Synchrotron Radiation: Production and Properties (Oxford 2000)

O X F ORD S ERIES ON S Y N C HROTRON R A D I A T I O N

Series Editors J. CHIKAWA J. R. HELLIWELL S. W. LOVESEY

O X F ORD S ER I E S ON S YN C HROTRON RADIATION

1. S.

W.

Lovesey and S. P . Collins:

materials

X-ray scattering and absorption by magnetic

2. J. R. Helliwell and P. M. Rentzepis: Time-resolved diffraction P. J. Duke: Synchrotron radiation: production and properties

3.

Synchrotron Radiation Production and Properties

Philip John Duke

Honorary Scientist Daresbury Laboratory

OXFORD UNIVERSITY PRESS

OXFORD UNIVERSITY PRESS

Great Clarendon Street, Oxford OX2 6DP Oxford University Press is a department of the University of Oxford, It furthers the University's objective of excellence in research, scholarship, and education by publishing worldwide in Oxford New York Athens Auckland Bangkok Bogota Buenos Aires Calcutta Cape Town Chennai Dar es Salaam Delhi Florence Hong Kong Istanbul Karachi KualaLumpur Madrid Melbourne MexicoCity Mumbai Nairobi Paris Sao Paulo Singapore Taipei Tokyo Toronto Warsaw with associated companies in Berlin Ibadan Oxford is a registered trade mark of Oxford University Press in the UK and in certain other countries Published in the United States

© Philip John

by Oxford University Press, Inc" New York Duke, 2000

The moral rights of the author have been asserted Database right Oxford University Press (maker) First published 2000 All rights reserved, No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, without the prior permission in writing of Oxford University Press, or as expressly permitted by law, or under terms agreed with the appropriate reprographics rights organisation, Enquiries concerning reproduction outside the scope of the above should be sent to the Rights Department, Oxford University Press, at the address above You must not circulate this book in any other binding or cover and you must impose this same condition on any acquirer A catalogue record for this book is available from the British Library Library of Congress Cataloging in Publication Data

851758

(Data applied for) ISBN 0 19

0

Typeset by Newgen Imaging Systems (P) Ltd" Chennai, India Printed in Great Britain on acid-free paper by TJ, International Ltd" Padstow, Cornwall

To Ian Hyslop Munro Friend, Colleague and Synchrotron Radiation Pioneer

A quella luce cotal si diventa, che volgersi da lei per altro aspetto

e impossibil che mai si consenta;

That light doth so transform a man's whole bent That never to another sight or thought Would he surrender, with his own consent.

Dante. The Divine Comedy. 3 Paradise. Canto 33, lines 1 00-102. Italian text taken from the Oxford University Press Edition 1 939. English translation by Dorothy L. Sayers and Barbara Reynolds, and published by Penguin Classics.

The European Synchrotron Radiation Fm:ility seen from the air. Photograph supplied

by the ESRF

and credited 10 ARTECHNIQlJE.

Preface

The purpose of this book is to provide the reader with a thorough introduction to the production of electromagnetic radiation by high energy electron storage rings. This radiation, called synchrotron radiation, has become a research tool of wide application. It is used by physicists, chemists, biologists, geologists, engineers, and other scientific disciplines. It is used as a structural probe for the study of surfaces, bulk materials, crystals, and viruses. The radiation is used for the spectroscopic analysis of solids, liquids, and gases. It provides illumination for the generation of images of special value in the biological and medical sciences. In order to provide this radiation, centres of excellence have sprung up across the world. At first, in the late 1 960s and early 70s the use of synchrotron radiation was a parasitic activity, riding on the back of nuclear and particle physics. The radiation was supplied by electron accelerators such as the synchrotron at Glasgow in Scotland, UK and the electron synchrotron at DESY in Hamburg, which, at that time was in West Germany. In the USA much early work with synchrotron radiation was carried out at three centres. At the high energy electron ring at the Stanford Linear Accelerator Centre in California and at the synchrotron centre at Madison, Wisconsin where a dedicated storage ring provided UV synchrotron radiation and at the National Bureau of Standards, Gaithersburg, Washington DC. An important step forward was the operation of the world's first centre dedicated to the production of synchrotron radiation in the X-ray region at the Daresbury Lab­ oratory, half way between Liverpool and Manchester in the UK. The electron storage ring, known as the Synchrotron Radiation Source, or SRS, constructed there, began operation in 1 9 8 1 and replaced a former electron accelerator, NINA which, in its last few years of operation, supplied a small synchrotron radiation laboratory. The SRS was upgraded in 1 989 to provide higher beam brightness and, at the time of writing, plans are being prepared for Diamond, a new high brightness source. Across the world, Europe, the USA, Japan, China, and Russia operate large scale centres for research using synchrotron radiation. Individual countries in Europe, Asia, and South America operate sources of their own. Alongside these sources there has developed an international interdisciplinary com­ munity of synchrotron radiation users and it is to this community that this book is addressed. This book does not lay claim to originality, rather its object is to educate synchrotron radiation users in the fundamental properties and production techniques of the radiation which they are all using. The book brings together a large amount of material which has already been published but which, in many cases, is not easily accessible and which the synchrotron radiation user may not even know about. The book begins by laying a foundation of electromagnetic theory which is used in later chapters to show how the production of the radiation and the operating principles

Vlll

Preface

of the radiation sources are grounded in this theory. The theory itself as based on certain well-known observations in electricity and magnetism so the aim is to relate these simple facts to the rather complex and far reaching conclusions which follow. One key observation is the constancy of the speed of light, independent of the motion ofthe observer. The consequent methodology, known as the special theory of relativity is essential to an understanding of the synchrotron radiation production process so this is treated in some detail. The main emphasis of this book is the production of synchrotron radiation from storage ring dipole magnets but additional chapters have been added to introduce the reader to insertion device sources of radiation. A fuller treatment of this important subject will be given in this series of books by other authors. This text grew out of a series of lectures given at King's College, London, as part of an MSc course entitled 'X-ray science and technology' . I am grateful to Professor R. E. Burge, Dr Alan Michette and the other teachers of this course for their constant encouragement over many years of collaboration. My interest in the development and uses of high energy particle accelerators began to flourish when I was an undergraduate at Birmingham University, where what was hoped to be the world's first accelerator to reach an energy of I GeV was under construction. My experience as a user of these machines continued at Brookhaven National Laboratory and, later, at CERN. It is impossible to mention all those from whom I learned over some 40 years as an accelerator user. I am particularly grateful to the Director of the Daresbury Laboratory and the Head of the Synchrotron Radiation Division for their hospitality during the writing of this book. I am most grateful to V. P. Suller and the past and present members of the SRS Accelerator Physics Group at Daresbury for answering my questions and providing me with much useful material. My special gratitude goes to my friend and colleague Ian Munro who was instru­ mental in bringing synchrotron radiation to the UK and to Daresbury. This book is dedicated to him. Most of all I would like the readers of this book to share my enthusiasm and delight in the remarkable phenomena of the natural world.

Bibliography An understanding of the production and properties of synchrotron radiation is based on the classical electromagnetic theory of radiation and the special theory of relativity. There is a massive literature already available on both these vast subjects and it is both impossible and unnecessary to refer to everything. There is an enormous amount of duplication and the books vary not so much in the material covered as in the clarity of presentation. Particularly useful is J. D. Jackson, Classical electrodynamics (2nd edition), John Wiley & Sons, New York which covers both electrodynamics and relativity. On relativity alone there is Wolfgang Rindler, Essential relativity (Revised 2nd edition), Springer Verlag, New York, Berlin, and Heidelberg.

Preface

IX

The literature available on the theory of particle accelerators is much more sparse. The most useful recent book is Helmut Wiedemann, Particle accelerator physics, published in 1993 by Springer Verlag. The first volume treats the linear theory of particle accelerators in detail and gives an introduction to synchrotron radiation. The second volume, published more recently, gives a full treatment of synchrotron radia­ tion production from dipole magnets and from insertion devices. There are a large number of "summer school" and conference proceedings which contain individual contributions on topics connected with storage rings and syn­ chrotron radiation production. Amongst there are: Synchrotron Radiation Sources and their Applications, Aberdeen, September 1985. Edinburgh University Press, 1 989. CERN Accelerator School, Chester, UK, April 1989. CERN 90-03. CERN Accelerator School, KFA, liilich, Germany, September 1990. CERN 91-04. The mathematical background can be found in Mary L. Boas, Mathematical methods in the physical sciences (2nd edition), John Wiley and Sons, New York. A world-wide survey of synchrotron radiation sources is available as H. Munro, C. A. Boardman, and J. C. Fuggle, World compendium ofsynchrotron radiationfacil­ ities, published by The European Synchrotron Radiation Society, c/o Batiment 209 D, LURE, Universite Paris Sud 9 1 405, Orsay, France.

I.

Acknowledgements Figures 1 2.8 and 1 2.9 were supplied by Lesley Welboume, formerly of the Dares­ bury Laboratory. Figure 1 2 . 1 0 was supplied by Rick Fenner, the Advanced Photon Source, Argonne National Laboratory. Figures 13.3 and 1 3.4 were supplied by James Clarke, Daresbury Laboratory . Figure 15. 1 was supplied by Chantal Argoud, and is reproduced by courtesy of the European Synchrotron Radiation Facility. Grateful thanks are expressed to these people and their organisations for permission to publish this material.

Helsby, Cheshire May, 1999

P. J. D .

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Contents

1 Synchrotron radiation and electromagnetic waves Prelude Elementary wave properties Electromagnetic waves The electromagnetic spectrum Electromagnetic waves around us The wave equation Analysis of the wave spectrum Electromagnetic wave pulses are called photons

2 Electromagnetic radiation is produced by electrons Introduction Electrons-source of electric field Electric fields can be described b y lines of force Moving electrons-source o f magnetic field Fields described b y potential functions The divergence equation The continuity equation The rotation or curl equation Changing magnetic fields produce electric fields Changing electric fields produce magnetic fields Maxwell's equations and electromagnetic radiation

3 Electromagnetic radiation-observed and imagined The special place of the velocity of light Relative motion-classical relativity Special Relativity Understanding the Lorentz transformation The Doppler effect-in the forward direction The Doppler effect-when the source is viewed from an oblique angle Lorentz transformation of four-vectors Transformation of velocities Transformation of accelerations

1 1 2 3 4 5 6 7 13

16 16 16 17 19 21 22 24 25 27 29 29

32 32 33 34 36 38 40 44 45 47

Contents

xii Energy and momentum The four-force

4 Radiation from moving electrons Electromagnetic waves in free space-no electric charges or currents Electromagnetic waves produced by current and charges Electromagnetic waves transfer energy-the Poynting vector Electromagnetic waves transfer momentum Electromagnetic waves generated by a distant source

5 Synchrotron radiation from dipole magnets Properties of circular motion Motion of a moving charge in a constant magnetic field Radiation of energy by a moving charge in a constant magnetic field Dependence of radiation on frequency and angle Fourier transform of the electric field Radiation from an electron moving along the arc of a circle

6 The spectral distribution of synchrotron radiation Properties of the modified Bessel functions Photon distribution as a function of energy and angle Photon polarization as a function of energy and angle

50 51

53 53 55 57 60 61

76 76 77 79 88 91 93

97 97 1 02 1 06

7 Photon spectral distribution integrated over vertical angles

1 14

8 Introduction to electron storage rings

127

Principles of operation Dipole magnets Quadrupole magnets Multipole magnets Total energy radiated Radio-frequency cavities Electron beam dimensions Beam lifetime and beam movement Ring injection Radiation shielding

1 27 127 1 30 134 1 34 136 141 141 142 143

Contents 9

Synchrotron radiation from electron storage rings Description of the electron beam Definition of synchrotron radiation brightness Calculation of synchrotron radiation brightness Use of the synchrotron radiation universal function Approximation to the photon angular distribution

10

Behaviour of the electron beam in a synchrotron radiation storage ring. The concept of phase space Introduction The magnetic field function The electron equations of motion-motion in the horizontal plane The electron equations o f motion-motion i n the vertical plane Electron beam trajectories Phase space The phase space ellipse

11

Behaviour of the electron beam in a synchrotron radiation storage ring. Betatron oscillations Introduction Betatron oscillations Betatron tune values Energy dispersion Orbit length and energy Isomagnetic guide field

12

Behaviour of the electron beam in a synchrotron radiation storage ring. Energy oscillations Introduction Damping of energy oscillations Minimizing the electron beam emittance Vertical beam emittance

13

Insertion devices-wigglers Introduction Single and multipole wigglers Electron trajectory in a multipole wiggler

xiii

144 144 146 148 148 1 50

156 1 56 1 56 158 161 1 62 1 65 1 68

174 1 74 174 1 75 181 1 82 1 84

186 1 86 1 88 200 203 206 206 206 2 09

xiv

14

15

Conlellis Insertion devices-undulators

213

Elementary theory of unuul ators Calculation of the photon flux

213 216

Total power output Realization of undulators

228

Recent developments and future prospects

231

Introduction High hrightness sources Radiation from insertion devices

230

231 231 232

Medium energy sources

233

Coherence of undulator radiation

234

Beam current and beam lifetime considerations

237

Concl usion Epilogue

238 238

Appendix I. Vector algebra Pre l ude Defi nitions��calars and vectors Vectors in a right-handed coordinate system

240 240 240

Vector m u l tiplication--scalar product

240 241

Vector mu ltiplication�vector product Vector m u l tiplication�scalar triple product

242 243

Vector mu ltiplication�vector tripl e product

243 244 244

Square of a vector product Differentiation of vectors The gradient operator

245

The gradient operator�divergence and rotation

246

Index

247

1 Synchrotron radiation and electromagnetic waves

Prelude 1 Synchrotron radiation was first observed i n 1947 and began to be used as a research tool in the mid-1960s. The first synchrotron radiation sources were low energy (several hundred MeV ) electron accelerators which wcre also used for nuc lcar and particl e physics. I t was 4uickly realized that the needs o f the growing number o f people who used synchrotron radiation for their research re4ui red dedicated sources of radiation. Many of these research programmes required X-radiation, which pointed to the need for these sources to aeccleratc electrons to GeV energies in order to provide a useful output of X-rays as well as lower energy radiation. The first such dedicated X-ray source was the SRS (Synchrotron Radiation Source ) at the Daresbury Laboratory in the UK, which operates at 2 Ge V electron energy. The SRS did not retain its unique status for long. The NSLS (National Syn­ chrotron Light Source) at Brookhaven, USA and the Photon Factory, Tsukuba. Japan quick l y fol l owed. There were similar developments in the then USSR: initial l y at Akade mgorodok. Novosibirsk, and l ater in Moscow. The most recent advances have inc luded the construction and operation of l arge scale facilities at Grenoble, France (ESRF. 6 GeV) Argonne. USA (APS, 7 GeV ) and Nishi-Harima. Japan (SPring-S. 8GeV). [n the ear l y days the emphasis in research applications was the expl oitation of VUV radiation for atomi c and molecular spectroscopy and surfaee'-,cience but the advent of the X-ray sources has stimu l ated a steady growth of research with harder rad iation in such areas as X-ray diffraction, absorption spectroscopy (EXAFS) . crystal l ography.

topography. and lithography. The users of the radiation. initial l y drawn from the physics community. have been augmented by chemists, biologists, geol ogi.'-,ts. and others ,0 that there is now a world-wide community of people using synchrotron radiation as an essential component of their rcsearch. It is the purpose of this book to provide the basic principles of synchrotron radi­ ation production and properties for a l l users of synchrotron radiation, whatever their research i nterest or disciplinary background. The material prescnted hen: covers thc radiation production process (Chapters 4-7) and the operational principles of the radiation source (Chaptl:rs �-12). Chapters 13 and 14 introduce the user to additional devices which are lIsed to enhance the output of the radiation source. Chapter l'i

S\'IICilmfmn mC/iafi()11 ([lid C/l'cfrollwRlIl'fic

2

H'(/\'I'S

contains a survcy of recent developments and an attempt to scan the horizon for the directions of future progress. The usc of mathcmatics is essenti a l for a preci se description of the radi ation and the manner of its production. Fortunately, ror our purpose the radiation production process can be described using the concepts of c lassical e lectrodynamics although a knowl edge of re lativity theory w i l l be required. These topics arc covcred to a sufficient

level in Chapters 2 and 3. I n this first chapter we consider some basic definitions and properties pertaining to electromagnetic radiation. Readers who arc already fam i l iar with a l l these things should proceed straight to Chapter 4. They wil l need o n l y to refer to the i ntervening material if they need to refresh their memory.

Elementary wave properties Synchrotron radiation i s one part i c u l ar form of energy transmitted from the source to the detector by an electromagnetic wave, known as e lectromagnetic radiation. Visible l i ght i s one example or thi s radiation. Radio waves and X-rays are others. In practice there is a whole spectrum of this radiation (see Fig, 1.3) which reaches us in the form

of wave motion. I n order to understand its properties . therefore. we must beg i n w ith the mathematics of waves. The wave model for energy transmission was introduced by Christiaan H uygens

(162lJ-95) fol l owing the discovery of the diffraction of light by Francesco Maria Grimaldi (161X-63). The wave theory of light was extended to cover the whole of e lectromagnetic radiation fol l owing the work of James Clerk Maxwe l l ( I X31-79). I n this theory. the passage of radiation through space generates an e lectric and a magnetic fie ld at each point traversed by the radiation. An analogous s ituat i on occurs

when a r i pple travels along the surface of a pool of water. Each point on the surface of the water is displaced upwards ( or downwards) as the ripple pa�ses across it. The si mplest form of wave is a sine wave in which the displacement (or amplitude ) of the wave at a certain poi nt is described by \"

=

Ao s i n 2rr(pf - k;:),

( 1.1 )

The displacement i s a periodic function of both time and position which repeats itself indefinitely i n both directions as shown in Fig. 1.1. A snapshot of the wave at a particular moment in t i me w i l l show peaks and troughs of d i s p l acement separated by

the wavelength A

=

I I k. The maximum disp l acement is Ao. At a parti c u l ar point

along the �-ax i s the di splacement w i l l vary w i th t i me i n the same way. C learly. the product 2rr (J!f k;:), which is the argumcnt of the s i ne function. has the form of an -

angle. measured in radians and is cal led the phase angle. q), or just the phase of the wave. The quantity \! is cal l ed the frequency of the wave and is measured in vibrations per second or s I. The unit or freq uency is Hi'. (hertl'. named after H e i nrich Rudolf HertL. 1 X57-lJ4) so that 50 Hz means 50 vi brations per second. The reciprocal of 1'. I Iv. i s the time taken for the completion of one v i brati on of the wave, i .e. for one

Elci·tromilgneth·

\I·({l'C.I'

3

__ A_

I'h,l'l: angie (radian, I

Wave motion (sinusoidal).

Fig. 1.1

crest to be replaced by the next. Often the angu lar frequency. per second, is used instead of

J'

and

(()

=

2rr I'.

The velocity by which the wave moves forward is just

II!,

(' =

measurcd in radians

Ii),. Strictly spcaking.

this is the phase vel ocity. the speed with which a particular feature on the wave (a crest for example) appears to move forward. The phase angle. rp. changes with position and ti me accordi ng to rp = 2rr (vI - k �). The rate at which a fixed phase angle is observed to advance along the I-axis is given by d�/dt = 1'/ k or I'A.

How does the wave transport energy from one place to another'! The average dis­ p l acement at right angles to the direction of the wave, i .e. along the x- (and y-) axes. is /ero so we might think that the energy carried by the wave averages to zero as wel l . However, this i s not so because we know from e lementary mechanics that energy is proportional to the sq uare of vel ocity so that at any point the energy stored in the wave is proportional to the square of the transverse velocity d y/dl. which is. by taking the derivative of eqn (1.1) with respect to time, at a fixed point ::1" d\'

-'-

dt

I

.. -- .. p __ _

=

.\.

=

2rrAol' cos 2rr(1'1

-

k::p)'

The energy stored is proportional to the square of this velocity and so is proportional to the square of the wave amplitude. We cou ld equa l l y we l l have chosen the cosine function instead of the s i ne function in eqn (1.1) to describe the variation of the amplitude with t i me. This wou l d produce a wave shifted in phase by l)() . Physicall y. this a lternative choice wou ld make little

dillerence because most detectors are only sensitive to the amplitude of the wave and

cannot detect the phase.

Electromagnetic waves In the case of electromagnetic waves. the wave motion consists of two waves. propa­ gating in phase with thc plancs of vibration mutua l l y perpendicu l ar (see Chapter 4 and Fig. 1.2).

Synchrotron radiation and electromagnetic waves

4

E '" -0

.€

1 " > '"

�

0.5

0 ---0.5 -I Phase angle (radians)

Fig. 1.2

Electromagnetic wave.

The displacement of one wave is proportional to the strength of the electric field (E) and the displacement of the other to the magnetic field (B) as is shown in Fig. 1 .2. The energy stored in the wave is proportional to the square of the field strength. The phase velocity of the wave is numerically equal to 3 x 1 0 8 mls so that the radiation trav­ els through a distance of 1 m in approximately 3 ns (Ins = 1 0- 9 s or one-thousandth of a microsecond). In a vacuum, the velocity does not depend on the wavelength so that all electromagnetic radiation travels at this same speed, often called the speed of light. This is a very high speed by human standards; for example, the time taken for a muscle to contract is about one-tenth of a second and during that time a flash of light or a radio signal will travel thirty thousand (3 x 1 0 5) km, which is about the same as the distance to a communication satellite in geostationary orbit above the surface of the Earth.

The electromagnetic spectrum Although all kinds of electromagnetic radiation (light, X-rays, microwaves, radio, and TV signals) travel at the same speed, they have an extremely wide range of frequencies and wavelengths. For example, the electromagnetic radiation which constitutes red light has a wavelength of about 1 p,m. The same quantity for violet light is about one-tenth of that. X-ray wavelengths are shorter still. A standard X-ray tube produces X-rays with a wavelength about 1 000 times shorter than the visible light. At the other extreme, a domestic microwave oven operates at a wavelength of 12 cm. TV signal wavelengths occupy a band around I-1 0 m and radio wavelengths can be even longer. The product of wavelength and frequency is equal to the speed of the wave so that radio waves oscillate with a frequency of a few megacycles per second and TV signals oscillate at a few gigacycles ( 1 0 9 cycles/s or 1 0 9 Hz). The shorter the wavelength, the higher the frequency so for visible light and X -rays, the frequencies become so huge as to be hardly useful as a unit of measurement.

Electromagnetic waves around us 12 1fm= 10- m

Wavelength

1fm lOfm 100fm Inm lOnm l 00nm l /Lm 10 /Lm 100/Lm I mm lOmm 100mm 1m 10m 100m I km 10km

9 1nm=lO- m

Energy

12.4 MeV 1. 24 MeV 124keY o. l A 12.4keY 1A 1.24keY lO A 100 A 124eV 1000 A 12 .4eV 124eV

-6 1/Lm=lO m

Electron

Fig. 1.3

Gamma rays

Hard Atom X-rays Molecule Soft Protein Ultraviolet Virus Cell

Frequency 300GHz 30GHz 3GHz 300 MHz 30 MHz 3 MHz 300kHz 30kHz

5

Visible Infrared

Ant Cat Human

Microwaves

Radio waves

City

The electromagnetic spectrum.

Figure 1 .3 shows (approximately) how the wavelengths of the electromagnetic spec­ trum relate to human experience of length. You should note at this stage that syn­ chrotron radiation contains all the wavelengths and kinds of electromagnetic radiation just mentioned so it is useful to have a picture in your mind of the wavelengths involved and how these lengths compare with the dimensions of everyday objects.

Electromagnetic waves around us At any particular point in space and time, the strength of the electric and magnetic fields is the sum of the amplitudes of the waves reaching that point, so that the time variation (or the variation in space as we move from one place to another) can be far removed from the simple expression given in eqn ( 1. 1 ). If you have the right equipment you will be able to distinguish between the different wavelengths (or frequencies) that are reaching you so that you become aware of the individual frequencies (or the spectrum). One such piece of equipment is a shower of rain which can spread out the visible part of the light that is coming from the Sun into the familiar colours of the rainbow. Sir Isaac Newton (1642-1727), in his book Opticks (1703), describes how he used a triangular prism of glass to do the same thing in a controlled way. This simple observation tells us that ordinary daylight, which would be difficult to describe mathematically, can be described as the addition of a series of elementary waves, which are the building blocks of light.

6

Synchrotron radiation and electromagnetic waves

The wave equation To investigate this in a formal way, we return to the original discussion of wave motion. We can determine the transverse acceleration of the wave at any point by taking the second derivative of eqn (1. 1). It is important to be careful about this because we can calculate both the time derivative and the position derivative. In order to allow for this, we must take the two partial derivatives, one derivative in which time is held constant while position along the z-axis changes and one for which position is held constant while the time coordinate changes. These derivatives (together with the original sine wave function) are set out in eqn ( 1.2) below:

y = Ao sin 2n (vt - kz); ay ay - = -2nkAo cos 2n(vt - kz), (1. 2) - = 2nvAo c os2n(vt - kz); az at a2 y . . 2n(vt - kz), a 2 y - 4n 2v2A2sm 2sm 2 = - 4n 2k 2Ao 2n(vt -kz). o az at2

-

=

We can use these equations to eliminate the sine and cosine functions t o give equations which contain only the first and second partial derivatives and the wave parameters v and k:

ay = az

kay vat

1 ay c at

(1.3)

and

k2 a2 y a2 y =az2 v2 at 2

a2 y . 2 at2

1

= -c

(1.4)

Equation (1.3) can be rearranged to tell us that c = v / k VA is the velocity of the wave along the z -axis, as we had deduced by examination of the original sine function in eqn (1.1). Equation (1 .4) is very important. It is called the wave equation and is a very general statement about any kind of disturbance. It has a multiplicity of solutions. This is clear from the way we have derived it. When we take the partial derivatives the original functional form of the wave has cancelled out. We could have equally well taken any function, ¢ which is linear in z and t, e.g. y = ¢(z, t). Provided the function is mathematically well-behaved so that it actually has a first and second derivative then it can be a solution of the wave equation. The form of the wave does not have to be a sine function (or a cosine function), almost anything will do. We can write the wave equation in a more general form to make it clear that a wave disturbance can transport energy throughout space. We write

=

Analysis of the wave spectrum

7

or V2 ¢

I a2 ¢

= - C2

at 2 '

(l.S)

The symbol V2 is called 'del squared' (see Appendix) and is shorthand for the sum of the second order partial derivatives along the three spatial axes.

Analysis of the wave spectrum

The discussion above is not merely of theoretical interest. The fact is that light from any kind of source almost never comes to us as a wave with a single unique wavelength, extending over the whole of time and space. The amplitudes of the electromagnetic waves that reach our eyes vary with time in an irregular way and contain many wavelengths. To put this very simply, our eyes detect an electromagnetic disturbance in the visible region of the electromagnetic spectrum. This is the function ¢ in the wave equation, a spatio-temporal electromagnetic disturbance which we can call a wave packet. The optical system in our eyes, the cornea and the lens, analyse the wave packet into an image on our retina. The light sensitive cells in the retina react differently to the optical wavelengths and, in combination with the processing power in the visual cortex of our brain, separates this image into a variety of colours. The process corresponds to an analysis of the original function into a series of spatial and temporal wave trains of the form shown in eqn (Ll). The idea that the time-varying amplitude of the electromagnetic radiation at any point in space can be resolved into a spectrum of frequency components was first put on a sound mathematical footing by the French engineer Jean Baptiste Joseph Fourier (Baron de Fourier) ( 1 768-1 830). He observed that any regular wave motion, no matter how complicated, could be expressed as the sum of a series of cosine and sine waves. This means that if you observe a regular variation of an amplitude A(t) as a function of time, that amplitude can be described as the sum of an infinite series of elementary frequency components in the following way:

A (t)

=

� Ao + A l cos(wt) + A 2 cos(2wt) + A 3 cos(3wt) + . . .

+ BJ sin(wt) + B2 sin(2wt) + B3 sin(3wt) + . . . .

( 1 .6)

Another way of looking at this expression is to say that the shape of the time distribu­ tion is being approximated as the sum of a series of cosine and sine functions so that 00

1 A (t) = lAo + L [Am cos(m wt ) + Bm sin(mwt)]. m

( 1 .7)

=J

The first coefficient, Ao, is zero so long as A (t) averages to zero. It is important to realize that the Fourier coefficients A m and Bm give the amplitude of the components of the frequency spectrum of A (t).

8

Synchrotron radiation and electromagnetic waves

The coefficients A and B are calculated by using the following property of the sine and cosine functions:

1

1-+]T]T +]T 1 1-+]T]T

7r 1 7r

1 -

7r

-

]T

sin(m u) sin(nu) du

=

cos(m u) cos(nu) du

=

sin(m u) sin(nu) du

=

( (0 I

0 I

n, i= n, whenm n, when m i= n, whenm whenm

=

=

( 1 . 8)

O.

I n eqn (1 .8), u i s any integration variable and m and n are integers. To use these identities, take either eqn ( 1 .6) or ( 1 .7), multiply each term by cos( w t) /7r , and integrate the whole thing from -7r to +7r (or in fact any interval spanning a total range of 27r) and you will obtain

Al

1

= 7r

1-+]T]T A(t)

cos (wt) dt

because [from eqn (1 .8)] the integral multiplying the constant A l is equal to unity and all the other integrals in the series reduce to zero. You can calculate the constant term Ao by mUltiplying eqn ( 1 .6) or ( 1 .7) by unity and integrating as before, remembering that cos(wt) = 1 when w = O. All the other coefficients can be obtained in the same way so that, in general, for the nth coefficient:

A (t) ]T 1+ 1 -]T 1-+]T]T B(t)

1 7r

cos(n wt) dt

Bn = 7r

sin (nwt ) dt

An

=

,

(1.9)

for n = 1 ,2, 3, etc. Once the amplitude function A(t) is known, the Fourier coefficients can be deter­ mined. It is worth noticing that, as a consequence of eqn (1.8), each of the coefficients A and B are independent of the others, since the trigonometric functions form what is called an orthogonal set. It is very important to understand that what is being done here is not an abstruse piece of mathematics but is a way of converting an observed wave amplitude, which may be a very complicated function of time or space coordinates into a sum of elementary cosine and sine waves. When the amplitude function depends on time in a regular repetitive way, the components consist of a fundamental frequency w and a series of harmonics whose frequencies are an integer multiple of the fundamental frequency. If the amplitude function depends on the space coordinates then the Fourier spectrum is a function of wavelength. The operation of obtaining the spectrum from the amplitude function is called the Fourier transformation [eqn (1 .9)] and the result

9

Analysis of the wave spectrum 1.5 1

-

71:1

-3 71:12 -71:

-

r--

0.5

2

0

-0.5 -\ -1.5

0

71:12

71:

371:/2

Phase angle (radians) Fig. 1.4

Square wave.

of the operation is called the Fourier transform. The transformation can be carried out in the opposite direction and a wave amplitude can be obtained from the sum of the Fourier components [eqn ( 1 .6) or (1.7)]. Very often, what we observe or calculate is a time-dependent spectrum but what we need to know is the same spectrum as a function of frequency. The process of Fourier transformation enables us to make that connection. As an example, consider an infinite series of regular square pulses, a brief portion of which is shown in Fig. 1 .4. The amplitude function A (t) is equal to 1 at times between 0 and +Jr /2, etc . , and - 1 between Jr/2 and 3Jr /2 and between -Jr/2 and -3Jr/2 (w = 1 ) . The pulse train repeats itself indefinitely at times 0. It is symmetric about t = O. It averages to zero over one period between -Jr and +Jr so that the constant term Ao must be zero. The train has the property that A(t) = A( -t) so that the only elementary waves in the Fourier expansion must be cosine waves whose argument is an odd number of periods between 0 and 2Jr . Cosine waves with an even number of periods and all sine waves have the opposite symmetry, namely A(t) = (-l)A( -t). The first three terms in the expansion eqn (1 .9) are then:

Al

j+7I: 1 j+Jr -Jr 1 j+Jr -Jr 1

= Jr

A3 = Jr

As =

Jr

-71:

4

cos(t) dt = -, Jr

4 3Jr

cos(3t) dt = --,

(1.10)

4

cos ( 5 t) dt = 5Jr

and the Fourier transform (or more correctly in this case, the Fourier series), which is the sum of (1. 10) and higher order terms, can be written as:

(1.11) In this expansion,

Am

=

0 when m i s even.

10

Synchrotron radiation and electromagnetic waves

Examination of the numbered curves in Fig. 1 . 5 shows how this works in practice. The first order term (curve 1 in the figure), known as the fundamental, is a cosine curve which is in phase with the square wave and represents a first approximation to the shape of the wave train. The addition of further terms (curve 7) improves the match between the square wave and the sum of the spectral components. In particular, the higher order harmonics are needed to generate the sharp corners present in the original wave. The more of these that are present (curves 1 3 and 19), the better is the fit to the original wave. The high frequency components are needed in the spectrum to generate rapid changes in the behaviour of the amplitude. In the case of the square wave, infinitely high frequencies are needed to generate the zero time duration of the rise and fall of the square wave amplitude. Figure 1.6 shows the relative importance of the fundamental and the harmonics up to m = 19. The spectrum contains only odd numbered harmonics. All even numbered harmonics are zero. We will see in Chapter 14 that the synchrotron radiation output from a magnetic undulator has a frequency spectrum with a similar form. Although regular pulse trains do occur in nature (from a laser for example), it often happens that the radiation is experienced as a train of brief pulses. Brief means that the time duration of each pulse is long enough to contain many oscillations of the fundamental frequency. In order to describe such a pulse as a series of elementary

19-L (At to At9) 13-L (At toAt3) 7-L(AttoA7) I-At

-I Phase angle (radians)

Fig. 1.5

.a

�

S -
directions. The contributions to the

ori g i n .

integral come from each of the six faces . Consider the face in the y-: plane

The area

of thi s face is

- lo'x fiy 8 z (the negative

s

i gn

face, E" has i nc reased by an amount E ,

inlegral from this face is two

faces together

+(E, +

inwards).

At the oppos i te

contribution to {he ( iJ Ex j iJx ) li x ) liyli;:: . The tot.al contribution from the

+

( J E , f iJ x ) li x so {he

add up the contrihutions from the o { h c r faces so

is the sum of the contributions from the i ndiv idual faces, namely

( IJ E , f lJ x )8x8v8.:: . Likewise, we can

thaI the ov eral l total comes

(

Fig. 2.5

(x = 0).

so the contrihution to the i ntegral from th is face is

is there heeause t.-x i s pointing

8y8 .:: ,

to

iJ E , ilx

--

+

Flu x of E from

a

iJ E

, + E: ) ,

i/ y

.

-'

volume

(J

iJ .::

-

r

lixo �r li z. .

through its enclosing surface

("f .

24

Electromagnetic radiation is produced by electrons j Y

Ey + (BEjBY)8y

z

Fig. 2.6 Dive rgence of a ve ct or E.

This is equal to div E, multiplied by the volume of the rectangular figure, which means that div E is the increase in the strength of E occurring within unit volume. We can make this volume as small as we please so that div E is the increase in strength of E at a particular point in space and, as stated above, in a region where there is a charge density p: div E = V' E .

=

.l!.­ So

(2. 1 0)

or, where there are n o charges div E

=

O.

(2. 1 1)

The continuity equation Suppose now that instead of an outward flux of E from the rectangular volume of Fig. 2.6, electric charge was flowing out through the enclosed surface at a rate given byaplat. This outflow of charge constitutes an electric current density, J, defined in Section 2.4. Because we know from experiment that electric charge can be neither created nor destroyed, it follows that

.

dlV J +

ap

- =

at

O.

Equation (2. 1 2) is known as the equation of continuity.

(2. 12)

The rotation or curl equation

25

The rotation or curl equation In the previous section we examined the divergence of the electric field. Let us now examine the curl or rotation of the magnetic field. We begin by calculating an integral similar to that for the divergence in eqn (2.lO) but here the integral is calculated not over a surface enclosing a volume but around a contour enclosing a surface (Fig. 2.7). In this figure, a quantity related to curl B at the point P is obtained when the integral of B is calculated round the closed loop of length s which encloses P. This is the integral B . ds. Figure 2.8 shows a simple case in which the surface is the y-z plane and the contour around which the integral is to be calculated is a rectangle in that plane whose sides are 8y and 8z. The integral is obtained by starting from the origin in the bottom left-hand comer of the rectangle and considering each side in tum until one returns to the starting point. The component of the field along the x-axis, Bx makes no contribution to the integral because it is everywhere at right angles to the contour. The contributions are then added as follows; By 8y + (Bz + (B BdBy) 8 y) 8 z - (By + ( B By/Bz) 8 z) 8y - Bz8z . The minus signs come about because the integral round the contour is being evaluated in the clockwise direction as viewed along the positive x-axis. This expression for

Is

Fig. 2.7

De fin it ion o f cur l B .

j y

Fig. 2.8

C alcu lation o f cur l B.

26

Electromagnetic radiation is produced by electrons

the integral simplifies to

r B . ds is

=

( _ ) aB y az

a BZ ay

8y8 z .

(2.13)

The right-hand side of eqn (2. 13) is the x -component of curl B multiplied by the area enclosed by the contour. This means that curl B is the total change in the value of the field as one goes round the contour, normalized to the area enclosed by the contour. In the case of a static magnetic field this quantity depends on the electric current density passing through the enclosed area so that, as stated previously, curl B

=

J1,o J.

(2. 14)

In the case of a static electric field, there are no loops in the lines of force so that the integral in eqn (2.13), when E is substituted for B , is always zero so that curl E = 0 everywhere. But the line integral E . ds is the work done when unit charge is carried around the loop, so it follows that if the loop is split into two sections as in Fig. 2.9, the work done (or the change in potential energy) in moving the unit charge from P to Q round one section of the loop is the same as the energy change in moving from P to Q round the other section (or indeed by any route). This argument shows that for an electrostatic field, the fact that curl E = 0 ensures that the field E can be represented by the gradient of a scalar potential which has a single value at each point in the field. In other words, from eqn (2.8), curl E and, from eqns

=

- V x Vv

(2.10) and (2. 1 1), div E

=

V · VV

=

V2 V

=

0

(2.15)

= .£cO ,

(2. 16)

which is known as Poisson equation, and, in the absence of electric charge,

which is the Laplace equation (Simeon Denis Poisson, 1781-1840; Marquis Pierre Simon Laplace, 1749- 1827). These equations, including the relation between field and potential, make it possible to determine the electric field distribution resulting from any arrangement of static electric charges. In the case of an electric field, the potential is a scalar function which makes it much simpler to use than the field, which is, of course, a vector. Contributions to the potential at a particular point, from charges located at different points, can be added arithmetically. The electric field components can be obtained from the potential by the use of the gradient operator. When we attempt to apply the concept of potential to the magnetic field we imme­ diately run into the difficulty that the curl of the magnetic field vector B is zero only when there is no current density at the point in question [eqn (2.14)]; so only in that

Changing magneticfields produce electricfields

27

situation can a scalar magnetic potential be defined. However, although, in general, the curl is non-zero, the divergence of the magnetic field must be zero everywhere because free magnetic poles (magnetic monopoles) have never been detected. Assum­ ing therefore that there are no free magnetic poles to act as sources or sinks for the magnetic lines of force we can write div B = V . B =

O.

Thus, although B cannot be written as the gradient of a scalar potential, it can be written as the curl of a vector potential A, because div curl A = V . (V x A) = 0

and B = V x A = curl A . Because the magnetic field can only be described b y a vector potential, this concept is not as useful in magnetic field calculations as is the scalar potential in the electrostatic case. However, we shall find that these potentials are very useful when electromagnetic radiation is being considered, so it is to this that we must now tum.

Changing magnetic fields produce electric fields So far we have thought about fields produced by static charges and steady currents. If the world contained these entities alone, everything would be very simple but much less interesting than the real world in which change is the rule rather than the exception. The rules of the game, developed so far, apply only to an ideal, static, steady situation which can exist only under very exceptional circumstances. Even if one of the charges moves to a new position, or one of the electric currents gets turned off, a new situation exists and the electric and magnetic fields assume a different configuration. The magnitude and direction of the fields at all points in the universe can change, at least in principle, and we would like to know how quickly these changes take effect.

Fig. 2.9

Ca lculat ion of J E dsa r oun da l oop . .

28

Electromagnetic radiation is produced by electrons

Suppose that a static electric charge is moved over a period of time to a different place where it is again held stationary. During that time of movement, the charge is first accelerated, then moves with constant velocity and finally is decelerated and comes to rest at its new position. Distant observers, with suitable detecting equipment would notice that, during the period of movement of the charge, a magnetic field had been generated, which fell back to zero once the new static situation had been reached. If the observers were themselves some distance apart (though in communication with each other) they would be able to measure the speed with which the new, transient magnetic field travelled from one to the other. If the observers were on other planets, their observations would be complicated by their movement relative to each other and to the moving charge. Indeed, moving observers would not necessarily know if the field had been generated by their own movements or by the movement of the charge or by a combination of both. At the very least, we cannot know if a static magnetic field is a consequence of our own movement at a constant speed past a stationary charge or a movement of the charge past our observation point. All of these ideas took a long time to develop but the starting point was the discov­ ery of magnetic induction by Michael Faraday (1791 - 1 867) at the Royal institution in Albemarle Street, London, in the summer of 1 83 1 . It was already known that a steady current would produce a magnetic field although the mathematical description, as described previously in the chapter, had not been formulated. It followed that if currents could produce fields then fields must produce currents. It was Faraday who first observed and understood that a change in a steady magnetic field would induce an electric current in a nearby electric circuit which was bathed in the lines of force of the magnetic field. The induced current was proportional to the rate of change of the magnetic flux, the direction of the current depended on whether the magnetic flux was increasing or decreasing. The change in flux could be produced by a moving magnet or by changing the current in a nearby electric circuit, or by moving the circuit to a new position. Faraday did not investigate the speed of propagation of the change in the magnetic field and in any case he was not equipped to carry out that kind of measurement. What he did observe was sufficient to start a new industrial revolu­ tion with the development of motors, dynamos and transformers and to propel our understanding of the natural world along a direction which would lead to a complete formulation of the theory of electromagnetism and electromagnetic radiation and to a description of relative motion known as the special theory of relativity. In order to incorporate the observations of Faraday which were confirmed and extended by many other people, all we need to do is to modify egn (2.15) so that curl E is no longer zero but

aB at '

curl E = - ­

(2.17)

which means that a magnetic field which changes with time at some point in space generates loops of electric lines of force. Ifthere is no magnetic field or if the field does not change, curl E is zero as before. Equation (2. 17) is the mathematical statement of the principle of the electrical generator. A rotating magnetic field, whose direction

Maxwell's equations and electromagnetic radiation

29

changes with time, produces an electric field which, in tum, generates an electric current in a conductor.

Changing electric fields produce magnetic fields We can ask the question: is there a reciprocal phenomenon whereby a changing electric field generates a magnetic field loop in addition to the field loop which is always associated with a steady electric current? The answer to this question is 'yes' and that it must be so can be deduced from the equation of continuity [eqn (2. 1 2)] and the statement about magnetic field loops and steady electric currents in eqn (2. 14). The second term in eqn (2. 1 2) is the rate of change of electric charge aplat which must be equal to a (div E) I at divided by co, from eqn (2. 1 6). Because a I at is a scalar operation, a (div E)lat is equal to div(aElat) so that eqn (2.25) may be written as div

(J

+

aE

� at cO

)

= o.

Suppose we now calculate div J from eqn (2. 14). It must be equal to div curl B, divided by /.Lo. But we have already seen that, from the definition of curl, div curl of any vector is always zero so that div J must always be zero. It appears that eqn (2. 14) is inconsistent with the continuity equation (2. 1 2) so something must be wrong. The correctness of eqn (2. 1 2) rests, ultimately, on the conservation of electric charge which is not a dogmatic principle but has been demonstrated by experiment to be correct to a high degree of accuracy. Maxwell understood that the charge conservation principle could be saved by modifying eqn (2. 1 4) (which, so far, had only been proved correct for steady electric currents) to read curl B

= /.LoJ

aE at

+ /.LOCO-.

This tells us that a time-varying electric field generates a contribution to the magnetic field loop which is additional to that generated by the steady current, as was suggested at the start of this section.

Maxwell's equations and electromagnetic radiation Let us now summarize the set of equations which must be satisfied by electric and magnetic fields. These are statements about the divergence and the rotation (or curl) of the fields at any point in space and are listed below as eqns (2. Sa-d). They are known as Maxwell's equations and represent both the culmination of classical physics and the gateway into the modem physics of the 20th century. They make a statement about electricity and magnetism as a universal phenomenon. We include the continuity equation (2. 1 2) because it constrains the movement of charge in space and time. We include the Lorentz equation (2.6) in the list because through it we can

I

30

Electromagnetic radiation is produced by electrons

relate electricity and magnetism to the mechanical phenomena described by Newton's laws of mechanical motion: Maxwell's equations: div E

= !!.... , 80

div B = 0, curl E curl B

(2. 1 8a) (2. 1 8b)

= --, aB

(2. I 8c)

at

aE = /Lo J + /L080 , at

(2. 1 8d)

ap . dIV J + - = 0, at

(2. 1 2)

F = e (E + v x B) .

(2.6)

Even if there are no sources of electric or magnetic fields at a point where measure­ ments are being made, the fields themselves continue to exist, at least in principle, generated by sources distant from the point of observation. If these distant sources are moving, the fields themselves change in accordance with eqns (2. 1 9). These are identical to eqns (2. 1 8) but with p and J both zero: Maxwell's equations when p = 0 and J = 0: div E = 0, div B = 0,

curl E = curl B

(2. 1 9a) (2. 1 9b) aB -

at

'

aE

= /L080 - . at

(2. 1 9c) (2. 1 9d)

Visual examination of these equations shows that they imply that changes in the electric and magnetic fields propagate through space because a change in the rate of change of B, through eqn (2. 1 9c), produces a change in the curl of E and, by implication a change in the value of E as well. This, through eqn (2. 1 9b), changes curl B and hence a B/at. While this mutual interaction of the fields is taking place, eqns (2. 1 9a) and (2. 1 9b) demand that any change at one point must be accompanied by an equivalent change at adjacent points so that the divergence of both E and B remain zero. In other words, any alteration in E and B, once begun not only continues but also propagates throughout the whole of space. This new phenomenon is predicted by the Maxwell equations and is known as electromagnetic radiation.

Reference

31

To express this precisely we must solve eqns (2. 1 9) in order to obtain expressions for

E and B separately.

Let us apply the curl operation to eqn (2. 1 9c). We write

curl curl

E

=

V

x

Vx E

=

-V

x

aB -

at

so that, from eqn (2. 19d)

We can expand the triple vector product to give

(V · E) V - (V · V)E From eqn (2. 1 9a) div E, which is

=

-/LoBo

a2 E at 2 .

V . E is zero so the solution is (2.20a)

and, by the same argument,

V2 B

=

a2 B /LOBO - . a t2

(2.20b)

Equations (2.20) are examples of the wave equation. We saw in Chapter 1 that an equation of this form describes how a disturbance travels outwards from a source with a phase velocity equal, in this case to

Jl / /LoBO .

In this case the equations describe

a disturbance which takes the form of changes in the field quantities

E and B which

propagate in a vacuum with a velocity given by the reciprocal of the product of the electric and magnetic constants. The value of [or Systeme Intemationale (SI)] to be

/LO

has been chosen, in MKSA units

4;rr x 10-7 Him and BO is a measured quantity

whose best value is x 1 0 1 2 F/m, in the same units, so that y'1//LoBo is equal 8 to 3 x 10 mls. This remarkable result, that the wave velocity of an electromagnetic

8.85

-

disturbance in a vacuum is equal to the measured value of the velocity of light, allowed Maxwell to infer that light and the electromagnetic disturbance are one and the same thing. W hat began as the study of the local effects of electric charges and currents has resulted in a description of a phenomenon which fills the whole of time and space and which surrounds us every moment of our lives.

Reference 1. J. D. Jackson, Classical electrodynamics (2nd edn). John Wiley & Sons, New York ( 1 975).

3 Electromagnetic radiation-observed and imagined

The special place of the velocity of light I n Chapter 2 it was shown that e lectromagnetic fields can bc propagated through space at the speed of light and that the field in this dynamic form is k nown as e l ectromag­ netic radiation. The identification of e l ectromagnetiL: radiation w i th light, combined with the direct observation of radio waves and the measurement of the velocity of these waves hy Hertz (Heinrich Rudol f Hertz, I S57-(4 ) , generated some serious conceptual problems. The speed at which the light was moving would be expected to depend on the rel ative movement of the SOUfL:e and the observer. On the other hand, this movement because c = 1 / jIL()C:() and in this ex pression the value of Po is defined by the system of units and EO is defined by the strength of the static e ledric field. This conclusion is backed by measurements of the speed of light from moving sources,

Maxwe l l 's equations would suggest that the speed of light would be independent of

made with i ncreasing accuracy and improved conceptual precision ( see ref. I ). The resol ution of this problem lcd, over a period of about 40 years, to the acceptance of the non-intuitive proposition that the speed of light is independent of the motion of the source and the observer. It is an absol ute veloL:ity. An important L:onsequence of the speed of l ight being an absol ute quantity is that the speed of light in vacuum is the maximum that any ohjeL:t can reach and is the same for a l l observers, whether they arc moving or stationary. This is another non­ intuitive idea, not something which we expect from our everyday experience, and the question i, often asked why the speed of light shou ld be singled out from a l l other rates o f motion for thi s distinction. The answer t o that question has t o b e that it is an experimental observation whose consequences we must take on board. Under everyday circumstances, we only encounter specds which are very much less than the speed of light so that thc constancy of the speed of light and its importance as the limiting speed is of no practical consequence. In contrast, when we consider the production of e lectromagncliL: radiation by e lectrons, the fact is that no malter how high the cncrgy of the e lectron" their veloc i ty cannot exceed that of the light they are producing. However, there is more to it than this hccause it is e lectrical forces which are used whenever information is transferred from nne ohserver to another. This is most ohvious when we consider that light enables us to sec objects and radio waves (long-wavelength e lectromagnetic radiation ) arc used to transmit information over vast distances. But also, on the microscopic scale, it is the e lectric forces between atoms which are cal led into action when we touch an object and our brains register

Refiltive

II/otioll-classical refatil'itv

33

that there is something there. Our experience tel l s us that there appears to be �ome­ t h i n g fundamental about e lectromagnetism and the appearance of i t s ve loc i ty i n the transformation equations, which we w i l l come to in a moment, i s j u �t one aspect of its u n i q ue p l ace i n our perception of the world around u s .

Relative moti on-classical rel at i v i ty It is often neces sary to compare experiments done i n d i fferent p l aces at d i fferent t i m e s . For e x a m p l e , we shal l find t h a t i t i s usefu l to calcu l ate t h e e m i ss ion of e lectromagnctie radiation as seen by an observer mov i n g along with an e lectron ( so that the e l ectron appears to be stat ionary ) and then to apply a tran sformat ion rule to the re s u l t so that we obta i n the characteri s t i c s of the rad iation from a mov i ng e l ectron as seen by a stationary observer. To make t h i s comparison we need a r u l e w h i c h w i l l take us from one s i t uation to the other. Fi g u re

shows two sets of coordinate axes ( two reference

frame s ) i n wh ich the X-ax i s o f one i s mov i n g with veloc i ty l! para l l c l to the x - ax i s of

3. 1

the other.

In one system, w h i c h we call L . the poi n t P has coordi nates ( x , \' , ,:: ) and in the

frame ( L ' ) which i s seen ( by an observer i n L ) to be mov i n g w i th a steady speed II,

paral l e l to the x - a x i s , the coordi nates of P are ( x ' , y ' , .::' ) , It then appears to be a

s i m p l e matter to write down the rule w h i c h al lows us to compare ohservatiom in the two frames: x'

=

(iO +

X

-

v' = ho + Y ,

lit ,

n. l )

::' = ('0 + ':: .

I n t h i s expres s i on ,

( ilo , ho ,

('o j are the coord i nates of the ori g i n of L ' in the frame L .

T h i s expression can be s i m p l i f i ed i f we su ppose that the ori g i ns of L and L ' coincide at the t i m e

t

=

O. At that t i m e the ohservers in the two frames synchronize the ir

• ( (/r).I)II' ! ' I ) )

Fi�. 3. 1

p . IJ

Reference frames.

EfectrtlllWRIlt'lic radiatioll-o/lserved ulld imaRilled

34

c locks to the same start i n g t i m e so that the coordinate tra n s formation rules from one frame to the other beCllme

x'

=

x

-

[It ,

-" = y , ,

t = ,

(3.2)

I.

These transforma t i on rules arc often cal led the G a l i lean transformat ions, after the I ta l i an scientist G a l i l e o G a l i l e i

in recogn i t i on o f h i s p l ac i ng: the idl:a of

( 1 564- 1 64 2 )

motion on a fi rm scientific foot i n g . We f i n d t he m very easy to write dow n , w i t h the benefit of 40( ) years o f h i ndsi ght, because we know i n t u i t i vely ( or t h i n k we do) that i n a time f. the ori g i n of the coord i n ates moves through a d i stance p I and we must

correct the x -coord i nate by this amount e i ther by subtraction l as i n egn

or by

addition depending on whether we are going from I:' to I: or from I: to I: ' . O n the

( 3 .2 ) 1

other hand. we know (or we t h i n k we know ) that we do not need to correct the t i m e

coord i n atl: because t h a t rl:mains thl: s a m e for everybody . The rules also tell us, by tak i n g t h e lirst derivative of t h e equation rel a t i n g x ' an d x

w i th respect to t i m e , d.\' -

dt '

dx ==

-

dt

�

(3.3)

1,1 .

In other words, the speed of an object along the x - a x i s in I: ' is j u st i ts speed in L:

m i n u s v . It seems from a l l t h i s that physic i sts have a roundabout way of stat i n g the

obv i o u s . If someone drops a bottle from the w i ndow of a train mov i ng at

1 00 km/h,

the bottle, as seen by a person i n the tra i n , has zero veloc i ty i n the d i recti on the train is movi ng but from the poi nt of view of a person standi ng: on the p l atform that same bottle moves at

100 km/h w i th , possibl y. lethal consequences.

B y the same token, the transformation e q u ation for the acceleration of the obj ect

can be obtained by taking the first derivative of egn eqns

( 3 .2 ),

(3.3)

( o r the second derivative o f

s o that 2 d \, dt '2

2 d x dt2

.

Agai n , t h i s is what we wou l d e xpect i n t u i tively. The acce leration of an object [ s t h e s a m e i n both coord i n ate systems, provided, of course t h a t t h e y are mov i n g w i th constant speed re lative to each other. w h i c h i s the cond i tion t h at the second term ( i n

v ) i n eqns

(3. 1 )

and

(3.2)

van i shes when w e take the second derivative. T h i s brief

d i scussion of c l as s ical motion has consi dered move ment only along the x - a x i s but the equations can easily be extended to the

y-

and ;:-axes with the same re s u l t .

Special Relativity W hat does t h i � have to do w i th electromagnctic rad i ation') Thc assertion that the speed of l i ght has a constant value and does not depend on the speed of the source means

SII('cio/

Relativity

35

t h a t emitting l i ght i s not l ike drop p i n g bottles from trai ns. i\ di fferent s e t o f rulcs arc req u i red to dcscrihc t h i s new s i t u ation. S u ppose that a l ight flash i s released at the

t

orig i n of the coord i n ates i n � at

O . Th i s flash moves through � w i th vcloc i ty

and is detectcd by an observer at p, whose space coord i nates are ( x .

l ater, where

t

=

is g iven, u s i n g the theorem of Pythagoras, by

C1

,

jx -

=

,

+

+

r. ;: )

c

at a t i me f

,

---- ---

r-

;:�

or

(3.4) I n � ' . t h e observer a t P sees t h e source of l i ght moving away w i t h speed

v

b u t t h i s does

not affect the speed of the l i gh t flash wh ich rem a i n s u n modi fied b y the movement of

the source. In consequem;c, the observer at p, whose coord i nates in � ' are ( x ' , detects the l ight fl ash at

it

t i me ,

,1

c-t -

t'

g i ven by

-

( 1'1 + y x

-

-

I'

+ ;: 17)

=

y', ;: ' )

O.

(3.5)

( 3 .4 ) and (3 . 5 ) can b e reconc i led w i th the transformation ( 3 . 2 ) w h i l e sti l l reta i n i n g the u nchanging ( or i nvariant) value of the speed of l i ght. A new set of rules are req u i red which can be derived from eqns ( 3 .4 ) and ( 3 . 5 ) . These rules are known a s the Lorentz transformat ions ( Hendrick Antoon Lorentl:, I g 5 3- 1 92 8 ) and can be wri tten as

There i s n o way that eqns rules o f eqn

,

x'

=

y (x -

v = v,

fiet ) , (3.6)

' et = y (et

- /-Jx l .

w i t h the deti ni t i o n s o f fi and y given by

fi = I n sertion o f eqns

v

( 3.7)

and

( 3 .6 ) and ( 3 . 7 ) i n eqns ( 3 .4 ) and ( 3 . 5 ) , fol l owed h y a h i t o f algehra, ( 3 .4 )

shows that this transformation o f the coordinate system is one w h i c h al lows both and

( 3 . 5 ) t o h e sat i s fi ed a n d therefore t o a l l ow t h c vclocity o f l ight t o b e the same i n

both � and � ' . These equations are a statement o f the Theory o f S pecial Relativity w h i c h was for m u l ated by A l bert E i n stein

( 1 879- 1 95 5 ) in 1 905. It i s cal led the S pecial

Theory because it app l i e s o n l y to comparisons bctwecn mcasurements in reference frames w h ic h are mov i n g at a steady speed relat ive t o each another. The i nversc transformation, from � ' to � , i s the same as eqn of

[I

( 3 . 8 ) hut w i t h the s i g n

reversed. T h i s i s because from t h e p o i n t of v i ew of an ohserver i n � ' , t h c frame

Electromagnetic radiation-observed and imagined

36

:E is moving with the same speed

v

but in the opposite direction along the x-axis. The

transformation equations are written for this case in eqns x y Z

ct When v «

c,

=

=

=

=

(3. 10):

' y (x' + j3ct ) , , y , / z ,

y(et '

(3 .8)

+ j3x ').

the binomial theorem can be used to approximate

y [from eqn (3.7)] by (3.9)

so that, when

v

(3.9) can be (3.7) reduce to eqns (3.2). Even for objects moving

is small enough, even the second order term in eqn

neglected completely and eqns

at quite substantial velocities, the equations of the Galilean transformation are suf­ ficiently accurate for all practical purposes. For example, a rocket, travelling with sufficient speed to leave the earth forever, is only moving at about 9 kmls so that is about 3

x

v /c 10-5 . This means that even an astronaut riding in the space shuttle can

ignore the equations of the Lorentz transformation and use the Galilean transforma­ tion when comparing measurements on the shuttle with a similar measurement at the ground station. In the case of electrons, things are quite different. An accelerated electron can

easily acquire a speed

v

close to the speed of light so that eqns (3.6) and (3.8) cannot

be ignored when its behaviour is being described. It is clear, from the definition of

(3.7) that y gets larger and larger (mathematicians would say that y tends v gets closer and closer to c. When v is exactly equal to c the value of y, defined by eqn (3.9), is indeterminate. For v > c, the term under the square-root

y in eqn

to infinity) as

sign becomes negative and y has no physical meaning so that c, the velocity of light, is the maximum that any object can reach. Whenever electrons (or other sub-atomic particles) are accelerated, this limiting speed is ob served as an experimental fact, consistent with the equations of the Lorentz transformation. Unlike the sound barrier which can be easily shattered, by Concorde for example, or by a speeding bullet, the barrier presented by the velocity of light has never been broken.

Understanding the Lorentz transformation The principal difficulty in understanding the implications of Special Relativity comes from the Lorentz transformation equation relating the time coordinates in two refer­ ence frames. Intuitively, we imagine that we can view the reference frames as though we were detached observers for whom time is an absolute quantity about which we can all agree. What the theory says to us is that we need to be quite specific about the state of motion of the observer relative to the reference frame in which the obser­ vations are taking place. The reason for this need for precision is because not only

Understanding the Lorentz transformation

37

the space coordinate but also the time coordinate depends on the reference frame in which it is being measured. This is stated mathematically in the fourth equation of (3.6) and (3.8). We can illustrate this by considering the phenomenon of time dilation and the corresponding length contraction. The idea that a moving body (such as a ruler or some more sophisticated equipment for measuring distance) can be reduced in length (as viewed by a stationary observer) was originally proposed to explain the null result of the Michelson-Morley experiment (Albert Abraham Michelson, 1 852- 1 93 1 ; Edward Williams Morley, 1 838-1 923). In that experiment the speed of light was compared along two directions at right angles to each other. Light was understood to be a disturbance of the electric and magnetic fields, which was propagating itself through some kind of elastic medium known as the ether (or the luminiferous ether). It appeared to follow from this picture that at any given moment the Earth must be moving through this ether so that a comparison of the velocity of light at the surface of the Earth along the two directions at right angles should be a measurement of the velocity of the Earth through the medium [according to eqns (3.2)]. No difference in the two light velocities could be detected, which implied that the Earth was stationary in the medium, no matter what time of the year the experiment was carried out. This result was scarcely credible. To account for it, Fitzgerald (George Francis Fitzgerald, 1 85 1 - 1 90 1 ) and, independently, Lorentz suggested that motion through the electromagnetic medium could contract the length of any body in the direction of motion by a factor equal to 1 / J 1 - / e) 2 . The effect would be very small but would be sufficient to account for the null result. The Special Theory of Relativity accounts for this contraction in a satisfactory way, as a consequence of the velocity of light having the same value in all reference frames even though these may be moving (at a steady speed) relative to each other. To be specific, let us consider an electron which is moving with a velocity f3e past a series of fixed alternating magnetic poles (this anticipates the theory of a magnetic undulator which we will encounter in more detail in Chapter 14). These poles, as shown in Fig. 3.2 are a distance AO apart in the reference frame 1: . In Fig. 3.2(a), the electron, which is at rest at a point Xo in its own reference frame (1:') 'sees' the first pole passing it at a time t; and the second [Fig. 3.2(b)] at a time t� at the same point xo. An observer in the laboratory tells a different story. That observer notes that the electron is moving (with the same velocity f3e) past a series of stationary poles and that the time interval in the laboratory frame (1:) is tl - t2 . These times are related by the equations for the inverse Lorentz transformation [eqns (3.8)] so that

(v

etl = y (et; + f3xo) , et2 = y (et;; + f3xo), t \ - t2 = y et; - t� ).

(3. 10)

This equation expresses the phenomenon of time dilation. A time interval in the rest frame of the electron (1:') (often called the proper time interval) transforms into a longer time interval in the frame 1: which is moving relative to 1:'. This surprising

38

Electromagnetic radiation-observed and imagined z

(a) z

'

z

( b)

�'

z

�

�

Pc

e.

xO

AD

'

�'

(!J'

�

tl

Pc

x'

Xl Xz � �

X

AD

X l XZ � �

t2'

8

x' tz

X

Fig. 3.2 Lorentz contraction.

result has been observed to be precisely correct in experiments where the lifetime of a sub-atomic particle, the IJ, meson, at rest has been compared with its lifetime when moving at high velocity. 2 If we return to Fig. 3 .2, in terms of distance, the observer in the laboratory sees that the electron is moving with velocity {3c and the observer in the electron rest frame sees the magnetic poles moving with the same speed (but in the opposite direction), so that, from eqn (3. 10), Now A = (3c(tz at rest and AO =

(3C(t2 - tl ) = (3cy(tz - ti).

- t{) i s the distance between the poles as 'observed' b y the electron (3 C(t2 - td is the distance between the poles in the laboratory so that AO (3. 1 1) A=y

Because y is always > 1 , the distance between the poles is shorter (contracted) com­ pared with the distance AO measured in the frame in which the poles are at rest. The same result would be obtained by using the relation between space coordinates in eqns (3.6). In this approach, the coordinates in 1: are related to those in 1:' by

X2 - Xl

=

{3cy(tz - tD,

giving eqn (3. 1 1) as before. We shall make use of these results in Chapter 14, where the production of X-rays from electrons moving through a magnetic undu1ator gives a precise confirmation of the correctness of this formalism. The Doppler effect-in the forward direction The frequency of the radiation emitted by the electron depends on the state of motion of the observer. This is the well-known Doppler effect (Christian Johann Doppler,

The Doppler effect-in the forward direction

39

1 803-53), familiar to all who have noticed that the pitch of the note emitted by an ambulance or a police siren depends on whether the vehicle carrying the siren is moving towards you or away from you. Qualitatively, what is happening is that as the siren moves towards you, the crests of the sound wave take less and less time to reach you as you stand on the pavement listening to the note. In other words, the number of waves per second arriving at your ear is higher than if the source had been at rest. As a consequence, the pitch of the note you hear is higher than that heard by the driver who is moving along with the vehicle carrying the siren. Conversely, if the siren is moving away from you, each successive crest takes a longer time to reach you because the source of the note is moving further away and so the pitch sounds lower to you. The numerical value of the frequency difference between the note you hear and the note the driver hears depends on the speed of the vehicle carrying the siren. The faster the vehicle is moving, the larger is the frequency difference. Figure 3.3 shows how this works in detail, where we consider an electron which is emitting an electromagnetic wave. The electron, at rest at the origin of coordinates in 1; ' , emits the crest of a wave at time t and the next crest at time t� so that t - ti = l /vo, where Vo is the frequency of the radiation emitted as measured by an observer who is also at rest in 1; ' . Consider now the frame 1; in which the electron and 1; ' are moving away from the observer with velocity {3c, as shown in Fig. 3.3. In 1; the emission of the two successive wave crests occurs at times tl and t2 and at positions Xl and X2 such that X2 - Xl = f3 C(t2 - tI). An observer located at the origin of the frame 1; detects these two wave crests at the later times TI and T2 which are related to the times of emission of the crests in 1; by

i

�

and

so that

(a) z

L

z

'

L'

(!/l

(b)

e

z

'

L'

/3c

/3c

(!)ti

Xl(!)t\

x

x'

X X2 " t 82

Fig. 3.3 The Doppler effect.

x'

Electromagnetic radiation-observed and imagined

40

In the non-relativistic case, the time interval t2 - t1 is identical to the time interval t� - t ; so that, if v is the frequency of the wave as measured by the observer in h , T - T1 = l / v, and 2 1 1 - = - ( 1 + ,8)

v

A = 1.0( 1 + ,8).

or

vo

(3. 1 2a)

To first order in ,8

v � vo( 1 - ,8).

(3. 1 2b)

Equations (3 . 12) are the familiar equations for the Doppler frequency shift when the speed of the moving source is much less than the speed of light. In the relativistic case, when ,8 becomes close to 1 , the time intervals t2 - t1 and t� - t are no longer equal but are related by the time dilation equation (3 . 1 0) so that

i

1 Y - = - ( 1 + ,8)

v

or

vo

1. =

AO y(1 + ,8).

(3. 1 3)

In this case, since, by definition [eqn (3.7)], Y = we can express this as

1

�

y(1 + ,8) =

,

1

(3. 1 4 )

y ( 1 - .8)

and we can write, from eqn (3. 1 3),

v = vOy( 1 - .8),

(3 . 1 5)

which is an exact expression, in contrast to eqn (3 . 12). Because ,8 is almost equal to unity for an electron whose speed is almost equal to the speed of light, the quantity ( 1 ,B) is difficult to calculate because it is the difference of two nearly equal quantities. However, from eqn (3. 14) it is easy to see that ( 1 - .8) = 1 /2y 2 to a good approximation, when .8 is almost equal to 1 , so that v = vo/2y. This expression will be useful later on when we shall show, by combining this result with that of the previous section, that the wavelength, A, of the radiation from a magnetic undulator with magnetic wavelength 1.0 is given by A Ao/2y2 in the forward direction. -

=

The Doppler effect-when the source is viewed from an oblique angle In the treatment of the Doppler effect in the previous section it has been assumed that the observer is looking straight at the source. But what happens when the source is

The Doppler effect-when the source is viewedfrom an oblique angle (a) Y y '

(b) Y

y'

fJc

fJc

c

p.

P' r'

41

r x' x

(e) Y

=

Fig. 3.4 Off-axis Doppler effect: (a) origins coincide at to = tb = 0; (b ) t; = r' le first crest reaches pI ; (c) t\ rIe first crest reaches P; (d) pI and P coincide at times t� and t, respectively.

being viewed at an oblique angle? Examine Fig. 3.4(a). An observer at a point pi who is moving along with the source (and for whom, therefore, the source is at rest), views the source at an angle () ' and detects radiation which has a frequency Vi . However, someone viewing the source at an angle () from the point P, which is at rest in the laboratory detects radiation with frequency v. From the point of view of the observer at P, the source is moving past P with velocity f3c times cos () . What is the relationship between v and Vi in this case? The answer can be found if each observer measures the time interval during which the same number of waves arrive at theirrespective observation points. This is the same thing as saying that each observer measures the time taken for the phase of the wave arriving at P or at pi to change by the same amount. Let us be more precise about this. The amplitude of the wave as observed at pi is given by A = Ao cos 2rr(v l t' k ' r') and the phase q/, of this wave is 2rr(v l t' k'r'). Here r' is the distance from the source to the observation point which can be written as

-

r'

= x

'

cos () ' + y ' sin () ' .

-

(3. 16)

It is useful to introduce the velocity of the wave, c, which, in the case of electro­ magnetic radiation, is the same for both observers, so that k' , which is equal to 1/)...1 , becomes V i /c and the phase of the wave is given by

¢'

= 2rrv' (tl - �) .

42

Electromagnetic radiation-observed and imagined

The rate of change of phase with time, which is depldt ' , is equal to 2n Vi , so that measuring the time taken for a known change of phase will tell us the frequency of the wave. In Fig. 3.4(a), the origins of the two coordinate frames coincide at t = t ' = 0 and at this moment the source begins to emit radiation. Neither of the two observers detect any radiation at this moment but some time t = r' / c later the observer at pi notes that the first wave crest has arrived (shown in the figure as a quadrant of a circle spreading out from 0') and starts measuring the change of phase with time [Fig. 3.4(b)]. The observer at P has still not seen anything but at a time t 1 = r/ c, that observer also notes the arrival of the first wave crest and starts the phase change measurement [Fig. 3 .4(c)]. While all this is going on, p i moves closer to P and sooner or later, at a time t = t ' + r'/ c according to pi and t2 = t + r/ c according to P, the two observers coincide [Fig. 3 .4(d)]. At that moment, both observers again measure the phase of the wave. Although the time (t' for pi and t for P) which has elapsed between the two phase measurements is different, both observers measure the same phase change and we can write

;

�

which, from (3. 1 6) can be expressed as v (et - x cos e - y sin e)

= Vi (et ' - x ' cos e ' - y ' sin e ') .

(3. 1 7)

Because points P and pi coincide in space, their coordinates in the two frames of reference are related through the Lorentz transformation. We can now apply this transformation to the right-hand side of eqn (3. 1 7) so that, with some rearrangement, v ' (y (et - f3x) - y (x - f3ct) cos e ' - y sin e ' ) = v ' (yet ( l + f3 cOS () I) - yx (cos e ' - f3) - y sin () l )

= v (et - x cos e - y sin e ) .

(3. 1 8)

Now eqn (3 . 1 8) must be true for all pairs of observers looking at the same source so that the coefficients of x ' , y' and t ' must be separately equal to each other. This gives the required relationships [eqns (3 . 1 9)] between the frequencies and angles of observation as seen by the two observers:

= vO y ( 1 + f3 cos ()o) , v cos e = vOy (cos ()o + f3 ) , v sin e = Vo sin eo, v

(3. 1 9)

where, with an obvious change of notation, Vo and eo have replaced V i and e' for the frequency and emission angle of the photon from the source at rest. We can rearrange

The Doppler effect-when the source is viewed/rom an oblique angle

43

these equations to give a relation [eqn (3.20)] between the angles of observation in the two reference frames: sin e

sin 80

= -----­

or

y ( 1 + f3 cos eo)

tan 8

sin 80

= ----­

y (cos eo + f3)

(3.20)

These equations for the Doppler effect are particularly important because of the rela­ tionship E hv, between frequency and energy. We can now express eqn (3 . 1 9) in terms of energy instead of frequency and obtain the energy of the radiation observed at an angle e when the source is emitting radiation with energy Eo at an angle 80 in its own rest frame. In particular [from eqn (3. 1 5)], the energy of a photon in the forward direction is given by

=

E

=

EOy ( l + f3)

=

Eo

(§ -

+ f3

­

I - f3

when cos e

=

+1

(3.2 1 )

-I.

(3.22)

and, in the backward direction, by E

=

Eo y ( 1 - f3)

=

Eo

� -f3 -

1 + f3

-

when cos 8

=

These results are an extension of our knowledge of the Doppler effect at low velocities. The observed energy of the radiation emitted by a source which is moving towards us at a speed approaching that of light (f3 � 1 ) is increased by a factor of approximately 2 y (compared with what would be observed when the source is at rest) and is reduced by the same factor if the source is moving away from us. A surprising result is that even when the source is viewed from the side, so that cos 8 = 0, the observed energy of the emitted radiation is increased by a factor y . These results are illustrated graphically in Fig. 3.5, in which a photon with energy Eo, frequency vo , emitted at an angle eo from an object at rest is observed as a photon with energy E, frequency v, at an angle 8 when the source is moving towards the observer in the laboratory. A moment's consideration of Fig. 3.5 may cause you to ask the question-where has the additional energy come from? After all, if an observer watching the source sees more energy emitted in the forward direction than it would emit if it were at rest, this extra energy must have come from somewhere. The answer must be that the total energy emitted, in a given time interval, summed over all angles, must be the same whether the source is at rest or whether it is moving. What the motion of the source does is to redistribute the energy while keeping the total amount the same.

Fig. 3.5

Transformation of photon energy.

44

Electromagnetic radiation-observed and imagined

Lorentz transformation of four-vectors The Lorentz transformation tells us how the space and time coordinates change as we go from one frame of reference to another. It also tells us how time intervals and the distances between points change when the transformation is applied. In the Galilean, or non-relativistic, transformation, distances between points did not change from one reference frame to another and a time noted by two observers was always the same regardless of their relative motion. In this new and different way of looking at the world, when the two reference frames are moving relative to each other at a speed approaching the speed of light, distances and time intervals are no longer invariant but are related by the eqns (3. 10) and (3.1 1). However, the equations of the Lorentz transformation are such that the quantity c2 t 2 - (x 2 y 2 + z 2 ) is unchanged when the transformation is applied, as can be seen from applying eqns (3.4) and (3.5) to this expression and using the definition which relates y and f3 . The term in brackets is just the distance from the point (x, y, z), obtained by applying the Pythagoras theorem to the distances to the origin of the coordinates along the three coordinate axes, x, 2 y , and z. Apart from the minus sign, the expression c2 t 2 - (x 2 + y + z2 ) i s what we would obtain if, in an analogous way, we applied the Pythagoras theorem in a four-dimensional space in which the fourth dimension is the distance obtained by multiplying the time coordinate by the velocity of light. We can imagine a vector in this four-dimensional space, whose components are ct and x, y and z. We call this vector a four-vector and write it as

+

s =

(Xo, x).

There are many such four-vectors in relativity theory (just as there are many ordinary, or three-vector quantities). Each four-vector is a combination of a scalar time com­ ponent (in this case, xo = ct), and a vector (a three-vector), space component which, in this case, is x = (x, y, z) or (Xl , x2 , X3 ). This four-vector, which we can call the four-distance, has four components each transforming according to the Lorentz . transformation. The length I I , of the four-vector is given by the scalar product In particular, for each component,

S,

S.S

S

xo = y (xb f3x i ) , X l = Y (x i - f3xb) , X2 = X2 ' X3 = x3 ' -

I

And, if we form the product then

S.S

(3.23)

I

by analogy with the scalar product in the Appendix,

(3.24) Because the length stays constant from one frame to another even though the com­ ponents of the four-vector change, the Lorentz transformation can be imagined as

45

Transformation of velocities

a rotation, which property changes theof alldirection of the that four-vector but undergo preservestheitsLorentz length. This is a general four-vectors, when they transformation, length remainson equations unchanged(3.23). but their four components change in accordance withtheir the transformati Transformation of velocities

Imagine now andxjdt object inwhich moves a distance inthisa timeframe,dt. This objectcanwillmove have aunder velocity some reference frame. In the object the infleration. uenceWeofwhatever forces arefirst. present so inexists addition to velocity it canframe also have accel deal with velocity If there a second reference relThiatis vsituation e to the fiisrst,shown how doin moving, as usual, at a uniformfromvelocity v alongto thethe x-axis components of transform one frame other? Fig.In3.6.general the motion of the object is described differently in the two reference frames, ofcourse these transformati two descriptions of position, velocity, and acceleration areframe,relatedthough through the Lorentz o n. There is however, one special reference moving with the object, in which, by defi n ition, the object is at rest. Inbythisan we will call the time coordinate T and the ti m e interval dT, measured frame, observer in thetherestsquare frameofofthetheinvariant object, thefour-distance, proper time and the2 , isproper timeas usual, interval.by Ineqnthis(3.24)frame d S given, 1 I so that dx

u =

u

R,

(3.25)

pointseqnalong(3.25),the is IatdSrestI 2 willso thatbe equal In thisaxis)frameandthetheobject Idx l to0c2(thedT2four-vector time l e ngth . Since, through for all ofreference frames, so is c2dT 2 . Because, by definition and by IdS 12 is the thesamevelocity experiment, light, c, is itself a Lorentz invariant, the scalar quantity dT, thequantiproper timeweincantervalmakemeasured in theto obtain object'thes restrateframe, is alsoofatheLorentz invariaS,nt t y and use of this of change four-vector dSjdT, which we can call the four-velocity. How do we do this? =

y

y'

v

�' R

x' x

Fig. 3.6

Addition of velocities.

46 Electromagnetic radiation-observed and imagined In(cdr,the rest frame of the object we can write the components of the four-distance dS asvector 0) and in the frame in which the object is moving the corresponding four­ is (cdt, dx). The individual components ofeqnany(3.four-vector transform accordingwe tohavethethatLorentz transformati o n so that, applying 2 3) to the time component, cd r, or dt / d r cd t which is just the equation for time dilation obtained earlier with 1/JO - luI 2 /c2 ). Now if we write S (ct, x) then, taking the first derivative, dS ( dt dX ) ( dt dx dt ) . dr dr dr dr ' dt dr Thus, the four-velocity U is given by (3.26) U ( c u ), where we have used the definition of given above. The 'length' ofthe four-velocity, U · U, or I U I 2 is calculated in the same way as the length of the four-distance I S l 2 so that = Yu

=

Yu = =

=

C

Yu ,

,

C

=

=

Yu

Yu

,

Yu

Yu .

is whatof light, we would expect because has thewithdimen­ from theof veldefiocity nitionandof theThis sions velocity is the only i n vari a nt quantity these dimensions. Nowvelocity reference frame (Fig. 3.6) which is moving we canv and findthethetransformati velocity u' oinn afactor with is y 1/ JI (v / c) 2 by applying the Lorentz equationstransformati are on [eqns (3.23)] to the components of the four-velocity U. The U

c,

-

=

for the components of U' parallel to the x-axis, and forequations the components of Ugi'vperpendi culraedr toexpressions the x-axis.forDivision of thecomponent, last three by the fi r st one es the requi each velocity namely, Ux + v U ,x (3.27a) + Ux v / c 2 ' uy u, (3.27b) y(1 + ux v/c2 ) ' Uz U, (3.27c) y(I + ux v/c2) When v c, y 1 and the term (u x v/c2 ) in the denominator of eqns (3. 27) is small as well so that, in this case, u� Ux + v, which is the simple formula for the addition -

Y z

«

:::::;

=

--,,­

-

I

- ---���� -

=

.

47 ofin velocities. However, in thetorelattheivistic case, evenUxif the0,object in question is moving a direction perpendicular x-axis so that the denomi n ator in (3.27a) isfactor equaly towhich 1 and u� the perpendicular components of Ux are still reduced by a is an(3.unexpected result. We can use eqns 2 7) to obtai n thethemagnitude andvelocity the direction ofobtained the velocity inlu' the1 2 moving frame. For example, size of the l u 'l is from uJ + uJ + uJ. If we remember that lul 2 u; + u� + u;, then (3.28) Consider the particasulaobserved r case of abyphoton whichwhois istravelling with velocity c in some arbi t rary direction someone moving along with the electron which has emitted the photon. That electron is travelling with velocity in the labo­ v the velocity ratory. What will an experimenter in the l a boratory expect to measure for ofso,thein thephoton? The electroncase,velocity mayinitself be close tocould the speed of lightwithanda non-relativistic the photon the laboratory be moving velocity almost equal tois what twicewethewould speedexpect of lightintuitively if the photon werealready emittedknowdirectly forward. Such a result but we from thewhether fundamental rule of rel a tivity that the vel o ci t y of l i g ht is the same for everybody areamoving alongforwithlu'1the2 electron or sitWetingcanin theobtain laboratory. Whattodoes in this case? the answer this thequestion theorythey ofrel ti v i t y predict by putting lul 2 c2 in eqn (3. 2 8). Because we do not know in what direction thethe component photon is going when it isvelocity emittedalong by thetheelectron, wether,dosonotweknow theU invalue(3.2of8) ofthe photon x-axis ei l e ave x just as it is. After a bi t of elementary al g ebra, and putting ( l /y 2 ) 1 (v/c) 2 , we findmatter that Uxinandwhat drop out completely and we arewhen left itwithemerges lu ' 1 2 c2 . In other words, nofrom direction the photon is going fromwhenthe electron (or anything else) and no matter what the speed of the electron it emits the photon, speed ofthelight.photon is always observed to travel at the same velocity c, which is the Transformation of accelerations

=

=

v,

=

=

=

v

=

=

-

Transformation of accelerations

What about the accel eratiwith on ofnothischange object?to itsNewton' s laws of amotion tellapplied us thattotheit. object continues for ever velocity unless force is a force is frame appliedintowhich the object, itsrest,velocity changes.willIf thesee object isngbeing viewed in reference it is at the observer it starti to move in the aIfdirection ofobject the applis viewed ied force.fromWe can askstandpoint the question-what wilframetheofacceleration when the the of some other reference?the We can answer this question quite easily, because, now that we have introduced idea of the four-velocity, we can say that the four-acceleration is the change of the be

48

Electromagnetic radiation-observed and imagined A =

A =

2 S /d.2 . More explicitly, four-velocity in unit time, so that dU /d. or d we can write dU � (Yuc, Yu U) [c dyu , d(Yu u) ] . d. d. d. d. As before, d is theabove invariant proper time intervalterm and asdt/thed deriYuvati. Ifveweof ainsert this into the expression and expand the second product, we find that dU [ dyu d(YuU) ] dr - yuc d! ' Yu � dyu ' Yu (Yu "dtdU + u d! dYu ) ] . [yuc d! Becauseoccurs the velocity ofthecomponents object underofinvestigation is changing,is notthezero quantity dYu /dt,be which in all four the four-acceleration and must calculated using the definition of yu : dyu d [ I 1 ] . d! dt ( 1 U . u/c2) /2 rulesofforthethefactors differenti tion ofproduct a functiondoeswithin a function and remembering thatsingthetheorder in aascalar not matter, =

=

T =

T

_

=

=

_

U

so that

dyu yJ du dt c2 dt Now we haveToallmakethethisingredients to put wetogether the theexpression for theforfour­the accel e ration. appear simpler substitute three-vector three-acceleration du /dt: [ y;4 U . (y; + :� u(u . ) ] . (3.29) You acanthree-vector see that, asandbefore, thefouracceleration four-vector combines a scalar quantity and these components transform according to the Lorentz transformation. We can use the same arguments as we used for the components of the velocity, to determine how the components of the acceleration three-vector change - = -u · -

a

A =

a,

a

a)

49 when webutswitch from find one reference frame to another. Equation (3.a2r 9)casesis rather com­are plicated we shall that there are really only two parti c ul which us, parallel toin whi andch perpendicular to consideration Let us lookisatmomentarily these now. imImagine portant fora reference frame the object under (or instantaneously) attherest.x-axis. In theIn laboratory, thisframe,frametheisobject seen hasto beanmoving with velocity parallel to its own rest acceleration parallel to the=sameandx-axis andso that therefore, in the sameofdirection as In the object' s rest frame, the components the four-acceleration can be = I written from eqn (3.38) as Transformation of accelerations u

a

a

v

Yu

u.

a

°

u

v.

A rest =

(0, a , 0, 0). Yu t = y, f3 a'

u

In thea component frame in which thealongobjectthe isx-axis, moving,so that . and f3a�, f3c.we can If wewrinotete forthatthef3 has only components of the four-acceleration in this frame: A move =

(

Y

=

=

4 f3 ax' , y 2ax, + y 4 f3f3ax' ' y 2 ay, y 2az' ) ' '

where primesofindicate therelacomponents of viewed in the laboratory. By definition the components are ted by the Lorentz transformation and fiIfnwed thatapply the transformation [eqn (3.23)] to the third and fourth components, we y 2az, = , y 2ay' shows thatofthemotion, transformati towhich the direction so thatoa'n doesa�,notandgenerate any acceleration perpendicular a,

A

IA rest l = IA move l .

°

= °, =

y 2 a ' + y4f3 2a , = ya.

y 4f3a ' = yf3a,

Thus, in this case, when the acceleration is parallel to the direction of motion, a, =

a

' 3 y

Inthethecomponents second case,of thethe acceleration is perpendicular to theofdirection four-acceleration in the rest frame the objectof motion are so that (0, 0, ay , az) and of so that are as before. Now apply the Lorentz transformation to thethefircomponents st two components 4 f3 ax' 0, y 2ax, + y4 f32 ax, = . It is clear immediately thatalonga� the0,direction so that theoftransformation does not generate any acceleration component motion. If we complete the exercise by writing down the transformation for the remaining two components we find A rest =

A move

Y

°

=

=

Y

2az, = Y az ,.

50 Electromagnetic radiation-observed and imagined so, when the2 acceleration is perpendicular to the direction of motion, a', which is just Ja� 2 a� a� 2 , and similarly for the unprimed quantities, is given by +

+

a a =y I

The properties of eltheectron) radiation emitted from anonaccelerating, electrically charged object (such as an depend very strongly the direction of the accel eration relative to the direction of motion of the object. Energy and momentum

In classical,y important non-relativistic mechanics the twoandquantities energy and momentum are extremel because the total energy the total momentum of any closed systemIfremain unchanged so longto theas system the system isthenotstrength acted ofuponthatbyforcean external force. such a force is applied then is equal tosamethe asratetheofdichange of momentum and the direction of the momentum change is theas r ecti o n of the force. For a si n gl e particle we defi n e the momentum theusual,masstheofusetheofobject times the speed withvector whichquantities. it is moving,In relativistic or = m mechanics, where, as bol d i t al i c type indicates wefour-momentum can use the defiasnition of four-velocity [eqn (3.35)] to define the Lorentz invariant v,

p

P

=

moU P,

=

( Yu moc, yu mou).

The three-vector component of yumOu is equal to the three-momentum of the particle mu, where the observed mass of the particle depends on the speed at which it is travelling. In general, when is the velocity of the particle, (3.30) = ymov. The has the dimension of energy/velocity so thatscalar we cancomponent equate thisof theto Efour-momentum / c and write v

p

so that m given by (3.42) as before. In non-relativistic mechanics, the energy of motion, orwith kinetic energy of a particle of mass m andc, Emovingmocwi2th+velocity is defined as imu 2 . Since, from eqn (3. 9 ), when imou 2 , the relativistic definition of energy reduces to the classical kinetic energy with an additional term u

«

�

u,

51

The four-force

equal to moc2 which acts as an energy baseline. In classical problems, the term moc2 drops out of the equation of energy conservation because the total mass of any closed system is assumed to remain constant. However, it is clear that even when u = the relativistic definition of energy demands that any object has a rest energy, Eo, equal to moc2 . When an object is moving, its total energy E is the sum of its rest energy and its kinetic energy T, so that E = Eo + T. Because mo is a Lorentz invariant so also is Eo and the length ofthe four-momentum vector, which is defined, as usual, by the quantity I P I2 , is equal to E5/c2, so that

0,

(3.31) It will be useful to notice, from eqn (3.31), that the Lorentz transformation factor and the particle velocity are given, respectively by

y

=

E

�

and

--

moc2

c

=

f3

=

pc

E.

In the particular case of a photon, which has zero rest mass, and E = pc.

f3

=

1,

Y is undefined

The four-force

We can continue further with the analogy with classical mechanics and define the relativistic four-force, often called the Minkowski four-Force (FM), after Hermann Minkowski In Newtonian mechanics force is often defined as mass times acceleration but it is more precise to take the definition as rate of change of momentum with time, which amounts to the same thing, so long as the mass is constant. In the mechanics of Special Relativity, we can define FM as dP / dr, so that FM is also a four-vector whose length does not change during a Lorentz transformation. We may write

(1864-1909).

FM =

as before, dt = y d r so that ,

FM =

(� dE dP ) . c dr ' dr

'

( � Y dE , Y dP ) . c dt

dt

(3.32)

However, if F is the three-vector force acting on the object, F=

dp dt

and, because, the force, applied over a distance ds, changes the energy of motion by an amount dT given by dT = F · ds,

52

Electromagnetic radiation-observed and imagined

so that dE dt

-

it follows that

= -t (Eo + -t d d

T) =

dT d

=

F · -ddst F =

.

u,

FM = (�F . u, F) .

y

(3.33)

What is the relationship between eqn (3.33) for the four-force and (3.29) for the four-acceleration? We would expect that, for the analogy to be complete, the four­ force would be equal to the rest mass times the four-acceleration, i.e.

That this is the case can be seen by calculating the four-acceleration from eqn (3.32): dE dt

- =

3 a, y (ya y3 a)) , a, (y2a :: ) )

-t (ym c2) d d

o

=

moy u ·

(3.34)

where we have used the expression for d /dt calculated earlier in this chapter. Similarly, dp dt

-

= - (ymou) d dt

so that, by insertion into (3.32):

=

FM = mo ( :4 u .

mo

+

+

2

c

u (u .

u(u . a)

.

Comparison with eqn (3.29) shows instantly that this result is exactly what we would expect to find.

References 1 . J. D. Jackson, Classical electrodynamics (2nd edn), Chapter 1 1 . John Wiley & Sons, New York, USA ( 1 975). 2. R. G. V. Rosser, Introductory relativity, Chapter 3, p. 70. Butterworths, London ( 1 967); D. H. Perkins, Introduction to high energy physics, Chapter 5, p. 1 92. Addison-Wesley, Reading, MA, USA ( 1 972).

4 Radiation from moving electrons Electromagnetic waves in free space-no electric charges or currents

In Chapter 2 we showed that Maxwell's equations for the electromagnetic field could be rearranged as two equations which described the progress of an electric and a magnetic field throughout the whole of space and time at a speed which experiment showed was equal to the speed of light. We call these two waves electromagnetic radiation. Let us consider their properties. In free space, where there are no electric charges or currents, we can put p and J equal to zero and Maxwell's equations take a very simple form:

V · E = O, V · B = O, aB V x E = --, at a B x B = at v

(4.1 )

/LOcO - ,

and the wave equations are

(4.2)

IIc2 .

where /LOcO has been replaced by These two wave equations can be written out in full as

(4.3) and

a2 B2 ax

-

+

a2 B2 + a2 B2 1 a2 B ay az c2 at2 -

-

-

-

- -

-

0

.

(4.4)

There are many possible solutions to these equations which give the values of E and at a particular point in space and time. The actual solution in any particular problem depends on the conditions which started the fields and in the first place. The most

B

E B

54

Radiation/rom moving electrons

general solutions have the form E(x, t) = / (k · x - wt) + g(k x + wt) and a similar equation for B, in which / and g are any function of x and t, the vector x stands for the space coordinates y , z), and k and w are constants such that w / k = c, the velocity of the disturbance through space. The function / represents a disturbance which has some value / at the origin of coordinates and for all points at which k . x = ct so that the wave travels outwards, away from the origin along the positive direction of the coordinate axes so that it gets further from its starting point at later times. The function g represents a disturbance which moves closer to the origin at later times. Because we can use Fourier's theorem to decompose any reasonable function into a sum of sine and cosine functions, we can make use of a solution which is easy to handle mathematically such as ·

(0)

(x ,

E(x, t) and

B(x, t)

Eo exp(ik x - iwt)

(4.5)

Bo exp(ik x - iwt) .

(4.6)

=

=

.

.

In eqns (4.5) and (4.6), Eo and Bo are constant vectors, independent of x and t, which describe the amplitude of the electric and magnetic waves. When these vectors are multiplied by the scalar exponential function, they give the magnitudes and directions of the vectors E and B at the point in space and time whose coordinates are x and t. As was explained in Chapter 1, the physical meaning of this solution is obtained by taking the real part of the expression for E or B. Equations (4.5) and (4.6) are solutions of (4.3) and (4.4), i.e. the three components of E and the three components of B, respectively, satisfy the wave equations provided that k . k = k2 = w2 /c2 so that k = 277:/A is the wavenumber and w = 277: V is the frequency of the periodic disturbance of the electric field. The direction of the vector k is the direction of propagation of the wave and the minus sign in the phase factor (k · x - wt) indicates that as time increases (gets later), the wave travels towards larger values of the space coordinates. The solutions of eqns (4.3) and (4.4) for the electric and magnetic field vectors must satisfy the Maxwell equations (4. 1), so that, for waves travelling in a region free of charges and currents, both div E and div B must be equal to zero everywhere . Let us calculate div E from eqn (4.3). We use the rule for the differentiation of a product, generalized to the vector calculus: div(¢A)

=

V . (¢A)

=

A . V¢ + ¢V . A ,

(4.7)

where ¢ is any scalar function and A is any vector field. In the case of the propagation of the electric field: V. which, since V .

Eo

=

E(x, t) = V . (exp(ik . x iwt)Eo) -

0,

reduces to ik ·

E(x, t)

=

0

=

0,

Electromagnetic waves produced by currents and charges y B

55

Phase angle (radians) Fig. 4.1

Electromagnetic wave.

so that E must be perpendicular to k. In other words, the component of E along the direction of wave propagation defined by k must be zero. similar argument shows that the magnetic field vector B must be perpendicular to k as well. The equations for the curl of E and B place further restrictions on these vectors. There is a rule similar to (4.7) for the curl of the product of a scalar function ¢ with a vector function A:

A

curl(¢A)

=Vx

(¢A)

= V¢ x A + ¢V x

A.

(4.8)

The application of this rule gives Vx which, from (4. 1), is equal to

E = ik x E,

-aBjat = iwB, so that wB = k x E,

(4.9)

which means that the magnetic field at any point along the wave is always pointing at right angles to both the electric field and the direction of propagation. Since wk = 1 /c , eqn (4.9) implies that B = E jc. The plane containing the vectors k and E is called the plane of polarization. This is illustrated in Fig. 4. 1 . The E and k vectors define the plane of polarization, which i s the x-z plane i n this case. The B vector is at right angles to the x-z plane, parallel to the y-axis. Equation (4.9) also implies that E and B have the same phase.

Electromagnetic waves produced by currents and charges

In order to proceed further, we need to consider the sources of the electromagnetic waves so that we must return to the full set of Maxwell's equations including charges

Radiation from moving electrons

56

and currents which we rewrite as

V . E = £" , £0

V · B = O,

aB --, at B = (J + 80 ��) .

(4. 10)

Vx E= Vx

/LO

Consider first what happens if charge is flowing at a uniform rate, so that no radiation is generated. From the first equation, this rate of flow of charge produces a rate of change of the divergence of the electric field given by

ap

at = 80

a (V · E). at

We can obtain the rate of change of the divergence of the electric field from the fourth equation as well:

(

V . V x B = /LO V . J + £0 +

a�) . a

The left-hand side of this equation is always zero because it is a scalar triple product containing two identical vectors. Since the vector operation V is time independent, we can combine these two equations to give

opat + V . J = O.

(4. 1 1 )

Equation (4. 1 1 ) tells u s that the rate of change of the charge density at any point in space is equal to the divergence of the electric current density at that point. Suppose we imagine a volume surrounding the point in question. We can integrate over this volume to obtain the rate at which the total charge is changing within the volume, so that

T

:t J pdT + J V . J dT = 0

T:

but, by the divergence theorem, the second integral can be written as the integral of the current density over the surface S which encloses the volume

:t J pdT J J . dS = O. +

(4. 12)

Equation (4. 1 2) is another statement of the continuity equation and expresses, in a formal way, the experimental observation that electric charge is conserved. It tells us that if the charge inside the volume is increasing, the rate of increase is equal to the inflow of electric current through the surface enclosing the volume.

Electromagnetic waves transfer energy-the Poynting vector

57

The process which has just been described implies an energy equation. Suppose the charge is being brought into the volume by the application of an electric field. This field does mechanical work to bring the charge from outside the volume, through the surface to the inside of the volume. Some of this energy is used to generate a magnetic field produced by the moving current. If the charge is being accelerated (as is inevitable when the process starts or finishes), some electromagnetic radiation is generated which could transfer energy (via the changing electric and magnetic fields) to some distant charges. We believe, as a result of experiment, that energy, as well as charge, is conserved during this process so there should exist a continuity equation for energy flow separate from that for charge. Suppose that U denotes the energy density located at some point in the volume T . Left to themselves, the charges adjust their position under the influence of the electric and magnetic fields in the volume so that the work done by the fields on the charges (conversion of field energy to energy stored in charge movement) is equal to the reduction in the energy density, U, stored in the fields, plus the field energy radiated through the surface. Let us express this statement as a continuity equation. We remember that the force acting on a charge q, produced by an electric field E, is equal to q E and the energy which must be supplied by the field to move the charge through a small distance dx will be q E . dx. If the field acts over a small time dt then the rate at which energy is being supplied by the field is q E . dx/dt which is q E . v. If there are N charges in unit volume, then this expression is equal to E . J because J Nq v . If U is the energy density residing in the field and S is the rate of outward flow of energy across unit surface area, then we can express the energy balance by

=

: (/ U . dT ) = / S · da + f E · J dT ,

- t

where the minus sign on the right-hand side indicates that this term represents the rate of loss of field energy. Application of the divergence theorem to the first term on the right-hand side, to transform it from an integral over a surface to an integral throughout a volume, enables us to write au - -

at

= \7

·

S

+

E

·

J

(4.13)

.

It is perhaps surprising that this equation contains no magnetic field term correspond­ ing to the electric field term E . J. The force acting on a charge q , moving with velocity v in a magnetic field B is q v x B and, by the same argument given above for the electric field, the rate at which energy is supplied to the moving charge by the magnetic field must be q v x B . v which is zero. Electromagnetic waves transfer energy-the Poynting vector

S

The next stage in the argument is to obtain expressions for U and in terms of E and B. We can do this by turning once again to the Maxwell equations and, in particular,

58

Radiation from moving electrons

to the second pairtiofmeequations which vari relaatetiothneofspatial variati onofofchange B to tofhe Bratewithof change of E with and t h e spatial E to t h e rate time. We rewrite these eqns (4.10) below:

V x B - -c12 -aEat = tto J, V x E+ -aBat = 0. Ifequation we takefortheEscalar product of the wifirstth ofB, thetakeabove equations with E (totheobtai nandan . J) and the second the di ff erence between two, group similar terms together, we obtain -at = ttoE · J. E · V x B - B·V x E- (E · -aBat + -c12 B · aE) . The first term in this equation is equal to . E x B and the second to V E x B; + we can rearrange the expression further to give Ex B - - ( - 80 E2 + -B 2 ) = V · -- + E · J (4.14) ata 21 2t1to tto toeqnbe(4.compared with eqn (4. 13) to give the required solution for U and S. We can write 14) in the form of an integral overthe surface area a, enclosing a volume r: E x B + f E· Jdr, - -ata f ( -&21 oE2 + -B 2t1to 2) dr = f V · --dr tto V

which,sidewithofthethe application of the divergence theorem to the first term on the right­ hand equation, yields -�at f ( �2 eo E 2 + 2_1tto_ B 2 ) dr = f E ttox B . da + f E· J dr. (4. 15) Equation it clear that ofthethevector quantity (E scalar x B)/tquantity equatedthe to can beunder tointegral the rateon(4.ofthe15)floleftmakes w across unit area surface, and the -hand sidebyoftheeqnsurface. (4. 15) can be regarded as the energy density within the volume enclosed In other words, 1 2 , U = -21 eo E2 + -B 2 t t o E x B = eoc2 E x B. S = -tto

(4. 16)

S1914,is oftenProfessor called ofthePhysics Poyntingatvector after itsUniversi discoverer, John Henry Poynting (1852Birmingham ty, England).

apply(4.9)] this tothatthetheplane electromagneti chapter.are WeLetsawus[eqn electric and magneticc waves fieldsdescribed at any poiearlier nt alonginthethiswave

Electromagnetic waves transfer energy-the Poynting vector

59

=

related by wB k x E. It is convenient to rewrite this expression in the form wB kn x E, where n is a unit vector pointing along the direction of wave propagation. Then, since k = wlc,

=

cB and, because, in tum,

c

=

n x

E

= I I../£0/10, we can write eqns (4. 16) in the form: V = £oE , 2

(4. l7)

The electromagnetic wave is transporting an energy density V with velocity c in the direction n along which the wave is being propagated. Reference to Fig. 4.1 shows that at any particular point, the values of V and S fluctuate in time so that an observer at any particular place (e.g. x 0), observes an electric field strength equal to the average of the real part of E, which is 1 Re( E) 1 and a wave energy proportional to the average of IRe(E) We can calculate this in the usual way, for example,

=

12.

(4. 1 8 ) and

(4. 19) In Fig. 4.2,

Vrms is the energy density in a volume

V of length

L and cross section

A The total energy in the volume is L Vrms. This amount of energy must flow out .

through the end face of the volume in a time L ie so that the rate of energy flow per unit area through the end face is equal to V Vrmsc

--= c rms , LA V

which is equal to Srms by definition, in agreement with eqns (4. 1 8) and (4.19).

v

S_ A __ rm_ s

�

L

Fig. 4.2

Energy density and flux.

60

Radiationfrom moving electrons

Electromagnetic waves transfer momentum

Besides carrying energy, the EM wave must also carry momentum. If we imagine a charge q within the volume r, the changing electric and magnetic fields generate a force F, on the charge given by the Lorentz equation:

= q (E + v x B). pv - = / (pE + J x B) dr.

(4.20)

F

p

If is the charge density in the volume, is the current density and the right-hand side ofeqn (4.20) can be written as an integral over the volume and set equal to dG jdt, the rate of change of momentum of the charges in the volume: dG dt

We now use the Maxwell equations (4. 10)

x B

to eliminate dG dt

P

\1 · E = -,EO \1 - f.1,oEO -aE = f.1,o J p J = / (EO(\1 . E)E + (\1 xf.1,OB)B 80 ( aE B)) and

at

so that

_

at

x

dr

.

We would like to eliminate the time derivatives from the right-hand side of this equation so that we can interpret these terms as the rate of change of momentum quantities. To do this, we note that

a B) = -aE x B + E aB -(E \1 E = -­aB at

x

x -

at

at

and, again from Maxwell equation

x

so that, by substitution,

at

+ � (/ 80(E x B) ) = f (80(\1 · E)E + (\1 f.1,oB)B +80(\1 x E) E)

dG dt

at

dr

x

x

dr.

(4.2 1 )

This equation expresses the conservation of linear momentum carried by the electro­ magnetic field. The first term on the left-hand side is the rate of change of momentum which the charges within the volume V possess by virtue of their motion. The sec­ ond term on the left-hand side is the rate of change of the momentum carried by the

Electromagnetic waves generated by a distant source

61

electromagnetic field. Even if there are no charges within the volume, this quantity need not be zero. The momentum density g is given by g

=

soE x

(4.22)

B.

The expression under the integral on the right-hand side of eqn (4. 2 1 ) can be expressed as the divergence of a quantity called the Maxwell stress tensor. The latter can be written as an array of nine quantities corresponding to the flow of momentum, per unit area, transmitted across the surface enclosing the volume V . The expression [eqn (4.22)] which we have obtained for the momentum density carried by an electromagnetic field is of more than just academic interest. It is obvious, from eqn (4. 1 6), that the definition of g is equivalent to g

S

= 2" c

or

g=

Uc

-

.

(4.23)

This result enables us to make the link between the energy and momentum of a photon, the quantum of electromagnetic radiation. Each photon carries an energy E = h v, so for the quantum theory to be consistent with the classical theory of electromagnetism, the photon must carry a momentum p = hvlc, from eqn (4.23), which is equal to h i),.. Furthermore, we know from the theory of relativity that for an object of mass m, its energy and momentum are related through the equation

E 2 = p 2c2 + m 2 c4 . In the case of the photon, the mass m is zero so that E = pc, which is again consistent

with eqn (4.23), just as it should be. These results also give us a prescription for calculating the number of photons emitted by an accelerating electron (or by any accelerating charge) . Equation (4. 1 6) relates the energy flow carried by an electromagnetic wave across unit area per second to the strength of the electric field at the point in question. It follows that if we can calculate this field strength, then we can obtain the energy flow and the photon flux. The calculation of the electric field at a point distant from an accelerating electric charge is not trivial. It is to that calculation we must now tum. Electromagnetic waves generated by a distant source

The calculation of the electric and magnetic field strengths is easier if we make use of the potential function. We showed in Chapter 2 that in the static case the electric field E could be described by a scalar potential function V with the relationship

E = -VV.

(4.24)

This is possible because, from Maxwell equations, V

x

aB

E = -­ at

(4.2 5)

which is zero in the static case, when all time derivatives are zero. To see this, replace E with -V V from eqn (4.24) in eqn (4.25). We are left with - V x V V , which is

Radiation from moving electrons

62

always zero. In the dynamic situation, when the charges and fields are changing with time as well as with position, we can replace the magnetic field B in eqn (4.25) by the curl of a vector potential function A , B = V' x A

(4.26)

and eqn (4.25) becomes (4.27)

so that grad V is now equal to the term in brackets in eqn (4.27) or aA E = - V' V - - . at

(4.28)

That eqn (4.26) is justified as a prescription for calculating B is clear from the Maxwell equation which states that V' . B = 0, which is always the case when B = V' x A because V' . B = V' . V' x A is always zero. The potential functions which we have chosen satisfy the two homogeneous Maxwell equations from eqns (4. 1 ) (so called because they have zero on the right-hand side and contain only first order derivatives), these are V' . B = 0, aB V' x E - - = 0, at

but what effect do they have on the two inhomogeneous eqns (4.10)? We can write these in the form P

V' · E = - ,

£0

1 aE V' x B - - - = fLo J . c2 at

(4.29)

Let us replace E and B by the potential functions according to the definitions above in eqns (4.26) and (4.27). This gives a p V'2 V + - (V' · A) = - - , at £0

1 V' x (V' x A) + 2" c

(-2 ) a2A at

1 a + 2" - (V' V ) = fLo J · c at

These two equations can now be simplified as follows. We know that V' x V' x A = V'V' . A - V'2 A by the usual rule for a triple vector product so we can insert this in

Electromagnetic waves generated by a distant source

63

the second equation, and, at the same time, rewrite the first of the above equations to obtain

If we now apply to the potential functions the condition that 1 OV V · A + -- = O c2 ot

(4.30)

everywhere [this restriction, eqn (4.30), is called the Lorentz condition], we have two separate inhomogeneous equations which relate the potential functions to their charge and current sources: (4.3 1 a) (4.3 l b) It is very convenient to invoke eqn (4.30) because its use decouples the equations and so simplifies the problem of determining V and A, but why are we allowed to do this? To answer this question we note first of all that in order to specify the vector potential function A we must specify both the curl and the divergence of A . The curl of A is given by eqn (4.26) and the divergence of A by (4.30) so that the use of the Lorentz condition completes the definition of A . However, this definition is not unique because, if Ao and Vo are solutions of eqns (4.3 1 ) then so are Ao + V 1/f and Vo - a 1/f / at, where 1/f is any scalar function. This can be seen by inserting these transformed values of the potential functions into eqns (4.26) and (4.28) so that B = V x A = V x Ao + V x V 1/f = V x Ao because V x V = 0, and aA av a AO 8(VV) E = - V V - - = - V Vo - V - - - +

at

a Ao = - V Vo - -

at

at

at

--

ot

because the gradient operator contains no time dependence. This means that there are a large number of solutions for the vector potential, all of them related by what is called a gauge transformation. We are at liberty to choose a gauge which is convenient for our purpose, in this case, the solution which yields eqns (4.3 1 ).

64

Radiation from moving electrons

The particular choice we make implies a choice of the scalar function 1/1. In this case, when we impose the Lorentz condition on Ao and the function must satisfy the homogeneous wave equation

Vo,

1/1

V21/1 - c21 aat221/1 0 V V . o + V2 1/1, 1 a¢ 1 a¢o 1 a 2 1/1 c2 at = c2 c2 at2 ; 2 1c2 a¢) V · V· + 1 a¢o) (V 2 1/1 - 1 a 1/1 ) o at ( c2 at c2 at2 ( 1/1 (4.3 1) (4.26) (4.28), aAat E = -VV - -. =V (4.3 1) aat22 v - - ­£p ' V2 V - -c21 0 2 1 a V2 c2 at2 = -fJ.,0]. =

because, with the definitions just given, .

A =

A

Tt

-

so, using the Lorentz condition we have A+

-

-

=

A

+

- -

- -

.

The transformed potentials satisfy the Lorentz condition provided that in the above equation the function makes the second term on the right-hand side equal to zero so that the gauge transformation does not depend on the charges and currents. The next stage in the argument is to solve eqns to obtain the scalar and vector potential functions which are related to the fields through eqns and namely, B

Let us examine eqns

x A,

which we rewrite below:

A

A

-

Both of these equations have the same general form. The left-hand side is the familiar wave equation and, when there is no time dependence, they reduce to those for static electricity and magnetism:

V2 V £0 V2 -/-to]. VCr) 41T£OI f Irpe-r'lr') r'. per') =

- £.. ,

A

(4.32)

=

The first of these is the Poisson equation and relates the electrostatic potential function to the source of the electric field. The solution of this static equation is =

--

---

dr,

r

(4.33)

which gives the value of the scalar potential at a point P defined by the vector which is generated by a charge density at the point The integral is carried out over the volume of space, r , occupied by electric charges as shown in Fig.

4.3.

Electromagnetic waves generated by a distant source

65

p

Fig. 4.3

Potential at P.

In free space, where there are neither charges nor currents, eqns (4.31) have the same fonn as the wave equations (4.2) for the electric and magnetic fields. A disturbance in the electromagnetic field, travelling through space with the velocity of light, can be represented by potential functions V and A moving with the same velocity. When charges and currents are present, these act as generators of the potentials which are now functions of time as well as position; so, by analogy with the solution of Poisson's equation (4.32) we must write

V(r, t)

=

I f per' t')

-4 rrso

Ir

"

-r

I d.

(4.34)

where / ' = t - Ir - r'lIe is called the retarded time, or the emission time. In other words, in order to calculate the potential at a point P, distance Ir - r' l from a charge p d. we must insert into the integral the value of the charge at the retarded time t ' . The value of the potential function at P at time t is produced by the charge p d. as it was at the earlier time t ' . The time difference t - t ' is the time taken for the electromagnetic disturbance to travel to P through the distance Ir - r' l. In the static case p (r' , t ') = p (r' , t) so that we get back the solution of Poisson's equation which is eqn (4.33). We also arrive back at eqn (4.33) in the limit when t' -+ t which is what we would expect. All this seems very satisfactory. It appears that eqn (4.34) is what we would expect intuitively. An observer at P sees the distribution of charge, not as it is at the time of observation but as it was at an earlier time, defined by the time taken for the signal to reach the point P. For example, we see the stars not as they are now but as they were many thousands or even millions of years ago, at the moment when the light was emitted. The precise time depends on their distance from us. However, as it sometimes happens, things are not quite what they seem. Why is that so in this case? We know that the solution (4.33) to Poisson's equation generates the inverse square law for the electric field from a spherical charge distribution occupying the volume .; in particular, for a point charge p d. = q at r, the strength of the electric field E at r' is equal to q l(4rrsor 2 ), where r = Ir - r' l. Suppose that q is suddenly moved through a distance small compared with r and returned to its fonner position at rest. The movement of the charge would be observed as a change in the electromagnetic field at P at a time rle later given by eqns (4.32), which, by

66

Rlil/iotioll ./iwll /I101'ill,!;

electrons

analogy w i t h the solut ion to Poi s son ' s equation, we wou l d expect to he observed as a momentary change in Il e l d strength w i t h an ampl i t u de decreas i n g a�

I 1 ,. 2 .

However,

the i ntensity of the e lectromagnetic wave generated hy the ch arge movement wou l d h e expected t o decrease a s t h e i nverse rourth power of t h e d i stance from t h e c h arge, because the intensity ( the energy carried by the wave ) is proportional to

£2 ,

Th i s

contradicts observat ion because w e know from experiment t h at the i ntensity o f a n e lectromagnetic wave from a p o i n t source decreases w i t h d i s tance accord ing to the inverse square law. The i n t u itive argument contradicts s i mp l e observation I T h i s contradiction is resolved later but it is i m portant at t h i s stage to be sure t h at the i n t u i t ive expression I eqn

(4.33) I for the potential generated by a charge d i stri bution

vary i n g w i t h t i me i s the correct one. M u c h e ffort ha� gone i n to a rigorous proof of this res u l t lI s i ng a method of sol u t i on l i rst developed by George Green ( m i l l er and mathemat i c i a n ,

1793- 1 X4 1 ) . [ n out l i ne , the proof proceeds as fo l lows .

We arc req u i red to Il nd the pote ntial function V ( I' , t ) for a known c h arge den s i ty d i stribution p e r , t ) , sat i s fy i n g the eq n

( 4 . 3 1 a ) , We beg i n by u � i n g the Fourier trans­ ( 4 . 3 1 a ) as an i ntegral

formation to express the t i me-dependent components of eqn

It f +-:>0 -

over frequency w so that. for example, V (r,

I) =

p e r, t ) =

-

y ::;t! ::;V _ If

and the i nverse transformation., V ( r , (v )

Ii

If

_

. p e r , w ) e x p ( � I Wt ) d l ,

�

It f +'X

= i) V If _

( r, w )

V ( r , (I) exp ( � i(vt ) dl ,

- ">0

(I- f +� _

V (r,

_�

(4.35 )

I ) ex p ( i (v t ) d l .

fI ( +oe p ( r , t) e x p ( i wt ) d t , 2lf L 'X

=V

in w h i c h V and Ii denote the Fourier transforms of V and p, respectively. When we

i n sert the Fourier transforms for V from eqn

( 4, 3 5 ) i nto eqn (4.3 1 ) , we obta i n , for the

terms on the left-hand s i de of the Ilrst equation, \7 2 V (I', t )

=

/-i� I� f1j' ! 'X,, ) -

i) 2 V ( r , 1 ) . ) = � I i ::;V _ If dt-

\7 2 V ( I' ,

_

�

w ) e x p ( � i iOl) d l .

.

(I)- V ( r. (v ) e xp( � I (j)t ) d t ,

so that, when we i nsert the rourier tran sform Ii we can e l i m i nate the i n tegrals and

write, for any value of (0,

(4.36) where

k =

wlc =

2lf/A .

(/

Electrollwgnf'fic WOFes generated hy

dis(({l1{ slluree

67

If the charge density at r' ( see F i g . 4 . 3 ) i s conta i n ed w i th i n a smal l sphere. rad i u s ro and Ir - r ' l » 1"0 . then outside t h e sphere. p = () a n d eqn (4.36) reduces to

( y 2 + k2 ) V == ().

Equation

(4.37)

(4.37). known as t h e H e l lll h ol Lr equation. i s s i m i lar to t h c Laplace equat i on

but w i th V2 replaced hy V 2

depending o n l y on

r

=

+ k 2 • and its solution must he spheri c a l l y sy mmetrical .

I r ! . In spherical coord i nates. the r dependence of the Laplacian

()perator. acting on any function

"

I

VI .

has the form

iJ

V � V/ = - r 2 ilr

so that eqn

(

, il Vi

r�

�� ilr

)

il 2

I

= - -- ( rVl )

(4.3S)

r iJ ,. 2

( 4 . 3 7 ) reduces to

- -, ( r V ) iJ 2

I

�

r iire.

+

"

k� V �

=

O.

The general solution to this equation is rV

=

A cx p ( i h ) + B ex p ( - ih l .

where A and B are constants whose values depend o n t h e ho undary condit ions and

r

= Ir

- r' / . A s we s h a l l show very soon. the second term wou l d correspond to an

e l ectromagnetic d i sturhance whose arrival at P wou ld precede the onset of the t i me vari ation w h i c h caused i t . v iolating causality. I t is reasonable to assume therefore that B = O. To obtai n the value of A. we exam i ne the first term and see that for ro

and

r

3:

ro. the solution for V must l ead to eqn ( 4. 3 3 ) . so A Il ( r' . (0 ) e x p ( i k l r

V i r . (0 ) = -

- 1" 1 )

4n f: o l r - 1" 1

=

fl/4n f o and

---'>

0

(4.39)

.

( 4 . 3 9 ) i n order t o ohtai n the (4.3 1 ) for the scalar pote n t i al . We suhsti tute

Now w e m u s t take the i nverse Fourier transform of e q n fu nction V which i s the solution of eqn eqn

( 4.39) i nto eqn ( 4 . 3 5 ) and write V (r. t ) =

=

where

/' =

= t -

Itf+0.. ;-I- f +"X. ; f 1 'Cv

! -

'v 2n

- 'Cv

V 2n

��0U

V 2n

-ex;

,

-

, .-

�I-

I I"

4n Fo

-

- r()

1' 1

fl ( r ' . (0) exp ( - i w ( t

4 nE O

II"

p ( r ' , ro ) e x p ( - i (o{ ' ) lr' - 1'1

4n E.()

ex p ( - l wt ) d t

.

- ( r' - r i le ) )

- 1' 1

dt

, dt .

12n . is j u s t the Fourier tra n s form of f! ( r ' . t ' ) . i d e n tical

Ir - r' l le i s the retarded t i m e . as before. But the expression under the

i n tegral sign. m u l t i p l i ed hy J I to eqn

fl ( r ' . w ) ex p ( i k ( r ' ----

( 4.34) so that

V (r. 1 ) = where the i ntegral over the volume

r

I

--

4n Fo

J

p ( r' . I ' ) l r - rl l

dr .

adds u p the contributions t o the scalar potential

function from a l l the charges at each separate p o i n t .

6X

Rw/i(/tioll /imn moving c/cctrollS

A s i m i l ar procedure can be appl ied to each component of the vector potential

function A, which i s a solution of eqn (4.3 1 b ) to show that A ( r , t ) i s givcn hy fl O

A ( r, t ) - 4][ _

f

J ( r' ,

' t )

I r - r' l

dT.

These two expre�sions for V and A are c a l l ed the retarded potentials and are often written in the form

V ir,

t)

A (r, t ) at time t but at the earl ier time t '

= - .�- f � dr, !!..!2 f � Ir - r l

4][ r:o

=

4][

I r - r' l

(4.40)

dr,

where thc square hrackets are used to denote quantities which must be calculated not =

t - I r - r' l /e.

Let us now use these expressions to calcu l ate the potentials at the point P generated

by the distrihution of mov i ng charges in Fig. (4.3 ) . In order to do this, we must obtain

the state of motion of the charge density at the time t ' and i ntegrate this over the vol ume whose charges contribute to the potential functions at P. This must be done

carefu ll y because although the observer is at a fi xed point i n �pace and t i me, the charge distribution fi l ls a region of space r so that each smal l vol ume of charge Ii dr in the distribution makes its contrihution to the potential at P at a di fferent time t ' .

The process o f adding these contributions c a n b e visualized h y al lowing a n i m agi­ nary �phere, centred on P, to contract onto the point P at a speed equal to that of l ight as shown i n Fig. 4.4.

The sphere has the fictitious property that poinh on its s urface can remember a l l the charges traversed b y the s urface and retain that memory u n t i l t h e sphere reaches the observer at P. A t the moment when the radius of the sphere is equal to Ir - r' l ,

those portions of the charge di stri hution i ntersecting the surface of the sphere at time ' t ' reach P at a time t t + I r - r' l /e. The sphere has hecome a detector of the =

retarded charge di stri bution I p i which features in the integral of eqns (4.40). The

process of integrat i on i s the same as al lowing the radius of the sphere to start at

"- ulr-r'l

r'

Fig. 4.4

p

r

Calcul atioll or t he Licnard-Wicchert potent i a l s .

Electroll1ognetic

W{/ \ 'CS

gellcrated hI'

(/

distlillt

SOIl)( '{'

69

infinity and end at Lero where the observer i s s ituated. The e lementary vol u me dy in

the integral i s equal to d(T d l r - 1" 1 and, if the charge distribution i s static. the amount o f charge traversed hy the sphere [ p [ d y will he equal to [ p J d(T d l r - 1' '1. However. if the charge distribu t i on is not static, some of the c harge is lost (or gained) hy the eleme ntary volume during the time d t ' taken hy the surface of the sphere to contract

through a distance d l r - 1" 1 . This time is d l r - r ' l /e. If [ v I is the velocity of the charge in this vol ume at I ' then the amount of charge lost from the vol ume through ' the surface d(T during the time d t i s the amount of charge i n the vo l u me t i mes the rate at which charge i s flowing across the s urface, This l ast q uantity i s the component

of [ v J a long the radius of the sphere and pointing towards the centre of the sphere, If n i s a u n i t vector i n t h i s direction, then

[ 11 [ -

[ I' - 1" [

I I I' - r ' l l

---

and the amount of charge lost through the surface area d(T is

[ p l ! n · v [ d(T d t ' , which i s equal to

[ p l l n · v l d (T

d l r - 1" 1

--­

('

an amount d l r 1" 1 i , equal to the amount present at radius I I' - r ' l reduced hy the ' amount l ost during the time d t , which is it fol lows that t h e amount of charge col lected h y t h e sphere a s i t s rad i us contracts hy

dq

=

=

d l r - 1" 1 , [ p l d(T d l r - 1' 1 - [ p l ! ll ' v l d (T (' [ II . v I [ p l d Y - [ p [ �--- d y .

--­

('

( 4.4 1 )

The i ntegral [ eq n ( 4.40 ) I which we have to determine i s

and, from e q n ( 4.4 1 ).

/'

, i�

[pI __

r' l

dY =

J

dq

___

I lr

r ' I I [ I - [ Il , v [ /c l

(4.42 )

In the case when the chargc distri hution reduces to it point charge at 1" then the denominator of the i ntegral on the right�hand side of eqn ( 4 ,42 ) i� constant and f dL!

Radiation from moving electrons

70

is equal to the value of the charge at r' which is q , so the expressions for the potential functions V Cr, t) and A (r, t) from eqns (4.40) are (remembering that [ j] = q [ v))

V(r, t)

=

A (r, t) =

1

{

--

{

q

r' l] [ 1 - [n . v]/c] 4;reo q [v] fLO 4;r [lr - r'l] [1 - [n . v]/c] [ Ir -

}

,

}.

(4.43a) (4.43b)

Equations (4.43) are expressions for the potential functions of a moving electric charge and are known as the Lienard-Wiechert potentials (A. Lienard, 1 869-?; Emil Wiechert, 1 861-1928). As before, the square brackets denote retarded quantities. The next stage in the argument is to use these expressions to obtain the electric and magnetic fields generated by the moving charge by making use of the relationship between the potential functions and the fields which are described by these functions through eqns (4.25) and (4.27) which are B=V

x

aA E = -V V - ­

A,

at

To determine the electric field E we must obtain the gradient of V and the time derivative of A at the point P, whose coordinates are (r, t). However, because the potentials of eqns (4.43) contain retarded quantities measured at the point (r' , t '), we must first obtain expressions which relate changes to r and t to changes in r' and t', where these coordinates are related by

t, = t -

I r - r' l

---

c

.

(4.44)

We start by obtaining aA/at. The use of the partial derivative implies that we are asking what change would be observed in the vector potential function at P, with a fixed position vector r, during the infinitesimal time interval at. From the definition of t' [eqn (4.44)], we have

at' at

- = I

-

1 a , - - Ir - r I =

c at

a at' 1 - -1 Ir - r, 1 c at' at

but

a - Ir - r, I = [n · v] at' [remember, r is fixed so a r/ a t' = 0 and the rate of change of the length of the vector linking the retarded position of the charge to the point P (Fig. 4.4) must be equal to the component of the velocity of the charge in the direction of the vector r - r'] so that

at' at

= -----

l - [n · v]/c

(4.45)

Electromagnetic waves generated by a distant source and the time derivative

aA/at is given by a A a A at' = 1at' at

aA

at =

Now, from eqn (4.43b)

aA at' =

J1,oq 4rr

a at'

so, it is useful to obtain separately

� [ Ir - r' l ( 1 at'

_

�)] c

[

I r - r' l ( 1 -

(4.46)

.

[n v]/c at' v n . vic ] .

71

)

;

(4.47)

v· ] [ at' c v· l = [ - n . v + v; c =

�

Ir

_

r' l

(r - r')

_

(r - r')

-

(4.48)

We can now make use of eqn (4.48) to evaluate eqn (4.47) and obtain aA/at with the help of eqn (4.46). The result, after some algebra, is

8A

at =

J1,oq

[

4rr

x

[

CV

] 3 n.v c 2 2 . ( n vic) -2 v /c ) v (1 - n · vic + v (n v = av/&t' = [av/at] 1

/ )

(l -

)

+

Ir - r' l

I r - r' l

. VIC)

]

.

(4.49)

In eqn (4.49), and elsewhere, . We must now obtain the gradient of the potential function, V' V or grad V, at the retarded time. If we imagine the detection point P to be moved through an infinitesimal distance dr, in some arbitrary direction, so that this movement can be denoted by the vector dr, then the gradient of any scalar function 1/1 in this direction is given, according to the definition of V'1/I or grad 1/1 , by dr d l rl

. grad 1/1

=

dr d lr l

.

V'1/I

=

d1/l

(4.50)

d l rl ·

In eqn (4.50), dr/d lrl is a unit vector in the direction dr so that d1/l/dlrl is the component of grad 1/1 in this same direction. In our case, the function we must insert for 1/1 in eqn (4.50) is the same one whose time derivative we obtained previously [eqn (4.48)], only now we need the spatial derivative at fixed time. We are required therefore to find ' a a I ._-_r....) ..:... .-'(r r r' l Ir r' l = c c a lr l a l rl

_[

_

(1

_

n_._v ) ]

[ v �a [I R I - _v�-Rl R.

_

=

rl

where the vector r - r' has been replaced by algebra is shown in Fig. 4.5.

_

_

_

]

(4.5 1 )

The physical basis of this piece of

Radiation from moving electrons

72

r

Determination of grad V

Fig. 4.5

.

The determination of the gradient implies the comparison of the potential function at points P and Q separated by dr, but at the same time t. An instrument at P measures the state of the field which started out from the point p' at position r' and time t', which are the retarded coordinates. An instrument at Q is detecting the field which starts out from Q'. Although P and Q have different space coordinates, they have the same time coordinate so that the time difference, dt', between p' and Q' is given by the difference in the path length between pp' and QQ', divided by the velocity of light, so that d I R I = -c dt ' . The minus sign comes about because Q' is earlier in time than P' . During the time dt', the electric charge, which is generating the fields observed at P and Q, moves from Q' to p', and its position vector changes from r' to r' + dr' so that the velocity of the charge is just given by v = dr' /dt ' . We can use these relationships to evaluate the expression in eqn (4.51). Let us look at each component of eqn (4.51) in tum: d l RI

-- = -c

dlrl

d d l rl

(R · V) = �

d

( = (

=

dt' d lr l

dl r l

,

, (r - r )

dr dlrl dr dlrl

-

( r - r')

v

.�+

dr' dt'

)

dt ' d l r l dt '

-v dlrl

)

.

v .

�

v

c

�

+

.

dv

dlrl

+ (r - r' ) R

.v

dt'

� d lr l '

v

dt'

. � dlr l

(4.52)

Examination of eqns (4.52) reveals that every term on the right-hand side, except one, contains the factor dt' /dlrl which relates a small change in the spatial position of P to a change in the retarded time. If we can express dt' /d l r l as a factor multiplying the unit vector dr/d l r l then we can solve eqn (4.5 1). To do this, we consider the quantity dR/d l r l which is equal to d(r - r')/dlr l : dR dlrl

=

dR dt' dt' dlrl

=

dr dl r l

-

dr' dlrl

dr dr' dt' dr dt' = - - - - = - - v- . d lrl dt' d l r l d l rl dlrl

73

Electromagnetic waves generated by a distant source

- r')/l r -ofr'lthewithvectoreachdR/dt' Now fonn the1) andscalarrememberthat product of then . dR/dt' unit vector n the(rcomponent tenn inin eqn (4. 6 is just the direction of R, which is d l R I /dt' = -c, so that dt' dr -n · v dt' = n · n · -dRdt' dl rl dl r l dl rl and 1 dt ' = dr (4.53) n · dlrl c(l -n · vic) d l r l which(4.is5the1) torelobtain ationship we need so that we can bring together the expressions in eqns � . V [I r - r' l (1 - � ) ] dlrl c = [ n -n· drldl. vic)rl ] - [ ( ddrl rl - � (l -ndr/dl. vic)rl ) . � ] [ , v c (1n-.-ndrldl. vic)rl - ] + (r - r ) . -c so that V [Ir - r' l (1 _ n � V ) ] = [ c(l -n · v/c) ] [ (1 _ vc22 ) n _ c (1 - �c ) �c + Ir - r' l �c n] , where for VVr-r' fromhaseqnbeen (4.4repl3): aced by Ir -r'l1 n . A bit more algebra leads to the expression VV(r, 41Teo -q- v { [Ir - r'l] [1 -n . vic] } = 41Teq o I [VI r[I-r r'-l]r'l[1(1--n[n · .v]/cv/c)]]2 I = 41Teq oc [1 - [n1. /c]3 . v/C)v/c + (n , v/c)n ] l . -n I[ C(1 - v2/c2)nIr -C(l Ir - r'l - r'I2 (4.54) Wespatialnote,vector in passing, that eqnti(4.me53)through is alsothea reldefinition ationshipvbetween at the retarded dr'/dthet, sochange that in any = dr' dr' dt' v dr dl rl = dt' dl r l = c (1 -n . vic) n · dl-rl . Towe obtain the expression componentwithof theeqnelectromagnetic now combine eqns (4for49)theandelectric (4.54)field in accordance (4.28) and makefield,use =

-

-

v n·

(I

--'---'-

1

t)

=

-

-

v]

x

- -

-

.

v

Radiation from moving electrons

74

We make of the expression ofproduct the factto simplify that c2 the second term on theuseright-hand side and arriforve,theatvector last, witrithple -v/c) n ] E(r, t) 4Jl'E'O [1 - 1 v/cp ![(I -v2/c2)( I r - r' I 2 . - vic) VIC}] +� [ c I r - r' l I . Acomponent. similar procedure canwebemust folloobtain wed tocurlobtaiorn theV expression for the magnetic field In this case v which is equal to V (a r' / at'). We can obtaiItn isthisconvenient quantity tobyuseexpanding operator terms of its three components. the determithenantcurlnotati on (seein Appendix). =

=

l / (E'O fLO ) .

-- ----__;;_

q

n

n x t en

x

v

Vx v=

a ax

Vx i

at' ax Vx

k

j

Vy

Vz

k

j

at' az

at' ay

Vz

Vy

x

x

i

j

k

Vx

Vy

Vz

a at' a at' a at' at' ax at' ay at' az

a az

a ay

(4.5 5)

=

(Vt' )

x V.

(4.56)

Notice thatmulwetiplier can use(a/ athet') from rules thefor second the evalrowuationandofappldeterminants tordfactor outthoutthe common y it to the thi row wi changi ngwetheobtain value ofVt'thebyresult. Next eqn (4.50) write using the definition of the gradient operator given in dr · Vt, dt' ' d lrl d l rl which, from eqn (4.53) is dr dr , d lr l Vt c ( 1 - vic) dl rl so that to

=

-·

and, from

=

-

Vt ' =

(4.56),

-

-

1 n.

n · -,

----__

n

c ( l - n · v/c)

(l - .

n x v V x v = - ----­ c n vic)

(4.57)

75 Electromagnetic waves generated by a distant source now have to use charge. eqns (4.We 26) can and write (4.43b) for evaluating theWemagneti c fieldthe ingredients generated bywetheneedmoving = V x = J-L4rrOq [V x ( Ir - r'l (1 v- [n . v c) )] = ��q [V x Cr r'l (1 � [n . ) + I r - r'l � : �n . v]jc) 1 The term inin theeqn square has already [eqn (4.54)] and V x first v is given (4.57) sobrackets that working throughbeenthecalculated algebra yields B(r, t) = _ 4rr [1 - n1· vlc]3 x (VIC))) ] ' x 1[(1 -v2I r/c-2)r'1(n2 x ] + �c [ n x {n xI r(vic - r'l I(4.58) the terms we replace n x (vic) by -n x (n - (vic which areif, inequal becausein then xcurlyn =brackets, 0, eqn (4.58) reduces to (4.59) B(r, t) = -c1 [n x E(r, As usualare toin betheseevaluated expressions theretarded squaretimbrackets indicatepreviously. that the quantities within them at the e t' as defined It is obvious thatknown the expressions (4.field55)component, and (4.58) decreases for and as thecontain twosquare com­ ponents. The first, as the near inverse offieldthewhen distance fromThethesecond chargeterm,andthereduces to component, the electrostatidecreases c expression forwiththe v far field linearly = to theas theelectromagnetic waveThiwhose intensity,depends proportionothenal distance toacceleration, x andorcorresponds Edv2/d, tdecreases inverse square. s expression so that, ifthetheelectric acceleration of the charge isarezero,dominated then thereby =rgeV,distances, isthenofarradicomponent. ation. At laThe and magnetic fields nearthat fieldthecomponent canfieldbevector, neglected. Equation (4. 5 9) indicates magnetic generated bygeneral the moving charge, is directed at right angles to the electric vector. This is the casein altheready noted in connection with the plane wave solutions of Maxwell's equations absence of charges andchapter currents.is that accelerating electric charges can produce The conclusion of this and magnetic fieeldsldsarewhicoupled ch can betogether observedandatarelargedetected distances from the point ofelectric generation. These fi as electromagnetic radiation. In the nextofchapter we willradishow observed properties synchrotron ation.how this radiative process leads to the B

A

B

]/

_

V]/C»

J-LM

VIC)

»

t)].

E

O.

E

B

B

5 Synchrotron radiation from dipole magnets Properties of circular motion

The mplest situation inawhipositively ch synchrotron radi ation isisproduced ismove wheninanaelectron (orpathasibypositron, which is charged electron) forced to circular the action of a magnetic field which is uniform (has the same value) along theLetentireus fipathrst establish of the electron. somenateelementary properti esmall s of circul arwith motiposition on. Figurevector5.1 shows a right-handed coordi system in which a object ngl with velocity vthewhosetrajectory magnit(see ude Fiis gconstant and equal to ds/dt, where dsR Ifiissthemovi a smal di s tance along . 5. 2 ). trajectory is part of a circle, so that I = R is constant, we can write R . R RI constant so thatalong R · dR/dt = O. Now dR/dt is equal topoint (dR/ds)(ds/dt) and dR/dssee isFig. a unit5.2)vector the tangent to the t r ajectory at the R (the unit tangent, time derivsoatithatve ofdR/dt R . v == v, isR · v = and R is always at right angles to v. Now the R · -dvdt + v · v = O so that, if j dv/dt denotes the acceleration, then R . j = -v . v = -v2 , (5.1) =

0,

0

=

z

y

x

Fig. 5.1

Circular motion.

Motion ofa moving charge in a constant magneticfield

77

v

s

Fig. 5.2

Unit tangent dR/ds.

and the magnitude of fis _v2 / R. What is the direction of f? Because v· v constant, . v dvldt v infthe direction so that fopposite is alwaysto atR,right angles tothev.centre The minus sign indicates that f points i. e . towards of the circul ar orbit. If w is the angular vel o city of the object, whose magnitude (equal to dB/dt) and direction are constant, then w is rel a ted to v and R through the vector equation v = w x R. The quantity w x v = w (w R) = -w2 R = f (byR areexpansion of the tripleascal aeach r product, remembering that w . Rfor= thebecause wand al w ays perpendicul r to other) is another expression accel e ration vector whose magnitude is _w2 R, equal to -v2 /R as before. .

=

=

=

°

x

x

°

Motion of a moving charge in a constant magnetic field

Ina constant order toaccel generate thisalong circulthear traject ovector. ry, a force mustobjectactisonantheeleobject toly produce e ration radius If the ctrical chargedy, particl e wi t h charge moving in el e ctric and magnetic fi e l d s E and respectivel then the force which acts on the charge is given by eqn = e(E v x The how theare momentum p and energy U of the particlequations e carryingofthemotion, chargewhich changedescribe with time, -dpdt = e(E v x dU = ev . (E v = ev . E. dt The magneti c field produces a forcev, andacting on thechange chargethewhich is alofwaysthe atcharge. rightconstant angles to its direction of motion so cannot energy When the electric field E is zero the equations of motion reduce to dp- = ev x dU = dt dt so that U is constant as expected. F

B

e,

(2.6):

F

B) .

+

B) ,

+

+

(5.2)

x B) F

B,

-

0,

(5.3)

78 Synchrotron radiationfrom dipole magnets Because wewearemustdealiuseng with a aparticle which may formoving at a speed closeandto that of l i g ht, the rel tivistic expressions the particle' s energy momentum, so that ym U ymOc (5.4) and the relationship between momentum and energy, (5.5) Asparticle usual,ismoat restis theandmassisofthetherelpart in a reference frame in which the ativiicle, stic measured Lorentz factor be

p=

=

Ov,

2

,

y

We canquantity now applBin y theseeqnequations to theto vB, motion of thethepartfieildcleisineverywhere a magnetiatc field. The (5. ) reduces because ght 3 2I R, from eqns ri(5.1) angl e s to the vel o ci t y of the particl e and d ld t becomes ymov and (5.4). and Equating these quantities trajectory the momentum of the gives particle:the relationship between the radius of the BeR. (5.6) Theseelectron, resultsandcanwhen be applweiedaretodeal anyingelectrical lyectron chargedenergies particle.which Whenarethemuch partilcarger le is anthan with el the eletoctronzero,restweenergy be equated may alsosothatwritthee term on the right-hand side of eqn (5.5) can U BeRc. (5.7) When theorquantities in eqns (5.6) ande) (5.units,7) arethemeasured intheMKS (metreis measured kilogram second) SI (Systeme Intemational energy of el e ctron injoules.l eVHowever, it is useful to bymeasure thectronenergy ofittheis accel electron in elbyectron volts, where is the energy gained an el e when e rated a potential difference ofthe1 V.elThe corresponding unit for6 the10-1momentum isis e1.V6Ic. Because the 9 9 -1 charge of e ctron i s numerical l y 1. eV 10 and -13 intoMeVeqnsis (5.1.66) and10 Ifwewefisubstitute nd that these values, along with the velocity oflight, p (MeVIe) U (MeV) 300BR (Tm) and,1.2forT, example, an electron with anwithenergy of 2ofGe5.V,56moving in a themagnetic fieanld ofelectron would describe a trajectory a radius m Because mass of . this case.at rest is 0.5 1 1 MeV, it is valid to take the approximation that U in vx

p

p =

pc

1

q

x

J. (5.7),

=

=

x

=

C, I

x

J

=

pc

=

Radiation ofenergy by a moving charge in a constant magneticfield Radiation of energy by a moving charge in a constant magnetic field

79

Now let usfieconsider how thisis electron radiatestoenergy as it moves ofin magnitude a constant magnetic ld. The electron being subjected an acceleration, v2/anR elsoectromagneti that a distantc wave observer detects theandelectric and magnetic fieldselectron in the .form oftotal [ e qns (4. 5 5) (4. 5 8)] from the radi a ting The amount of energy transported by the wave to the observer, per second, per unit area, is given by the val u e of the Poynting vector S [eqn (4. 1 6)] at the observation point, S E x B /m2), and since [eqn (4.59)] B -nc1 x E, n is a unit vector line from the point at which the radiation is emitted towhere the observation point,along we canthewrite S focE x (n x E) foc{nE2 - (n . E)E} . In the far field, from eqn (4.55) v/C}3 ] E(r, t) 4rrfoc [nI r x- {(n - v/c)(n . xv)/c) (5.8) so that n . E and V,

-

=

flO

(W

=

=

=

q

= --

=

r' l ( 1

0,

,

-

(5 .9)

This togeneral. be true for plane electromagnetic waves in eqn (4.18). Here it is seenSuppose towasbeshown truethe inradiation isorigin beingofgenerated by an system, electron,thenwhich, at thedescribe momentwhatof emission, is located at the the coordinate we can is happeni ngfirst,usingthespheri cal coordinates as in Figvelocity . ofthe electron at the origin Consider case when the instantaneous isofvery much lessduring than thethespeed oflight. In thison case wethatcantheforgetpower,aboutdUthe/dt,movement the electron ti m e of observati dt, so radiated into a solid angle dQ in sr s) is given by 5.3.

J/(

(5. 10)

eqn (5.8) reduces to from eqn (5.9). In this case, when Iv/cl 1, and I r E t ) - 4rrfoc (n x (n x v/c») (5.11) -

«

(r,

_

q

_

_

r

r' l

= r,

,

Synchrotron radiationfrom dipole magnets 80 and inserting this into (5.10), we find that

(5.12) Ifation,is iJthe, then anglen between the (ndirection of observation, n, and the direction of acceler­ n m)iJ sin where m is a unit vector perpendicular to (and (see Fig. 5.4), so that In m l2 and (5.1 3) This is a maximum in ofthetheplaneorbit.defined bypltheanetangent 2 hasto theits elmaximum ectronangular orbivaltdistribution and the normal to the plane In this sin uevector, of 1. Figureof the5.4radiation shows thepoints shapeinofathisdirectiangular distribution.to both The el e ctric o n perpendicular n and m [from eqn (5.12)] and is therefore plane polarized with the electric vector e

n

iJ)

x

x

iJ =

x

x

e,

=

1

e

E,

z

y

x

Fig. 5.3

""'"_�_

rsin 8dl{>

Spherical coordinate system.

Fig. 5.4 The sin2 e radiation distribution.

Radiation ofenergy by a moving charge in a constant magneticfield 81 lying both landooking v. Thealongdirections of theseof vectors are shown, aseleanctronininset,the(e inplaFig.neseescontaining 5.4no. radiati observer the direction acceleration of theon o n. An observer looking at ri g ht angles to thi s di r ecti detects maximum radiated energy. Moreover, this radiation isvisualized polarizedinparallel to4 theby direction of v. The ¢ distribution of the radiation can be Fig. 5. 2 e around the v axis to form a doughtnut-shaped surface in three rotation of the sin dimensions. Equation (5.13) can radi be integrated over the whole charge of the 4rrin unisolidt time. angleTheto give the total amount of power a ted by a non-relativistic element of solid angle dQ sin () de d¢ so that, by symmetry dU 8q2 i;2 3 i7r/2 sin2 esined() i7r/2 d¢ dt (4rr) 2EOC 0 0 and, after carrying out the integration dU (5.14) dt which is often called theectron Larmoris moving formulainaftera circular Sir Josephorbit,Larmor (1857-1942). this case, the el with angular In 2 and dU/dt isconstant frequency w/2rr and radius R, so that Ivl Rw then given by 4 dU q2 R2w . (5.15) dt 3 4rrEo C3 We must nowthisgeneralize thisreturning result totoinclude allexpression electrons,forwhatevert)thei[eqnr velocities. We can do either by the ful l (5.8)]thisor bylatterexpressing eqn (5.14) in a Lorentz i n vari a nt form. It is i n structi v e to follow first. Theevenquantity dUthe/dtexpression in the Larmor formula [eqnonly(5.a14)Jnon-relativistic is a Lorentz iapproximation. nvariaroute nt quantity though in eqn (5.15) is In other words, the rate atiswhich energy isobvious, radiatedbecause is independent of theof energy, referenceL'l. Uframe of the observer. This by no means an amount radiated inaatime time interval L'l.second t in oneframe reference frame will become an amount L'l.U', radi a ted in L'l. t in a rel a ted to it by the Lorentz factor The questiisongiven is whether L'l. U' ! WeL'l.willU. Anassume indicitatitoonbeofso.the proof that this assertion is correct by Jackson. With this theassumption we can express left-hand side as av,Lorentz substituting four-acceleration for thethethree-acceleration so that invariant by dU 2 q 2 3 1 A 1 2 (5.16) dt - 3 4rr£oc The minus sign istheneeded tosign,makecomes dU /dtfroma ratetheofLorentz energyinloss. Inntaenergy-momentum rigorous treatment eqn (5.16), wi t h minus vari a tensor. =

An

0)

n

=

- =

v

=

=

=

2

E(r,

y

= y

y.

A

=

82 Synchrotron radiationfrom dipole magnets momentsowhen theaccel electron isninstantaneousl yesattorestthe difromrectitheon poiof Consider nmotion t of viewofnow ofthetheeltheectron observer that the e rati o i s at right angl the acceleration four­ vector are obtained fromalongeqnthe(3 .2x9)-axis.whicTheh, sicomponents nce v . v = 0,ofgives A

and

dU = ---2 qZ (5 . 1 ) dt 3 4rr£oc3 y4 v. Z Now, sincebythetheenergy and field, the magnitude of thefrommomentum of theofelmomentum ectron are unchanged magnetic we can write, the definition in eqn (5.4), 1 dp v. = ---(5.18) ym dt and dp/dt = p deand/dinserting t pw soeqnthat(5.18) dp/dintto=eqnpv/(5.R.17)Substitution ofthis expression into eqn (5.18) gives the upgraded Larmor formula which is valid for electrons of any energy: dU = 2 qZ yZ p1v1 (5.19) dt "3 4rr£o c3 m o1 cz R z ' As(5.14).we Inwould expect,energyin thelimit,low£energymocz,limit,pZczwhen £zy and1, eqn (5.19) reduces to the high dU = 2 q1c y4 = 2 qZc £4 (5.20) dt "3 4rr£0 RZ "3 4rr£0 (mocZ)4RZ' particul ainrlyunitsignificant feature of eqnto(5.the20)fourth is thatpower the total power radiaenergy ted by theOne el e ctron time is proportional of the el e ctron and inverselconsequence y proportionalof tothisthepower fourthlapower of thecharged electronparti(orclesposiheavi tron)ermass. Anthe ielmeportant w i s that than ctron areradiates very poor radiatators byreduced comparison. For the(1/18 same energy and orbiwith t radius,the 4 , compared aelproton energy a rate by a factor 6) 3 eSuppose ctron. that the electron is being accelerated in the direction of its motion, foraccelexampl eIn, inthisa lcase, inearvaccel erator or inlel,thealoradi o-frequency cavity of a( circular e rator. and v are paral ng the -axis and, from eqn x 3 .29) as before, with Ivl = f3c, (y4f3V, (y 2 V + y4f3Z V) , 0, 0) = (y4f3V, y4v, 0, 0) 7

=

O

I

»

A =

l

=

=

Radiation ofenergy by a moving charge in a constant magneticfield 83 since,I Afrom Weresul now tevaluate 1 2 , andtoremember . A = (A6of-y , 1 + y 2and{32 =inserty 2 . the that 1 2 =theA definition into eqnIA(5.16) give dU = -2 -q2 - y6 v. 2 . (5.21) dt 3 4JTeOC3 Atgreater first than sight,when it appears thateration the rateis perpendicul of loss of energy indirection this caseofwould be butmuchin the accel a r to the motion, fact this is not so, because, from eqn (3 .34), dEdt = mOy 3 vv. so that 1 dE Y V = -­ mo and dU = q2 1 ( dE ) 2 (5.22) dt :3 4JTeoc3 m 6 dx which is byindependent ofmake the energy of synchrotron the electronradiation so that there is noloss.energy limit imposed the need to up for the energy Of course, , the energy gain perofunitits distance, is justelethectricaccelfielderating force acting onto xctron thedE/d el e i n the direction motion. What woul d be required provi(5.de2a2),force which wouldfor equal give theradisame oss of radi Fromwilleqnsbe(5.20) and the condition ated lpower lossaintedthepower? two cases dE E2 1 dx - moc2 R ' For example, the Daresbury SRS has E = GeV and R = 5. 5 7 m, so, with moc2 = = 1.4would10be6 MeV/m. 0.gradient 5 11 MeV,of 1400 dE /dMVx /mm Inwhich orderistoabout provide105 such athatforce,whicha ficaneld required, times beWereached insopractice. can al eration four­ .29) whicase ch weincanthe foll writoe,wing withway.=The {3c foraccelconvenience: vector A is givenexamine by eqnthe(3general A = (y4{3 . iJ , (y2iJ + y4{3({3 . iJ) )) . From this we can form the length of this four-vector: IA 1 2 = y 8 ({3 . iJ)2 - (y2iJ + y4{3({3 . iJ) )2 . After some vector productelementary which in althisgebracaseandis the use of the vector identity for the square of a A . A)

3.

d.x

2

x

2

v

Synchrotron radiationfrom dipole magnets 84 we obtain IA e y 6 C2 [(f3 iJ) 2 - iJ2 ]. The Larmor formula forformula an electron ocitysomustthat,reduce toeqnthe(5.16), non-relativistic whentravelling is smallwithandanyy isallcloowedse tovelunity from dU -2 --q 2 y 6 (v. 2 - -1 (v v). 2 ) (5.23) dt 3 4nBOC3 c2 byexpressions Lienard (1898). For the two cases examined above, eqn (5.23) reduces toas We theobtained same as before. havetheobtai ned theofabove result [eaqnnce(5.on23the)] using an intuifortivthee approach which imposes condition Lorentz i n vari expression radi a ted power the Larmorfor formula, eqnc and(5.14). Forcafields rigorous treatmentthewecondi musttioreturn tothegiventhespeedbyexpressions the el e ctri magneti and remove n thatthe of the electron i s much l e ss than the speed of light. If we are detecting radiation time ta during a timedt'interval dt, tthis radiation isobserver emittedisbystationary the electron atthetimeelectron t',atduring time interval and t' / The and is moving so that is a function of t' onl y and d t /dt ' 1 - (v · n)/e [eqn (4.45)].of theAsvector usual, vlinking is the velocity of theatelectron and then iselectron a unit vector along the direction the observer ti m e t and at the retarded timIfewet'. These vectors aretheshown in Fig. 5.of5.radiated energy U which will be detected wish to compute total amount atntegral the posioftidUon /dtgivenoverbythatthe vector over aButtimdUe in/dttervalis expressed Ilt, we mustas acalculate theof iquantities time i n terval. functi o n which must be calculated at the retarded time t' so we must write 1tO+61 d U dt BOC I t�+M Ir[E(r, t)] 1 2 -dt dt'. dt to dt tb as noted Equationso (5.that,8) following gives the expression for t)as andin eqns dt/dt(5.10)-(5.12), ' 1 - (v · n)/c, above, the same treatment but with vic taking any value 1, d2 U q 2 In { en - v/e) iJ/c} 1 2 (5.24) dQdt (4n)2Boc (1 - (n . v)/c)5 Equation If we consider(5.2fi4)rstisthea completely case where vgeneral and areexpression parallel twhich o eachappl otheriesthento allv situations. iJ = 0, and x

=

f3

- =

x

=

r

-

Ir l

e.

=

r

r

-

=

I

E(r,


, and (5.25)

andItintegration givesto express eqn (5.17).eqn (5.25) in the small angle approximation, for which is very useful we write - f3 cos e) /2y2 ) (l + y 2e2) , so that q2 d2 U y 61 v l2 ( 4 y 2 e 2 cos2 3 dQ dt 4rr2cQc (1 + y 2e2)3 1 - (1 + y 2e2 )2 ' which shows that the important scaling parameter for the angular distribution is not e butFigure(ye). 5.7inshows the shape of4>the distribution, plotted against yevinandthevy-z(theplane 90°), the plane where 45° and the plane containing x-z = plane, where 4> 0°). It is clear that the distribution is strongly peaked in the direction (1

� (I

4»

=

(4)

=

=

Synchrotron radiationfrom dipole magnets

86

v

n y

Fig. 5.6 Coordinate system for synchrotron radiation production.

ofthemotion of the el e ctron. The mean square val u e (& )2 is a2 measure of the width of angulaer expression distributionofwhich formidabl eqn is shown to be equal to l/y by evaluating the rather 1000 e2e de IoJr/2 d¢ - 1000 4y 2e2e2e de IoJr/2 cos2 ¢d¢ (e )2 = (1 + y2e2)3 (1 + y2 e 2)5 1000 e de IoJr/2 d¢ - 1000 4y 2e2e de IoJr/2 cos2 ¢d¢ (1 + y2e2)3 0 0 (1 + y2e2)5 (5.26):

0

0

0

0

0

o

(5.26)

Because the angularwhendistribution is lconfi nofedtheto smal l valuover es ofee ,isthere is no appreci able lDoing oss ofthisaccuracy the upper i m i t i n tegral extended to infinity. that the substitution ye = totan ex can be used, with ex running from the expression reduces further to / , andmeans (e ) 2 = If:12 sin3 ex cos ex dex - If:12 sin5 ex cos3 ex dex . y2 JrOJr/2 sm. ex cos3 ex dex - JrOJr/2 sm. 3 ex cos5 ex dex The integralsbutinit eqn can beusievalng uiated in terms ofparts,factorial functionsevaluation, (gamma functions) is easy to show, n tegrati o n by and without that [JrP [JrP 10 sinn - 1 ex cosn +1 ex dex 10 sinn +1 ex cosn - 1 ex dex. o

7r

2

I

(5.27)

(5.27)

=

Radiation of energy by a moving charge in a constant magneticfield c:

� "0

"0

c:

� "0 "0

c:

� "0

"0

1 0.9 0.8 0.7 0.6 0.5 0.4 OJ 0.2 0. 1 0

0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0. 1 0

1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

0

,

0.5

0

0.5

0

0.5

yO

yB

yO

1.5

2

1.5

2

1 .5

2

87

Fig. S.7 Angular distribution of synchrotron radiation from a point on the circular orbit of an electron in a magnetic field.

Either way, eqn (5.27) reduces to (5.28)

which is the characteristic opening angle of synchrotron radiation.

Synchrotron radiation from dipole magnets

88

Dependence of radiation on frequency and angle

Throughout discussionwhenso thefar weelectron haveconsidered the raditheatioriongiton ofbetheemicoordi tted byntheate elsystem. ectronThiat thesourtelmoment passes through ls whatthatanassumption observer woulin order d see duri ng an infinhow itelytheshortradilength offrom time.a We must now relax to determine a ti o n singlea electron appears toIn anotherobserver whose detection systhowemthecollects ionnformati on over l o ng ti m e period. words, we need to know radi a ti from the eleQual ctronitatichanges witobserver h time asviseenewinbyg aa stati onaryelectron observer. v el y , an single is presented with a cone of radiation with cone angle of order 1/y 2 . Figure 5.8 shows part of the trajectory of an travelling along an arc of a circle of radius R with a velocity f3c. A detector atelectron a di s tance D from the origin of the coordinate system at registers the arrival fitedrstasphotons to bepasses registered atnthet A,detector, ofproduced a photonwitathina aticone me toangleD8/arec. The emi t the el e ctron the poi to arrive atemithetteddetector at ti m e fa and the last to be registered arrive at a time tb, having been atthepoinlength t B onofthethetrajectory. The pulse atlength of the photon pulse, or, more precisely, electric fi e ld pulse the detector i s tb - tao In order to generate a pulse of this duration, the electric field must rise and fall again in the time tb - ta, so that the wavelength of the electric field must be in the region C(tb - ta) and thehigher frequency l /(tb - ta). The frequency can be lower than this, but it cannot be much without a corresponding decrease inelectron the coneenergy angle of theyradi ation. The frequency of the radiation is determi n ed by the moc2 . pulse length taken fortothetravelelectron AOB,Thereduced by theistiequal me takento thefortithmeeradiation directltoy travel from Aalong to B, theso thatarc 0

=

E

=

2R8 2R sin8

tb - ta = -- f3c

---

c

x

D

h.---i-A-""""',-----I .. Z �----'�-+----'�'-

A

� Detector I

Jf\

ta 0 tb Time course of photon pulse at detector Fig. 5.8

Time variation of synchrotron radiation pulse.

89 and whenmated8 isbygisivenn 8 by= eqn- 8(5.3 /628)thenand is sufficiently small so that sin 8 can be approxi 1 .!. 1 ) = 2R ( _ fb c yf3 y + 6y 3 which reduces to 4R fb - fa = 3c y 3 with approxiismgiativoenn byl/yf3 = l/y + 1/2y 3 , so that the corresponding frequency of thetheradiation Dependence of radiation on frequency and angle 8

_

_ _

fa

v=

and the wavelength by

,

_

3cy 3 -4R

4R3 ' 3y

(5.29) The argument which has thejustcone beenanglgivene ofisthemeant totorshow thatsmalforl thattheancaseobserver of an ulseestra-rel a tivistic radi a tor, radi a is so a pulse ofpresent radiatioinn whose durati oFigure n in tim5.e 8determines thetheordereleofctronmagnitude of thein afrequency the radi a tion. assumes that is moving magneti c fiedistance, ld whichtheextends over othen attrajectory AOB.is shorter However,as wellif theandfieldthe covers a shorter pulse durati the detector minimum frequency is sweeps even higround her. Thia bend s mayinseem strange. After all, theoncolour of a carof theheadlight as the car the road does not depend the radius bend.is large In thatsocase theC(tbcar is moving slowlycompared and the cone anglewavelength of the radiation produced that is very long wi t h the of theat light produced by the car headlight and so does not restri c t the radi a ti o n observed wavelengths in the visible regioclose n of theto thespectrum. Onlight, the other hand, for the case of anin theelectron, moving at a speed speed of C(tb - is comparable, region forconstraint �2 GeV elonectrons, to the wavelthatengthcanresponse of the detector andThissoX-ray exerts a strong the wavelengths be observed. intuitivetheargument indicates that theChapter frequency spectrum may be obtained byelectricalculating Fouri e r transform (see 1) of the ti m e distribution ofittheis c fi e l d at the observer. When we follow thi s procedure we shal l see that conveniparameter ent to useis aoften wavelcalength ch is tismticeswavel that giength ven byAc eqn This led theparameter critical orwhicharacteri given(5.by29). 4rrR (5.30) Ac = 3 . 3y What exampl above,is inR the= 5.56m, E 2GeV so thatvalue y is ofAc 3914;canthenweAcexpect? wouldInbetheabout O.4enm,givenwhich X-ray region of the A=

- fa )

fa )

rr

=

90

Synchrotron radiation from dipole magnets

3 so that doubling the electron elenergy ectromagnetic spectrum. Also, varies inversely as y reduces the characteristic wavelength by a factor of 8, moving it further into theWeX-raymust region. now proceed to thebycalanculobject ation ofin citherculfrequency spectrum offrom the elec­ tromagneti c radi a ti o n produced a r motion. We know eqns (5.9) and (5.10) that, in general, for any kind of motion (5.3 1 ) Inat theeqn observati (5.31), E 2 is the square of the modulus of the electromagnetic field vector explicitly, as on point. We can write eqn (5.31), showing the time dependence Ac

(5.32) =

where G(t)is emittedJ8QC[r E],thecalelceulctron. ated The at thestatiretarded time at who whichis thedetecting radiationthe observed from o nary observer radi on emittedfromintothethemoving solid angl total energy at theatidetector chargee dQdurimeasures ng a timetheinterval dt as received, in Jlsr dU 1+00 I G (t)1 2 dt. dQ -00 From the odiscussion above, wethe know thatnitethecontribution radiation pulto sthie sisinemitted over a from very short peri d of ti m e so that only fi tegral comes times close towhich t Extension ofthethephysical integralresult. to infinite times is a mathematical conveni e nce does not affect time course of the elofectric field vector buttwowhat we needareisrelG(w), thea elFourier eG(t) ctric transformation, fiisetheld vector as awhich function frequency. These quantities a ted by expandby the continuous function G (t) as a Fourier integral, so that G(t)means is relathat ted toweG(w) 1 1 +00 G (t) V2K -00 G(w) exp(-iwt) dw and G(w) to G(t) by the inverse transform G(w) V2K1 1+-0000 G(t) exp(+iwt) dt (5.33) Using thiscareprocedure, webecause can calalctulate G(w) using eqn what (5.33). There is need for special at this point hough we understand negati vchange e times tomeanthe physically and we can, if we wish, make all times posi t i v e by a si m pl e coordinate system, a negative frequency has no physical meaning and we must obtain =

=

O.

=

=

r,:c

r,:c

.

Fourier transfonn of the electric field

91

theposienergy radiated intoonlyunitso thatsolid angle by integrating the frequency spectrum over tive frequencies =

dU dQ

('X) d2 I (w, n) dw.

Jo

(5.34)

dQ dw

Onintothefrequencies other hand,whithech formalism demands that thereso,istoactual power beingincluradide athatted l i e between zero and make sure we power, we must write -

= I G(w)1 2

d2 I (w, n) dQ dw

00 ,

+

IG(-w) 1 2 ,

where w is always positive, as in eqn (5.34). However, from eqn (5.33) + 00 G(t) exp(-iwt) dt � G( -w) 1 2]( the complex G* (w) (see Chapter 1) is related to the complex conjugate ofandG(t), which isconjugate G*(t) by 1 +00 G*(t) exp( iwt) dt G * (w) 2]( 1 but, since G(t) must be real, G(t) = G* (t) so that G(-w) G*(w), so that d2 / (w, n) (5.35) dQ dw 2IG(w) 1 2 and dU (5.36) 100+00 2IG(w)2 1 dw. dQ =

=

'\f

- 00

� '\f

-

- 00

=

=

=

Fourier transform of the electric field

From the definition of G (t) eqn (5.8)] G (t)

=

[ given at eqn (5.32), we write [from

y'6OC r E],

il/C} ] = 4]( y'6OC [ n (l{(n- (nvic) . v)/c) 3 q

x

-

x

.

=

must be evaluated at the retarded time t' t - r(t')/c soThethatquantity we mustin square write eqnbrackets (5.33) with t ' as the continuous variable instead of t, so that n {en - vic) il/C} ] ret '» )) ' [ + ( ( W 00 dt t' exp G(w) i + 1 3 2 c ( 1 - (n · v)/c) J32]( 80 C because dt dt'(l n . vic), from eqn (4.45). =

q

=

- 00

-

x

x

+

92

Synchrotron radiation from dipole magnets

Figure 5.9 shows the relationship between an observer at a fixed point P whose coordinates are (x, t), x being the position vector of P and the radiating electron at (R, t'). Since the distance ret') is much larger than I RI, the zero order approximation would make Ir(t')1 = Ixl. However, because the electron is moving we must take the next approximation and write Ir(t')1 = I x l - n . R(t') so that we can write the exponential factor in eqn (5.36) as exp

(+IW o X ) exp (+1(0 (t .

-;;-

'

n o R(t'» c

-

)) 0

The factor exp(iwx I c) is a constant phase angle which can be placed outside the integral and when, at the end of the calculation, we take IGI 2 this multiplies out to unity. Physically, this corresponds to a constant phase shift of the electric field vector which is of no interest to the observer and eqn (5.36) can be written as x

x

] [ n { en - vic) q f+OO (1 - (n . v)/c) 2 J-'=32=7r�3:=80=C -(+iw (t' n . �(t'» )) dt' .

G (w)

_

-00

x exp

VIC}

_

In this form G (w) can be evaluated if we note that

� n x {n dt'

[

1

-

x

VIC}

(n 0 v)/c

=

] [n

x

{en - vic) VIC} (1 - (n 0 v)/c) 2 x

]

and d dt'

( (+ w (t' - n . R(t'» ))) ( n . v ) (+iw (t' iw 1 - -exp

=

i

C

c

exp

-

n o R(t' » c

)) 0

n·R x

Note:

o

r ='

n it-I

Fig. 5.9 Position vectors for evaluation of integral.

(5037)

93

Radiation from an electron moving along the arc of a circle ) ] ( (t' n · c t') )) dt' 1-+0000 [ n ( {en1 - -(nvic. v)lc) 2 ) t'=+OO n (n (vic )) ( (t' n . R(t' = 1 1 - (n v)/c c )) I t ' = oo 00 - iw £: n (n �) (+iw (t' n . �(t') ) ) t' 00 ( (t' n · �(t') )) dt'. = - w £: n (n �)

Integration by parts yields x

x

x

x .

VIC}

exp

x

x

+iW

R(

_

_

exp

x

x

i

exp +iW

d

_

exp +iW

_

Because the radiation is confined to a forward cone close to t' = 0, the contribution from the first term, which is to be evaluated at t' = ±oo, must be zero and we can write, from eqn (5.35)

d2 I (w, n) d Q dw q2 16rr 3 , -0

c3", .0§ �'" �0

'':;u

-a� t!:: �

�

.0 (l.

'"

b�

><X ....1 " "0 c: " '"

1

-Q 3 Qt--= ' Q � �� 2.5

2

1"":--:::.....1. ---__

p

�-r�--�-L-�

�

.8 sin B sin1/!

�p

Fig. 6.14 Directions of acceleration vector components as seen by an observer at P, drawn in plane view (top) and elevation (bottom).

�

When the trajectory of the electron has turned through an angle () can be resolved into two components as shown: cos (), at right angles to the line of sight which generates the horizontally polarized component E 1 ; and sin () which points along the line of sight. When P is in the horizontal plane, this component cannot contribute to the radiation field at P, and the observer sees only the field generated by the component cos () but as soon as P moves out of the horizontal plane to an angle 1/1 above or below the plane, the observer sees an electric field at right angles to the line of sight, generated by the acceleration component sin () sin 1/1 and pointing predominantly upwards or downwards depending on whether the electron is moving towards 0 or away from O. It follows that, over the time period of the pulse, the observer sees an electric vector generated by the resultant of the acceleration vectors cos () and sin () sin 1/1 which appears to rotate through 2n: radians during the duration of the pulse. Of course, this is exactly what we would expect from eqn (6. 14), where the quantity G 2 depends on 1/1 , the angle of observation. It is also clear from eqn (6.14) (and from Fig. 6. 14) that the direction of rotation of polarization depends on the sign of 1/1 so that an observer above the median plane detects right-handed elliptical polarization whereas an observer below the plane observes left-handed polarization. Let us calculate the degree of elliptical polarization. To do this, we separate the elliptically polarized radiation into two new vector components, one which is rotating, with frequency (j) in the clockwise direction (as viewed by the observer) and the other which is rotating, with the same frequency, in the opposite (anticlockwise) direction. These components are often called right-handed and left-handed, respectively and will be denoted by ER and EL . Figure 6. 15 shows a vector diagram in which the total electric field vector E is shown resolved into its components Ex and Ey or ER and EL.

�

�

�

,

�

�

�

111

Photon polarization as a function of energy and angle In terms of these vectors, we can write =

Ex EO,R EO,L Eo,x Ey EO,R -EO,L Eo,y EO,x EO,R EO,L , EO,y EO,R - EO,L EO,R ! (Eo,x Eo.y), EO,L = ! (Eo,x -EO,y) . =

cos wt +

cos wt

sin wt

sin wt

=

cos wt,

=

sin wt

(6. 1 6)

so that, from eqn (6. 1 6),

=

=

+

(6. 1 7)

and, from eqn (6. 1 7)

=

+

y

x

Fig. 6.15 Time-varying electric vector E expressed in terms of the vectors Ex and Ey or ER and EL.

::: 0

�

0.9 0.8 0.7 0.6

.� N 0.5 .� 0.4 '0 0..

'"

0. 3

0.2 0. 1 °

I

I

I

j

I

\. /

'" '" '"

-- ---- PHnear

- - - - Pcircular

"'"

'"

�

1/

o

2

Fig. 6.16

I

I

--I--

y * 1/f

3

4

Linear and circular polarization for wjwc

5

=

0. 1 .

6

112

The spectral distribution of synchrotron radiation ---

, 0. 9 � ... """ 0.8 !'v/ 0.7 " "'0.6 '" 0.5 , ._- Plincar I'--. , 0.4 �I- - - Pcircular , 0.3 .... r--. .. 0.2 I---0.1 / 0 0.5 1 .5 2 2.5 o y * 1/f ---

,,-

0: 0

.� N ·5 '0 Il.

/

/

I

/

-

I

Fig. 6.17

3

Linear and circular polarization for w/wc

=

----

N

I

I

Fig. 6.18

1 0. 9 0.8 0.7 .i 0.6 .g 0.5 C 0.4 0.3 0.2 0.1 o

0.5.

--

., 0.9 � ", '" '" 0.8 -- Plinear "'�/ L 0.7 - - - - Pcircular / "", 0.6 .� / '" 0.5 "-·5 0.4 / � 0.3 � �c-. 0.2 / ---t---.... / 0. 1 0 0.5 2 o 1 .5 y* 1/f 0: 0

3.5

--

Il.

o

"

;"

Linear

�

,,'

;F

0.2 Fig. 6.19

'"

'"

and circular polarization for w/wc

...

""-,,, '" ," �

... - -

-

� I·-- Pljnear )- - - - Pcircular

0. 4

0.6

y * 1/f

--�

2.5 =

1 .0.

_- - - -

I , ...... ----

Linear and circular polarization for w/wc

f--

�I-

1 .2

0.8 =

3

5.0.

1 .4

1 13

References ---- -

I

c: 0 '.:;:1 oj N

'0

.� c..

0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0. 1 0

D

,;'

/"

/ ""

,;'

....

-.....

.... '

0.2

0.1

-

f.- .....

....

""""--

-.-

I I -- Punear

i

l- - - - Pcircular

0.3

y * 1/I

".. -

-

=----I J

0.5

0.4

- .,.. - -

So that, defining

p .

Clfcuiar

-

_

-------

0.6

Fig. 6.20 Linear and circular polarization for w/wc

_ .....

=

0.7

.

-

0.8

1 0.0.

I Eo,R I22 - IEo, L I22 I EO,R 1 + IED,L 1

we have the required expression for 2EO,c-'-x-':'::EO,y = ---:: IEo,x l2 + I Eo,�y l2 2 K2/3 (�) Kl/3 (� ) (y1jrl)l y 2 1jr2) . Ki/3 (� ) (y 2 1jr2j( 1 y 2 1jr2)) K f/3 (� ) is zeroonwhen We 1jrgn=of0 1jrand. its sign (which corresponds to theFigures dinoterectiagain o6.16-6.20 n ofthatrotation), depends the si show the result of a polarization calculation for various values of j Pcireular :

Pcireular

+

+

+

Pcircular

W

We.

References 1 . Mary L. Boas, Mathematical methods in the physical sciences (2nd edn). Chapter 1 1 , p. 462. John Wiley & Sons, New York, NY, USA ( 1 983). 2. 1. Schwinger, Physical Review 75, 1 9 1 2 ( 1 949). 3. J. R. Helliwell, Macromolecular crystallography and synchrotron radiation. Cambridge University Press, Cambridge, England ( \ 992).

7 Photon spectral distribution integrated over vertical angles

The fact that the radiation from the accelerated electron is emitted into a narrow forward cone means that quite often an observer is interested in the photon energy (or the total number of photons) emitted over all vertical angles. This is the case when the target being irradiated subtends a vertical angle which is significantly larger than the opening angle of the forward cone from the emission point. The energy radiated by the passage of one electron into unit solid angle, located at an angle 1jf above the plane of the electron orbit, and unit frequency interval is given as usual by eqn (7. 1 ) which i s identical to (5.44); 1

--- = - -- y 2 ( -Wew ) 2 ( l + y 1jf ) Kif3 (� ) + :2�: K f/3 (� )

d2 1 (w, n ) dQ dw

q2

3

2

4 47r3 COC

X

[

1

2

2

2

]

(7. 1 )

3/2 as before and the units are J s/rad2 into unit where � = 1 + y 2 1jf 2 ) frequency bandwidth. When we use eqn (7. 1 ) to calculate the energy emitted by the electron we multiply the result by the frequency bandwidth of the detector to obtain the energy (in J/rad2 ) incident on the detector resulting from that one electron. We can simplify the equation and make calculations easier by noting that the fine structure constant, which is a dimensionless number approximately equal to 1 / 1 37 is given by

! (W/We) (

ot =

47rcohc

so we can rewrite eqn (7. 1 ) in the form

and h which is the Planck constant divided by 27r is equal to 1 .054 x 1 0 - 3 J s or 6.582 x 10-22 MeV s.

4

Photon spectral distribution integrated over vertical angles

1 15

Integrati one)n of, gieqnves(7.1), overredtheresul anglet forl/f (tboth he observati onpolanglariezatiaboveon. The or belinotegra­ w the medi a n pl a t h e requi states of tiroeatn misentnotgistraivenghere htforward because of theeddependence of the2 Inparamet etro simonpll/fif.yThethe tprobl fol l o ws t h at present by Wi e demann. order we notquanti e thattithese isoln anid angl from ethm,e other obvieousdQwayde[eqndl/f(7.and2)]:we separate the l/f dependence �

=

and replace the modified Bessel functions with Airy functions defined as -3-2 ) (7.3a) Ai(x) ;y(X3"KI/3 (2x3/ and its derivative is x 2/3 (2x3/ Ai/(x) --1 -K --3 2 ) (7.3b) so thatthemaki3/2npower g this ofrepl(1acement tr)ansforms thedependence. dependenceInspecti of the oBessel functi(7.o2n) 2 2 from y l/f t o a l i n ear n of eqns and (7.3) shows that �x2/3, so that =

=

�

=

1

7T

-J3

+

With this substitution we can rewrite eqn (7.2) in terms of the Airy functions:

(7.4)

(7.5) Equati statement he radiartiatoenfispectrum 2) buttheis in a moreon (7.sui5t)ablis eanformequiforvaleinnttegrati on. Weof tconcent rst on thetoineqntegral(7.over

Photon spectral distribution integrated over vertical angles squareto make of theaAiry is theitselfsecond 3 and startfunction, on this wemultiplied note thatbythe1/12,Airywhich function can term be wrioftteneqnas(7.5) Ai(x) -;1 1r0oo cos ( 3"u3 +xu) du so that Ai2 (X) :2 1000 cos ( u; xu) du 1000 cos ( v; + xv) dv (7.6) and make the substitution (7.7) s u + v, t = u - v so that 3ts2 + xt)1] 3st2 + xs) + ( t3 + 12 cos ( 3"u3 + xu ) cos [ 21 (( s123 12 12 and 3st2 + xs) ( t3 + 3ts2 + xt)1] . cos (3"v3 x v) cos [ 21 (( s123 + 12 12 12 completethattheeqnssubstitution we need the area element du dv and we can obtain this byToscalednoting (7.7) can be written as a rotation of the (s, t) coordinate system, by a factor 1 /V2, through angle of 45° [eqns (7.8)], i.e. s . t . e V2 u cos e + v sm e , V2 u cos e - v sm (7.8) where e 45° and cos e sm. e V21 so that du dv (ds /V2 dt/V2 and eqn (7.6) becomes 3 + 3st2 +xs) + cos ( t3 3ts2 +xt) 1 ds dt 1 1 0010 00 s ( Ai2 (x) -( cos 12 12 12 + 12 47l' 2 0 0 (7.9) where we have made use of the identity a+b a-b (7.10) 2 cos -- cos -- cos a + cos b. 2 2 t so thetheintegrati over thetocosine theEquation same for(7.9)bothis symmetric terms and wein scanandsimplify integralon further give function is Ai2 (x) 2�2 1000 1000 cos ( 112 (s3 + 3st2) + xs) ds dt (7. 1 1) for the square of the Airy function. 1 16

=

=

+

=

=

+

+

=

-

an

=

=

=

)(

=

=

=

)

-

--

-

=

=

=

--

Photon spectral distribution integrated over vertical angles (7.4)], (7.5):

117

We now insert the l/r dependence explicitly and obtain [from the definition of x in eqn the form of the second integral in eqn

=

:2 1000 1000 1000 l/r2 ( 112 (s 3 + 3st2) 3 1 + y 2 2) ) d dtdl/r. / Y (:: s ( l/r S + l/r2 , (7.12) s cos

(7. 1 2 )

Because the integral depends o n w e need to integrate over positive values of only. Next we rearrange eqn by collecting terms depending on the variable of integration so that

l/r

(7.1 3) Now make the substitution

t r

2"

=

3W ) 1/3 yl/r r (4wc

cos X ,

=

With this substitution

sin X .

(7.14) (7.15)

and

2 ( 4w3wc ) 1 /3 yl/rt (7.14)

tan x =

-

-

(7. 1 6)

so that the integration of l/r from 0 to 00 is replaced by the integration of X from 0 to /2. The substitution defined in eqn is formally equivalent to a transformation from rectangular to polar coordinates so that the area element r dX dr is given by

7r

(7.17) so that

t

d d l/r -

d x dr

2r 3 y (3w/4wc)I/ ' ---'-----:-'-=

--

(7.18)

118 Photon spectral distribution integrated over vertical angles Inserting this into eqn (7 .13) gives all

(7. 19)

and we can integrate over X [eqn (7.20)]: sin22X )] T{/2 = -4 1oT{/2 sin2 XdX = 1oT{/2 -(I21 -cos2X)dX = [12 ( -0 0 (7.20) to obtain o

-

X

lr

-

The finalingstanep Aiis ryto functi transform thceh argument ofbythesetticosingnew3functi ons3in/12to asoformthat resembl o n whi we can do /3 = w = s/22/3 , ds = 22/3dw and x = 22/3 (r2 ( 4wc3w )2/3) sothatdx = 2Sj3 r dr. The new limits ofintegration are: when r = 0, x = given by xO = ( 2wc3w )2/3 , (7.22) and when r = x = Also 3w ) 2/3 ) r2 = ( 22x/3 ( 4wc (7.23) +

Xo

00,

00.

-

Photon spectral distribution integrated over vertical angles and 2r dr ds dx d The integral now becomes =

1 19

w.

where Xo ( 23wwc ) 2/3 (7.24) The Airy functions are solutions of the equation (7.25) Ai"(x) - xAi(x) 0 so that (7.26) (Xl x Ai(x )dx [00 Ai"(x )dx [Ai'(x )]� - Ai'(xo) Jxo Jxo and inserting this result into eqn (7.24) gives the final result for this integral: =

=

=

=

=

1 1 /3 ( Ai'(Xo)o r oo Ai(X)dX) (7.27) 4y3 (3w/4wc) x Jxo 2/3 witInh xorder as before o (3w/2wc) . over the first derivative of the Airy function in to compute the integral eqn (7.5)itself, we usebyeqnanal(7.11) function ogy withandeqnwrit(e7.down 1 2): the integral over the square of the Airy =

+

_

=

use the same [eqns (7.13)-(7. 1 9)] to reduce this integral to aandmanageable form.argument Now eqnas(7.before 20) becomes Jr/2 dX 2 Ioo = ­ 7r

Photon spectral distribution integrated over vertical angles by making the same substitutions in the integral as detailed in eqns (7.21)-(7.26), weand,obtain 1 +00 Ai2 ( ( 4w3we ) 2/3 ( l y21J!2) ) d1J! = 2y (3w/4w1 e) 1 /3 100 Ai(x) dx

120

-

-00

+

Xo

=

(7.28)

withFinally xo (3w/2wc) 2/3 as before. we can now obtain the integral over Ai/ 2 (x). The trick is to carry out a double differentiation of eqn (7. 2 8) with respect to the vari a bl e (3w/4we ) 2/ 3 . To do this weanneed the rulsign.es forThecarrying which involves differentiation under integral integral outis ofthithes operati generalonform V(X) f(x, t) F(x, t) t=V (X) F(x, v ex»� - F(x, u (x» (7.29) I I t=u(x) Iu(x) and we requiquantity re the total81, which differentiis asome l dl/dfunction first of all that if the small x . We note incremental of x, u, and v, is imagined as a vectorandinu-directions a multidimensional space withcosines components ax, 8v, and 8u along the wi t h the direction ax/aI , av/aI, and au/aI, then the magnitude writtensoasthatthe sum over each component divided by the correspondingof 8diIrecticanonbecosine, al al al 8 I -8 -av + -8u ax x av au and, in the limit, when 81, 8x, etc. tend to zero, we can write dI aI aI dv a I du (7.30) + dx ax av dx a u dx Weall other remember that the partiheldal deri vative and a 1/ ax is the derivative of I with respect to x, vari a bles being constant, al derivatives, and with this in mind we can apply eqn (7.3likewise 0) to eqnfor(7.2the9) other with thepartiresult 1 =

=

dt =

X-,

V-,

=

-

=

+

-

+

- -

- - .

V (X) f(x, t) dt Iu(x) V(X) 8f(x, t) dt f(x, v(x» -av(x) 8u(x) - f(x, u (x» ax ax Iu(x) ax

- = -

dI dx

d dx

=

(7.3 1) which isapply often Leibnitz' known ass Leibnitz' s rul(7.e (Got tTofriedclarify Wilhelm Leibnitz,we1646-1716). 4 Now rule to eqn 2 8). the algebra fi r st express eqn (7.28) in terms of two new variables p and q , where � 1 /3 � /3 --

+

P

=

( We ) 2 4

'

q

=

Y

( 4We )

so we can rewrite eqn (7.28) as [+00 Ai2 (p + q 2 1J!2) d1J! �2q 1[00 122/3 00

=

P

Ai(x) dx.

(7.32)

121 Photon spectral distribution integrated over vertical angles Weitselnow diffeorentn ofiap,te weeqncan(7.32)ignoretwithice swibecause th respectwetareo theapplvariyinablg thee p.same Although q is f a functi operati one tofo botchange h sidesof ofeachthesiequati o n. The operat i o n i s bei n g appl i e d, not t o det e rmi n e t h e rat e 3w/4wcwe butwritase an operation dwhie consequent ch generateupon s for auschange the functiin tohen underl Ai/2 (x).yinIng vari otherablwords, d2 { d2 {l+OO 2 � roo Ai(x) dX } i A p q 2 1/J 2) d1/J } ( 2 dp 2 2q 122/3 p dp +

- 00

=

and make use of Leibnitz's rule. The first differentiation gives and the second 2 i: [Ai'2 ( p + q 2 1/J 2 ) Ai ( p =

+

21 /3 -qAi' (22/3 p) .

+

q 2 1/J 2

) Ail ( p

+

q 2 1/J 2

)] d1/J

(7.33)

Next, we make use of the identity of eqn (7.25) and rewrite eqn (7.33) as 2 i: [Ai/2 (p + q2 1/J2) (p q2 1/J2) Ai2 (p q2 1/J2)] d1/J +

+

+

2 1 /3 (7.34) -qAi' (22/3 p) . The integralso,s overinsertithensquare readyt known from eqns (7.27) and (7.28); g theseofintheqne Airy(7.34)functgiivoesn arethealresul =

i:oo Ai'2 (( ::J 2/3 ( 1 + y21/J2) ) d1/J (3w/4wc) 1 /3 [ 3 Ai/ (XO) 100 . ( AI X)dx] (7.35) Xo 4y 2/ 3 as usual. wiWeth xonow (3w/2wc) substi t ute thesespectrum integralsof[etheqnssynchrotron (7.27) and (7.35)] in eqn (7.5) to give expressi o n for t h e energy radiation, integrated over the whole range of vertical angles: d2 I (w) 3 y � (�) 3 eoc de dw 4{ 4rr[3Ail(XO) 100 . ] Ail(XO) 100 . ] = -

=

+

x o

an

=

We

X

-

Xo

+

o

X

Al (X)dx - [ � + Al(X)dx } . o x

Photon spectral distribution integrated over vertical angles The lasteqnstage(7.3)isand to transform this expression into the form involving Bessel functions. From the definition of Xo we have Ai'-(xo) I K2/3 ( - ) (7.36a) xo = --7r.J3 and (Xl A (x) dx = 1 ;;:;- (XI ,JX K l/3 (� x 3 /2) dx (7.36b) 3 ho 7rv 3 ho ifx thexvariable of integration y is given by y �x 3/2 so that dy = Jx dx then, when the Airy integral [eqn (7.36)] can be rewritten as 1 22

w

We

i

=

o, YO

=

w /we,

1Xu00 Ai(x) dx

=

=

I

;;:;-

7r v 3

1W00/Wc Kl/3 (y) dy .

We now make use of the recurrence relation for the Bessel functions:5 K v- l (Y) - Kv+ l (Y) = -2 K� (y) , K-v (Y) Kv (Y); =

(7.37)

(7.3 8)

so, using eqn (7.38) we may write the Airy integral in eqn (7.37) as 1Xo00 Ai(x) dx - 7r 2v 3 1W00/Wc K�j3 (Y) dy 7rv 3 100 K5j3 (Y) dy. (7.39) The in eqn (7.39) can be integrated if we remember that K2/3 (Y) 0 as first termso that =

Y

--+

00,

;;:;-

-

I

M

W/Wc

--+

and we reach the result for the photon spectrum integrated over all vertical angles: (7.40)

Intioneqnand(7.40) the firsttermtermthedescribes they polarized spectrumradiation. of the horizontally polarized radia­ the second vertical l The integral in eqn (7.40) culated numerically. Values of6 arethe tabul integralateditselin fTablandethe7. 1.integral multiplied bymust be calcalculated using Mathematica w / we

Photon spectral distribution integrated over vertical angles 7.1 Values of J::(wc KSj3 ( Y) dy and (w/wc) J::(wc KSj3 (Y ) dy as a function of w/wc and are shown graphically in Fig. 7.1

1 23

TABLE

0.2 / 3 1 39 0.267 196 0.304575 0.333962 0.358497 0.3797 1 5 0.398497 0.4154 0.430802 0.444973 0.547239 0.61 3607 0.662796 0.70 1 572 0.733248 0.759722 0.782199 0.801493 0.8 1 8 1 86 0.903386 0.9 1 7705 0.901937 0.8708 1 9 0.83 1 475 0.787875 0.74241 3 0.696603 0.65 1423 0.301636 0.1 28566 0.052827 0.02 1 248 0.008426 0.003307 0.001 288 0.000498 0.0001 9224

0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0. 1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 2 3 4 5 6 7 8 9 10

0.0002 1 3 1 4 0.00053439 0.00091 373 0.00 1 33585 0.001 79249 0.00227829 0.00278948 0.0033232 0.00387722 0.00444973 0.01094478 0.01 840821 0.0265 1 1 84 0.0350786 0.04399488 0.0531 8054 0.06257592 0.0721 3437 0.08 1 8 1 86 0. 1 806772 0.2753 1 15 0.3607748 0.4354095 0.498885 0.55 1 5 1 25 0.5939304 0.6269427 0.65 1423 0.603272 0.385698 0.2 1 1 308 0. 1 0624 0.050556 0.023 149 0.010304 0.004482 0.001 9224

The spectrum summed over both states of polarization is given by eqn (7.41 ) (7.41) d�2:w = ;; Y ( 4::0C ) (: ) L:c KSj3 (y) dy where the units are now s/rad into unit frequency bandwidth. J

Photon spectral distribution integrated over vertical angles

124 e bfJ

10-1

0

10-2

B .S

05

OJ

'-

� ;> OJ ;:I

'"

-

(a)

/

(a) fK

5/3 (x)dx

V (b)

10-3 10-4

10-3

10-2

Fig. 7.1 Values of (a) of wjwc.

10-1

10

J::;wc K5/3 (X) . E Y ax -

'

The increment f>.s i s equal to the increment f>.l along the orbit times the cosine o f y which is J 1 - yl2 which can be set equal to 1 because y l2 is a second order quantity so that dl/ds � 1 and, in the same approximation, y

" = d2 y = Ky (s)y. ds 2

( 1 0. 1 2)

As we would expect, the focusing introduced by the gradient function is opposite in sign in the x- and y-directions. What is, perhaps, unexpected is that the dipole magnetic field generates a focusing force, proportional to G (s) 2 in the x-s plane as seen in eqn ( 1 0. 1 0). In general this force is smaller than that produced by the quadrupole magnets.

Electron beam traj ectories

We are now in a position to trace the progress of a particle through the magnetic fields of the storage ring. Equations ( 1 0. 1 0) and ( 1 0. 1 2) can both be written in the form

where, in the

u " + K (s)u = 0,

( 10. 1 3)

f>.E u " = x" - G (s) -

( 1 0. 14)

x-s plane, Eo

and

K (s)

=

_ [G(s) 2 + KAs) ]

( 1 0. 1 5)

and in the y-s plane

u " = y"

and

K(s) = - Ky (s).

In any particular element of the storage ring, whether it be a quadrupole magnet, a dipole magnet or a drift space, K (s) is constant and eqn ( 1 0. 1 3) has the form of the harmonic oscillator equation in which -JK plays the role of the frequency of the oscillation. For a focusing quadrupole, for which K > 0, two particular solutions for u are of the form Ao cos -JKs and Bo sin -JKs. In these solutions, Ao and Bo are

Electron beam trajectories

1 63

constants which depend on the position and direction of the particle as it enters the region of magnetic field defined by K (s). We would like to obtain a general solution which would apply to any particle in the region, so we combine these two independent solutions and write, for the generalized position of the particle at s,

u = Ao cos .JKs + Bo sin .JKs. Clearly the generalized direction of the particle at s will be

u ' = -Ao.JK sin .JKs + Bo.JK cos .JKs. At the entrance to the field region s = 0, the boundary conditions u serve to fix the values of Ao and Bo so that

u(s) = Uo cos .JKs +

u' �

= Uo and u' = u�

sin .JKs,

( 1 0. 1 6)

u ' (s) = -uo.JK sin .JKs + u� cos .JKs.

When K < 0, the quadrupole is defocusing and .jK must be replaced by i.JTKT and the corresponding solutions are

u (s) = uo cosh /iKTs +

u' � sinh /iKTs, K

vl I u ' (s) = uo/iKT sinh /iKTs + u� cosh /iKTs. If we denote the position and direction of the particle by the matrix

( 10. 17)

[�,��]

then we

can write the solutions ( 10. 1 6) and ( 10. 17) in the form of a matrix equation

U (S) ] uo(S) [u'(s) = [u�(s) ] M

and M is given by M

=

[

cos -JKs

- .jK sin -JKs

for a focusing quadrupole and M

=

[

cosh .JTKTs

.JTKT sinh .JTKTs

( 10. 1 8)

]

�

sin -JKs

�

sinh .JTKTs

cos -JKs

cosh .JTKTs

( 1 0. 1 9)

]

for a defocusing quadrupole. In the simple case where the energy of the particle is equal to the ideal energy Eo, !:J,.E = 0 and u and u' can be replaced by x and x ' or y and y ' [see eqn ( 1 0. 14)] and K by Kx or K y to obtain the expressions for the horizontal or the vertical motion.

Behaviour of the electron beam in a synchrotron radiation storage ring

1 64

The problem of tracing an individual particle through the magnetic lattice of the storage ring has been reduced to the multiplication of the matrix for a given initial position and direction of the particle by a succession of matrices describing the mag­ netic fields and drift spaces between them. In such a drift space between magnets the appropriate transformation matrix is ( 10.20)

To illustrate the effect of this matrix on the particle trajectory, we expand the equation

U(s ) ] [ s ] [uo(UO (S)s)] [u'(s) =

1 0

1

to give =

u(s) uo(s) + uo(s)s

and

=

u'(s) uo(s)

showing, as we would expect, that the effect of a drift space is to leave the gradient unchanged and to shift the position of the particle laterally by an amount depending on the product of the gradient of the particle multiplied by the length of the drift space. The transformation matrix for a dipole magnet can be obtained by substitution of eqn ( 1 0. 1 5) with Kx = 0 into the matrix of eqn ( 1 0. 19) to give M

=

[ -G Gs

�

cos

cos

sin Gs

GS ] Gs

sin

in the horizontal plane. In the vertical plane the matrix for the dipole magnet is the same as that for a drift space. In general, bearing in mind eqns ( 1 0. 1 6) and ( 10. 1 7), the transformation matrices are often written as M

=

[C'(C(Ss)) S'S((Ss)) ]

C(s) S(s)

( 10.2 1 )

C' S'

where and are cosine-like and sine-like functions and and are their first derivatives. These functions correspond, in the case of a focusing quadrupole, to the standard ray-tracing trajectories shown in Fig. 8.5. An important property of all these matrices is that their determinant is equal to unity. In other words IMI

=

l.

( 10.22)

This is a consequence of the fact that the forces acting on the particle in the storage ring are non-dissipative, i.e. they do not lead to any loss of energy by the particle.

1 65

Phase space

Phase space

The matrices which transform the panicle traj ecto ry

from one point in the storage ring

to another operate on the position and direction of the particle, denoted by

It i s useful to describe th es e operat ions graphically using

u

and

u'.

the concept of phase spac e .

We are a l l familiar w ith depiction of a particle a s a straight l i ne starting at the posi t io n

(u . s) of the p artic le and whose directi on is (u I , s ) . Thi s pictorial representation is easy to visualize but does not take into account an important propert y of the particle

motion as we s h a l l see.

When

we make

point whose can

usc

of t he

coordinates

are

phasc s pace ( u , U ) . The

description, the particle is represented by

a

ad vantage of lh is representation is that we

usc it to desc ribe the motion of an assemblage of part icles at a particular position '

around the ring. If we consider a plane which intersects the ideal orbit at any arbitrary

point around the ring then we can provide a

plotting the

Figure

u

and

1 0.4,

u'

picture

of the particle distribution by

coordinates of each panicle at that point.

on the left, shows an extended source of partic les. These can be pho­

tons, electrons, or any kind of radiation. Their n at u re does not matter. The so u rc e has a w idth 2w and emits radiation over a l l

forward tingles. The radiation emission is l i m i ­

ted h y a n aperture. also 2w wide (th i � is a co n ve ni ent simpl i fication. any

width would ce ntre of

do), placed downstream from the source, symmetrical ly with respect to the

the source al

C.

Again, the symmetry doc� not matter, it just makes the description

simpler. The right-hand side of Fig. 1 0.4 shows the

phase space representation or this .� ource, comprising the radiating surface and the aperture. How i s this representation constructed? Particles emitted from the point A of the surface must follow a traj ectory which l ies hetween AA' and AA!' if they are to pass through the aperture. Consi der the extreme ray AA' which starts out from th e top edge of the source and skirts the edge of the aperture. This ray has the posi lion coordi nate {I' =

u

n. In the phase diagram this ray i s plotted a� the point P

B"

S

/I

'

=

+w

( + 11: . 0)

+2111/d

p

t'ig. 10.4

-21V1d

Q

Phase space �kscrip\i(ln 01