The Flight of a Relativistic Charge in Matter: Insights, Calculations and Practical Applications of Classical Electromagnetism [1 ed.] 3031234456, 9783031234453, 9783031234460

Table of contents :
Preface
Acknowledgements
Contents
Part I Building on Simple Ideas
1 Waves and sources
1.1 Energy Loss, Cherenkov, and Transition Radiation
1.2 Considering a Two-Dimensional Scalar Field
1.2.1 Real Transverse Wave Vector
1.2.2 Imaginary Transverse Wave Vector
1.2.3 The Field of a Charge in Vacuum by Analogy
1.2.4 The Field of a Charge in Matter by Analogy
1.2.4.1 In the Optical Region
1.2.4.2 In the X-Ray Region
1.2.5 Conclusions from the Two-Dimensional Field Study
1.3 Considering a Photon with Effective Mass
1.3.1 A Field Mediated by Massive Photon Exchange
1.3.1.1 Photon Mass in the Optical Region
1.4 Considering Free Photon Emission
1.4.1 Kinematic Conditions
1.4.2 Emission with Recoil in a Periodic Medium
1.5 Considering Diffracted Cherenkov Radiation
1.5.1 Radiation Emitted Passing Through a Slab
1.5.1.1 But a Term Is Missing from This Description
1.5.1.2 Integral and Differential Fields
1.5.2 Transition Radiation in the X-Ray Range
1.5.2.1 Two-Sided Slab and Single Interface
1.5.3 X-Ray Transition Radiation Dependence on Energy and Angle
1.5.3.1 Angular Distribution
1.5.3.2 Energy Distribution
1.5.3.3 Total Energy Flux
1.5.4 X-Ray Transition Radiation Limit for Thin Slabs
1.5.5 Summary of Conditions for X-Ray Transition Radiation
References
Part II Calculations in Classical Electromagnetism
2 The Influence of a Passing Charge
2.1 Electric and Magnetic Fields in Vacuum
2.1.1 The Force on a Charge
2.1.2 Maxwell's Field Equations
2.1.3 A Short Field Pulse from a Moving Charge
2.1.3.1 The Pulse in the Semiclassical Picture
2.2 Equations Modified for Media
2.2.1 Physical Interpretation of Resonant Dispersion
2.3 Equations for Linear Media
2.3.1 Transparent Media Approximation
2.3.2 An Ionised Plasma
2.4 Practical Measurement of a Moving Charge
2.4.1 Relativistic Kinematics and Particle Identification
2.4.2 Cherenkov Effect
2.4.3 Transition Radiation
2.4.3.1 A Moving Slab of Dielectric
2.4.4 Energy Deposited in the Medium
2.4.4.1 Fluctuations
References
3 The Field of a Moving Charge
3.1 Field Equations in Vacuum or Non-dispersive Media
3.2 Potentials and Field Solutions in Vacuum
3.2.1 Wave Equations for the Potentials
3.2.2 Retarded and Advanced Potentials
3.2.3 The Liénard–Wiechert Potentials for a Point Charge
3.2.4 Solution for the E and B Fields
3.3 Field Solutions in Media
3.3.1 Non-local Relations Between Fields
3.3.1.1 Non-locality in Time
3.3.2 Maxwell's Equations and Potentials in Fourier Space
3.3.3 The Solution for a Charge Movingwith Constant Velocity
4 Radiation by the Apparent Angular Acceleration of Charge
4.1 Apparent Angular Acceleration in Vacuum
4.1.1 Non-relativistic Motion
4.1.2 Relativistic Motion and the Feynman–Heaviside Form
4.1.2.1 Radiated Photon Flux According to Feynman–Heaviside
4.2 Apparent Acceleration and Cherenkov Radiation
4.2.1 Calculation of the Cherenkov Flux
4.3 Apparent Acceleration and Transition Radiation
4.3.1 Radiation from a Discontinuity in ApparentAngular Velocity
References
5 The Dispersion and Absorption of Electromagnetic Waves
5.1 Refraction and Attenuation
5.1.1 Phenomenology of Absorption
5.1.2 Response of a Classically Bound Electron
5.1.3 A Medium of Bound Electrons
5.2 General Form of Dielectric Permittivity
5.2.1 Oscillator Strength and Its Sum Rule
5.2.2 Effects of Finite Density
5.2.3 Causality and Dispersion
5.3 Photon Cross Section
5.3.1 Resonance Collisions
5.3.2 Electron Constituent Scattering
5.3.2.1 Thomson Scattering
5.3.2.2 Compton Scattering
5.3.3 Pair Production
References
6 Energy Loss of a Charge Moving in a Medium
6.1 Basic Ideas
6.1.1 The Force That Slows the Particle
6.1.2 The Electric Field That Provides That Force
6.1.3 Planck Quantisation and the Cross Section
6.1.3.1 Scattering
6.2 Energy and Momentum Transferred to the Medium
6.2.1 The Generalised Dielectric Permittivity
6.2.1.1 The Generalised Oscillator Strength Density
6.2.1.2 Oscillator Strength in the Resonance Region
6.2.1.3 Oscillator Strength in the Constituent Scattering Region
6.3 Mean Energy Loss
6.4 Energy Loss Cross Section
6.4.1 The Cross Section Evaluated in Argon
6.4.2 Terms in the General Energy Loss Cross Section
6.4.2.1 The First Term
6.4.2.2 The Second Term Below Cherenkov Threshold
6.4.2.3 The Second Term Above Cherenkov Threshold
6.4.2.4 The Accelerator Solution
6.4.2.5 The Third Term
6.4.2.6 The Fourth Term
6.5 Distributions in Energy Loss
6.6 Comparison with Experimental Data
6.6.1 The Energy Loss Fluctuations
6.6.2 An Optimal Estimator for Energy Loss
6.7 The Bethe–Bloch Approximation
6.7.1 Four Assumptions in the Bethe–Bloch Formula
6.7.1.1 The First Assumption
6.7.1.2 The Second Assumption
6.7.1.3 The Third Assumption
6.7.1.4 The Fourth Assumption
6.7.1.5 Conclusions on the Use of the Bethe–Bloch Formula
6.8 Bremsstrahlung
References
7 Scattering of a Charge Moving in a Medium
7.1 The Scattering and Energy Loss Cross Section
7.1.1 Cross Section in the Resonance Region
7.1.2 Scattering by a Constituent Charge
7.1.3 Modifications of Point-Like Nuclear Scattering
7.1.3.1 Thomas–Fermi Form Factors for Nuclear Scattering at Low Q2
7.1.3.2 Hydrogenic Form Factor for Nuclear Scattering at Low Q2
7.1.3.3 Form Factor for a General Nuclear Target at High Q2
7.1.3.4 Form Factor for a Proton Target at High Q2
7.1.4 Form Factors for Electron Constituents
7.1.5 Cross Sections dkTdkL with Form Factors
7.1.5.1 Checking That Cherenkov Radiation Is Included
7.2 Scattering and Energy Loss Distributions
7.2.1 Single and Multiple Scattering
7.2.2 Evaluating Probability Maps in pT and pL
7.2.3 Multiple Scattering and Energy Lossfor Various Elements
References
Part III Two Practical Applications
8 Relativistic Particle Identification by dEdx
8.1 Design Criteria for a Detector
8.1.1 Charged Particles to Be Distinguished, P/K/π/e
8.1.2 Choice of Medium to Maximise Discrimination
8.1.3 Estimator and Sampling Required
8.1.3.1 Preliminary Experiments and Calculations
8.2 Identification of Secondaries by Ionisation Sampling, the Detector Design
8.3 The Performance of ISIS in the EHS Spectrometer
8.3.1 Identification Performance by ISIS
8.3.2 Quantitative Check on Calculated Ionisation Errors
References
9 Ionisation Beam Cooling
9.1 The Need to Compress Charged Particle Beams
9.1.1 Unique Experiments with High Energy Muon Beams
9.1.1.1 Stochastic Cooling
9.1.1.2 Electron Cooling
9.1.1.3 Ionisation Cooling
9.2 Ionisation Cooling of Muon Beams
9.3 Validation of Scattering Calculation for Hydrogen
9.3.1 Probability Maps for Energy Loss and Scattering
9.3.2 Correlation Between Energy Loss and Scattering Distributions
References
List of Symbols
Constants
Scalars
Vectors
Index

Citation preview

Lecture Notes in Physics

Wade Allison

The Flight of a Relativistic Charge in Matter Insights, Calculations and Practical Applications of Classical Electromagnetism

Lecture Notes in Physics Founding Editors Wolf Beiglböck Jürgen Ehlers Klaus Hepp Hans-Arwed Weidenmüller

Volume 1014

Series Editors Roberta Citro, Salerno, Italy Peter Hänggi, Augsburg, Germany Morten Hjorth-Jensen, Oslo, Norway Maciej Lewenstein, Barcelona, Spain Luciano Rezzolla, Frankfurt am Main, Germany Angel Rubio, Hamburg, Germany Wolfgang Schleich, Ulm, Germany Stefan Theisen, Potsdam, Germany James D. Wells, Ann Arbor, MI, USA Gary P. Zank, Huntsville, AL, USA

The series Lecture Notes in Physics (LNP), founded in 1969, reports new developments in physics research and teaching - quickly and informally, but with a high quality and the explicit aim to summarize and communicate current knowledge in an accessible way. Books published in this series are conceived as bridging material between advanced graduate textbooks and the forefront of research and to serve three purposes: • to be a compact and modern up-to-date source of reference on a well-defined topic; • to serve as an accessible introduction to the field to postgraduate students and non-specialist researchers from related areas; • to be a source of advanced teaching material for specialized seminars, courses and schools. Both monographs and multi-author volumes will be considered for publication. Edited volumes should however consist of a very limited number of contributions only. Proceedings will not be considered for LNP. Volumes published in LNP are disseminated both in print and in electronic formats, the electronic archive being available at springerlink.com. The series content is indexed, abstracted and referenced by many abstracting and information services, bibliographic networks, subscription agencies, library networks, and consortia. Proposals should be sent to a member of the Editorial Board, or directly to the responsible editor at Springer: Dr Lisa Scalone [email protected]

Wade Allison

The Flight of a Relativistic Charge in Matter Insights, Calculations and Practical Applications of Classical Electromagnetism

Wade Allison Department of Physics and Keble College University of Oxford Oxford, UK

ISSN 0075-8450 ISSN 1616-6361 (electronic) Lecture Notes in Physics ISBN 978-3-031-23445-3 ISBN 978-3-031-23446-0 (eBook) https://doi.org/10.1007/978-3-031-23446-0 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

To Kate

Preface

This book is about the energy loss and the coherent radiation emitted by a relativistic charge in matter. These phenomena—locally deposited energy, Cherenkov radiation and transition radiation—are the basis of any charged particle detector able to discriminate charges by their velocity. The absorptive and dispersive properties of materials have a first-order role in the classical electrodynamics that determines the signals, their size and fluctuations, yielded by each particle in an experiment. As a result, these apparently disparate phenomena are closely related. The cross-sections for these phenomena derived from electrodynamics are used to generate distributions in thin absorbers, first for energy loss and then for scattering, including correlations between the two. Two specific applications are then followed: the first shows the energy loss resolution and particle identification achieved in practice with a multi-particle detector in the course of an experiment at CERN; the second shows how, by including scattering as well as energy loss, the technique of ionisation cooling of accelerator beams may be reliably simulated. The treatment assumes some knowledge of mathematical physics at an undergraduate level, specifically Maxwell’s Equations and classical optics. It is based on a series of lectures given at the University of Oxford to graduate students in experimental particle physics. Both as a student myself and later in my career too, I always found the omission of steps in the derivation of a result most unhelpful in my search for real understanding. So, in this book, instead of appealing to an authoritative reference or leaving a derivation as an “exercise for the student”, mathematical details are worked explicitly for the student to follow and establish his own satisfaction. For sound historical and theoretical reasons, the standard works on the subject are based on the principles of special relativity. The pages of books by JD Jackson, Landau and Lifshitz, and others are well studied by students today, as they were by their parents and grandparents before them. These start from a picture of the fields of an electric charge in vacuum, which is then modified for the effects of dispersion and absorption in real material media. However, in doing so, they omit the intuitive understanding that comes from considering dispersion and absorption as first-order effects. For instance, at speeds of a charge close to that of light, dispersion causes major changes to the form of the electromagnetic field. And then, frequency-dependent variations give a picture and results that are not obvious as simple extensions of special relativity. vii

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Preface

While fundamental theoretical physics concerns itself first with behaviour at the hard end of the spectrum, the most relevant response for radiation detectors is at the soft end of the spectrum. For these there is a preferred frame of reference—that of the medium—that undercuts the axiomatic relevance of special relativity. For signals detected by instruments in a high-energy experiment, absorption is a first-order effect. Since the charge of each particle is very small, 1.6 × 10−19 Coulombs, signals are also small, so that signal-to-noise ratios and detection efficiencies are always critical. In an interaction, the need is to measure the velocities of many such charges simultaneously, each within a thousandth of the speed of light in vacuum. The calculation of these signals and their fluctuations is central to understanding, analysing and designing experiments. Part I of the book introduces some simple ideas that are relevant to the influence of a source moving at constant velocity. These are applied to the electromagnetic field and used to describe the relationship between different phenomena. Part II builds up a precise description of the electromagnetic fields of a charge in steady motion in terms of the properties of the medium. The energy loss and scattering cross-sections are deduced from these and used to predict distributions in finite thicknesses. The analysis is validated by comparing with experimental measurements. Two practical applications are discussed in Part III. First, the ISIS detector was designed to identify the decay products of charmed particles by energy loss and used successfully in experiments at CERN. Second, the cooling of beams of highenergy muons in liquid hydrogen was simulated using the method of Part II. In both cases, the improvement in simulation compared with simple traditional formulae was crucial. Oxford, UK December 2022

Wade Allison

Acknowledgements

This study grew out of a discussion with Lou Voyvodic when we were both at Argonne National Laboratory in 1970, namely how might relativistic charged particles of known momentum be distinguished experimentally. Over the following decade, the theoretical understanding that grew was a collaboration with John Cobb. Many of his ideas proved essential to this work. The experimental work benefited from the encouragement of John Mulvey at Oxford and involved many people, at Oxford, the Rutherford Appleton Laboratory, CERN and in the United States. Nothing can be done without mutual confidence, and I acknowledge the trust that others in the international collaboration that built the European Hybrid Spectrometer showed in the ISIS Project, an apparently speculative instrument built on theoretical calculations that lacked initial verification. In addition to John Cobb, there are several people in the Oxford group that I should thank for sharing their skills, judgement and mutual confidence: Barney Brooks for his broad knowledge of electronics and readiness to engage new and unfamiliar hardware problems; Peter Shield for the crucial design of the ISIS electronics; Nick West for the development of the ISIS software; also the staff of the mechanical and electronic workshops in the Oxford Physics Department who built the ISIS chamber. I acknowledge with thanks the essential contributions of Simon Holmes and John Cobb to the work on Ionisation Cooling described in Chap. 9. Recently, following a stimulating discussion in Vienna with Martin Tazreiter, I resolved to write up the whole story, a work that would not have been concluded without the support of my wife and family. I am grateful to Georg Viehhauser for his interest and careful working through the mathematics, and also to my editor, Lisa Scalone of Springer Nature Switzerland, for guiding me through the labyrinth of publication.

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Contents

Part I

Building on Simple Ideas

1 Waves and sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.1 Energy Loss, Cherenkov, and Transition Radiation . . . . . . . . . . . . . . . . . . . 1.2 Considering a Two-Dimensional Scalar Field . . . . .. . . . . . . . . . . . . . . . . . . . 1.2.1 Real Transverse Wave Vector . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.2.2 Imaginary Transverse Wave Vector . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.2.3 The Field of a Charge in Vacuum by Analogy . . . . . . . . . . . . . . . . 1.2.4 The Field of a Charge in Matter by Analogy .. . . . . . . . . . . . . . . . . 1.2.5 Conclusions from the Two-Dimensional Field Study .. . . . . . . . 1.3 Considering a Photon with Effective Mass . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.3.1 A Field Mediated by Massive Photon Exchange .. . . . . . . . . . . . . 1.4 Considering Free Photon Emission . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.4.1 Kinematic Conditions . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.4.2 Emission with Recoil in a Periodic Medium . . . . . . . . . . . . . . . . . . 1.5 Considering Diffracted Cherenkov Radiation . . . . .. . . . . . . . . . . . . . . . . . . . 1.5.1 Radiation Emitted Passing Through a Slab .. . . . . . . . . . . . . . . . . . . 1.5.2 Transition Radiation in the X-Ray Range .. . . . . . . . . . . . . . . . . . . . 1.5.3 X-Ray Transition Radiation Dependence on Energy and Angle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.5.4 X-Ray Transition Radiation Limit for Thin Slabs.. . . . . . . . . . . . 1.5.5 Summary of Conditions for X-Ray Transition Radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Part II

3 3 4 6 6 6 8 9 10 11 12 12 13 14 14 17 18 20 20 21

Calculations in Classical Electromagnetism

2 The Influence of a Passing Charge . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.1 Electric and Magnetic Fields in Vacuum.. . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.1.1 The Force on a Charge.. . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.1.2 Maxwell’s Field Equations .. . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.1.3 A Short Field Pulse from a Moving Charge . . . . . . . . . . . . . . . . . . . 2.2 Equations Modified for Media.. . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.2.1 Physical Interpretation of Resonant Dispersion . . . . . . . . . . . . . . .

25 25 25 26 27 28 29 xi

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2.3 Equations for Linear Media . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.3.1 Transparent Media Approximation . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.3.2 An Ionised Plasma .. . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.4 Practical Measurement of a Moving Charge .. . . . . .. . . . . . . . . . . . . . . . . . . . 2.4.1 Relativistic Kinematics and Particle Identification .. . . . . . . . . . . 2.4.2 Cherenkov Effect . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.4.3 Transition Radiation . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.4.4 Energy Deposited in the Medium .. . . . . . . . .. . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

30 31 31 32 32 33 33 35 36

3 The Field of a Moving Charge .. . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.1 Field Equations in Vacuum or Non-dispersive Media .. . . . . . . . . . . . . . . . 3.2 Potentials and Field Solutions in Vacuum . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.2.1 Wave Equations for the Potentials . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.2.2 Retarded and Advanced Potentials .. . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.2.3 The Liénard–Wiechert Potentials for a Point Charge . . . . . . . . . 3.2.4 Solution for the E and B Fields . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.3 Field Solutions in Media . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.3.1 Non-local Relations Between Fields . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.3.2 Maxwell’s Equations and Potentials in Fourier Space . . . . . . . . 3.3.3 The Solution for a Charge Moving with Constant Velocity . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

39 39 40 40 41 43 43 44 44 45

4 Radiation by the Apparent Angular Acceleration of Charge . . . . . . . . . . . 4.1 Apparent Angular Acceleration in Vacuum.. . . . . . .. . . . . . . . . . . . . . . . . . . . 4.1.1 Non-relativistic Motion .. . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.1.2 Relativistic Motion and the Feynman–Heaviside Form .. . . . . . 4.2 Apparent Acceleration and Cherenkov Radiation .. . . . . . . . . . . . . . . . . . . . 4.2.1 Calculation of the Cherenkov Flux . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.3 Apparent Acceleration and Transition Radiation ... . . . . . . . . . . . . . . . . . . . 4.3.1 Radiation from a Discontinuity in Apparent Angular Velocity .. . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

49 49 49 49 51 52 55

5 The Dispersion and Absorption of Electromagnetic Waves .. . . . . . . . . . . . 5.1 Refraction and Attenuation . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.1.1 Phenomenology of Absorption .. . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.1.2 Response of a Classically Bound Electron . . . . . . . . . . . . . . . . . . . . 5.1.3 A Medium of Bound Electrons.. . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.2 General Form of Dielectric Permittivity . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.2.1 Oscillator Strength and Its Sum Rule . . . . . .. . . . . . . . . . . . . . . . . . . . 5.2.2 Effects of Finite Density. . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.2.3 Causality and Dispersion . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.3 Photon Cross Section . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.3.1 Resonance Collisions .. . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.3.2 Electron Constituent Scattering . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

59 59 59 60 61 62 62 63 63 65 65 65

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5.3.3 Pair Production.. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

67 68

6 Energy Loss of a Charge Moving in a Medium . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.1 Basic Ideas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.1.1 The Force That Slows the Particle . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.1.2 The Electric Field That Provides That Force . . . . . . . . . . . . . . . . . . 6.1.3 Planck Quantisation and the Cross Section .. . . . . . . . . . . . . . . . . . . 6.2 Energy and Momentum Transferred to the Medium . . . . . . . . . . . . . . . . . . 6.2.1 The Generalised Dielectric Permittivity . . .. . . . . . . . . . . . . . . . . . . . 6.3 Mean Energy Loss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.4 Energy Loss Cross Section . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.4.1 The Cross Section Evaluated in Argon . . . .. . . . . . . . . . . . . . . . . . . . 6.4.2 Terms in the General Energy Loss Cross Section . . . . . . . . . . . . . 6.5 Distributions in Energy Loss . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.6 Comparison with Experimental Data . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.6.1 The Energy Loss Fluctuations.. . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.6.2 An Optimal Estimator for Energy Loss . . . .. . . . . . . . . . . . . . . . . . . . 6.7 The Bethe–Bloch Approximation . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.7.1 Four Assumptions in the Bethe–Bloch Formula . . . . . . . . . . . . . . 6.8 Bremsstrahlung.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

69 69 69 70 70 70 72 73 75 75 76 79 80 81 82 83 85 88 88

7 Scattering of a Charge Moving in a Medium . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 89 7.1 The Scattering and Energy Loss Cross Section .. . .. . . . . . . . . . . . . . . . . . . . 89 7.1.1 Cross Section in the Resonance Region . . .. . . . . . . . . . . . . . . . . . . . 91 7.1.2 Scattering by a Constituent Charge . . . . . . . .. . . . . . . . . . . . . . . . . . . . 92 7.1.3 Modifications of Point-Like Nuclear Scattering.. . . . . . . . . . . . . . 93 7.1.4 Form Factors for Electron Constituents. . . .. . . . . . . . . . . . . . . . . . . . 95 7.1.5 Cross Sections dkT dkL with Form Factors . . . . . . . . . . . . . . . . . . . . 95 7.2 Scattering and Energy Loss Distributions.. . . . . . . . .. . . . . . . . . . . . . . . . . . . . 97 7.2.1 Single and Multiple Scattering .. . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 97 7.2.2 Evaluating Probability Maps in pT and pL . . . . . . . . . . . . . . . . . . . 98 7.2.3 Multiple Scattering and Energy Loss for Various Elements . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 99 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 100 Part III

Two Practical Applications

8 Relativistic Particle Identification by dEdx . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 8.1 Design Criteria for a Detector . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 8.1.1 Charged Particles to Be Distinguished, P /K/π/e . . . . . . . . . . . . 8.1.2 Choice of Medium to Maximise Discrimination . . . . . . . . . . . . . . 8.1.3 Estimator and Sampling Required . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 8.2 Identification of Secondaries by Ionisation Sampling, the Detector Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

103 103 103 104 105 106

xiv

Contents

8.3 The Performance of ISIS in the EHS Spectrometer . . . . . . . . . . . . . . . . . . . 8.3.1 Identification Performance by ISIS . . . . . . . .. . . . . . . . . . . . . . . . . . . . 8.3.2 Quantitative Check on Calculated Ionisation Errors . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

108 110 110 112

9 Ionisation Beam Cooling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 9.1 The Need to Compress Charged Particle Beams . .. . . . . . . . . . . . . . . . . . . . 9.1.1 Unique Experiments with High Energy Muon Beams . . . . . . . . 9.2 Ionisation Cooling of Muon Beams . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 9.3 Validation of Scattering Calculation for Hydrogen .. . . . . . . . . . . . . . . . . . . 9.3.1 Probability Maps for Energy Loss and Scattering.. . . . . . . . . . . . 9.3.2 Correlation Between Energy Loss and Scattering Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

115 115 115 116 117 119 119 120

List of Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 121 Index . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 125

Part I Building on Simple Ideas

1

Waves and sources

1.1

Energy Loss, Cherenkov, and Transition Radiation

As a charged particle moves through a material, it loses energy in various ways. Most obviously this happens as a result of direct electrical interactions with the nearby atoms and molecules of the material, leaving a track of excited and broken debris. Alternatively this may involve the emission of radiation that can travel away and be absorbed at a distance. An example of such radiation is shown in Fig. 1.1. Whether in the form of radiation or localised disruption, the loss of energy is not smooth but involves a sequence of discrete processes with probabilities determined by Quantum Mechanics. The broad spectrum of deposited energy is usually manifested as ionisation. Typically the scattering of the incident charge is small and the energy loss is much less than the energy of the charge. Thus the phenomenology of energy loss is better described in terms of the rate of energy loss along the track. Then there is Cherenkov radiation [1], a phenomenon first considered by Heaviside [2] in 1888/9 and by Sommerfeld in 1904. It was first observed by Cherenkov in 1934 [3] and its flux calculated by Frank and Tamm [4]. This is emitted in a uniform medium as a coherent cone of optical radiation, if the speed of the charge exceeds that of light in the medium at the optical frequency concerned. Finally, there is transition radiation emitted when a high velocity charge crosses the boundary between two different media. This was described by Ginzburg and Frank in 1946 as the radiation emitted in the optical region as a result of the need to satisfy boundary conditions at the interface [5]. Then in 1958 Garbyan showed that radiation should also be emitted in the X-ray region [6]. This turns out to be the most effective method of velocity measurement at the highest energies. Further references are given in Cobb’s thesis [7].

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 W. Allison, The Flight of a Relativistic Charge in Matter, Lecture Notes in Physics 1014, https://doi.org/10.1007/978-3-031-23446-0_1

3

4

1 Waves and sources

Fig. 1.1 The characteristic blue light of Cherenkov radiation emitted in water by relativistic electrons from the beta decay of radioactive nuclei. [Credit “ATR in Operation” by Idaho National Laboratory is licenced under CC BY 2.0]

It is easiest to understand these phenomena and the relationship between them by considering some rather simple ideas before engaging with the full details of the electromagnetic field [8].

1.2

Considering a Two-Dimensional Scalar Field

For maximum simplicity, let us think about a scalar field in two dimensions, φ(x, y, t), generated by a source moving steadily along the x-axis with velocity v. What can we learn? Many important features are already revealed in this grossly simplified model. A field that illustrates this generic model is the amplitude of the wake, the twodimensional water wave, emitted by a boat travelling at a constant speed over the surface of water that is more or less calm. (A full description of the physics of water waves including tidal waves and tsunamis is given in Chap. 6 of Fundamental Physics for Probing and Imaging [9].) There is something peculiar about this wave pattern. To an observer onboard the boat, the pattern of water waves appears stationary—it is constant and does not change with time. In this frame the frequency is zero, ω = 0, although there is a pattern—the wavelength λ is not zero. The wave vector k has magnitude 2π/λ with a direction perpendicular to the wavefront.

1.2 Considering a Two-Dimensional Scalar Field

5

On the other hand, in the stationary frame of the water, the wave is moving and has a non-zero frequency ω, related to kx and ky by the phase velocity, u(ω) = ω/ | k | .

(1.1)

The wake amplitude pattern may be expressed as an integral over Fourier compoφ (kx , ky , ω), thus nents,     φ(x, y, t) =

 φ (kx , ky , ω) eı(kx x+ky y−ωt ) dkx dky dω.

(1.2)

The phase relation, the one that maintains ω = 0 in the boat frame, requires that the phase velocity along the x-axis is equal to the boat velocity. In this way all Fourier components are phase-locked to the source. Thus ω/kx =| v |,

(1.3)

ω − k · v = 0.

(1.4)

or

This very general condition applies to any field due to a source that moves at constant velocity and is static in its own frame. In particular, it applies to every Fourier component of the electromagnetic field of a moving charge. It appears frequently in the algebra of more detailed descriptions in later chapters in the form of a delta function factor, δ(ω − k · v). In fact, the phase velocity for water waves depends strongly on ω, so that the group velocity, dω/dk, is different. The same is true in electromagnetism. In the simple two-dimensional case we can write down the component of the wave vector transverse to the motion using Pythagoras, ky =





 | k 2 | −kx2 =

ω2 ω2 ω − 2 = 2 v u v

v2 − 1. u2

(1.5)

There are two cases to consider, according to whether the resulting square root is real or imaginary.

6

1.2.1

1 Waves and sources

Real Transverse Wave Vector

If the source speed v is greater than the phase velocity u(ω), then ky is real. Since kx is also real, this means that a real wave is emitted at an angle θ between k, the normal to the wavefront, and the vector v with cos θ =

u kx = . |k| v

(1.6)

Since u depends on ω, so will the angle and the transverse wave vector. As a result, the bow wave of a speedboat is not so simple.

1.2.2

Imaginary Transverse Wave Vector

On the other hand, if the source speed v is less than the phase velocity u(ω), then ky is purely imaginary. Since kx is real, the tangent of the angle is imaginary too. A field component that propagates harmonically in time and in the x-direction, exp [ı(kx x − ωt)], varies exponentially in the transverse direction, exp (−y/y0). This is an evanescent wave of the kind that occurs in the forbidden region in the phenomenon of the total internal reflection of light. Its exponential range is −1/2  v 1 λ   v2 = βγ . y0 = = 1− 2 | ky | ω u 2π

(1.7)

Here, λ is the wavelength of a free wave of frequency ω, and we have used two new variables, β =

 −1/2 v v2 and γ  = 1 − 2 . u u

(1.8)

These familiar-looking variables appear here naturally without any reference to Relativity. The amplitude of this two-dimensional wave is shown in Fig. 1.2, growing and expanding rapidly for increasing values of v/u.

1.2.3

The Field of a Charge in Vacuum by Analogy

This picture can be applied to the emission of a two-dimensional field for the case where u(ω) = c, the velocity of light in vacuum. Since the source velocity v may not exceed c for a real mass, there is no free field emission. But for v < c, the evanescent field is described in Eq. 1.7 with transverse range y0 =

−1/2  λ v λ v2 βγ , = 1− 2 2π c c 2π

(1.9)

1.2 Considering a Two-Dimensional Scalar Field

7

Fig. 1.2 Field expansion in a 2D model for increasing values of the source velocity relative to the phase velocity, v/u

where β and γ have their conventional meanings, as in Special Relativity, β=

 −1/2 v2 v and γ = 1 − 2 . c c

(1.10)

The transverse expansion of the actual 3D electromagnetic field with the γ of a moving source is often described as relativistic in origin. It is usually derived by applying the Lorentz transformation to the field components of a static charge.

8

1 Waves and sources

The above simple picture suggests that it has a more basic origin. This is not surprising. Maxwell’s field equations themselves were not affected as a result of the modification of Newtonian Mechanics by the principles of Special Relativity. The important conclusion is that such a transverse field expansion is a general property of wave motion for any virtual field as the source velocity approaches the phase velocity. In the real world of three dimensions, the Cartesian form of the simple field is replaced by one with cylindrical symmetry and the expansion is described by a Bessel function rather than a simple exponential form, but the principle of the twodimensional result survives.

1.2.4

The Field of a Charge in Matter by Analogy

In a material medium the phase velocity of electromagnetic waves may be described by the refractive index n(ω), and thus u(ω) = c/n(ω), related to the dielectric √ relative permittivity r and magnetic relative permeability μr as n = r μr . The medium is often transparent, and n real, in the optical or X-ray region (Sect. 2.3.1).

1.2.4.1 In the Optical Region With n greater than one, the velocity of the source charge v can be greater than c/n. Then the conditions are the same as for the emission of a real wave in the simple 2D scalar model. The emission angle is given by cos θ = 1/β  = 1/nβ.

(1.11)

This corresponds to the emission of Cherenkov radiation for an electromagnetic field in three dimensions. On the other hand, if the source velocity v is lower than c/n, the field is an evanescent transverse wave with range y0 =

λ   β γ where β  = nβ. 2π

(1.12)

This virtual field expands as higher velocities are considered and the range goes to infinity as β  approaches one, which is the threshold condition for the emission of the real field just considered. At low velocities cos θ is greater than one. At higher velocities it is lower. When it falls below unity, the values of the angle θ and sin θ become real and physical, instead of imaginary.

1.2.4.2 In the X-Ray Region In this region the wave velocity is greater than c so that the threshold for the emission of free radiation is at a physically unattainable source velocity. The field is therefore virtual, a transverse evanescent wave, whatever the source velocity.

1.2 Considering a Two-Dimensional Scalar Field

9

The range is still given by Eq. 1.12. The greatest value of β  is when the source moves at kinematic limiting velocity v = c. Then β  = n and the limiting range, y0 =

1 λ n√ . 2π 1 − n2

(1.13)

For all frequencies in the X-ray range, the relation between n and the plasma frequency, ωp , is given by (see page 31)  n=

1−

ωp2 ω2

 where ωp =

ne e2 .

0 me

The limiting range then reduces to y0 =

c . ωp

(1.14)

This asymptotic range limits the expansion of the field with increasing velocity and is responsible for the Fermi plateau and the density effect [10]. Comparing Eqs. 1.7 and 1.14, this saturation sets in above γ ≈

ω . ωp

(1.15)

This limit is on the order of the Mean Ionisation Potential of the medium divided by the plasma energy. What kind of medium might show the most significant effect? The mean ionisation potential increases roughly linearly with the atomic number like 10Z, as shown on page 85. So high Z is beneficial and so is low plasma density which favours gases, in fact noble gases. Argon at standard density has a plasma energy 0.8 eV so that saturation is expected at γ ≈ 220. This back-of-the-envelope estimate varies from γ ≈ 70 for helium to γ ≈ 400 for xenon. In Chaps. 6 and 8 we shall see whether such crude guestimates are wide of the mark in practise.

1.2.5

Conclusions from the Two-Dimensional Field Study

It is remarkable that this simple picture is able to describe so many features, usually described as complicated consequences of the particular problem of an electric charge described by Special Relativity. There are differences of course. The extension to three dimensions involves the use of Bessel Functions which are closely related to sine, cosine, and exponential functions of radius away from r = 0. The replacement of the scalar by the vector electromagnetic field introduces further features, as shown in later chapters. As it turns out, the longitudinal electric

10

1 Waves and sources

field polarisation does not show some of the interesting features suggested by the scalar model. However, the transverse polarisation field does behave in this interesting way, and that is responsible for the features that turn out to be most significant. Real materials are necessarily absorptive and no description is complete if it ignores this. In Chap. 6 where this is taken into account in full, it will be seen that the transverse attenuation, as described by this simple model, appears in quadrature with the effect of absorption, as one might naively have guessed.

1.3

Considering a Photon with Effective Mass

Light passing through matter does so with a velocity different from its value for vacuum. Phenomenologically this is described by the refractive index n, such that the velocity u = c/n. But this is an unsatisfactory explanation. What actually happens, physically? This is a significant question. In free space every photon is massless and this property is directly related to its universal velocity c in all frames which are equivalent in this respect. But the principles of Special Relativity do not apply so simply within a material. The material rest frame permits a distinction to be made between observers.1 An electromagnetic wave in matter interacts with it and energy is exchanged with the moving charges within the material. Basically the energy of the wave is shared between being electronic motion and being a simple electromagnetic field. Depending on phase synchronisation, this interaction can delay or hasten the photon phase. This involves a defined coherent mix of the electromagnetic wave and the related electronic motion, coupled together in a normal mode, as with similar purely mechanical coupled oscillators. If the frequency of the electromagnetic wave is near to a natural electronic resonance, the electronic amplitude will be large and the mixing or coupling strong. This results in a large refractive index and also strong absorption if the electronic oscillation dissipates energy. The refractive index n may then depart from unity and have an imaginary component to describe absorption. In this view we ignore absorption for the sake of simplicity. The relationship between the energy and momentum of this renormalised wave is different from that of a free electromagnetic wave. The phenomenology can still be described by a refractive index, but this necessarily has a significant dependency on frequency. The phase velocity is u=

ω c = , n k

(1.16)

where the energy of a quantum of the renormalised field is h¯ ω with momentum h¯ k. 1

The principles of Special Relativity can only be established by considering the material medium itself as having a mass capable of recoil. However, this is a distraction. The photon still has to interact with the medium.

1.3 Considering a Photon with Effective Mass

11

In terms of n, the momentum is p=

h¯ ωn . c

(1.17)

Then we may use mass to parametrise the relation between energy and momentum, thus  h¯ ω  E2 p2 m∗ = − = 1 − n2 . (1.18) c4 c2 c2 Such a mass is not an invariant. It depends on ω in a non-trivial way and that is why we label it with an asterisk. This m∗ is the effective mass of a photon in a material. The picture is closely parallel to the use of the similar idea in the treatment of the eigenstates, that is normal modes, of electrons in crystals. In that case the normal mode is coupled state of many similar electrons whose wave functions are mixed by the crystal structure. There the effective mass, m∗ , is not equal to the free electron mass, me , and may be negative characteristic of hole states, as well as positive. The effective mass of a photon in a crystal differs from the electron case in that the relativistic parametrisation is appropriate.

1.3.1

A Field Mediated by Massive Photon Exchange

Electric charges interact through an exchange of photons, and this is true in a material as in vacuum. The wave equation for this photon field, simplified as a scalar field φ, is − h¯ 2 c2 ∇ 2 φ + m∗2 c4 φ = −h¯ 2 ∂ 2 φ/∂t 2 .

(1.19)

In the static approximation, solutions for a point source at the origin have the form φ(r) = A/r exp (−r/r0 ),

(1.20)

which gives rise to a potential of the same form with range r0 =

h¯ . m∗ c

(1.21)

12

1 Waves and sources

This is an example of the Yukawa potential with range equal to λc /2π, the reduced Compton wavelength of the exchanged quantum. We have an expression for m∗ in terms of n, Eq. 1.18, so that r0 =

c2 n h¯ λ = √ . √ c h¯ ω 1 − n2 2π 1 − n2

(1.22)

This is the same result that we found for the evanescent range in Eq. 1.13.

1.3.1.1 Photon Mass in the Optical Region In the X-ray region where n is less than one, the photon effective mass is real and positive, so that the Yukawa potential has the anticipated exponential form. But what happens in the optical region where n is greater than one? The algebra suggests that the effective mass is purely imaginary, but how may this be interpreted? In any elastic scattering process, the magnitude of the momentum exchanged (multiplied by c) is always greater than the energy exchanged. In other words, the energy–momentum four-vector of the exchange is space like. Assuming that the exchanged quantum has a real mass, the singularity in the scattering associated with it is outside the region of physical scattering angle. If the exchange quantum is described by an imaginary mass, the singularity associated with its exchange lies within the physical scattering region. This seems alarming until one realises that this is the Cherenkov condition. At the Cherenkov angle the virtual field propagates to infinity without evanescence. It is not wrong to picture Cherenkov photons being exchanged between a relativistic particle in a material and a photomultiplier many metres away. The corollary is that photons at the right angle and phase may be sent to contribute to the field of a moving particle, that is to accelerate it. In any case the conditions are frequency dependent, and, since at high frequency n tends to one from below, there are no unacceptable consequences of this appearance of singularities in the physical scattering region.

1.4

Considering Free Photon Emission

Describing a photon as “free” is only useful if the medium is approximately transparent, as discussed in Sect. 2.3.1. This only applies in the optical and the X-ray regions. Otherwise, the general discussion of later chapters is required.

1.4.1

Kinematic Conditions

If the incident charged particle of mass Mi , energy E, and momentum p emits a photon of wave vector k and frequency ω, energy and momentum conservation requires that Mi2 c4 = E 2 − p2 c2 = (E − h¯ ω)2 − (p − h¯ k)2 c2 .

(1.23)

1.4 Considering Free Photon Emission

13

Thence we have the relation − 2h¯ ωE + 2h¯ k · pc2 + (h¯ ω)2 (1 − n2 ) = 0,

(1.24)

where we have taken k = ωn/c. At low frequencies the last term of Eq. 1.24 is negligible because h¯ ω  E. At high frequencies (1 − n2 ) tends to (ωp /ω)2 and the whole term tends to (h¯ ωp )2 , which is negligible compared with E 2 . So this term can be dropped for all values of ω. Using the conventional relativity symbols β and γ , E = γ Mi c2 and p = βγ c so that −ωγ Mi c2 + k · βγ Mi c2 = 0, that is ω = k · βc.

(1.25)

This is the phase or Cherenkov condition that we saw previously in View A but now derived in a different way.

1.4.2

Emission with Recoil in a Periodic Medium

Figure 1.3 and Eq. 1.25 describe the kinematics of photon emission in the absence of crystal recoil. Any non-uniformity of the charge distribution in the medium or crystal can be described either in space F (r) or by its Fourier transform f (K 2 ). If the material structure is periodic in space, the transform will only have values for discrete values of wave vector K. This makes it possible to extend the condition for energy and momentum conservation, Eq. 1.25, by emitting or absorbing multiples of crystal momentum h¯ K. This picture is familiar from the usual description of energy and momentum conservation in Bragg scattering of X-rays, for example. In both cases the crystal momentum is carried by a huge mass M, so that the related energy transfer, (h¯ K)2 /2M, may be neglected. This suggests a generalisation of Eq. 1.25 of the form, ω = (k + nK) · βc, where n is an integer.

Fig. 1.3 Energy–momentum four-vectors for the emission of a photon

(1.26)

14

1 Waves and sources

This equation applies whether the spatial non-uniformity is atomic (crystalline) or macroscopic (foils or grains). With Bragg scattering, it is the crystalline structure that is important. With transition radiation, a variant of Cherenkov radiation, it is the macroscopic variations that matter, as we shall see in the next section.

1.5

Considering Diffracted Cherenkov Radiation

1.5.1

Radiation Emitted Passing Through a Slab

In Fig. 1.4 a charge with velocity βc is pictured passing normally through a slab of thickness and refractive index n, immersed in vacuum, for example. Between its entry at point A and its exit at point B, the charge may emit Cherenkov radiation in a certain frequency range dω with angle θ where cos θ (ω) = 1/βn(ω). The number of photons emitted by this two-sided slab, N2 , is given by the well-known formula derived on page 55, dN2 =

α 2 sin θ dω. c

Fig. 1.4 An observer sees a charge passing through the front and back of a slab

(1.27)

1.5 Considering Diffracted Cherenkov Radiation

15

Fig. 1.5 The finite wave front BC, shown in Fig. 1.4, and the related Fraunhofer radiation diffraction pattern in cos θ

We may re-express this in terms of solid angle d = 2π d cos θ at the Cherenkov angle with cylindrical symmetry   α 1 d2 N2 = sin2 θ × δ cos θ − . dω d 2πc βn

(1.28)

The finite thickness of the slab means that the radiation is emitted with a limited wavefront, BC, shown in Figs. 1.4 and 1.5. This limit causes the wave to be diffracted around the Cherenkov angle, as in the Fraunhofer pattern for a single slit in regular wave optics. So the wave elements should be integrated across the wavefront with linearly increasing relative phase from 0 to 2, where   π 1 = cos θ − . λ βn

(1.29)

The result is to replace the δ-function in Eq. 1.28 by sin2  , λ 2 leading to α sin2  α d2 N2 = sin2 θ × = 2 sin2 θ dω d 2πc λ 2 π ω

  1 −2 2 cos θ − sin . βn (1.30)

If is large, BC is wide and the broadening of the Cherenkov peak is small. But if is small, the peak is very broad in terms of cos θ . We should also make a change of variable. The emission angle of Cherenkov radiation is the internal emission angle inside the medium, θc . However, transition radiation is seen outside the slab at the external angle φ, as shown in Fig. 1.4. (This

16

1 Waves and sources

is drawn for the optical range with n > 1. In the X-ray range, n is less than 1, so that the angle φ is less than θc .) Indeed it can be so broad that, even if the Cherenkov condition corresponds to an unphysical angle with cos θ > 1, the radiation spreads into the physical region, cos θ ≤ 1. This sub-threshold Cherenkov radiation from a thin slab is called optical transition radiation.

1.5.1.1 But a Term Is Missing from This Description Where is the mistake? When a slab of matter is inserted into the path of the charge, we should consider that we have exchanged a slab of vacuum for an equivalent slab of matter.2 So the field amplitude should be the difference of amplitudes (as derived above), for the slab of material and the slab of vacuum. By Snell’s law,  sin θc = sin φ/n and cos θc = (1 − sin2 φ/n2 ). There is also an extra overall factor of order n2 arising from the redefinition of solid angle, d, that we ignore. With these corrections, the flux of transition radiation from a slab of material, thickness , is given by

 α sin2 φ d2 N2 2 ω 2 − sin2 φ − 1/β) = sin ( n dω d π 2ω 2c ⎞2 ⎛ 1 1 ⎠ . × ⎝ − cos φ − 1/β 1 − sin2 φ/n2 − 1/(βn)

(1.31)

The first term in the big round bracket has a singularity at the condition for Cherenkov radiation in the material of the slab. This term is responsible for Optical transition radiation, as above. The second term in this bracket is the most interesting. It has a singularity at the condition for Cherenkov radiation in a vacuum, the kinematic limit β = 1. This is responsible for X-ray transition radiation with its linear dependence on the relativistic γ of the charge. For this reason, it is of great importance for the identification of highly relativistic charged particles with γ of 1000 or more [7] [11]. This flux, 1.31, is of the form 4A2 sin2 . The sine-squared term with its argument in square brackets that multiplies the big round bracket represents the interference between two waves of amplitude A and −A emitted from the front and back surfaces of the slab with phase difference 2, as shown in Fig. 1.6. In the limit of a thin slab, the phase difference  goes to zero with , ensuring that an infinitesimally thin slab does not radiate, as physically required. 2 Slabs of vacuum and matter are equivalent if they appear the same, phase-wise. The only dependence on in Eq. 1.30 is in the phase factor, sin2 , so we may assume that this factor is the same for the two slabs.

1.5 Considering Diffracted Cherenkov Radiation

17

Fig. 1.6 The vectors from the front and back interfaces that interfere to give the transition radiation pattern

Later on page 20, other phenomenological consequences of the interference factor 4 sin2  will be explored. The transition radiation emitted from one surface of the slab alone, A2 , would be ⎛ α sin2 φ d2 N1 ⎝ = dω d 4π 2 ω

1 1 − sin2 φ/n2 − 1/βn

⎞2 1 ⎠ . − cos φ − 1/β

(1.32)

1.5.1.2 Integral and Differential Fields It is instructive to compare the meaning of the Cherenkov and transition radiation fields. The transition picture is an integral one in which the total is seen as the difference between the values of an indefinite integral field evaluated at the two ends of the path. The Cherenkov picture is a differential one in which the field as an integrand is to be summed along the track. They are equivalent.

1.5.2

Transition Radiation in the X-Ray Range

In this high frequency range, the refractive index is n(ω) = 1 − ωp2 /2ω2 .

(1.33)

Furthermore, because the emission is directly related to the Cherenkov condition in vacuum at φ = θc = 0 and β = 1, radiation is only observed at very small angles φ to the charged particle path and for β close to unity. sin φ = φ, cos φ = 1 − φ 2 /2 and

1 1 =1+ . β 2γ 2

(1.34)

These forms simplify the algebra and give many of the formulae traditionally used [7, 11].

18

1 Waves and sources

1.5.2.1 Two-Sided Slab and Single Interface With these substitutions, the flux for the two-sided slab, Eq. 1.31, becomes

αφ 2 d2 N2 2 ω 2 2 2 2 = 2 4 sin (ω /ω + φ + 1/γ ) dω d π ω 4c p 2  1 1 × − 2 , ωp2 /ω2 + φ 2 + 1/γ 2 φ + 1/γ 2

(1.35)

where the interference factor combining the two interfaces is 4 sin2  = 4 sin2

ω 2 2 (ωp /ω + φ 2 + 1/γ 2 ) , 4c

(1.36)

and the flux from a single interface, Eq. 1.32, with the approximations 1.34 becomes αφ 2 d2 N1 = 2 dω d π ω



1 1 − 2 2 2 2 2 ωp /ω + φ + 1/γ φ + 1/γ 2

2 .

(1.37)

This is an important result because it is identical to the formula most quoted in the literature for the flux of transition radiation. The fact that we have derived it by considering the effect of diffraction on Cherenkov radiation proves that transition radiation and diffracted Cherenkov radiation are one and the same. In his book Jelley has worried that it is difficult to tell them apart. He need not have worried. They are the same.

1.5.3

X-Ray Transition Radiation Dependence on Energy and Angle

We study the distributions of transition radiation in angle and frequency emitted at a single interface, as given by Eq. 1.37.

1.5.3.1 Angular Distribution To get the distribution of energy flux as a function of polar angle φ, we integrate over h¯ dω. At small angle, an element of solid angle is d = 2πφ dφ. The angular distribution of energy flux, S, is  −5/2 d2 N α dS = 2πφ × h¯ ω dω = h¯ ωp φ 2 × φ 2 + 1/γ 2 . dφ dω d 2

(1.38)

In Fig. 1.7, this distribution is plotted as a function of φ for various values of γ on a log–log scale. The flux has a sharp peak at an angle near 1/γ and the height of the peak increases with γ 2 . So the total energy flux increases linearly with γ .

1.5 Considering Diffracted Cherenkov Radiation

19

Fig. 1.7 The angular distribution of the flux of transition radiation emitted from a single interface

1.5.3.2 Energy Distribution Integrating over the angle, we get the energy flux distribution S as a function of photon energy E or rather of a universal scaling parameter ω E = . γ ωp γ Ep

(1.39)

 d2 N α ω dφ = −2 + (1 + 2a 2) ln(1 + 1/a 2) . dω d π

(1.40)

a= Thus dS = dE

 2πφ

This function is plotted as a function of a in Fig. 1.8. Most of the energy flux is emitted with quantum energy E < 0.5γ Ep . The plasma energy,  Ep = h¯

ne e2 ,

0 me

(1.41)

depends on the electron density ne . Higher values of γ give linearly greater energy fluxes by expanding the scale of E.

1.5.3.3 Total Energy Flux The total energy flux from a single interface (ignoring absorption) may be found by integrating the distribution in φ, S=

α γ Ep . 3

This increases linearly with γ as expected.

(1.42)

20

1 Waves and sources

Fig. 1.8 The frequency distribution of transition radiation emitted from a single interface

The typical photon emitted has an energy of order γ Ep /4, as is evident from the spectrum, Fig. 1.8. It follows that the number of photons emitted by an interface is a constant of order α. The individual photon energy increases linearly with γ and also with the plasma energy of the foil — that is with the square root of the electron density of the material.

1.5.4

X-Ray Transition Radiation Limit for Thin Slabs

The flux of radiation increases with the thickness of the slab, , as given by the interference factor, Eq. 1.36. This is a maximum when the phase  is π/2, that is when = 2πc/ω(ωp2 /ω2 + φ 2 + 1/γ 2 )−1 . Now 2πc/ω = λ the wavelength of the emitted X-ray radiation. Also φ ≈ 1/γ and ωp /ω is of the same order. The interference factor is of order 4 sin2 (γ 2 λ/ ), and for thicknesses greater than γ 2 λ, the flux is reduced by destructive interference. This distance is called the Formation Zone of transition radiation. It squares with the simple picture of the slab seen in the rest frame of the charge as a pulse of width /γ . When scattered and detected in the laboratory frame, this pulse has a width /γ 2 .

1.5.5

Summary of Conditions for X-Ray Transition Radiation

In principle, whenever a charged particle crosses a boundary between two materials, the continuity conditions for the electromagnetic field need to be satisfied and so

References Table 1.1 The properties of various materials that have been used to generate X-ray transition radiation

21

Density kg/m3 Z/A Plasma (eV) K shell (eV) K shell energy/plasma energy

Li 534 0.432 12.15 54 4.5

Be 1848 0.449 25.35 112 4.4

C 2210 0.499 29.22 284 9.7

Mylar 1400 0.520 23.74 284 12

transition radiation is always emitted. However, its energy, of order γ times the plasma energy of the foil h¯ ωp , should be well above the K-shell absorption band of the material; otherwise, it will be absorbed before it can escape. This is the reason that transition radiation is only detected at high γ . In summary, transition radiation is emitted (and not re-absorbed within the foil) if the following conditions pertain: • • • •

Photon energies of order γ times the plasma energy Photon energies far above the K-edge of the foil About α photons per interface Photon wavelength of order the foil thickness over γ 2 .

As a result, a transition radiator should be composed of very many foils made of the dense material of the lowest atomic number. The threshold for generating a measurable flux of X-ray transition radiation is a value of γ far above the ratio of the K shell energy to the plasma energy. The need for thick foils just adds to the problem of absorption. Equally, the foils have to be spaced sufficiently to avoid destructive interference between the back of one foil and the front of the next. However, at least this requirement does not increase absorption. Some numbers are given in Table 1.1. Lithium is best and was used by Willis et al. in the experiment in which they detected evidence for bound charm particles at the CERN ISR [11]. Carbon fibres have also been used and mylar foils were used in the early experiment by Harris et al. [7, 12].

References 1. Wikipedia, Cherenkov Radiation. https://en.wikipedia.org/wiki/Cherenkov_radiation 2. P.J. Nahin, Oliver Heaviside: The Life, Work, and Times of an Electrical Genius of the Victorian Age (1988), pp. 125–126. ISBN 978-0-8018-6909-9 3. P.A. Cherenkov, Dokl. Acad. Nauk SSSR 2, 457 (1934) 4. I. Frank, I. Tamm, Dokl. Acad. Nauk SSSR 14, 107 (1937) 5. V.L. Ginzburg, I.M. Frank, Sov. Phys. JETP 16, 15 (1946) 6. G.M. Garibyan, Sov. Phys. 6, 1079 (1958) 7. J.H. Cobb, A Study of Some Electromagnetic Interactions of High Velocity Particles with Matter Oxford DPhil. Thesis (1975) 8. W.W.M. Allison, P.R.S. Wright, The Physics of Charged Particle Identification, Contributed to Formulae and Methods in Experimental Data Evaluation, vol. 2, ed. by R.K. Bock et al. (European Physical Society, CERN, 1984)

22

1 Waves and sources

9. W.W.M. Allison, Fundamental Physics for Probing and Imaging (Oxford University Press, 2006). ISBN 0-19-920388-1 10. E. Fermi, The Ionization Loss of Energy in Gases and in Condensed Materials. Phys. Rev. 57, 485 (1940) 11. J.H. Cobb et al., Transition Radiators for Electron Identification at the CERN ISR (1977) https://doi.org/10.1016/0029-554X(77)90355-X 12. F. Harris et al., The Experimental Identification of Individual Particles by the Observation of Transition Radiation in the X-ray Region (1973). https://www.sciencedirect.com/science/ article/abs/pii/0029554X73903753

Part II Calculations in Classical Electromagnetism

2

The Influence of a Passing Charge

2.1

Electric and Magnetic Fields in Vacuum

2.1.1

The Force on a Charge

Electric charge q is defined using Coulomb’s inverse square law. This says that the repulsive force F between two equal stationary charges q separated by a distance R is proportional to q 2 /R 2 . In the SI system of units, if F is in Newtons, R in metres, and q in Coulombs, then F =

1 q2 , 4π 0 R 2

(2.1)

where the constant 0 = 8.86 × 10−12 Farads per metre. It is found that every charge in nature is an integer multiple of e = 1.6 × 10−19 Coulombs. Furthermore the repulsive force between any two charges, q and q  , is found to be F =

1 qq  . 4π 0 R 2

(2.2)

This factorisation means that the electric field of q at a distance R may be written as E=−

1 q s, 4π 0 R 2

(2.3)

where s is a unit vector pointing towards q.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 W. Allison, The Flight of a Relativistic Charge in Matter, Lecture Notes in Physics 1014, https://doi.org/10.1007/978-3-031-23446-0_2

25

26

2 The Influence of a Passing Charge

Then the mutual force F = q E is equal and opposite for any q  , provided both charges are stationary. Any charge q that is moving with a velocity v experiences a magnetic force in addition, F = q(E + v × B),

(2.4)

where B is the magnetic field (strictly, the magnetic flux density) at that point.1 This is called the Lorentz force. How do these E and B fields depend on the charge and current densities, in general?

2.1.2

Maxwell’s Field Equations

A point charge q with velocity v that passes through the origin at time t = 0 has charge and current densities, ρ and J, given by ρ(r, t) = q δ 3 (r − vt),

J(r, t) = v ρ(r, t),

(2.5)

where δ 3 is understood as the product of three Dirac delta functions, one for each Cartesian component of its vector argument. Within any local empty region, so-called free space, the E and B fields are related to one another and to the local densities there by Maxwell’s equations,2 div B = 0,

0 div E = ρ, ∂B 1 ∂E , curl E = − , curl B = J + 0 ∂t μ0 ∂t

(2.6)

where the constant μ0 is 4π × 10−7 SI units (Newtons per Ampere2) and 0 = 1/(μ0 c2 ), c being the speed of light in free space. The two divergence equations are known as Gauss’s law and express the fact that the lines of electric flux begin or end on elements of charge only. Since magnetic monopole charge is not observed, lines of magnetic flux are endless.

1

The sloppier name “magnetic field” is often used, but this can cause confusion with the H field. The description “B field” is recommended. 2 Although we have introduced the definitions of sources and fields quite carefully, we skip over the justification for Ampere’s law and Faraday’s law that are well described in standard books on Electricity and Magnetism [1, 2].

2.1 Electric and Magnetic Fields in Vacuum

27

The third equation is the differential form of Faraday’s law of electromagnetic induction. The last equation is the differential form of Ampere’s law with the addition of the term called the displacement current density, introduced by Maxwell himself. It ensures that electric charge is locally conserved. This is proved by taking the divergence of both sides and substituting for div E div J = −

2.1.3

∂ρ . ∂t

(2.7)

A Short Field Pulse from a Moving Charge

What does a stationary observer detect when a charge passes by? The components of the electric field are seen to rise as the charge approaches and then fall away again as it passes on its way, as sketched in Fig. 2.1. The various electric and magnetic field components form a pulse that can be analysed into frequency components. Classically, these have a smooth spectrum as a function of angular frequency, ω.

2.1.3.1 The Pulse in the Semiclassical Picture In the semiclassical quantum approximation, this spectrum is reinterpreted as a flux of photons, each with energy h¯ ω. This follows the simple reinterpretation of classical fields, originally made by Max Planck in his quantum description of black

Fig. 2.1 (a) Sketch of E field lines for a stationary charge. (b) and (c) The longitudinal and transverse E field strength pulses from a passing charge seen by a stationary observer

28

2 The Influence of a Passing Charge

body radiation. The statistical fluctuations in the absorption of these photons can be described by a spectrum of “collisions” each with a cross section. In the same way each of these photons with wave number k carries a momentum h¯ k. The photons also have a polarisation that encodes the direction of the vector fields, E and B. The significance of the prefix semi is that, in this approximation, the radiation reaction can be neglected. This means that the energy and momentum of each photon is very small compared to the energy and momentum of the moving charge itself. Phenomena for which this approximation is not true, such as Bremsstrahlung and Synchrotron radiation, are discussed by Ginzburg [3] but omitted from this book. This is not a disadvantage in general because it is the information from the quieter and softer photons that are most useful in non-destructive particle detection. Processes with harder photons are infrequent, statistically noisy, and also more disruptive to the energy and momentum variables of the charge, which are the quantities of interest. They are important in detectors designed to absorb or scatter the incident charge, like shower counters and muon absorbers. These semiclassical photons may be virtual and tethered to the charge itself, they may be freely emitted, or they may be absorbed at a smaller or greater distance from the path of the charge. Which of these is possible depends essentially on the medium that the charge is traversing. In fact the vacuum case is rather sterile. Responsive electromagnetic media, like those in detectors, give the detectable phenomena that depend crucially on dispersion and absorption.

2.2

Equations Modified for Media

If materials are present, the fields generally induce distributions of charge and current. These are called bound charges and currents. However, they are not distinguishable in principle from the imposed free charges and currents. The distinction is useful for practical application, calculational convenience, and visual interpretation. Thus ρ = ρfree + ρbound and J = Jfree + Jbound.

(2.8)

It is important to realise that any problem analysed making use of this distinction can also be analysed without it, in principle. In that case, sources of all kinds are treated as being free and embedded in vacuum. Comparison between these two equally valid descriptions raises an apparent contradiction.

2.2 Equations Modified for Media

2.2.1

29

Physical Interpretation of Resonant Dispersion

An electromagnetic wave travelling in a medium, glass for example, moves with speed c/n, where n is the refractive index. If the same phenomenon is viewed with the induced atomic charges and currents in the glass embedded in vacuum, then the wave moves with speed c. How can that be? Although it moves between atoms at c, it is absorbed coherently by every atomic electron it passes and re-emitted with a phase change that depends on how close the wave is to the resonant frequency of the atomic electron. This phase change retards or advances the net movement of the wave resulting in the wave velocity given by the refractive index. So the two descriptions match. However, it would be much more involved to solve explicitly for the motion of both free and bound charges and currents. Drawing the artificial distinction between them makes calculations simpler, that is all. In a material medium the electric field E may induce a separation of charge, the dipole moment per unit volume, or polarisation P. Thence ρbound = −div P ρfree = div( 0 E + P).

(2.9)

Defining a new field that Maxwell called the Displacement, D = 0 E + P,

(2.10)

Gauss’s law of electrostatics in a medium becomes div D = ρfree .

(2.11)

Similarly the magnetic field B may induce a magnetisation, a magnetic moment per unit volume M. In addition, there is the bound current from the rate of change of polarisation P, J = Jfree + curl M +

∂P . ∂t

(2.12)

Defining the new field, H=

B − M, μ0

(2.13)

the final Maxwell equation becomes curl H = Jfree +

∂D . ∂t

(2.14)

30

2 The Influence of a Passing Charge

The second term on the right, the Displacement Current Density, now has two parts: the polarisation current density which is a flow of real charge, and the vacuum displacement current density which is a flow of “charge on credit.” The fields D and H depend on the option to distinguish free and bound charges and currents. As such they are not physically measurable, and this is true for each term in the two Maxwell equations that involve them. The other two Maxwell Equations are unaffected by media and each term describes measurable quantities. Gathering together the four general Maxwell equations div B = 0,

div D = ρfree , ∂D ∂B . curl E = − , curl H = Jfree + ∂t ∂t

2.3

(2.15)

Equations for Linear Media

The expressions in Eq. 2.15 do not assume that the medium is linear. However, the distinction between free and bound charges or currents is particularly useful if linearity applies. In general, a linear relationship between two vector fields, like E and D or B and H, is a second rank tensor field. However, in an isotropic material it reduces to a scalar field and in the following we will assume that to be the case for simplicity. Thus D = r 0 E and B = μr μ0 H,

(2.16)

with the relative dielectric permittivity r = 1.0 in vacuum, and the same for the relative magnetic permeability, μr = 1.0 in vacuum. Furthermore, if r and μr are independent of position and time, they can be brought through the differential operators. Maxwell’s equations may then be written as div B = 0,

r 0 div E = ρfree , ∂B ∂E . curl E = − , curl B = μr μ0 Jfree + μr μ0 r 0 ∂t ∂t

(2.17)

It is not physically possible for the induced charge and current in a material to respond instantaneously to an incident electric or magnetic field. As a result these two scalars are functions of frequency ω. In the limit of high frequency, there can be no response and they both must tend to unity. Furthermore, both scalars are described by complex functions of frequency in which the imaginary part of the response describes absorption of energy by the medium, that is by the induced charge and current being out of phase with the applied fields.

2.3 Equations for Linear Media

31

Indeed, this absorbed energy is of great interest. It is what drives the signal measured in any radiation detector. Here we look at the transparent approximation where there is no energy transfer.

2.3.1

Transparent Media Approximation

In so far as r and μr are real, photons travel in a medium without attenuation, much √ as in free space, but with a speed c/n, where n = r μr is the refractive index. Broadly speaking, all materials are highly absorptive in the ultraviolet region which covers the frequencies of electronic resonance. In practise this extends up to and including the K-shell absorption edge of the highest atomic number, Z, in the material. Above this there is the X-ray region in which the material is transparent and phenomenology in terms of free photons is useful. At frequencies below the ultraviolet absorption bands, a material may, or may not, be transparent, depending on the effect of molecular resonances. Free photons in this range are said to be in the optical region. As in any resonant system, like a pendulum, when driven at frequencies below resonance, the response has the same phase as the driving field. The refractive index in the optical region is generally greater than unity and free photons travel with a phase velocity speed less than c. Conversely, in the X-ray region the phase of the response is negative, n is less than unity, and the phase velocity of photons is greater than c. In this frequency range all electrons in the medium behave as if they were essentially free and unbound, as in an ionised plasma.

2.3.2

An Ionised Plasma

If unbound electrons with uniform density ne in a stationary matrix of positive charge all move a distance x, there will be a polarisation and a related electric field E P = ne ex = ( r − 1) 0 E. The harmonic field will accelerate each electron, eE = me

d2 x = −me ω2 x dt 2

for motion of frequency ω. In terms of the plasma frequency, ωp2 = ne e2 /( 0 me ), the relative permittivity is

r (ω) = 1 −

ωp2 ω2

.

(2.18)

32

2 The Influence of a Passing Charge

Since r is less than unity, the phase velocity is greater than c, which might initially seem unphysical. However, the dependence on ω ensures that the group velocity is less than c. Photon motion is governed by the speed of a pulse which is the group velocity, not the phase velocity. For ω less than ωp , r is negative and the refractive index is pure imaginary. In that case harmonic waves decay exponentially with distance. This is not a situation in which energy is absorbed—rather it is repelled from a forbidden region, as in the phenomenon of total internal reflection in optics and the reflection of radio waves by the ionosphere. Next we should explain the relevance of this electromagnetism to experimental measurements in physics.

2.4

Practical Measurement of a Moving Charge

2.4.1

Relativistic Kinematics and Particle Identification

The practical problem in a laboratory experiment is to detect and analyse a single charged particle of unknown mass m and electronic charge q, usually ±1.6 × 10−19 coulombs, moving at an unknown relativistic velocity v. The kinematics of such a particle is fully described by its energy E and 2 momentum vector  p. These are related to m and v by E = γ mc and p = γ mv, 2 2 where γ = 1/ 1 − v /c and c is the speed of light in vacuum. (Since these relationships are kinematic, not electrodynamic, in origin, they are independent of the presence of materials and their effect on the speed of light.) The momentum vector p is comparatively easily measured by tracking the passage of the particle through a region with static magnetic field B. The trajectory follows a helical path with radius of curvature R which can be measured. Experimentally this calls for a sequence of accurate position measurements to be made with least disturbance to the particle’s energy and direction. In practice this is made substantially more difficult because it needs to be done for a very large number of tracks simultaneously. The radial force on the charge, q(v × B), is equal to the inertial mass γ m times the radial acceleration v 2 /R. Thence R = p/(qB sin θ ) with all quantities in SI units and θ the angle between p and B. If q is the electronic charge and p is expressed in GeV/c, R in metres, and B in tesla, then R = p/(0.3B sin θ ). The numerical factor here, 1/0.3, is actually 109 /c. That leaves the more difficult task of completing the measurement of the fourvector that will give the mass m. And it is the mass and charge of a particle that characterise many of its other quantum numbers of most interest to the physics experiment. This task is called particle identification. In the non-relativistic range particle identification can be achieved by measuring kinetic energy using absorption calorimetry or v using time of flight. These techniques are relatively straightforward in principle.

2.4 Practical Measurement of a Moving Charge

33

Fig. 2.2 A diagrammatic illustration of the Cherenkov effect, the wavefronts emitted at an angle θc by a super-luminous charge passing through a medium

However in the relativistic range, as v approaches c asymptotically, E also approaches pc, and these methods fail. As a result, less destructive methods of particle identification that exploit the electromagnetic effects of relativistic charges must be used. These are the subject of this book. There are three basic effects and we outline the phenomenology of each in turn.

2.4.2

Cherenkov Effect

The first is the emission of radiation by a free charge as it moves through a material at a constant speed greater than the speed of light in that material. The emission of the electromagnetic shock wave is similar to a supersonic shockwave and its surface water–wave equivalent. In the right angle triangle ABC shown in Fig. 2.2, AB is the distance, βct, travelled by the charge in the time, t, that the radiation has travelled the shorter distance BC, ct/n, where n is the refractive index. Then the Cherenkov radiation is emitted at the angle ABC with cos θc = 1/nβ. Note that at a velocity below emission threshold, cos θc is greater than one, while sin θc and θc itself are given by pure imaginary numbers. A quantitative derivation of the flux of emitted photons is less easily derived but is calculated later, first on page 78 and again by a quite different method on page 55.

2.4.3

Transition Radiation

Transition radiation is emitted from the point where a relativistic charge crosses the boundary between two different media, as shown in Fig. 2.3. Essentially the two fields of the charge, that in the first and that in the second medium, do not satisfy the electromagnetic boundary conditions at the interface. We are familiar with the idea that when a free field passes from one medium to another, an additional field, the reflected wave, leaves the interface to match the boundary

34

2 The Influence of a Passing Charge

Fig. 2.3 A sketch suggesting the transition radiation emitted in the forward and backward directions as a relativistic charge crosses the boundary between two different dielectric media

conditions. In the same way in this case a new free radiation field is emitted at the interface. The optical component of transition radiation was predicted by Ginzburg and Frank [5]. However, there is also a component in the X-ray range, first studied by Garibyan [6]. This has remarkable and unexpected properties which can be used to discriminate between charge velocities in the ultra-relativistic velocity range. The original calculations derived the flux from the details of matching fields at the boundary of the two media. However, in Sect. 1.5 we already identified transition radiation as diffracted Cherenkov radiation. Then quite separately again, in Sect. 4.2 we will relate it quantitatively to the apparent acceleration of the charge as seen by a stationary observer. These three methods all give the same results.

2.4.3.1 A Moving Slab of Dielectric Here we ask what an observer co-moving with the charge would see when it passes through a slab of dielectric of thickness . The slab would appear to have a physical thickness /γ to that observer and so generate frequencies of order βγ c/ in that frame. This pulse may be scattered by the charge, in particular in the forward direction with a frequency that will be seen as blue-shifted in the laboratory frame with frequencies of order γ 2 c/ . The probability of emitting such a photon will be of order α with factors depending on the size of the pulse related to the electron density in the slab. At high values of γ , of order 1000, the factor γ 2 throws the radiation frequencies up into the X-ray range. At lower values of γ , the radiation fails to escape the K-shell absorption edge of the slab atoms and so is not observed.

2.4 Practical Measurement of a Moving Charge

35

Fig. 2.4 A sketch of the energy loss, dEdx, by a charge in a material as a function of p/mc

The width of the pulse gives a limited range to the γ -dependence of the radiation. This is the formation zone effect discussed already on page 20.

2.4.4

Energy Deposited in the Medium

The third effect that can be used to identify relativistic particles is the energy loss deposited locally in the material media through which the charge passes. It is often called dEdx, although it is not a true mathematical differential coefficient, as there are random fluctuations in the energy deposited, dE, in each step, dx, along the path of the charge. It includes energy lost as excitation of the medium, as well as its ionisation.3 The dependence of dEdx on p/mc or βγ on a logarithmic scale is sketched in Fig. 2.4. At non-relativistic velocities, dEdx falls rapidly as 1/v 2 in all materials. This is a very pronounced effect that facilitates simple particle identification in that velocity range. In low density materials in the relativistic range, there is a gentle logarithmic velocity dependence giving a total increase of 30–40%, at most, between p/mc of 4 and a few hundreds. This is called the relativistic rise. At higher velocity the electromagnetic field becomes asymptotic, as suggested qualitatively in the discussion of Sect. 1.2.4.2. Then dEdx saturates as first

3

In real materials any energy that is initially deposited as excitation often becomes ionisation if it is above the ionisation threshold of molecular impurities. In fact in noble gases such impurities are usually added to the gas mixture to ensure that excited states and ultraviolet photons are quenched by such transfer to secondary ionisation.

36

2 The Influence of a Passing Charge

described by Fermi in 1940 [4]. The effect of this Fermi Plateau with its saturated electromagnetic field is termed as the density effect. In most materials, however, the density effect suppresses any useful variation of measured dEdx in the relativistic range.

2.4.4.1 Fluctuations Measurements of ionisation and excitation are dominated by large statistical fluctuations. These arise from the quantisation of the electromagnetic field. The absorption of a quantum by the material medium may be described as a collision process with a cross section that relates to the probability of occurrence. These fluctuations are called Landau Fluctuations, after Landau who first attempted to calculate them, rather unsuccessfully as it happened [5]. The effect of these fluctuations may be calculated by Monte Carlo process or more precisely by folding collision probability distributions. Anyway, the ability to measure velocities in the region of the relativistic rise depends on overcoming the poor resolution due to these fluctuations. The first study to suggest how this might be done was published by Alikhanov in 1956 [6]. It was evident that a hundred or more separate measurements would be necessary for each particle. The energy spectrum of individual collisions contributing to dEdx is very broad. It stretches from soft collisions that occur with high probability and smooth statistics, all the way up to very hard collisions, such as those describing Rutherford scattering from nuclei, which are rare, but cause disruptive statistical behaviour. Attempts to calculate the relativistic rise and the shape of the Landau distribution have proved difficult and early attempts failed to match measurements in either case [7]. The huge range of collision energies and cross sections made Monte Carlo techniques difficult to use to make reliable predictions. Meanwhile on the experimental side the fluctuations and their long tail hampered attempts to determine a simple estimator like the mean dEdx signal, even from a large number of measurements. Limited progress was made using ad hoc estimators, such as the mean of the lowest 60% of a number of measurements, as discussed by Harris [8] and Cobb [9].

References 1. B.I. Bleaney, B. Bleaney, Electricity and Magnetism, 3rd edn. (Oxford, 1976) 2. P. Lorrain, D.P. Corson, F. Lorrain, Electromagnetic Fields and Waves, 3rd edn. (Freeman, New York, 1988) 3. V.L. Ginzburg, Applications of Electrodynamics in Theoretical Physics and Astrophysics (Gordon and Breach, 1989) 4. E. Fermi, The Ionization Loss of Energy in Gases and in Condensed Materials. Phys. Rev. 57, 485 (1940) 5. L.D. Landau, J. Phys. (USSR) 8, 201 (1944). Reproduced in Collected Papers of Landau ed. by ter Haar (Gordon and Breach, 1965) 6. A.I. Alikhanov et al., High Precision Measurement of the Ionizing Power of Fast Charged Particles with the Help of Multilayer Proportional Counters, in Proc CERN Symposium, vol. 2 (1956), pp. 87–98

References

37

7. J.H. Cobb et al., The ionisation loss of relativistic charged particles in thin gas samples and its use for particle identification. I. Theoretical prediction (1976). https://doi.org/10.1016/0029554X(76)90625-X 8. F. Harris et al., The Experimental Identification of Individual Particles by the Observation of transition radiation in the X-ray region (1973). https://www.sciencedirect.com/science/article/ abs/pii/0029554X73903753 9. J.H. Cobb, A Study of Some Electromagnetic Interactions of High Velocity Particles with Matter Oxford DPhil. Thesis (1975)

3

The Field of a Moving Charge

3.1

Field Equations in Vacuum or Non-dispersive Media

The general form of Maxwell’s equations in a medium is div B = 0,

div D = ρ, ∂D ∂B , curl E = − , curl H = J + ∂t ∂t

(3.1)

where, here and in the following, the subscripts of ρf ree and Jf ree are understood but have been dropped. With the assumptions that the magnetic and dielectric properties are linear, isotropic, homogeneous, and time independent, D = r 0 E, H = B/(μr μ0 ).

(3.2)

With the assumed properties of homogeneity and time independence, the constants can be brought outside the differential operators, div B = 0, curl E = −

div E =

ρ ,

r 0

∂B ∂E , curl B = μr μ0 J + μr μ0 r 0 . ∂t ∂t

(3.3)

The solutions of these equations describe electric and magnetic waves generated and received by free charge and current densities as sources and sinks. However, in this form they do not appear to have any features in common with a simple wave equation for a scalar field ψ with wave velocity c in the absence of a source, ∇ 2ψ −

1 ∂ 2ψ = 0. c2 ∂t 2

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 W. Allison, The Flight of a Relativistic Charge in Matter, Lecture Notes in Physics 1014, https://doi.org/10.1007/978-3-031-23446-0_3

(3.4) 39

40

3 The Field of a Moving Charge

There are three reasons for this apparently arcane formulation: • The description of the polarisation by the entwined E and B fields • The influence of the material response • The language used historically to describe the electromagnetic field The two vector fields E(r, t) and B(r, t) with six components can be reduced to four by describing them using a vector field A(r, t) and a scalar field φ(r, t). These are called the potential fields. It is worth remembering that the E and B fields were introduced by Faraday and others only as an aid to understand practical problems. Ultimately, it is the forces that have the greater reality. The role of A and φ is similar, although we have to re-educate our intuition to appreciate their significance. We postpone the discussion of solutions in the presence of materials to Sect. 3.3 and consider first vacuum solutions, with constants r = 1 and μr = 1.

3.2

Potentials and Field Solutions in Vacuum

There are four steps to solving Eq. 3.3 in vacuum.

3.2.1

Wave Equations for the Potentials

We can find recognisable wave equations by changing variables to the magnetic vector potential A and electric scalar potential φ. The substitution is B = curl A, E = −grad φ −

∂A . ∂t

(3.5)

These substitutions ensure that the Maxwell equations for div B and curl E are satisfied identically, leaving the remaining two Maxwell equations to determine A and φ in terms of J and ρ. In fact, this does not fix A and φ uniquely—meaning that there are different solutions for A and φ that yield the same physics, an ambiguity that should be resolved. Taking care of this ambiguity is termed choosing the gauge and any transformation that changes the gauge leaves physical reality unchanged. One solution to this ambiguity is to choose the Lorenz gauge. This imposes the subsidiary condition div A + 0 μ0

∂φ = 0. ∂t

(3.6)

3.2 Potentials and Field Solutions in Vacuum

41

The remaining two Maxwell equations then become − ∇ 2 A + 0 μ0

∂ 2A = μ0 J, ∂t 2

(3.7)

∂ 2φ ρ = , 2

0 ∂t

(3.8)

−∇ 2 φ + 0 μ0

a set of four separate inhomogeneous wave equations for the potentials Ax Ay Az φ with source μ0 Jx μ0 Jy μ0 Jz ρ/ 0 . In each case the wave velocity is 1 c= √ .

0 μ0

3.2.2

(3.9)

Retarded and Advanced Potentials

The second task is to solve these four equations. It will be sufficient to solve one of them and then simply write down the result for the other three by analogy. Figure 3.1 shows a region V1 containing a charge density ρ that gives rise to a potential φ at a field point P, position r relative to the origin at O. The potential field is built up by the superposition of contributions from elements dV1 at position r1 . R is the distance from P to an element dV1 and s is a unit vector in the direction of dV1 . Outside the region V1 , there is no charge, so that φ satisfies the source-free wave equation − ∇ 2 φ + 0 μ0

∂ 2φ = 0. ∂t 2

(3.10)

If R is large compared with any wavelength involved and if V1 is small compared with any wavelength, the solution will be dominated by spherically symmetric terms

Fig. 3.1 The potential at P due to a charge in a region V1 , shown shaded

42

3 The Field of a Moving Charge

of the form φ=

f (R − ct) g(R + ct) + . R R

(3.11)

The first term is known as the retarded potential and represents an outgoing spherical wave emitted by the charges in V1 . The second term is known as the advanced potential and represents an incoming spherical wave, that is one absorbed by the charges in V1 . The two cases (with their initial and final conditions) are related by time reversal. There may also be waves that are neither emitted nor absorbed by the charges. Their presence is always allowed, but they are not coupled to the problem. The effects of non-spherical terms at small R fall off faster than 1/R and so become negligible at large distances. In general non-spherical distributions arise by the linear superposition of solutions for different elements dV1 and the polarisation of their sources. In the static approximation we have the simple Coulomb result for the element of potential due to the charge ρ(r1 )dV1 dφ =

1 ρ(ri )dV1 . 4π 0 R

(3.12)

For the retarded potential, it is straightforward to extend this in the required form, that is as a function of R − ct, dφ(r, t) =

1 ρ(ri , t − R/c) dV1 . 4π 0 R

(3.13)

To obtain the advanced potential, the sign of time should be inverted. In the following we look at the retarded case only. This contribution to the retarded potential has then to be integrated over the whole source volume V1 , being careful that the density ρ is to be evaluated at the retarded time t −R/c that varies with R. This is conventionally indicated by the use of square brackets, thus  1 [ρ] dV1 . φret (r, t) = (3.14) 4π 0 R Then the retarded potentials, Ax , Ay , and Az , are related to the corresponding components of J integrated at the retarded time in the same way. In each case the integration is less simple than it appears because of the retardation. The influence of each different source element, evaluated at an earlier time t − R/c, reaches the field point P at the same time t.

3.2 Potentials and Field Solutions in Vacuum

3.2.3

43

The Liénard–Wiechert Potentials for a Point Charge

For the case of a point charge, further simplification is possible. Assuming that such a charge q moving at constant velocity βc passes through the origin at time t = 0, the source terms are then the charge density, ρ = q δ 3 (r1 − βct), and the current density, j = βc × ρ. In terms of retarded time, the charge density is   | r − r1 | ) . ρ(r1 , t − R/c) = q δ 3 r1 − βc(t − c

(3.15)

To get the retarded potential φ(r, t), the δ-function should be integrated over r1 . Using the identity δ(az1 ) = δ(z1 )/a with a = (1 + β · s), where s is the unit vector in the apparent direction of the charge at time t (that is to say the actual direction at the earlier time t − R/c), we get

1 q 4π 0 R(1 + β · s) ret

(3.16)



βc qμ0 A(r, t) = . 4π R(1 + β · s) ret

(3.17)

φ(r, t) = and by analogy

These are the Liénard–Wiechert potentials at point P due to a point charge q at Q in terms of its distance R and direction s, evaluated at the earlier time, t − R/c. They are physically reasonable in that they relate to the charge as it appears to the observer at P at time t.

3.2.4

Solution for the E and B Fields

The E and B fields for a point charge are derived from these Liénard–Wiechert potentials using equations, B = curl A E = −

∂A − grad φ. ∂t

(3.18)

Although taking the gradients and time derivatives required is complicated by the retardation condition, it turns out that there is a simpler way to obtain the radiation fields, as discussed in the next chapter.

44

3 The Field of a Moving Charge

3.3

Field Solutions in Media

In a dispersive medium the velocity of light no longer has a single value but varies with frequency. Retarded and advanced prescriptions are not well defined when the propagation time depends on frequency. Absorption too cannot be ignored. These necessary first-order features have a major impact on the electromagnetic field and all its manifestations. We need to go back a step to the Maxwell field equations in isotropic and linear media, as on page 30, div B = 0,

r 0 div E = ρ ∂B ∂E curl E = − , curl B = μr μ0 J + μr μ0 r 0 . ∂t ∂t

(3.19)

Finding dynamic solutions to these equations is not simple. The substitutions, written conventionally as D = r 0 E and H = B/(μr μ0 ), are deceptive shorthand for something rather more complicated that we now explore. The scalars r and μr may be independent of position and time, but they do depend on frequency and even on wave number, in general.

3.3.1

Non-local Relations Between Fields

3.3.1.1 Non-locality in Time  (r, ω) with Consider the Fourier transforms  E (r, ω) and D  ∞ 1  (r, ω) exp(−ıωt) dω, E(r, t) = √ E 2π −∞  ∞ 1  (r, ω) exp(−ıωt) dω. D(r, t) = √ D 2π −∞

(3.20)

Now we can relate the elements of the two fields, frequency by frequency, thus  (r, ω) = r (ω) 0 E  (r, ω). D

(3.21)

This implies a relationship between D and E in time of the form, 1 D(r, t) = 2π





∞ −∞

r (ω) 0

∞ −∞





E(r, t ) exp(ıωt ) dt



exp(−ıωt) dω. (3.22)

There are two points to note about Eq. 3.22. The first is the symmetry of r = 1 + ı 2 as a function of ω. Since E(r, t  ) and D(r, t) are both real field values independent of ω, the only functions in the integration over ω are exp[ıω(t  − t)] and r . When integrated

3.3 Field Solutions in Media

45

over all ω, the result can only be real if their symmetries in ω match. That is,

1 (ω) is symmetric because cos[ω(t  − t)] is an even function. Similarly, 2 (ω) is antisymmetric because sin[ω(t  − t)] is an odd function. Second is that Eq. 3.22 describes a medium that takes time to respond to an electric field. Its polarisation P (and therefore the D field) at a time t depends on the electric field at earlier times t  (but not later ones).1 In other words the relation between P and E (also D and E) is not local in time. In fact, in a spatially structured medium, the relationship may not be local in space either. In particular, for an atomic or molecular medium, the polarisation at one point generally depends on the electric field at other places, as well as at other times. Therefore it is appropriate to take the 4-dimensional Fourier transform using (k, ω) space with  D(k, ω) = r (k, ω) 0  E(k, ω).

(3.23)

That means Maxwell’s equations should be expressed in (k, ω) space altogether. At this point it might seem that the picture is becoming hard to visualise, but fortunately great simplifications follow.

3.3.2

Maxwell’s Equations and Potentials in Fourier Space

The 4D Fourier transform of every field quantity and its components, labeled with a tilde, are related by taking the transform of every equation. (This is valid because the equations are true for all space and time, so that integrands may be equated.)  we have For each field X and its transform, X X(r, t) =

1 (2π)2

 ω) = X(k,

1 (2π)2



∞



∞



∞



∞

−∞ −∞ −∞ −∞  ∞ ∞ ∞ ∞

 ω) exp(ı(k ·r −ωt)) dω d3 k; X(k,

−∞ −∞ −∞ −∞

X(r, t) exp(−ı(k ·r −ωt)) dt d3 r. (3.24)

In Maxwell’s equations, the field operators in time and space become simple factors in (k, ω), thus ∇ 2 becomes − k 2 , curl becomes ık×, ∂/∂t becomes − ıω, grad becomes ık, div becomes ık · .

1

We return to this question on page 63.

(3.25)

46

3 The Field of a Moving Charge

As a result the equations for the transforms are ordinary equations without differentials. B = curl A becomes  B = ık ×  A; E=−

∂A − grad φ becomes  E = ıω A − ık φ; ∂t

(3.27)

∂ 2φ ρ  ρ 1 = = becomes φ ; 2 2 2 ∂t

0 k − r 0 μr μ0 ω r 0

(3.28)

∂ 2A 1 = μ0 J becomes  A= 2 μr μ0 J 2 ∂t k − r 0 μr μ0 ω2

(3.29)

−∇ 2 φ + 0 μ0 −∇ 2 A + 0 μ0

(3.26)

and the Lorenz Condition, Eq. 3.6: divA + r 0 μr μ0

∂φ = 0 becomes k ·  A − r 0 μr μ0 ω  φ = 0. ∂t

(3.30)

Suddenly the equations seem easier to solve. Their solution takes four steps: 1. Find the 4-dimensional Fourier transforms, ρ (k, ω) and  J(k, ω), of the charge and current densities, ρ(r, t) and J(r, t). 2. Apply Eqs. 3.28 and 3.29 to get  φ(k, ω) and  A(k, ω).  3. Apply Eqs. 3.27 and 3.26 to get E(k, ω) and  B(k, ω). 4. Invert these transforms to get the required fields E(r,t) and B(r,t).

3.3.3

The Solution for a Charge Moving with Constant Velocity

For the charge and current densities, ρ(r, t) = δ 3 (r − βct) and J(r, t) = βcρ(r, t), the transforms are  ∞ ∞ ∞ ∞ 1 ρ (k, ω) = qδ 3 (r − βct) exp(ı(k · r − ωt)) d3 r dt (2π)2 −∞ −∞ −∞ −∞ q δ(ω − k ·β c); = (3.31) 2π q  (3.32) δ(ω − k ·β c), J(k, ω) = βc 2π where the spatial integrations are trivial and that over time uses the property 

∞ −∞

exp(ıαt) dt = 2πδ(α).

(3.33)

3.3 Field Solutions in Media

47

Step 2 gives the transformed potentials: 1/ r 0 q δ(ω − k ·β c); 2 2π k − r 0 μr μ0 ω2

(3.34)

μr μ0 βc q  δ(ω − k ·β c). A(k, ω) = 2π k 2 − r 0 μr μ0 ω2

(3.35)

 φ (k, ω) =

In terms of these, the fields are 1 E(r, t) = (2π)2 B(r, t) =

1 (2π)2

 

∞



∞



∞



∞

  ı ω A − k φ exp ı(k · r − ωt) d3 k dω,

−∞ −∞ −∞ −∞ ∞  ∞  ∞  ∞  −∞ −∞ −∞ −∞

 ık ×  A exp ı(k · r − ωt) d3 k dω. (3.36)

If the dielectric and magnetic dispersion functions, r and μr , are known, the fields are then defined. These complex functions are studied in Chap. 5. Nevertheless, the physical interpretation of these solutions is not obvious by inspection. Some help comes from noting that every potential and field component is subject to the constraint ω = k ·β c. This is just the universal phase condition that we already understood in Sect. 1.2 by considering the scalar wakefield of a source in two dimensions, such as a speedboat. In fact for transparent media, there is a more intuitive way to find the radiation field of a moving charge, known as the Feynman–Heaviside picture. We discuss this in Chap. 4.

4

Radiation by the Apparent Angular Acceleration of Charge

4.1

Apparent Angular Acceleration in Vacuum

4.1.1

Non-relativistic Motion

In the non-relativistic approximation the factor a = (1 + β · s) is unity and the Liénard–Wiechert Potentials, Eqs. 3.16 and 3.17, become φ=

q 4π 0 R

A=

qμ0 βc . 4πR

(4.1)

The transverse electric field is minus the time derivative of the transverse component of A. The transverse part of βc/R is the angular velocity of the charge as seen from P, that is ds/dt. So the transverse electric field is simply related to the angular acceleration. Picking up the constant factors, E=−

d2 s qμ0 d2 s q = − . 4π dt 2 4π 0 c2 dt 2

(4.2)

So the radiation field is simply related to the transverse angular acceleration. There is no explicit dependence on the observation distance R, except that it should be large enough that the local virtual (or induction) field may be neglected.

4.1.2

Relativistic Motion and the Feynman–Heaviside Form

In the relativistic case all we need to do is to consider the apparent direction, s , of the charge observed at time t. This depends on the actual direction at the earlier time t − R/c. In general then the radiated electric field is fixed by the apparent angular acceleration observed at P. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 W. Allison, The Flight of a Relativistic Charge in Matter, Lecture Notes in Physics 1014, https://doi.org/10.1007/978-3-031-23446-0_4

49

50

4 Radiation by the Apparent Angular Acceleration of Charge

This extraordinarily simple result is known as the Feynman–Heaviside formulation of Classical Electrodynamics. In Volume 1 of his lectures Feynman writes of the radiation field [1] Therefore, we find the following rule: Imagine that we look at the moving charge and that everything that we see is delayed - like a painter trying to paint a scene on a screen at a unit distance. A real painter, of course, does not take into account the fact that light is going at a certain speed, but paints the world as he sees it. We want to see what his picture would look like. So we see a dot, representing the charge, moving about in the picture. The acceleration of that dot is proportional to the electric field. That is all – all we need. This is the complete and correct formula for radiation; even relativity effects are all contained in it.

Including the short range parts of the field, the general expression is q E=− 4π 0



R d s + 2 R c dt



s R2



1 d2 s + 2 2 . c dt

(4.3)

The first term is the Coulomb field which falls away as 1/R 2 . The second term is the induction field that expresses the effect of Faraday’s law at close range. The third term is the radiation field at large R that we have been talking about. It does not depend on R, only on s —as Feynman says, that is all – all we need. So for the radiation field in the relativistic case we have E=−

q d2 s , 4π 0 c2 dt 2

(4.4)

the same as the non-relativistic case but with s as a reminder that the unit vector is in the apparent direction of the charge at observer time t. The radiated magnetic field is related to the electric field in the same way as for any electromagnetic travelling wave E=−

d2 s q s × E . , B = 4π 0 c2 dt 2 c

(4.5)

The form for the field radiated by a relativistic charge found in most texts looks rather different because it is expressed in terms the actual velocity and acceleration. This brings in the relation between the observation time t with retarded time t  , described by the Doppler factor, 1/(1 + β · s). The description in terms of apparent angular charge acceleration is technically elegant, simple, and intuitive. Discussed here for sources of charge in vacuum, it becomes particularly revealing when applied to situations involving radiation in material media.

4.2 Apparent Acceleration and Cherenkov Radiation

51

4.1.2.1 Radiated Photon Flux According to Feynman–Heaviside In a vacuum the Feynman–Heaviside formulation for the radiation seen at P by an observer watching a charge q at Q is calculated as follows: 1. Calculate s (t), the unit vector in the apparent direction of Q as seen by P at time t. 2. Calculate its second time derivative with respect to observer time t. 3. The observed radiated E and B fields are then E=−

q s × E d2 s . and B = − 4π 0 c2 dt 2 c

(4.6)

4. Fourier analyse in time these E and B fields seen by P. 5. Calculate the energy flux per unit area seen by P as a function of ω. 6. Calculate the photon flux by dividing the energy flux by h¯ ω.

4.2

Apparent Acceleration and Cherenkov Radiation

The Feynman–Heaviside method can be applied when transparent material media are present—the observer P may be visualised as being in a small bubble of free space [2]. All we need are the apparent movements of the apparent charges, as seen at P. Figure 4.1 shows three timeframes in the motion of a rapidly moving charge Q being observed by a stationary observer at P. In the first frame no electromagnetic wave has yet reached P and, as a result, the observer is unaware of the approaching charge; in the second the shock wave has already passed P and there are now two waves just reaching him, the waves emitted from A and those from B; in the third the waves apparently from A and B are just arriving, where A’ is to the right of A and B’ is to the left of B. So the sequence A,A ,... is of a charge seen as progressing forward along the world line of Q. However, the sequence B,B .. is seen as retracing the earlier history of Q. In this region, if t is observer time and t  is time on the charge, the Doppler factor, the derivative dt/dt  , is negative. So, at first the observer sees nothing and then two charges moving away from one another. Since charge is locally conserved, it appears conserved on any spacetime surface—if the forward-going charge appears as +q, then the backward-going charge is −q. In short, the observer sees, or thinks that he sees, the creation of an expanding charge dipole—and, therefore, detects the corresponding associated radiation according to the Feynman–Heaviside formulation. As we have seen appearances are all important in electrodynamics and the physical picture is clear. Fundamentally, it is the bound charges in the medium that are being accelerated by the shock wave, and, therefore, radiating.

52

4 Radiation by the Apparent Angular Acceleration of Charge

a ×P

Q

b

×P

Q B

A

c

×P

Q B'

A'

Fig. 4.1 The spherical wavefronts observed at P coming from a charge at Q moving faster than the phase velocity: (a) before, (b) after, and (c) longer after the passage of the shockwave

4.2.1

Calculation of the Cherenkov Flux

The Cherenkov flux observed at P is calculated from the apparent creation of the +q-q charged pair seen at that point. The observer at P in Fig. 4.2, a distance b from the line of the trajectory, is first influenced when the charge is at Q0 at time t = 0. She or he sees it at position B0 at the Cherenkov angle with cos θC = 1/(βn).

4.2 Apparent Acceleration and Cherenkov Radiation

53

C

×P

T B

T0

Q0

Q

B0

Fig. 4.2 An observer at P sees radiation from a source which is actually at Q0 when the shock wave arrives at P and the apparent point of pair creation is B0 . A time t later its real position is Q and its apparent backward-going position is B (The forward-going solution is not drawn)

A time t later, the influence appears to come from B at an angle θ at which point the actual position of the charge is Q. From geometry B0 P sin θC = BP sin θ = b. During the time t the charge has moved a distance Q0 Q = βct. BQ = BB0 + B0 Q0 + Q0 Q = b(cot θC − cot θ ) + b csc θC sec θC + βct.

(4.7)

This distance is also BC sec θC with BC = BP = b csc θC . Putting these together we have an expression for t in terms of θ , βct = b(cot θC − cot θ + csc θ sec θC − csc θC sec θC ).

(4.8)

Note that there are two values of θ for each value of t greater than zero, the apparently forward moving charge as well as the backward moving one. To keep the signs consistent, we concentrate on the backward moving solution. To get the apparent angular velocity we differentiate with respect to t βc = b(csc2 θ − csc θ cot θ sec θC ) × βc sin2 θ cos θC dθ = . dt b cos θC − cos θ

dθ . dt

(4.9)

(4.10)

54

4 Radiation by the Apparent Angular Acceleration of Charge

Following Eq. 4.6 the radiated E field is E=−

q d2 θ 4π 0 c2 dt 2

(4.11)

and its Fourier transform q  = − √1 E 2π 4π 0 c2



d2 θ exp(ıωt) dt. dt 2

(4.12)

This integral may be evaluated by parts to give  −

dθ ıω exp(ıωt) dt = − dt

 ıω exp(ıωt) dθ,

(4.13)

where the constant term has been omitted because the rate of change of angle goes to zero at large time.  q  = √1 (4.14) ıω exp(ıωt) dθ. E 2π 4π 0 c2  This integral is of the form exp ıωt (θ ) dθ and may be evaluated by the method of stationary phase [3]. It is dominated by the neighbourhood θ = θC and we expand the phase ωt (θ ) around this point by Taylor’s Theorem, ωt ≈ k(θ − θC )2 where the parabolic expansion coefficient is k = ıω

1 d2 t . 2 dθ 2 θ=θC

Taking the reciprocal of Eq. 4.10 b dt = (cosec2 θ − cosecθ cot θ sec θC ). dθ βc Differentiating with respect to θ and evaluating at θ = θC , b d2 t = . 2 dθ θ=θC βc sin θC cos θC Using the standard form, valid as k becomes large, 

 2

exp(ık(θ − θC ) ) dθ =

 ıπ  π exp k 4

(4.15)

4.3 Apparent Acceleration and Transition Radiation

q 1+ı 1  × ıω √ × E(ω) = √ 2π 4π 0 c2 2



55

2πβc sin θC cos θC . ωb

(4.16)

Then q 2ω β sin θC cos θC . × 2 2 3 b 16π 0 c

 | E(ω) |2 =

(4.17)

Using Parseval’s theorem the number of photons per unit frequency lost in direction θC in all azimuthal directions per unit length is the cylindrical area × energy density × radial flow rate divided by the energy per photon,  Nc (ω) = 2πb × r 0

 |2 dω × c sin θC 1 . |E n h¯ ω

(4.18)

Since nβ = 1/ cos θC , q2 dNc = sin2 θC , dω 4π 0 hc ¯ 2

(4.19)

where we have multiplied by two to take account of the contribution of negative frequencies ([4], p. 674). For q = e, the flux of photons per unit angular frequency per unit path length is α dNc = sin2 θC . dω c

(4.20)

This is the well known flux of Cherenkov radiation [4, 5], here considered as emitted by the apparent acceleration of a charge. This flux is also derived, quite separately, on page 78 as the component of general energy loss in a transparent medium.

4.3

Apparent Acceleration and Transition Radiation

The method of apparent angular charge acceleration can also be used to derive the flux of transition radiation emitted at the interface between two media [2]. Figure 4.3 shows an observer looking at a charge, e, coming out of a medium of refractive index n into free space. When the charge is in vacuum the observer sees it moving with an apparent transverse velocity vT vacuum =

βc sin φ . 1 − β cos φ

(4.21)

56

4 Radiation by the Apparent Angular Acceleration of Charge

f

q bc

Q

Fig. 4.3 An observer sees a charge apparently accelerate as it crosses a boundary between two media

However, when still in the medium its apparent transverse velocity (by geometry and the use of Snell’s law) is vT medium = 

βc cos φ sin φ n2 − sin2 φ − β(n2 − sin2 φ)

.

(4.22)

Therefore, if at a distance r from the observer, it crosses into the vacuum, there is a discontinuity in apparent transverse angular velocity =

4.3.1

vT vacuum − vT medium ds  = . dt r

(4.23)

Radiation from a Discontinuity in Apparent Angular Velocity

If a charge q = e undergoes a jump in apparent transverse angular velocity of  at t = 0, then d2 s =  δ(t). dt 2 From Eq. 4.6, the transverse electric and magnetic fields, E(t) and H (t), are E=−

e  δ(t) 1 and H = − E. 4π 0 c2 cμ0

(4.24)

The corresponding emitted energy per unit area is given by the Poynting vector along the line of sight, EH , integrated over time. This is related to the number of

References

57

photons per unit area, dN(ω)/dω, with energy h¯ ω, dN h¯ ω dω = dω



∞

1 e 2 2 E(t)H (t) dt = 2 2 c 4 cμ0 16π −∞ 0



∞ −∞

δ(t)2 dt.

(4.25)

∞ Substituting δ(t) = 1/(2π) −∞ exp(iωt) dω for one of the δ-functions, this last ∞ ∞ integral becomes 1/(2π) −∞ dω = 1/π 0 dω. In terms of the fine structure constant, α, the observer should see a radiated flux of N1 photons per unit area per unit frequency for this single interface, dN1 =

α 4π 2 c2 ω

2 dω,

(4.26)

with the value of  calculated in Eq. 4.23. Setting area dA = r 2 d, the number of photons per unit solid angle αβ 2 d2 N1 = cos2 φ sin2 φ dω d 4π 2 ω  ×

1  − cos φ − β cos2 φ n2 − sin2 φ − β(n2 − sin2 φ) 1

2 (4.27)

.

With the usual X-ray approximations, 1.33 and 1.34, this reduces to α d2 N1 = 2 φ2 × dω d π ω



1 1 − 2 2 2 2 2 ωp /ω + φ + 1/γ φ + 1/γ 2

2 .

(4.28)

This is the same result for the transition radiation flux from a single interface as deduced by considering transition radiation as diffracted Cherenkov radiation in Eq. 1.37 on page 18.

References 1. R.P. Feynman, R.B. Leighton, M. Sands, The Feynman Lectures on Physics, vol. 1 (AddisonWesley, Boston, 1963), pp. 28–34 2. W.W.M. Allison, Paper presented at the First International Conference on transition radiation Erevan, Armenia SSR (1977) 3. Wikipedia, Stationary Phase Approximation. https://en.wikipedia.org/wiki/Stationary_phase_ approximation 4. J.D. Jackson, Classical Electrodynamics, 3rd edn., ed. by J. Wiley (1998). ISBN 0-471-30932-X 5. I. Frank, I. Tamm, Dokl. Acad. Nauk SSSR 14, 107 (1937)

5

The Dispersion and Absorption of Electromagnetic Waves

The contributions to the photoabsorption cross section are discussed. At low frequencies there is the resonant structure of atoms and molecules as a whole. Then at higher frequencies there is scattering of the photon by a constituent electron (Thomson scattering). At even higher energies this is modified by recoil (Compton effect). Finally, there is the production of electron-positron pairs in the field of the nucleus.

5.1

Refraction and Attenuation

5.1.1

Phenomenology of Absorption

An electromagnetic wave in a medium, E exp ı(k · r − ωt), has a modified phase √ velocity c/n, with a refractive index, n = r μr . Such a wave may decay and lose energy either with distance or with time. A wave with real frequency that decays as a function of position has complex k. Alternatively it may have real k and decay with time, ω being complex.1 If attenuated in space, its energy is absorbed with a mean range, λabs , so that its amplitude, for motion in the z-direction, falls as exp(−z/2λabs). This attenuation may also be described in other ways: • By the imaginary part of k = 1/2λabs in the direction of propagation • By σγ , the absorption cross section per electron, with λabs = 1/ne σγ , where ne is the density of electrons

1

The distinction is illustrative only. Any finite wave in a finite region can be expressed as an integral over Fourier components with real k and ω.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 W. Allison, The Flight of a Relativistic Charge in Matter, Lecture Notes in Physics 1014, https://doi.org/10.1007/978-3-031-23446-0_5

59

60

5 The Dispersion and Absorption of Electromagnetic Waves

• By the imaginary part of n. Taking ω as real Im k = Im n ω/c, so that σγ = 2ω Im n ne c • (in the absence of magnetic dispersion, μr = 1) by the imaginary part of r =

1 + ı 2 . If λabs λ and 1 2 , then σγ =

ω 2 √ ne c 1

(5.1)

The energy dissipated in the medium is then the Joule heating of the electric field working on the bound current in the medium Jbound = ∂P/∂t = −ıω( 1 + ı 2 ) 0 E. Then the absorption length is this energy loss divided by the energy flux, λabs =

E · Jbound |E×H |

(5.2)

Generally, if magnetic dispersion is included, λ = 2π λabs

5.1.2



2 μ2 +

r μr

 .

(5.3)

Response of a Classically Bound Electron

An electromagnetic wave interacts with a medium through both its constituent components and composite structures. At the energies of interest here, the constituents are electrons and nuclei, while the composites are atoms and molecules. While constituent scattering typically has a small energy-independent cross section, scattering from composite structures may involve a number of resonances with large energy-dependent cross sections. Ideally these resonances behave like classical simple harmonic oscillators. (In Sect. 5.2 we relax this classical requirement.) The differential equation for the classical motion of an electron held in an atom by a restoring force (resonant frequency ω0 and damping ) and driven by an electromagnetic wave is dr e dr d2 r × B), +  + ω02 r = − (E + dt 2 dt me dt

(5.4)

where the wave is E = E0 exp ı(k · r − ωt) and B = 1/c E. Assuming that the speed of the electron, dr/dt, is very small compared to c, we can make two simplifications. Firstly, the magnetic force on the electron due to B can be neglected. Secondly, the position of the electron relative to equilibrium is always very small compared to λ. Then the factor exp(ık · r), that is exp(ı 2πr/λ), is very close to unity.

5.1 Refraction and Attenuation

61

This is called the dipole approximation. Generally, it assumes that the phase of electric field of the wave at any time is effectively the same over the whole region, usually the atom or molecule, occupied by the electron. In this approximation the problem no longer depends explicitly on k. The equation of motion is then d2 r dr e +  + ω02 r = − E0 exp(−ıωt). dt 2 dt me

(5.5)

The steady state solution for the position r is r=

5.1.3

me (ω02

−eE0 exp(−ıωt). − ω2 − ıω)

(5.6)

A Medium of Bound Electrons

The polarisation of a medium in which there are ne such electrons per unit volume displaced from their equilibrium position in this way is P = ne er =

ne e2 E0 exp(−ıωt). me (ω02 − ω2 − ıω)

(5.7)

But P = ( r − 1) 0 E0 and, therefore,

r (ω) = 1 +

ωp2 ne e2 = 1 + ,

0 me (ω02 − ω2 − ıω) ω02 − ω2 − ıω

(5.8)

where ωp is the plasma frequency, as defined in Eq. 2.3.2. Equation 5.8 relates the real and imaginary parts of the dielectric response function, r , and their dependence on frequency, as plotted in Fig. 5.1. Such equations are called dispersion relations and this particular form applies to a classical oscillator. We shall find the more general form in Sect. 5.2. The imaginary part is given by

2 (ω) = ωp2

(ω02

ω . − ω 2 )2 + ω 2  2

(5.9)

This absorptive part determines the resonant cross section near ω = ω0 and is known as the Breit Wigner Formula in Nuclear and Particle Physics, and the Lorentzian in Atomic Physics. Its dependence on ω is the square of the Fourier transform of the time dependence of the resonant state decay. Inspection of the shape 5.9 shows that the curve is half-

62

5 The Dispersion and Absorption of Electromagnetic Waves

Fig. 5.1 The real and imaginary parts, 1 and 2 , of the dielectric response of a medium composed of electrons with a single frequency ω0 , as described by Eq. 5.8

height at ω = ω0 ± /2, assuming   ω0 . So the Full Width at Half Maximum (FWHM) is equal to the damping constant . In this simple case of a single resonant frequency the phenomenology as a function of ω is easily summarised. Much below ω0 , the absorption is small, the medium is transparent and r is greater than one, and the refractive index likewise. This is the optical region. Much above ω0 , the absorption is also small, the medium is transparent and r is less than one, and the refractive index likewise. This is the X-ray region. Near ω0 , the absorption is large and r varies rapidly.

5.2

General Form of Dielectric Permittivity

5.2.1

Oscillator Strength and Its Sum Rule

Electrons in real materials have more than one resonant frequency. The spectrum of these is the set of absorption lines of the material, ω1 , ω2 , ω3 , . . . with their widths 1 , 2 , 3 , . . .. Each frequency has a weighting factor called its oscillator strength, f1 , f2 , f3 , . . ., and these obey the sum rule, fi = 1. In this more realistic picture the dielectric permittivity is

r (ω) = 1 + ωp2

 i

(ωi2

fi . − ω2 − ıωi )

(5.10)

5.2 General Form of Dielectric Permittivity

5.2.2

63

Effects of Finite Density

At low density each oscillator responds independently and dispersion behaves linearly, as given by Eq. 5.10. However, at higher density the oscillators influence one another, and this is most marked at lower frequencies. There are five distinct effects. Plasma oscillation. The basic mutual effect of unbound electrons, described on page 31. Chemical shift. The electron density affects the resonant frequencies, ωi . As atoms are brought together their frequencies shift, either down or up according to symmetry. This change in resonant frequency is most pronounced in chemical bonding. Line broadening. The line widths, i , are affected likewise, such that each discrete narrow resonance broadens into a band that is no longer describable by a simple width. The details are a principal topic in Condensed Matter  Physics. Under these conditions the spectrum of discrete oscillators, ωi with fi = 1, is described as a continuum of oscillator strength density, f (ω ) dω with the sum  rule, f (ω ) dω = 1. Then 

r (ω) = 1 + ωp2

0

∞

f (ω ) dω . − ω2 − ıω)

(ω2

(5.11)

In this complex integral the significance of  is reduced to an infinitesimal positive number—the role of its magnitude, which spreads out the oscillator strength in the discrete picture, is subsumed into the distributed nature of the continuous function f (ω). Photon cross section. The density modifies the photon cross section by a factor √ 1/ 1 . This is the product of two terms, first from the reaction rate and second from the photon flux. The photon density depends on 1 E 2 , so the absorption √ rate scales as 1/ 1 . The flux, c/n, gives a further factor 1 . The reduction in energy loss of a charged particle at relativistic velocity. This density effect, introduced earlier on page 9, is caused by the polarisation of the medium. It is unrelated to the other effects of density described above. It is discussed further in Chap. 6.

5.2.3

Causality and Dispersion

The use of the distributed oscillator strength density no longer involves separate simple harmonic oscillators. The most general form of dispersion relation depends on the principle of causality. This makes the point that bound charges (and currents) cannot be induced by fields that occur later in time. This point was already noted in passing when discussing Eq. 3.22.

64

5 The Dispersion and Absorption of Electromagnetic Waves

Considering fields at time t, there can be no contribution from the integral over t  in Eq. 3.22 for any t  > t. That would involve a response preceding its cause. This leads to the Kramers–Kronig relations discussed by Jackson [1] 3rd edn., Section 7.10. These relate the real and imaginary parts of r by the complex integrals

1 (ω) = 1 +



1 P π

1

2 (ω) = − P π



∞ −∞

∞

2 (ω ) dω ω − ω

1 (ω ) − 1 dω , ω − ω

−∞

(5.12)

(5.13)

where P indicates principal part of the integral in the complex ω’ plane—and this is where the vestigial effect of  with its small positive value plays a role. These integrals extend over all ω , both positive and negative. Since 1 is an even function and 2 is an odd function, as shown an page 45, the integration may be taken over positive ω only

1 (ω) = 1 +

2 (ω) = −



2 P π

2ω P π

∞ 0



∞ 0

ω 2 (ω ) dω ω2 − ω2

(5.14)

1 (ω ) − 1 dω . ω2 − ω2

(5.15)

Then, as discussed by Landau and Lifshitz [2] 1st edn., Section 62, f (ω) =

2ω 2 (ω) . πωp2

(5.16)

So 2 (ω) is required to obey the sum rule 

∞ 0

2ω 2 (ω) dω = 1. πωp2

(5.17)

From Eq. 5.1 the photon cross section per electron is σγ =

1 e2 f (ω) ω 2 f (ω) √ = πcαλc √ , √ = 2π 2 ne c 1 4π 0 mc

1

1

(5.18)

where α is the fine structure constant and λc is h/(me c), the electron Compton wavelength. The result is that the photoabsorption cross section should satisfy the sum rule 

∞ 0

√ √  ∞ 2ne c 1 σγ 2 0 me c 1 σγ dω = dω = 1. πωp2 πe2 0

(5.19)

5.3 Photon Cross Section

65

To get the atomic cross section for an element this cross section per electron σγ is multiplied by Z. Tabulated values of atomic cross sections are usually measured √ and stated for low density in the vapour phase for which the factor 1 can be taken as unity [3]. At higher density the other effects of density, chemical shifts, and pressure broadening invalidate the simple picture of separate collisions with independent cross sections.

5.3

Photon Cross Section

5.3.1

Resonance Collisions

From the measured absorption spectrum as a function of energy values for 2 can be calculated. This shows sharp edges corresponding to the contribution of successive electron shells. Figure 5.2a shows this for Argon, as an example, in terms of photon range. From left to right three edges of M, L, and K shell electrons are evident. Then the real part 1 is determined as a function of energy by Eq. 5.14. The result is plotted in Fig. 5.2b as 1 − 1. It is this small difference and its sign that determines the difference between the photon wave velocity and c. In the absorption bands and X-ray region 1 is very slightly less than one. In a noble gas like Argon there are no low energy molecular bands in the infrared region and the whole optical region is transparent with 1 just above one. Molecular media have absorption in the infrared also, but the real and imaginary parts of may be calculated from absorption data in the same way.

5.3.2

Electron Constituent Scattering

5.3.2.1 Thomson Scattering At higher energies above any resonance the photon cross section falls to a small value. The mechanism becomes one of scattering by single constituent electrons that are effectively free, with the emission of a scattered photon of the same frequency. This is called Thomson scattering and its cross section may be calculated classically. The acceleration of such an electron due to the incident electric field is e/me E0 exp(−ıωt). The angular acceleration seen by an observer at a distance R and at angle θ between the line-of-sight s and E0 is d2 s sin θ e exp(−ıωt). = E0 dt 2 me R

(5.20)

66

5 The Dispersion and Absorption of Electromagnetic Waves

Fig. 5.2 The absorption and dispersion in Argon gas at normal density as a function of energy, shown as (a) photon range and (b) the value of 1 − 1 with positive and negative branches on separate log scales

The scattered electric field is, therefore, Er = −

e d2 s sin θ exp(−ıωt), = re E0 4π 0 c2 dt 2 R

(5.21)

where re = e2 /(4π 0 me c2 ) is the so-called classical radius of the electron.2 The differential scattering cross section into d is dσγ scattered flux in d R 2 Er2 = = = re2 sin2 θ. d incident flux per unit area E02 2

(5.22)

This is the radius an electron would have if its rest mass was equal to its electrostatic energy as a charged spherical shell.

5.3 Photon Cross Section

67

Integrating over all directions to get the total cross section σThomson =

8π 2 r = 6.65 × 10−29 m2 . 3 e

(5.23)

This is very small compared with the absorption cross section at resonance (order λ2 , in the range 10−12 to 10−20 m2 ). As such, its contribution to the evaluation of

1 is negligible. We already remarked how constituent scattering is smaller and energy independent compared to resonant scattering from composites on page 60. This is an example. Although the atomic nuclei are also constituents of matter, their contribution to photon scattering is smaller by a factor of order a million. That is because the cross section depends inversely on the constituent mass squared, see Eqs. 5.20 and 5.21.

5.3.2.2 Compton Scattering For a photon with energy h¯ ω of order me c2 the cross section is modified. This is because the motion of the recoil electron may be relativistic and the magnetic effects that were explicitly dropped in the derivation of Thomson scattering become important. Compton scattering also incorporates the electron recoil and the related frequency shift of the scattered photon. (In the centre-of-mass frame of the electron and photon collision there is no recoil or shift, only the magnetic effect.) In the lab frame the frequency change η=

h¯ ω ωin =1+ (1 − cos θ ), ωout me c 2

where θ is the scattering angle. There is also a factor 1/η2 to transform the solid angle into the laboratory frame. The resulting Compton cross section falls slowly with energy relative to the Thomson value.   dσγ dσγ 1 (η − 1)2 = × × 1+ (5.24) . d Compton d Thomson η2 η(1 + cos2 θ )

5.3.3

Pair Production

It is also possible for the scattered gamma ray to manifest itself as an electronpositron pair. The threshold energy is 2me c2 and its cross section rises above the Compton cross section. The mechanism is γ + γ  → e+ + e− , where the virtual γ  is a component of the static electric field, not of the target electron, but of the more highly charged target nucleus. The density of these virtual photons increases rapidly with Z, and so too does the cross section.

68

5 The Dispersion and Absorption of Electromagnetic Waves

Fig. 5.3 The photon cross section per electron for three gases (Hydrogen, Oxygen, and Xenon) in units of 10−28 m2 (barns)

The contributions of the three different mechanisms to the photon cross section per electron – a) resonance collisions, b) electron constituent scattering, and c) pair production – are shown graphically for three elements in Fig. 5.3. Note that, because of the sum rule, on a linear-linear plot the area under the curves a) would be the same for every element. The curve b) is identical for all elements, but the curve c) is heavily dependent on the nuclear charge, Z.

References 1. J.D. Jackson, Classical Electrodynamics, 3rd edn., ed. by J. Wiley (1998). ISBN 0-471-30932-X 2. L.D. Landau, E.M. Lifshitz, Electrodynamics of Continuous Media (Pergamon, Oxford, 1963) 3. J. Berkowitz, Atomic and Molecular Photoabsorption (Academic Press, Cambridge, 2002). ISBN 0-12-091841-2

6

Energy Loss of a Charge Moving in a Medium

The distribution of energy loss in a finite thickness is derived from the cross section by considering the distribution in an elemental thickness and folding it, progressively. The shape of the distribution and its dependence on βγ are compared with experimental measurements for Argon gas. An estimator for βγ that minimises the effect of data fluctuations is chosen. A detailed comparison is made with the simpler Bethe–Bloch analysis. The effect of Bremsstrahlung is discussed briefly.

6.1

Basic Ideas

To calculate dEdx, the energy loss of a particle with charge e moving through a uniform absorptive medium with velocity βc, we build on several basic ideas. In a unified way these cover the full range of processes contributing to dEdx, from Cherenkov radiation to fast electron emission, also known as delta rays. For simplicity we assume the path of the charged particle is described by the equation r = βct.

6.1.1

The Force That Slows the Particle

The only force that can slow the particle at time t is that of the electric field at its position, r = βct at time t. The magnetic field can only apply a transverse force, βc × B, that cannot deliver any loss of energy. The mean rate of energy loss with distance is equal to that force dEdx = −

e E(βct, t)·β. β

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 W. Allison, The Flight of a Relativistic Charge in Matter, Lecture Notes in Physics 1014, https://doi.org/10.1007/978-3-031-23446-0_6

(6.1)

69

70

6.1.2

6 Energy Loss of a Charge Moving in a Medium

The Electric Field That Provides That Force

The electric field required is given by Eq. 3.36 on page 47 E(βct, t) =

1 (2π)2



∞



∞



∞



∞

−∞ −∞ −∞ −∞

  ı ω A − k φ exp ı(k·βc − ω)t d3 k dω, (6.2)

where the transform potentials with q = e are 1/ r 0 e δ(ω − k ·β c) 2 2π k − r 0 μr μ0 ω2

(6.3)

μr μ0 βc e  δ(ω − k ·β c). A(k, ω) = 2π k 2 − r 0 μr μ0 ω2

(6.4)

 φ (k, ω) =

6.1.3

Planck Quantisation and the Cross Section

This mean energy loss with distance is an integral over ω. We follow Planck in his reinterpretation of such a frequency integration that led to his successful description of black body radiation. Thus the smooth integral leading to a mean is re-expressed as an integral of collision probabilities for energy transfers h¯ ω. This probability is the target density ne multiplied by the differential cross section as a function of ω.  ∞  ∞ dσγ dEdx = . . . . . . dω = −ne (6.5) hω ¯ dω. dω 0 0 The energy loss differential cross section can then be written down by equating the integrands. From that we can derive the statistical distribution of individual energy loss measurements. That is needed because the experimental resolution of dEdx is dominated by statistical fluctuations.

6.1.3.1 Scattering The same technique can be applied to derive the cross section for a momentum transfer h¯ k and we study the general case in the next chapter. In this chapter we consider only the energy loss for which it is sufficient to integrate over k in Eq. 6.2.

6.2

Energy and Momentum Transferred to the Medium

This three-dimensional integration covers the magnitude, k, and its direction relative to βc which satisfies the phase condition ω = k · βc. The minimum value of k is, therefore, ω/βc.

6.2 Energy and Momentum Transferred to the Medium

71

Fig. 6.1 Typical kinematic regions in energy and momentum transfer sketched on log base-10 scales, in units of eV for energy and 1/a0 for momentum transfer (a0 being the Bohr Radius)

The relation between the energy transfer h¯ ω and the vector momentum transfer h¯ k is sketched qualitatively in Fig. 6.1, where each axis covers about five orders of magnitude, shown on log scales. The dashed lines show the polar angle of k set by the universal factor δ(ω − k · βc). Four kinematic features are marked: (a) The relation between k and ω for a free photon (the refractive index has a small influence not shown here) (b) The collision with a stationary target electron (c) The effect of Fermi momentum in spreading line (b) into a band, a resonance region that extends down to the edge of the physical region described by the photon line (a) (d) A collision with a stationary nucleus contributing to effects discussed further in Chap. 7

72

6 Energy Loss of a Charge Moving in a Medium

6.2.1

The Generalised Dielectric Permittivity

A more general description of the effect of dielectric permittivity is needed. In Chap. 5 the dielectric response of the medium was described in terms of the complex function r (ω). However, on small spatial scales, that is atomic scales, the medium is not uniform. As a result r should be taken as a function k as well as ω, thus r (k, ω). Note that at low ω it is sufficient to ignore the k dependence because the dipole approximation holds there.

6.2.1.1 The Generalised Oscillator Strength Density Equation 5.16 on page 64 can be generalised, too, in terms of the generalised oscillator strength density, f (k, ω),

2 (k, ω) =

π ωp2 f (k, ω). 2 ω

(6.6)

The sum rule as it applies to f (k, ω)  f (k, ω)dω = 1

(6.7)

was first derived by Bethe and has been discussed more recently by Inokuti [1]. This Bethe sum rule says that the electron has the same charge on every spatial scale—that is, it is point-like. To determine the function f (k, ω), we first evaluate it in the resonance region and then in the quasi-free electron scattering region, while matching the two so as to saturate the Bethe sum rule.

6.2.1.2 Oscillator Strength in the Resonance Region In the resonance region the dipole approximation holds good. That means, if the exponential in the quantum field operator describing the electromagnetic field is expanded, 1 exp ık · r = 1 + ık · r − (k · r)2 + . . . , 2 the first term contributes nothing to the transition matrix element since it corresponds to no transition. The third term is smaller in amplitude than the second term by a factor of order r/λ. This factor is typically 1/100 or smaller, so that the second term is dominant. This is the dipole term. In first order perturbation theory the dependence of the operator on k cancels in the expression for f (k, ω), so that in the resonance region f (k, ω) is independent of k. From Eqs. 6.6 and 5.1 (on page 60) it is given by f (k, ω) =

2 0 me c √

1 σγ (ω). π e2

(6.8)

6.3 Mean Energy Loss

73

Data for atomic and molecular photoabsorption cross sections, σγ (ω), are given for √ low density, as noted on page 63. So the factor 1 may be dropped.

6.2.1.3 Oscillator Strength in the Constituent Scattering Region Outside the resonance region the target electron may be treated as initially stationary and free. Then non-relativistic mechanics for the recoiling electron requires the energy transfer h¯ ω to be given by (h¯ k)2 /2me . Some initial momentum of the electron (Fermi momentum) may broaden this peak value, but the Bethe sum rule requires that the total oscillator strength be accounted for. The combined result is

 ω   2 0 me c h¯ k 2  h¯ k 2    + , σγ (ω) × H ω− f (k, ω) = σγ (ω )dω × δ ω− π e2 2me 2me 0 (6.9) where the step function H (x) = 0 for x < 0 and H (x) = 1 for x > 0. The first term covers the resonance region with H = 1. The second is the constituent scattering with the integration over frequencies ω < ω counting the fraction of electrons that are effectively free at frequency ω. The effect of the Bethe sum rule is to match the two terms. From Eqs. 6.6 and 6.9 we find

 ω    ne c hk h¯ k 2  ¯ 2   +

2 (k, ω) = σγ (ω )dω × δ ω − . σγ (ω) × H ω − ω 2me 2me 0 (6.10) Thence 1 (k, ω) is found from the Kramers–Kronig relation, Eq. 5.14.

6.3

Mean Energy Loss

Recalling Eqs. 6.1 to 6.4: dEdx = − E(βct, t) =

1 (2π)2



∞



∞



∞



e E(βct, t)·β; β

∞

−∞ −∞ −∞ −∞

 φ (k, ω) =

  ı ω A − k φ exp ı(k·βc − ω)t d3 k dω;

1/ r 0 e δ(ω − k ·β c); 2π k 2 − r 0 μr μ0 ω2

μr μ0 βc e  δ(ω − k ·β c), A(k, ω) = 2π k 2 − r 0 μr μ0 ω2

74

6 Energy Loss of a Charge Moving in a Medium

the mean energy loss is given by e2 ı dEdx = − β (2π)3



∞



∞



∞



∞

−∞ −∞ −∞ −∞



k·β ωμr μ0 β 2 ck 2 − ωk·β − k 2 k 2 − r 0 μr μ0 ω2

r 0 k 2

× δ(ω − k·βc) exp ı(k·βc − ω)t d3 k dω.



(6.11)

Writing d3 k = 2πk 2 d(cos θ )dk and integrating over cos θ using the δ function,   ω 1 δ cos θ − , δ(ω − k·βc) = kβc kβc we obtain dEdx = −

ı e2 2 2 β c (2π)2



∞



∞

−∞ ω/βc

ω β 2 c 2 k 2 − ω2 1 − dk dω. μr μ0 2 k k − r 0 μr μ0 ω2

r 0 (6.12)

As required the dependence on time t has dropped out. From here onward, we ignore magnetic dispersion and assume that μr = 1. We combine the integral over positive and negative ω by using the symmetry

r (−ω) = r∗ (ω),  ∞ ∞ e2 ω dEdx = 2π 2 β 2 c2 0 ω/βc k  × μ0 (β 2 c2 k 2 − ω2 )Im

  1 

1 − Im dk dω. k 2 − r 0 μ0 ω2

r 0 (6.13)

We can put in the expressions for the real and imaginary parts of r and integrate over k using the delta function which may be rewritten1       h¯ k 2 me 2me ω δ k− = δ ω− . 2me 2h¯ ω h¯

1

The mathematics also gives a further δ term for negative k outside the range of the integration that we ignore.

6.4 Energy Loss Cross Section

75

The integrals may look involved, but all turn out to be standard. e2 dEdx = 4π 2 β 2 c2 0



ω max

 dω × ne cσγ (ω) ln 

0



1 (1 − β 2 1 )2 + β 4 22

 2m β 2 c2  ne c

1  e  + σ (ω) ln γ h¯ ω | r |2 | r |2

 ω ne c   + σ (ω ) dω , γ ω | r |2 0  + ω β2 −



2 β 2 where  = arctan 1 − 1 β 2 and hω ¯ max

(6.14)



 −1 m2e 2me γ = 2me β γ c 1 + 2 + . Mi Mi 2 2 2

(6.15)

(6.16)

h¯ ωmax is the largest kinematically possible energy transfer in a collision between the incident mass Mi and a stationary electron me .

6.4

Energy Loss Cross Section

Using the Planck reinterpretation, Eq. 6.5, we may extract the energy loss cross section from Eq. 6.14.   1  2 α σγ (ω) 1 dσ

1   +  = ln β − dω πβ 2 ω ne c | r |2 (1 − β 2 )2 + β 4 2 1

2

 ω  2m β 2 c2  σγ (ω) 1 e   + ln σ (ω ) dω + . γ ω | r |2 ω2 | r |2 0 h¯ ω

(6.17)

Equation 6.17 gives the energy loss cross section per electron for an energy transfer between h¯ ω and h¯ (ω + dω) where ne is the electron density.

6.4.1

The Cross Section Evaluated in Argon

This can now be evaluated from σγ (ω), the known photon cross section per electron, which is plotted in Fig. 6.2a for Argon gas. Shown on a log scale and multiplied by E, equal areas under the curve make equal contributions to the sum rule. Hence, the K,L,M shells have areas in the

76

6 Energy Loss of a Charge Moving in a Medium

a 3 arb. scale 2 1

Esg

M

L

K

0

b

3 arb. scale

100

1000

10000

E eV

100

1000

10000

E eV

E 2ds/dE

2

1

0

Fig. 6.2 (a) The measured photoabsorption cross section of Argon, weighted by E. (b) The energy loss cross section of Argon, weighted by E 2 , for β = 1

ratios 2:8:8, corresponding to the number of electrons. This is compatible with the spectrum shown. Figure 6.2b shows the energy loss cross section of Argon calculated from Eq. 6.17, for β = 1 as an example. Values are weighted by E 2 , such that on a log scale equal areas make equal contributions to the average energy loss, dEdx.

6.4.2

Terms in the General Energy Loss Cross Section

The four terms in Eq. 6.17 have distinct physical interpretations. Terms 1 and 2 give the shaded area in Fig. 6.2b, including the small contribution below ionisation threshold which turns out to be due to Cherenkov radiation. The upper unshaded area comes from term 3 and the lower unshaded area from term 4 that persists at high energy. The meaning of the terms is checked below in a quantitative comparison with known phenomena, such as the emission of Cherenkov radiation and Rutherford scattering by constituent electrons. This shows some important insights familiar from the simplified ideas explored in Chap. 1.

6.4 Energy Loss Cross Section

77

6.4.2.1 The First Term This term is due to transverse polarisation of the electric field dσ dω

= 1

  1 α σγ (ω)  ln . πβ 2 ω (1 − β 2 )2 + β 4 2 1

2

√ It involves the absorption of a photon with phase velocity u = c/ 1 . In the transparent approximation ( 2 ≈ 0), the argument of the logarithm is 1 1 = = γ 2 2 1 − β 1 1 − v 2 /u2  with γ  = 1/ 1 − v 2 /u2 , as defined in Chap. 1 on page 6. This limits the photon range, as suggested by the simple scalar two-dimensional model discussed there. The evanescent range appears here in the argument of the logarithm in quadrature with the contribution of regular absorption, which makes intuitive sense. This dependence on log γ  is responsible, therefore, for both the so-called relativistic rise of the ionisation energy loss and also of its saturation, as the difference between γ  and γ cuts in. This saturation gives the Fermi plateau due to the density effect [2].

6.4.2.2 The Second Term Below Cherenkov Threshold This term is also due to transverse polarisation dσ dω

2

  α 1

1 2 = . β − πβ 2 ne c | r |2

Below threshold tan  is positive and small, so that  may be replaced by its tangent  ≈ tan  =

2 β 2 . 1 − 1 β 2

The resulting contribution to the cross section dσ dω

2

  α 1

1

2 β 2 2 = − β πβ 2 ne c | r |2 1 − 1 β 2

may be negative. However, this is not unphysical because, at short range without the emission of real photons, no meaningful physical distinction can be made between the first and second terms. The physical requirement is that they should be positive, taken together.

78

6 Energy Loss of a Charge Moving in a Medium

6.4.2.3 The Second Term Above Cherenkov Threshold The value of tan  factor above Cherenkov threshold, β 2 1 > 1, is small and negative. Then  ≈ π and the differential cross section for the second term is dσ dω

2

  1 α 1− 2 . = ne c β 1

The flux of Cherenkov photons Nc emitted per unit path length is ne times this cross section.   α dNc α 1 = (6.18) = sin2 θc . 1− 2 dω c β 1 c This result for the flux is the same as that derived historically by Frank and Tamm [3], except that they include the effect of relative magnetic permeability that we have ignored. It is also derived, quite independently on page 55, from the apparent acceleration of charge according to the Feynman–Heaviside formulation.

6.4.2.4 The Accelerator Solution There is further class of solutions of interest. Equation 6.15 always has two solutions for  with the same tangent, one positive and one negative, differing by π. Which applies is determined by the boundary conditions, in particular whether the electromagnetic field of the charge is incoming or outgoing. Below the Cherenkov condition there is a value of  just greater than −π that describes the charge gaining energy from an incoming field. This corresponds to acceleration by an applied field in a wave guide, for example. 6.4.2.5 The Third Term This is due to the longitudinal electric field and has no special velocity dependence. Because such fields are necessarily virtual, they are unable to spread their wings like transversely polarised ones. This is why this term does not exhibit the interesting features suggested in Chap. 1. dσ dω

= 3

 2m β 2 c2  α σγ (ω) e ln . πβ 2 ω | r |2 h¯ ω

6.4.2.6 The Fourth Term This term is due to constituent electron scattering and is non zero even for energies above that at which the photoabsorption cross section has fallen to zero. dσ dω

4

1 α = πβ 2 ω2 | r |2



ω 0

σγ (ω ) dω .

6.5 Distributions in Energy Loss

79

The integral over ω is simply counting the number of electrons that are effectively free when considering collisions of energy h¯ ω. At values of energy transfer above the absorption region this is the only term contributing to energy loss. Then r ≈ 1 and the sum rule saturates with 

∞

σγ (ω )dω = πe2 /(2 0 me c).

0

In this region dσ dω

= 4

2πα 2 h¯ πe2 α = 2 2 . 2 2 πβ 2ω 0 me c β ω me

Expressing this cross section in terms of energy transfer, E = h¯ ω, 2πα 2 h¯ 2 dσ = . dE me β 2 E 2

(6.19)

This is the Rutherford cross section and can be expressed to describe relativistic recoil of the electron. If a stationary electron mass me receives an energy E and 3-momentum transfer h¯ k, the relationship for the recoiling mass is (mc2 + E)2 − h¯ 2 k 2 = m2 c4 . Defining the 4-momentum transfer squared as h¯ 2 Q2 = h¯ 2 k 2 − E 2 , we may replace the expression 2mc2 E in the cross Sect. 6.19 by the explicit Lorentz invariant −h¯ 2 Q2 . As a result the constituent cross section for any stationary point charge target of any mass with unit electronic charge e is 4πα 2 dσ = . dQ2 β 2 Q4

(6.20)

This is the relativistic expression of the Rutherford scattering formula. At the highest Q2 there are magnetic corrections that depend on the spins of the incident charge and the target. These are not significant for measurable energy loss, though noted again on page 85 and 94.

6.5

Distributions in Energy Loss

Given the cross section per electron and the electron density, ne , it is simple in principle to calculate the distribution of energy loss for a given charge velocity. A monoenergetic beam of particles incident on a certain thickness, , may be tracked by Monte Carlo to find the energy distribution on exit. However, when there is a long tail of large but improbable events, as in this case, it is both faster and less

80

6 Energy Loss of a Charge Moving in a Medium

prone to statistical uncertainty to integrate by folding probability distributions, as follows. If g(E, ) is the distribution of energy losses E in a thickness , then the distribution in a thickness 2 is 

E

g(E, 2) =

g(E − η, ) × g(η, ) dη.

0

Start with an initial thickness 1 so thin that τ , the probability of any collision, is less than 10−5 , say. Then, to that accuracy, g(E, 1 ) = (1 − τ )δ(E) + ne 1

dσ dE

with  τ = ne 1

dσ dE. dE

Folding each distribution with itself to double the thickness a dozen times or so, the true distribution for a finite thickness can be derived quickly (with an error of 10−5 ), provided that the particle does not lose a significant fraction of its energy (thereby changing the cross section) or scatter through a significant angle (thereby changing its path length through the medium). Such a technique makes it unnecessary to consider the form of the distribution actually suggested by Landau who was working without a computer in 1944 [4]. He was forced to make statistical assumptions that are unjustified and, typically, his results for the width of the distribution differ from experiment by factors of order two. Examples of distributions derived by folding are shown in Fig. 6.3. As the thickness increases the features associated with the electronic shell structure evident in Fig. 6.2 get smeared out. Thus, in Argon where the mean free path between M shell ionisations is about 300 µm, and for L shell 5 mm, these structures play a major role in the shape of Fig. 6.3a, the distribution for 3 mm thickness, but are no longer seen in Fig. 6.3b, the distribution for 15 mm. The shape of such distributions is critical in determining the velocity resolution of real detectors, such as the ISIS Chamber described in Chap. 8.

6.6

Comparison with Experimental Data

Practical measurements of energy loss can only be made for gases that form the working media in real detectors. Argon gas at normal density is the gas that has been assumed as an example for the theoretical simulations thus far. However, a detector such as that discussed in Chap. 8 would not work if filled with pure Argon

6.6 Comparison with Experimental Data

81

Fig. 6.3 Calculated energy loss distributions in pure Argon for a charged particle with different values of p/mc = βγ , (a) for 3 mm thickness; (b) for 15 mm thickness

gas. The deposited ionisation electrons have to be cooled and ultraviolet photons have to be absorbed to suppress the unwanted propagation of deposited energy. The addition of a polyatomic gas with low ionisation potential can fulfil both requirements, incidentally, converting deposited energy in the form of excitation into ionisation. This is why energy loss can be described as ionisation while simply ignoring the fact that some is initially deposited as excitation. Most such gases are effective provided that they are not electronegative and attach electrons. Methane and carbon dioxide are frequently chosen as additives to Argon, Krypton, or Xenon as the main constituent. The calculations have then to be modified. The photoabsorption cross section per electron is taken to be linear, that is the weighted superposition for the gases present. However, the energy loss cross section is not linear although the non-linear effects are included in the Eq. 6.17. The comparison of calculations with data discussed here is for Argon with 20% carbon dioxide at ambient density, this being the mixture used in the ISIS detector discussed in Chap. 8.

6.6.1

The Energy Loss Fluctuations

It is not experimentally possible to make measurements for very finely divided samplings of deposited energy. Attempts to measure data to match spectra as in

82

6 Energy Loss of a Charge Moving in a Medium

Fig. 6.4 The histogram is experimental data for 16 mm samples in Argon/20% CO2 [5]. Note that the energy scale of the data is not absolute but is normalised to the calculation given by the dashed curve [Reprinted from [5] credit: © 1984 Published by Elsevier B.V. Reproduced with permissions. All rights reserved]

Fig. 6.3a would be frustrated by tiny signals and the diffusion of ionisation between samples. It turns out that the task of measuring thicker samples as in Fig. 6.3b is more realistic. First, the generic shape that describes the calculated fluctuations of energy loss is checked against those observed experimentally in Fig. 6.4. The shape is seen to be a reasonable match to the data, thereby confirming the physics of the fluctuations. More difficult is the experimental confirmation of the dependence of energy loss signals on βγ . It is not immediately clear how to measure it.

6.6.2

An Optimal Estimator for Energy Loss

The energy loss estimator should have the best resolution by using all data and taking proper account of statistics.

6.7 The Bethe–Bloch Approximation

83

Consider the energy loss cross section. The resonance region covers 10 eV to a few KeV. The constituent scattering cross section stretches all the way up to the maximum energy transfer, given by Eq. 6.16. Taking the example of a 10 GeV muon, this maximum is 2.46 GeV, six orders of magnitude above the resonance region. The constituent differential cross section falls as energy to the minus two. So its contribution to the energy loss falls as energy to the minus one. Consequently the integrated energy loss depends on the logarithm of the maximum energy transfer. So high energy collisions, though very rare, still contribute towards the mean. The mean free path for such a muon to suffer a loss of more than 1 GeV in a single collision is 275 m in water—a very rare event. The conclusion is that the energy loss distribution is very badly behaved statistically on account of this long tail. As a result the true mean is unmeasurable, at least for relativistic charges. Experimental measurements inevitably include a number of overflow signals for pulse heights that saturate, physically or digitally. These cannot be included in any moment of the distribution in a reliable way. Suggestions such as calculating the mean of the smallest 60% of pulse heights are more stable but not optimised. The family of distributions shown in Fig. 6.3b all have a similar shape. They may be seen as having a single generic form with a variable scaling factor along the abscissa relative to a fixed origin. Such a scaling factor and its error, determined by a maximum likelihood fit, have the advantage that it is straightforward to include the number of saturated measurements in the likelihood analysis. So this is the best single-parameter estimator for dEdx and the one we use. With the generic shape of Fig. 6.4 providing the likelihood function, the most probable scaling factor can be derived from a set of track measurements. The dependence of such an estimator on βγ described by the calculation in this chapter should be validated by comparison with experimental data. This check is made in Fig. 6.5 and the dependence is confirmed. Consequently, this procedure may be used to estimate the measured energy loss and its statistical error, track by track, in a practical physics experiment. The probability of different mass assignments may then be calculated for each track of known momentum, as discussed further in Chap. 8. As the final vindication of the statistical method, for those tracks whose mass is established by other means, the distribution of ionisation probability for that mass should be flat. This validation of the error analysis is made in Chap. 8.

6.7

The Bethe–Bloch Approximation

A traditional discussion of dEdx, or stopping power as it is often called, is based on the original work of Bethe in 1930 that leads to the Bethe–Bloch formula [7], dEdx =

4πα 2 h¯ 2 ne me β 2



 ln

2me β 2 γ 2 c2 IP



− β2 .

(6.21)

84

6 Energy Loss of a Charge Moving in a Medium

Fig. 6.5 The dependence of relative ionisation in Argon/20% CO2 on p/mc = βγ . The curve is based on calculation. The measurements with error flags from ISIS [5] fitted with a single linear calibration. Similar measurements by Lehraus et al. are shown as simple crosses [6] [Reprinted from [5] credit: © 1984 Published by Elsevier B.V. Reproduced with permissions. All rights reserved]

It makes no reference to the photoabsorption data on which this chapter is based. As a result it looks quite different. We should understand this difference. In the Bethe–Bloch formula, apart from the electron density ne , the only dependence on the medium is through the quantity IP . This is the mean ionisation potential, defined logarithmically as,  ln IP =

ln(h¯ ω) σγ (ω) dω  . σγ (ω) dω

(6.22)

For all atoms, IP is roughly proportional to the atomic number, Z, thus IP = IP 0 Z with IP 0 of around 10 to 12 eV, as shown in Fig. 6.6. Since dEdx depends only on the logarithm of IP , this variation has little effect and the Bethe–Bloch formula is widely used with this linear assumption. When the Bethe–Bloch formula was first derived, precise answers were not needed to interpret experiments and most applications involved energies far lower than those encountered in high energy physics today. As a result, the shortcomings of the Bethe–Bloch formula were not so evident or important then. On page 35 of their book Blum and Rolandi ask how calculations based on photoabsorption data, such as those described here, are related to the Bethe–Bloch formula [8].

6.7 The Bethe–Bloch Approximation

85

Fig. 6.6 Plot of IP /Z against Z

6.7.1

Four Assumptions in the Bethe–Bloch Formula

We identify four assumptions made in the Bethe–Bloch formula that distinguish it from the analysis given here. • The incident velocity is assumed below Cherenkov threshold at all ω and 1 ≈1. • The attenuation length is assumed to be small compared to the wavelength of the electromagnetic field. • The Bethe–Bloch formula includes a magnetic factor in the constituent scattering from target electrons—that is it assumes that the incident charged particle is spinless, so that its scattering by a spin- 12 constituent electron should be described by Mott scattering, rather than the spinless electric or Rutherford scattering assumed in this chapter. • The maximum energy transfer used by Bethe–Bloch ignores the quantity γ me relative to the incident particle mass Mi in the full formula 6.16. If the calculations derived in this chapter were to make those same assumptions, agreement with Bethe–Bloch would be achieved. However, that raises the question of whether those assumptions are justified.

86

6 Energy Loss of a Charge Moving in a Medium

6.7.1.1 The First Assumption On this assumption the second term of Eq. 6.17 simplifies to   dσ2 α α dEdx2 = − ne hω

2 hω h¯ ωp2 ¯ dω = ¯ dω = dω πc 2c =

α hn 4πα 2 h¯ 2 ¯ e e2 = ne . 2c 0 me 2me

This is exactly one half of the non-logarithmic term in Bethe–Bloch, Eq. 6.21. Observe that in both cases the term is positive, meaning that it reduces the loss of energy with distance.

6.7.1.2 The Second Assumption This assumption amounts to ignoring the density effect. This may be seen as taking γ  = γ , as defined in the discussion on pages 6 to 9. It also ignores 2 relative to 1/γ . The logarithm in the first term in Eq. 6.14 is then just ln(γ 2 ), making the contribution of the first term to dEdx  α h¯ dEdx1 = −ne 2 ln(γ 2 )σ (ω) dω. πβ In the same approximation the longitudinal polarisation contribution becomes α h¯ dEdx3 = −ne 2 πβ





2me β 2 c2 ln h¯ ω

 σ (ω) dω.

The contributions from the first and third terms may then be simply combined. dEdx1+3

α h¯ = −ne 2 πβ





2me β 2 γ 2 c2 ln hω ¯

 σ (ω) dω.

Using the sum rule and the mean ionisation potential, previously defined, the integration can be completed dEdx1+3

  2me β 2 γ 2 c2 2πα 2 h¯ 2 = −ne ln . me β 2 IP

Curiously perhaps, this is just half of the first term of Bethe–Bloch.

6.7.1.3 The Third Assumption The assumption of Mott scattering instead of Rutherford scattering modifies the contribution to constituent electron scattering by including a factor (1−β 2 ω/ωmax ). This is not significant except for collisions where a large fraction of the maximum energy transfer h¯ ωmax occurs.

6.7 The Bethe–Bloch Approximation

87

With this modification the fourth term becomes    dσ4 β 2ω dEdx4 = −ne h¯ ω dω 1− dω ωmax  

 ω max  β 2 ω h¯ ω α σ (ω ) dω dω. 1− = −ne 2 πβ 0 ωmax ω 0

(6.23)

This integral can be evaluated by parts to give dEdx4 = −

2πα 2 h¯ 2 ne me β 2



 ln

h¯ ωmax IP



− β2 .

Combining all four terms we have 4πα 2 h¯ 2 dEdx = ne me β 2



√ 

βγ 2me hω ¯ max 2 ln − β . IP 

6.7.1.4 The Fourth Assumption From Eq. 6.16 we can substitute for ωmax 4πα 2 h¯ 2 dEdx = ne me β 2





2me β 2 γ 2 c2 ln IP



 

m2e 1 2γ me 2 − ln 1 + 2 + − β . 2 Mi Mi

The fourth assumption requires that γ me is neglected compared with the incident charge mass Mi . If that is true, m2e /Mi2 can be neglected too, and so the second log term in the above equation vanishes. The Bethe–Bloch formula is then recovered.

6.7.1.5 Conclusions on the Use of the Bethe–Bloch Formula It has long been acknowledged that the relativistic increase of energy loss included in the Bethe–Bloch formula is not observed except to a limited extent in gases. Part of this theoretical increase comes from the increase in the growth of the maximum energy transfer which is irrelevant instrumentally at relativistic energies in thin counters. Anyway, the mean energy loss that the Bethe–Bloch attempts to describe is not experimentally accessible for relativistic incident charges. The broad distribution and its high loss tail caused by exceptional collisions make the mean value statistically and instrumentally irrelevant. The other part of the relativistic increase refers to the soft collisions discussed here. Traditionally it is said that a density effect correction should be made that describes the reduction of these according to the work of Sternheimer et al. [9]. That work assumes that this is due to the shift of oscillator strength with density. But as already pointed out on page 63 this is only one of the several effects of density. Not surprisingly, the predictions of Sternheimer do not agree with the work described here or with experimental observations in gases.

88

6 Energy Loss of a Charge Moving in a Medium

The Bethe–Bloch formula is undoubtedly the Best Buy for simple calculations, but it is not suitable for critical applications in the relativistic range. In those cases, the less convenient methods of this chapter can be used successfully, as discussed further in Chaps. 8 and 9.

6.8

Bremsstrahlung

For completeness we note that relativistic particles of low mass in media of high Z can also emit Bremsstrahlung radiation. The physics of this is interesting and simply explained but turns out to be irrelevant to the ionisation signals discussed in this book. An observer moving with the velocity of the incident charge sees the electrons and nuclei coming towards her (or him) with the same speed that the charge has in the laboratory frame. The electric field pulse of the passing relativistic nuclei with their charge Ze, in particular, can be represented as a dense flux of photons that can be scattered by the incident charge—the phenomenon appears as Compton scattering to the co-moving observer. This analysis of the incident virtual field as a superposition of quasi-free photons is the Weizsacker-Williams picture. Those photons scattered in the forward hemisphere appear Doppler-shifted as seen in the laboratory frame and so are very energetic. These have a long range and so do not contribute to the locally deposited ionisation energy, except in media with high density and high Z chosen to act as a calorimeter. In effect, even for electrons where Bremsstrahlung may contribute to energy loss, this phenomenon does not affect measured dEdx track signals in a low density detector.

References 1. M. Inokuti, Inelastic Collisions of Fast Charged Particles with Atoms and Molecules—Bethe Theory Revisited. Rev. Mod. Phys. 43, 297 (1971). https://doi.org/10.1103/RevModPhys.43.297 2. E. Fermi, The Ionization Loss of Energy in Gases and in Condensed Materials. Phys. Rev. 57, 485 (1940) 3. I. Frank, I. Tamm, Dokl. Acad. Nauk SSSR 14, 107 (1937) 4. L.D. Landau, J. Phys. 8, 201 (1944). Reproduced in Collected Papers of Landau ed. ter Haar, Gordon and Breach (1965) 5. W.W.M. Allison et al., Relativistic Charged Particle Identification with ISIS2. Nucl. Instrum. Methods Phys. Res. 224, 396–407 (1984). https://doi.org/10.1016/0167-5087(84)90030-9 6. I. Lehraus, NIM 153, 347 (1978) 7. H. Bethe, Annalen der Physik 397(3), 325–400 (1930) 8. W. Blum, L. Rolandi, Particle Detection with Drift Chambers (Springer-Verlag, Berlin, 1994). ISBN 3-540-58322-X 9. R.M. Sternheimer, R.F. Peierls, Phys. Rev. B3, 3681 (1971)

7

Scattering of a Charge Moving in a Medium

7.1

The Scattering and Energy Loss Cross Section

In Chap. 6 the energy loss was derived by integrating over transverse momentum transfer and the effect of scattering by atomic nuclei was ignored. Here the analysis is broadened to describe both transverse scattering and energy loss. Specifically we look at collisions in terms of pT = h¯ kT and pL = h¯ kL , the transverse and longitudinal momentum transfer, and derive the double differential cross section d2 σ . dkT dkL Three different underlying physical processes are evident in the cross section plotted in Fig. 7.1, namely resonant scattering from the atom as a whole, constituent scattering by the atomic electrons, and constituent scattering by the atomic nucleus. In Fig. 7.1 scattering by the atom as a whole is marked by “A” in a blue circle. This broad region of resonant scattering at moderate kL and limited kT was discussed in Chap. 6. Scattering by constituent electrons follows the locus marked “B”. As discussed in Chap. 6 it joins the resonance region and describes scattering only for the fraction of electrons that are effectively free at a given energy transfer. It reaches a maximum value of kT when the scattering in the centre of mass is 90 degrees. At greater kL it falls back towards kT = 0 at 180 degrees, marked “C”.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 W. Allison, The Flight of a Relativistic Charge in Matter, Lecture Notes in Physics 1014, https://doi.org/10.1007/978-3-031-23446-0_7

89

90

7 Scattering of a Charge Moving in a Medium

Fig. 7.1 The calculated cross section for scattering of 500 MeV/c muons by Argon (without correction for finite size and other form factor effects). The scales on all three axes are very large and plotted logarithmically. [Note that what is actually plotted is the cross section integrated over bins in log(kT ) and log(kL ). This integration generates two artifacts in the plot. Firstly, all elements appear multiplied by a factor kT kL . Secondly, the singular cross section for constituent scattering creates an artificial “stockade effect” depending on how the delta-function contributions fall]

Nuclear scattering contributes collisions with a large cross section but small momentum transfer. This contributes the phenomenon of multiple scattering discussed below in Sect. 7.2.2. Otherwise, the nuclear curve marked “D” is similar to the electron one, except that it extends to much greater values of kL and kT because of the higher momentum in the centre of mass. At “E” the incident particle is reflected backwards at large angles. This is the extreme case, first explained by Rutherford as a consequence of the tiny size of the nucleus relative to the atom.

7.1 The Scattering and Energy Loss Cross Section

7.1.1

91

Cross Section in the Resonance Region

This was already derived as the energy loss rate, Eq. 6.11 on page 74,  ∞ ∞ ∞ e2 ı dEdx = − β (2π)3 −∞ −∞ −∞

 ∞ k·β ωμr μ0 β 2 ck 2 − ωk · β × − k 2 k 2 − r 0 μr μ0 ω2

r 0 k 2 −∞ ×δ(ω − k · βc) exp ı(k · βc − ω)t d3 k dω.

(7.1)

This now needs to be recast in terms of kL and kT . Combining positive and negative ω using (−k, −ω) = ∗ (k, ω)  ∞ ∞ ∞ ∞ e2 ı δ(ω − k · βc) β (2π)3 −∞ −∞ −∞ −∞  ωμr μ0 β 2 ck 2 − ωk · β k·β  × exp ı(k · βc − ω)t × − k 2 k 2 − r 0 μr μ0 ω2

r 0 k 2

dEdx = −

 ωμ μ β 2 ck 2 − ωk · β k·β  r 0 − ∗ 2 2 2 ∗ 2 k k − r 0 μr μ0 ω

r 0 k

× exp −ı(k · βc − ω)t d3 k dω.

−

(7.2)

Instead of using the δ function to integrate over k, as in Chap. 6, we integrate over ω, 2e2 dEdx = − (2π)3 β



2 2 β ck − (k · β)2 c k · βcμr μ0 Im 2 k2 k − r μr (k · β)2



k·β 1 − Im d3 k.

0 k 2

r

(7.3)

Using d3 k = d2 kT dkL , the energy relation h¯ ω = h¯ kL βc and    dEdx = −

d3 σ d2 kT dkL

ne h¯ ω d2 kT dkL ,

92

7 Scattering of a Charge Moving in a Medium

we extract the cross section in the resonance region, d3 σ 2e2 = d2 kT dkL (2π)3 β 2 cne h¯ kL



  β 2 k 2 − β 2 kL2 kL βμr Im

0 k 2 k 2 − r μr β 2 kL2



1 kL β Im 2

r

0 k     α 1 1 2 2 = 2 β kT Im 2 − Im π βne k 2

r k − r β 2 kL2  β 4 kL2 kT2 α

2 = ne π 2 β(kL2 + kT2 ) kL2 (1 − 1 β 2 ) + kT2 )2 + ( 2 β 2 kL2 )2  1 − , | r |2 −

(7.4)

where 1 and 2 are evaluated at ω = kL βc. With d2 kT = 2πkT dkT and 2 = ne cσγ /ω = ne σγ /βkL, 2αkT σγ d2 σ = 2 dkT dkL πβ kL (kL2 + kT2 )

7.1.2



β 4 kL2 kT2 kL2 (1 − 1 β 2 ) + kT2 )2 + ( 2 β 2 kL2 )2

1 . − | r |2 (7.5)

Scattering by a Constituent Charge

Point scattering by target electrons and nuclei is shown in Fig. 7.1. This is described by the Rutherford scattering cross section, Eq. 6.20 on page 79, applied to the elastic collision of an incident point mass Mi with velocity β by a stationary point electron, mass me with charge e, and similarly by a nuclear target mass, mass M, atomic number Z, and charge Ze. The Lorentz scalar Q2 is defined by Q2 = kL2 + k2T − ω2 /c2 . As such, it may be evaluated in any Lorentz frame and is most usefully visualised in the centre of mass. In that frame ω = 0 for elastic scattering and Q is equal to the modulus of the 3-momentum transfer k. So the maximum value of the modulus kT in this frame is at 90 deg, is equal to Q, and has the same value in the laboratory frame. This point scattering is modified if the effective charge is reduced by screening at large distances (low Q2 ) or by a diffuse distribution of charge at short distances (high Q2 ). For charged particles with spin and magnetic moments there may also be magnetic effects at high Q2 , but these tend to be small. These modifications are different for nuclear and electron scattering.

7.1 The Scattering and Energy Loss Cross Section

7.1.3

93

Modifications of Point-Like Nuclear Scattering

If, instead of being point-like, the target nucleus has a distributed charge, ρ(r), then  2 dσ 4πZ 2 α 2 2 = × F (Q ) , dQ2 β 2 Q4

(7.6)

where the form factor F (Q2 ) is the Fourier transform of ρ(r). This reduces scattering by the nucleus at high Q2 where its finite size is important. At very low Q2 the effective nuclear charge is reduced by the screening effect of the atomic electron cloud, such that at large distances the atom appears to have net charge zero.

7.1.3.1 Thomas–Fermi Form Factors for Nuclear Scattering at Low Q2 At low Q2 the effect of atomic electron shielding of the nucleus is described by the Thomas–Fermi model, in general. This defines χ(r), the Thomas–Fermi screening function, V (r) =

Ze χ(r) 4π 0 r

with boundary conditions χ(0) = 1 and χ(∞) = 0. The screened potential V (r) is related to the atomic electron density ρ(r) taken to be a Fermi fluid that just fills the atomic potential well with maximum kinetic energy −eV . The differential equation for χ(r) is derived by relating V (r) to the charge density in a spherical atom by Poisson’s equation, ∇ 2V =

1 d2 ρ (rV ) = − . r dr 2

0

As a result χ(r) should satisfy the differential equation  4 dχ = 2 dr 3π

8Zχ 3 , r

where r is measured in units of a0 , the Bohr Radius. Its solution is shown in Fig. 7.2. This matches the original hand calculation of Condon and Shortley [1]. The form factor required is found by taking the Fourier transform,

ρ(r) exp(ıQr cos θ ) 1 + 2πr 2 d(cos θ ) dr Ze 0 −1  √  8 2Z ∞ χ 3 sin(Qr) dr. = 1− 3πQ 0 r 

F (Q2 ) =

∞ 1

(7.7)

94

7 Scattering of a Charge Moving in a Medium

Fig. 7.2 The Thomas-Fermi screening function

However, this description of atomic shielding by a statistical model is weakest at low Z. Fortunately, a better treatment is possible for Hydrogen.

7.1.3.2 Hydrogenic Form Factor for Nuclear Scattering at Low Q2 In the case of atomic Hydrogen we know the exact electron distribution from the solution to Schrodinger’s equation, ρ(r) = −(e/π) exp(−2r) in units of a0 . Comparing this with the general formula for a Thomas–Fermi atom we can define an equivalent   r 4r . χH = (3π)2/3 exp − 2 3 7.1.3.3 Form Factor for a General Nuclear Target at High Q2 For a nucleus of atomic weight A, the radial nuclear charge density is assumed to be described by the Woods-Saxon form in general [2],   r − r0 −1 ρ(r) = 1 + exp , s

(7.8)

with r0 = 1.07 10−15A1/3 metres and s = 0.545 10−15 metres. For nuclei with spin there is the possible effect of magnetic scattering which we ignore in general.

7.1.3.4 Form Factor for a Proton Target at High Q2 In this case the scattering is described by the Rosenbluth Formula which describes the effect of both the size and the magnetic interaction [3]. F (Q2 ) = (1 + τ κ 2 ) G2 (1 − Q2 /Q2max ) + 2τ (1 + κ)2 G2 Q2 /Q2max ,

(7.9)

where G = (1 + Q2 h¯ 2 /MF2 c2 )2 , formfactor mass MF = 0.9 GeV/c2 , κ = 1.79 (the anomalous magnetic moment of the proton) and τ = −Q2 h¯ 2 /4M 2 c2 .

7.1 The Scattering and Energy Loss Cross Section

95

Fig. 7.3 Calculated nuclear form factors on a log Q2 scale. Dashed curve for Hydrogen and the solid curve for Iron

Plots of the form factors are shown in Fig. 7.3 for Hydrogen and Iron, as examples. Q2 is in units of m−2 and the range of the form factor is about 10 units in log Q2 . This corresponds to the five orders of magnitude in the relative size of an atom and its nucleus. Other atoms are similar to Iron. Hydrogen is slightly different having the smallest nucleus and so the largest form factor at high Q2 , as given by the Rosenbluth Formula. At low Q2 Hydrogen also has a high form factor because its atomic wavefunction is more diffuse than atoms with inner and outer electrons.

7.1.4

Form Factors for Electron Constituents

Rutherford scattering by an electron target is also modified at both low and high Q2 . At low Q2 the scattering is described by the resonance cross section that joins the point scattering for the fraction of electrons that are effectively free for an energy transfer h¯ ω = h¯ 2 Q2 /2m, as discussed in Chap. 6. At high Q2 scattering by an electron is point-like and unmodified, unless the incident charge is not point-like. There may also be spin factors. In particular, for an incident muon and a constituent electron there is point-like Dirac scattering.

7.1.5

Cross Sections dkT dkL with Form Factors

The effect of form factors has a significant effect on the cross section, even at 500 MeV/c. This is illustrated by the difference between Fig. 7.4a, b, without and

96

7 Scattering of a Charge Moving in a Medium

Fig. 7.4 Calculated log cross sections against logkL , logkT for muons in Argon: (a) At 500 MeV/c without form factors (the same as Fig. 7.1 shown for comparison). (b), (c), (d) at 500 MeV/c, 5 GeV/c, 50 GeV/c, respectively, including form factors

with form factors, respectively, at 500 MeV/c. The trajectory for nuclear scattering is unchanged but the magnitude of the cross section is drastically reduced above 90 degrees in the centre of mass. At low Q2 the cross section is also seen to be reduced by atomic screening. Electron scattering at high Q2 is modified by the inclusion of spin effects (Dirac scattering) but the effect is evidently small. A comparison of Fig. 7.4b, c, and d shows the effect of a higher incident momentum, increased from 500 MeV/c to 5 GeV/c and 50 GeV/c, respectively. The resonance cross section is barely changed. Nuclear scattering at high Q2 moves to higher kL but its cross section is reduced progressively at higher incident momentum. Indeed, there is very little backward scattering at 5 GeV/c and none at all at 50 GeV/c. However, the high-Q2 electron constituent scattering is quite different. While it moves to progressively higher values of kL , the cross section is not reduced because the scattering remains point-like.

7.1.5.1 Checking That Cherenkov Radiation Is Included Inevitably, it is not possible to see detail in the particular plots shown. However, by changing scale we can check for expected features.

7.2 Scattering and Energy Loss Distributions

97

Fig. 7.5 Cross section plots for 500 MeV/c, 5 GeV/c, and 50 GeV/c, respectively, for a region near ionisation threshold on a much expanded scale showing emission of Cherenkov radiation at higher momentum

For example, Cherenkov radiation should appear below ionisation threshold, evident at an angle between kL and kT . Figure 7.5 shows this region of the cross section on the same much expanded scale for each of the three incident momenta. At 500 MeV/c no contribution to the cross section appears below ionisation threshold. However, at 5 GeV/c there is such a line of elements. At 50 GeV/c it is stronger and at a larger angle. Of course these plots serve only to indicate qualitatively that the calculation of the cross section has the features we expect. It is all part of checking that we have a complete picture of the cross section before considering distributions.

7.2

Scattering and Energy Loss Distributions

7.2.1

Single and Multiple Scattering

In passing through a finite thickness of material a charged particle suffers many collisions, each contributing to changes in transverse and longitudinal momentum, pT = h¯ kT and pL = h¯ kL . Whereas the effect of successive changes in longitudinal momentum is simply additive, each change in transverse momentum is a 2-D vector with random azimuth. The result, transversely, of many such scatters with high probability is diffusive with zero mean and a characteristic RMS deviation in two dimensions. This is known as multiple scattering. But this is not the whole story because of the significant chance of much larger transfers of momentum in one (or more) collisions that are not well described by the statistical average. This is called Single scattering. These exceptional collisions occur for both transverse and longitudinal momentum transfer and are correlated in principle. Anyway, the cross section in the two

98

7 Scattering of a Charge Moving in a Medium

dimensions provides the necessary input for evaluating the combined distribution in energy loss and scattering for small thicknesses of material.

7.2.2

Evaluating Probability Maps in pT and pL

The value of the different probability elements varies over many orders of magnitude. Some occur very rarely while others occur so many times that fluctuations in their occurrence are less important and sampling them all is unnecessary. A time efficient but careful procedure is required. This is provided by the programme ELMS, Energy Loss, and Multiple Scattering. The 2-D distribution in energy loss and scattering in a thickness for, say, 105 traversals may be derived by taking the following steps: 1. Choose a granularity, pT and pL on a logarithmic scale, such as 5%. 2. Consider a full list of contiguous fine-grained probability elements P robi,j of changes pT i to pT i + pT i and pL j to pL j + pL j for the passage of a charge through an infinitesimal elemental thickness  with electron density ne . In terms of the cross section, P robi,j = ne 

  dσ pT i , pL j pT i pL j . dpT dpL

(7.10)

3. Find the value of n such that in a sub thickness  = × 2−n , the chance of any collision at all is less than unity, say. 4. Calculate the list of probability elements using Eq. 7.10 and sort it in order of decreasing probability. 5. Make a 2-D pL − pT probability map for 105 traversals of this sub thickness, omitting those elements whose probability for all the traversals is less than 1.0. Add to the map a δ function at the origin, pL = pT = 0, such that the probability map is normalised. The list of less probable elements is held over until greater thicknesses are considered. 6. Generate the probability map 105 traversals of twice the above thickness by sampling at random two elements from the already-derived half thickness map, combining them vectorially with a random azimuth in pT , and then also randomly sampling those probability elements in the held-over list whose probability for all the traversals of the new thickness exceeds 1.0. Reduce the elements in the heldover list that remained small. 7. Repeat the previous step until the full thickness is sampled. In the final iteration all remaining elements in the list are sampled at random. This method assumes that neither the cross section nor the path length in the material is changed significantly as a result of the collisions. This is the sense in which it applies for samples described as thin.

7.2 Scattering and Energy Loss Distributions

7.2.3

99

Multiple Scattering and Energy Loss for Various Elements

Any simple mean scattering value is unstable because of the bad statistical behaviour of outliers, that is very large scatters with very low probability. The traditional solution is to define a measure of multiple scattering, RMS98, from which the disruptive contribution of the largest 2% of scattered trajectories is discarded and the root mean square scattering angle of the remaining 98% is evaluated. We can calculate such a value for RMS98 from a probability map and compare with the traditional value. An approximate formula for RMS98 is given by the Particle Data Group (PDG) based on a fit to Molière scattering in terms of the ratio of the material thickness, , to the radiation Length of the medium, X0 [4].  13.6 RMS98 = βp(MeV/c)

 

× 1 + 0.038 ln radians. X0 X0

(7.11)

Not too much should be expected of this one size fits all formula, but the PDG states that it is reliable to ±11% for β = 1 and 10−3 < /X0 < 100. Note that the treatment in this chapter does not use X0 explicitly at all. However, Table 7.1 shows values for RMS98 for 180 MeV/c muons in various materials and thicknesses evaluated by the method described in this chapter and compares them with values calculated by formula 7.11. The differences are small and generally within the margin stated by PDG. The largest difference is for liquid Hydrogen, as is to be expected, given the unusual balance between electron and nuclear scattering for the case Z = 1. Table 7.1 Comparison of calculated scattering with that based on the simple use of radiation Length, X0 , according to formula 7.11 Element Liquid Hydrogen Lithium Beryllium Carbon Aluminium Iron Lithium Beryllium Carbon Aluminium Iron

mm 500 1 1 1 1 1 10 10 10 10 10

This calculation RMS98 mrad 16.11 1.58 3.49 4.94 7.68 18.81 5.44 12.01 17.06 26.53 63.62

PDG /X0 0.0578 0.0006 0.0028 0.0053 0.0113 0.0569 0.0065 0.0284 0.0530 0.1125 0.5686

RMS98 mrad 18.78 1.60 3.63 5.11 7.71 18.62 5.69 12.75 17.93 26.94 64.65

100

7 Scattering of a Charge Moving in a Medium

Table 7.2 Comparison of resonance, electron and nuclear mechanisms to scattering and energy loss in Hydrogen and Aluminium

Mechanism Resonance Electron only Nuclear only

Liquid Hydrogen 500 mm mean pT mean dEdx MeV/c MeV cm2 gm−1 0.586 1.831 2.523 3.037 2.939 0.017

Aluminium 10 mm mean pT mean dEdx MeV/c MeV cm2 gm−1 0.449 1.144 1.278 1.103 6.291 0.071

This effect is examined further in Table 7.2 where the contribution of the three major mechanisms of scattering and energy loss are separated out and compared for Hydrogen and Aluminium. It shows how scattering is dominated by the nucleus for Aluminium but receives a similar contribution from electron scattering in the case of Hydrogen. As expected for both elements, nuclear collisions contribute little to the mean energy loss. In Hydrogen, since scattering by electrons contributes significantly to both energy loss and scattering, it is true that fluctuations in energy loss and scattering should be correlated to some extent. In any other materials this effect is predicted to be smaller. A detailed comparison of the whole scattering distribution of experimental data for Hydrogen with ELMS and other calculations is covered in Chap. 9.

References 1. E.U. Condon, G.H. Shortley, The Theory of Atomic Spectra (Cambridge University, Cambridge, 1935) 2. Wikipedia, The Woods-Saxon Potential. https://en.wikipedia.org/wiki/Woods-Saxon_potential 3. M.N. Rosenbluth, High Energy Elastic scattering of Electrons on Protons (1950). https://doi. org/10.1103/PhysRev.79.615 4. Particle Data Group, Review of Particle Physics. https://pdg.lbl.gov/index.html

Part III Two Practical Applications

8

Relativistic Particle Identification by dEdx

8.1

Design Criteria for a Detector

8.1.1

Charged Particles to Be Distinguished, P /K/π/e

A high energy particle physics experiment involves the statistical analysis of a huge number of events. In each of these two particles collide with sufficient energy to create a large number of secondary particles, from a few to many dozen depending on the energy available and quantum fluctuations in the interaction. Some of these particles are uncharged and others are electrically charged, plus or minus the electronic charge. The energy of collision is chosen to be sufficient to create the extra mass of unstable particles, whose lifetime may be too short for measurements to be made on them directly. These unfamiliar particles may be studied and identified by their instability, their mass, and the more familiar particles into which they decay. Their mass is determined by measuring their decay products—ideally that requires the energy and momentum of each of them. Then, if the decay products themselves are identified by their mass and charge, it is possible to deduce the quantum numbers characteristic of the quark content of the new unstable particle being studied. Typically, a given charged decay product will have its charge and momentum vector, p, well measured by magnetic deflection. Its mass might be that of a proton (938 MeV/c2 ), kaon (494 MeV/c2 ), pion (140 MeV/c2 ), muon (106 MeV/c2 ), or electron (0.511 MeV/c2 ). So the detected charge should have one of a discrete set of p/mc or βγ values. The need in a particle physics experiment is to measure the likelihood of each of these so that an identification with well defined confidence can be made. In fact, the difference in mass between pion and muon is too small for this distinction to be made electromagnetically. So we ignore muons as a category separate from pions in the following. This chapter describes the design and performance of a device, the ISIS Chamber (Identification of Secondaries by Ionisation Sampling), that provided such © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 W. Allison, The Flight of a Relativistic Charge in Matter, Lecture Notes in Physics 1014, https://doi.org/10.1007/978-3-031-23446-0_8

103

104

8 Relativistic Particle Identification by dEdx

identification by measuring the distribution of energy loss for each decay product. ISIS was a critical component of the magnetic European Hybrid Spectrometer (EHS) built at CERN and used in the early 1980s, in particular to study charmed particles [1–3]. As well as the magnetic spectrometer and vertex detector, other components of the spectrometer included lead-glass shower counters and Cherenkov counters that provided particle identification independent of ISIS. In many instances these facilitated the validation of ISIS results, as reported in Sect. 8.3.

8.1.2

Choice of Medium to Maximise Discrimination

Since dEdx depends on velocity, measurements of deposited energy in the form of normalised ionisation, I /I0 , should give information about the velocity, even in the relativistic range. This possibility was first discussed seriously by Alikhanov [4]. However, this is difficult in practice because the magnitude of the relativistic dependence is small and reduced further by the density effect. The velocity dependence of the electromagnetic field is saturated at frequencies below γ ωp , as discussed on page 9. dEdx involves the broad spectrum with typical frequencies around ω = IP /h¯ , where IP is the mean ionisation potential (page 85). So we should expect the dependence of dEdx on γ to saturate in the region γ = IP /h¯ ωp . This ratio gives a rough indication of the value of γ at which the electromagnetic field becomes independent of velocity. Data for various materials are shown in Table 8.1. The ratios shown in the bottom row indicate the rough values of γ at which saturation is expected. This suggests that for condensed materials no increase in dEdx in the relativistic range is likely. For gases, saturation is expected around γ = 50 for Hydrogen, rising with atomic number to 400 for Xenon. The expectation for Argon is broadly confirmed in Fig. 6.5 on page 84. This is a convenient choice of sensitive medium for a practical detector. However, even with this choice the weak dependence on p/mc makes discrimination demanding. The calculated curves for different masses are illustrated Table 8.1 Plasma energy, mean ionisation potential and K-shell energy of some detector materials Material density, kg/m3 Z/A plasma energy, eV IP , eV K shell, eV ratio, IP /plasma energy

H2 gas 0.084 1.000 0.26 14 14 54

Li

Be

C

Mylar

534 0.432 12.2 30 54 2.5

1848 0.449 25.4 40 112 1.6

2210 0.499 29.2 60 284 2.0

1400 0.520 23.7 284

Ar gas 1.66 0.451 0.761 180 3206 240

Xe gas 5.495 0.411 1.322 540 34,600 410

8.1 Design Criteria for a Detector

105

Fig. 8.1 Simple dependence of mean dEdx on momentum, calculated by truncating large energy collisions greater than 10 KeV. The curves show the estimated values for electron, pion, kaon, and proton masses at the given momentum

in Fig. 8.1. These show that the difference between candidate masses at a given momentum is typically 10% or less.

8.1.3

Estimator and Sampling Required

Evidently measurements of dEdx should be reliable at the level of a few percent, although the intrinsic variation of single measurements is very broad, as shown in Fig. 6.4 on page 82. To achieve the resolution to distinguish different mass assignments, this scatter of about 70% should be reduced by a factor of ten, at least. Because of the high energy tail of this distribution, the resolution is not improved usefully, simply by measuring the energy loss in thicker gas samples. So, to achieve the required resolution, more than 100 separate measurements of ionisation loss need to be made for each individual particle, and their distribution compared statistically for each candidate mass assignment. A related question is the choice of estimator to be derived from each distribution. This should be relatively insensitive to the largest energy losses that are inevitably truncated by analogue saturation in the detector and digital saturation in the electronics. The design of the detector is made more challenging by the other secondary particles produced in the same primary interaction. At the energy required to produce the heavy quark particles under study, these are inevitably numerous. Since quark particles are produced in pairs and there may also be other beam particles passing through the gas in the same period of time, the density of contemporaneous signals in the detector is high.

106

8 Relativistic Particle Identification by dEdx

Consequently, hundreds of measurements need to be made on dozens of tracks simultaneously with sufficient spatial resolution to enable their signals to be disentangled offline. In addition, the calibration of signal amplitudes has to be tracked and systematic effects monitored at the level of 1%, so as to achieve the required resolution.

8.1.3.1 Preliminary Experiments and Calculations Early in the 1970s it was evident that an altogether more rigorous approach to the calculations, the experimental technique and the data analysis would be required if relativistic energy loss was ever to become a viable method of individual particle identification [5, 6]. A preliminary experiment [7] confirmed the departure of energy loss spectra from the historical prediction of Landau [8]. It also revealed a total relativistic increase in dEdx of about 60% in both Argon and Krypton, some 10–15% less than predicted by the only available calculation at that time [9]. In 1976 calculations of the velocity dependence and density effect by Cobb using the Photo Absorption Ionisation model gave improved agreement with experimental measurements [10–12]. In 1980 the cross section and folding technique described in Sect. 6.5 completed the ability to predict results and simulate measurements [13].

8.2

Identification of Secondaries by Ionisation Sampling, the Detector Design

The idea for ISIS was conceived in April 1972 [5]. Its sensitive volume, 4 m high, 2 m wide and 5.12 m deep in the downstream direction, was filled with Argon/20%CO2 at ambient pressure and separated into symmetrical upper and lower drift regions. It is shown photographically in Fig. 8.2 and diagrammatically from the side in Fig. 8.3. At mid-height a single horizontal wire plane composed alternately of 25 µm diameter anodes and 250 µm diameter cathodes is shown separating the two regions. The anode wires at ground potential were connected in pairs to 320 channels of preamplifiers and readout electronics. The voltage of the cathode wires was set to control the gas amplification at the anodes in the linear range. The top and bottom plates of the chamber were maintained at −100 KV. The uniformity of the electric drift field gradient (50 KV per metre) was maintained between these and the wire plane by the surrounding double walls of horizontal tubes connected to a resistor chain. To prevent arcing of the high voltage at the edges and corners electrostatic shields were linked to the resistor chain, as may be seen in the photograph. The charged particles passing through the gas liberated ionisation electrons that formed a picture that then drifted towards the central wire plane with that of the upper and lower drift regions superimposed. Each channel of electronics picked off a 16 mm slice and recorded an independent vertical line of this picture [16]. A

8.2 The design of the ISIS Detector

107

Fig. 8.2 A photograph of the ISIS Chamber looking 5.12 m downstream into the 2m-wide lower 2m-drift space. Light reflected by the horizontal wire plane is seen separating the upper and lower drift spaces. The lower front field-shaping tubes and the double mylar window have been removed for the photograph

shortened prototype with 80 channels was constructed first and operated in a test beam [17, 18]. An example of the raw timing data in the full-sized detector is shown in Fig. 8.4. For each track hit in the picture a pulse height was also recorded. At the top of the picture an electronic “track” is seen, injected into the electronics at the end of the drift time for calibration purposes. Also visible are some superimposed track vectors, reconstructed offline by combined spatial fitting with other elements of the magnetic spectrometer. The presence of the CO2 in the gas mixture kept the ionised electrons cool so that they did not suffer significant diffusion during the drift time of up to 100 µs [15]. It also ensured that the gas amplification at the anodes remained in the linear region, essentially by quenching ultraviolet species. To prevent electrons being captured by oxygen contamination during the drift, the gas was continuously recirculated and purified to maintain the oxygen concentration at the level of a part per million or less. Each of the 320 channels of electronics recorded, in addition to the drift time, the integrated pulse height for up to 32 tracks. Signal discrimination and baseline restoration electronics ran continuously and independently on each channel. The

108

8 Relativistic Particle Identification by dEdx

Fig. 8.3 A diagram of the ISIS Chamber transverse to the particle flux, with its upper and lower 2 m drift spaces, 5.12 m in length

discriminated pulse heights were stored on analogue capacitors and digitised by an 8-bit ADC during the readout phase that followed a trigger of the whole magnetic spectrometer [16].

8.3

The Performance of ISIS in the EHS Spectrometer

The spectrometer target was a liquid Hydrogen bubble chamber that cycled at 30 Hz and was sensitive for 800 µs. Gas amplification at the ISIS anode wire plane was gated on, only during this window to prevent drift field distortion by released positive ion space charge. The data showed that track hits closer than 1.5 cm in the plane of Fig. 8.4 were not resolved spatially and had typical overlapping pulse heights. Such hits were excluded from further analysis [19]. The histogram of all other pulse heights associated with a spatially reconstructed track were analysed to find an optimum figure for energy loss estimator and its error by maximum likelihood method, as discussed on page 82. The slowly changing calibration of the energy loss estimator was followed for large numbers of tracks over periods of hours. The main cause of variation was atmospheric pressure, but some steady attenuation with drift distance also occurred. Other details are described elsewhere [14, 19]. In this way the absolute value of calibrated energy loss, I /I0 and its error from the likelihood analysis, were calculated for each track linked to the spectrometer. Then, knowing its momentum, the probability of each possible mass assignment was found.

8.3 The Performance of ISIS in the EHS Spectrometer

109

Fig. 8.4 A display of track-hit drift time data from upper and lower drift spaces superimposed. Hits linked together offline into segments are shown weighted, and some decay trajectories in the spectrometer are superposed as straight lines. Two bars indicate the different horizontal and vertical scales. [Reprinted from [14] credit: © 1984 Published by Elsevier B.V. Reproduced with permissions. All rights reserved]

Beam particles incident on the spectrometer without interacting passed through the ISIS detector volume within a narrow vertical slot below the signal wire plane. In the transverse horizontal direction, that is perpendicular to the plane shown in Fig. 8.4, it was found that all tracks within ±2 cm of this beam slot in the lower drift space had pulse heights systematically reduced by 5–15% due to the effect of space charge on gas amplification at the anode wires. These were not used for identification.

110

8 Relativistic Particle Identification by dEdx

No other systematic distorting effect was identified for the remaining 90% of tracks [19].

8.3.1

Identification Performance by ISIS

This was checked by asking three questions: • Does the ionisation resolution match theoretical calculation? • Are tracks of known mass correctly identified? • Are the ionisation errors correctly evaluated? √ The RMS ionisation resolution was shown to be 56%/ N where N is the number of samples measured. If N = 250, for example, this is 8.3% FWHM, which may be compared with a theoretical value of 6.5% calculated from the theory of Chap. 6. Some particles measured by ISIS were also independently identified in the spectrometer [20]. Specifically, pions and protons (or anti-protons) were identified being the reconstructed tracks from decays of K0 and Λ (or anti-Λ) observed in the bubble chamber, and electrons (or positrons) were identified by tracking to one of the two lead-glass shower counters. The scatter plot in Fig. 8.5 shows the normalised ionisation, I /I0 , against momentum for these identified tracks, confirming visually that the identification by ionisation is broadly correct. The truth table below shows that the mass preferred by ionisation is confirmed as correct in 558 cases (89%) (Table 8.2).

8.3.2

Quantitative Check on Calculated Ionisation Errors

A comparison of the shape of the pulse height spectrum has already been shown in Fig. 6.4 and the velocity dependence has been compared with theory in Fig. 6.5. As a result the error on each set of ionisation measurements, calculated as described on page 82, should be correctly related to the statistical probability of the true mass assignment. To test this, in Fig. 8.6 the ionisation probability is plotted for the independently verified mass assignment. It should be flat if errors have been correctly evaluated. Evidently this critical quantitative test is well satisfied. This assessment was not made with special data but with those acquired during physics runs dedicated to the study of charm particle decays. This underlines the simultaneous theoretical and instrumental success of this work with dEdx in the relativistic range.

8.3 The Performance of ISIS in the EHS Spectrometer

111

Fig. 8.5 Scatter plot of the measured track ionisation versus momentum for each independently identified track with more than 100 measured ionisation samples. Filled circle = known electron, cross = known pion, open square = known proton. The full lines show the expected I /I0 dependence for electron, pion, and proton. The dashed line shows that expected for kaon—there are none in this sample. The error bars indicate one standard deviation resolution for tracks with 250 samples. [Reprinted from [14] credit: © 1984 Published by Elsevier B.V. Reproduced with permissions. All rights reserved]

Table 8.2 The truth table of ionisation identification for the 625 independently identified tracks. [Reprinted from [14] credit: © 1984 Published by Elsevier B.V. Reproduced with permissions. All rights reserved]

Identity preferred by ionisation

electron pion kaon proton

Independently established identity electron or positron pion kaon 313 12 0 26 228 0 3 19 0 0 3 0

proton or anti-proton 0 0 4 17

112 60

No.of tracks

Fig. 8.6 Distribution of ionisation probability for the correct mass, as independently identified. [Reprinted from [14] credit: © 1984 Published by Elsevier B.V. Reproduced with permissions. All rights reserved]

8 Relativistic Particle Identification by dEdx Ionisation probability

40

20

0

0

50

100%

Probability

References 1. LEBC-EHS Collaboration, The European Hybrid Spectrometer—a facility to study multihadron events produced in high energy interactions (1983). https://doi.org/10.1016/01675087(83)90176-X 2. B. Adeva et al., Observation of the fully reconstructed D0D0 pair with long lifetimes in a high resolution Hydrogen bubble chamber and the European Hybrid Spectrometer (1981). https:// www.researchgate.net/publication/29511458 3. M. Aguilar-Benitz et al., Charm Hadron Properties in 400 GeV/c pp Interactions. Zeitschrift für Physik C 40(3), 321–346 (1983). https://doi.org/10.1007/BF01548848 4. A.I. Alikhanov et al., High Precision Measurement of the Ionizing Power of Fast Charged Particles with the Help of Multilayer Proportional Counters, in Proc CERN Symposium, vol. 2 (1956), pp. 87–98 5. W.W.M. Allison et al., The Identification of Secondary Particles by Ionisation Sampling ISIS (1974). https://doi.org/10.1016/0029-554X(74)90800-3 6. W.W.M. Allison et al., Relativistic Charged Particle Identification by Ionisation Loss Counters (1978). https://doi.org/10.1016/0029-554X(78)90619-5 7. F. Harris et al., The Experimental Identification of Individual Particles by the Observation of Transition Radiation in the X-ray Region (1973). https://www.sciencedirect.com/science/ article/abs/pii/0029554X73903753 8. L.D. Landau, J. Phys. (USSR) 8, 201 (1944). Reproduced in Collected Papers of Landau ed. by ter Haar (Gordon and Breach, 1965) 9. R.M. Sternheimer, R.F. Peierls, Phys. Rev. B3, 3681 (1971) 10. J.H. Cobb et al., The Ionisation Loss of Relativistic Charged Particles in Thin Gas Samples and Its Use for Particle Identification. I. Theoretical Prediction (1976). https://doi.org/10.1016/ 0029-554X(76)90625-X 11. W.W.M. Allison, et al., The Velocity Dependence of Ionisation in Gases (1981). http:// iopscience.iop.org/1402-4896/23/4A/005 12. I. Lehraus, NIM 153, 347 (1978) 13. W.W.M. Allison, J.H. Cobb, Relativistic Charged Particle Identification by Energy Loss (1980). https://doi.org/10.1146/annurev.ns.30.120180.001345 14. W.W.M. Allison et al., Relativistic Charged Particle Identification with ISIS2 (1984). https:// doi.org/10.1016/0167-5087(84)90030-9 15. J.H. Cobb, A Study of Some Electromagnetic Interactions of High Velocity Particles with Matter. Oxford DPhil. Thesis (1975)

References

113

16. C.B. Brooks et al., The Design and Construction of Multitrack Pulse Height Electronics for ISIS (1978). https://doi.org/10.1016/0029-554X(78)90727-9 17. W.W.M. Allison et al., The Construction of ISIS1 A 4 m × 2 m × 1.6 mm Drift Chamber for Particle Identification (1978) https://doi.org/10.1016/0029-554X(78)90709-7 18. W.W.M. Allison et al., Multiparticle Identification with ISIS Tests with the Full Aperture ISIS1 (1979). https://doi.org/10.1016/0029-554X(79)90116-2 19. W.W.M. Allison et al., Relativistic Particle Identification by dE/dx: The Fruits of Experience with ISIS (1982). https://www.researchgate.net/publication/253146769 20. M. Aguilar-Benitez, et al., The European Hybrid Spectrometer. Nuclear Instruments and Methods in Physics Research Section A Accelerators Spectrometers Detectors and Associated Equipment 258(1), 26–50 (1987). https://doi.org/10.1016/0168-9002(87)90077-5

9

Ionisation Beam Cooling

9.1

The Need to Compress Charged Particle Beams

The interaction rate between colliding beams of charged particles depends crucially on their density. This is increased by focussing, but the random motion of individual charges within the beams sets a limit described by the phase space density. To compress the density, this random motion must be cooled. This is a major requirement in modern accelerators and can be achieved in three different ways, depending on the application.

9.1.1

Unique Experiments with High Energy Muon Beams

Muon beams have unique advantages for fundamental physics experiments. Because they are purely leptonic point-like particles, their interactions are less dependent on the complications of strong interactions in first order. This property is shared by other leptons, including electrons and neutrinos. However, neutrinos cannot be accelerated, and work with electron beams is severely disrupted by radiation effects at high energy. However, muon beams suffer from two severe difficulties that have to be overcome. Muons do not exist in the wild. Significant fluxes are only available from the decay of intense beams of charged pions. This decay process releases heat in the form of a randomly aligned decay momentum of 29.8 MeV/c. If the resulting beam of muons could be cooled to remove this random 29.8 MeV/c, it could be accelerated further for use in a muon–muon colliding beam experiment. Alternatively, such a beam could provide a high brightness source of neutrinos from muon decays, but that too would require cooling.

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A major difficulty is that the muon lifetime is only 2.2 µs. Even enhanced by time dilation for relativistic motion, this means that only the fastest beam cooling method can be effective.

9.1.1.1 Stochastic Cooling This is a statistical method that uses fast feedback loops and kicker magnets to progressively correct deviating charges within a bunch [1]. It may be described as employing a Maxwell Demon to do the work of compressing the bunch [2]. It was developed by Simon van der Meer at CERN to cool beams of anti-protons. It was an essential step in the discovery of the W and Z carriers of the weak force for which he and Carlo Rubbia were awarded the Nobel Prize in 1984. But it takes a considerable time to cool a beam in this way, and so it is not suitable for beams of unstable particles such as muons with a short lifetime. 9.1.1.2 Electron Cooling This was invented by Budker in Novosibirsk and uses a beam of electrons co-travelling with the ion beam of interest [3]. As seen in the co-travelling centreof-mass frame, elastic coulomb collisions occur between ions and electrons due to their random velocities. If the electrons are well bunched, the random energy of the ions is progressively shared with the lighter electrons through multiple collisions. In this thermalising process, the random motion of the ions is reduced at the expense of the electrons. 9.1.1.3 Ionisation Cooling In principle, this method works in the same way as electron cooling. But instead of the diffuse charges in an electron beam, it uses the electrons in stationary matter, in particular liquid Hydrogen. Because of the greater density of electrons, the cooling is much faster than that with electron beams. So this is the method favoured for cooling muons.

9.2

Ionisation Cooling of Muon Beams

If the energy of each incident muon is reduced by passing through a medium, such as Hydrogen, then it will lose momentum in its own direction of travel on average, including that transverse to the beam as a whole. If the whole beam is then accelerated in the uniform longitudinal electric field of an RF accelerator pulse, the longitudinal momentum will be re-established without the transverse component. The beam is thereby be cooled in a relatively short time. However, each ion in the beam undergoes scattering collisions in the medium as well as energy loss, and these work to increase the transverse motion and heat the beam. There is therefore a delicate balance between scattering and energy loss. To understand this balance, we need the statistical behaviour of both processes and their correlation, as we derived in Chap. 7. Since dEdx contributes cooling and

9.3 Validation of Scattering Calculation for Hydrogen

117

scattering contributes heating, the preferred absorber medium is liquid Hydrogen for which the ratio is greatest. Cooling is then achieved by passing the muon beam through a sequence of Hydrogen absorbers and RF cavities. Circular accelerators have been designed in this way to progressively cool the muons, as described by the study ICOOL [4]. Because ionisation cooling is an iterative process, any error in its description is likely to have disproportionate consequence on the performance of the whole accelerator. The simulation of the energy loss and scattering of muons in liquid Hydrogen are critical to the design of such a muon cooling channel [5], and confirmation that such simulations are correct is important.

9.3

Validation of Scattering Calculation for Hydrogen

Hydrogen differs from other elements because of the relative prominence of constituent scattering by electrons. This was already evident in Tables 7.1 and 7.2 where RMS98 for Hydrogen is predicted by ELMS to be significantly smaller than by the conventional formula based on radiation length, X0 . As a result, the validation of the ELMS simulation by experimental scattering data for Hydrogen is important. Relevant data have been measured by the MUSCAT Collaboration for 172 MeV/c muons [6]. These are shown for two thicknesses of liquid Hydrogen in Table 9.1 with values calculated with ELMS [7]. These are plotted graphically in Fig. 9.1, also with two sets of values calculated with Moliére scattering and others with GEANT [8]. The agreement with ELMS is evident. The plot shows that GEANT gives a poor match to the data. When the contribution of electron scattering is included in the standard Molière calculation [9], it fits the data, but only at small angles. At large

Table 9.1 Experimental probability values of folded binned projected scattering angle [6] for 172 MeV/c muons compared with ELMS calculation [7] for the density 0.0755, determined by the experiment [Reprinted from [7] credit: © 2007 IOP Publishing Ltd. Reproduced with permissions. All rights reserved] Range mrad 0–2.69 2.69–8.95 8.95–16.2 16.2–24.8 24.8–34.7 34.7–46.3 46.3–59.7 59.7–75.4 75.4–93.8 93.8–115.1

Thickness 109 mm Measured ELMS 49.5 ± 2.7 48.4 35.8 ± 1.5 37.2 15.7 ± 0.41 15.0 2.70 ± 0.39 2.75 0.34 ± 0.061 0.298 0.061 ± 0.012 0.0571 0.022 ± 0.006 0.0224 0.013 ± 0.003 0.0106 0.008 ± 0.004 0.0050 0.007 ± 0.005 0.0024

Ratio 1.02 0.96 1.05 0.98 1.14 1.07 0.98 1.23 1.59 2.9

Thickness 159 mm Measured ELMS 40.6 ± 1.7 39.7 32.3 ± 0.9 33.2 18.3 ± 0.18 17.8 5.10 ± 0.38 5.27 0.88 ± 0.014 0.831 0.117 ± 0.028 0.123 0.039 ± 0.009 0.036 0.017 ± 0.004 0.017 0.010 ± 0.002 0.0078 0.008 ± 0.003 0.0034

Ratio 1.02 0.97 1.03 0.97 1.06 0.95 1.09 1.01 1.28 2.33

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Fig. 9.1 Experimental projected scattering angle distributions for (a) 109 mm and (b) 159 mm of liquid H2 from the MUSCAT collaboration, compared with ELMS and other calculations [7]. The data are shown with error bars. The continuous curves are the ELMS predictions, the circles are those of GEANT4, and the two dashed curves are for Molière calculations, including and excluding the electron contribution [Reprinted from [7] credit: © 2007 IOP Publishing Ltd. Reproduced with permissions. All rights reserved]

angles beyond the limited μ-e centre-of-mass momentum, the Molière calculation only fits when electron scattering is excluded. This kinematic effect is incorporated naturally in ELMS.

9.3 Validation of Scattering Calculation for Hydrogen

9.3.1

119

Probability Maps for Energy Loss and Scattering

In Sect. 7.2.2 on page 98, the generation of a map for a thin sample at a given momentum was described. To provide such maps for arbitrary momenta and thicknesses of liquid Hydrogen, a database of maps has been constructed for thicknesses of one mm or less for 138 momenta on a logarithmic scale from 5 MeV/c to 53 GeV/c, close enough to permit interpolation to 1% accuracy. Each map is on a 5% logarithmic grid in energy loss and pT . This database of normalised probability maps may be sampled and interpolated for any required thickness and momentum. The energy loss simulation has been validated by Holmes who checked against the known range-momentum table for muons in Hydrogen [10].

9.3.2

Correlation Between Energy Loss and Scattering Distributions

Fluctuations in energy loss are inevitably correlated with those in scattering because they arise from the same collisions. This is particularly true for those with electrons that play a major role in Hydrogen. Such correlations are described by ELMS but are not included in the standard ICOOL simulation with multiple scattering based on radiation length and of energy loss based on the Bethe–Bloch formula. How big is the correlation in 10 cm of Hydrogen and does it influence the cooling performance? Figure 9.2 shows the mean transverse momentum for each energy

Fig. 9.2 The mean scattered momentum plotted against energy loss at 200 MeV/c on passing through 10 cm of H2 . Simulations by standard ICOOL and by ELMS-ICOOL [10]. [Reprinted from [7] credit: © 2007 IOP Publishing Ltd. Reproduced with permissions. All rights reserved]

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loss decile. The ELMS simulation shows the expected positive correlation between scattering and energy loss in contrast to the standard simulation. However, Holmes has studied the effect of suppressing this correlation in ELMS-ICOOL simulations and confirmed that it has no effect on the cooling characteristics [7, 10].

References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10.

Wikipedia, Stochastic Cooling. https://en.wikipedia.org/wiki/Stochastic_cooling Wikipedia, Maxwell’s Demon. https://en.wikipedia.org/wiki/Maxwell’s_demon Wikipedia, Electron Cooling. https://en.wikipedia.org/wiki/Electron_cooling R.C. Fernow, Proceedings of the 2005 Particle Accelerator Conference (Knoxville TN) (IEEE, Piscataway, 2005), pp. 2651–2653 W.W.M. Allison, Calculations of Energy Loss and Multiple Scattering (ELMS) in Molecular Hydrogen (2003). https://doi.org/10.1088/0954-3899/29/8/334 MUSCAT collaboration, The Scattering of Muons in Low Z Materials (2005). https://doi.org/ 10.1016/j.nimb.2006.05.006 W.W.M. Allison et al., Ab Initio Liquid Hydrogen Muon Cooling Simulations with ELMS (2007). https://iopscience.iop.org/article/10.1088/0954-3899/34/4/007/pdf GEANT4, The Simulation of the Passage of Particles Through Matter Using Monte Carlo Methods. Nucl. Inst. Meth. A 506, 250–303 (2003) H.A. Bethe, Molière’s Theory of Multiple Scattering. Phys. Rev. 89, 1256 (1953). https:// journals.aps.org/pr/abstract/10.1103/PhysRev.89.1256 S.J. Holmes, The Physics of Muon Cooling for a Neutrino Factory, DPhil thesis (Oxford University, Oxford, 2006)

List of Symbols

Constants

re /α 2

a0 c e e me re h¯ α

0 κ μ0

e2 /(4π 0 me c2 ) e2 /(4π 0 h¯ c) 1/(μ0 c2 )

0.529 × 10−10 m 2.998 × 108 ms−1 −1.602 × 10−19 2.7183 9.109 × 10−31 kg 2.818 × 10−15 m 1.054 × 10−34 Js 7.297 × 10−3 8.854 × 10−12 SI 1.79 4π × 10−7 SI

Bohr radius of hydrogen atom Velocity of light in free space Electron charge Exponential constant Electron mass Classical radius of the electron Planck’s constant h over 2π Fine structure constant Dielectric permittivity of free space Anomalous magnetic moment of proton Magnetic permeability of free space

Scalars

a a b A, A’, B, B’, B0 B dEdx E E

Energy–frequency scaling parameter for transition radiation Doppler factor appearing in the Lienard–Wiechert potentials A generic distance Position points on a charge trajectory Modulus of magnetic field B Mean rate of (fluctuating) energy loss along path Energy  Modulus of E field, and its transform E

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122

List of Symbols Er Ep E

ELMS f , fi f (ω) H H IP I /I0 k kL , kT m also m∗ M Mi n ne N Nc N1 N2 P p pL , pT q = ze Q, Q0 Q2 R R r0 RMS98 S t u v V1 y0 vT X X0 z Z

The modulus of the scattered electric field Plasma energy Energy loss in a single collision A code mapping distributions in energy loss and multiple scattering The oscillator strength of an electron in the medium The oscillator strength density per electron in the medium Modulus of H field Heaviside function, indefinite integral of Dirac δ function Mean ionisation potential of a material ≈ IP 0 Z Ionisation density normalised to its minimum at βγ ≈ 4 Modulus of wave vector k Components of k transfer Thickness of a slice or layer of material Mass of charged particle Generic target mass, me or M Photon effective mass (variable) Mass of charged nucleus Mass of incident charged particle √ Refractive index, r μr Electron density Generic number of photons Number of Cherenkov photons per unit track length Number of photons emitted from a single interface Number of photons emitted from a 2-sided slab Field point label Modulus of momentum Components of 3-momentum transfer Charge (Coulombs) Position label of a charge 4-momentum transfer Radius of curvature of charge trajectory Distance of the charge as seen at point P Range of a Yukawa potential Root mean square scattering angle of least 98 % Energy flux Time Group velocity Modulus of charge velocity Name of a certain region of space Transverse 2D field range for a source moving along x-axis Apparent transverse component of velocity  A generic field, with its transform X Radiation length Generic charge in units of e Nuclear charge in units of electron charge

List of Symbols α β γ  also δ  also

1

2

r η η θ, θ0 θC  λ λabs λc μr ρ σ σγ τ φ φ φ  ψ ω ω0 ωp 

123 A generic constant in definition of Dirac δ function Modulus of β  Factor 1/ (1 − β 2 ) Classical damping constant of a harmonically bound electron, Full width at half maximum of resonance Dirac delta function Discontinuity in apparent angular velocity A layer thickness (dEdx calculation) Real part of dielectric permittivity relative to free space Imaginary part of dielectric permittivity Complex relative dielectric permittivity ( 1 + ı 2 ) Frequency shift factor in Compton scattering An energy loss in a layer (dEdx calculation) Polar angle Cherenkov angle Angular parameter arising in energy loss calculation Radiation wavelength Radiation energy absorption length Compton wavelength h/mc Complex relative magnetic permeability (μ1 + ıμ2 ) Charge density field, with its 4D-transform ρ  Energy loss collision cross section per electron (or atom) Photon absorption cross section per electron (or atom) A tiny probability (in dEdx calculation) A generic scalar field Emission angle of radiation observed outside the medium Electric scalar potential field, with its 4D-transform  φ Half the phase difference of waves across slab A generic scalar field Angular frequency The resonant angular frequency Plasma frequency Solid angle, thus d = 2πd cos θ

124

List of Symbols

Vectors

A B D  D E  E H J k K M p pT P r s v β

A Magnetic vector potential field, with 4D-transform  B Magnetic flux density (or B field), with 4D-transform  D Displacement field density, with 4D-transform  Time deriv. of Fourier transform of D E Electric field (amplitude E0 ), with 4D-transform  Time deriv. of Fourier transform of E Magnetic field J Current density field, with 4D-transform  Wave vector or number, magnitude 2π/λ Crystal wave vector (Bragg) Magnetic moment per unit volume, magnetisation field Vector momentum Transverse momentum Electric dipole moment per unit volume, polarisation field Position vector, e.g. of an electron relative to equilibrium Unit vector in the direction of the charge as seen at a point P Charge velocity Charge velocity relative to c

Index

A Absorption length, 59, 60 Advanced potential, 42 Anomalous magnetic moment, 94 Apparent angular velocity, 53 Apparent charge dipole creation, 52 Argon, 96, 106 Attenuation, 59

B Baseline restoration, 107 Beam heating, 116 Bethe-Bloch, 83 Bethe sum rule, 72 B-field, 26 Blum, W., 84 Bohr Radius, 71, 93 Bound charge, 28, 29 Bound current, 28 Bragg scattering, 13 Bremsstrahlung, 28, 88

C Calorimetry, 32 Centre of mass, 89 CERN ISR, 21 Chemical bonding, 63 Cherenkov radiation, 3, 8, 14, 33, 51, 77, 78, 97 Classically bound electron, 60 Classical oscillator, 61 Classical radius of the electron, 66 Collision, 36 Compton scattering, 67 Compton wavelength, 12, 64 Conservation of electric charge, 27 Constituent scattering, 60, 83, 89 Cooling, 115

Correlated cross section, 98 Correlated distributions, 119 Coulomb field, 50 Coulomb’s law, 25 Cross section, 59

D Decaying wave, 59 DEdx distributions, 81 Delta function, 46, 56 Density, 63 Density effect, 9, 36, 77, 87, 104 Dielectric permittivity, 30 Dielectric polarisation, 29 Diffraction, 14 Diffuse charge, 93 Diffusion, 82 Dipole approximation, 61, 72 Dirac scattering, 95 Discrimination, 107 Dispersion relations, 61 Displacement, 29 Displacement current density, 27, 30 Doppler, 50, 51

E Effectively free electrons, 95 Effective mass, 10 Electric field, 25 Electric scalar potential, 40 Electron cooling, 116 Electron scattering, 92, 95 Energy absorption, 30 Energy loss, 3 Energy Loss, and Multiple Scattering (ELMS), 98 Energy loss distribution, 82 Energy loss estimator, 82

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126 European Hybrid Spectrometer (EHS), 104 Excitation, 35

F Faraday’s law of electromagnetic induction, 27 Fermi plateau, 9, 36, 77 Feynman–Heaviside, 49, 51, 54, 78 Folding, 79, 80 Folding distributions, 80, 106 Formation zone, 20 Form factor effects, 96 Form factors, 93 Fourier transform fields, 44 Fraunhofer diffraction, 15

G Gas amplification, 107 Gas mixtures, 81 Gauge transformation, 40 Gauss’s law, 26, 29 Group velocity, 5, 32

H Heaviside function, 73 H-field, 29 Hydrogen form factors, 94, 95

I Identification of Secondaries by Ionisation Sampling (ISIS), 81, 104, 106, 108 Induction field, 50 Integral and differential fields, 17 Integrated energy loss, 83 Ionisation cooling, 116 Ionisation probability, 83, 110 Ionisation resolution, 110 Ionisation truth table, 111 Iron form factors, 95 Isotropic medium, 30

Index Krypton, 106 K-shell, 104 K-shell absorption edge, 31

L Landau distribution, 80 Landau fluctuations, 36 Leptons, 115 Liénard–Wiechert Potentials, 43, 49 Linearity, 30 Liquid Hydrogen, 117 Lorentz force, 26 Lorenz gauge, 40, 46

M Magnetic deflection, 32 Magnetic dispersion, 60, 74 Magnetic flux density, 26 Magnetic permeability, 30 Magnetic vector potential, 40 Magnetisation, 29 Maximum energy transfer, 75, 83, 85 Maxwell’s equations, 26, 39 Mean dEdx, 105 Mean energy loss, 73 Mean ionisation potential, 84, 104 Mean range, 59 Moliére scattering, 99, 118 Monte Carlo tracking, 79 Mott scattering, 85 Moving slab, 34 Multiple scattering, 90, 97 Muon, 103 Muon beam cooling, 116 Muon lifetime, 116 Muon-muon colliding beams, 115

N Neutrinos, 115 Normalised ionisation, 104 Normal mode, 10 Nuclear scattering, 92

J Joule heating, 60

K k L , 90 k T , 90 Kaon, 103 Kinematics, 32

O Optical region, 31, 62 Optical transition radiation, 16, 34 Oscillator strength, 62 Oscillator strength density, 63 Oxygen contamination, 107

Index P Pair production, 67 Particle Data Group (PDG), 99, 119 Particle identification, 32 Phase lock, 4, 47, 70 Phase velocity, 5, 32 Pion, 103 Planck quantisation, 27, 70 Plasma frequency, 31, 61, 104 Plasma oscillations, 31 Poisson’s equation, 93 Polarisation, 29, 61 Potential fields, 40 Poynting vector, 57 Probability maps, 98 Proton, 103

Q Q2 , 92 Quantum numbers, 103 Quarks, 103

R Radiation Length, 99, 119 Refraction, 59 Refractive index, 8, 31, 59 Relative dielectric permittivity, 61 Relativistic rise, 35, 77, 84, 104 Resolution, 106 Resonance region, 83 Resonance scattering, 60, 89 Retarded potential, 42 Retarded time, 42 RMS98, 99 Rolandi, L., 84 Rosenbluth scattering, 94 Rutherford scattering, 79, 90

S Saturation, 83 Scattering, 89 Screening, 92 Semiclassical quantum approximation, 27 Simple harmonic oscillators, 60

127 Single scattering, 97 Snell’s law, 16 Space charge effect, 109 Special relativity, 6 Spin effects, 92 Steady state solution, 61 Step function, 73 Sternheimer, 106 Stochastic cooling, 116 Sum rule, 62, 64 Synchrotron radiation, 28

T Thin samples, 98 Thomas–Fermi atom, 93 Thomson scattering, 65 Time dilation, 116 Time of flight, 32 Total internal reflection, 6 Transition radiation, 3, 14, 33, 55 angular distribution, 18 energy distribution, 19 total energy, 19

U Ultraviolet absorption, 31, 81

V Validation, 83 Velocity resolution, 80

W Wake field, 4 Water waves, 4 Wave vector, 4 Weissacker–Williams, 88 Woods-Saxon nuclear shape, 94

X X-ray region, 31, 62 X-ray transition radiation, 3, 16, 17, 34, 57