Table Of ContentPRINCIPLES OF CYCLIC
PARTICLE ACCELERATORS
ARCOXKE IUATlO&.AL LABORATORY
Operated by the C’nhwsity of C’laicayo
for the United States Atomic ihcrgy Commission
PRINCIPLES OF CYCLIC
PARTICLE ACCELERATORS
k?
JOHN J. LIVINGOOD, Ph.D.
Argonne National Laboratory
D. VAN NOSTRAND COMPANY, INC.
PRINCETON, NEW JERSEY
LONDON
TORONTO
NEW YORK
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COPYRIGHT @ 1961 BY
D. VAN NOSTRAND COMPAKY, INC.
The publishers assign copyright to the General Manager of the
United States Atomic Energy Commission to be held by him.
All royalties from the sale of this book accrue to Argonne National
Laboratory operated by the University of Chicago, under contract
with the Unit,ed States Government.
Published simuhaneously in Canada by
D. VAN NOSTRAKD COMPANY (Canada), LTD.
.-
All Rights Reserved
l’his book, or any parts thereof, rnav not be
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approved by the Atomic Energy Commission.
PRINTED IN THE UNITED STATES OF AMERICA
PKEFACE
This book has been written for the novice. It has been planned to fill the gap
bctwecn the existing monographs which give a qualitative picture of cyclic
accelerators and the comprehensive review a,rticlcs which can best be appreci-
ated by those who already have somewhat more than a speaking acquaintance
with such machines. The purpose of the book has been to introduce the subject
from the very beginning and to give the reader a quantitative understanding of
the basic principles of the large variety of particle accelerators now existing.
The emphasis is largely on the dynamics of the particle, in the linear approxi-
mation.
Clarity of exposition has been uppermost in my mind. The aim has been to
present the subject in such a way that the reader knows at all times where the
argument started, where it is going, and when it has arrived at its goal. Sym-
bolism should act as an aid rather than a hindrance, so pains have been taken
t’o make each mathematical step as evident as possible, in order to avoid the
dircrsion of attention from the main objective which occurs when the reader
is puzzled as to how one equation follows from another. No further mathe-
matical skill is required beyond the ability to differentiate and integrate simple
expressions.
Chapter 1 sets the stage and introduces much of the terminology. The pur-
loose of accelerators is stated in a few \vvords! and tile difficulties and limitations
of dc dcviccs are pointed out. The fundamental ideas behind the cyclotron,
synchrocyclotron, synchrotron, bctatron, and linear accelerator are introduced.
The important influence on accelerator behavior of the increase of mass with
velocity is pointed out, and the relations between momentum, velocity, and
total energy are developed in relativist,ic form.
The problem of orbit stability in weak-focusing machines is introduced in
Chapter 2, argued first qualitatively, then quantitatively. The WKB method
is used in discussing adiabatic damping. Chapter 3 is devoted entirely to the
matrix method for determining the criteria for stability and for calculating the
bctatron frequcncitls. This powerful technique finds applications in later pages.
Chapter 4 t,reats of edge focusing, and Chapter 5 considers resonances, largely
from a qualitative standpoint. elciiicntary discussion of the effect of space
lZ~l
rlrarge is iiicludctl.
Phase stability, momentum c’omlrac~tion, and synchrotron oscillations are
the subject matter of Chapter 6. Chaptors 7 through 11 are devoted to the weak-
V
vi PREFACE
focusing cyclotron, synchrocyclotron, synchrotron, betatron, and microtron.
The problems peculiar to each are discussed, quantitatively for the most part.
The description of hardware is only sufficient to give some appreciation of the
problems involved and the magnitude of the structures; no attempt has been
made to write a handbook on machine design.
The alternating-gradient synchrotron is considered in Chapter 12, the em-
phasis being on the criteria for stability, the determination of the betatron
frequencies, and the calculation of the momentum compaction. Chapter 13
deals with fixed-field alternating-gradient machines. Expressions for the beta-
tron frequencies are developed in a very approximate manner, since a rigorous
and detailed analysis would be beyond the planned tenor of the book. This is
followed by a discussion of isochronous cyclotrons, radial-ridge and spiral-
ridge annular accelerators, and the FFAG betatron. The problem of intersect-
qualitative treatment.
Linear accelerators are considered in Chapter 14, and the basic incompat-
ibility of radial stability and phase stability is brought out in detail. The
palliative effects of grids are developed in the paraxial approximation, and
mention is made of the focusing properties of magnetic solenoids and of
quadrupoles and includes a brief discussion of their use in matching the
tance of one accelerator to the acceptance of another. Stochastic acceleration
is considered in the final chapter.
Many items on the subject of cyclic accelerators have been omitted, in
particular nonlinear forces and the effects of errors in construction. It has
been my feeling that with a basic understanding of linear theory, the inter-
ested student will be equipped to consider these more detailed problems by
perusal of the many advanced papers with which the scientific literature
abounds. Nor has any mention been made of the circuitry of oscillators, am-
plifiers, and dee systems. Although many problems of resonating circuits are
peculiar to accelerators, the basic theory of the generation and distribution of
a
vacuum, nothing has been said other than that a good one is required; this
important subject has been handled exhaustively elsewhere. The necessity for
radiation shielding is mentioned, but no theory is given. The speculations on
plasma guide fields and on the novel means of acceleration such as have been
proposed recently by Soviet scientists have all been omitted, since these topics
have not yet solidified sufficiently to warrant inclusion in a book of this
character.
The problem of references is a vexatious one. The existing bibliographies on
accelerators resemble telephone directories, so it would be impractical to re-
PKEF.\(:G vii
by scanning some of the original literature, I have listed most of the articles
which describe existing machines and have included the basic theoretical
~~al~~~rs and many of the general rcvieus. Refercnccs arc listed according to the
type of k)rea,kdown is possible) and, again when
frasible, with further subdivision into p:tl~rs on theory and on hardware. Tbc
title and number of pages are included, so the reader may know whether the
subject is to his interest and may liar-e ;omc idea of the length of the reading
task before him. Listings, in the main, are chronologicnl, since this gives
some clue as to the sophisticat,ion of the argument.
This book had its origin in a set of notes prepared shortly the war,
when several associates and I were engaged in a commercial venture in cyclo-
tron construction at Collins Radio Company. The effort was expanded later
when a series of lectures was given by Dr. &lorton Hamermesh and myself to
arouse enthusiasm among our collc~aguc~s at hrgonnv for the acquisition of a
large proton synrhrotron. A considerably abbreviated volume was prepared
by me in 1957 in order to brief nvw recruits to the Particle Accelerator Divi-
sion then being staffed for the design of the 12-I3cv machine which is now
under construction. I make no claim for originality in what is here presented,
and gratefully acknowledge the help of all the authors listed in the bibliog-
raphy, of the unnamed writers of multitudinous memoranda which have been
distributed privately, and of many people with whom I have had personal con-
ycrsations and correspondence. In partirular. I have borrow-cd heavily from
Dr. Hamermesh’s notes on the alternating gradient principle, and I am obliged
to Dr. E. A. Crosbic and Dr. R/I. H. Foss for My most
heavy debt is to Dr. L. C. Tcng for his many hour,c; of patient explanation and
discussion of knotty problems. Dr. Frarlcis Tl~row has been especially kind in
his careful reading of the manusrript. He has trapped many a dangling par-
ticiple, pointed out many Crrors and ni:.idc numerous ~aluabl(~ contriljutions to
clarity of expression. Finally, I am sincc>rcly al)prcv3ative of the cooperation
of Dr. Norman Hilbcrry, Director of Argonne Laboratory, and of Dr. Louis
Turner, Deputy Director, in affording me the opportunity of carrying this
work to its completion.
CONTENTS
PAGIC
CHAPTER
PRELNISARY CON 81 DERATIOSK
1
l-1. The Reason for Particle Accelerators
2
l-2. -DC Arcclerators
4
l-3. Cyclic Accelerators
6
1-4 . The Linear Accelerator
7
1-5 . The Cyclotron
11
l-6. The
11
Synchrotron
12
1-8 . The Betat’ron
13
1-9 . Relativity
18
l-10 . The Electron -Volt
18
1-I 1. The Rest Energy of Electrons and Protons
19
1-12 . The Size of the Magnet
2. ORBIT STABILITY
22
2- . Introduction
24
2-2 . The Field Index
28
2-3 . Qualitative Stability of Orbits
31
2-4 . Basic Assumptions
32
2-5 . Quantitative Axial Stability
34
2-6 . Quantitative Radial Stability
37
2-7 . Radial Oscillation of an Ion with Momentum p + dp
38
2-8 . The Initial Amplitudes of Betatron Oscillations
39
2-9 . Adiabatic Damping of Betatron Oscillations
3. MATRIX METHOD OF CALCULATING STABILITY
3-l. Introduction 42
3-2. I,inear Transformations by the TTse of Matrices 43
3-3. The Criterion of St,ability and tlw Betatron Frequency 44
3-4. Application to a Synchrotron with Straight Sections 52
3-5. Approximate Values of the Botatron Frequencies r3
3-6, Values of (T and v for Circular hlachines Z4
4. EDGE: FOCTJSING
4-l. Introduction 55
4-2. The Axial Focal Length of an Edge 57
ix
CONTENTS
X
CHAPTER PAGE
4-3. The Radial Focal Length of an Edge 59
4-4. Matrix Representation of a Magnet Edge 60
4-5. The Zero-Gradient Synchrotron 61
4-6. Axial Betatron Frequency in a Zero-Gradient Synchrotron 62
4-7, Radial Betatron Frequency in a Zero-Gradient Synchrotron 63
5. RESONANCES
5-l. Introduction 66
5-2. Coupled Resonances 67
5-3. Imperfection Resonances 69
5-4. Sum Resonances 69
5-5. Resonances in a Race-Track Synchrotron 70
5-6. The Effect of Space Charge 71
6. PHASE STABILITY
6-l. Introduction 76
6-2. The Principle of Phase Stability 77
6-3. Momentum Compaction 80
6-4. The Relation Between Period and Momentum 84
6-5. The Phase Equation 87
6-6. The Analogy of the Biased Pendulum 88
6-7. The Phase Diagram 90
6-8. Permissible Error of Injection Energy 94
6-9. Frequency of Synchrotron Oscillations of Small Amplitude 96
6-10. Adiabatic Damping of Synchrotron Oscillations 96
6-11. Over-all Motion of Ions 100
6-12. Synchrotron Oscillations in a Fixed-Frequency Cyclotron 101
7. FIXED-FREQUENCY CYCLOTRONS
7-l. Energy and Types of Projectiles 102
7-2. Operating Frequency 102
7-3. Output Current of Ions 103
7-4. Magnets 103
7-5. Exciting Coils 107
7-6. Vacuum Chambers 108
7-7. Dees and Drivers 109
7-8. Ion Sources 118
7-9. Early Orbits in Cyclotrons 120
7-10. Transit Time 123
7-11. External Beams 125
7-12. Deflector Calculations 129
7-13. The Separation of Equilibrium Orbits 133
7-14. The Minimum Dee Voltage in a Cyclotron 134
