The Principles of Circular Accelerators and Storage Rings

This book is a basic introduction to the principles of circular particle accelerators and storage rings, for scientists, engineers and mathematicians. Particle accelerators used to be the exclusive province of physicists exploring the structure of the most fundamental constituents of matter. Nowadays, particle accelerators have also found uses as tools in many other areas, including materials science, chemistry, and medical science. Many people from these fields of study, as well as from particle physics, have learned about accelerators at various courses organised by CERN, the European Organisation for Nuclear Research, which has established a reputation as the world's top accelerator facility. Kjell Johnsen and Phil Bryant, the authors of this book, are distinguished accelerator physicists who have also run the CERN Accelerator School. The text they present here starts with a historical introduction to the field and an outline of the basic concepts of particle accelerators. It goes on to give more details of how the transverse and longitudinal motions of the particle beams can be analysed, including treatments of lattice design, compensation schemes, phase focusing, transition crossing, and other radio frequency effects. Operational and diagnostic techniques and the optimisation of luminosity are discussed in detail. One chapter is devoted to synchrotron radiation and the special features of synchrotron light sources. Although the book emphasises circular machines, much of the treatment applies equally to linear machines and transfer lines. The book will be an essential reference for anyone working with particle accelerators as a designer, operator or user, as well as being a good preparation for those intending to go to the frontiers of accelerator physics.

The Principles of

Circular Accelerators and Storage Rings Philip J. Bryant CERN, Geneva, Switzerland

Kjell Johnsen Formerly of CERN, Geneva, Switzerland

CAMBRIDGE UNIVERSITY PRESS

PUBLISHED BY THE PRESS SYNDICATE OF THE UNIVERSITY OF CAMBRIDGE The Pitt Building, Trumpington Street, Cambridge, United Kingdom CAMBRIDGE UNIVERSITY PRESS The Edinburgh Building, Cambridge CB2 2RU, UK 40 West 20th Street, New York NY 10011-4211, USA 477 Williamstown Road, Port Melbourne, VIC 3207, Australia Ruiz de Alarcon 13, 28014 Madrid, Spain Dock House, The Waterfront, Cape Town 8001, South Africa http ://www. cambridge.org © Cambridge University Press 1993 This book is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 1993 First paperback edition 2005 A catalogue record for this book is available from the British Library Library of Congress cataloguing in publication data Bryant, Philip J. The principles of circular accelerators and storage rings / Philip J. Bryant, Kjell Johnsen. p. cm. ISBN 0 521 35578 8 hardback 1. Particle accelerators. 2. Storage rings. I. Johnsen, Kjell. II. Title. QC787.P3B79 1993 539.7'3—dc20 92-17737 C1P ISBN 0 521 35578 8 hardback ISBN 0 521619696 paperback

The Principles of Circular Accelerators and Storage Rings

CONTENTS

Foreword Coordinate system Symbols Useful constants 1 Introduction

xv xvii xix xxv 1

1.1 Direct-voltage accelerators 1.1.1 Cockcroft-Walton rectifier generator 1.1.2 Van de Graaff generator 1.1.3 Tandem electrostatic accelerator 1.2 Accelerators that use time-varying fields 1.2.1 Principle of the Wideroe rf linear accelerator 1.2.2 Fixed-frequency cyclotron 1.2.3 Synchro-cyclotron or frequency-modulated cyclotron 1.2.4 Betatron 1.2.5 Synchrotron 1.2.6 The linear accelerator revisited

3 3 3 4 6 6 7 7 9 10 12

1.2.7 Other accelerators 1.3 Storage rings 1.4 Linear colliders 1.5 Concluding remarks

13 13 15 15

2 Basic concepts and constant-gradient focusing

17

2.1 Cyclotron motion

17

2.2 Transverse motion

19

2.2.1 Radial motion

19

2.2.2 Vertical motion

21

2.3 Solutions

21

vn

Contents 2.4 Stability

22

2.5 Acceptance and emittance 2.6 Momentum compaction 2.7 Historical note

23 25 26

3 Alternating-gradient focusing 3.1 A segment of a magnet as a focusing element 3.1.1 Relation between vertical and horizontal focusing 3.1.2 Point lens 3.1.3 Doublet 3.2 Simple description of an alternating-gradient accelerator

28 30 32 34 36

3.2.1 Stability criterion 3.2.2 'Necktie' stability plot 3.3 Edge focusing 3.3.1 Hard-edge model 3.3.2 Edge focusing in the plane of bending 3.3.3 Edge focusing perpendicular to the plane of bending 3.3.4 General formulation of edge focusing 3.4 Motion with a momentum deviation 3.4.1 Equations of motion 3.4.2 Local dispersion function 3.4.3 Momentum compaction in alternating-gradient lattices 3.5 General remarks

37 38 40 41 42 44 46 46 47 49 50 52

4 Parameterisation of the transverse motion

Vlll

28

53

4.1 Parameterisation 4.1.1 Generalised transfer matrix

53 55

4.1.2 The transfer matrix for a periodic lattice 4.2 Invariant of the unperturbed motion 4.3 Propagation of the Courant and Snyder parameters

56 57 58

4.3.1 3 x 3 matrix solution 4.3.2 2 x 2 matrix solution 4.4 Emittance and acceptance 4.5 Distinctions between circular machines and transfer lines 4.5.1 Circular machines 4.5.2 Transfer lines 4.6 Motion with a momentum deviation 4.6.1 Principal trajectories 4.6.2 Use of principal trajectories to express the dispersion function for a circular machine 4.6.3 Dispersion invariant

58 58 59 61 61 63 63 63 64 66

Contents 4.7 A simple approach to lattice design

67

4.7.1 FODO matched cells in arcs and straight sections 4.7.2 Dispersion suppressors embedded in a FODO lattice 4.7.3 Matching 4.7.4 Injection and extraction 4.8 General remarks

68 73 79 87 89

5 Imperfections and resonances 5.1 Closed-orbit distortion from dipole kicks 5.1.1 Qualitative description of a closed orbit 5.1.2 Closed-orbit bumps 5.1.3 Harmonic response of the closed orbit 5.2 Gradient deviations 5.2.1 Stability and tune shifts 5.2.2 Betatron amplitude modulation 5.3 Weak linear coupling 5.3.1 Coupling in uniform skew quadrupole and longitudinal fields: basic equations 5.3.2 Normal modes 5.3.3 Observations with coupling 5.3.4 Results from a more exact analysis 5.3.5 Vertical dispersion and compensation schemes 5.4 Non-linear resonances 6 Chromaticity 6.1 Chromatic effects 6.2 Evaluation of the chromaticity 6.2.1 Chromaticity and natural chromaticity 6.2.2 A more careful evaluation of the chromaticity 6.2.3 Higher orders of the chromaticity 6.3 Adjusting the chromaticity 6.3.1 Compensation of the natural chromaticity of a quadrupole 6.3.2 A simple scheme for chromaticity control 6.4 The w-vector formulation of chromatic effects 6.4.1 Basic theory 6.4.2 The w-vector in a FODO lattice 6.4.3 Strategy for the chromaticity correction in a collider 6.5 Analytic expressions for the chromatic variables 6.5.1 Approximate expressions for a and b 6.5.2 Exact expressions for a and b 6.5.3 Summation over a series of sextupoles

91 91 93 95 99 100 100 101 103 104 109 112 114 115 117 120 120 120 120 123 124 124 124 125 126 126 129 129 131 131 131 132 ix

Contents 6.6 Planning sextupole families

134

6.6.1 Interleaved schemes 6.6.2 Non-interleaved schemes 6.7 Non-linearities

134 137 138

7 Longitudinal beam dynamics 7.1 7.2 7.3 7.4

139

Betatron acceleration Basic accelerating cavity for synchrotrons Travelling wave representation of the accelerating Equations for the longitudinal motion

7.5 Phase stability 7.5.1 General considerations for phase stability 7.5.2 Separatrices; the boundaries of stability 7.5.3 Longitudinal acceptance and emittance 7.6 Small-amplitude deviations 7.7 Close to transition 7.7.1 7.7.2 7.7.3 7.7.4

Above transition and after the phase has been shifted Below transition and before the phase has been shifted Phase shift at the right moment What can typically go wrong?

field

140 142 143 145 148 148 150 154 156 160 161 161 161 162

7.7.5 Some practical examples 7.7.6 Special use of transition energy 7.8 General remarks

165 166 168

8 Image and space-charge forces (transverse)

170

8.1 Theoretical context 8.2 Components of the space-charge force

170 173

8.2.1. Self 8.2.2 Image

fields fields

173 175

8.2.3 Boundary conditions and coefficients 8.2.4 Tune shifts in a coasting beam 8.3 Bunching, neutralisation and practical structures 8.3.1 Bunching 8.3.2 Neutralisation 8.3.3 Practical structures 8.4 Evaluation of the image coefficients 8.4.1 General relationships 8.4.2 Parallel-plate geometry 8.4.3 Coefficients for simple geometries

175 179 181 181 181 182 182 182 183 184

8.5 Measurements and applications

186

Contents 9 Coherent instabilities

188

9.1 General description 9.2 Classification of the coherent beam modes 9.2.1 Classification

188 191 191

9.2.2 Fast and slow waves 9.3 Wake fields and coupling impedance 9.3.1 Delta pulse excitation and wake potential 9.3.2 Sinusoidal excitation 9.4 Longitudinal instability in a coasting beam 9.4.1 Description of the negative-mass instability 9.4.2 Specimen calculation of the longitudinal impedance

194 195 197 198 200 200 201

9.4.3 Longitudinal stability with a momentum spread 9.4.4 Longitudinal stability criteria 9.5 Transverse instability in a coasting beam 9.5.1 Transverse stability with a momentum spread 9.5.2 Transverse stability criteria 9.6 General remarks

204 210 212 212 215 217

10 Radiating particles 10.1 10.2 10.3 10.4 10.5

10.6 10.7

10.8

10.9

220

Power radiated by a relativistic charge Angular distribution Frequency spectrum Quantum emission Damping of synchrotron oscillations 10.5.1 Damped phase equation

221 223 224 226 228 228

10.5.2 Evaluation of damping constant Quantum excitation of synchrotron oscillations Damping of betatron oscillations

230 233 235

10.7.1 Zero dispersion at cavity 10.7.2 Finite dispersion at cavity 10.7.3 Anti-damping due to the macro-effect of the energy loss Quantum excitation of the betatron motion 10.8.1 In the absence of dispersion (vertical motion) 10.8.2 In the presence of dispersion (horizontal motion) Damping partition numbers 10.9.1 Increase of U7

235 238 238 242 242 244 247

10.9.2 Design a suitable value of D 10.10 Radiation integrals 10.11 High-brightness lattices

248 248 248 249 xi

Contents 11 Diagnosis and compensation 11.1

Closed orbit 11.1.1 Prognosis for random errors 11.1.2 Obtaining the first turn 11.1.3 Algorithms for correcting circulating beams 11.1.4 Measurements and diagnosis Tune measurement 11.2.1 The'kick method'in the time domain

253 253 253 257 261 262 262

11.2.2 The 'kick method' in the frequency domain 11.2.3 RF knockout, swept-frequency and g-diagram meter

263 265

11.2.4 Schottky noise Beam transfer function Profiles 11.4.1 Emittance 11.4.2 Scraper scans 11.4.3 RF scans

266 270 270 270 272 275

11.5

Lack of reproducibility

276

11.6

11.5.1 Hysteresis errors 11.5.2 Fringe field effects 11.5.3 On-line field display system RF manipulations

276 278 279 280

11.6.1 Beam control by phase lock and radial steering 11.6.2 RF matching 11.6.3 Bunch length measurements

280 281 284

11.2

11.3 11.4

12 Special aspects of circular colliders

285

12.1

Energy relations

286

12.2

Luminosity 12.2.1 Two ribbon-shaped beams crossing at an angle 12.2.2 Bunched beams in head-on collision

288 288 292

12.2.3 12.2.4 12.2.5 12.2.6 12.2.7

295 295 297 300 301

12.3

xn

252

Some general remarks on luminosity formulae The ISR approach Proton-antiproton colliders Future hadron colliders Electron-positron colliders

12.2.8 Electron-proton colliders Summary of some other effects 12.3.1 Current and luminosity lifetime 12.3.2 Vacuum 12.3.3 Intra-beam scattering

302 303 303 303 305

12.3.4 Beam-beam effect

306

Contents

12.4

12.3.5 Aperture

307

12.3.6 Stored beam energy Conclusion

307 308

Appendix A Transverse particle motion in an accelerator

309

(i) General formulation

309

(ii) Tailoring the Hamiltonian to a circular accelerator

311

(iii) Conservation of phase space (iv) Effects of small terms and approximations on the motion

313 314

Appendix B Accelerator magnets (i) (ii) (iii) (iv) (v)

Multipole expansion of a 2-dimensional magnetic Dealing with a 3-dimensional magnet Rotating-coil measurements Nomenclature for magnet measurements Practical lenses (a) Conventional accelerator magnets (b) Superconducting accelerator magnets (c) Twin-bore superconducting magnets

317 field

317 318 320 322 323 323 323 326

Appendix C Closed orbits

328

Appendix D Phase equation

329

(i) General derivation of the phase equation (ii) Choice of alternative variables (T, E) (iii) Choice of alternative variables (©, W) Appendix E Vlasov equation

329 332 333 335

References

337

Index

353

xni

FOREWORD

A dictionary definition of acceleration is an increase in speed* from which one understands that a charged-particle accelerator would increase the speed of charged particles - as indeed it does. However, today's accelerators work at ultra-relativistic energies and it is not so much the particle's speed that increases as its mass. For example, between 1 MeV and 1 GeV an electron gains speed modestly from approximately 95% of the speed of light to what is virtually the full value, but its mass leaps forward from approximately three times its rest value to around 2000 times. This anomaly led Ginzton, Hansen and Kennedy! to propose the names mass aggrandiser or ponderator, but neither became fashionable. More strictly one should speak of a momentum aggrandiser, but since this is sure to be as unfashionable as the others, we are left with the simple name accelerator. The accelerator family is, however, very large, so the authors will concentrate on synchrotrons and storage rings with only brief references to linear accelerators and many of the early circular machines. Although universities often include some lectures on accelerators in their physics courses, there are very few courses which can claim to be principally about accelerators. The machines and the expertise in this field are mainly in national and international laboratories. Since these laboratories have a more mission-orientated approach than universities, relatively few books have been written and the accelerator community has relied heavily on a 'learning-by-working apprenticeship' for newcomers and on personal contacts and conferences for the dissemination of knowledge. The subject has not been stationary. Machine designs continuously develop, industrial applications increase and the need for more trained personnel is ever-present. The progress in this field has been such that even at an elementary level there are many differences between a text book written now and one produced 10 years ago. This is a basic book on circular accelerators and storage rings, but as basic * The new national dictionary, W. Collins Sons & Co. Ltd., London & Glasgow (1966). t E. L. Ginzton, W. W. Hansen and W. R. Kennedy, Rev. Sci. Instr., vol. 19, No. 2 (1948), p. 89.

XV

Foreword is a relative term we shall try to describe what is meant by this. It is assumed that the reader starts with either a first degree knowledge of physics or electrical engineering. Using simple mathematical tools, the book then aims to treat mainly the single particle and linear theory of accelerators for the transverse and longitudinal phase planes. However, some of the simpler collective effects are also included and in the Appendixes some effort is made to prepare the way for those readers who will wish to carry on to more advanced topics. For example, the basic equations for the transverse and longitudinal motions are derived in an intuitively straightforward manner in the main part of the book, but in Appendixes A and D these derivations are repeated using the Hamiltonian formalism. This adds little to the results of the elementary treatment for the newcomer to the field, but it is the stepping stone needed for analysing non-linear resonances, dynamic aperture, stochasticity, chaos, etc. The validity of certain commonly-made approximations is also discussed. The understanding of these finer points is essential when preparing for more advanced work. Space charge and image forces and coherent instabilities in coasting beams are included, but the extremely complex and large field of coherent instabilities in bunched beams is omitted. The Vlasov equation is mentioned, but it is not widely applied. Collective processes with diffusion, such as the action of radiofrequency noise on full buckets and stochastic cooling, which can be treated by the Fokker-Planck equation, are also omitted as being beyond elementary. We have written this book with the main idea of helping those who are just entering the field of accelerators and have no previous experience of the subject. However, it should also be helpful to those who have experience in some accelerator speciality and now wish to broaden their knowledge and much of the book should be useful to accelerator users. Whatever the situation may be, we hope that this book will be of interest and of use to you and that you will derive as much satisfaction from reading it as we have had from writing it. In producing this book, we have been greatly helped by our colleagues in CERN and the many contacts we made through the CERN Accelerator School. We are especially grateful to W. Hardt for his thorough and critical reading of the manuscript and the many important improvements that he proposed. H. G. Hereward, in particular, contributed significantly to the clarification of specific problems and G. Guignard made many constructive suggestions. We also wish to acknowledge the helpful discussions we have had with O. Grobner, A. Hofmann and G. Dome. The labour of producing and labelling the diagrams was shared by S. M. Bryant to whom we give our thanks. We are indebted to CERN and its Directors General for the support given during the period of preparation of the manuscript and for permission to make generous use of some extracts from CERN Reports. The authors

xvi

COORDINATE SYSTEM

The curvilinear coordinate system following the central orbit (x, s, z) is shown in the diagram below.

Average azimuthal angle

Machine centre

Local centre

of gyration

Radius of gyration ft x Radial

Equilibrium orbit i 3 Tangential to beam direction

The particle motion is defined with respect to an equilibrium orbit by means of a right-handed curvilinear coordinate system x, s, z. The azimuthal coordinate s is directed along the tangent of the orbit. The local radius of curvature p and the bending angle s/p subtended by the equilibrium orbit are defined as positive for anticlockwise rotation when viewed from positive z. In the vertical plane, radii of curvature and bending angles are defined as positive for anticlockwise rotation viewed from positive x. Anticlockwise rotation is assumed throughout the book unless otherwise stated. The above can be summarised as: p is positive when bending to the left or upwards when looking along the beam direction. This has the consequence that the local, radial xvn

Coordinate system coordinate x is positive to the outside of a positive bend, while the local, vertical coordinate z is positive to the inside of a positive bend. Special care is needed to account for the effect of this sign asymmetry in the transfer matrices of the accelerator elements. When relating the beam to an external reference system, or when defining a transfer line, the sign convention must be rigorously applied. However, when dealing with only the optical parameters of a ring the sense of rotation is unimportant and it is usual to use a simplified convention that can be expressed as: p is positive when bending towards the ring centre or upwards when looking along the beam direction. The general transverse coordinate y is frequently used to represent x or z. In some applications, an average azimuthal angle 0 = 2ns/C is used where C is the circumference of the equilibrium orbit and 0 is defined as positive for anticlockwise rotation.

xvin

SYMBOLS

SPECIAL SYMBOLS AND ±, +

' < > Av A

* © O

CONVENTIONS

the upper sign corresponds to the horizontal (x, s) plane and the lower sign to the vertical (z, s) plane in expressions containing y the general transverse coordinate differentiation w.r.t. to time, t differentiation w.r.t. a specified variable average value average over a distribution maximum and minimum values also indicates the complex amplitude of a phasor an extreme value (maximum or minimum), also used for the Fourier Transform, and in accordance with common usage for D the lattice damping constant complex conjugate field vector or particle motion entering or leaving the plane of the paper rounded brackets denote a matrix and T denotes transpose

Tr suffix 0 bold type PI PV Re, Im

trace of a matrix denotes a reference value denotes vector or matrix particular integral Cauchy principal value of an integral real, or imaginary, part of a parameter or function xix

Symbols ARABIC SYMBOLS a

chromatic variable, also used as horizontal beam radius for an elliptical

beam [m] magnetic vector potential [T • m] or [V • s • m ~ x ] transverse acceptance [m], also used as an amplitude function [m], and as a constant At longitudinal acceptance [eV/c] or [eV • s] b chromatic variable, also used as vertical beam radius for an elliptical beam [m] B magnetic induction [T], also used unscripted as a constant Bp magnetic rigidity [T • m] c speed of light [m • s ~ x ] C = Cq + j Cb complex coupling coefficient C = 2KR circumference of an equilibrium orbit [m], also used as capacitance [Farad] # 'Cosine-like' principal trajectory

A A

D =

transverse dispersion function [m]

D D = (dnn) e E £, Eo / f(0) f(p)

the lattice damping constant diagonal matrix unit electronic charge [A • s] electric field strength [V-m" 1 ] total energy and rest energy of a particle [eV] focal length of a lens [m] dipole error distribution p3/2AB/(B0p0) [m*] beam distribution function in momentum space, j f(p) dp = N, number of particles in the beam revolution frequency [Hz] force [N] harmonic coefficients of the closed orbit distortion and the dipole error function /(oL

0(s0) =

critical wavelength [m] or [A] modified betatron wavelength [m] bunching factor (average current/peak current), also used as the charge in a bunch [ A s ] , and as a switch between sector (A = 1) and straight (A = 0) dipoles

fso ds

phase advance of the betatron oscillation [rad]

betatron phase advance in a lattice period [rad], also used as the permeability of vacuum [ H m " 1 ] image space-charge coefficients for the electric and magnetic images for the coherent motion local bending radius of trajectory [m], also used as resistivity [Q • m] distance (similar to s) [m], also used as standard deviation of a distribution

propagation coefficient for chromatic variables, also used as crosssection for a particle interaction damping time constant [s], also used for lead or lag of a particle w.r.t. the rf phase [s] electric scalar potential [V], or magnetic scalar potential [T-m], also used as a phase angle

fso ds

X = (Y2 + Y'2)*

normalised betatron phase advance [rad] an amplitude in normalised coordinates, also used as an integration variable

\jj

normalised distribution function,

i/^(x) Ax = 1

co corf a>s CDC Q

general angular frequency [ s " 1 ] rf angular frequency [ s ~ x ] synchrotron oscillation angular frequency [s" 1 ] critical angular frequency [s" 1 ] general revolution angular frequency [ s " 1 ] , also used for solid angles [steradian] cyclotron angular frequency [s" 1 ]

J — 00

Qc

Where symbols do not conform to the above lists they are defined in the text. xxiv

USEFUL CONSTANTS

+ 1.602 177 x 10

19

[As]

e, unit electronic charge (upper sign protons, lower sign electrons) 8 1 2.997 924 58 x 10 [ m s " ] c, speed of light in vacuum (exact) m0, rest mass of electron 9.109 390 x 10" 3 1 [kg] 1.672 623 x 10~ 27 [kg] m0, rest mass of proton 0.510 999 [MeV] equiv. energy of rest mass of electron 938.272 3 [MeV] equiv. energy of rest mass of proton 15 re, classical radius of electron 2.817 941 x 10~ [m] 1.534 699 x 10" 1 8 [m] rp, classical radius of proton [rCtP = e2/(4nsomoc2)~] 7 1 fi0, permeability of vacuum An x 10~ [ H - m " ] 12 1 8.854 188 x 10" [ F - m " ] e0, permittivity of vacuum [£0 = l/(juoc2)] 376.730 [Q] Z o , impedance of vacuum [Z o = l/(e o c )] 34 1.054 573 x 10" [J-s] ft, Planck's constant divided by 2n

xxv

CHAPTER 1

Introduction

Nuclear physics research was the birth-place of charged-particle accelerators and for many decades their main 'raison d'etre'. This has given them a somewhat specialised and academic image in the eyes of the general public and indeed accelerators and storage rings do provide an extremely rich field for the study of fundamental physics principles. However, this academic image is fast changing as the applications for accelerators become more diversified. They are already well established in radiation therapy, ion implantation and isotope production. Synchrotron light sources form a large and rapidly growing branch of the accelerator family. The spallation neutron source is based on an accelerator and there are many small storage rings for research around the world relying on sophisticated accelerator technologies such as stochastic and electron cooling. In time accelerators may be used for the bulk sterilisation of food and waste products, for the cleaning of exhaust gases from factories, or as the drivers in inertial fusion devices. During the first third of our century, natural radioactivity furnished the main source of energetic particles for research in atomic physics. Let us mention a famous example. At McGill University, Montreal, Canada, in 1906, Rutherford bombarded a thin mica sheet with alpha particles from a natural radioactive source. He observed occasional scattering, but most of the alpha particles traversed the mica without deviating or damaging the sheet. He continued this work at Manchester University, UK, and in 1911 Geiger and Marsden, under Rutherford's guidance, verified Rutherford's theory for atomic scattering. The RBS technique (Rutherford BackScattering) is still a standard technique used at modern particle accelerators. In 1908 Rutherford received the Nobel Prize in chemistry for his investigations into the disintegration of elements and the chemistry of radioactive substances. Natural sources are limited in energy and intensity and it is not surprising that in 1928, Cocker oft and Walton, encouraged by Rutherford, set about the task of building a particle accelerator (see Section 1.1.1) at the Cavendish Laboratory, UK. By 1932 the apparatus was finished and used to split lithium nuclei with 400 keV protons. This was the first fully man-controlled splitting of the atom. From the measurement 1

Introduction

1PeV

synchrotrons

L "near electron accelerators

+ocv$ and Sf\ or vice versa. Consider the identities, (4.35)

—

ds \y\

(4.36)

=

which must be unity, since | 7 | =(95

>•

1 »0

(Values at entry to elements) ax a* az Cm]

CmD

0.0000 19.996 -0.9998 19.996 -7.9452 35.432-10.6127 35.432 0.1838 34.274 -0.0017

i o.ooo

34.350

0.0000

5. 000 0.0000 24.991 -1.9996 24.991 6.6811 15.140 5.1622 15.140 0.5487 . 11.637 0.0024 11.640

0.0000

I

•

1-05 A / f Dx Cm:

'*

0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

Figure 4.10. Search for a low-/? solution with a thin-lens doublet. (Matching: low-/?, fix = 10 m, fiz = 5 m, a xz = 0 to half-strength F-quadrupole terminating FODO cell with fix = 34.35 m, 'j5z = 11.64 m, ax,z = 0.)

a reduced region finds a good solution. A standard lattice program with a minimisation routine will quickly optimise the parameters of the thin-lens solution into the equivalent thick-lens one. Reference 15 describes an application of doublet matching. (ii) Variable-geometry triplet {I) for a, j? matching in a dispersion-free region The quadrupole triplet shown in Figure 4.11 is similar to the doublet analysed in the previous section, inasmuch as it has the same number of free parameters. In this case, it is better to treat the insertion in two stages and to first evaluate the transfer matrix P of the central section and then the overall matrix Q. The details of the derivation will be left to the reader as an exercise, but it is useful to note the following relationships for constructing the sum and difference terms for the matrix Q. 82

A simple approach to lattice design

Main term

Sam term + •

9r)9

n ij • m^ij —'

u..

Difference term Q~.. Q Q+"

Q Q~ •

^n^m^ ij

9n9mQij

Entry

Exit

Pzi.CCzi

Dz2=D'z2=O

Beam Figure 4.11. Variable-geometry triplet I. (The parameters are numbered from g2 in order that the results can be directly used in the next section where drift spaces will be added.)

The final equations for determining the parameters of the insertion are: Ql = (