E 1

(4.187)

and

where all the zero subscripted quantities that represent the equilibrium state of the two fluids do not depend on time. Again, in order to first deal with the simplest case, we put V e0 = 0, V i0 = 0, E 0 = 0

(4.188)

It is easy to check that these conditions satisfy the equilibrium state, which is defined by putting all time variations equal to zero and ne0 = ni0 = n0 . Here ρe = ne me and ρi = ni mi . The linearized equations are ∂ ne1     (4.189) + ∇ · n0V e1 = 0 ∂t

∂ V e1 n0 e  E1 =− ∂t me ∂ ni1     + ∇ · n0V i1 = 0 ∂t

n0

(4.190) (4.191)

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n0

∂ V i1 n0 e  = E1 ∂t mi

125 (4.192)

∇ · E 1 = 4πσ1 ,

(4.193)

σ1 = e (ni1 − ne1 )

(4.194)

Assuming the plane wave dependence for the first-order quantities as     ne1 = ne exp ik ·r − iω t , ni1 = ni exp ik ·r − iω t     V e1 = V e exp ik ·r − iω t , V i1 = V i exp ik ·r − iω t   E 1 = E 1 exp ik ·r − iω t

(4.195)

and substituting in Eqs. (4.179–4.184) begets ne

k · V e = n0 ω

(4.196)



k · V i ni = n0 ω

(4.197)

 V e = −ie E me ω 1

(4.198)

V i = ie E 1 mi ω

(4.199)

k · E  = −4π iσ1 1

(4.200)

What remains to be done is to eliminate in these equations all quantities except one. Let us begin with the Poisson equation: k · E 1 = −i4π e (ni1 − ne1 )

(4.201)

Substitute for the density perturbations from Eqs. (4.196) and (4.197) and get  k · V  k · V   i e k · E = −i4π e n0 − n0 (4.202) 1 ω ω Now take the scalar product of Eqs. (4.198) and (4.199) with k and substitute in Eq. (4.202) and find   k · E  = −i4π ek · E  n0 ie − n0 −ie (4.203) 1 1 mi ω 2 me ω 2

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Plasmas

We can cancel the scalar product on both sides as it is not zero. The dispersion relation of the electron plasma waves is found to be 2 2 ω 2 = ωep + ωip

(4.204)

where 

ωep =

4π n0 e2 me

1/2 (4.205)

is the electron plasma frequency and 

ωip =

4π n0 e2 mi

1/2 (4.206)

is the ion plasma frequency. If the dispersion relation, Eq. (4.204) contains both the electron and the ion plasma frequencies, why do we call it a dispersion relation of the electron plasma waves? Note that the ion plasma frequency is much smaller than the electron plasma frequency. Therefore, the wave frequency is essentially the electron plasma frequency and is written as

ω ≈ ωep

(4.207)

The frequency of the electron plasma wave is directly proportional to the square root of the plasma density and inversely proportional to the square root of the electron mass. Since all factors except n0 are constants in ωep , it can be expressed as

ω ≈ ωep ≈ 5.47 × 104n0 sec−1

(4.208)

where n0 is in per cubic centimeter. The frequency of the electron plasma wave does not depend on the wave vector k. The group velocity is therefore zero. This means that the waves are localized in the plasma and the associated electrostatic energy cannot be transported out of the plasma. The phase velocity is inversely proportional to the wavevector. Mind you all these features would change as we add more forces in the momentum conservation law of the electron fluid as we shall see later.

4.23

Polarization of the Electron Plasma Wave

It is essentially an electrostatic longitudinal wave with k   E 1 . There is no wave magnetic field, i.e., B1 = 0. The density perturbation is oscillating at the electron plasma frequency. The electron velocity perturbations

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127

have a phase difference of −90 degrees with the electric field and the ion velocity perturbation has a phase difference of 90 degrees with the electric field. The electron velocity perturbation has a phase difference of 180 degrees with the ion velocity perturbation.

4.24

Energy Partition in the Electron Plasma Wave

There are the kinetic energy densities of the electron and the ion fluids and the electrostatic energy density associated with the wave electric field. The total kinetic energy density of the two fluids is  n0 1  2 + miVi12 = WK = n0 meVe1 2 2



e2 e2 + 2 me ω mi ω 2



E12

 2 + ω2 ωep ip E 2 E2 1 = ≈ 1 = WE 2 ω 8π 8π 

(4.209)

where WE is the electrostatic energy density. Thus there is equipartition between the kinetic energy density of the two fluids and the electrostatic energy density of the electron plasma wave.

4.25

Dissipation of the Electron Plasma Wave

There are various ways by which an electron plasma wave could suffer dissipation. The simplest of them all is the collisional dissipation. As the electrons execute oscillations in order to maintain charge neutrality in the system, their oscillatory motion could be disrupted due to their collisions with other electrons and ions in the plasma. The wave loses energy to the colliding particles and gets damped. A simple way of taking this effect into account is to add the electron–ion collisional force to the electron fluid momentum Eq. (4.180). Further, as we have seen that the ion dynamics plays a negligible role in the excitation of the electron plasma wave, we can neglect the ion dynamics. It is very reasonable to assume that the ions provide a static positive background on the time −1 . The new set of equations, then, scale of the electron plasma waves, ωep is the following:

∂ ρe     + ∇ · ρeV e = 0 ∂t

(4.210)

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Plasmas



ρe

∂ V e + (V e · ∇)V e ∂t

= −ne eE − ρe νeiV e

(4.211)

∇ · E = 4πσ ,

(4.212)

σ = e (ni − ne )

(4.213)

The linearized forms of the above equations are

∂ ne1     + ∇ · n0V e1 = 0 ∂t

(4.214)

∂ V e1 e = − E 1 − νeiV e1 ∂t me

(4.215)

∇ · E 1 = 4πσ1 ,

(4.216)

σ1 = −ene1

(4.217)

Thus we have assumed V i1 = 0, ni1 = 0 in addition to the equilibrium conditions in Eq. (4.188). Assuming the plane wave dependence for the first-order quantities as      ne1 = ne exp ik ·r − iω t , V e1 = V e exp ik ·r − iω t     E 1 = E 1 exp ik ·r − iω t

(4.218)

and substituting in Eqs. (4.214–4.216) begets ne

k · V e = n0 ω

V e =

(4.219)

−ie E  me (ω + iνei) 1

k · E  = 4π iene1 1

(4.220) (4.221)

What remains to be done is to eliminate in these equations all quantities except one. Substitute Eqs. (4.219) and (4.220) in Eq. (4.221) to get k · E 1 =k · E 1

2 ωep ω (ω + iνei)

(4.222)

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129

We can cancel the scalar product on both sides as it is not zero. The dispersion relation of the electron plasma waves including the collisional force is found to be 1−

2 ωep =0 ω (ω + iνei )

(4.223)

Clearly, now the wave frequency is a complex number. Therefore, writing ω as

ω = ωR + iωI

(4.224)

assuming ω >> νei , a condition for weak damping, we find

ωR ≈ ωep

(4.225)

and

ωI = −

νei 2

(4.226)

Thus the time variation of the amplitude  E 1 is given by   E 1 = E 1 e−iωept e−

νei t 2

(4.227)

and the e fall time of the field is (2/νei). Thus the electron–ion collisions diminish the electrostatic energy density of the electron plasma wave by an amount e in νei−1 seconds.

4.26

Inclusion of Thermal Pressure

Inclusion of the thermal pressure in the momentum equation of the electron fluid brings in the new feature of dispersion and the attendant changes in the phase and the group velocities of the electron plasma waves. The electron fluid equations modify to ∂ ρe     (4.228) + ∇ · ρeV e = 0 ∂t  ∂ V e ρe (4.229) + (V e · ∇)V e = −ne eE − ∇Pe ∂t which for the isothermal equation of state becomes  ∂ V e ρe + (V e · ∇)V e = −ne eE − KB Te∇ne ∂t

(4.230)

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Plasmas

The linearization process fetches

∂ V e1 KB Te  e ∇ne1 = − E 1 − ∂t me n 0 me

(4.231)

The Poisson equation remains unaltered. Carrying out the much practiced procedure by now provides the new dispersion relation: 2 ω 2 = ωep +

KB Te 2 k me

(4.232)

From the kinetic theory of gases, one recalls that the average kinetic energy of an electron < E >, for three-dimensional Maxwellian velocity distribution, is related to the temperature T as < E >=

2 3KBTe meVeT = 2 2

(4.233)

2 is the mean square speed of the electrons. The dispersion where VeT relation becomes 2 ω 2 = ωep +

2 k2VeT 3

(4.234)

clearly bringing in the effects due to the thermal motion of the electrons. 1.05 1.03 w/w ep 1.01

w ep

0.99 kl eD

0.97 0.

0.05

0.1

0.15

0.2

0.25

0.3

)LJXUH  The dispersion relation, Eq. (4.235), for the electron plasma wave is displayed as a plot of (ω /ωep ) vs. kλeD . The dashed line marks the value of ωep .

The dispersion relation, Eq. (4.234), can also be expressed as   2 2 ω 2 = ωep 1 + k2 λeD where

λeD =



KB Te 4π no e2

(4.235)

1/2 (4.236)

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131

is the electron Debye length. The temperature-dependent term introduces dispersion. The wave frequency, the phase, and the group speeds are now a function of the wavevector k. The dispersion relation, Eq. (4.235), is displayed in Fig.(4.8). The waves transport energy by their group speed which is found to be for kλeD > 1, we find   Ti 2 2 WK = k λeD + 1 WE (4.269) Te which again shows that WK >> WE for Ti ≈ Te .

4.32

Dissipation of the Ion Acoustic Wave

We have seen so far that the collisions between the plasma particles are the main cause of the dissipation of waves. Is it true of the ion acoustic

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Plasmas

waves too? It is true that the ion–electron collisions would dissipate the ion acoustic waves. However, there is a rider! The ion–electron collision frequency is found to be proportional to the square root of electron mass me . We have determined the dispersion relation of the ion acoustic waves by neglecting the electron inertia, i.e., in the limit me ≈ 0. So if we are consistent and we should be with our theoretical framework and the assumptions made therein, we must neglect this source of damping of the waves. At the most we can claim that the waves suffer negligible damping. So what else can damp the waves? Recall that we briefly mentioned the mechanism of the Landau damping for the electron plasma waves. This mechanism applies to the ion acoustic waves too. The ion acoustic waves lose energy to the ions through the wave–particle interactions. The interaction process is governed by the momentum and the energy conservation laws. The waves damp exponentially as exp(−ViT2 /2v2ph ) where ViT is the thermal speed of the ions and v ph is the phase speed of the ion acoustic wave. Thus, the waves suffer strong damping if ViT2 ≥ v2ph which translates to Ti ≥ Te . In other words, the ion acoustic waves are well defined (weakly damped) if the electron temperature is larger than the ion temperature. The two-fluid nature of the waves is at its best!

4.33

Detection of the Ion Acoustic Wave

The ion acoustic waves are low-frequency waves. They cannot be detected from the ground-based telescopes even if they are converted into electromagnetic waves as they cannot pass through the ionosphere. Most space missions such as Ulysses and Cluster have enabled the detection of the electric fields associated with the ion acoustic waves. The laboratory detection, of course, is possible by inserting probes in a plasma carrying these waves such as a Laser-plasma system. A Laser entering a plasma excites both the electron plasma as well as the ion acoustic waves under favourable conditions. The ion acoustic waves can be identified by measuring the phase speed of the propagating low-frequency electric field fluctuations. A phase speed value close to the sound speed in the plasma is a good indicator of the presence of the ion acoustic waves.

4.34

Electrostatic Waves in Magnetized Fluids

Inclusion of a magnetic field in a plasma creates conditions for the excitation of additional waves both electrostatic and electromagnetic. It is essentially due to the cyclotron motion of the electrons and the ions in the presence of the magnetic field. The two new frequencies, viz., the electron cyclotron frequency ωec and the ion cyclotron frequency ωic appear. We

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139

now have four characteristic frequencies in the system: ωep , ωip , ωec , ωic . Their combinations give rise to three new electrostatic wave modes: (1) the upper hybrid waves, (2) the lower hybrid waves, and (3) the ion cyclotron waves. The upper hybrid wave is essentially the electron plasma wave modified by the magnetic field. The lower hybrid wave is the magnetic counterpart of the ion acoustic wave. The ion cyclotron wave arises due to the cyclotron motion of the ions. The electron cyclotron waves at ω = ωec are not electrostatic as we shall see. They are electromagnetic in nature and will be studied later. Let us, here, study the three electrostatic waves.

4.35

The Upper Hybrid Wave

Let us include a zero order or an equilibrium magnetic field in the twofluid plasma. Let the magnetic field B0 be independent of space and time. The dynamics of the two-fluid plasma is described by the set of Eqs. (4.151) to (4.164). The electron plasma waves were studied using the set of Eqs. (4.179) to (4.184). Since the upper hybrid waves are the magnetic counterpart of the electron plasma waves, we can neglect the dynamics of the ion fluid. The ion fluid can be treated as providing a uniform positively charged background since its response is too slow to affect the high-frequency upper hybrid waves. This amounts to neglecting all the ion fluid equations. The electron fluid equations on inclusion of the uniform magnetic field B0 become ∂ ρe     (4.270) + ∇ · ρeV e = 0 ∂t   V e × B ∂ V e     ρe + (V e · ∇)V e = −ne e E + (4.271) ∂t c ∇ · E = 4πσ ,

(4.272)

σ = e (ni − ne )

(4.273)

We linearize the equations by writing ne = n0 + ne1 , V e = V e0 + V e1

(4.274)

ni = n0

(4.275)

 E = E 0 + E 1 , B = B0 + B1

(4.276)

and

where all the zero subscripted quantities that represent the equilibrium state of the two fluids do not depend on space and time. Again in order

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to first deal with the simplest case, we put V e0 = 0, V i0 = 0, E 0 = 0

(4.277)

B0 = 0

(4.278)

but

It is easy to check that these conditions satisfy the equilibrium state, which is defined by putting all time variations equal to zero and ne0 = ni0 = n0 . Here ρe = ne me and ρi = ni mi . For the electrostatic waves B1 = 0. The linearized equations are ∂ ne1     (4.279) + ∇ · n0V e1 = 0 ∂t  V e1 × B0 ∂ V e1 e  =− (4.28) E1 + ∂t me c ∇ · E 1 = 4πσ1 ,

(4.281)

σ1 = −ene1

(4.282)

Assuming the plane wave dependence for the first-order quantities as      ne1 = ne exp ik ·r − iω t V e1 = V e exp ik ·r − iω t   E 1 = E 1 exp ik ·r − iω t (4.283) and substituting in Eqs. (4.279-4.282) begets k · V  e ne = n0 ω and the components of the momentum equation

ωec  −ie   E −i V me ω 1x ω ey ωec  −ie   E +i V Vey = me ω 1y ω ex

Vex =

Vez =

−ie   E me ω 1z

(4.284)

(4.285) (4.286) (4.287)

where the zero-order magnetic field has been taken to be in the z direction and ωec = (eB0 /me c) is the electron–cyclotron frequency. The Poisson equation is k · E  = i4π en e 1 We can consider two special cases.

(4.288)

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141

Let us first take the propagation vector k to be parallel to the ambient magnetic field B0 , so that k = (0, 0, kz). Equation (4.284) gives kzVez ω Substituting for Vez from Eq. (4.287) begets    −iekz E1z  ne = n0 me ω 2 ne = n0

(4.289)

(4.290)

Substituting for ne in Eq. (4.288) furnishes 2 ω 2 = ωep

(4.291)

which is the dispersion relation of the electron plasma waves. Thus the magnetic field has no effect on the waves propagating parallel to it. The transverse components of the electron velocity are completely decoupled from the parallel component Vez and do not participate in waves propagating parallel to B0 . This is easily understood as the cyclotron motion of the charged particles is always in a plane perpendicular to the magnetic field. Since B0 is in the z direction, only the x and the y components of the electron velocity are affected by it. In the second case, let us consider waves propagating perpendicular to ambient magnetic field so that k = (kx , 0, 0). Equation (4.284) gives ne = n0

kxVex ω

(4.292)

Equation (4.288) becomes  kx E1x = i4π en0

kxVex ω

We should solve for Vex from Eqs. (4.285) and (4.286) to get  2  ωec −ie    E Vex 1 − 2 = ω me ω 1x

(4.293)

(4.294)

where for the electrostatic waves, k  E 1 , and therefore  E 1 = (E1x , 0, 0). Substituting for Vex in Eq. (4.293) gives 2 2 2 ω 2 = ωep + ωec = ωuh

(4.295)

which is the dispersion relation of the upper hybrid waves with frequency ωuh . The nomenclature is self-explanatory. It is a hybrid wave incorporating the motion of the electron fluid in the electric field  E 1 and the magnetic field B0 . In the absence of the electric field E1x , the electrons

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execute the perfectly circular motion at the cyclotron frequency ωec . The electric field E1x stretches the circular motion into an ellipse. There are two restoring forces and hence the larger frequency ωuh . The oblique propagation at an angle θ to the ambient magnetic field would furnish the following quadratic equation in ω 2 : 2 2 2 ω 4 − ω 2 ωuh + ωec ωep cos2 θ = 0

(4.296)

which has the two roots

ω±2 =

2 1/2 ωuh 1 4 2 2 − 4ωec ωep cos2 θ ± ωuh 2 2

(4.297)

In the limit θ tending to zero, the plus root corresponds to the electron 2 = ω 2 . This is a spurious root. plasma wave and the minus root gives ω− ec If the wave frequency ω is completely determined by the cyclotron motion of the electrons, the electric field E 1 has no role to play and must vanish as it is evident in Eq. (4.294). Similarly in the limit θ tending to 90 degrees, the plus root corresponds to the upper hybrid wave and the minus root gives a zero frequency spurious mode. The dispersion relation 2 = 3000MHz2 of the upper hybrid wave is displayed in Fig.(4.10) for ωep 2 2 and ωec = 1000MHz . These values are typical for Earth’s ionosphere. 1.5 w 2 /w 2uh 1

w 2ep

0.5

q

0 0

10

)LJXUH 

4.36

20

30

40

50

60

70

80

90

2 /ω 2 vs. the angle θ for the upper hybrid wave. The plot of the ω+ uh

The Lower Hybrid Wave

The lower hybrid waves arise due to the coupled motion of the electron fluid and the ion fluid in the ambient magnetic field  B0 . As emphasized while studying the ion acoustic waves, the electrons play an im-

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143

portant dynamical role in exciting even these low-frequency waves. We shall take the propagation vector k to be perpendicular to B0 as in the previous section. The mass conservation equations of the electron and the ion fluids give ne = n0

kxVex kxV  , ni = n0 ix ω ω

(4.298)

We shall work under the plasma approximation, which permits ne = ni

(4.299)

and therefore Vex = Vix

(4.300)

We can use Eq. (4.294) for the perturbed electron fluid velocity as  2  ωec −ie   Vex 1 − 2 = E (4.301) ω me ω 1x The ion fluid velocity can be straight away written by replacing the electron charge −e with the ion charge e and the electron mass me with the ion mass mi so that   ωic2 ie   E (4.302) Vix 1 − 2 = ω mi ω 1x Equating the two speeds gives     ω2 ω2 me 1 − ec2 = −mi 1 − ic2 ω ω

(4.303)

or 2 ω 2 = ωec ωic = ωlh

(4.304)

This is the dispersion relation of the lower hybrid waves of frequency ωlh propagating perpendicular to the magnetic field. Recall that while studying the ion acoustic waves, the electrons were assumed to be distributed with a Boltzmann distribution. But here we have taken into account the full momentum equation for the electron fluid without the pressure gradient force. What is the difference in the two formulations? The Boltzmann distribution of the electrons arises when they are free to move under the combined action of the electric field and the pressure gradient force. In the presence of a magnetic field, the motion of the electrons is confined to a plane perpendicular to the magnetic field and, even in this plane, they can move only in a circular

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orbit of radius equal to the electron cyclotron radius. In the combined electric field E 1 and the magnetic field B0 , the electrons have only the  E 1 × B0 drift speed. Thus in the present case with E1x and B0 in the z direction, the electrons move back and forth in the y direction. So the electrons are not free to move to establish the Boltzmann distribution in the presence of a magnetic field. However, if the wave propagation is not strictly perpendicular to B0 , the electrons have freedom to move and they establish the Boltzmann distribution. We shall study the consequences of this picture in the next section.

4.37

Electrostatic Magneto-Ion Acoustic Waves in Magnetized Plasma

These waves are the magnetic counterpart of the ion acoustic waves. They propagate nearly perpendicular to the magnetic field B0 . For the nearly perpendicular propagation of the waves, i.e., for small but nonzero kz the electrons attain the Boltzmann distribution under the balance of the electric force due to E1z and the the pressure gradient force in the z direction. The ions are too massive to move much in the z direction and cannot establish the Boltzmann distribution. Thus we can use Eq. (4.248) for the electron fluid, i.e.,   eφ1 ne1 = n0 (4.305) KB Te The ion density perturbation is given as before ni1 = n0

kxVix ω

(4.306)

where kz ≈ 0 for the ion fluid. The ion fluid equation of motion including the Lorentz force can be obtained from Eqs. (4.285) to (4.287) by replacing the electron charge and mass by the ion charge and mass. This has been done to get Eq. (4.301) as   ωic2 ie   E (4.307) Vix 1 − 2 = ω mi ω 1x Using the plasma approximation ne1 = ni1 then provides us with all that we need to find the dispersion relation of the magneto-ion acoustic or the ion cyclotron waves as follows: ne1 = ni1 or

(4.308)

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kxVix eφ1 = ω KB Te

145 (4.309)

 gives Substituting for Vix and φ1 in terms of E1x

ω 2 = ωic2 + kx2Cs2

(4.310)

This is the modified dispersion relation of the ion acoustic waves or the dispersion relation of the electrostatic ion cyclotron waves. The mechanism is rather similar to the upper hybrid waves. There are two restoring forces, the pressure gradient force and the Lorentz force, and hence the larger frequency.

4.38

Electromagnetic Waves in an Unmagnetized Plasma

The electromagnetic waves have a wave magnetic field in addition to the electric field. They are transverse waves. The plasma behaves like a medium with its refractive index, which controls the propagation of the electromagnetic waves. The wave equation for the transverse waves can be derived by combining the Ampere law and the Faraday law, Eqs. (4.37) and (4.38). Take the curl of Eq. (4.37) as    ∇ × ∂ B = −c∇ × ∇ × E ∂t Take the time derivative of Eq. (4.38) as   ∂ ∇ × B 4π ∂ J 1 ∂ 2 E = + ∂t c ∂t c ∂t2

(4.311)

(4.312)

and substitute in Eq. (4.311) to get   2  ∇ × ∇ × E + 4π ∂ J + 1 ∂ E = 0 c2 ∂ t c2 ∂ t 2

(4.313)

 4π ∂ J 1 ∂ 2E =0 − 2 2 c ∂t c ∂t2

(4.314)

Or ∇2E −

since for transverse waves ∇ · E = 0

(4.315)

which also says that there is no net charge density associated with the electromagnetic waves. Equation (4.314) is the wave equation for the

146

Plasmas

transverse waves. The magnetic field B of the wave also satisfies an identical equation. The wave equation is a linear equation in the absence of the current density J or when J is a linear function of the electric  Since there is no net charge density field. We need to determine J. for the transverse waves, i.e., ni = n0 , ne = n0 We need to determine the first-order velocities of the electron and the ion fluids. We first consider the case of an unmagnetized plasma, B0 = 0. The linearized electron fluid momentum equation becomes n0

∂ V e1 en0  E1 =− ∂t me

(4.316)

and the linearized ion fluid momentum equation is n0

∂ V i1 en0  E1 = ∂t mi

(4.317)

With the plane wave variations of the perturbed quantities, we get V e1 = − ie E 1 me ω

(4.318)

V i1 = ie E 1 mi ω

(4.319)

and

The linearized current density is   in0 e2 1 1   J1 = E1 + ω mi me Substitute in the linearized wave equation to get   ω2 4πω n0 e2 1 1  2 −k E 1 − 2 E1 = 0 E1 + 2  + c ω mi me c

(4.320)

(4.321)

Or 2 2 ω 2 = ωep + ωip + k2 c2

(4.322)

 1 = 0. We have found the dispersion relation of the electromagsince E netic waves in an unmagnetized plasma. The dispersion relation of the electromagnetic waves in a vacuum can be recovered by putting the electron and the ion plasma frequencies to be zero. The major modification to the vacuum dispersion relation comes from the electron plasma frequency or the electron number density.

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147

The phase velocity  1/2 2 ω ω ep v ph = ≈ c2 + 2 k k

(4.323)

of the electromagnetic waves becomes larger than its vacuum value c. The group velocity  −1/2 2 ωep dω 2 2 vg = ≈c c + 2 dk k

(4.324)

becomes smaller than its vacuum value c. One can define a refractive index μ for a plasma as

μ2 =

2 ωep c2 k2 c2 = ≈ 1 − ω2 ω2 v2ph

(4.325)

and we find that vg = μ c,

(4.326)

c μ

(4.327)

v ph = so that

vg v ph = c2

(4.328)

The refractive index of a plasma is less than unity unlike most other common media. It is also a function of the frequency that makes plasma a dispersive medium. Some of the consequences of the characteristic refractive index of a plasma will be discussed in Chapter 5. The polarization and the absorption properties of the electromagnetic radiation will also be discussed in Chapter 5.

4.39

Electromagnetic Waves in Magnetized Plasmas

It is clear that the form of the current density J would change in the presence of the ambient magnetic field  B0 in a plasma. We also have to define the direction of propagation of the electromagnetic wave with respect to the direction of B0 . Naturally the direction of the wave electric field would also have to be fixed accordingly.

148

4.40

Plasmas

Ordinary Wave, k ⊥ B0 , E 1  B0

Let us take the wave propagation vector k to be perpendicular to the ambient field B0 . As before, let B0 be in the z direction and k to be in the x direction. Since the electric field E 1 of the wave must be perpendicular to k, it can be in the y or in the z direction. Let us take  E 1 to be in the z direction, i.e., parallel to B0 . The linearized wave equation, now with only the z component of the wave electric field, becomes ∇2 E1z −

4π ∂ J1z 1 ∂ 2 E1z =0 − c2 ∂ t c2 ∂ t 2

(4.329)

On substituting the plane wave spatial and time variation of the firstorder quantities as exp(ikx x − iω t), the wave equation becomes −kx2 E1z +

ω2 4π i ω J + E1z = 0 1z c2 c2

(4.330)

We need to determine only the z component of the current density and therefore only the z component of the electron fluid velocity. We shall neglect the ion dynamics from now onwards in the study of these highfrequency electromagnetic waves. The linearized electron fluid momentum equation can be written as  V e1 × B0 ∂ V e1 e  =− (4.331) E1 + ∂t me c and the z component of the first-order electron fluid velocity is found to be Ve1z = −

ie E1z me ω

(4.332)

The current density is given by J1z = −n0 eVe1z =

in0 e2 E1z me ω

(4.333)

Substitute in the wave equation to get 2 ω 2 = ωep + kx2 c2

(4.334)

which is identical to the dispersion relation, Eq. (4.322), obtained in the absence of the ambient magnetic field except that the direction of propagation is now fixed to be perpendicular to B0 . The ion plasma frequency

149

The Waving Plasmas

does not show up since we have neglected the contribution of the ion fluid to the current density. In conclusion, an ambient magnetic field has no effect on the dispersion relation of the electromagnetic waves propagating in a direction perpendicular to B0 with the wave electric field parallel to B0 . That is why this wave is also called the ordinary wave.

4.41

Extraordinary Wave, k ⊥ B0 , E 1 ≈⊥ B0

 1 being preWe now consider the possibility of the wave electric field E dominantly in the y direction, however, allowing also a component parallel to k, in addition. Thus E 1 = (E1x , E1y , 0),k = (kx , 0, 0), B0 = (0, 0, B0). Hence this is not a purely transverse wave. It is partly longitudinal in its polarization. Therefore, the divergence of the wave electric field is nonzero. The full wave equation, Eq. (4.313), should be used as   4π ∂ J 1 ∂ 2 E E + ∇ ∇ · E + 2 −∇2  + 2 2 =0 c ∂t c ∂t

(4.335)

The x and the y components of the linearized wave equation read as

ω2 4π i ω J1x + 2 E1x = 0 2 c c

(4.336)

and kx2 E1y −

ω2 4π i ω J − E1y = 0 1y c2 c2

(4.337)

The components of the electron fluid momentum equation read as, Eqs. (4.285) and (4.286), Vex =

ωec  −ie  E 1x − i V me ω ω ey

(4.338)

Vey =

ωec  −ie  E 1y + i V me ω ω ex

(4.339)

where the spatial and the time variation of the first-order quantities is taken to be exp(ikx x − iω t). Solve for the electron velocity components to find   ω2 −ie eωec Vex 1 − ec2 = E1x − E1y (4.340) ω me ω me ω 2

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and Vey

 2  ωec eωec −ie E1y + E1x 1− 2 = ω me ω me ω 2

(4.341)

Therefore, the x and the y components of the current density can be determined from J1x = −en0Vex , J1y = −en0Vey

(4.342)

Equation (4.336) becomes 2  2  ωec ωep 2 2 ω − ωec − ωep E1y = 0 E1x + i ω

Equation (4.337) becomes   2     2 2 2 2 ω − kx2 c2 ω 2 − ωec ω E1y + −iωec ωωep − ωep E1x = 0

(4.343)

(4.344)

Eliminating, let us say, E1y from the preceding two equations gives   2     2 2 2 2 2 2 4 ω − ωec − ωep ω − kx2 c2 ω 2 − ωec ω 2 = ωec ωep (4.345) − ωep Can you identify the upper hybrid frequency? Equation (4.345) can be put into a simplified form as follows: 2 ω4    2 ωec ep 2 2  + ωep ω − kx2 c2 ω 2 − ωec ω2 = 2 2 ω − ωuh

Or

ω

2

− kx2 c2

   2 4 2 ω2 ω2 − ω2 ωec ωep + ωep uh   = 2 ) ω2 − ω2 (ω 2 − ωec uh

so that 2 ωep kx2 c2 = 1 − ω2 ω2



  2 2 2 ωec ωep + ω 2 ω 2 − ωuh   2 (ω 2 − ω 2 ) ω 2 − ωuh ec

which can be recast as  2 2 ωep ω 2 − ωep kx2 c2 = 1− 2 2 ω2 ω ω 2 − ωuh This is the dispersion relation of the extraordinary wave.

(4.346)

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4.42

151

Polarization of the Extraordinary Wave

The polarization of the extraordinary wave can be determined from either of the Eqs. (4.343) and (4.344). The ratio of the two components of the wave electric field is found, from Eq. (4.344), to be

    2 − ω2 ω2 −i ω 2 − kx2 c2 ω 2 − ωec E1x ep   = (4.347) 2 E1y ωec ωωep which, on substituting from the dispersion relation, simplifies to 2 −iωec ωep E1x  =  2 2 E1y ω ω − ωuh

(4.348)

We note that the extraordinary wave is elliptically polarized. It has a mixed polarization of both the longitudinal and the transverse waves. The longitudinal component E1x vanishes for B0 = 0, i.e., for an unmagnetized plasma. It also vanishes for ωep = 0, i.e., in a vacuum. Thus, the mixed state of polarization exists only in a magnetized plasma. It is easy to see that for ωec = 0 and E1x = 0, the dispersion relation, Eq. (4.345), reduces to the dispersion relation of the ordinary wave. One should also note that for ωep = 0 one obtains from Eq. (4.345)  2   2  2 2 ω − ωec ω − kx2 c2 = 0 (4.349) This gives two wave modes: (1) at ω 2 = kx2 c2 , which is the dispersion 2, relation of the electromagnetic radiation in vacuum and (2) at ω 2 = ωec which is the dispersion relation of the electromagnetic electron cyclotron waves in vacuum. The electrons gyrating in a magetic field produce cyclotron radiation as we shall see in Chapter 5.

4.43

Electromagnetic Waves Propagating along B0

Let us take the propagation vector k = (0, 0, kz) and as before B0 = (0, 0, B0) so that the wave electric field can be  E 1 = (E1x , E1y , 0). The x and the y components of the wave equation (4.335) can be written as kz2 E1x −

ω2 4π i ω J − E1x = 0 1x c2 c2

(4.350)

and

ω2 4π i ω J1y − 2 E1y = 0 (4.351) 2 c c The electron fluid momentum equation does not change. We can use Eqs. (4.340)–(4.342) to determine the components of the current density kz2 E1y −

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Plasmas

and substitute in the wave equation. We find  2 2 ω iωep ωep ec 2 2 2 ω − kz c − E1y = 0 E1x + a aω and



ω

2

− kz2 c2 −

2 ωep a

E1y −

2 ω iωep ec E1x = 0 aω

(4.352)

(4.353)

where 2 ωec ω2 The elimination process furnishes 2  2 4 ω2 ωep ωep ec 2 2 2 ω − kz c − = 2 2 a a ω

a = 1−

Or



2 ωep ω 2 − kz2 c2 − a

=±

2 ω ωep ec aω

(4.354)

(4.355)

(4.356)

which can be put in a simple form as (for the plus sign) 2  2 ωep ωep kz2 c2 ωec  = 1 − 1 + = 1 − ω2 a ω ω (ω − ωec )

(4.357)

We shall see in the next section that this is the dispersion relation of the right circularly polarized wave. Similarly for the negative sign in Eq. (4.356), we find 2 ωep kz2 c2 = 1 − ω2 ω (ω + ωec )

(4.358)

We shall see in the next section that this is the dispersion relation of the left circularly polarized wave.

4.44

Circularly Polarized Radiation

We determine the ratio of the two components of the wave electric field from Eq. (4.353) to be 2 ω iωep E1y ec =   ω2 E1x aω ω 2 − kz2 c2 − aep

(4.359)

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153

Substituting the dispersion relation for the plus sign from Eq. (4.356), we find E1y =i (4.360) E1x This ratio of the wave electric field components represents a right circularly polarized wave propagating along the magnetic field in the z direction. In this mode the tip of the electric vector rotates in an anticlockwise direction with the frequency ω . Thus Eq. (4.357) represents the dispersion relation of the right circularly polarized wave. The dispersion relation for the minus sign in Eq. (4.356) represents the left circularly polarized radiation (with a clockwise rotation of the tip of the electric vector) since the ratio of the wave electric field components is found to be E1y = −i (4.361) E1x Thus Eq. (4.358) represents the dispersion relation of the left circularly polarized wave. A more detailed discussion of the polarization characteristics of radiation can be found in Chapter 5.

4.45

The Whistler Wave

The dispersion relation of the right circularly polarized radiation is found to be, Eq. (4.357), 2 ωep kz2 c2 = 1− 2 ω ω (ω − ωec )

(4.362)

Let us see how the phase velocity v ph varies with the frequency of the wave by differentiating v ph with respect to ω :  1/2 −ω 2 + ωωec v ph = c (4.363) 2 −ω 2 + ωωec + ωep −1/2 dv ph c 2  = ωep −ω 2 + ωωec dω 2   2 −3/2 × −ω 2 + ωωec + ωep (ωec − 2ω )

(4.364)

It is seen that the phase velocity increases with the frequency until 2ω < ωec . Convince yourself that all the other factors remain positive. It can be shown that the group velocity also increases with the frequency. This implies that high-frequency waves travel with higher velocity and

154

Plasmas

would reach a receiver earlier than the low-frequency waves generating a whistling effect. Such waves are called whistlers. They have been detected in Earth’s atmosphere traveling along Earth’s magnetic field. For 2ω > ωec , the phase velocity begins to decreases with the frequency.

4.46

The Faraday Rotation

The Faraday rotation is the rotation of the plane of polarization of an electromagnetic wave while propagating through a magnetized plasma. The rotation of the plane of polarization of the radiation emitted by the cosmic radio sources is attributed to the magnetized plasma of the interstellar medium through which the radiation propagates. The angle of rotation offers a good diagnostic of the conditions in the interstellar medium. The rotation is caused by the difference in the phase velocities of the right and the left circularly polarized radiations. The electric field of a linearly polarized electromagnetic wave traveling along the ambient magnetic field in the z direction can be written as  E(z, t) =ex Ex exp (ikz z − iω t)

(4.365)

The linearly polarized wave can be split into two oppositly polarized circular waves as  E (z, t) = E R + E L

(4.366)

where the right circularly polarized field E R is given by 1  E R (z, t) = (ex Ex + iey Ey ) exp(ikz z − iω t) 2  R is given by and the left circularly polarized field E

(4.367)

1  E L (z, t) = (ex Ex − iey Ey )exp(ikz z − iω t) (4.368) 2 Here ex ,ey are the unit vectors in the x and the y directions. Let the linearly polarized field travels for a time duration T in a magnetized plasma. The two fields E R and E L , with their respective phase speeds vR and vL , would traverse different distances in the time duration T (Fig. 4.11). The field E R traverses the distance S from an initial point z = 0, t = 0 in time duration T and becomes 1  E R (S, T) = (ex Ex + iey Ey ) exp (ikz vR T − iω T ) 2

(4.369)

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155

B0

=

+ EL

ER B0

=

+ ER

EL

)LJXUH  Top: a plane polarized wave is split into a right circularly polarized wave and a left circularly polarized wave. Below: after a time interval T , the two waves traverse different distances and develop a phase difference. Their sum gives the rotated plane polarized wave.

where S = vR T

(4.370)

The field E L traverses the distance vL T in the time duration T , from an initial point z = 0, t = 0 and becomes: 1  E L (vL T, T) = (ex Ex − iey Ey )exp (ikz vL T − iω T ) 2

(4.371)

vL = vR + v

(4.372)

Let

Thus adding the two fields at the end of the interval T , we find: 1  E (T ) = (ex Ex + iey Ey ) exp (ikz vR T − iω T ) 2 1 + (ex Ex − iey Ey )exp (ikzvL T − iω T ) 2

(4.373)

Or ex Ex  E (T ) = exp (ikz vR T − iω T ) (1 + exp (ikz vT )) + 2 iey Ey exp (ikz vR T − iω T ) (1 − exp (ikz vT )) 2

(4.374)

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This can be rewritten as      kz vT kz vT  +ey Ey sin × exp (ikz vR T − iω T ) (4.375) E (T ) = ex Ex cos 2 2 First observe that for v = 0 we recover the incident linearly polarized wave after it has traversed a distance S in time T . However, in a magnetized plasma vR = vL due to which the left and the right circularly polarized waves have undergone different number of their wave periods in the duration T . The incident field has been rotated to the new field given by Eq. (4.375), Fig.(4.11). The angle of rotation φ is found to be kz vT (4.376) 2 The difference v can be determined from the dispersion relations, Eqs. (4.357) and (4.358) as   1/2 1/2 −ω 2 − ωωec −ω 2 + ωωec v=c −c (4.377) 2 2 −ω 2 − ωωec + ωep −ω 2 + ωωec + ωep

φ=

which can be simplified to 2 ωec ωep (4.378) 3 2ω under the assumption that ω is much larger than both ωec and ωep . The rotation angle of the plane of polarization can now be expressed as

v=c

2 2 S kz cT ωec ωep ωec ωep ≈ (4.379) 4ω 3 4cω 2 where for ω >> ωep , ω >> ωec , ω ≈ kz c, and S ≈ cT have been used. Thus the rotation angle is directly proportional to the product of the electron fluid density, the distance traveled S, and the component of the magnetic field B0 parallel to the direction of propagation of the wave. The rotation angle is inversely proportional to the square of the frequency ω . It is for this reason that rotation angles, in astrophysics, are measured for radiation at radio frequencies. A common practice in astrophysics is to define the physical parameters in terms of their representative values to get a feel for the value of the rotation angle. Thus, for the interstellar medium, the electron density, ne ≈ 0.1cm−3 , the magnetic field B0 ≈ 10−6 Gauss, the distance to Earth S ≈ 3 × 1018 cm. The rotation angle at a frequency ν = 109 Hz turns out to be ≈ 0.3 radians. Here the angular and the circular frequencies are related as ω = 2πν . The measurement of the rotation angle gives valuable clues about the conditions in the interstellar medium.

φ=

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4.47

157

Cutoff Frequencies of the Electromagnetic Waves

The astrophysical and the space plasmas are far from uniform in their properties. The plasma density and/or magnetic field vary from region to region. A familiar example is Earth’s ionosphere. It is essential to know what type of waves at what frequencies can propagate through a plasma. This need is met by determining the cutoff and the resonance frequencies of all the waves. The cutoffs and the resonances can be determined from the dispersion relations. Let us take the first example. The electromagnetic wave in an unmagnetized plasma for which the refractive index, given by Eq. (4.325), is

μ2 =

2 ωep c2 k2 c2 = ≈ 1 − ω2 ω2 v2ph

(4.380)

Observe that at ω = ωep the refractive index μ = 0, the wave vector k = 0 or the wavelength becomes infinitely large, the phase speed becomes infinitely large and the group speed vanishes. This is the circumstance of a cutoff. An electromagnetic wave propagating in an unmagnetized plasma is said to be reflected when it reaches a plasma region where its frequency ω becomes equal to the electron plasma frequency ωep . The ordinary wave travels in a direction perpendicular to the magnetic field in a magnetized plasma. It is also reflected at its cutoff frequency ω = ωep . The cutoff frequency of the extraordinary wave is found from Eq. (4.346) to be  2 2 ωep ω 2 − ωep 1− 2 =0 (4.381) 2 ω ω 2 − ωuh Or 4 2 2 ω 4 + ωep − 2ω 2 ωep − ω 2 ωec =0

(4.382)

which gives two quadratic equations 2 ω 2 − ωωec − ωep =0

(4.383)

2 ω 2 + ωωec − ωep =0

(4.384)

and

Notice that Eq. (4.383) gives the cutoff frequency of the right circularly polarized wave, the dispersion relation of the right circularly polarized wave being given by Eq. (4.357). We call this cutoff frequency ωR , which is found to be

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Plasmas

  2 2 1/2 ωec + 4ωep ωec + ωR = 2 2

(4.385)

We also notice that Eq. (4.384) gives the cutoff frequency of the left circularly polarized wave, the dispersion relation of the left circularly polarized wave being given by Eq. (4.358). We call this cutoff frequency ωL , which is found to be   2 2 1/2 ωec + 4ωep ωec ωL = − + 2 2

(4.386)

Thus ωR and ωL are the two cutoff frequencies of the extraordinary wave and ωR is larger than ωL . We have taken only the plus sign in front of the square root to conform with our choice of only positive values of ω .

4.48

Resonances of the Electromagnetic Waves

A resonance occurs when the refractive index becomes infinite and the wavelength becomes zero. The wave is absorbed in a plasma at the resonance frequency. Let us take the first example. The electromagnetic wave in an unmagnetized plasma for which the refractive index, given by Eq. (4.325), is

μ2 =

2 ωep c2 k2 c2 = ≈ 1 − ω2 ω2 v2ph

(4.387)

Observe that at ω = 0 the refractive index μ = ∞, the wave vector k becomes an imaginary number, the phase speed and the group speed are not defined. Since the resonance occurs at ω = 0, it is usually not listed as a resonance. What one must appreciate is that the waves of frequency smaller than the electron plasma wave cannot propagate through a plasma. The aforementioned remarks also apply to the ordinary wave. The resonance frequency of the extraordinary wave is found from Eq. (4.346) to be: 2 ω 2 = ωuh

(4.388)

One recognizes that this is the dispersion relation of the upper hybrid waves propagating perpendicular to the magnetic field. As the extraordinary wave of frequency ω approaches a region in a plasma where ω becomes equal to the upper hybrid frequency, the electromagnetic wave

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159

actually becomes the electrostatic upper hybrid wave. Thus there is a change of the nature of a wave at the resonance point. The cutoffs and the resonances mark the regions of propagation of the waves. We are now ready to display the regions of propagation for all the electromagnetic waves.

4.49

Propagation Bands of the Electromagnetic Waves

We shall display the propagation regions of the electromagnetic waves by taking the example of Earth’s ionosphere. The representative values of the ionospheric plasma are: the electron density ≈ 106 per cubic centimeter and the magnetic field ≈ 0.3 Gauss. The electron plasma frequency and the electron cyclotron frequency are found to be 1

ω ep

0 –1 2 μ 0 –2

100

200

ω

300

400

500

–3 –4 –5 )LJXUH  The ordinary wave propagates in the region where μ O > 0. The cutoff frequency is ωep where the refractive index vanishes. 2 ωep = 3 × 103 (MHz)2 , ωep ≈ 55MHz, ωec ≈ 5MHz

(4.389)

We see that ωec 0. The two cutoff frequencies are at ωL and ωR . The resonance occurs at ωuh . This wave cannot propagate for ω < ωL and for ω lying between ωuh and ωR . The refractive index becomes unity at the electron plasma frequency and at frequencies much larger than ωR .

The refractive index for the extraordinary wave (Eq. 4.346) is given as  2 2 ωep ω 2 − ωep 2 μEx = 1 − 2 (4.391) 2 ω ω 2 − ωuh 2 vs ω is shown in Fig.(4.13). The cutoff at ω = ωL is easily The plot of μEx located since μEx goes to zero at the cutoff frequency. Recall that ωL is the frequency of the left circularly polarized wave (Eq. 4.386). This cutoff frequency for the ionosphere is about 41 megahertz. The refractive index approaches the value unity at the electron plasma frequency ωep . The refractive index becomes larger than one as the frequency of the extraordinary wave becomes larger than the electron plasma frequency but remains smaller than the upper hybrid frequency. Further on, the refractive index approaches an infinitely large value as the wave frequency

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161

nears the value of the upper hybrid frequency ωuh . At ω = ωuh , μEx becomes infinite and this is the resonance frequency of the extraordinary wave. This resonance frequency for the ionosphere is about 63 megahertz. The second cutoff of the extraordinary wave is located at ω = ωR given by Eq. (4.385). This cutoff frequency for the ionosphere is about 73 megahertz. The refractive index approaches the value unity at frequencies much larger than the upper hybrid frequency. The extraordinary waves cannot propagate in the region ω < ωL and between ωuh and ωR since the refractive index of the plasma becomes imaginary for these ranges of the wave frequency. The typical value of ωL for the ionosphere is ≈ 41 MHz. 15 13 11 9

w ec 2

μ

R

7

wR

(wec)/2

5 3 1 –1

10

20

30

–3 –5

40

50

60

70

80

90

ω MHz

–7 –9 –11 –13 )LJXUH  The right circularly polarized wave propagates in the region where μR > 0. The frequency region between ω2ec and ωec is the propagation region of the whistlers. The cutoff frequency is at ωR . The resonance occurs at ωec . This wave cannot propagate for ω lying between ωec and ωR .The refractive index approaches unity at frequencies much larger than ωR .

The refractive index for the right circularly polarized wave, Eq. (4.357), is given by

μR2 = 1 −

2 ωep ω (ω − ωec )

(4.392)

The plot of μR2 vs ω is shown in Fig.(4.14). One observes on the upper left corner that the refractive index decreases as the wave frequency increases from zero to a value equal to half the electron cyclotron frequency. For

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Plasmas

wave frequencies larger than half the electron cyclotron frequency, the refractive index increases with the frequency. There is a corresponding change of gradient of the phase velocity. As we have discussed earlier this region of propagation of the right circularly polarized wave is identified with the whistler waves that propagate until ω = ωec . The refractive index becomes infinite at ω = ωec . This is the resonance point of the right circularly polarized wave and it lies at ωec = 31.6 MHz. for the ionosphere. The cutoff at ω = ωR is easily located since μR goes to zero at the cutoff frequency. Recall that ωR is the frequency of the right circularly polarized wave (Eq. 4.385). This cutoff frequency for the ionosphere is about 73 megahertz. The refractive index approaches the value unity at frequencies much larger than ωR . 5 4

2

μ

L

3

wL

2 1 0 0

10

20

30

40

50

60

70

80

90

–1 w MHz

–2 –3 –4 –5

)LJXUH  The left circularly polarized wave propagates in the region where μL > 0. The cutoff frequency is at ωL . The resonance does not occur for this wave. The wave cannot propagate for ω < ωL . The refractive index approaches unity at frequencies much larger than ωL .

The dispersion relation of the left circularly polarized wave, Eq. (4.358), is

μL2 = 1 −

2 ωep ω (ω + ωec )

(4.393)

The plot of μL2 vs ω is shown in Fig.(4.15). The propagation region of the left circularly polarized wave lies beyond the cutoff frequency ωL and

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163

approaches a value of unity at frequencies much larger than ωL . The value of ωL for the ionosphere is ≈ 41 MHz. The knowledge of the region of propagation of the various waves is absolutely necessary in order to investigate the properties of the ionosphere. With this information one is able to devise experiments to study the ionosphere as well as to study the radiative properties of those objects whose radiation may pass through the ionosphere. The forbidden bands of radiation in the ionosphere has necessitated the advent of the space telescopes.

4.50

Summary

Plasmas can support a variety of waves, both electrostatic as well as electromagnetic. We have studied the excitation of these waves in the linear regime. First, the equilibrium configuration of a plasma is determined by setting all time derivatives equal to zero. The equilibrium state of the plasma could be inhomogeneous wherein the plasma parameters such as the density, the temperature, and the magnetic field could be functions of the space coordinate. The equilibrium state is then disturbed by a small but finite amount. This entails space, and timedependent variations in the plasma parameters. The variations are of the plane wave type for linear waves. Incorporating the variations in the linearized forms of the mass, momentum, and energy conservation laws furnishes a self-consistent solution, which is the dispersion relation of the particular wave under study. It is a straightforward exercise to determine the polarization of the waves. The amplitude of the wave is one quantity that cannot be determined in the linear regime. Using this methodology, the Alfv´en and the fast and the slow magnetosonic waves that exist in a single fluid description of a plasma have been studied. In the two-fluid picture, we have studied transverse Hall waves, the electrostatic electron plasma waves, the ion acoustic waves, the upper hybrid waves, the lower hybrid waves, and the ion cyclotron waves. In addition, the electromagnetic waves propagating parallel as well as perpendicular to the ambient magnetic field have also been studied. Here we come across electromagnetic waves that are not purely transverse and have mixed polarization. This is the novelty of the plasmas. The propagation regions of the electromagnetic waves are determined by locating their cutoff and the resonance frequencies.

Problems 4.1 Find the directions of V 1 , B1 , E 1 ,k for the dispersion relation ω = −kVA for the counter-propagating Alfv´en wave.

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Plasmas

4.2 Solve Eq. (4.89) to obtain the value of ωI 4.3 Derive Eq. (4.108). 4.4 Show that the spatial damping rate kI of the fast wave is given by η k3

kI ≈ 2ωR where kR and ω are the real parts of the wave vector and the frequency of the fast wave. Find kR too. 4.5 Find the relative magnitudes of V1x , B1x ;V1z , B1z for the fast wave for θ = 45 degrees. 4.6 Determine the value of the ratio of WB to WK from Eq. (4.132) for the angle θ = π2 . 4.7 Derive Eq. (4.142) and find the values of ωI for θ = 0 and θ = 90 degrees. Compare it with the damping rates of the Alfv´en and the fast waves. 4.8 Derive the dispersion relation of the fast wave, for θ = 90 degrees, including the displacement current. 4.9 Redraw Fig.(4.5) for β = 2. 4.10 Estimate the ion inertial scale for (1) solar photospheric plasma and (2) solar wind plasma. 4.11 What does the linearized form of the Ampere law give on substituting for the first-order current density J1 ? 4.12 Derive the dispersion relation of the ion acoustic waves under the plasma approximation. 4.13 Show that the product of the group and the phase speeds of the ion acoustic waves is nearly the square of the sound speed. 4.14 Derive the dispersion relation of the ion acoustic waves assuming the electron fluid to be in the isothermal state and the ion fluid to be in the adiabatic state. 4.15 Estimate the frequency of the upper hybrid wave in the solar corona. 4.16 Estimate the frequency of the lower hybrid wave in Earth’s magnetosphere. 4.17 Learn about the short wave radio transmission in Earth’s ionosphere. 4.18 Show the propagation region of the extraordinary wave in the magnetosphere of a pulsar. 4.19 Show that Eq. (4.345) reduces to the dispersion relation of the ordinary wave for E1x = 0. 4.20 Show that the group velocity of a whistler wave increases with its frequency.

5 5.1

The Radiating Plasmas

Radiation and Plasmas

Plasmas and radiation have a symbiotic relationship. We learnt in Chapter 1 that the universe evolved from a soup of radiation and plasma. High-energy radiation creates electron–positron plasma. The right kind of radiation can ionize a gas to produce a plasma. A plasma consists of energetic electrically charged particles. Electrically charged particles undergoing acceleration or deceleration produce radiation. Radiation is a generic term for electric and magnetic fields fluctuating with space and time in a wide variety of ways. In fact, extremely energetic particles such as obtained in the laboratory accelerators are often referred to as particle radiation. Extremely energetic particles observed in the cosmos are called cosmic rays. This is because the quantum mechanics revealed the dual character of radiation as well as of material particles. Radiation can act like a particle called the photon, and it can also act like a wave called the electromagnetic wave. Similarly, an electron possesses the wave– particle duality. We shall, in this chapter, learn about the essentials of radiation, how it can be generated from a single charged particle, and what difference would a collection of charged particles, a plasma, make.

5.2

A Quick Revisit of Waves

Waves are disturbances that vary periodically with space and time. A wave is characterized by an amplitude that represents its strength and a phase that represents its space–time location. A linear sinusoidal wave can be represented by a sine or a cosine function or a combination of the two, an exponential function as Ψ = ψ exp [iφ (r, t)]

(5.1)

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Plasmas

The amplitude ψ of a linear wave does not depend on space and time whereas the phase φ is a function of the position vector r and time t. The time rate of change of phase is the frequency, ω , of the wave and the space rate of change of the phase is the wavevector, k, which gives the direction of propagation of the wave:

∂φ = ω, ∂t

(5.2)

∂φ  =k ∂r

(5.3)

The time period T of the wave and the wavelength λ are given by T=

2π , ω

(5.4)

λ=

2π k

(5.5)

where k is the magnitude of the wavevector k. The phase factor can be expressed in terms of the frequency ω and the wavevector k by integrating Eqs. (5.2) and (5.3) to find

φ (r, t) =k ·r + ω t

(5.6)

Thus ψ is the amplitude of the wave at the space–time point r = 0, t = 0. The frequency ω , in general, is a function of the wavevector k. The relation between the frequency, ω , and the magnitude of the wavevector, k, called the dispersion relation of the wave, describes the dispersive properties of the wave. The orientation of the amplitude with respect to the direction of propagation defines the polarization of a wave. All types of waves have two characteristic velocities, the phase velocity and the group velocity. The phase velocity V p describing the velocity of propagation of phase is defined as V p = ω . k

(5.7)

and it is directed along the propagation vector k. The group velocity V g , obviously, refers to a group of waves and describes the rate at which the waves transport energy from one space point to another. It is defined as V g = d ω . dk

(5.8)

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167

The group velocity is also directed along the propagation vector k but under certain conditions, especially in a plasma, it could be different. The characteristics of the waves depend on the medium in which they propagate.

5.3

Electromagnetic Radiation

As the name implies, the electromagnetic radiation is composed of an electric field E and a magnetic field B. These are related to each other in a well-defined manner and represent the amplitude of the radiation. The electric and the magnetic fields oscillate in time at a common frequency ω . The fields vary in space with a common spatial period, the wavelength, λ . The orientations of the electric and the magnetic fields with respect to the direction of propagation define the polarization of the radiation. The most familiar form of the electromagnetic radiation is, light, the visible radiation. We receive it from the sun and other stars. We receive it, in myriad forms, from the good old electric bulb to the state of the art lasers. The nature of the electric and the magnetic fields of the electromagnetic radiation can be understood from the fundamental laws of electromagnetism established by Charles Austin de Coulomb, Lord Michael Faraday, Andre Marie Ampere, and James Clerk Maxwell. The Coulomb law describes the electric field  E produced by a charge distribution. It can be written in its differential form as: ∇ · E = 4πσ

(5.9)

where σ is the charge per unit volume, the charge density. The Ampere law relates the current density J and its associated magnetic field as: ∇ × B = 4π J c

(5.10)

Maxwell made the most important addition to the Ampere law by adding the displacement current density JD arising from a time varying electric  With this addition the Ampere field to the conduction current density J. law reads:   ∇ × B = 4π J + JD , (5.11) c 1 ∂ E JD = 4π ∂ t

(5.12)

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Plasmas

We shall see that it is the displacement current density JD that is responsible for the existence of the electromagnetic radiation. Besides the inclusion of the displacement current density is absolutely essential to ensure the conservation of electric charge which can be seen as follows. Take the divergence of Eq. (5.11)     E ∂ 1 ∇ · ∇ × B  = 0 = ∇ · J + (5.13) 4π ∂ t Take time derivative of Eq. (5.9) and substitute in Eq. (5.13) to find

∂σ   +∇·J = 0 ∂t

(5.14)

which spells the law of charge conservation: the rate of change of charge density is equal to the net charge flux density lost. The charge flux density is nothing but the product σV which is equal to the current density  J , V being the charge velocity. The Faraday law of induction describes the electric field induced by the time varying magnetic field. Its differential form can be written as 1 ∂ B ∇ ×  E =− c ∂t

(5.15)

The statement of the absence of magnetic monopoles is contained in ∇ · B = 0

(5.16)

The set of Eqs. (5.9–5.16) contain all there is to know about electromagnetic radiation within the framework of the classical mechanics. The set of Eqs. (5.9), (5.11), (5.15), and (5.16) is called Maxwell’s equations. We are now well equipped to study electromagnetic radiation in all its glory. One of the most important properties of the electromagnetic radiation is that it can exist and propagate in vacuum. This can be appreciated as follows. The charge density σ and the conduction current density J are both zero in the classical vacuum as there are no charged or for that matter uncharged particles in the classical vacuum. Take the curl of Eq. (5.15) and substitute for curl of B from Eqs. (5.11–5.12) to get  ∇ × ∇ × E = − 1 ∇ × ∂ B c ∂t Or

∇ × ∇ × E = − 1 ∂ c2 ∂ t



∂ E ∂t

(5.17)  (5.18)

The Radiating Plasmas

since J = 0 in the classical vacuum. expanded as   ∇ × ∇ × E = −∇2 E + ∇ ∇ · E

169

The double curl of E can be

(5.19)

But Eq. (5.9) says that ∇ · E = 0

(5.20)

since σ = 0 in vacuum. Therefore Eq. (5.18) simplifies to 2 ∇2 E − 1 ∂ E = 0 2 c ∂t2

(5.21)

This is the wave equation of an electromagnetic wave with electric field  E in vacuum. Be assured that an identical wave equation is obtained in terms of the magnetic field B. Let us consider one dimensional wave propagation and solve this equation in cartesian coordinates (x, y, z). Let us assume that the wave is traveling in the z direction so that the propagation vector has only a z component, i.e., k = (0, 0, k). The electric field,  E, (also  B) is a function only of z and t. The wave equation becomes:

∂ 2E 1 ∂ 2 E − =0 ∂ z2 c2 ∂ t 2

(5.22)

This is a linear equation in E. The solution of this equation can be easily verified to be  E = E 0 exp (ikz − iω t)

(5.23)

where

ω 2 = k2 c2

(5.24)

ω = ±kc

(5.25)

Or

We keep the frequency to be positive. Therefore,

ω = kc

(5.26)

for propagation in the z direction, k > 0. The minus sign in Eq. (5.25) corresponds to a wave propagating in the (−z) direction with k < 0. So we have obtained the dispersion relation, a relation between the frequency ω and the wavevector, also called the propagation vector, k.

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Plasmas

The phase velocity Vp of the electromagnetic wave in vacuum is found to be Vp =

ω =c k

(5.27)

the velocity of light in vacuum, Vp is in the z direction. Thus the Maxwell equations have correctly given the speed of the electromagnetic waves in vacuum. Note the role of the displacement current without which there would be no time variation of E and hence no radiation. The group velocity is Vg =

dω =c dk

(5.28)

which is also the speed of light and equal to the phase velocity; Vg is in the z direction. The electromagnetic radiation transports energy with the speed of light c in vacuum.

5.4

Polarization of Electromagnetic Waves

We still have to determine the polarization of the electromagnetic wave. It is seen from Eq. (5.9) that

∂ Ez =0 ∂z

(5.29)

 can have which can be satisfied if Ez = 0. Therefore the electric field E only x and y components, i.e., E = (Ex , Ey , 0). The magnetic field of the wave should also satisfy the divergence-free condition (Eq. 5.16), which becomes

∂ Bz =0 ∂z

(5.30)

As in the case of the electric field, the magnetic field also has only the x and the y components, i.e., B = (Bx , By , 0). To find the relative orientation of  E and B, examine the three components of Eq. (5.15). The x component gives −

∂ Ey 1 ∂ Bx =− ∂z c ∂t

(5.31)

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171

Or substituting from Eq. (5.26) Ey = −Bx

(5.32)

The y component of Eq. (5.15) gives

∂ Ex 1 ∂ By =− ∂z c ∂t

(5.33)

Or substituting from Eq. (5.26) Ex = By

(5.34)

The z component of Eq. (5.15) does not give anything new since Bz = 0 and there is no spatial variation with respect to x and y. The electromagnetic wave, therefore, has E = (Ex , Ey , 0) and B = (Bx , By , 0) such that  E · B = Ex Bx + Ey By = 0

(5.35)

which shows that the electric vector E is perpendicular to the magnetic vector B and both are perpendicular to the propagation vector k. Such a wave is said to have a transverse polarization since the wave displacement E is transverse to its direction of propagation. We can define the polarization with either of the fields  E and B since they have a definite relationship between them. An electromagnetic wave with k = (0, 0, k), E = (Ex , Ey , 0), B = (Bx , By , 0) is called a plane polarized wave since the field lies in the transverse (x, y) plane (Fig. 5.1). z k x

y

E

E for a )LJXUH  Orientations of the propagation vector k and the electric field  plane polarized electromagnetic wave.

172

Plasmas

An electromagnetic wave with k = (0, 0, k), E = (Ex , 0, 0), B = (0, By , 0) is called a linearly polarized wave. Another possible case of linear polarization is k = (0, 0, k), E = (0, Ey , 0), B = (Bx , 0, 0). Thus there are two linearly polarized waves for a fixed direction of propagation k, Fig.(5.2). z

z k

k

Ex

x

Bx

By y

x

Ey y

E and  B for a fixed k for a )LJXUH  The two possible orientations of the fields  linearly polarized electromagnetic wave. A plane wave propagating in the direction k can, therefore, be more generally described as a superposition of the two independent linearly polarized waves. Let us represent the two linearly polarized waves as  E 1 =ex E1 eikz−iω t

(5.36)

 E 2 =ey E2 eikz−iω t eiφ

(5.37)

with their electric fields pointing in directions ex and ey and have a phase difference φ between them. The amplitudes E1 and E2 are real numbers. The linear combination E of the two waves is then    E = ex E1 +ey E2 eiφ eikz−iω t (5.38) Now for φ = 0, the field E represents a linearly polarized wave with its polarization vector E making an angle θ = tan−1 (E2 /E1 ) with the x axis, Fig.(5.3). The magnitude of the electric field,  E , is (E12 + E22 )1/2 . E becomes For φ = (π /2) and E1 = E2 = E0 , the field   E = E0 (ex + iey ) eikz−iω t

(5.39)

The components of the actual real field are recovered by taking the real part of Eq. (5.39): Ex = E0 cos (kz − ω t) Ey = −E0 sin (kz − ω t )

(5.40) (5.41)

The Radiating Plasmas

173

E2

E ey

θ ex

E1

E ) obtained by the superposition of the two )LJXUH  A linearly polarized wave ( linearly polarized waves,  E 1 and E 2 propagating in phase with each other. The magnitude of the electric field, E0 , is a constant. The electric vector  E makes an angle (θ = −ω t) with the x axis at a fixed point z in space since Ey = − tan ω t. Ex

(5.42)

The electric vector E rotates in a circle of radius E0 in a time period T = (2π /ω ) and since the rate of change of the inclination, θ , of E with the x axis is negative, the rotation is clockwise. Such a wave is said to have left circular polarization when viewed along the z direction (Fig. 5.4). E2

E2

E

ey

ωt ex

E

ey

E1

ωt ex

E1

)LJXUH  A right (anticlockwise) and a left (clockwise) circularly polarized waves ( E ) when viewed along the z direction.

For φ = − π2 and E1 = E2 = E0 , the field  E becomes  E = E0 (ex − iey ) eikz−iω t

(5.43)

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Plasmas

The components of the actual real field are recovered by taking the real part of Eq. (5.39): Ex = E0 cos (kz − ω t)

(5.44)

Ey = E0 sin (kz − ω t)

(5.45)

The magnitude of the electric field E0 is a constant. The electric vector E makes an angle (θ = ω t) with the x axis at a fixed point z in space since Ey = tan ω t. Ex

(5.46)

The electric vector E rotates in a circle of radius E0 in a time period E with the x T = (2π /ω ) and since the rate of change of inclination, θ , of  axis is positive, the rotation is anticlockwise. Such a wave is said to have right circular polarization when viewed along the z direction, Fig.(5.4). For φ = (π /2) but unequal amplitudes E1 and E2 the field  E becomes  E = (ex E1 + iey E2 ) eikz−iω t

(5.47)

The components of the actual real field are recovered by taking the real part of Eq. (5.39): Ex = E1 cos (kz − ω t)

(5.48)

Ey = −E2 sin (kz − ω t )

(5.49)

The field components satisfy the equation of an ellipse since Ex2 Ey2 + = 1. E12 E22

(5.50)

Therefore, the electric vector E traces an ellipse with its principal axes in the x and y directions. The semi-major axis is E1 , the semi-minor axis is E2 . Such an electromagnetic wave is said to be elliptically polarized. Again it could be left elliptically polarized (φ = −π /2) or it could be right elliptically polarized (φ = π /2). It is easy to see that the ellipse would collapse to a line (linear polarization) for φ = 0 and swell to a circle (circular polarization) for E1 = E2 = E0 . Finally it would suffice here to mention that the semi-major axis of the ellipse would get inclined to the x axis by an angle α2 if there is a phase difference of α between E1 and E2 . This completes the discussion of the polarization of the electromagnetic radiation.

175

The Radiating Plasmas

The flux of energy carried by an electromagnetic wave is estimated to be the real part of the complex Poynting vector S defined as   S = 1 c E × B∗ . 2 4π

(5.51)

∗  The where B is the complex conjugate of the magnetic induction B. electromagnetic energy per unit area per unit time is

S = c | E02 | ek . 8π

(5.52)

where we have substituted for B in terms of E from the Ampere law. The direction of flow of energy is along the wavevector kek , ek being the unit vector in the direction of propagation. The time-averaged energy density U of the radiation field is found to be U=

1   ∗   ∗  E ·E +B·B . 16π

(5.53)

U=

1 | E02 | . 8π

(5.54)

Or

Equations (5.52) and (5.54) show that the flow velocity of the energy in vacuum is the speed of light c. Having learnt the essential characteristics such as the phase and group velocities, amplitudes and polarization and energy density and energy flux of electromagnetic waves in vacuum, the next task is to see how these characteristics change when the radiation propagates in a plasma. It is by understanding the generation, the propagation, the absorption and the scattering of the electromagnetic radiation in plasmas that one aspires to know the cosmos.

5.5

Propagation of Electromagnetic Waves in a Plasma

The most important difference that a plasma makes to the electromagnetic radiation is through the presence of electric charges and electric currents. A current is driven through a plasma on the application of an electric field. This is the conduction current J, which is zero in vacuum. The electric field and the current density are related through Ohm’s law, which in its simplified form can be written as J = ΣE

(5.55)

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Plasmas

where Σ is the electrical conductivity of the plasma. Equation(5.55) is just another way of writing the much familiar form V = IR of Ohm’s law where V is the electric potential, I the electric current, and R is the electric resistance. To see what happens when the electromagnetic waves propagate in a plasma, we shall again deploy Maxwell’s equations including the conduction current density this time. Take the curl of Eq. (5.15) and substitute for the curl of B from Eq. (5.11) to get     2 E 1 ∂ 4π Σ  1 ∂       E+ (5.56) −∇ E + ∇ ∇ · E = − c ∂t c c ∂t This is the wave equation in a plasma. It is still a linear equation provided the electrical conductivity is independent of the electric field. We can assume a plane wave, one-dimensional, solution, as before, i.e.,  E = Ex exp (ikz − iω t)

(5.57)

Recalling that the electromagnetic radiation is a transverse wave (∇ · E = k · E = 0) we find

∂ 2 Ex 4π Σ ∂ Ex 1 ∂ 2 Ex − =0 − ∂ z2 c2 ∂ t c2 ∂ t 2

(5.58)

Substitution of Eq. (5.57) in Eq. (5.58) furnishes the dispersion relation:

ω 2 = k2 c2 − 4π iω Σ

(5.59)

To find the electrical conductivity Σ of a plasma, we must determine the conduction current density J driven by the electromagnetic wave while traversing a plasma. The current density J is defined as J = ni qiV i + ne qeV e

(5.60)

where (n, q, V ) represent, respectively, the density, the electric charge, and the velocities of the plasma particles, the ions, and the electrons. We take a hydrogen plasma for which qi = e, qe = −e, ni = ne = n. The velocities are to be determined from Newton’s second law of motion. The forces acting on the plasma particles in the presence of an electromagnetic wave with fields (E, B) are the Lorentz force and the force due to the collisions among the particles. The equation of motion of the electron fluid is   V e × B ∂ V e  ei + n e me = −ene E +F (5.61) ∂t c

The Radiating Plasmas

177

where  F ei is the collisional force on the electrons due to the electron–ion collisions. The collisional force causing the momentum transfer due to collisions is modelled by assuming: (1) a constant collision rate (νei−1 ) and (2) that the collisional force is proportional to the relative velocity between the electrons and the ions. Thus,    (5.62) F ei = −ne me νei V e − V i The equation of motion of the ion fluid is     V i × B ∂ V i  n i mi = eni E + − ni mi νie V i − V e ∂t c

(5.63)

where the last term on the right-hand side is the collisional force on the ions due to ion–electron collisions, which is proportional to the relative velocity between the ions and the electrons. Observe the symmetry between the electron and the ion equations of motion. The collisional force is momentum conserving by which is meant that ne me νei = ni mi νie . We assume that the electromagnetic wave is linearly polarized with  E = (Ex , 0, 0) and it is not strong enough to accelerate electrons and ions to relativistic velocities so that Ve νei , one finds 2 ω 2 = k2 c2 + ωep

(5.76)

The dispersion relation of an electromagnetic wave in a plasma is definitely different from the one in vacuum and the difference depends on the plasma density n, the electron charge e, and the electron mass me . Clearly the phase and the group velocities would also be different. The phase velocity of an electromagnetic wave in a plasma  1/2 2 ωep ω Vp = = c 1 + 2 2 k k c

(5.77)

is larger than c. There is no violation of special theory of relativity as the electromagnetic energy is transported by the group velocity and not the phase velocity. The group velocity of an electromagnetic wave in a plasma −1/2  2 ωep dω = c 1+ 2 2 Vg = dk k c

(5.78)

is smaller than c and all is well. It is interesting to note that the product VpVg = c2 is the same as it is in a vacuum.

(5.79)

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Plasmas

One can also define a refractive index μ = (kc/ω ) for a plasma medium as the ratio of the phase speed in vacuum to the phase speed in a plasma. It is convenient to write the square of the refractive index as

μ2 ≈ 1 −

2 ωep k2 c2

μ2 ≈ 1 −

2 ωep ω2

for ωep > νei .

(5.81)

and

The refractive index of a plasma is less than unity. This is new since most material media have indices of refraction larger than unity. Moreover, a plasma is a dispersive medium since its refractive index varies with the frequency of the electromagnetic radiation. The refractive index is nearly one at high frequencies, ω >> ωep ; a plasma is transparent to electromagnetic radiation at frequencies much higher than the electron plasma frequency. The refractive index is zero at ω = ωep ; a plasma reflects radiation at the plasma frequency. This is the mechanism of short-wave transmission around the world, by the reflection of radio waves in the ionospheric plasma. The typical electron density in the ionosphere is in the range of 104 − 106 electrons per cubic centimeter and the corresponding electron plasma frequency lies in the range of 6 × 106 − 6 × 107 s−1 . Thus the ionosphere reflects radio waves of frequencies of tens of megahertz. We shall see the effect of including the electron–ion collisions in the dispersion relation, a little later. The square of the refractive index becomes negative and refractive index becomes purely imaginary at ω < ωep . The propagation vector k is also purely imaginary. Let us write k ≡ ±ik. The radiation electric field, therefore, becomes    E = Ex exp ±i2 kz − iω t (5.82) Or  E = Ex exp (∓kz − iω t)

(5.83)

The field decays exponentially as it enters the plasma. The penetration depth zd of the field is determined from zd k ≈ 1 Or

(5.84)

181

The Radiating Plasmas

 2 −1/2 c zd ≈ c ωep − ω2 ≈ ωep

(5.85)

The low-frequency radiation cannot propagate through the plasma. The fields decay exponentially within a distance zd from the surface of the plasma. The distance zd is called the skin depth of the plasma and it is predominantly due to the electrons.

5.6

Absorption of Electromagnetic Waves in a Plasma

The simplest process by which the electromagnetic radiation is absorbed in a plasma is the electron–ion collisions. It has been seen earlier that in the presence of radiation the electrons oscillate with the frequency of the radiation while also undergoing collisions with the electrons. The oscillatory energy of the electrons is transferred to the ions via the electron–ion collisions. The electromagnetic radiation loses energy and the wave is damped. We wish to find out the absorption rate of the electromagnetic radiation of frequency ω as it propagates, say, in the z direction in a plasma. This can be easily seen by recalling the dispersion relation, Eq. (5.74), as k2 c2 = ω 2 −

2 ωωep (ω + iνei)

(5.86)

The spatial absorption rate is related to the imaginary part of the wavevector k. Let us write the real (R) and the imaginary (I) parts of k as k = kR + ikI

(5.87)

and substitute in the dispersion relation Eq. (5.86). Separating the real and the imaginary parts, we find: 2 kR2 c2 = ω 2 − ωep

and

 kI = kR

2 ωep 2 ω 2 − ωep

(5.88) 

ν  ei

2ω

(5.89)

where we have neglected a term in kI2 . One notices that the absorption rate increases as ω approaches ωep . In fact, this relation breaks down at ω = ωep . That is where the nature of the electromagnetic wave undergoes a drastic change. These changes can be fathomed only when one is prepared to venture into the realm of wave–wave interaction processes,

182

Plasmas

2 , the absorption rate which we shall not. Alternatively, for ω 2 >> ωep simplifies to   2 ν  ωep ei . (5.90) kI = kR ω2 2ω

It is important to keep these assumptions in mind while estimating the numerical value of kI for a particular case. The wave undergoes exponential damping as it propagates since  E = Ex eikR z−iω t e−kI z

(5.91)

kI is the absorption rate per unit length. The characteristic distance d over which the amplitude of the wave falls by a factor e is d = kI−1

(5.92)

It is seen that for a given electron plasma frequency (read electron density) and the collision frequency, the high-frequency waves suffer less damping than the low-frequency waves. The quantitative dependence of the damping rate on the plasma parameters, the density and the temperature, can be found only after we have learnt how to determine the electron–ion collision frequency. There is another way of estimating the damping rate of radiation propagating in a plasma. As mentioned earlier, the radiation loses energy by imparting the oscillatory energy to electrons that transfer it to ions via collisions. Therefore, we can write  ne meVe2 d E2 ≈ νei (5.93) dt 8π 2 where (E 2 /8π ) = (EE ∗ /8π ) is the energy density of radiation. The velocity of electrons is determined as  Ve2

=

eE me ω

2 (5.94)

The absorption rate per second, γA , is defined as

γA = Or

 1 d 2 E /8π (E 2 /8π ) dt

(5.95)

The Radiating Plasmas

γA =

2 νei ωep ω2

183 (5.96)

Observe the similarity with the expression for kI , Eq. (5.90). But the two are different, one is the damping rate per second and the other the damping rate per unit length. There is also a difference of a factor of 2. Here γA is the absorption rate per second of the energy density (E 2 /8π ) whereas kI is the absorption rate per unit length of the amplitude E. The absorption rate per unit length of E 2 is 2kI since E 2 = Ex2 e−2kI z

(5.97)

And the two rates are related as γA ≈ 2kI c where the approximate relation ω ≈ kR c has been used, which is valid for high values of ω . Thus the two approximate methods of calculating the absorption rate furnish nearly equal values. The collisional absorption of radiation is the first process one employs, for example, to study the heating of plasmas by lasers. This process is also known as the inverse bremsstrahlung, the bremsstrahlung being the process by which radiation is emitted. There are other more complex and more efficient absorption processes based on the existence of plasma instabilities, a rather advanced topic that will not be discussed here.

5.7

Electron–Ion Collision Frequency

The electrons and the ions in a plasma are in incessant motion. However, they are never completely free to move around. There is the Coulomb force between them. The electrons being attracted by the positively charged ions would often traverse in the close vicinity of the ions. Their trajectories are then defleced by the Coulomb force. Such a process is described as the electrons undergoing collisions with the ions. They, however, do not collide like billiard balls. The electrons being highly energetic, at the temperature of the plasma, would not just go and sit over the ions but rather undergo acceleration and deflection (Fig. 5.5). The collision frequency νei is defined as the number of collisions that an electron suffers with the ions of number density ni in a time duration of one second. If the collision cross section is χc and the electron velocity is Ve , the collision frequency is given by

νei = ni < χcVe >

(5.98)

where the angular brackets represent the average over the velocity distribution of electrons. The collision cross section is the square of the

184

Plasmas

electron

electron (-e) b ion (+e) )LJXUH 

Deflection of an electron in the Coulomb field of an ion.

impact parameter b, which is the closest distance of approach of an electron to the ion. The impact parameter can be approximately estimated by equating the kinetic energy and the potential energy of an electron in the Coulomb field of an ion. Thus 1 e2 meVe2 ≈ 2 b

(5.99)

Or b≈

2e2 meVe2

(5.100)

and the cross section is 4e4 m2e Ve4

χc = b2 ≈

(5.101)

The collision frequency is therefore given by 4e4 > m2e Ve3

νei = ni
=

3KBT me

(5.103)

The collision frequency becomes 

νei =

3 me

1/2

ni e4 (KBT )−3/2

(5.104)

The Radiating Plasmas

185

The direct dependence of νei on the number density ni of the ions is understandable: the more the number of ions, the more the number of collisions. The dependence on the temperature is rather counterintuitive. One would expect that the number of collisions would increase with an increase in temperature. But it is not so. This unexpected behavior is due to the nature of the Coulomb force. The higher the temperature the higher the electron velocity and the shorter is the duration of time spent by the electron in the field of influence of the ion and hence the lower the impulse felt by the electron. This results in a smaller momentum transfer to the ions and hence the lower collision frequency. Thus a high-temperature plasma is nearly collisionless. One way of putting it quantitatively is that the electron–ion collision frequency is much smaller than the electron plasma frequency ωep . It also follows then that there is very little absorption of an electromagnetic wave in a high-temperature plasma. This is the difficulty in trying to heat a plasma by the absorption of a laser radiation. One has to devise other ways of heating plasmas in a laser-fusion application. The solar corona and the solar wind are the other familiar examples of collisionless plasmas. The approximate (Eq. 5.104) expression for the collision frequency is made more accurate by including the effects of small angle deflections of the electron trajectory, which may increase the collision frequency by a factor of ten or so for most plasmas one encounters in the cosmos or in a laboratory. Now that we have a fair idea of radiation and its propagation and absorption in a plasma, we can embark upon the generation of radiation. And it is wise to first become familiar with the generation of radiation by a single particle. The role of a plasma as a coherent and intense source of radiation could then be better appreciated.

5.8

Generation of the Electromagnetic Radiation

Electromagnetic radiation is composed of electric and magnetic fields. Only electrically charged particles can create electric and magnetic fields. A charged particle at rest creates an electric field  E around it. The field at a point P, a distance r from the particle of charge q, is given by the Coulomb law as q  E = 2 e r

(5.105)

→ where − e is the unit vector pointing from the position of the charge to the point P at which the field is determined (Fig. 5.6).

186

Plasmas

P e

r

q )LJXUH 

E at a point P due to a static charge q. Electric field 

Since the charge is not in motion, there is no conduction current and no magnetic field either, according to the Ampere law. There is no displacement current since the electric field does not vary with time. So there is no electromagnetic radiation. A static electric charge does not produce any radiation. A charged particle moving with a uniform velocity does not produce any radiation either. It is only when the electric charges are in accelerated motion that they produce radiation. Let a particle of charge q and mass m have an acceleration a. The Newton law relating the force and the acceleration determines the electric field produced by the accelerated charged particle as ma = qE

(5.106)

where the magnetic force has not been included. This is justified if the particle velocity remains much less than the speed of light c. Thus the electric field is proportional to the acceleration of the charged particle. The electromagnetic flux or the electromagnetic energy density, being proportional to the square of the electric field, is proportional to the square of the acceleration of the charged particle. This is the essence of the Larmor radiation formula, which will be derived a little later.

5.9

Radiation from an Oscillating Electric Dipole

Let us first study the generation of radiation from an oscillating charge  These density σ associated with which is an oscillating current density J. can be written as

σ (r, t) = σ (r)e−iω t

(5.107)

J (r, t) = J (r) e−iω t

(5.108)

where ω is the frequency of oscillation and also the frequency of the emitted radiation. As always, the physical quantities are extracted by taking the real parts of the equations. It is instructive to calculate the electromagnetic potentials from which the fields would be determined.

The Radiating Plasmas

187

It is shown in Chapter 6 that the vector potential, in the Lorentz gauge, can be expressed as   



J r , t  |r −r |  A (r, t) = 1 d3 r dt  t δ + − t (5.109) c c |r −r | After substituting for the current density and integrating over time t  making use of the Dirac delta function, the vector potential takes the form  

 J r ik|r−r | −iω t A (r, t) = 1 d3 r e (5.110)  e c |r −r | where the dispersion relation, ω = kc, of an electromagnetic wave in vacuum has been used. The magnetic induction is determined from the curl of the vector potential as B = ∇ × A

(5.111)

and the electric field from the Maxwell equation  ∇ × B = 1 ∂ E = −ikE. c ∂t

(5.112)

Here the term containing the conduction current density has not been included in the Maxwell equation. This is done when the source of radiation, the charge, and the current densities are far removed from the region in which the radiation is determined. The radiation is essentially determined in a source-free region or in a vacuum and hence ω = kc. As a matter of fact, one can identify three regions in which the properties of the electric and the magnetic fields are distinctly different. We have the source size ls, the wavelength of the radiation λ , and the distance r at which the radiation is determined. The source size is assumed to be the smallest, i.e., it is much smaller than the wavelength λ = (2π c/ω ). The three regions are then defined in comparison with the wavelength. The near static region is defined by r > 1 and the exponential function in Eq. (5.110) oscillates violently and determines the character of the fields. In this region the denominator in Eq. (5.110) is simplified as   1/2 r2 r ·r ≈ r −n ·r ≈ r 1+ 2 − 2 |r −r |= r2 + r2 − 2r ·r 2r r

(5.116)

where n is the unit vector in the direction of the vector r. The expansion in Eq. (5.116) is permissible since the size of the source ls ≈ r λ ), retaining terms only of the order of (1/r), the magnetic induction reduces to ikr

B = k2 e

r

  n × P e−iω t

(5.135)

and the electric field becomes   eikr  E = −k2 n × n × P e−iω t r

(5.136)

The Poynting vector describing the flow of energy per unit area per unit time is by definition   S = c E × B 4π

(5.137)

The time-averaged electromagnetic power L received per unit solid angle at the distance r is defined as  dL c ∗ (5.138) Re r2n · E × B = dΩ 8π ∗ where the area element vector at r is defined as dA = r2 dΩn and B is the complex conjugate of the complex vector  B. The time average is performed as

1 T

T 0

eiω t e−iω t dt = 1

(5.139)

Substituting for the E and B fields for the radiation zone, the radiated power becomes     dL c 4  k n · n × P × n × n × P = dΩ 8π which, on using the vector identity,     X · Y × Z = X ×Y · Z

(5.140)

(5.141)

can be expressed as   dL c 4 k |n × n × P |2 = dΩ 8π

(5.142)

192

Plasmas

Let θ be the angle between the vectors n and  P (Fig. 5.7). The cross product can be simplified as     n × n × P = n · P n − P = P cos θn − P cos θn − P sin θn p = −P sin θn p (5.143)

np

P n

q

)LJXUH  Orientations of the Poynting vector S ≡ n ≡ k, the electric dipole moment  P and the vector n p .

where the vector P has been decomposed into components along n and perpendicular to the vector n, n p . The radiated power is given by dL c 4 2 2 = k P sin θ dΩ 8π

(5.144)

which describes the angular distribution of the power. The emitted power is maximum for θ = (±π /2) or in directions perpendicular to the direction of the electric dipole moment vector  P and no power is emitted in the direction of P(θ = 0). This angular distribution is known as the dipole pattern since the emission is maximum at the two poles θ = ± π2 . The total power can be estimated by integrating over the solid angle dΩ = sin θ dθ dφ as L=

2π 0

dφ

π 0

dθ sin θ

dL c 4 2 k P (2π ) = dΩ 8π

1 −1

 ck4 2 P 1 − μ 2 dμ  = 3

(5.145)

where the substitutions μ  = cos θ and dμ  = − sin θ dθ have been used to evaluate the integral over θ . The polarization state of the radiation can be retrieved from the direction of the electric field. For example, if the electric dipole moment is in the z direction and the radius vector rn is in the x direction so that n · P = 0, the electric field is in the direction of the electric dipole moment P and the radiation is linearly polarized. In conclusion, an electric dipole moment oscillating at a frequency ω produces linearly polarized radiation at the same frequency ω with the dipole angular distribution pattern and total power proportional to the fourth power of the radiation frequency ω since ω = kc.

The Radiating Plasmas

5.10

193

Radiation from an Accelerated Single Charged Particle

A point charge such as an electron in acceleration emits electromagnetic radiation. The properties of the emitted radiation such as the angular and the spectral distribution of power, the total power and the polarization are uniquely related to the trajectory of the point charge. Before one can learn these characteristics of the radiation, the electric and the magnetic fields due to a single accelerated charged particle must be determined. The charged particle could be energetic enough to be in the relativistic motion. The four-vector relativistic notation will be used in this section. The vector and the scalar potentials due to a localized charge and current distribution without the surface effects due to boundaries can be written as    



1 R 3   Jα r , t  d r dt δ t + −t (5.146) Aα (r, t) = c R c

where R = r −r and the index α = 1, 2, 3, 4, the first three values represent the three components of a vector and the fourth represents a scalar. Thus A1 , A2 , A3 are the three components of the vector potential A and A4 = iφ stands for the scalar potential φ ; J1 , J2 , J3 are the three components of the current density vector J and J4 = iσ with σ as the charge density. The Dirac-delta function takes care of the causality, i.e., the signal emitted at time t  is received at a distance R at a later time t such that t = t  + (R/c). This is also known as the retarded behavior and results due to a finite travel time (R/c) taken by the signal. The current density distribution due to a point charge at the position re (t) with the four velocity Vα can be expressed as Jα (r, t) = eVα δ [r −re (t)]

(5.147)

with e the electric charge of the point charge. Ve( t )

e

n(t )

R( t )

P

re (t ) r(t) O )LJXUH  Trajectory of the accelerated charged particle e and the point P where the emitted radiation is observed.

194

Plasmas

Here V1 ,V2 ,V3 are the three components of the velocity and the fourth component V4 = ic. The charge density is found from J4 = iσ = ieδ [r −r e (t)]

(5.148)

The spatial integration in Eq. (5.146), by substituting for Jα , gives 

Vα (t  ) e R (t  )  t dt  δ + − t (5.149) Aα (r, t) = c R (t ) c where R (t  ) = |r −re (t )|. The integration over t  can be carried out by using the following property of the Dirac delta function, viz.,  

      g (x )  dx g x δ f x − a = (5.150) df dx

f (x )=a

The function f in the delta function of Eq. (5.149) can be easily identified to be   R (t  ) f t = t + c

(5.151)

and its derivative with respect to t  is dR df = 1+  dt cdt 

(5.152)

The second term can be simplified by noting     R t  =r −re t 

(5.153)

  dR d    d     R t =n  = −  re t  = −V e t  dt  dt dt

(5.154)

and

where n = (R/R) is the unit vector directed from the position of the charge re to the observation point r. Taking the scalar product with n on both sides, one gets   dR = −n · V e t  dt  and Eq. (5.152) becomes

(5.155)

The Radiating Plasmas

df n · V e (t  ) ≡Γ = 1−  dt c

195 (5.156)

The function Γ reduces to unity for nonrelativistic speeds, i.e., for V e dτ c dτ

(5.200)

Thus neglecting the energy derivative term, the radiated power can be written as  2 dpe 2e2 2e2 = (ωγ pe )2 (5.201) L= 3m2 c3 dτ 3m2 c3 where ω is the circular frequency of the charged particle. The radius Rc of the circular orbit is given by 2π Rc = Ve

2π ω

(5.202)

The power can be expressed as L=

2ce2 γ 4 β 4 3R2c

(5.203)

The radiative energy δ U emitted by the charged particle in one revolution or in time duration (2π /ω ) is

δU = L

2π 4π e2 γ 4 β 3 = ω 3Rc

(5.204)

Putting in the numerical values of the various constants, we find that an electron of energy Ee = 0.5γ MeV loses

δ U ≈ 10−18

γ4 erg Rc

(5.205)

of energy in the form of radiation. Here Rc is in centimeters. For a typical low-energy circular accelerator γ = 103 and Rc = 100 cm and the typical energy lost by an electron in one revolution is about 0.6 KeV. Thus the electron loses a fraction

δU ≈ 10−5 Ee

(5.206)

of its energy in the form of radiation. We learnt that a charged particle executes circular motion in the presence of the Lorentz force caused by a magnetic field B. The frequency of

The Radiating Plasmas

203

the circular motion is the cyclotron frequency ω = qB/mc where q and m are the charge and the mass of the particle. The radius of the circular orbit Rc = Vq /ω . The charged particle emits radiation called the synchrotron radiation with a power determined by Eq. (5.203) as L=

2ce2 γ 4 β 4 3R2c

(5.207)

The radiative energy emitted in one revolution of the charged particle is

δU = L

2π 4π e2 γ 4 β 3 = ω 3Rc

(5.208)

In a typical low-energy synchrotron, Rc is about 100 cm, electron energy is about 300 MeV and the corresponding Lorentz factor is γ = 600. It turns out that the electron loses about a KeV of energy per revolution in the form of radiation.

5.12

Radiation Spectrum

An accelerated charged particle emits radiation at different frequencies depending on its state of acceleration. The spectral composition of radiation can be determined by the use of the Fourier transform of the timedependent electric field generated by the charged particle. The power radiated per unit solid angle can be written as (Eq. 5.183): dL c | RE |2 =| A (t) |2 = dΩ 4π

(5.209)

where A (t) =

 c 1/2   RE 4π ret

(5.210)

here A(t) is written in observers’s time frame since we wish to know the observed radiation spectrum. As discussed earlier, the spatial extent of the radiating charge density region is much smaller than the distance of the observer from the radiating region. The total radiated energy per unit solid angle is obtained by integrating over time as dU = dΩ

∞ −∞

dt | A (t) |2

(5.211)

204

Plasmas

The fourier transform A (ω ) of A (t) is, by definition, given by A (ω ) = √1 2π

∞ −∞

dt A(t)eiω t

(5.212)

along with the inverse relation A (t) = √1 2π

∞ −∞

dω A (ω ) e−iω t

(5.213)

Now substitute for A (t) in Eq. (5.211) to find dU 1 = dΩ 2π

∞ −∞

dt

∞ −∞

dω

∞ −∞

∗

dω A

  ω  · A(ω ) ei(ω −ω )t

(5.214)

Integration over t gives the Dirac-delta function since

   1 ∞ δ ω − ω = dtei(ω −ω )t 2π −∞

(5.215)

Using the Dirac-delta function to integrate over ω  gives dU = dΩ

∞ −∞

dω | A (ω ) |2

(5.216)

Compare the forms of the total energy per unit solid angle in Eqs. (5.211) and (5.216). Keeping the frequencies of the radiation to be positive, one can define the energy radiated per unit solid angle per unit frequency interval as dU = dΩ

∞ 0

dω

dI (ω ) dΩ

(5.217)

with dI (ω ) =| A (ω ) |2 + | A (−ω ) |2 dΩ

(5.218)

∗ It is clear from Eq. (5.213) that for real A (t), A (ω ) = A (−ω ) and the radiated energy per unit solid angle per unit frequency interval becomes

dI (ω ) = 2 | A (ω ) |2 dΩ

(5.219)

Taking the radiative part, E, from Eq. (5.177) and using Eq. (5.210) for A, the Fourier transform, Eq. (5.212) can be evaluated as

The Radiating Plasmas

A (ω ) =



e2 8π 2 c

1/2

∞ −∞

iω t

dte



   n    × n − β × β Γ2 ret

205

(5.220)

As before ret reminds us that the expression is to be evaluated at t  = t − (R (t  ) /c). Changing the integration variable t to t , one gets A (ω ) =



  1/2 ∞ e2 R (t  )   t dt exp i ω + 8π 2 c c −∞     n  β × β  ×  n − Γ2 ret

(5.221)

Since the point of observation of radiation is far from the region of acceleration of the charged particle, the vectors R andr are approximately parallel and in the direction of n. The vector n can be assumed to remain constant in time. Recalling Eq. (5.165), the distance R can be approximated as    1/2     ≈ r −n ·re t  R t  =|r (t) −re t  |= r2 + re2 − 2r ·re

(5.222)

where it is clear that R and r are both much larger than re . Equation (5.221) takes the form A (ω ) =



  1/2 ∞ e2 r n ·re (t )   t dt exp i ω + − 8π 2 c c c −∞     n  β × β  ×  n − Γ2 ret

(5.223)

It can be easily checked that      n  β × β  = d n × n × β ×  n − Γ2 dt  Γ

(5.224)

Integrating Eq. (5.223) by parts fetches A (ω ) = −



1/2 ∞    e2  β  n ×  n × dt 8π 2 c −∞   r n ·re (t  )  exp iω t + − c c

(5.225)

where the first term of the integration vanishes at the limits of integration and

206

Plasmas

   d r n ·re (t )  exp iω t + − = dt  c c   r n ·re (t  )  iω Γ exp iω t + − c c

(5.226)

has been used. The energy radiated per unit solid angle per unit frequency is now found to be   

∞   dI (ω ) e2 ω 2 n ·re (t  )    dt exp iω t − n × n × β |2 (5.227) = | 2 dΩ 4π c −∞ c The state of polarization of the emitted radiation is given by the direction of the electric field. The polarization should be determined from the vector integral before taking the absolute square.

5.13

In a Plasma

There are a large number of charged particles in acceleration in a plasma. Therefore, the total radiation emitted can be estimated by adding the contribution of each particle. In other words, the single particle term  n ·re (t  )  eV e exp −iω c

(5.228)

in Eq. (5.225) must be replaced by the sum  n ·re j (t  )  ∑ e jV e j exp −iω c j=1 N

(5.229)

where N is the total number of charged particles, e j and V e j being the charge and the velocity of the particle j. In the continuum limit, i.e., when there is a continuous distribution of charge instead of a large number of discrete particles, the sum in Eq. (5.229) can be replaced by an integral over volume as  

N n ·re j (t  ) n ·x(t  ) 3   e exp −i ω d x J ( x, t)exp −i ω V → (5.230) j e j ∑ c c j=1 where J is the current density. Thus the emitted radiation per unit solid angle per unit frequency becomes

The Radiating Plasmas

 

∞ dI (ω ) ω2 n ·x (t  )    = 2 3 | dt exp iω t − dΩ 4π c c −∞      |2 n × n × J x, t 

5.14

207

(5.231)

Radiation from Collisions between Charged Particles

The charged particles in a plasma are in a continuous state of collisions under the influence of the Coulomb force that they exert on each other. An electron gets accelerated in the attractive electrostatic force due to a positive ion and so does the positive ion in the field of an electron. Both the charges are accelerated and both must emit radiation. Since the heavier of the two charges would have a lower acceleration, it would generate much less radiation than the lighter particle. This is true of, for example, an electron–proton plasma. To estimate the amount of radiation generated in a collision process, we shall make certain simplifying assumptions. Recall Eq. (5.223). We shall now simplify the form of the function A for non-relativistic motion so that terms only of first order in β are retained since β bmin. This gives an upper limit to the radiation frequency

ωmax ≈

5.17

me v3e 2Ze2

(5.249)

Bremsstrahlung in a Thermal Plasma

It has been assumed above that all the electrons move with a common speed ve . In a plasma at a temperature T , however, the electrons have a thermal or a Gaussian distribution F (v) of speeds defined as  F (v) = ne

me 2π KBT

3/2



me v2 exp − 2KBT

(5.250)

for an isotropic distribution. The function Fdv gives the number of electrons per cubic centimeter with velocity between v and v +dv. The total number of electrons per cubic centimeter are found from the integral ne =

∞ −∞

d3 vF (v)

(5.251)

For an isotropic distribution, the volume element in the velocity space is given by d3 v = 4π v2 dv. The average of any quantity g, which is a function of v, is defined as < g (v) >=

∞

2 −∞ dv4π v F (v) g (v) ∞ 2 −∞ dv4π v F (v)

(5.252)

The process of taking the velocity average of the radiated energy, Eq. (5.247), over the distribution function, Eq. (5.249), would entail integrals of the form

212

Plasmas

∞



me v2 dvv exp − 2KBT −∞



2

1 2KB T = v me =

∞



me v2 d 2KBT vmin



 me v2 exp − 2KB T

 hν 2KBT exp − me KB T

(5.253)

and   ∞   1  1/2  me v2 d 2 dvv2 exp − d u v exp −uv = −2 2KBT du 0 u1/2 −∞    −3/2 me d π 1/2 1/2 (5.254) =π = −2 du u1/2 2KBT

∞

where u = (me /2KBT ) has been used. In the integral in Eq. (5.253) the lower limit of integration over v is vmin where vmin is the minimum electron speed required to generate a quantum of radiation, a photon of energy h¯ ω , i.e., me v2min = h¯ ω 2

(5.255)

Here h = 2π h¯ is the Planck constant and ω is the radiation frequency. For velocities lower than vmin there is no emission. In the integral in Eq. (5.253), there is no such requirement. Thus replacing the electron flux ne ve by vF (v) in Eq. (5.247) and integrating over the electron velocities, one finds emission from a plasma at temperature T to be dU 64π Z 2 e2 ni ≈ dV dω dt 3



e2 me c2

2

 c 2 me v3 dv v F (v) ln v 2Ze2 ω vmin ∞

(5.256)

Putting in the value of the integral, the emitted radiative energy from a thermal plasma at temperature T is found to be 16Z 2 e2 neni dU ≈ dV dω dt 3



e2 me c 2

2

  h¯ ω me v3e < ln > exp −  1/2 KB T 2Ze2ω 2π KB T c

me

(5.257) The velocity average of the logarithmic factor is known as the gaunt factor and its value of unity at h¯ ω ≈ KB T is considered to be a good approximation. The emission drops exponentially at h¯ ω > KBT .

The Radiating Plasmas

213

The bremsstrahlung is an integral part of the radiation mechanisms of every cosmic source. The plasma by virtue of being hot always produces bremsstrahlung radiation whatever else may or may not happen. The T −1/2 dependence along with the exponential fall with the frequency is the signatures of this process. It is by noting these features that the bremsstrahlung is identified from the host of other radiation processes that might be operative simultaneously in a cosmic object.

5.18

Scattering of Radiation by Plasma Particles

Electromagnetic radiation is scattered by electrically charged particles such as the electrons and the ions of a plasma. The process of scattering can change, the direction of propagation, the frequency, the polarization, and the intensity of the incident radiation. A scattering process can also be considered as a radiative process. The incident radiation exerts a force on a charged particle, which gets accelerated and generates radiation. Here, we shall consider a simple case of scattering of radiation by electrons.

5.19

Thomson Scattering

The acceleration of an electron by the force exerted by the radiation, the Lorentz force, can be written as   V × B dV + = −e E (5.258) me dt c where E and B are, respectively, the electric and the magnetic fields of the linearly polarized radiation. Let the radiation be weak enough so that the motion of the electron remains nonrelativistic. We can then ignore the magnetic force. Assuming a plane wave form of the radiation field as  E = ε E0 sin ω0 t under the dipole approximation (kr = NL2m f p

(5.274)

218

Plasmas

where Li is the distance covered in the ith scattering and the mean value of which is the mean free path Lm f p . The average distance covered in any one scattering is the same. Thus the mean distance traversed by a photon is proportional to the square root of the number of scatterings and L = N 1/2 Lm f p

(5.275)

The average of the cross terms in Eq. (5.274) vanishes for isotropic scattering or for symmetric scattering. If a photon travels a distance L before escaping from the medium, i.e., L = R, the number of scatterings is found to be N=

R2 = ι2 L2m f p

(5.276)

since ne χs = L−1 m f p . One should realize that the (vector) sum of the distances covered in a large number of scatterings is zero. Therefore, we considered the mean of the square of the distance. However, for small N or in other words for small optical depth, the sum of the (vector) distances does not vanish. We can write the total distance covered by a photon in a small number of scatterings as L = NLm f p

(5.277)

and thus for L = R N=

R =ι Lm f p

(5.278)

For most purposes it suffices to take the larger of the ι and ι 2 for the value of N. The intensity of the scattered radiation can be determined by the techniques of radiative transfer, which will not be discussed here. So we have learnt a way to include the effect of scattering or emission of radiation from a plasma. But this is not all. The most spectacular effect of scattering of radiation in a plasma arises when the electrons and the ions in a plasma collectively scatter the radiation. This is also called the coherent scattering. The collective behavior of a plasma is best manifested through the excitation of plasma waves, for example, the electron plasma wave near the electron plasma frequency ωep and the ion acoustic wave near the ion plasma frequency ωip discussed in Chapter 4. The electromagnetic radiation is now scattered by these plasma waves instead of by single charged particles, electrons, and ions. The process of scattering becomes a wave–wave scattering process involving three or more number of waves.

The Radiating Plasmas

219

One of the most important three wave scattering processes is the Raman scattering. In this process, an incident electromagnetic wave of frequency ω0 is scattered by an electron plasma wave of frequency ωe to generate the scattered wave at the frequency ω0 ± ωe (Fig. 5.12). ω e , ke

we, k e w 0 – we, k 0 – k e

w0, k0 )LJXUH 

w0 + we, k 0 + ke w0, k 0

Raman Scattering of radiation.

Thus the scattering agency, which is an electron in the Thomson scattering and the Compton scattering, is replaced by an electron plasma wave. This process turns out to be much more efficient than the single particle scattering, which is essentially an incoherent scattering by the plasma electrons. The scattering process is governed by the momentum and the energy conservation laws as indicated in Fig.(5.12). There are new features associated with the Raman scattering such as the scattered radiation could be at much larger frequency if the electron plasma wave has sufficient energy. The scattering also produces a change of polarization of the incident radiation. Since waves suffer damping in a plasma, the damping of all the three waves must be taken into account. The inclusion of the damping mechanisms brings in a threshold for the intensity of the incident radiation below which the process would not take place. The three wave scattering processes are nonlinear in nature and a plasma under the circumstances can exhibit completely unexpected behavior for large intensities of the incident radiation. A plasma that reflects a low-intensity incident radiation of frequency near the electron plasma frequency can turn into a completely absorbing medium for large-intensity incident radiation (Fig. 5.13). we wi w0 > we )LJXUH  frequency.

Nonlinear absorption of an intense radiation near the electron plasma

220

Plasmas

This happens because an intense electromagnetic wave near the electron plasma frequency decays into the electrostatic electron plasma and the ion acoustic waves both of which cannot leave the plasma. The electrostatic waves transfer their energies to the plasma particles via wave–particle interaction processes. Thus the incident radiation is absorbed by the plasma. This process is known as the parametric decay instability. It is one of the most efficient ways by which a laser can be absorbed in a plasma. Alternatively, a plasma that is completely transparent to a low-intensity incident radiation of frequency much larger than the electron plasma frequency can turn into a reflecting plasma at high intensity of the incident radiation (Fig. 5.14). w 0 >> w e

we

ws )LJXUH  Nonlinear reflection of intense radiation at a frequency much larger than the electron plasma frequency.

This happens because an intense electromagnetic wave of frequency much larger than the electron plasma frequency decays into another electromagnetic wave (ωs ) and an electrostatic electron plasma wave (ωe ) or an electrostatic ion acoustic wave (ωi ); the former process is called the stimulated Raman scattering and the latter as stimulated Brillouin scattering. These processes have the maximum cross section for scattering in the direction opposite to the direction of propagation of the incident radiation and hence results in reflection. The electrostatic waves are absorbed in the plasma. These nonlinear radiation–plasma interaction processes have been observed in laser–plasma experiments. Most of these processes have also been shown to be operative in high-energy cosmic objects such as the quasars. The many and varied characteristics of the solar radio bursts find their explanations in the nonlinear plasma–radiation interactions. The theoretical development of these nonlinear phenomena is beyond the scope of this book.

5.21

Summary

It is amply clear that the electromagnetic radiation and plasmas share an intimate relationship. The essential characteristics of radiation in a vacuum have been reviewed before considering the propagation of ra-

221

The Radiating Plasmas

diation through a plasma. A plasma is a dispersive medium with its refractive index a function of the frequency of radiation passing through it. A single charged particle in acceleration generates radiation. However, more often than not we need to find the radiation generated by a plasma at a given temperature T . This is the process of bremsstrahlung wherein electrons undergoing acceleration in the Coulomb field of the ions produce radiation. It is one of the most prevalent radiation mechanisms in the cosmos. Scattering of radiation by a single charge, the Thomson scattering, is another common process that can polarize an initially unpolarized radiation without any spectral change. The relativistic version of the Thomson scattering is the Compton scattering in which the radiation loses energy to the scattering electron. The frequency of the scattered radiation is lower than that of the incident radiation. In the inverse Compton scattering, however the radiation gains energy from the highly relativistic electrons. The frequency of the scattered radiation can be much larger than that of the incident radiation. The nature of these scattering processes changes drastically in a plasma. Since the collective behavior of a plasma manifests itself in the form of the excitation of waves, there is scattering among the waves in addition to the wave–particle scattering. The wave–wave scattering lies in the domain of the nonlinear plasma–radiation interaction. These processes exhibit completely novel features of the plasma–radiation system.

Problems 5.1 Estimate the low-frequency ionospheric plasma.

electrical

conductivity

of

the

5.2 Estimate the electron skin depth of the ionospheric plasma. 5.3 What is the absorption rate of a radiation beam of 30 MHz frequency in the ionosphere? 5.4 Derive Eq. (5.209). 5.5 Determine the radiation energy per unit volume per unit frequency interval per unit time (Eq. 5.247) for a screened Coulomb potential (Ze2 /r)e−r/λ . 5.6 Describe the physical meaning of the dipole approximation. 5.7 Estimate the radiation emitted by the plasma contained in an accretion disk around a supermassive black hole via the bremsstrahlung mechanism. 5.8 Estimate the optical depth of the plasma between the solar corona and the Earth for the Thomson scattering of the solar radiation.

222

Plasmas

5.9 Find the change in the wavelength of a photon during the Compton scattering. 5.10 An electromagnetic wave of frequency and wavevector (ω0 ,k0 ) is scattered by an electron plasma wave (ωe ,ke ). Find the frequency and the wavevector (ωs ,ks ) for a scattering angle θ from the momentum and the energy conservation laws. Use the appropriate dispersion relation for each wave.

6

6.1

Supplementary Matter

Derivation of Eq. (3.11)

Equation (3.11) is a single fluid equation describing the momentum conservation. It is derived by combining the electron fluid and the ion fluid momentum conservation laws as shown below. We begin with the momentum conservation law for the electron fluid, Eq. (4.11):     V e × B ∂ V e ρe + (V e · ∇)V e = −ne e E + − ∇Pe ∂t c   + ρeg − ρe νei V e − V i

(6.1)

where the viscosity term has been neglected. The momentum conservation law for the ion fluid, Eq. (4.12) without the viscosity term, is     V i × B ∂ V i     ρi + (V i · ∇)V i = ni e E + − ∇Pi ∂t c   + ρig − ρi νie V i − V e

(6.2)

We must add these two equations in order to find the momentum conservation law of the single fluid. We define the center of mass velocity V , Eq. (4.5), as   V = meV e + miV i me + mi

(6.3)

224

Plasmas

since ne = ni = n. Let us first add the nonlinear terms:     me V e · ∇ V e + mi V i · ∇ V i

(6.4)

Solve for V i from Eq. (6.3)   V ≈ meV e + miV i ≈ V i + me V e mi mi

(6.5)

We now substitute for V i in the sum (Eq. 6.4) to get         me V e · ∇ V e + mi V i · ∇ V i = mi V · ∇ V − me V · ∇ V e      m2  − me V e · ∇ V + e V e · ∇ V e ≈ mi V · ∇ V mi

(6.6)

where the terms multiplied by me have been neglected. Now the addition of Eqs. (6.1) and (6.2) becomes     V i − V e × B   ∂ miV i + meV e n + nmi V · ∇ V = ne − ∇ (Pi + Pe ) + (ρi + ρe )g ∂t c      (6.7) − ρi νie V i − V e + ρe νei V e − V i Substitute from Eq. (6.3) to get

ρ

  J × B  ∂ V + ρ V · ∇ V = − ∇P + ρg ∂t c

(6.8)

Where

ρ = ρe + ρi ≈ ρi

(6.9)

has been used. The collisional force terms in Eq. (6.7) vanish since       ρi νie V i − V e + ρe νei V e − V i = V i − V e (ρi νie − ρe νei ) = 0 (6.10) where the condition of momentum conservation for elastic collisions between the electrons and the ions, i.e., (ρi νie − ρe νei ) = 0

(6.11)

is applied. We thus obtain the single fluid momentum conservation law used in Chapters 3 and 4.

Supplementary Matter

6.2

225

Collisional Processes

Several different binary collisional processes occur in a plasma. There are: (1) electron–electron collisions (2) electron–ion collisions (3) ion– electron collisions and (4) ion–ion collisions. All of these are caused by the Coulomb force between the colliding particles. We have studied the electron–ion collision process in section (5.15). A charged particle of charge and mass (q1 , m1 ) is deflected while passing by another charged particle of charge and mass (q2 , m2 ) due to the Coulomb force between them. The magnitude of the Coulomb force on particle 1 due to the particle 2 at a distance r from it is given by F1,2 =

q1 q2 r2

(6.12)

The two-particle collision problem can be reduced to the problem of the relative motion between the two particles and their center of momentum motion. The relative motion is described by the motion of a single particle of reduced mass mR = (m1 m2 )/(m1 + m2 ) moving with a relative velocity (V R = V 1 − V 2 ) in the Coulomb force field. The center of momentum motion consists of both the particles moving with a common velocity V = (m1V 1 + m2V 2 )/(m1 + m2 ). Let us work in the center of momentum frame (V = 0). Then the relative velocity is given by   V R = V e − V i = V e 1 + me ≈ V e . (6.13) mi and the ion velocity is V i = me V e . mi

(6.14)

During the collision process, the particle of mass mR feels the Coulomb force for the duration δ t it spends in the vicinity of the force center. This is the time needed to cross the closest distance of approach b, also called the impact parameter. Thus

δt =

b VR

(6.15)

The change δ P in the momentum of the particle is therefore given by the impulse so that

δ P = δ (mRVR ) = F1,2 δ t =

q1 q2 b b2 VR

(6.16)

226

Plasmas

Now we know that a particle reverses its direction of motion for a deflection of 180 degrees and undergoes a change of momentum twice the value of its initial momentum. Therefore we expect that for large deflections the change in the momentum is of the order of the initial momentum, i.e.,

δ P ≈ (mRVR )

(6.17)

The impact parameter b then turns out to be b=

q1 q2 mRVR2

(6.18)

The effect of all the small angle deflections is collectively contained in a parameter called the Coulomb logarithm, lnΛ, where Λ is the ratio of the maximum and the minimum values of the impact parameter. The maximum value of b is of the order of the Debye length since the Coulomb potential in a plasma varies as exp(−r/λeD ). The particle would hardly feel any force beyond a distance of the order of the Debye length. The minimum value of b corresponds to the maximum deflection and Eq. (6.17) provides the minimum value of b. However, in the classical regime, the minimum value must exceed the de Broglie wavelength of the particle. Thus  Λ = λeD

q1 q2 mRVR2

−1 (6.19)

The collision cross section for a binary collision is the area of a circle of radius b. The total cross section χc in a plasma for large and small deflections, therefore, becomes

χc = π b2 ln Λ

(6.20)

The mean free path of the particle 1 passing through a medium containing n2 particles per cubic centimeter is given by L1 =

1 n2 χc

(6.21)

The mean time, τ12 , between collisions of particle 1 with particle 2 is given by

τ12 =

L1 V1

(6.22)

Supplementary Matter

227

The mean collision frequency, ν12 , of the particle 1 colliding with the particle 2 is given by

ν12 =
=< n2 χcV1 >=< n2 21 42 V1 ln Λ > L1 mRVR

(6.23)

where represents the average over all possible velocities of the particle 2 in a Maxwellian distribution. One can now find the collision frequencies between the electrons and the singly charged ions. The electron–ion collision frequency νei , in a plasma at a temperature T is determined by replacing the relative velocity VR ≈ Ve ,V1 ≈ Ve by its root mean square value; (q1 , q2) by (−e, e), n2 by ni which is also equal to ne , and mR = (me mi )/(me + mi ) ≈ me . We find:  VR =

1/2

3KBT mR

, m2RVR4 = 9KB2 T 2

(6.24)

 3 1/2 (3KB T )−3/2 ln 12π ni λeD

(6.25)

and

νei =

ni π e4 1/2

me

The electron–electron collision frequency, νee is determined by replacing the relative velocity VR ≈ Ve ≈ V1 by its root mean square value, (q1 , q2) by (−e, −e) and mR = (me me )/(me + me ) ≈ (me /2). We find

νee =

ne π e4 1/2

(me /2)

 3 1/2 (3KBT )−3/2 ln 12π ne λeD

(6.26)

The ion–ion collision frequency, νii is determined by replacing the relative velocity VR ≈ Vi ≈ V1 by its root mean square value, (q1 , q2) by (e, e), n2 by ni and mR = (mi mi )/(mi + mi ) ≈ (mi /2). We find

νii =

ni π e4 1/2

(mi /2)

 3 1/2 (3KB T )−3/2 ln 12π ne λeD

(6.27)

There is the fourth collision frequency, νie , for collisions between ions and electrons. At first it may appear that νie should be equal to νei . It is not as we shall see. The ion–electron collision frequency, νie is determined by replacing the relative velocity VR ≈ Ve ,V1 = Vi = (me /mi )Ve , (q1 , q2) by (e, −e), n2 by ne and mR = (mi me )/(me + mi ) ≈ (me ).

228

Plasmas

We find

νie = ne χcVi = ne =

π e4 meVe ln Λ m2e Ve4 mi

   3KBT 1/2 π e4 me me 3 1/2 ne ln 12π ne λeD = νei 2 2 mi me mi 9KBT

(6.28)

Thus we see that mi νie = me νei , as was discussed in the previous section. The collision frequencies between a pair of particles tell us how fast or how slow the two species of particles would equilibrate between themselves and attain a common temperature.

6.3

Derivation of Eq. (5.109)

The divergence-free nature of the magnetic induction B leads to the introduction of the vector potential A: ∇ · B = 0, B = ∇ × A The Faraday law can be written as   A 1 ∂ ∇ × E + =0 c ∂t which implies that   A ∂ 1  = −∇Φ E+ c ∂t

(6.29)

(6.30)

(6.31)

Or 1 ∂ A   − ∇Φ E=− c ∂t

(6.32)

where Φ is the electrostatic potential. This is Eq. (5.160). The Poisson equation ∇ · E = 4πσ can be written in terms of the potentials as   ∂ ∇ · A 1 ∇2 Φ + = −4πσ c ∂t

(6.33)

(6.34)

229

Supplementary Matter

The Ampere law  ∇ × B = 4π J + 1 ∂ E c c ∂t

(6.35)

can be written in terms of the potentials as   1 ∂ 2A    1 ∂ Φ 4π − ∇ ∇ · A + = − J c2 ∂ t 2 c ∂t c

∇2A −

(6.36)

Now since B is the curl of A, the gradient of a scalar can be added to A without any change in B. That is the transformation A → A = A + ∇Ξ

(6.37)

leaves B unaltered. The electric field becomes, under the transformation,  E=−

  A + ∇Ξ ∂ 1 c

∂t

− ∇Φ

(6.38)

If E has to remain unchanged, we must impose the transformation Φ → Φ = Φ −

1 ∂ Ξ c ∂t

(6.39)

Combining Eqs. (6.37) and (6.39) then furnishes  ∇ · A + 1 ∂ Φ c ∂t

(6.40)

and Eqs. (6.34) and (6.36) respectively become ∇2 Φ −

1 ∂ 2Φ = −4πσ c2 ∂ t 2

(6.41)

∇2A −

1 ∂ 2A 4π = − J 2 2 c ∂t c

(6.42)

The transformation, Eqs. (6.37) and (6.39), is called a gauge transformation. The fields E and B remaining unaltered under the gauge transformation are said to be gauge invariant. The condition, Eq. (6.40), is called the Lorentz condition. The Maxwell equations are the partial differential equations. These have been converted to ordinary differential equations by introducing the electromagnetic potentials A, Φ.

230

Plasmas

The solutions of Eqs. (6.41) and (6.42) can be found by the well-known method of Green’s function. This method is well described in Classical Electrodynamics by J.D. Jackson. Here we reproduce some salient features of this method. The time-independent Green functions (used in electrostatics, for example) G are a class of functions that are the solutions of the equation   ∇2 G x,x = −4πδ x −x (6.43) where the prime on the gradient operator reminds us that the gradient operation is with respect to x and δ is the Dirac-delta function. One example of the functions G is  G x,x =

  1 x,x  +F  |x −x |

(6.44)

where F satisfies the Laplace equation:  ∇2 F x,x = 0

(6.45)

In a time-dependent problem, the Green functions depend on (x, t;x , t ) and satisfy the following equation      1 ∂2 2 ∇ − 2 2 G x, t;x , t  = −4πδ x −x δ t − t  (6.46) c ∂t The solutions of the wave Eqs. (6.41) and (6.42), for an infinite medium, can now be, respectively, found from Φ (x, t) =



  G x, t;x , t  σ x , t  d3 x dt 

(6.47)

and A (x, t) =



  x , t     J  G x, t;x , t d3 x dt  c

(6.48)

It must be borne in mind that G must satisfy the boundary conditions imposed by the physical problem under consideration. To find G, take the Fourier transform of Eq. (6.46). The Dirac-delta function can be represented as   δ x −x δ t − t  =

1 (2π )4





d3 k

  exp iω t − t 

   dω exp ik · x −x (6.49)

Supplementary Matter

231

The Fourier transform g(k, ω ) of G can be written as  G x, t;x , t  =

1





d3 k

(2π )4

   dω exp ik · x −x

    g k, ω exp iω t − t 

(6.50)

Using Eq. (6.50) carry out the following calculations:       ω2 1 ∂2 ∇2 − 2 2 G x, t;x , t  = −k2 + 2 G x, t;x , t  c ∂t c

(6.51)

Substitute Eqs. (6.49) and (6.51) in Eq. (6.46) to get   g k, ω =

1 4π 2 (k2 − ω 2 /c2 )

(6.52)

Note that one encounters a singularity in the integrand at ω = ±kc when one tries to carry out the integrations in Eq. (6.50) after substituting for g. The contribution to the integral, therefore, comes from the residues of the poles. Now G in Eq. (6.46) represents the wave disturbance at x caused by a point source at x . The point source exists only for an infinitesimal time duration at t = t . Recall throwing a stone in a lake! The disturbance propagates outwards as a spherically diverging wave (here with speed c). Therefore, G must be zero for t < t  and represent a spherical wave for t > t  . The ω integration has to be done as a Cauchy integral in the complex ω plane. The integral over ω along the real axis is equivalent to the integral along the contour C enclosing the singularities. Since the integral must be nonzero for τ > 0, the poles must be enclosed by a contour for τ > 0. C′ t < 0

–ck –ie

ck–ie

Ct>0 )LJXUH 



Contours (C for τ > 0, C for τ < 0) in the complex ω plane.

232

Plasmas

The exponential factor exp (−iωτ ) = exp (−i (kc + iε ) τ )

(6.53)

blows up for τ > 0, ε > 0. Therefore, we must have the poles below the real axis at kc − iε and −kc − iε so that the integral remains finite for all times t > t . The singularities can be pushed by a positive infinitesimal amount ε below the real axis (Fig.6.1). This is done by replacing ω by (ω + iε ) so that Eq. (5.60) becomes  



exp ik · R exp (−iωτ )  1 G x, t;x , t  = 2 d3 k dω 4π k2 − (ω + iε )2 /c2

(6.54)

 Where  R = x −x , τ = t − t  . The integration over ω for τ > 0 can be carried out by the Cauchy theorem as follows:

dω

exp (−iωτ ) k2 − (ω + iε )2 /c2  = πi = 2π

exp (−ikcτ ) exp (ikcτ ) + −ikc ikc



sin (kcτ ) kc

(6.55)

and

  sin (kcτ )  c G x, t;x , t  = 2 d3 k exp ik · R 2π k

(6.56)

Integration over k can be carried out as

∞

1    sin (kcτ ) c G x, t;x , t  = 2 2π k 2 d (cos θ )exp ikR cos θ dk × 2π 0 k −1

=

2c πR

∞ 0

dk sin (kcτ ) sin (kR)

(6.57)

Note that the integrand is even in k. The integral can be written over the whole interval, −∞ < k < ∞. Eq. (6.56) can be rewritten as

∞  1   G x, t;x , t = dx (exp (ix (τ − R/c)) − exp (ix (τ + R/c))) (6.58) 2π R −∞

Supplementary Matter

233

Note that the integrals are just the Dirac-delta functions. The argument of the second Dirac-delta function never vanishes as τ > 0, therefore only the first integral survives to give the Green function  δ (τ − R/c) G x, t;x , t  = R

=

   | x − x |  δ t −t + c (6.59)

| x −x |

This is called the retarded Green function. It demonstrates that the signal produced at x at time t  = t− | x −x | /c is received at x at a later time t. The scalar (Eq. 6.47) and the vector (Eq. 6.48) potentials can now be written as   | x−x | 

δ t −t + c  σ x , t  d3 x dt  Φ (x, t) =  |x −x |

(6.60)

and    | x − x |  

δ t −t + c J x , t  3   A (x, t) = d x dt c | x −x |

(6.61)

This is Eq. (5.109). The integration over t  can be performed to get the retarded solutions: Φ (x, t) =



  |x −x | 3  σ x , t − d x c

(6.62)

and 

A (x, t) =



J x , t −

c



| x−x | c

 d3 x

(6.63)

Note the time argument of the charge density and the current density. | x−x | They are to be evaluated at the earlier time t − c .

234

6.4

Plasmas

Physical Constants

Speed of light in vacuum

c

=

3 × 1010 cm/s

Gravitational constant

G

=

6.67 × 10−8 dyne-cm2/gram2

Planck constant

h

=

6.62 × 10−27 erg-s

Electron mass

me

=

9.1 × 10−28 gram

Electron charge

e

=

−4.8 × 10−10 statcoulomb.

Classical electron radius

r0

=

e2 /me c2 = 2.8 × 10−13 cm

Thomson cross section

8π r02 /3

=

6.6 × 10−25 cm2

Compton wavelength of electron

h/me c

=

2.4 × 10−10 cm

Proton mass

mp

=

1.67 × 10−24 gram

Bohr radius

a0

=

h¯ 2 /me e2 = 5.2 × 10−9 cm

Atomic cross section

π a20

=

8.8 × 10−17 cm2

Boltzmann constant

KB

=

1.38 × 10−16 erg/degree K

Avogadro number

NA

=

6 × 1023 mol−1

Loschmidt number, number density at STP,

n0

=

2.7 × 1019 cm−3

Standard temperature

T0

=

273 degree K

Atmospheric pressure

P0

=

106 dyne/cm2

Supplementary Matter

6.5

Electromagnetic Spectrum

Designation

Frequency range

Wavelength range

ULF

0–30 Hz

–10 Mm

VF

30–300 Hz

10–1 Mm

ELF

300 Hz–3KHz

1 Mm–100 Km

VLF

3–30 KHz

100–10 Km

LF

30–300 KHz

10–1 Km

MF

300 KHz–3MHz

1 Km–100 m

HF

3–30 MHz

100–10 m

VHF

30–300 MHz

10–1 m

UHF

300 MHz–3 GHz

1 m–10 cm

Microwave

3–300 GHz

10 cm–1 Mm

Submillimeter

300 GHz–3 THz

1 mm–100 μ m

Infrared

3–430 THz

100 μ m–700 nm

Visible

430–750 THz

700–400 nm

Ultraviolet

750 THz–30 PHz

400–10 nm

X ray

30 PHz–3 EHz

10 nm–100 pm

Gamma ray

3 EHz–

100 pm–

235

236

6.6 6.6.1

Plasmas

Astrophysical Quantities Planets Planets

Mass

Mercury Venus Earth Mars Jupiter Saturn Uranus Neptune Pluto

0.06 0.82 1.00 0.11 317.84 95.15 14.60 17.21 0.18

ME

Mean density gm cm−3

Rotational period days

Distance from Sun AU

Orbital period years

5.4 5.2 5.5 3.9 1.3 0.7 1.3 1.7

58.66 242.98 1.00 1.03 0.40 0.43 0.89 0.53 6.39

0.39 0.72 1.00 1.52 5.20 9.54 19.19 30.06 39.53

0.24 0.61 1.00 1.87 11.86 29.47 84.06 164.81 248.54

The mass of the planets is normalized by the mass of the Earth ME = 6 × 1027 gram. One AU = 1.5 × 1013 cm is the distance between the Sun and the Earth. One AU≈ 200 times the radius of the sun. Radius of the Earth RE = 6371 Km. Magnetic field of the Earth ≈ 0.3 gauss. 6.6.2

The Sun

Mass M = 1.99 × 1033 gm Radius R = 6.96 × 1010 cm Luminosity L = 3.83 × 1033 erg/s Age = 4.6 × 109 year Surface gravity g = 2.74 × 104 cm/s2 Radiation flux density = 6.28 × 1010 erg/cm2 /s Temperature of the solar photosphere T = 5780 K Escape speed from sun = 6.18 × 107 cm/s Radiation flux density at 1AU = solar constant = 1.36 ×106 erg / cm2 /s.

Supplementary Matter

6.6.3

The Milky Way

Mass = 5 × 1010 M Rotation period = 3 × 108 year. 6.6.4

The Hubble Constant

H = 50h km sec−1 Mpc−1 , 1 < h < 2. 6.6.5

Electron Density and Temperature of some of the Astrophysical Plasmas

Ionosphere, E Region F Region Solar corona Solar wind Cometary tail Planetary Nebulae HII Regions Pulsar atmosphere Relativistic electron– positron plasma in pulsar magnetosphere Interstellar medium Active galactic Nuclei Broad emission line regions Extragalactic jets Accretion disks around compact objects Accretion disks around black holes Herbig–Haro objects Intracluster region of galaxies

Electron density (cm−3 )

Temperature (Kelvin)

1.5 × 104 − 3 × 104 5 × 104 − 2 × 105 109 − 1011 5 − 10 10 − 100 106 10 − 104 1010

104 − 105 106 − 107 105 − 106 104 − 105 105 105 105

1013 − 1015

γ  102 − 103

10−1 1014 − 1010

108 − 1011

104 − 106 105 − 108 4 10 − 5 × 105

10−2 − 10−4

105 − 107

1018 − 1021

106 − 107

1010 − 1011 108 10−3

105 − 106 105 − 107 1 − 3 ×108

Lorentz factor

237

238

6.7

Plasmas

Vector Identities

 ,W  are vectors; and T is a tensor. Here, S1 , S2 are scalars; V , U  · V × W  =U  × V · W  = V · W  ×U  = U V × W  ·U  =W  ·U  × V = W  ×U  · V

(6.64)

         × V × W  =W  × V × U  = U  ·W  V − U  · V W  U

(6.65)

       × V × W  + V × W  ×U  +W  × U  × V = 0 U

(6.66)

           × V · W  × Z = U  ·W  V · Z − U  · Z V · W  U

(6.67)

         × V · Z W  − U  × V · W  Z  × V × W  × Z = U U

(6.68)

∇ (S1 S2 ) = S1∇S2 + S2∇S1

(6.69)

  ∇ · SU  = S∇ · U  +U  · ∇S

(6.70)

  ∇ × SU  = S∇ × U  +U  × ∇S

(6.71)

  ∇ · U  × V = V · ∇ × V − U  · ∇ × U 

(6.72)

          ∇ × U  + V · ∇ U − U  · ∇ V  × V = U  ∇ · V − V ∇ · U

(6.73)

          ∇ U  · V = U  × ∇ × V + V × ∇ × U  − V · ∇ U + U  · ∇ V (6.74) ∇2 S = ∇ · ∇S

(6.75)

   = ∇ ∇ · U  − ∇ × ∇ × U  ∇2U

(6.76)

∇ × ∇S = 0

(6.77)

∇ · ∇ × U  =0

(6.78)

Supplementary Matter

239

If e1 ,e2 ,e3 are orthonormal unit vectors, a second-order tensor T can be written as T = ∑ Ti j ei e j

(6.79)

The divergence of a tensor is a vector in cartesian coordinates. The components are written as   ∇ · T = ∂ T ji i ∂xj

(6.80)

∇ · (ST ) = ∇S · T + S∇ · T

(6.81)

      ∇ · U  V = ∇ · U  V + U  · ∇ V

(6.82)

If r = xiˆ + y jˆ + zkˆ be the radius vector of magnitude r from the origin to the point (x, y, z), then ∇ ·r = 3

(6.83)

∇ ×r = 0

(6.84)

∇r = r r   ∇ 1 = − r r r3 ∇ ·



r r3

(6.85) (6.86)

 = 4πδ (r)

(6.87)

If Ω is a volume enclosed by a surface Σ and dΣ =ndΣ where n is a unit normal outward from Σ

Ω

Ω

Ω

dΩ∇S =

Σ

 = dΩ∇ · U dΩ∇ · T =

dΣS

Σ

Σ

(6.88)

 dΣ · U

(6.89)

dΣ · T

(6.90)

240

Plasmas



 = dΩ∇ × U

Ω



Σ

 dΣ × U

(6.91)

  

2 2 dΩ S1 ∇ S2 − S2 ∇ S1 = dΣ · S1∇S2 − S2∇S1

Ω

Σ



(6.92)

   · ∇ × ∇ × V − V · ∇ × ∇ × U  = dΩ U

Ω

Σ

   −U  × ∇ × V dΣ · V × ∇ × U

(6.93)

If Σ is an open surface bounded by the contour C, of which the line element is dl,

Σ

Σ

dΣ × ∇S =

Σ

Σ

6.8 6.8.1

C

 = dΣ · ∇ × U





dlS

 C

(6.94)

 dl · U

(6.95)

   = dl × U  dΣ × ∇ × U

(6.96)

C

    dΣ · ∇S1 × ∇S2 = S1 dS2 = − S2 dS1 C

C

(6.97)

Differential Operations Cartesian coordinates (x, y, z)

Gradient ∇S = ∂ S xˆ + ∂ S yˆ + ∂ S zˆ ∂x ∂y ∂z

(6.98)

Divergence ∇ · V = ∂ Vx + ∂ Vy + ∂ Vz ∂x ∂y ∂z

(6.99)

Curl ∇ × V =



∂ Vz ∂ Vy − ∂y ∂z





   ∂ Vx ∂ Vz ∂ Vy ∂ Vx xˆ + − yˆ + − zˆ ∂z ∂x ∂x ∂y

(6.100)

Supplementary Matter

241

Laplacian of a scalar ∇2 S =

∂ 2 S ∂ 2S ∂ 2S + + ∂ x2 ∂ y2 ∂ z2

(6.101)

Laplacian of a vector ∇2V = ∇2Vx xˆ + ∇2Vy yˆ + ∇2Vz zˆ

(6.102)

 Components of (V .∇)U  = Vx ∂ Ux +Vy ∂ Ux +Vz ∂ Ux (V · ∇)U x ∂x ∂y ∂z

(6.103)

 = Vx ∂ Uy +Vy ∂ Uy +Vz ∂ Uy (V · ∇)U y ∂x ∂y ∂z

(6.104)

 = Vx ∂ Uz +Vy ∂ Uz +Vz ∂ Uz (V · ∇)U z ∂x ∂y ∂z

(6.105)

Divergence of a tensor

6.8.2

∇ · T = ∂ Txx + ∂ Tyx + ∂ Tzx x ∂x ∂y ∂z

(6.106)

∇ · T = ∂ Txy + ∂ Tyy + ∂ Tzy y ∂x ∂y ∂z

(6.107)

∇ · T = ∂ Txz + ∂ Tyz + ∂ Tzz z ∂x ∂y ∂z

(6.108)

Cylindrical coordinates (r, θ , z)

Gradient ∇S = ∂ S rˆ + 1 ∂ S θˆ + ∂ S zˆ ∂r r ∂θ ∂z

(6.109)

Divergence ∇ · V = 1 ∂ (rVr ) + 1 ∂ Vθ + ∂ Vz r ∂r r ∂θ ∂z

(6.110)

242

Plasmas

Curl ∇ × V =



1 ∂ Vz ∂ Vθ − r ∂θ ∂z 





 ∂ Vr ∂ Vz ˆ θ+ rˆ + − ∂z ∂r

1 ∂ (rVθ ) 1 ∂ Vr − r ∂r r ∂θ

 zˆ

(6.111)

Laplacian of a scalar   ∂S 1 ∂ 1 ∂ 2S ∂ 2S + ∇ S= r + 2 r ∂r ∂r r ∂ θ 2 ∂ z2 2

(6.112)

Laplacian of a vector   2 ∂ Vθ Vr − 2 rˆ+ ∇2V = ∇2Vr − 2 r ∂θ r   2 ∂ Vr Vθ ˆ  2 2 ∇ Vθ + 2 − 2 θ + ∇ Vz zˆ r ∂θ r

(6.113)

 Components of (V .∇)U  = Vr ∂ Ur + Vθ ∂ Ur +Vz ∂ Ur − Vθ Uθ (V · ∇)U r ∂r r ∂θ ∂z r

(6.114)

 = Vr ∂ Uθ + Vθ ∂ Uθ +Vz ∂ Uθ + Vθ Ur (V · ∇)U θ ∂r r ∂θ ∂z r

(6.115)

 = Vr ∂ Uz + Vθ ∂ Uz +Vz ∂ Uz (V · ∇)U z ∂r r ∂θ ∂z

(6.116)

Divergence of a tensor ∇ · T = 1 ∂ (rTrr ) + 1 ∂ Tθ r + ∂ Tzr − Tθ θ r r ∂r r ∂θ ∂z r

(6.117)

∇ · T = 1 ∂ (rTrθ ) + 1 ∂ Tθ θ + ∂ Tzθ + Tθ r θ r ∂r r ∂θ ∂z r

(6.118)

∇ · T = 1 ∂ (rTrz) + 1 ∂ Tθ z + ∂ Tzz z r ∂r r ∂θ ∂z

(6.119)

Supplementary Matter

6.8.3

243

Spherical coordinates (r, θ , ϕ )

Gradient ∇S = ∂ S rˆ + 1 ∂ S θˆ + 1 ∂ S ϕˆ ∂r r ∂θ r sin θ ∂ ϕ

(6.120)

Divergence 1 ∂ Vϕ ∂ ∇ · V = 1 ∂ (r2Vr ) + 1 (sin θ Vθ ) + 2 r ∂r r sin θ ∂ θ r sin θ ∂ ϕ

(6.121)

Curl ∇ × V =

1 1 ∂ Vθ ∂ (sin θ Vϕ ) − r sin θ ∂ θ r sin θ ∂ ϕ

(6.122)

∇ × V |θ =

1 ∂ Vr 1 ∂ − (rVϕ ) r sin θ ∂ ϕ r ∂r

(6.123)

r

∇ × V |ϕ = 1 ∂ (rVθ ) − 1 ∂ Vr r ∂r r ∂θ Laplacian of a scalar     ∂ 2S ∂S 1 1 ∂ ∂S 1 ∂ sin θ + ∇2 S = 2 r2 + 2 r ∂r ∂r r sin θ ∂ θ ∂θ r2 sin2 θ ∂ ϕ 2

(6.124)

(6.125)

Laplacian of a vector 2Vr 2 ∂ Vθ 2 cot θ Vθ 2 ∂ Vϕ − − 2 ∇2V = ∇2Vr − 2 − 2 r r r ∂θ r2 r sin θ ∂ ϕ 2 ∂ Vr Vθ 2 cos θ ∂ Vϕ − − ∇2V = ∇2Vθ + 2 2 2 θ r ∂θ r sin θ r2 sin2 θ ∂ ϕ Vϕ 2 ∂ Vr 2 cos θ ∂ Vθ + + ∇2V = ∇2Vϕ − ϕ r2 sin2 θ r2 sin θ ∂ ϕ r2 sin2 θ ∂ ϕ

(6.126) (6.127) (6.128)

 Components of (V .∇)U V V U +Vϕ Uϕ  = Vr ∂ Ur + Vθ ∂ Ur + ϕ ∂ Ur − θ θ (V · ∇)U r ∂r r ∂θ r sin θ ∂ ϕ r

(6.129)

V cot θ Vϕ Uϕ  |θ = Vr ∂ Uθ + Vθ ∂ Uθ + ϕ ∂ Uθ + Vθ Ur − (V · ∇)U ∂r r ∂θ r sin θ ∂ ϕ r r

(6.130)

244

Plasmas

Vϕ ∂ Uϕ Vϕ Ur cot θ Vϕ Uθ ∂U ∂ Uϕ  |ϕ = Vr ϕ + Vθ (V · ∇)U + + + ∂r r ∂θ r sin θ ∂ ϕ r r

(6.131)

Divergence of a tensor 1 ∂ 1 ∂ (sin θ Tθ r) (∇ · T )r = 2 (r2 Trr ) + r ∂r r sin θ ∂ θ +

1 ∂ Tϕ r Tθ θ + Tϕ ϕ − r sin θ ∂ ϕ r

(6.132)

1 ∂ 1 1 ∂ Tϕ θ ∂ (sin θ Tθ θ ) + + (∇ · T )θ = 2 (r2 Trθ ) + r ∂r r sin θ ∂ θ r sin θ ∂ ϕ Tθ r cot θ Tϕ ϕ − r r

(6.133)

1 ∂ 1 1 ∂ Tϕ ϕ ∂ (∇ · T )ϕ = 2 (r2 Trϕ ) + (sin θ Tθ ϕ ) + r ∂r r sin θ ∂ θ r sin θ ∂ ϕ +

Tϕ r cot θ Tϕ θ + r r

Source: NRL Plasma Formulary (2002) J. D. Huba.

(6.134)

Select Bibliography

Bittencourt, J.A. 2004. Springer.

Fundamentals of Plasma Physics.

New York:

Chen, F.F. 1974. Introduction to Plasma Physics. New York: Plenum Press. Dendy, Richard. 1993. Plasma Physics: An Introductory Course. New York: Cambridge University Press. Frank-Kamenetskii, D.A. 1972. Plasma: The Fourth State of Matter. New York: Plenum Press. Ichimaru, S. 1973. Basic Principles of Plasma Physics: A Statistical Approach. New York: W.A. Benjamin, INC. Jackson, J.D. 1962. Classical Electrodynamics. New York: John Wiley & Sons. Krishan, Vinod. 1999. Astrophysical Plasmas and Fluids. Dordrecht:Kluwer Academic Publishers. Nicholson, D.R.1983. Introduction to Plasma Theory. New York: John Wiley & Sons Schmidt, G. 1966. Physics of High Temperature Plasmas. New York: Academic Press. Tanenbaum, B.S.1967. Plasma Physics. New York: McGraw-Hill Book Company

.

Index Abell supercluster, 6 Ablation, 30, 85 Absorption electromagnetic waves, 181 Accelerated single, 193 ACIS (Advanced CCD Imaging Spectrometer), 7 Accretion disk, 61−62, 221, 237 Active galactic nuclei, 237 Advanced CCD Imaging Spectrometer (ACIS), see ACIS Alfv´en speeds, 112 velocity, 101 wave fields, 103 Ambipolar diffusion, 36 Ampere’s law, 66 Anisotropy, 217 Astrophysical quantities, 236 Avogadro number, 234 Azimuthal magnetic field, 66, 70, 87 Bessel functions, 73−74 Bohr radius, 234 Boltzmann constant, 43−44, 94, 234 distribution, 45, 133, 143−44 factor, 45 law, 43, 49 relation, 136

Bremsstrahlung mechanism, 221 radiation, 213 inverse, 183 Cauchy theorem, 232 Cesium atoms, 1 Charged dust, 37−38 Classical electrodynamics, 230, 245 CMEs (Coronal Mass Ejections), 14−15, 22−23 Collisionless damping, 131 Collisional ionization, 42 Collision time, 43, 207 Coulomb logarithm, 226 Compton scattering, 217, 219, 221−22 Conjugate aurorae, 23 Convective derivative, 58 Coronal Mass Ejections (CMEs), see CMEs Cosmic rays, 24, 165 Coulomb collisions, 46, 209 Critical velocity ionization, 46 Cyclotron radius, 63, 79 Debye cloud, 48−50, 53 screening, 49 sphere, 54 Deuterium nucleus, 3−4

248

Plasmas

Dipole pattern, 192 Dirac delta function, 187, 194 Dispersion relation of the wave, 115, 166 Doppler shift, 98 Eddington luminosity, 82, 84, 87 Einstein’s mass energy relation, 3 Electromagnetic waves, 132, 138, 145−48, 154, 157−60, 163, 165, 174−77, 181, 185, 222 Electron debye length, 50−52 sphere, 54 Electron density and temperature, 237 Electron fluid, 57−58, 83, 89−92, 94, 112, 117−19, 126, 129, 132, 136, 142−44, 164, 176, 223 plasma waves, 33, 53, 123, 126−27, 129−32, 135, 138−39, 141−42, 158, 218−19, 222 radius, 83, 201 trajectory, 185 Electrostatic waves in magnetized fluids, 138 European Space Agency, 22 Extragalactic jets, 87, 237

Gases, kinetic theory of, 39−40, 43, 57, 130 Gaunt factor, 212 Gaussian distribution, 211 Gluons, 2, 38 Green functions retarded, 233 time-independent, 230 Grip of gravity, 56 Hall effect, 120−22 term, 120 velocity, 120 wave, 119−21, 123 Hubble constant, 237

Impact parameter, 184, 209, 226 Implosion, 85−86 Incident polarization vector, 215 Inclusion of thermal pressure, 129 Inductive electric field, 94, 96 Inertial confinement fusion, 30 Intergalactic plasma, 6, 8−9 International Thermonuclear Experimental Reactor (ITER), see ITER Intracluster region, 237 Ion inertial scale, 121, 164 thermal velocity, 55, 60 Faraday law of induction, 168 viscosity, 91, 223 Faraday rotation, 47, 154 Ionization Fermi-Dirac statistics, 45 avalanche, 46 Fluids, ideal magnetohydrodynamic, 98 potentials, 43 Force-free magnetic field, 72−73 thermal, 43−44 Ionospheric electron density, 25 Form deuterium, 3 Ions, fluorescing, 47 Fourier transform, 203−04, 231 Irving Langmuir, 1, 20−21 Isotropic distribution, 211 Galactic plasma, 9 ITER (International Thermonuclear Galaxy cluster Abell, 7 Experimental Reactor), 30 Gamma ray, energetic, 13

Index

ITER organization, 30 Keplerian flow, 61−62 velocity, 87 Kepler’s laws, 61 Landau damping, 131, 138 Langmuir probe, 21, 48 waves, 21 Laplace equation, 230 Larmor formula, 199 radiation formula, 186, 214 Length, screening, 49 Linear Alfv´en wave, 99 Linearization process, 99, 130 Linearly polarized wave, 154 Linearized mass conservation law, 99 Lorentz condition, 229 force, 29, 62−63, 66, 119, 144−45, 176−77, 202, 213 gauge, 187 Loschmidt number, 234 Magnetic bottles, 62 Magnetic confinement fusion, 29 Magnetic field configuration bipolar, 9 closed dipole, 13 Magnetic mirrors, 75, 79 Magnetic monopoles, 168 Magnetic pressure, 22, 68, 70−71, 106, 112 Magnetized fluids, 93, 112, 138 Magnetoacoustic waves, 110 Magnetohydrodynamic waves, 89, 95, 100, 115, 117 Magnetosphere, 15−16, 19−22, 24

249

Magnetostatic force free equilibrium, 71 Maltese cross, 25−26 Maxwell equation, 87, 198, 229 Mirror ratio, 82 Monopole field, 76 National Ignition Facility (NIF), see NIF Neutral atomic hydrogen density, 12 NIF (National Ignition Facility), 31 Nonlinear Alfv´en waves, 104 Nuclear fission, 28 Oblique propagation of magnetoacoustic waves, 110 Ohm’s law, 175−76 Oscillating electric dipole, 186, 214 Parametric decay instability, 220 Particle cyclotron frequency, 63 Particle energy, 201 Particle orbit, 78 Pauli’s exclusion principle, 45 Penning ionization, 43 Periodic table, 5 Perturbed sinusoidal form, 100 Photoionization, 41−42, 55 Photons, 2, 5, 42, 82, 217 Physics, solar-terrestrial, 15 Planck constant, 44, 212, 234 Plane polarized wave, 155, 171 Plasma collisionless, 36, 185 deuterium-tritium, 30, 85−86 electron-ion, 12, 35− 37, 43, 91 electrons, 21, 33, 46−47, 112, 219 formation, 41 frequency, 180 ions, 46, 112

250

Plasmas

ionospheric, 25, 55, 159, 180, 221 isothermal, 112 magnetospheric, 22 quark-gluon, 2−3, 5, 38 Poynting vector, 175, 191−92 Pressure ionization, 19, 45, 55 Primordial nucleosynthesis, 4 Propagation vector, 53, 100, 113, 141, 143, 148, 151, 166−67 Quantum plasma, 38−39 Quick revisit of waves, 165 Radiation cross section, 210 cyclotron, 151 isotropic, 217 spectrum, 203 synchrotron, 9−10, 203 Radio galaxy hercules, 70 Radio Signals (RSS), 38 Raman scattering of radiation, 219 Raman scattering, stimulated, 220 Relativistic generalization, 199 Reversed field pinches, 71−72, 74 Saha equation, 44 Satellite operations, 14 Scattering coherent, 218−19 isotropic, 217−18 Single fluid dynamics, 95 Solar corona, 12−13, 36, 47, 55, 70, 84, 89, 106, 112, 117, 132, 164, 185, 221

Solar wind, 10, 14−19, 21−22, 62, 82, 84, 117, 164, 185 Spatial absorption rate, 181 Structures, double-helix, 38 Tensor, 238−39, 243−44 Terrestrial plasma, 20 Thomson cross section, 83, 215, 234 Thomson scattering, 47, 83, 213−19, 221 Tokamak, 29−30, 75 Transformation, gauge, 229 Transverse MHD Waves, 99 Ultracold plasmas, 36 Upper hybrid waves, 138−39, 145, 163 Van allen radiation belts, 82 Velocity amplitude, 100, 102 Wave amplitudes, 106, 110, 113−14 Wave equation, 145−46, 148−49, 151, 169, 176 Waves dispersionless, 102, 108 electron cyclotron, 139 ion cyclotron, 139, 144 magnetohydrodynamic, 95 polarized circular, 154 Whistlers, 154, 161 X-ray band, 217 Z pinch, 66, 69−71, 74−75