Principles of Cyclic Particle Accelerators

PRINCIPLES 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 TORONTO

NEW YORK

LONDON

D. VAN NOSTRAND COMPANY, INC. 120 Alexander St., Princeton, New tJersey (Principnl ofice) 24 West 40 Street, New York 18, New York D. VAN NOSTRAND COMPANY , LTD. 358, Kensington High Street, London, W.14, England D. VAN NOSTRAND COMPANY (Canada), LTD. 25 Hollinger Road, Toronto 16, Canada

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.

D.

Published simuhaneously in Canada by VAN NOSTRAKD COMPANY (Canada), LTD.

.All Rights Reserved l’his book, or any parts thereof, reproduced in ang form without mission from the author and the approved by the Atomic Energy

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IN

THE

UNITED

rnav not be written perpublisher as Commission.

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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 appreciated 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 approximation. 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. Symbolism 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 mathematical skill is required beyond the ability to differentiate and integrate simple expressions. Chapter 1 sets the stage and introduces much of the terminology. The purloose 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 find s applications in later pages. Chapter 4 t,reats of edge focusing, and Chapter 5 considers resonances, largely from a qualitative standpoint. lZ~l elciiicntary discussion of the effect of space 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 weakV

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 emphasis 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 betatron 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 spiralridge annular accelerators, and the FFAG betatron. The problem of intersecting beams, obtained by two-way machines and by storage rings, receives only qualitative treatment. Linear accelerators are considered in Chapter 14, and the basic incompatibility 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 quadrupole lenses. Chapter 15 develops the theory of electrostatic and magnetic quadrupoles and includes a brief discussion of their use in matching the emittance 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 interested 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, amplifiers, and dee systems. Although many problems of resonating circuits are peculiar to accelerators, the basic theory of the generation and distribution of radiofrequency power is well treated in many standard texts. With regard to 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 rel)roduce them. On the other hand, they are not available in all libraries. With the hope that the reader of the present volume will gain pleasure and insight

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 arcelerator (when such a 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 after the war, when several associates and I were engaged in a commercial venture in cyclotron 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 Division 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 bibliography, of the unnamed writers of multitudinous memoranda which have been distributed privately, and of many people with whom I have had personal conycrsations 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 nlany informative talks. 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 participle, 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

1. PRELNISARY CON81DERATIOSK l-1. The Reason for Particle Accelerators l-2. -DC Arcclerators l-3. Cyclic Accelerators 1-4. The Linear Accelerator 1-5. The Cyclotron l - 6 . The Synchrocyclotron l-7. The Synchrotron 1-8. The Betat’ron 1-9. Relativity l-10. The Electron-Volt 1-I 1. The Rest Energy of Electrons and Protons 1-12. The Size of the Magnet

1 2 4 6 7 11 11 12 13 18 18 19

2. ORBIT 2-l. 2-2. 2-3. 2-4. 2-5. 2-6. 2-7. 2-8. 2-9.

22 24 28 31 32 34 37 38 39

STABILITY Introduction The Field Index Qualitative Stability of Orbits Basic Assumptions Quantitative Axial Stability Quantitative Radial Stability Radial Oscillation of an Ion with Momentum p + dp The Initial Amplitudes of Betatron Oscillations Adiabatic Damping of Betatron Oscillations

3. MATRIX METHOD OF CALCULATING STABILITY 3-l. Introduction 3-2. I,inear Transformations by the TTse of Matrices 3-3. The Criterion of St,ability and tlw Betatron Frequency 3-4. Application to a Synchrotron with Straight Sections 3-5. Approximate Values of the Botatron Frequencies 3-6, Values of (T and v for Circular hlachines 4 . EDGE: FOCTJSING 4-l. Introduction 4-2. The Axial Focal Length of an Edge ix

42 43 44 52 r3 Z4 55 57

CONTENTS

X

PAGE

CHAPTER

4-3. 4-4. 4-5. 4-6. 4-7, 5.

RESONANCES 5-l. Introduction 5-2. Coupled Resonances 5-3. Imperfection Resonances 5-4. Sum Resonances 5-5. Resonances in a Race-Track Synchrotron 5-6. The Effect of Space Charge

6. PHASE 6-l. 6-2. 6-3. 6-4. 6-5. 6-6. 6-7. 6-8. 6-9. 6-10. 6-11. 6-12. 7.

The Radial Focal Length of an Edge Matrix Representation of a Magnet Edge The Zero-Gradient Synchrotron Axial Betatron Frequency in a Zero-Gradient Synchrotron Radial Betatron Frequency in a Zero-Gradient Synchrotron

STABILITY Introduction The Principle of Phase Stability Momentum Compaction The Relation Between Period and Momentum The Phase Equation The Analogy of the Biased Pendulum The Phase Diagram Permissible Error of Injection Energy Frequency of Synchrotron Oscillations of Small Amplitude Adiabatic Damping of Synchrotron Oscillations Over-all Motion of Ions Synchrotron Oscillations in a Fixed-Frequency Cyclotron

FIXED -FREQUENCY CYCLOTRONS 7-l. Energy and Types of Projectiles 7-2. Operating Frequency 7-3. Output Current of Ions 7-4. Magnets 7-5. Exciting Coils 7-6. Vacuum Chambers 7-7. Dees and Drivers 7-8. Ion Sources 7-9. Early Orbits in Cyclotrons 7-10. Transit Time 7-11. External Beams 7-12. Deflector Calculations 7-13. The Separation of Equilibrium Orbits 7-14. The Minimum Dee Voltage in a Cyclotron

59 60 61 62 63 66 67 69 69 70 71 76 77 80 84 87 88 90 94 96 96 100 101 102 102 103 103 107 108 109 118 120 123 125 129 133 134

CONTENTS

xi P.4GF.

CHAPTER

7-15. Electric Focusing and Defocusing Forces at the Dee Gap 140 142 7-l(j. Variable-Energy Fixed-Frequency Cyclotrons 7-17. Harmonic ()perations of a Cyclotron: Acceleration of 143 Heavy Ions 145 7-18. Multiple Dces and Mode (.Q)eration 147 7-l 9. Shielding 8.

9.

SYNCHROCYCLOTRONS 8 - 1 . [ntroduction 8-2. Frequency Range 8-3. Dee Voltage 8-4. Rate of Frequency Modulation 8-5. Frequency of Synchrotron Oscillations 8-6. External Beams 8-7. Shielding

148 149 150 150 151 152 153

SYNCHROTRONS 9-1. Proton Synchrotrons, Operating and K’ndcr Construction 9-2. Magnets 9-3. Vacuum Chambers 9-4. Frequency Range 9-5. Acceleration by Cavities 9-6. Acceleration by Drift Tubes 9-7. Frcqucncy of Synchrotron Oscillations 9-8. Betatron Action in Synchrotrons 9-9. Influence of Injection Energy 9-10. Injection of Particles into Synchrotrons 9-11. Synchrotron Targets 9-12. External Beams 9-13. Shielding 9-14. Electron Synchrot,rons 9-15. Radiation by Electrons in Circular Accelerators

1.54 154 164 165 167 168 171 172 173 173 178 179 181 181 182

1 0 . BETATROSS 10-l. Introduction 10-2. The Two-to-One Rule 10-3. Flux Change and Energy Gained 10-4. General Description 10-5. The Biased Betatron 11.

MICROTRONS 11-l. Introduction 11-2. Conditions for Resonance

184 184 186 186 187 189 190

xii

CONTENTS PAGE

CHAPTER

11-3. Variability of Energy I l-4. Speculative Elaborations 12.

ALTERNATING-GRADIENT SYNCHROTRONS 12-1. Introduction 12-2. The Stability Diagram 12-3. Betatron Frequencies 12-4. Phase Stability 12-5. The Shape of Equilibrium Orbits 12-6. Momentum Compaction 12-7. The Use of “Half” Magnets 12-8. Existing and Future A G Synchrotrons

193 194 195 197 201 205 208 212 213 214

13.

FIXED -FIELD ALTERNATING-GRADIENT ACCELERATORS 219 13-l. Introduction 222 13-2. Flutter 224 13-3. Thomas Focusing 226 13-4. Spiral Focusing 230 13-5. The Average Field Index k 232 13-6. Betatron Frequencies 233 13-7. Isochronous Cyclotrons 235 13-8. The Conditions for Isochronism 13-9. Radial Stability and Energy Limits in Isochronous Cyclo239 trons 240 13-10. Axial Stability in Isochronous Cyclotrons 13-11. Variable-Energy Multiple-Particle Isochronous Cyclotrons 241 243 13-12. Variable Energy by Means of Harmonic Operation 244 13-13. Existing and Planned Isochronous Cyclotrons 248 13-14. Spiral-Sector Ring Accelerators 250 13-15. Phase Stability in Spiral-Sector Ring Accelerators 251 13-16. Rotation Frequency in a Spiral-Sector Ring Accelerator 252 13-17. Existing and Potential Spiral-Sector Ring Accelerators 255 13-18. Radial-Sector Ring Accelerators 257 13-19. FFAG Betatrons 259 13-20. Center-of-Mass Energy 13-21. Intersecting Beams of Particles 263 264 13-22. Two-Beam Accelerator

14.

LINEAR ACCELERATORS 14-1. The Widerae Linear Accelerator 14-2. The Alvarez Linear Accelerator 14-3. Economics of Cavity Design 14-4. Phase Stability 14-5. Transverse Stability

267 270 276 278 278

COlr;TE:NTS

...

x 111

PAGE

CHAPTER

14-6. 14-7. 14-8. 14-9. 14-10. 14-11. 14-12. 14-13. 14-14.

Transverse Focusing by Grids Bunchers Debunchers Typical Proton Linacs Linear Accelerators for Heavy Ions .Electron Linear Accelerators Stability in a Traveling-Wave Electron Linac Typical Traveling-Wave Electron Linacs Standing-Wave Electron Linacs

284 285 286 286 290 291 295 296 297

15. QUADRIJPOLE LENSES 15- 1. Introduction 15-2. nlagnetic Quadrupole Lenses 15-3. Focal Length of a Converging Lens 15-4. Focal Length of a Diverging Lens 15-5. Focal Length of a Converging-Diverging Pair 15-6. Focal Length of a Diverging-Converging Pair 15-7. Image-Object Relat,ions of a Converging-Diverging Pair 15-8. Image-Object Relations of a Diverging-Converging Pair 15-9. Stigmatic and Astigmatic Images 15-10. Electrostatic Quadrupoles 15-11. The Effects of Fringing Fields 15-12. The Use of Lenses in Matching Two Accelerators

299 299 302 305 306 307 308 310 310 311 313 314

16. STOCHASTIC ACCELERATORS 16-1. Introduction 16-2. Hybrid Accelerators 16-3. Stochastic Ejection 16-4. The Equations of Stochastic Motion 16-5. The Band Width of RF Noise 16-6. The Application of Diffusion Theory 16-7. The Calculation of the Current

319 322 323 323 327 328 333

BIBLIOGRAPHY

337

AUTHOR INDEX FOR BIBLIOGRAPHY AND REFERENCES 375 SUBJECT

INDEX

385

PRELIMINARY COKSIL~ERATIONS

l-l.

The Reason for Particle Accelerators

Particle accelerators are built for the sole purpose of endowing nuclear particles with large quantities of kinctic energy so that’ they may serve as projectiles t,o alter the st,ructure of other nuclei or to act as probes to discover relevant information on the forces that hold nuclei together. The goal to be accomplished determines the necessary energy, which is conveniently measured in a unit called the electron-volt (see 3 l-10, below). Valuable information can be obtained by studying the scattering of projectiles with energies ranging from a few thousand electron-volts to some tens of millions. To produce artificial radioactivity, or to eject neutrons which can be employed as secondary projcctilcs, requires a span from a few million to pcrliaps 30 million. If mesons are to be created, some hundreds of millions of electron-volts are necessary: while to produce antiprotons and antihypcrons the figure is 6 billion and up. The attainment of such energies has been a gradual one, lasting about a generation; on the whole, the energy has gone up by about a factor of 10 every six years. It is the purpose of this book to discuss the accelerators themselves, rather than the experiments performed with their aid or the problems and the thcorics of the structure of matter which have developed concurrently. Emphasis will be primarily on positive-ion accelerators, particularly those in which magnetic fields play a crucial role both in returning the particles to t,hc region of the accelerating electric field and in keeping the projectiles properly on course during their enoriiiously long paths. The projrctilcs used in particle acrellcrators must be electrically charged so as to be subject’ to the influence of electric and magnetic fields. The negative projectiles arc usually electrons, while the positive ones are most often ionized atoms of light hydrogen (protons) or of heavy hydrogen (dcutcrons) or of doubly ionized helium (often called alphas, in correspondence with the name originally assigned to the helium ions emitted from certain radioactive nuclei). To an increasing extent, ions of heavier atoms are coming into use. Historically, the first man-made attack on the nucleus of an atom occurred in 1919 when Rutherford bombarded nitrogen with the alpha particles emittccl by radium. This work resulted in important information on nuclear properties

2

PRELIMINARY

COXSIDERATIONS

and suggested the need for a more copious source of projectiles. Consequently efforts were made to find m&hods of accelerating nuclei by artificial means to energies equal to or surpassing those of the alpha particles from radioactive clcments. 1-2. DC Accelerators The technique of endowing charged particles with kinetic energy by allowing them to fall in a vacuum through an electrostatic potential had been known for years, in the production of x-rays and in the early work on the measurement of the charge-to-mass ratio of electrons and of positively charged isotopes. In principle it was only necessary to apply these techniques at higher and higher voltages. But there are difficulties, primarily of insulation, both with respect to the voltage-producing device and to the discharge tube through which the particles pass. This tube must be of insulating material, for the full voltage is applied to it from end to end, the ion source being at one end and the target at the other. It must be well evacuated, not only to prevent an arc or glow discharge within it, but also to permit free passage for the ions as they gain energy in falling down the potential hill. The first device of this sort to effect a nuclear disintegration was that of Cockcroft and Walton, in 1932. A DC potential of 600,000 volts was generated by a succession of voltage-multiplying rectifier circuits, the discharge tube being a glass cylinder several feet long. Copies, modifications, and improvements soon followed, many incorporating the results of earlier studies on the best way to maintain a high potential without the occurrence of miniature lightning bolts along the outside surface of the discharge tube, down supporting structural members, or directly from the high-voltage electrode to ground. These problems ‘became more pronounced with the advent of the Van de Graaff electrostatic generator in 1931, which could, at that date, produce a potential of 14 million volts on an insulated electrode by bodily transporting charges up to it on a moving belt, though no adequate discharge tube was immediately available to take advantage of this voltage. The harnessing and control of multimillion-volt potentials, produced either by cascaded voltage-multiplying circuits, cascaded transformers, or the Van de Graaff technique, made considerable progress during the 1930’s. It became evident that, left to itself, a high voltage generally will not be distributed to form a uniform gradient to ground across intervening space or insulating materials. It is much more likely for a gentle gradient to develop across the majority of the distance, followed by a large potential drop across the remaining small space, thus initiating flash-over. By forcing intermediate positions to be held at intermediate potentials, a distribution of voltage much nearer to the uniform ideal can be attained, with consequent improvement in voltageholding ability. This condition may be reached by breaking up insulating col-

Coortrpy of High Voltage Engineering C o r p

PLAr'E 1 A 6-Mv Van de Graaff accelerator with pressure vessel removed. The supporting members, charging belt and clischarge tube lie inside thc potential-dividing hoops. The ion source is within the high-voltage e1t:ctrodc at the top.

4

PRELIMINARY

CONSIDERATIONS

umns and discharge tubes into short segments separated by metal pieces which are connected to appropriate points on a potential-dividing system between the high-voltage terminal and ground. This system may be a series array of high resistances or its equivalent, such as a succession of point-to-plane corona discharges. Such schemes waste a certain fraction of the available current but they are essential to high-voltage control. In modern Van de Graaff machines a cylindrical volume of relatively uniform gradient is produced by a large number of coaxial metal hoops maintained in this manner at successively graded potentials. Inside this region are located the segmented structurai members, the segmented discharge tube, and the charging belt. Finally, the entire assembly is encased in a steel shell filled with several atmospheres of nitrogen or a mixture of other gases which inhibit breakdown. By these means, developed over the years with much painstaking effort, the Van de Graaff technique now yields nuclear projectiles of unsurpassed precision and stability of energy up to 10 million electron-volts. This seems to be just about the limit, for insulation problems become intractable at higher voltage. Quite recently the two-stage (‘tandem” Van de Graaff has made its appearance as indicated in Fig. l-la. Positive ions produced at ground potential are accelerated to a few kilovolts and sent through a channel containing a gas, where an appreciable fraction of the positive ions pick up two electrons and become negatively charged. These are then accelerated by perhaps 5 million volts to the high-voltage terminal, where they pass through another gas-filled chamber. Both electrons become stripped off, leaving a positive ion which is further accelerated (through a second discharge tube) back to ground potential, thereby gaining, over all, an energy twice that of the potential of the high-voltage terminal. In a possible extension of this method to three stages, shown in Fig. l-lb, positive ions produced at ground will be neutralized and allowed to coast up to a terminal at 5 million volts negative; then the neutrals will gain an electron and be accelerated as negative ions to ground and then undergo another 5-million-volt acceleration by a second voltage generator with a positively charged electrode. Here the negative ions will be stripped of two electrons to become positively charged and will receive a third 5-million-volt acceleration as they travel back to ground. A fourth stage can be added by the inclusion of a pair of bending magnets so that the particles will be re-introduced into the first voltage generator to receive a fourth 5-million-volt increment as they move to the target, which in this case is at high potential; this is shown in Fig. 1-1~. l-3. Cyclic Accelerators With the exception of the tandem Van de Graaff (often, and appropriately, dubbed the “Swindletron”) and its possible four-stage elaboration, all the accelerators mentioned earlier produce a single value of potential. In falling across this, the projectile acquires its full kinetic energy. On the other hand,

CYCLIC hCCISLEllArrC)E:S POSITIVE ION SOURCE ELECTRON A D D I N G CANAI.

STRIPPING CANAL *CHARGING BELT P O S I T I V E ION BEAM

1

\ \

SO A N A I Y Z I N G M A G N E T

TARGET

POSITIVE ION SOURCE POSITIVE ION BEAM ANALYZING MAGNET

I ,/ 1 1 1 I11 1

NEUTRALIZING CANAL N E U T R A L BEAM ELECTRON ADDING CANAI. NEGATIVE HIGH-VOLTAGE TERMINAL i I 1 CHARGINGBELT ;EGATlVEIONBEAM STRIPPING CANAI. \ POSITIVE HIGH VOLTAGE TERMINAL

,

I

1

;

POSITIVE ION S O U R C E POSITIVE ION BEAM ANALYZING MAGNET i " E u TN RE AU LT I ZRIANLG B cA EA NM AL

1 11 1

/

/

1

I

1

ELECTRON A D D I N j C A N A L 1 NEGATIVEHIGH VOLTAGETERMINAL

STRIPPINGCANAL

Fig. 1-1. 'I'andem Van de C;raaff accelerators.

6

PRELIMINARY CONSIDERATIONS

the cyclotron and its descendants, which form the main subjects of this discussion, are devices in which a low voltage is used over and over again with the result t h a t the final energy equals that gained in one passage through the voltage generator multiplied by the number of such traversals. The developnlent of such cyclic acceleration (or, as it is sometimes called, resonant or synchronous acceleration) has opened up a whole new world of laboratory exploration, for particles are now available with energies u p t o 3LbW thousand times as great as can be produced by the application of steady electrostatic potentials, and the end has not yet been reached. It is true that man-made projectiles are still a long way from the potency of the most energetic cosmic rays, but it is not yet established t h a t any reaction induced by cosmic rays has a threshold energy above that which will be reached by some of the accelerators now under construction; furthermore, the quantity of machine-produced projectiles vastly exceeds t h a t supplied by nature. I n order to introduce the terminology and to avoid repetition of certain concepts which are common to all these devices, i t will be worth while first to give a brief qualitative description of a few of the different types, with an indication of how each variety developed out of its predecessor. We will then be in a position to discuss stability problems and other matters which will lead to a more detailed description of these machines and prepare the way for a discussion of the more recent accelerators.

1-4. The Linear Accelerator Devices in which high energy is attained by the repeated application of a small accelerative force, and in which the particles travel in a straight line, are known as linear accelerators or linacs. An excellent example is the original Wideroe type first constructed in 1928, in which a series of hollow metal 'drift tubes" are aligned along the axis of a cylindrical glass vacuum envelope, as shown in Fig. 1-2. Alternate tubes are connected to the terminals of an AC generator running a t constant frequency, so that even-numbered tubes are positively charged when the odd-numbered tubes are negative, and vice versa. Suppose a positive ion is emitted from a source a t one end of this array a t the moment when the first drift tube is a t its peak negative potential.

Fig. 1-2. The Wideroe linear accelerator. Each drift tube becomes charged to the opposite sign from that of its neighbors. The distance from gap to gap is traveled in a half-period of the oscillator.

The electric field betwcen sourcc and tube will accelerate the ion until i t plungcs into the interior of the tube, whereupon the pwticle finds itself in :I firld-free rcgion and coasts on a t steady velocity. \\iith tlie propor ratio of t u l ) ~ length to p:trticle velocity, tlic ion will reacli the gap between the first and w ond tubes exactly one half cyclv later when tllc potentials of all tubw I~avc, reversed, so that a second acce1cr:ition is experienced in crossing tlie sccontl gap. This process continues, the final energy being the sum of all the increments gained in all the gaps. Since the projcctiltl goes faster and faster, while the time for reversal of the field is fixed a t a constant value, it is apparent that the successive drift tubes must increase in length in an appropriate innnner. A more reccnt varicty of linear accelerator is tlie Alvarez type (1946) indicated in Fig. 1-3. Here the drift tubes arc enclosed by a cylindrical tube of

ion source

1

F

'copper

envelope

RF oscillator

Fig. 1-3. The Alvarez linear accelerator. All drift tubes become simultaneously polarized in the same direction because of the electric field in the resonant cylinder. The distance from gap to gap is traveled in a full period of the oscillator. copper, rather than of glass, and the interior of the structure acts as a resonant cavity. When excited by a source of AC power, it oscillates in a mode in which the electric ficld is entirely longitudinal, first point~ngone way and then the other, half a cycle later, thereby polarizing tlie drift tuhes. Tlie projectiles coast within thc tuhes when the field betwcen thein is pointing the wrong way. Largely because of its rugged coristruct~on (tlrc. only noniilct:~l parts arc thc vacuum-seal supports for the input power conductors) and because of the recent availability of adequate oscillator and amplificbr tubes, this type of linac has almost entirely superseded the original varitty. It has been used not only as an instrument of research on nuclear problems, thc most powerful installation producing protons with an energy of 68 million electron-volts, but it is also employed as an injector, that is, as :in auxiliary device to give the necessary initial energy to projectiles before they are introduced into a main accelerator known as a synchrotron, a s will be explained in due course. 1-5. The Cyclotron

The concept of the cyclotron was originated in 1929 by the late E. 0. Lawrence. The first model was constructed a year later by Lawrence and Edlefsen, and definite proof of the acceleration o f particles was established by M. S. Livingston in 1931.

8

PRELIMINARY CONSIDERATIONS

The cyclotron (Fig. 1-4) may be considered as a Wideroe linear accelerator which has been wrapped u p into a flat spiral and encased in an evacuated chamber, with the addition of a steady magnetic field perpendicular to its plane. The effect of the field on the moving ions is to cause their paths to become circular, with greater radius after each energy increment, as will be explained shortly. The dozens of drift tubes can be dispensed with and their

rlindricol poles

RF driver

Fig. 1-4. The cyclotron. The dees, driven by an R F oscillator, lie between the poles of a magnet excited by direct current. The ion source between the dees and the vacuum chamber which surrounds them are not shown. function fulfilled by two semicircular hollow electrodes shaped like the two halves of a pill box which has been cut in two along a diameter. (Because of the resemblance of these structures, in the earliest models, to the capital letter Dl they were called dees, and the name has continued to be used, even though the resemblance may have disappeared with constructional changes.) These two electrodes are connected to a source of alternating voltage of fixed frequency, as were the drift tubes in the linac. The action is much as before. A positive ion released from a centrally located source between the dees is acted on by the electric field between them, heing attracted towards whichever dee is negatively charged. The ion then coasts a t constant speed in the region within the dee, free of electric forces. The magnetic field bends the path into a semicircle, and when the interelectrode gap is again reached, the potentials have reversed, leading to a second acceleration. This process continues, the projectile spiraling outward towards the bounds

THE CYCLOTRON

9

of the niagnetic field and the resulting final encxrgy being the sum of the individual contributions. It is important to realize th:tt tlic stvacly 111agnetic field docs not add anything to tlic particle's energy, hut nicrcly stcerh i t brwk to t h e gap b c t w e m t l ~ e dces where the energy-su1)j)lying electric field mists. I n a linac, each d r l f t tube niust be of a particular lcngth so t h a t the spccding-up ions will traverse each in the s:unc t h e . I n a cyclotron, there are no specific drift tulxs, the lrngth of time a particle is sheltered froin the electric field dcpcmling on t h e length of tlw pat11 i t follows insicle a (lee. T h e success of the innchiilc as an accelerator depends o n the fortunat(, eircuiiistancc t h a t fast particles t a k r a long path and slow 11,urticlcs :t short one, so t h a t t h e time rcquired for e i t l ~ c ris t h e same (though t l ~ c wis a 1imit:ztion t o the validity of this statement, a s will soon hc c x p l a m c t l ~ .This prol)crty of ions inoving in a circular pat11 In a magnetic field is extrcwcbly important and also is easy to understand. A t every instant, the. clcctroinagnetic forcc qzjB supplies the centripetal forcc Mv"r required for a circular path, whcrc K is the magnetic field strength, q is tlic cliargc on tlie ion, v is its velocity, :tnd r is the radius of curvaturc of tlie path. Thus,

Hence,

Mu

==-

qBr.

(In the Jfk'S system, M is in ki!ogr:mb, u in m e t ( ~ r s / ~ e qc , in coulolnhs, r in meters, and B in wcbcrh/~ti~.In the ('GS sy&m, the units a r c grams, cm, cm/sec, elect roniagnetic unith of charge and gauss. F o r a singly ionized atom, y = e = 1 6 :< lo-'!' coulomh = 1.6 x 10-"' e.111.u.'I?lie rest mass of a n electron is 9.1 x 10 -:" kilogram, t h a t of a proton is 1.67 x lo-" kilogram. 1 weher/m2 = lo4 gauss.) Equation (1-21 s h o ~ mt h a t :is the iriomcnturn increases, t h e radius of curvaturr rises, sincc 13 is assuincd constant; a m t g n e t with poles of large diameter is necessary if the morncntum is to rise t o a high value. Also from t h e last equation r/Kr 2,

=

--'

M

B u t the t h e t o complete one circular orbit is

10

PRELIMINARY CONSIDERATIONS

Thus the period of revolution does not depend on either the velocity or the size of the orbit, but only on the field strength and on the charge-to-mass ratio of the projectile; if these are all constant, then so is the period of rotation. T o make a cyclotron work under these conditions, one has merely to tune the oscillator which drives the dees until its electric period coincides with the period of revolution, or else t o adjust the magnetic field until this resonant condition holds. Ordinarily, one speaks less often of the period than the frequency f = 1 / ~ ; in even more customary use is the radian frequency (i.e., angular velocity) w = 2 ~ f generally , referred to simply as the "frequency." The revolution or cyclotron frequency is therefore given by the all-important relation

An ordinary cyclotron of this sort is known as a fixed-frequency cyclotron, the niodifier referring to the fixed frequency of the oscillator. At every cycle a batch of particles is extracted from the ion source. After some dozens of revolutions the ions acquire high energy, reaching the radius a t which the target is located. The target is bombarded with a group of projectiles a t each cycle, much as though it were being struck with the pellets from a rapidly fired automatic shotgun. But one cannot in this manner indefinitely increase the energy (or momenturn) by an unending succession of small energy increments gained in passing between the dees, even if the radius of the magnetic field and the size of the dees are correspondingly enlarged; a halt is brought about by a fundamental phenomenon of nature described by the theory of relativity. The kinetic energy of any object is associated both with its mass and with its velocity, and since the latter cannot exceed that of light, an increase in energy of a body nloving a t a speed near this limit shows u p largely as an increase in its mass. This is outstandingly true in particle accelerators, in which ions attain substantial fractions of the velocity of light. Consequently the rotation frequency of a fast particle in a cyclotron is less than that of a slow one, as is shown by Eq. (1-4), and if the oscillator has been adjusted to be in synchronism with ions just starting out, it will be a t the wrong frequency for the more energetic ions which have gained somewhat in mass. The onset of this difficulty is gradual, the ions slowly slipping out of phase until ultimately traversal of the dee-to-dee gap brings about deceleration. The techniques used to delay this limit until the highest possible energy has been obtained (20 to 25 million electron-volts, for protons and deuterons) will be described in due course. (The reason one cannot sidestep the difficulty entirely by making the magnetic field increase in proportion to the rising mass, so as to hold constant the revolution time of particles of all energies, will be made clear when we discuss the stabil-

ity of orbits. Tllc recent dcveloplncnts wliic.h lwrniit us to h a w our cnlw ant1 e:it i t too, nil1 nppc:lr ~nucllI:~tcro n in Chapter 13.)

1-7. The Synchrotron

As has hecw wen in Eq. (1-21, the niomcntul~i ( a n d tllcreforc the energy) of a particle t l ~ : can ~ t I)e iwung in n circu1:ir orl)it dclwntlh h t l l on the lnngnitude of the ficllil :inti on the radlns of crrrvnt~u-c:.If11 = q B r . Since tlicre arc, practical Ilnlit\ to :ittiiin:iI)l~field T : L ~ I I (i~y~~, i ( ~ l ~ r o r ~ y o Iofo t ~g ~~o ~~ ~~sand t e r grc,:~tt,rcncrgy I ~ n wrc~llured1:irgcr and I:LI gcr iri:iqucct\, tllcl nr:~sbof iron risinq 7'11~ ( ~ ( ~ r qlilnit:ttio~~ y so011 [I('roughly with tlrc cuhc of tllc polp (ii:t~llc~tor.. colnc. on(%of clcononiics. T h r synchrotron sitlc.tc'l)- tllis dificl~lty.Insttt:\tl c ~ fl)cvlrittirl~ion< to spir:iI outn artl, thus necewtating a Iargc, area ovctr nllicl~:I con\t:l.nt ficltl i- n1:iintaincd, the, ~xirticlcs:ire conitrained alw:iys to rotat c a t :I. constant ~ : L ( I I I I < l)y i n i ~ n c r h gtlrcwl in :I ~ n a g n c t i cficl(1 which grows in time along ~ n t l ltlrc, ric~nci e n t q y . 3Ioit of tlrc iron in t h t ccmtcr or the machine can I)c oinittcil, ho til:it the magnet t)cwinc.: a na13rowring of large radius, thcrehy reducing the n-cigllt by a n cnorunous f:wtor. T h e dccs, ~ h i c hotller~visewo111tl have to he monstrou. in size, re1 crt lx\c>lito c h f t t u l ~ c ior . hy inqcniouq methods are rcducetl even f u r tl1c.r in size to -1nal1 units located nt onr or more points on the circumfcrcncc Since t l ~ c~):lrticlcvelocity incrc\:l.cs while the circular path is fixed in lengtll, the frequency of the altrrnating :icwlerating voltage must now he made to rise, so as to s t a y in synclironism with the ions. T h e present record for encrgy is the attainment of protons a t 30 hillion rlcctron-volt., which is ahout 1000 times t h a t rrnchc7d in a cyclotron. R u t such supcrenergy again is bought at n price. T h e t i ~ n r - a v e r a g eyield of high-speed ions is cut by another large factor, sincc after one huncli of ions has 1)ern

PRELIMINARY CONSIDERATIONS

12

brought to full energy, both the magnetic field and the oscillator frequency must be returned to their initial values before another group can be carried up. For the larger synchrotrons the repetition rate may be as little as once every few seconds because of the great inductance and stored energy of the magnet. (Carrying on the analogy, we are now hunting with a muzzle-loading fowling piece of tremendous range.) The rate a t which the magnetic field rises is subject only t o very gross control, so this field iiiust be taken as the independent variable. This means that the frequency of the accelerating voltage must be increased very accurately and the magnitude of the accelerating voltage continually adjusted, since the rising magnetic field demands that the projectiles have always a specified amount of energy; with too much or too little they will run into the walls of the vacuum chamber and be lost. As a result, the control system becomes extraordinarily elaborate. Since it is impossible from a practical standpoint to control the magnetic field precisely all the way from zero upwards, it is necessary to inject the ions into the synchrotron a t an energy which corresponds to the magnet's radius and to the minimum value of field which can be obtained with reliability. The attainment of this injection energy requires the further complication of using a separate particle accelerator, such as a linac or a Van de Graaff machine, as the "injector." Some synchrotrons have been built to accelerate electrons while others are constructed to produce high-energy protons. Often the n~agnetis divided into a number of sections with straight vacuum chambers interposed in the fieldfree regions. Such machines are sometimes called race tracks, since the first of this type had two straight sections, as in a race track for horses.

TABLE1-1 SIMILARITIES AND DIFFERENCES

Magnetic field Orhit radius Frequency of oscillator

Fixed-frequency cyclotron

Frequencymodulated cyclotron

Synchrotron

Constant Increases Constant

Constant Increases Decreases

Incrmses Constant Increases

1-8. The Betatron

This is a device for the acceleration of electrons, and it was in connection with such a machine that certain types of orbit perturbations first received comprehensive study and hence became known as betatron oscillations. The expression still clings to such motions, even though the accelerator in which they occur may not be a betatron. A betatron resembles a synchrotron to the extent that a rising magnetic

field causes particles of increasing t,ncrgy to rot:tte :tt a fixed radius; but the riletliod of supplying energy to t h c n ~is ( p t c different. The region of growing field includes :l large part of the urea inslde thc o r h t so that, just as in a transfor~ncr,an elcctronlotive force whicli tlepentls on the rate of rise of the cncloscd flux is developed :wound the o u t l ~ u tcircuit. This circuit here is not a copper c~onduci,or,but siml~lythe stream of electrons, wl~ichare consequently accelerated. I t may be ~nentioncda t this pomt that, a certain amount of "betatron acceleration" takeb pl:ice in a synclirotron I)ecausc of that portion of the rising field which lies inside the particles' orhit: hut since this area is a very narrow annulus, rather tlian the entire c~rcularinterior area as in the true betatron, the encrgy change protluced hy tlns 1ne:ms i.j :tl~llo,jtnegligible and appears only in very refined calculations. 1-9. Relativity

Tllc amount of the theory of relativity required to understarid particle accelerators ib w r y liiiiitcd but :~bsolutelyI-rucial, so it, will be well to introduce it at this point. Elementary t~xt1)oolistell us t h a t h i e t i c cnergy is given by thc expression ~ J l , , v 2so , a body of mass M,, wdl quatlruplc its kinetic energy if the velocity v is doubled; or putting i t the other way around, quadrupling tlic kinetic energy will double tile velocity. Sinctl there is nothing to prevent us frorn adding as niucli energy as we plcasc, i t looks as though any velocity whatsoever should be attainable. But the thelory of relativity dec1:tres t11:tt the velocity of light is the ultinlnte and that mass arid cncrgy are intcrconvert~ble.An increase in kinetic cncrgy is associatrd w t h an increase in both Illass and vc>locity,the change in the fornier bemg s111al1when the veloc~tyis low; but as the liniiting velocity of light is appro:tched, a larger and larger fraction of tlic energy increment appears as a gain in muss. This concept is difficult to swallow a t first sight, for it appears contrary t o experience. But ordinary expcriencc~is limited to sn1:tll velocities and for then1 the mass invrclase is below the lirnit of perception. [f we broaden our experience (say by building particle accclcrators, where velocities of 10, 20, 50, or even over 99 percent of that of light arc. common) it will be found that the relativistic concept is indeed correct, exen if other evidence is disrcgardecl. T h a t is to say, if tllc machine is h i l t without ptiying attention to relativity, it simply won't work; but if t,hc increase of mass wuth velocity is allowed for, then it will. The relntiv~sticst:ttement for the mass as a function of velocity will not be derived here but is merely stated. I t is

14

PRELIMINARY CONSIDERATIONS

Here c is the velocity of light, M o is the rest mass (the mass as measured by an observer with respect to whom the body is a t rest) while M is the mass as measured by an observer with respect to whom the body has velocity v. The difference between M and M o is utterly negligible for ordinary man-made projectiles, but it becomes distinctly noticeable for electrons and ions; even in small accelerators truly high velocities can be reached. For example, a t v = 0 . 8 6 6 ~we find from the above equation that M = 2Mo. Einstein introduced the concept that mass and energy are interconvertible, the ratio of energy to mass being c2. Thus a body of mass Mo, a t rest with respect t o an observer, represents a supply of energy (its so-called rest-mass energy or rest energy for short) given by Eo, where

Eo

=

Moc2.

(1-6)

Since c has the value 3 X lo8 m/sec, 1 kilogram has an energy value of 9 x 1016 joules. (Whether or not an actual conversion of matter into energy takes place, or vice versa, is quite irrelevant, just as one paper dollar always represents one hundred copper pennies, irrespective of whether the conversion is carried out.) Now let this mass be given some kinetic energy T, as by throwing it across the room. The total energy is then

It is possible to express this total energy in terms of a total mass M through thc cspression E = Mc~, (1-9) so that Mc2 = Moc3 T; whcnce T M = Mo -'

+

+ C!

This inass M of a body in motion with respect to an observer is identical with its inertial mass, as seen by the same observer. Owing to the cnornlous value of c" the kinetic energy must be very large indeed in order for M to be noticeably different from Mo. Solving the last equation for T gives and on using for 11.1the rclntivistic expression of Eq. (1-5), there results

I f u is w r y sn1:ill coinpard with c, this may be expanded in a power series in v/c :

wliicli is the exl)ression found in e1einent:try textbooks. As long as tlie vclocity is vcry small coinl~aredwith that of light, tlic simple expression for liinctic energy is an atl(quxtc approxiir~ationto tlie truth, hut it fails utterly wlicn tl~c, vclocity is high; tlie relativistic c>xpressionnii~stthen bc used, for i t is :lln:\p correct. It is convenient to measure velority as s fraction of that of light by tlefininq the dimensionless quantity

This is the equivalent of tlie airplane pilot's Mach number, which measures speed in terms of that of sound. B u t whereas Mach numbers greater than unity can be attained, this is not true of P ; nothing hut light can reach /3 = 1, though this figure can bc approached very closely by material bodies of sufficient energy. The relativistic expression for kinetic energy then becomes

Solving for P2 yields

which shows that if a particle has a kinetic energy equal to its rest energy, then /3 = 0.866. This is a useful fact to reniemhrr, to get it feeling for the behavior of part,icles in accclcrators. Recall also the previous statement t h a t a t this velocity the total mass is twice the rest niass. This equation also shows what uphill work it is to approach /3 = I ; for 1' = 2 E , , , /3 = 0.94; for T = 3Eo, p = 0.97; for T = 10Eo,/3 = 0.996. Since T = E - E n , the last equation can be written in the vcry useful form

There are two other important dimensionless qumtitics of which the definitions should be k q ~ in t mind. The first, which measures total energy in units of the rest energy, is

16

PRELIMINARY CONSIDERATIONS

By the use of Eqs. (1-6) and (1-9) this can also be written as

Momentum, generally indicated by the symbol p, is defined by p E MU, (1-17) and the other dimensionless number, which indicates momentum in units of the rest mass multiplied by the velocity of light, is

There are many relations between velocity, momentum and energy which are useful in the analysis of particle dynamics. The more important ones are derived bclow, although we shall not have occasion to use all of them. A general relation between p, y, and is found by use of Eqs. (1-18) and (1-17). These show that q = p/(Moc) = Mv/(Mac); so by use of Eqs. (1-16) and ( 1 - l l ) , The connection between y and (1-11) :

T = Pr. P may be obtained from Eqs.

(1-19) (1-16), (1-5), and

On rearrangement this gives

The relation between y and 7 is found by rearranging Eq. (1-20) into the form y" 1 y2p%nd then using Eq. (1-19) to obtain

+

y2 = 1

To find the connection between givc the result that

+ q2.

(1-22)

p and 7, Eq. (1-22) is used in Eq. (1-21) to

By rearrangement this gives

It is often convenient to know the relation between fractional changes in p, and q . If we take the logarithmic derivative of Eq. (1-22)-alternatively

y,

speaking, if we take the derivative of Eq. (1-22) and tlmi divide by Eq. (1-22) -we obtain

By the use of Eq. (1-22) this can be written as

i? - _ 17

7' -

-..

rly

(1-26)

y 2 - l r

T o convert the right side of Eq. (1-25) into a function of P, we first use Eq. (1-19) to show that drl/vl = d y / y @/P and then employ Eq. (1-23). This yields

+

Use of Eqs. (1-20) and (1-21 ) gerniits writing this as

The right side of this can be cliungcd to a function of 7 by noting, from ICq. (1-19), t h a t d y / y = d r l / ~- dP//3, iind 197 eml)loying Ey. (1-22) to convert "y 1 into v'. Then Eq. (1-28) becomes

By Eq. (1-24) this can be written as

These relations betwcen frnc.tiona1 chmgcs in P, y , and 7 are sometimes more convenient wl~enexpressed in fornis in~.olvingall thwe of these quantities. Thus if we use Eq. (1-23) in Eq. (1-25), the latter becomes

Use Eq. (1-24) in Eq. (1-27) to find that

When Eq. (1-20) is put in Ey. (1-301 there results

Note t h a t in all these expressions we may set d Y l y -= d E / E , d r l h = d p / p , and d p / p = dv/v.

PRELIMINARY CONSIDERATIONS

18

1-10. T h e Electron-Volt I n dealing with accelerators and in discussing the energy of nuclear or atomic phenon~ena,it has been found convenient to introduce the electron-volt (ev) as a unit of energy. The amount of energy gained by any particle bearing a charge as large as that of an electron in falling freely across a 1-volt potential difference is called 1 electron-volt. Since an electron has 1.6 x 10-l9 coulomb of charge, then 1 ev

=

1.6 X 10-l9 coulomb X 1 volt

=

1.6 X 10-l9 joule

=

1.6 X 10-l2 erg.

Mev is the abbreviation for a million (10" ev, while Bev represents a billion (10" ev. (But note that in Europe 1 0 % ~is called Gev, to avoid use of the word "billion," which in the United States and France means lo", whereas in England and Germany it means 10'" Gev comes from 'lgiga-electron volts," derived presumably from Graeco-Latin gigas as in "gigantic.") Since a proton has a charge exactly as large as does an electron (though of positive rather than of negative electricity) one could equally well speak of "proton-volts" as of "electron-volts"; but custom has settled on the latter terminology. If either a proton or an electron falls freely through 1 volt, it will acquire 1 ev = 1.6 x 10-l9 joule. A doubly ionized helium atom, bearing two electron charges, in falling through 1 volt will gain 3.2 x 10-l"oule, which is 2 ev. Even if no voltage is present in the energy-gaining process, the energy may be expressed in electron-volts, if desired. 1-11. T h e Rest Energy of Electrons and Protons An electron a t rest has a mass Mo = 9.1 x kilogram. Its rest mass energy is EO = Mot" 9.1 X X (3 X 108)2= 81.9 X 10-l5 joule

- 81.9 X 10-l5 joule

1.6 X 10-l9 joule/ev

=

=

0.51 X lo6ev

0.51 Mev.

The electron's rest energy is thus about half a million electron-volts. If we give i t a kinetic energy of this magnitude, it will double its mass and will attain a velocity of ,b' = 0.866. Modern technology readily makes electrons become "extremely relativistic" because it is quite easy to bring them almost to the velocity of light so that a further increase in energy must be accompanied mostly by an increase in mass. A proton ( a singly charged nucleus of light hydrogen) has a rest mass Mo = 1.67 x lo-" kilogram; by going through the same conversion steps we find

THE SIZE OF TliE ALXGXET ,To = 938 Nev. Protons are iiiucli inore difficult to get mto the "e:ctren~ely rclativistic" condition. Slmost, a billion electron-volts of kinetic energy inust be added to double tlicir mass an([ to bring tlicir velocity up to 0.866~. An unfortunate habit of using words loosely 1e:tds occasionally to expressions like "a billion-~oltaccclcrator" or 'Larest energy of half a million volts," d m properly speaking "\)illion-electron-volt accelerator" should be said, and "half a inillion electron-volts." 1-12. The Size of the Magnet

Fro111 Eq. (2-2) it is seen that the parttick's ri~omenturna t any time is givcri by (1-34) p = Mu := qBr. The maxinluln monienturi~of particles that can I)(. rctained hy thc magnet is therefore li~iiitetl by the product of tlic raclius ant1 field strength. iTlic quantity IZr is often referred to as the n y l ' t k t y of :I hc:m of ions, for it mcasurcs the difficulty of kmtling it.) It is now of intcrcst to see how B r is related t o the particle's energy. By Eqs. ( 1-34), ( 1-91, (1-1 I ) , and (1-14) we find

Hence

This is a very important relation. I t hecomes merv useful for calculations when cxpresstd in terms of tlic kinetic ericrgy, sinre tlifftwnccs between alinoht c r l ~ ~ na ln u l l ~ ~:ire s :~voiclcdITsing ( I -7 1 , tv.vcl 11i:~kethe following transfor111:itions: 7,3!2 - lJo! = (E,, + 2')' -= 'l'J+ 2T]to, SO

There are occasions tvlien the relati\,istic mass increase is of little iiiiportancc, as in preliminary studies of the size of a magnet rcquired for a chosen energy; in this case a simplified approximate relation may be used if T is small compared to 2 Eo. Then on writing Eo = iMoc" Eq. (1-36) becomcs

This emphasizes the fact that the kinetic energy rises approximately with the square of B and the square of r , so a w r y profitable prcinium is put on high ficlds and large magnets. The energy also rises with the square of the cl~arge

20

PRELIMINARY CONSIDERATIONS

on the ion. Since the radian frequency of rotation is given by = Bq/M, particles with the same charge-to-mass ratio may be accelerated in the same field without readjusting the oscillator frequency. Consequently a cyclotron adjusted for deuterons, where q/Mo = 1/2 (in units of the electronic charge and the mass of a nucleon), will also accelerate doubly charged helium ions (q/iVIo = 2/4) t o an energy twice t h a t of the deuteron, the energy being multiplied by 4 because of the doubled charge and halved because of the doubled mass. (The not quite integral relation between the masses can be taken care of by a very sinall change in oscillator frequency or magnetic field strength, such changes being well within the flexibility required in seeking for resonance t o begin with.) I n Eq. (1-36), the field is in webers per square meter, the radius in meters, the energies in joules, and the charge in coulombs. Converting joules to Mev, and setting q = Z x 1.6 x 10-l%oulomb, for an ion bearing Z charges, we find that

Br

=

1 (T2 + 2TEO)>$ (webers/m2, m, Mev). 3002

(1-38)

In the literature on accelerators, field strengths are almost invariably quoted in gauss or kilogauss, and in the United States lengths are usually given in feet or inches, so t h a t the following expressions are convenient.

Br

=

1

+

(1-39)

1

+

(1-40)

-(TY 2TEo)ti (kilogauss, cm, Mev), 0.3002

Br = -(T2 2TE0)s (kilogauss, inches, Mev), 0.7822

Br

I 9.152

= -(7':

+ 2TIC0)4' (kilogauss, feet, Mev).

(1-41)

Note that these equations are valid for :ill charged particles, the mass being implicit in the quantity Eo. The following numbers will give an idea of the magnitude of Br, calculated for protons, using the relativistic expression to three significant figures. TABLE1-2 VALUES O F Br FOR PROTONS O F VARIOUS ENERGIES T (Mev) Br(kg-ft)

10 20 50 100 500 1000 10,000 12,500 25,000 14.5 21.2 33.8 47.5 119 185 1190 1470 2900

Since there are practical and economic limits to the magnitude of the field which can be obtained in iron or steel (ordinarily somewhere between 8 and 20 kilogauss, depending on circumstances) it is apparent that very high energy can be reached only by using orbits of large radius of curvature. A 20-Mev proton cyclotron may operate a t 18 kilogauss, so the final radius is

21.2 kg-ft/l8 kg = 1.18 ft, wllercns a 25-Rcv proton synchrotron with a peak field of 10 kilog:uiss lilust have a ~ ~ t l l of u s2900 kg-ft/ 10 lig = 290 ft. The mor1, t l ~ ekinetic energy is in exces5 of s c v c ~ x rest-energy l units (scvcrd Mev for electrons, several Bev for 1)rot01is, etc ) the morc we are justified in neglecting 2TEo in cornpanson wit11 T%n Eq. (1-36). Under hucll extrcrncly relativistic circ~irnst:mces, the kinttic enc.rgy of :my particle niay be expressed approximately :is T w Brqc. (1 -42) K h c n electrornagnctic units are used and the cncrgy is cxprcsscd in clcctronvolts, this 1)ecornes (for singly clmrgcd particles) the simple expression T T 300Br (ev, gauss, cm).

(1-43)

It is also worth noting that with the use of Eqs. (1-6), (1-15), and (1-18)) we may convert Eq. (1-22) into the form

This makes i t possible to measure ii~oinenturnin units of "Mev/c" or "Bev/c," as is custoniary in discussions of high-energy physics. For highly relativistic energy this becomes

Attention is callcd to the lists of references at th13 hack of this book. These will servc as a source of col1:tter:tl reading for those who wish to pursue the subject further. I n connection with thtl presmt chapter, thew are general bibliographies :md refcrenccs to tabulations of :melvrators and to introductory hooks and articles, which are listed on pp. 337-338. Gcmral review articles and 1j:tsic papers on DC: and noncyclic AC inacliines appear on pp. 338-340.

O R B I T STABILITY

2-1. Introduction

Particles in magnetically guided accelerators travel a long way before reaching final energy, from a hundred feet or so in small cyclotrons up to several hundred thousand miles in very large synchrotrons. Practically none of thc projectiles are launched so accurately in the desired direction that they could travel this distance unguided, and even these very few will suffer many deflecting collisions with residual gas atoms. Corrective steering forces must be applied, most vigorously to particles farthest off course, and the mechanism must be automatic, for i t is not possible to keep track of individual ions nor will all of them a t the same azimuth need the same adjustment in trajectory. Corrections must be supplied for both up-down and right-left deviations, for if the ions once hit the surfaces of the dees or the walls of the vacuum chamber, they are lost. Consider a bowling alley (Fig. 2-1). A truly flat surface is bad enough,

Fig. 2-1. Bowling alley analogy of systems of forces which produce neutral, unstable. or stable orbits. while a convex one would be impossible. But if the alley is concave, a certain tolerance in the direction of launching is permissible, for the ball will oscillate to right and left but none the less advance without falling off. It is indeed fortunate that the magnetic field of an accelerator can be so shaped that i t not only returns the particles again and again to the accelerating unit, but also influences their paths so that a misdirected ion will oscillate back and forth about the desired trajectory, not only sideways but also up and down. 22

T h e orbit therefore can be inadc to 11:tve the property of two-dimensional stability. T h e undisturbed path about which t h e deviations occur is known as the equilzbriurn orbit. It is associated with an equilibriztm particle of fixcd energy, so tliat the orbit closes on itself. F o r this reason i t is often referred t o as the closed orbit.

-

-

/

equilibrium o r b ~ t

Fig. 2-2. Schematic representation of betatron oscillations about a circular equilibrium orbit. Radial motion is shown occurring on the midplane of the magnetic ficld, while axial motion is shown on a cylindrical surface. In actuality, both motions occur simultaneously. With weak-focusing machines, there is always less than one oscillation of either kind per turn. ,\s is indicated in Fig. 2-2, a cylincirical coordinate systeiii is used, t l ~ c axis of s y m n ~ e t r yof the inagnetic field being nonnal to the equilibrium orbit which lies in the mid-plane of the space betacell the pole tips. A motion occurring in the mid-plane is given thc adjective ~ ~ d ! r n(or l sometimes horlzontal, for in lnost accelerators, tliouglt not in all, t h ~ splane is horizontal). Notion p a ~ d l t~ol the axis of tllc ~nagneticlfield is called axial or vertacctl. \-e will a d a l ~ tthe axial-radial tcrniinolugy, with -=r 1ue:lsuring the displacement t o one or tlic other side of the n ~ d - p l a n e ,n l d e --fx indicates a radial displaccnicnt outside or inside of the equilibrium orbit. For simplicity of analysis tlic t n o motions will be c o n s i d e r d a s intlependcnt; t h a t is, axial motion is considered as occurring a t constant radius, while racilul motion is imagincd to t a k e p h c e entirely in the niedlan plane. I n p r a c t ~ c c of , course, both motions m a y occur sirriultaneously. I n some ac.celcrators there a r e field-free regions called straight sectzons or simply straights between portions of the inagnet, so the ecpilibrium orbit is a series of curves connected by tangent.. A 111ultil)lc-cylintlricalgeometry is then used, with a styarate axis for each c u r w d portion of thc orbit. I n such cases, a n y "radial dependence of field," such 21s will soon hc encountered, rcfcls to thc r:ttllus of the local cylinder, :tnd the ~ u y i n grlistnnw of tlic' clo.c~l orbit from tlic center of tllc machine is of no particular significance and innrely appears in the calculations.

24

ORBIT STABILITY

The oscillatory motions about the closed orbit are known as betatron oscillations or free oscillations and the control of their frequencies and amplitudes constitutes one of the major problerns of accelerator design. Two general methods are in use, the earlier being known as weak focusing, while the later development is called strong focusing. These techniques are sometimes also known as constant-gradient ( C G ) focusing and alternating-gradient (AG) focusing, respectively, the words "constant" and "alternating" implying that the radial gradient of the magnetic field either maintains a steady value or else alternates in sign as the azimuth changes a t any given radius in regions where the magnetic field is not zero. Although widely used, the CG and AG terminology is not too happy a choice, for there are many subtle exceptions and variations; some CG machines actually enlploy gradients of different values and in some AG accelerators the gradient not only changes sign but also alters in magnitude. The study of strong focusing will be postponed to a later portion of this book; for the present, attention will be devoted wholly towards weak-focusing devices. 2-2. The Field Index

Weak-focusing forces are obtained by using a magnetic field which decreases slightly with increasing radius, as will be demonstrated in § 2-3. This contour is approximately obtained with pole faces that are plane and parallel, for the tendency of the field to extend for some distance outside of the gap produces just such a weakening of field with increasing radius. The importance of this with regard to axial focusing was realized in a qualitative manner from the days of the earliest experiments in building cyclotrons. A quantitative analysis of this and a detailed appreciation of the field's influence on radial motion did not come for some years, after many cyclotrons had been built and used successfully, just as many ships sailed the seas long before Archimedes announced the principle of buoyancy. It has been said that had the full theory of orbit stability been developed initially, in all the con~plexitywith which it now exists, no one would have had the courage to attempt to build a cyclotron. On the other hand, it can be argued that the rewards to be obtained from properly shaped fields are so enticing that earlier understanding would have precipitated even faster development. The way in which the strength of the magnetic field varies with radius, on the median plane between the poles of the accelerator, generally cannot be described by any simple functional relation which is valid over the full radial width of the magnet aperture. However, it has been found in practice that the radial betatron oscillation amplitudes are fairly small; so in order to correlate these motions with the field, it is only necessary to have a knowledge of the field-vs.-radius relation which is valid over the small range of radius necessary to encompass these oscillations. This can be obtained with

of the ratio of the fraction:il cliangc In ficltl associated with any fractional e l ~ a n g cIn r a t l i ~ ~Til.~ u h :

This can be writ ten altcrri:itivc~ly:is

T h e introduction of the m i ~ l u ssign is l)u~.clyco11ventiona1, in order to yield a positivc index for n wc:ik-focusing fiel(1 (one that c l c c l - c : ~ ~with s increasing r a d ~ u s,)for tlicn the graclicnt tlR/dr is inllcwntly ncg:rtivc. F o r exuml)l~.,if the ficld is R :rt ratlliii r :inti if it is found to be reduccd by $9 : ~:1t r:idius 15'r grcatcr, tlicm dll/ll = - 0.003 a n d dr/r = 0.01 and the inclcx in tllc r~c~glik)urliooti of r liab the, \xlutl 0.5. ( 'onrcrsely, if we know n a t rxtlius r w11cl.e the field i i R, tlle~i21 c~l1:ingc~of ratilus hy dr causes a cliangc of ficltl given by dl3 = - nHdr/r. Over t l ~ er n t l d range en1l)racctl by a radial betatron osci1l:ltion wllicl~is ccmtt~rcdon some l):irtic,ular r a l u c of r, the index of t l ~ coscillation a r e is considered to I)($ constant :inti the cll:wactr~r~st~cs calculated on tli:lt I):/& At n diffcrcmt rucliw, n 111.ty 11:~r-ea tl~fferentvalue,, again tnkcn :is :L ctonst:mt. B y intvgrntion of Eq. (2-1) l)c"cvccn thc limits rl and r wllcre tlie field is R , and R , n-it11 n fixed, thew is o1)t:lincd

Here r1 is :my rcfercnre radius where tlltl field is Ill, d l i l e B is the ficld a t a r:dius r suffici~ntlyclose to r1 that over t l ~ cintcrvvning slmce tlrc indcx has not cli:tngcd apl)rccia\)ly.This expression can l)e writtcn in the form

where K =- B l r l n is a constant. This shows t h a t over a restricted range t h e ficld can be considcrtd as inversely 1)ropoitional to thc nth I m w r of the radius, a new value of n pcrllnps h i n g nssoci:ltcd w t l l :I grosily chfferent value of r. From this c~xpression i t is clrar that tlw ficld gets n c : ~ k e r with increasing r:tdiub if n > 0 , wl~ilcthe opposite iitl~ntionis drscribcd ~f n < 0. A u n i f o r ~ u field is reprcsented 1)y 1, = 0. I n studying l)ctntro~ioscill:itions, thc wferencc r:td~uq r, and its associated field B , a r e interpreted as the radius r,, :tnd thc field R, of the equilibriunl orbit, wliilc tlic r:tdi:d displaccrnent of a particale following some noncquilibriurn path of varying radius r is denoted by n: = r - r,. I n cyclotrons and synchrocyclotrons the field R f d l s with increasing radius, but hy only a few pcrcent from its central value during a largc increase in r, 50 t h a t t h e q w n t i t y Br increase with r U l t i ~ n a t e l ya s the pole edge is np-

26

ORBIT STABILITY

proached, l3 falls more rapidly than r increases, so that Br begins to decline after passing through a maximum. At the radius a t which this maximum occurs the magnet can retain ions of greatest momentum. Kow consider what happens to n as we move outward in these niachines. I t starts a t zcro a t the axis of the poles, bccause from symmetry the gradient there is nil. As r increases, B falls a little and dB/& acquires a small negative value so that n starts to rise, very gradually a t first but extremely rapidly as the pole edge is approached, owing both to the precipitous fall of B and to the associated augmentation of gradient. The index passes through the value unity just a t that radius a t which Br reaches a maximum value, as is easily demonstrated. Thus:

Since n = -rdB/(Bdr) from Eq. (2-1)) then B - nB = 0 or n = 1. As will be seen before long, the entire region from where n = 0 to where n = 1 happens to fulfill the requirements of orbital stability (so if an ion sonlcliow gets off course it tends to return to where it sllould be) and consequently it appears a t first sight that particles of greatest moinentunl and hence greatest energy could be obtained by letting thein be accelerated outward to where n = 1. Most unfortunately there are reasons why this is not practical. In fixed-frequency cyclotrons it would be necessary to apply dee voltages of entirely unmanageable magnitude if this final energy were to be obtained in so few revolutions that the ions with rising relativistic mass had not yet slipped so far in phase with respect to the fixed-frequency voltage between the dees as to become decelerated. The largest radius that is ordinarily reached in cyclotrons is that corresponding to n = 0.3 or thereabouts, and the projectiles are used a t that point. With synchrocyclotrons a disastrous axial spreading of the beam occurs a t n = 0.2 (see Chapter 5, below, Resonances) ; the particles are driven up and down into the surfaces of the dees, so they must be put to use just before this cataclysm occurs. A good design of synchrocyclotron magnet is one in which n = 0.2 is reached a t as large a radius as possible; ordinarily this is a t a distance from the pole edge about equal to the height of the gap, though a closer approach to the periphery has been obtained hy a special configuration of the steel. Control over the value of n a t different radii in cyclotrons and synchrocyclotrons is obtained by altering the air gap between the magnet's poles. Within limits, the appropriate value can be computed by plotting the flux lines through the iron and making use of tabulated values of the permeability as a function of flux density for the particular steel under consideration. More generally, empirical corrections are made, either on a scaled model or on the full-size magnet, by altering the contour of the pole tips a t appropriate radii. Rose rings, so called after their originator, are annuli of iron screwed to the flat

THE FIELD I S D E S

27

faces of cyliiidrical poles. If placed n e w tlie periphery they postpone the ultimate r:tpiti drop of field and hence keep n a t w lower value. Discs of sheet iron of varying radii, hyi~imetricallyaligned on tlie poles' axis, can be used t o generate a gentle gradient in the central region which without them would have an alrnost uniform ficld. It is important to note that a particular contour of polc face tlius brought about will produce a specified contour of ficld a t various field levels only if the flux density in the iron is so low that tbe perrnewbility is constant over the field range in question. For reasons of economy this is seldom the case; the tlcsigncr works a t high fields and high flux dcnsitios to get the most for his money. As a result the chosen pole face contour ici correct only a t that one average field strength. Should one wish to operate the cyclotron a t an appreciably lower field, a different pole contour would be required to obtain the same field shape, since the permeability a t 5:irious regions within the rnagnet will have altered. I n betatrons and synchrotrons the magnetic ficld rises during the acccleration process and, as will be sccn later, i t is highly desirable t h a t the field index sliould remtin a t a fixed value throughout. Since by Eq. (2-1) we have n = - r d R / ( l i r l r ) , :t constant value of n will result if tlie gradient clB/dr increases as rapidly :is does R , since r is fixcd in accelerators with pulscd magnets. Il'e nil1 now examine the situation to see if there is any reason to suppose t h a t n will remain constant. We make the assumption (would that it were rcally true!) that the pcrinexhility p of iron is a constant, irrespective of tlic flux density. I n keeping with this naive point of view, we assume the effective are:$ il of the gap to equal the area of the iron transverse to the path of the flux, so tliat in :I prirtiitivc. way we write

where I; is the 1):itli Irngtli in tlir iron, assumed coristant for all lines, G is the g:ip licight 1)etwecn polcs, X I is the number of ampere-turns in the exciting coils, po is thc permeability of space, and p is the relative permeability of iron. Then

where C anti D are constants for a given magnet. Let the subscripts 1 and 2 refer to the inner and outer bountlaries of the vucuurn cha~nher.Hence

I n Eq. (2-1 I , where we have n = - r d B / ( B d r ) , the ficld 13 is the value a t the iioininal radius in whose neighborhood n is assumed constant. In tlic prvscnt

ORBIT STABILITY

28

example we take for this B the value halfway across the chamber where the gap height is fr (GI G2). Therefore

+

Since the current I cancels out, n depends only on fixed parameters and hence should be constant a t all field strengths. Most lamentably, the assumptions made hold only in the crudcst way, since the path lengths are not the same for all lines, and the permeability does change with flux density; so the result is only approximate. But a constant index is so essential for accelerators with pulsed fields that it is well worth while to try to make the field behave in the desired manner by judicious shaping of the pole faces and alteration of the quantity of iron in various parts of the yoke. This is a major experilnental and computational effort which can run into many man-years of work and which becomes more difficult the further the iron is driven towards saturation, i.e., the higher the peak field that is desired. The problem is further complicated by residual fields left over from the previous pulse and by the fact that eddy currents in the iron produce distorting fields. Even though pulsed magnets are made of laminated iron, eddy currents are still troublesome a t low values of the guide field, so the magnet designer must take into account the lamination thickness and the time rate of rise of field. Help in holding the index fixed must often be obtained by dynamic means such as pole-face windings which are activated to different degrees throughout the acceleration cycle. These consist of loops of conductors attached to the pole faces a t different radii. They cause local variations of field and produce corrective gradients. A passive closed loop will create a "bucking" field, since the rising flux within it engenders a current tending to resist the change. An active loop may be driven by a programed generator so as either to "buck" or to "boost," or a large bucking loop a t one location may drive a smaller boosting circuit a t another. 2-3. Qualitative Stability of Orbits

It has been mentioned that the magnetic field in a weak-focusing accelerator must decrease as the edges of the poles are approached. Accompanying this condition is a curvature of the lines of the field-a bulging outward, although the field remains vertical on the median plane of symmetry between the pole tips. This fall-off of field and the bulging can be made greater by increasing the distance between the poles as the radius increases. The force which acts on the projectile is perpendicular both to the field and to the velocity of the ion. As is indicated in Fig. 2-3 by the arrows, the force on the median plane is wholly radial and directed towards the axis, but for particles traveling above or below this plane, the forces have components which drive the ions back towards the mid-plane. It is the axial component of field which produces

++

Q~7A\T,ITA\TIT'T;: PTABITJTT O F ORBITS

20

North Pcle

protons approaching

protons receding

/

/ /

Cyclotron

/,,

,

Synchrotron

Fig. 2-3. A field which decreases radially ( n > 0) produces axially focusing forces because of the outward bowing of the field lines. Arrows indicate force on projectiles.

the radial force, which, in turn, causes the circuhr o r l i t , and it is the r:ttlial component of field tliat suppllcs the :~xl:~lfocusing force. So, if the field decreases radi:tlly, as occurs if n > 0, vertical qtahiliiy will exist and the ion's path oscillatrs above and below the nlctli:~n plane. Puch a motion is cnlled a n axial betatron oscillation, sincc it was first dcwrihed in connection with tlie betatron twcc~lcrator. It is :~p1):rrcnt t l ~ i tthe a x d focusing force will be stronger the more rapidly the ficld drops off-that is, the larger the value of the index n. On the o t l m hand, if tlic opening lwtwecn the pol(>tips is made to tlccrcase as the radius rises, SO that the ficld increaseh outward (71 < O ) , the lines of force bulge t o w ~ r d stlic center and tlie forces arc suc'h as to produce axial instability, as is indiratcd in Fig. 2-4. Wc must also inquire into thc nccci5ary contfitions for radial stability, that is, the circun~stnnces under which an ion, if tlisl)l:tced rnclially from the cquilil~riurnorl)it, will tend to icturn to it rfit1lc.r than move fnrthcr away. It will be recalled that n circular orl)it iq clctcrnlinctl by the equality of Eq. (1-1):

Now suppose B varies in the way described by Eq. (2-4) : B = K / P , whcrc K is a constnnt. Thcn for equilibrium we must have

A plot of thc left side of this against r will give a hyperbola. How the right

30

ORBIT STABILITY

Impractical Cyclotron

impractical Synchrotron

Fig. 2-4. A field which increases radially forces.

(n

< 0)

produces axially defocusing

side plots depends on the choice of n. If n > 1, then qvK/rn falls off with r more rapidly than does Mv2/r and Fig. 2-5a is obtained. The intersection point determines the radius re of the equilibrium orbit. At greater values of r the inward force qvK/rn is insufficient to supply the required centripetal force Mv2/r, SO ions beyond re continually move further out. Correspondingly, a t radii less than re the inward force is excessive and the particles move steadily towards the center. Hence if n > 1 the system is radially unstable. But if n < 1, the situation is as shown in Fig. 2-5b. At r > re the magnetic force exceeds the centripetal force needed for an orbit of radius r so the ion is driven inward towards the equilibrium path, while a t r < re the particle is forced outward towards re. This is the condition for radial stability. Indeed

Fig. 2-5. The system is radially unstable if n

> I,

but radially stable if n

< 1.

BASIC ASSUMPTIONS

31

the index n profitably could be negative (which corresponds to a field that increases with distance from the center) for then the restoring force would be even stronger. In general, for radial stability n, must be less than $1. Recapitulating, axial stability is obtained if n lies between 0 and + 00, while radial stability occurs if n lies between - ~0 and + 1. Stnbilitz~ in both directions can be obtained simultaneously only if n lies between 0 cand +l. The field must not be uniform; it must decrease radially, but not at too fast a rate. This is a fundamental characteristic of all weak-focusing accelerators. 2-4. Basic Assumptions In a quantitative analysis of orbital stability, many approximations are made, as will become evident as the argument progresses. Two of these must be emphasized at the start. The first is that the particles are assumed to travel at constant velocity, the process and the results of acceleration being entirely neglected. This may seem odd, for the purpose of a particle accelerator is to accelerate particles; but it turns out that this aspect of the situation is irrelevant to the problem under consideration, which is simply to discover if the orbits are dynamically stable. It will appear that the periods of betatron motions are comparable to the period of revolution, and a great many turns must occur before any appreciable change in energy can develop; so it is justifiable to treat the velocity as constant, thereby considerably simplifying the analysis. This does not mean, however, that we completely close our eyes to the accelerative process. Procedures exist, as will be demonstrated later, that take into account the effect on the betatron oscillations of a change in velocity which occurs over a relatively long time. It is simply more convenient to study the situation in two stages rather than all a,t once. The second basic assumption is that the radial and axial motions may be treated separately, just as though either could exist without the other. This is by no means as well justified as the first assumption and, in some of the most recently devised machines with complex field contours, its use may give erroneous results if a very detailed picture is wanted of what happens. But even in such machines the assumption yields results which are at least approximately correct, while for weak-focusing devices and the simpler kinds of strong-focusing accelerators, calculations made on this basis are considerably better than mere approximations, provided the amplitudes of the motions do not become excessive. One further preliminary remark must be made. It will soon appear that t,he deviations from t’he closed equilibrium orbit, eit8her radial or axial, can be esprcsscd approximately by an equation of the form id$ + K’y = 0.

32

ORBIT

STABILITY

This is the equation of simple harmonic motion, with a sinusoidal solution for the deviation y. The constant K2 is the restoring force per unit displacement. In the usual mechanical case such an equation implies a mechanism wherein potential energy is stored, to be exchanged for the kinetic form as the velocity dy/dt increases. In the case of particle accelerators, although restoring forces are present, no storage device for potential energy exists and a question may arise as to the origin of the kinetic energy which appears periodically in the transverse motion. The answer lies in the statement that it is actually the total velocity of the particle that remains constant, its azimuthal component dropping when the transverse motion rises from zero. This is much the same as the way in which the over-all eastward motion of a ship must fluctuate if its captain steers a zigzag course which deviates to north and south, the speed through the water remaining fixed. In particle accelerators these variations in azimuthal velocity are so small as to be entirely negligible, so that it is justifiable to assume this velocity as constant on this score, in addition to the previous supposition that no acceleration occurs during the interval in which a few betatron oscillations can take place. 2-S.

Quantitative

Axial

Stability

A simple, first-order analysis of axial stability is as follows, under the assumptions that the particle does not vary in radial position and that it maintains a const,ant azimuthal velocity. In a machine with no straight sections (this limitation will be removed later) consider an ion rotating in a circle with radius r about the axis of a magnetic field that weakens radially. The field lines bulge outward; and although on the median plane the field is entirely axial, there is a radial component B, at regions above and below which increases with increasing axial displacement x measured from the mid-plane, as shown in Fig. 2-6. It is legitimate to express this z-dependence in a power

Fig. 2-6.

Field components for n > 0.

Q~:ANTITATI\‘E

ASIAI,

STAI:IT,ITY

33

srrics with coefficients evaluated at that niid-pi:lne where their values are constant for a given rittliu?, i.e., (2-7) For smaI1 values of z onIy the first term need be retained, which implies that II, yarics linearly with x in this approxinlation. Now in a static magnetic field the curl of 13 is zero in regions where there is no current. The static condition ccrtninlp hol~ls in cyclotrons and synchrocyclotrone and is npproximatcly tile case in a l&ccl-field synchrotron or bctatron, if the situation is limited to the exceedingly short interval in which a few betatron oscillations occur. Thercfore

so that Eq. (2-7)) approximated by its first terq becolncs

whrrc i)R/i)r is written for (i)R,/iIrI. _ (, since nt i := 0 tlicrc is no radial comp o n e n t a n d & = B. By virtue of Eq. (2-1’1, where dli/dr = -n B/r, this becomes B, = --nB;.

(2-10)

A particle, distant z from the mid-plane and moving with constant velocity V, is acted on by an axial force ql)K,. If the sign of the particle’s charge, the direction of its motion, and the orient>ntion of the field are sucll that an orbit about the magnetic axis is produced, then it, is readily apparent that the dir&ion of K,. in a radially falling field is such that qvlZ, is always towards the mid-plane. (Sate from Eq. (2-10) that B,. changes sign with 2.) The equation of axial motion is therefore (2-l 1) with the mass taken as const,nnt since t hc wlocity is nssumcd not to change. From this, 1’ may be eliminated through the relnt.ion u = UT, where w is the radian frequency of rotation, and M is rcl)lawd by USC of the ryclotron rclation Eq. (l-4): MU = pB. With these subst,it,utions t,here results (2-12) This is the well-known Kcrrt-Serlw equation for axial bctatron mot,ion. One possible solution is x = 0, reprwenting the axially unperturbed orbit. Another

34

ORBIT

STABILITY

solution describes simple harmonic motion as long as the coefficient of x is positive, that is, for positive values of n, in accord with the previous remarks on the conditions for axial stability. Such a solution is 2 = zm sin nwwt 7 (2-13) which indicates that the particle oscillates about the mid-plane with some amplitude zm and that the radian frequency of this axial motion is w. = db.

(2-14)

The number of axial betatron oscillations per revolution (i.e., the axial radian frequency measured in terms of the rotation radian frequency) is

y, = sw = n’“.

(2-15)

This is an important relation that holds in all purely circular weak-focusing accelerators. (The corresponding expression if straight sections are included will be derived later.) Since n is less than unity, there is less than one axial oscillation per turn. It may be noted that while Y, (or vu) is the conventional symbol in the United States for the axial (i.e., vertical) betatron oscillation frequency, European authors employ Qz or Q,. The corresponding frequency of radial motion, soon to be described, is denoted by vZ or a+ in the West and by QZ or Qr abroad. Sometimes the value of vZ or vz is referred to as the tune. Equation (2-12) puts no limit on the magnitude of n, provided it be positive, the “restoring force” w’nz becoming more powerful with increasing index. Equation (2-15) shows that the axial betatron frequency rises without limit as n is increased. Thus, as far as axial oscillations are concerned, a large positive value of the index is desirable. It is only because radial instability arises if n > 1 that a value between 0 and +l is mandatory in weak-focusing accelerators. 2-6. Quantitative Radial Stability In polar coordinates, radial acceleration is given by the expression A, = (d%+/dP) --T (de/&) 2. With particle orbits, the radius is always almost perpendicular to the path so that, with good accuracy, it may be said that rdO/dt = T-W = v, where v is the azimuthal velocity. Therefore A, = (dY/dt”) 2/*/r. The radial equation of motion is found by multiplying t,his acceleration by the mass and equating to the force qvB, produced by the motion with velocity v through the axial component B, of the magnetic field. By restricting the motion to lie in the median plane, B, becomes identical with the total field B. Hence the equation of motion is dzr Mdt:

MU? --= r

- qvB.

The minus sign is used on the right bccnuw the force. acts to reduce r, and since t l ~ evclocity is assunled fixed, it is legitimate to thus write Newton's second law wtli constant rims. A possible solution to this equation is r = constant, i.e., x circular path. Then the first term vanishes and

so Mw

=

qB

and the momentum is p z Mu = qBr.

These three relations have Been derived earlier as Eqs. ( 1 - I ) , (1-4), and (1-2) on very elementary considerations. Such a circular path (the equilibriunl closcd orbit for an ion of momentum pl can exist only if the projectile is launched tangtmtially on the orhit with just the right velocity. I s such an orbit stable? That is, if the particle 1)ccomes displaced from the circle by some circumstance, will it tend to rcturn to its former path or will it deviate further? T o determine this, let the radial coordinate be re -t rc, where x is the deviation from the equilibrium circular orhit of constant radius of curvature r,, with x r,. Tlien the equation of motion is

<

Md"(r,+x)

- Mu?

dt'

where B.,, is the field a t radius r, be expressed as

r,

+ x + qvB,.

=

0,

+ rc. Bnt s < re, so the factor l / ( r , + s) can

Therefore, since r, is constant, the equation of' motion hecomes

( );

s A!ll)Z nf 8 - - -- 1 -- + dt' r,. Rut Mv2/r, = qvB,, where R,. is t l ~ efield :it r,, so

=

q z l ~ ,

0.

and

For small displacements, the radial dependence of the field can be expressed in a Taylor's expansion: R, = £3, xtiB/dr - . so that B, - B, m xdB/dr. wr, for the off-course ion, Eliminate v through the approximation that v divide hy dl, arid recall that q / M = w/B,. Then

+

36

ORBIT STABILITY

By Eq. (2-1) the field index in the neighborhood of re is n = -r,dB/(B,dr). So

This is the Kerst-Scrber equation for radial motion. A possible solution is x = 0, corresponding to the equilibrium orbit. An alternate solution indicates simple harmonic motion of the deviation x, prorided the index n is less than unity, in agreement with the qualitative argument given earlier. Such a solution is x = xmsin (1 - n)%t, (2-22) where x, is the amplitude, and the radian frequency of motion is given by Consequently the number of such oscillations per turn (that is, the radial l~etatronfrequency in units of the revolution frequency) is given by the important relation

This is valid for all weak-focusing machines without straight sections. Note that Eq. (2-21) still represents oscillatory motion even if the index n is negative, which corresponds to a magnetic field that increases with rising radius. Note also that Eq. (2-24) shows that the radial betatron frequency becomes greater the more negative the value of n. A field of this sort would be highly desirable as far as radial motion is concerned, were it not for the fact that it would produce axial instability. The successful performance of weak-focusing accelerators depends on the fortunate circumstance that there is an overlap in the permissible values of n appropriate for axial and radial oscillatory motions. Orbits that are stable in both directions can exist only if n lies between 0 and +l. The relevance of the field index to a certain variety of mass spectrometer is worth noting. If n = 3, then by Eqs. (2-15) and (2-24) it is seen that v, = v, = V T ; that is, there is 4 of a wavelengtl~of both radial and axial motion in 2 r radians of orbit. Consequcntly, the angular path in which 4 wavelength occurs is ~ - \ / 2 radians. This fact forms the basis of the "Q,@' spectrometer, for if a point source of charged particles is placed on the mid-plane of such a field, particles of the same momentum which are emitted with radial and axial divergence are brought to a. point focus rV5 radians (254.5") further around." *See N. Svartholm and K. Siegbahn, "An Inhomogeneous Ring-Shaped Magnetic Field for Two-Dimensional Forusing of Electrons and Its Applications t o p-Spectroscopy." A~lc.f . M n t . Astr. o . Fysik 33A, No. 21 : 1-28 (1917).

RADIAL RIOTION O F IONS \\'ITI-I MOMENTU?\I p i - d p 2-7. Radial Oscillation of an Ion with Momentum p

37

+ dp

It has t)ren seen that a circular path of radius r, is possible for an ion of n~oinentump :md that if i t is displaccd i t will oscillate about this circle. It is now of intcrcst to inquire about tile ort)it of a particle of greater rnoincntum p dp. It is clear that it also can describe a circul.tr path and describe stable oscillntions about it, but it will bc instructive to disc-over the form of the equation representing tliis motion in ternis of par:uneters based on the circular orbit of the ion with ioomentum p , for tliis will g i w some insight into what happens wlicn a particle's velocity is suticlenly increased, as when it crosses the accclcrating gap. Bincc the existence of a nloinentuin p d p iinp1ic.s increased velocity and increased mass, the new orbit will lie a t a greater radius and in a different field. \Ire rewrite Eq. (2-20) in the form

+

+

The new wlocity, and hence the new mass, are both assumed to be constant. .4s before, the radial deviation from the circular orbit of an ion with momcnturn p is g i w n by r = r - r,. Again expand l / ( r , 4-x) into (1 - r / r , /r, ant3 then p c d o n n the indicated operations, dropping all terms abovc first order in small quantities x, d M , (?I,, and dB. Since qB, = M u , the term B,qv becomes Muzl, which is also e.;prcssihle :is Mv2/r,, and this cancels with a corresponding negative term. \LTe obtain the intermediate expression

Kow v / ~ , ? w and @, = Mu, while Ixy Eq. (2-1) n-e have dl? = -nB,x,'r,, nhere x is written for dr, so that q~ldB== -qunB,x/r, Hence d 'z - 2Mwdv - vwdM clt"

M --

=

=

-nB,dr/r, --Mw"nx.

+ h!lw2x - Mw'nx + Mwdv = 0.

On collecting, wt: find d2x Mdt'

Rut wdv Hence

=

+ MwY(l - n)x = Mw dv -t- vw dM.

w~dv/v = r,w2dv/v and since p

=

Mz:, then d ~ / u= dp/p - dM/hil

tlt'

+ w ' ( ~- n)x = r,w2 ($ - '$)+ reu! tlM M

dt'

+ w (I - n)r = r.w- rliv -.P

d'x

-

-7

$8

ORBIT STABILITY

38

+

This is the equation of motion of an ion with momentum p dp, its displacement x being measured from the circular orbit of a particle with momentum p which rotates with radian frequency w a t radius re. The quantities p, dp, w, and r, are all constant. A solution may be found by writing the equation in ax - b = 0, where a = w'(1 - n) and b = r,w2dp/p. Set u = the form x ax - b to obtain ii au = 0, of which a solution is u = u, cos (a"t 3- K ) where the constant K is zero if u = u, a t t = 0. Then x = (b/a) (u,/a) cos a x t , and if x = 0 when t = 0, then u, = - b. This gives x = (b/a) (1 cos a"t). Substitute for a and b to obtain

+

+

+

-

The significance of this may be seen as follows. Suppose an equilibrium ion is rotating in a circle of radius re, and a t t = 0 it receives a single momentum increment dp as the result of crossing the dee-to-dee gap, assumed infinitely narrow. The dee voltage is then turned off. The motion becomes a cosine oscillation of amplitude x, = re d p / ( l - n ) p about a reference circle of radius r, x,. It is clear that the succession of "semicircular" orbits mentioned in elementary descriptions of a cyclotron is somewhat of an oversimplification. They should be considered as portions of sinusoidal radial betatron oscillations described about a reference circle. A new portion of such an oscillation is generated a t each acceleration. The accuracy of this description depends, of course, on the validity of the approximations used in obtaining Eq. (2-26).

+

2-8. The Initial Amplitudes of Betatron Oscillations

In order to design the vacuum chamber and magnet gap in which it lies so as to be large enough and yet not too large, it is desirable to inquire about the amplitudes of the betatron oscillations. Unfortunately there are no definite answcrs, any more than there is to the query: "What is the amplitude of a pendulum?" It all depends on the starting conditions-the initial displacement or transverse velocity or both. Information as to these parameters of the ions must be obtained by independent means, very often past experience, pious hopes, or sheer guesswork. For proton synchrotrons the problem involves knowledge of the angular spread of the beam as it is injected into the machine, plus information as to just where its center lies with respect to the ideal orbit. I n cyclotrons and synchrocyclotrons the ions usually emerge from a small hole in the side of the "chimney" of the arc source near the center of the magnet. The vertical component of electric field between the chimney and neighboring dee can be calculated (but usually is not) in order to estimate the initial vertical velocity of the ions. But even if such information is lacking, it. is interesting to compute its influence on the amplitudes of the resulting oscillations.

ADL4BATIC DALIPISC, O F BETATRON OSCILLATIONS

39

Using 74 as the coordinate for either axial or radial motion, the governing equation has h e n shown to 1)e of the form

where K2 = 7~or (1 - n ) . The solution is y = y, sin Kwt from which the transverse velocity is y = y,Kw cos Kwt, so that the maxtmum t,ransverse velocity is y, = y,Kw. Hcnce, specifically, 2,

=

i, and ~ 7 ~ 4 5

s, =

x, w(l - n)$$

(2-28)

These results may be converted into alternative forms involving the maximum angle of divergence 0 , measured with respect to the desired direction. If v and L arc tlw tangential velocity and the distance traversed, then y

=

dy/dt = (dy/dL)(dL/dt)

=

v tan 0 zr: u0 for small 0.

The maximum divergence, 0,) occurs when the orbit is crossing the axis with t,he maximum transverse velocity y,, so y, = v0,. If T is the period of revolution and C = VT is tihe orbit's circumference then

But K

=

n44 or ( 1

- n))4,so that

Thus for given initial conditions of transverse velocity or divergence in angle, the initial amplitudes of the betatron oscillations am dependent inversely on the restoring forces, that is, inversely on the square root of n or of (1 - n ) . The field index lies between 0 and + I for constant,-gradient (weak-focusing) accelerators; as will he seen later, there is a class of alternating-gradient (strong-focusing) machines which makes use of n values ranging up into the hundreds, so that greatly reduced amplitudes are possible for the same initial conditions antl accelerators hecome cheaper or more cnergetic ones can be built for the same cost. 2-9. Adiabatic Damping of Betatron Oscillations

As it stands, the axial betatron oscillation expression, Eq. (2-12),

ORBIT STABILITY

40

implies a sinusoidal motion of constant amplitude. It must not be forgotten, however, that in deriving the equation in this form the simplifying assumption was made that the tangential velocity did not change; this implies constancy of the mass, of the radian frequency, and of the field index. Such a procedure was legitimate, for interest lay in a solution valid only over a relatively short interval (say a few dozen revolutions) during which time none of these quantities change appreciably. But for the long-time behavior during the entire acceleration interval, which may be a second or more in a proton synchrotron, the changes must be taken into account. The equation is therefore now written in more general terms, in accord with Newton's second law, in the form

Here is a differential equation in which t,he coefficients of both z and 2 are time dependent,; the mass increases always, w drops, and n rises in cyclot,rons (both fixed-frequency and frequency-m~dulat~ed), while w rises in a synchrotron, where n is fixed, as will be shown later. By performing the indicated differentiation and then dividing by M there is obtained

where b = d n varies with time. The S term with positive coefficient indicates that the motion is damped, the reduction in amplitude being dependent on the rate of increaee of mass. Since the changes in M and in b occur slowly with re~ p e c to t the period of oscillation, an approximate solution may be found by the FO-calledWKB method (abbreviated from the initials of its originators, Wentzel, Kramers, and Rrillouin). This goes as follows. Take as a trial solution 2 =

1

(2-33)

zJ(t) exp [=t3 ~'~(t)dt],

where f ( t ) is some as yet undetermined function of time and j = d-1. That this is a reasonable solution can be seen by imagining that both f and b are constant; the assumed solution then becomes a simple sinusoid, which is correct if the coefficients in the differential equation are constant. We now form the first and second derivatives of z :

Khen t h e ~ eexpressions for z, i , and I are substituted in Eq. (2-32), and z, and the exponential term are canceled out, the result is f =J= 2fjW

- fb

fjb

=L- -

$b4$

A f. f --fib$$ $1 ++ fb M M

=

0.

XII1~WAITIC DAMPIT\;(: OF BETATRON OSCILLATIOKS

41

The third and last turn5 cancrl. Since thr, time \ariation\ off arid M arr assumed .mall, the j and the A?j terms may he dropped as iucollsequential. Then divide 11y2 j V j to obtain

RIultiply by

tlt

t o find

Hence In j = In b-'~

+

+

111 ~l/l-'-'In

K

=

In (b-'"1-WK),

here K is the. constant of integration. 'l'hci~

I3ut M

=

@/w,

and using this value of .f in Eq. (2-33), we find 2 =

collst ant rb'i]j'5

crp

[+

/'

on54<1t].

'I'his shous thv motion to hc a damped wine wave with the amplitude dying out as l/(n'Q12). h similar analybis for the radial motion obviously results in

I n synchrotrons w l w c the field rise5 with t h e by :I factor of pe11l:tps 39, ~ ~ l i inl cis held constant (or nearly s o ) , the :rmplitudt~sdecrease in inverse proportion to tlic squ:ire root of the field. Therefore although a large cross-wctional region of properly shaped field is essential st injection tinw, only a much snlallcr spacc h t m e n the pole> need have :r "good" field a t the top cncrgy (provided thc beat11 is kept c,cntrred in the apcrturc by an appropriate energy gain pcr turn anti by an adequately controllctl frcqucncy of the accelerating voltage). This ulti~naterelaxation of tolcrancc. is of great help to the magnet designer, wllilc the small cross section of the high-emcrgy bean1 is of advantage to tlic, cxperimcntalist. I n cyclotrons and synchroryt~lotronstlic field is alinost const:tnt, dropping only a few pcrccnt a t thr largeqt orbits, whilc the index n riqcs fro111 close to zero ( a t tlic point just off t h r axis where the ion source is located) to a. value around 0 2 or 0.3 a t top energy. C o n m p c n t l y tlic vertical oscillations arc) damped, h t the ratiial anq)litutlcr arc increased during the acceleration process. For refcrenceh to general rcvimv articles on cyclic machines, see pp. 341-343; for the theory of orbits, see pp. 343-344.

Chapter 3 MATRIX METHOD OF CALCULATING STABILITY

3-1. Introduction Thus far the discussion of orbit stability has been limited to circular accelerators. But, as has been mentioned, many synchrotrons are built with straight sections interposed between portions of thc magnet. I t is apparent that in such field-free regions immediate control of the particle's trajectory is lost, the ion flying in a straight line until it comes under the influence of the guiding and stabilizing field of the next magnet. Obviously the straight sections must not be too long or many particles will miss the succeeding magnet entirely. Intuitively it can be seen that the field-free regions will have some effect on the criteria for stability and on the values of the radial and axial betatron frequencies. A method of investigating this more involved shape of orbit is the subject of the present chapter. The procedure to be expounded has considerable generality and will be found useful in the study of the even more conlplicated guide fields of certain other types of accelerators. The exposition is necessarily somcwliat lengthy but it results in a simple prescription, quite easily applied in a wide variety of cases, permitting determination of stability criteria and betatron oscillation frequencies in a matter of minutes instead of hours or days. Thc method may be employed in the study of orbits in those cases in which the restoring force per unit displacement remains constant for a finite azimuthal length. The particular procedure to be described determines the particle's transverse displacement and transverse velocity a t the boundaries of these regions, somewhat as though one examined the path of a bouncing ball by looking a t every tenth frame of a motion picture film of the event. If the particle's behavior is satisfactory a t these check points, an argument of continuity gives assurance that nothing unsuspected or catastrophic occurred in between. If desirable, the grain of the investigation can be made much finer, but ordinarily the procedure to be described is adequate. Since matrices will bc employcd to reduce the labor of performing successive l i n ~ a rtransformations, the initial step will be to recall this process to mind. 42

3-2. Linear Transformations by the Use of Matrices Assume two pairs of linear equations with constant coefficients, the first pair the second pair transforming x, and ! j 2 transforniing .rl and y 1 into x1 and into x, and 7~~ xz = allxl a w l (3-1) yz = ~ 2 1 x 1 a22yl (3-2) XR = b i i . ~ ~ b i z ~ 2 (3-3) y3 = h215? b22?~2. (3-3) hy snhstitution: Eliminate a:, and

+ + + +

w l d e Eqs. (3-3) and (3-4) take the f o m

and Eqs. (3-5) and (3-61 become

where the c's are abbreviations defined as

c2?

=

+

bzlal?

b2?02?.

If formal substitution of Eq. (3-7 I is made in Eq. (3-8) thcre rcsults

By comparison of Eq. (3-11) with Eqs. (3-9) and (:3-10) it is clear that

44

MATRIX METHOD OF CALCULATING STABILITY

This formal pattern can be used in multiplying together any pair of matrices (that is, in substituting one set of equations in another set). It will be observed that each element is expressed in the form

Care must be taken not to commute any two matrices. That is, for example, in finding the product of four matrices M4M3M2M,, the operation is carried out in pairs such as (1M4M,) (M2M1) or as M4[Ma(M2M1)] or as [ (M4M3)M2]M1 etc.; but one must not interchange the order and write (M3M4)(M2M1) etc. 3-3. The Criterion of Stability and the Betatron Frequency The first-order equation of axial motion, applicable in a synchrotron to the parts of the orbits that lie within a magnet, has been seen to be

where w is the local ra.dian frequency of revolution given by w = v/r, with r a constant and v assumed so to be. The distance traveled in time t is L = v t so that dt = dL/v. Consequently Eq. (3-14) can bc written as

A general solution of this is

n55L

z = A cos r

+ B sin n$$L r -1

(3-16)

where A and B are onst st ants. The slope of the orbit is $ = -dZ= dL

n54L +B cos -. r r

dP n%L -A sin r

r

4,1

(3-17)

Let the subscript 1 refer to that edge of a magnet a t which the ion initially enters the system; this will serve as a fiducial mark. At this point where L = L1 = 0, assume as initial conditions z = zl and z' = zl'. The values of A and B are then found to be A = zl and B = rzl' -. (3-18) n $5 Let m be the length of path within a magnet, measured along the equilibrium orbit. (The extremely small change of path due to the sinusoidal character of the betatron oscillations is neglected.) The position and slope of the orbit a t the end of the magnet where I, = L:! = m is then given by

By employing the short-cut notation of nmtrices these two equations iiiay be written in the form r (3-21)

n!L.m

-sin

--

cos --

r r On leaving tlie magnet the particle enters a straight section where there is no magnetic field antl a t the point of entry the disl~laccmentand slol)e are z 2 and 2.,'. ( I t is here assunled t h a t the fare of tllc niagi~etend is perpendicular to the equilibrium orblt, so tliat thc fringing field a t t11e end has no effect on the radial and axial inotions \ \ k i t 1l:tppens nlien this is not the case will be dcscribed in duc coursc.i \Yliile crossing tllc field-free region the slope remains constant but t h e displacement increases linearly with distance, the proportionality factor being tlie slope xz'. \\-l~enthe straight scrtion of azimuthal length s has been traversed, the position antl slope are given by

These equations can be written in matrix form a s

T h e particle has now traversed one complete set of the different elements which make u p its 1)atll: one magnet :inti one straight s c c t ~ o n Such . a complete set is called a sector and some integral nurnbcr S of identical sectors forms the cntire circurrlftwnce. T h e p:rrnnletc~rs2:( :mtl z:(' a t the end of the sector are given by Eq. (3-24 I in terms of a:! and z z f , and thcxv in turn arc cxl)ressed by Eq. (3-21) as functions of the initial values 2 1 a n d zl'. \Ye can eliminate x2 and x2' by substituting E q . (3-21) in Eq. (3-24) :

. T

sin --

T

(3-25)

cos -

This indicates tliat we s t a r t with 21 and x l l , pass tlirougll thc n1:iguet and tlwn tl~rouglithe straight section and finally obtain the x:~lucbs -, and i:<'. S o w n ~ u l tiply the matrices togcther in the manner described c w l i w to find

,

46

MATRIX METHOD OF CALCULATING STABILITY

where we have used the abbreviations

Mu

=

n%"m n% n%m cos - - s - sin r r r

1

It is very important to realize that the matrix M has been derived for any sector and hence has the same value in all sectors. The differences, if any, between the orbits in successive sectors all result from changes in the initial values of z and z' appropriate to each sector. I t will be convenient to specify the displacement and slope of the orbit a t a given azimuth as a single point on the zx' plane, so that a line from the origin to this point represents a vector Z given by

In this notation Eq. (3-26) becomes 2 3

=

MZ1.

(3-29)

But the azimuthal distance from index 1 to indcx 3 is that of a coinplcte scctor of length S, consisting of one magnet and one straight section, so that we may also write the last equation as When the particle traverses a second sector and is then a t a distance 2S from the fiducial azimuth, its displacement and slope a t the end of the second sector are determined by the values of these parameters a t its beginning, and these are the same as the values a t the end of the first sector. Hence we write where the matrix M is the same as for the first sector, as was pointed out earlier. Substitution for Zgfrom Eq. (3-30) gives Zts = M(MZ1) = M2Z1.

(3-32) That is, the matrix 144, specified by Eqs. (3-26) and (3-27), must be multiplied by itself in order to yield x and z', after traversal of two sectors, in terms of zl and 2,'. Similarly the displacement and slope after one trip around the machine composed of N sectors is described by

CRITERIOX OF' ST.4HI1,I'I'Y

47

while the values a t the final energy, rearhcd after perhaps lOVurns, arc ohtnined from

Zhnal= 11'05 NZ1.

(3-34)

'l'o evaluate AT', with ,V = 4 say, is no trivial task, but to carry out the program specified by Eq. (3-34) is utterly inlpossiblc. Fortunately such Herculean labor is not necessary, for a mathematical stratagem exists which leads directly to a critcrion of stability and we do not have to spend our lives following the path of a projectile through hundreds of thousands of sectors. This stratagem will now be developed. In general when a vector Z with components x and z' is multiplied by an arbitrary matrix M, the resulting vector M Z has a different direction and a different magnitude than the original vector Z. For esample, suppose z = 3 and z' = 4 and that we give M a special value so that me write

Then

for M Z represents the two ecluations z = O X 3 + l X 4 = 4 zf=2X3+0X4=6

Therefore M Z has a different length and direction than Z . For reasons that will appear later, it will be extremely convenient if one (or more) particular vectors can be found with the property that multiplication by a matrix leaves the direction unchanged and simply alters the length by some factor A. We postulate that this does occur and endeavor to discover the necessary circumstances. Hence assume that we may write

MZ

=

XZ.

(3-35)

Here M is the matrix transforming through a single sector and is some n u n bcr as yet undetermined. Written out in full, this expression represents the two equations Mllz MI2z' = XZ (3-36)

+ Mzlz + M&

=

XZ'.

(3-37)

These can also be given as

Such a pair of linear homogeneous equat,ions can have values of z and z' different from zero only if the determinant vanishes. (Zero values of z and z'

48

MATRIX METHOD OF CALCULATING STABILITY

represent the equilibrium orbit, which is of no present interest.) On setting the determinant equal to zero, we find (Mil

- h)(Mzz - A )

-

MiZMzl

=

0.

When multiplied out, this gives

+

+

X2 - (Mil M22)X (MiIMzz - M12M21) = 0. (3-40) A digression must now be made to prove that the last term in this expression is equal to unity. Suppose there are two solutions z , and z , of the original equation of motion, Eq. (3-15)) so that we have

and

Multiply the first by z , and the second by z, and then subtract. This gives

Hence

d

dL ( z v

dz,

- z, h) = 0, dL

and so z,

dz,

--

dL

- z, dz, - = constanb. dL

The quantity on the left of this expression is known as the Wronskian of Eq. (3-15) and is seen to be independent of the variable L. Therefore the Wronskian has the same value a t all points along the orbit. We evaluate it a t the start of the sector (subscript 1) and a t the sector end (subscript 3) : Z ~ B Z , ~ '-

~ ~ ~ , l1z ,'l ' .

(3-44) Now write each expression on the left in terms of its value a t the start of the sector, by use of the expressions represented by Eq. (3-26). (M1lzvl

+

M1zzvll)(MBAI

zuazv3' =

~

~

1

+ Mmzull) - (M1lzul +

M12zUl1) (M21zv1

-

-I- M22zvll) zvlzul'

-

zulzvll.

Multiply out the left side, cancel, and collect, to find (MiiMzz

- M12M21)(z~1~~1' - ~ ~ l z ,= ~ 'Z ~) I X , I '

-~

~

1

~

~

1

~

Hence MiiM22

- Mi2Mzi

= 1,

which was to be proved. As a consequence, Eq. (3-40) becomes

(3-45)

.

CRITERION OF STARII,ITY

49

This quadratic shows tliat if Eq. (3-35) is t o be valid, the factor ~ i i u s tbe intiniately depc~ntlcnton the vahrcs of t l ~ et1i:tgonal eleiiicnts of the matrix M. I n addition it sliows t h a t there are two values of A:

+ hf22) + +[(MI1+ M22)' - 419' +(lMi~+ 4122) - a[ (Mi, + M22)' - 41)'. +(MI1

and A,

=

(3-17) (3-46)

Consequently there arc two v:tlucbs of tlic vector Z (cull them Z, and ZI,) wliicli have the desired property t2i:it when niultiplicd by the matrix ;I1 they are cliangcd not in direction but only in magl~itudeby t l ~ cfactors A,, and XI, respectively, providrtl that A, and All I ~ a v etho values specified by Eqs. (3-47) and (3-481. T l ~ e(1xistcncc of such spcc3ial vc.ctors is of great convcnicncc, for no nmttcr what the value of the original vector Z (\vliicli represents the clisplaccnient a n d slope of the orbit a t t h r beginning of a sector), i t can be considcrcd as t h e resultant of Z, and Zbwlien each is multiplied by some appropriate numbcr A a n d R. Thus: (3-49) RZb. Z = AX,

+

Consequently when Z is multiplictl by tlic. niatrix 4 1 there results

MZ

=

AMZ,,

+ BMZb.

(3-50)

Passage tlirougli a n y number N of sectors is then given by nlultiplying both sides by M Z - l to yield M N Z = ,4MNZ, BMNZ,$ (3-5 1 )

+

a n d by t h e special characteristics of Z, and Z b given by Eq. (3-35) this beconic~s

Now if i t tiappcms tliat either A,, or A,, exceeds unity, tlicm i t is obvious tliat % will grow :IS c w h stlctor is t r a v e r s d , so t l ~ a tz and z' also grow and tliercfore the orbit is ur~st:ible.On tli(3 other 1i:tnd if i t turns out t1i:it hot11 A,, arid All are and x' do not grow and the orbit is stable. equal to o r lcqs than unity, thcn Z, i, So let us ex:iinine Eqs. (3-47) and (3-48)) which specify the A's. It is clear t h a t two conclusions m a y be drawn: and

+

( T h e quantity Mil M2ais the sum of tlic c>l(mcntsof the principil t1i:rqonal of the niatrix ,ZI and is called the T ~ CofQM , a h t ~ r w i a t e da s Tr J f ) \\ liatever m a y be the values of A, and At,, the relation oE Eq, (3-53) will he iiiaintained if thc tlrfinitions are niadc t h a t A,

= elD and X b = e-W,

(3-55)

50

MATRIX METHOD OF CALCULATING STABILITY

where w is some unspecified quantity. Eq. (3-52) can hence be written MNZ

+ BCNWZb.

AeNwZ,

(3-56) If w is real and of no matter what sign, one or the other of the exponentials exceeds unity and the orbit is unstable, for the reasons given above. But if w is imaginary, it may be written as =

w = ju,

(3-57)

where j = 1 / q a n d u is some real number. Equation (3-55) becomes A, = eia and

Ab

=

e-1".

(3-58)

Therefore Eq. (3-56) may be expressed as MNZ

=

+ Be-iNaZb.

AeiNcZa

(3-59)

Here Z , and Zb determine the initial displacement zl and slope zl', while MKZ specifies these parameters after the particle has passed through N sectors. Equation (3-59) can be written more explicitly as

where the elements of the matrices on the right represent the initial values of the con~ponentsof Z , and Z b into which the true initial displacement and slope have been resolved. Written as two equations, this becomes ZN

ZN'

+ = AzlaleiNa+ B z ~ ~ ' ~ - ~ ~ ~ . =

AxlaeiNu

(3-61) (3-62)

Now we know that ZAT is real, so the right side of Eq. (3-61) must be real, and hence Bzlh must be the complex conjugate of Az,,. Therefore write Azla

=

p +jq

Bzlb

=

p - jq,

(3-63) (3-64)

where p and q are real numbers. Then Eq. (3-61) becomes ZN

=

(p

+ .jq)e3Nc+ ( p - jq)e-iN"

= p(eiNr =

+ e - i N ~ ) + jq(eiNa - e-nVc 1

21, cos N u - 2q sin N u

Define the angle 8 such that this equation may be written in the form ZN

ZN

+ q2)b4[cos8 cos N u - sin 8 sin N u ] = 2(p2 + q2)t%os (0 + Nu). =

2(p2

(3-66)

CRITERION O F STABI1,ITX'

51

From the definitions of the symbols in\-olved, it is clear that 0 and the amplitude of the above cosirle function are real and constant, since they depend on the initial displacement of thc particle. If N,the nurnher of scctors traversed, is consitlt red as a running v:~riable,then Eq. (3-66) specifies the value of z at the crid of each sector as X takes on successive integral values. I t is apparent that thcse particular valurs of z p ~ ~ r t a kofe a cosine oscillation, provided that a is real, as assumed in Eq. (5-38),and that a is the change in phase of the oscillation w l i c ~:my singlc scctor ih trawrsed. A similar argunlcnt m.ty be n ~ : ~ (wit11 t ( ~ rchpect to Eq. (3-62). \Y(,know tlmt 2,' is real, so that 1 3 ~ ~nlust ; be the coniplex conjugate of A Z , ~ ; Set . AzLU;I= p' jq' and R s ~ ~= , ' p' - jqf, where p' and q' are real. The argument proceed.; as before to yield (3-67) zNf = 2(pf2 p ) $ l cos (8' N U ) .

+

+

+

Here again 0' and the amplitude are real and constarlt so that a is again the phase change per sector. The last two expressions, Eqs. (3-66) and (3-671, indicate a cyclic, stable behavior of the orbit provided that a is real in the definitions A, = eio and A b = e-la, Eq. (:3-58). But t,he A's are related to the magnetic and geometric properties of :t sector by Eq. (3-54) which reads ha

+ hb

= M11

+ Mzz.

By the use of Eq. (3-58) this becomes which can be written cos a,

=

+

+ ( M I , Mz.J

=

$Tr M .

(3-69)

Here the subsvript i has en added to indicate that the discussion refers to axial motion. T h i s expression states the criterion for cczzal stab7lity: if MI1 and Af22, :LS given by Eq. (3-27), are such that half their sm~tzzs a n u m b e r which lies between +1 and -1, then a, is real and the orbit i s axially stable. This conchsion is valid for all accelerators in which the axial forces per unit displacement are constant over finite azinnuthal distances so tha,t matrices may be used to calculate tlie position and slope of the orbit a t one end of such a region in t e r m of thc values of these parameters a t the other end. Since a, is the change of phase of the betatmn oscillation per sector, then in one turn of N sectors Nu, gives the phase shift per turn. Consequently the number v, of axial betatron oscillations per turn is given by

A similar argument is applicable in determining tlie criterion for radial stability. The cquations of radial motion are set up for each portion of a sector,

52

MATRIX METHOD OF CALCULATING STABILITY

solutions are found and a matrix is constructed which transforms the initial displacement and slope XI and xs' into the corresponding values a t the end of the sector. The orbit will be radially stable provided a, is real in the expression

where now M is the matrix appropriate for radial motion. The angle a, is the phase change pcr sector of the radial betatron motion and is related to the number of radial betatron oscillations per turn v, by

where, as before, N is the number of sectors making up the machine. 3-4. Application to a Synchrotron with Straight Sections

We may now quickly apply the results of this lengthy derivation to the weakfocusing synchrotron with straight sections. Equation (3-27) gives the elements of the matrix which transforms z and z' across a sector, but we need be concerned only with MI1 and M22. From these expressions we eliminate the radius of curvature r through the relation since N is the number of magnets in the machine and m is their common length. Equation (3-69) takes the form

n s n96 cos u, = cos 2n - - - 1- sin 21r -*n45 N m N N If the magnetic and geometric properties of the synchrotron are such that the right-hand side of this expression lies between 1 and - 1, the system is stable and a numerical value for u, may be obtained, from which the axial betatron frequency v, can be calculated by Eq. (3-70). To find the criterion for radial stability, write the equation for radial motion applicable within a magnet:

+

dt"

+ u2(1 - nlz = 0.

This differs from the expression for axial motion, Eq. (3-14), only in that (1 - n) replaces n; so with that change, the matrix of Eq. (3-21) takes the particle through a magnet. The straight-section matrix is given by Eq. (3-24) as before. It is clear that the argument will proceed in the same way to give the analog of Eq. (3-74). The criterion of radial stability is contained in the expression cos uz = cos 2n

(

1 N

)

s (1-n)$s (1 -sin 2 1 m= N

- n)54 N

(3-76)

APPROXIMATE VALVES OF BETATRON FREQUENCIES

53

From the form of Eqs. (3-741 and (3-76) it is seen that the presence of straight sections reduces the values of cos u, and cos a, and therefore raises u, and u, so that the betatron frequencies v, and v, are increased. This latter conclusion can be seen directly on physical grounds, for the longer path nlakcs one betatron wavelength a smaller fraction of the circumference, so a, greater number of oscillations occur in one turn. The amplitudes of the motions are also increased because the field is not present in straight sections to steer a divergent ion hack towards the equilibrium orbit. Consequently the particle gets farther off course before the field can apply corrective measures. I t is clear that the ratio s / m has an influence on the limits of stability, but in gencral t l k is of theoretical rather than of prac?tical importance. Thus for the customary values in a weak-focusing synchrot~ron,N = 4 and n = 0.6, we see by Eq. (3-74) that cos u, would become rnore negative than - 1 only if s / m exceeded 2.37, while cos u, would be in similar trouble only if s / m were greater than 3.72, as follows from Eq. (3-76). In practice, straight sections are always considerably shorter than thc magnets, so there limits are far from being approached. On the other hand, it may be noted that the presence of s&aight sections in no way alters the restriction that n must lie between 0 and 1. For suppose n = -6 where 6 is some positive number; then nlh = (-a)% = $3" and Eq. (3-74) becorncs cos a,

=

21 cos - '6)"

iV3

sx . mLV.

-@%!in

2~

j6s N

But in general if 0 > 0, then cosh 6 > 1 and sinh 0 > 0 ; so the right-hand side of the above equation exceeds unity, a, is not real, and the orbit is axially unstable Similarly if n = 1 6, then ( I - n)b4 = ( - 6))' = ,i6'$ and Eq. (3-76) for cos u, becomes identical with the equation above for cos u,, and a, is not real and radial instability will occur.

+

3-5. Approximate Values of the Betatron Frequencies

Approximate expressions which directly exhibit the connection between the machine parameters and the betatron frequencies can be derived by expansion of Eqs. (3-74) and (3-76). Thus cos a , may be replaced by 1 - ut2/2, the error committed decreasing from 26y0 for a , = 70" to below 1% if u, is 40" or less. On the right side of Eq. (3-74) or (3-76), expansion is carried to several terms (that is, sin x = x - r8/3! 5"/5! and cos x == 1 - x 2 / 2 ! x4/4! - x 6 / 6 ! ) in order to obtain full contributions to terms involving l/N%nd l/iV4. After such expansion, v, and V, are introduced by means of Eqs. (3-70) and (3-72) with the following results:

+

+

54

MATRIX METHOD OF CALCULATING STABILITY

3-6. Values of u and v for Circular Machines

If there are no straight sections, s = 0 and Eq. (3-74) simplifies to

which indicates that the reality of cr, depends on the reality of n4" so that n may have any positive value. Similarly Eq. (3-76) reduces to

and u, is real if (1 - n)M is real, which occurs for n lying between minus infinity and +l. Further, by the application of Eqs. (3-70) and (3-72), it is seen that v, = n44 and v, = (1 - n)$P. (3-81) These conclusions are in complete accord with those derived earlier by less sophisticated arguments.

EDGE FOCUSING

4-1. Introduction

T h e mechanisms for producing orbit stability so f a r considered have employed forces t h a t act continuously on the particles when they are within t h e confines of t h e magnetic field, with the result t h a t the nonequilibrium trajectories in these regions :ire approxiniately sinusoidal paths about the equilibrium orbit. There is no simple optical analogy to this behavior unless one postulates a medium with a continually varying index of refraction. Most optical devices guide light rays hy abrupt changes of the index of refraction (and hence in the direction of the path) a t specially shaped surfaces. \Ye will now discuss the application of t h i ~latter teclinique t o particle orbits, by the use of radial or axial forccs which act for only a small fraction of thc cntire p a t h :tnd thereby introduce sudden changes in the directions of the orbits. F o r a qu:tlitativo understanding of tllis, consider one magnet of a race-track synchrotron in which the c ~ p i l i h r i u n i orbit is horizontal. A t each ~ n a g n c t end the fitlld docs not ccase abruptly, hut t h e line5 of force bulge outward into the straight section, forming a so-called fringing field. If the ends of thc magnet arc cut "square," so t h a t the plane of the magnet's face is not only vertical but, also ih nornlal to the equil~briumorbit in the straight section, then the lincs of force in the fringing fiel(1 hnv13 no radlal component-that is, no component in the direction of thc local radius of curvature a t the end of the magnet. On the other hand, if tlic. ends of t h c magnet are cut a t a slant angle, so t h a t tlic plane of the magnet's face, although still vertical, is rotated so as not t o be perpendicular to thc orbit in t h e straight section, then the fringing field does h a w a radial component whcli m t y be directed outward or inmr:trcl. If outw:ird, the extended pl:mes of the two faces of a magnet intersect a t a point on the far side of the local centcr of curvature; if inward, the intersection is on the near side. T h e tllrcc 1msil)ilitics are indicated in Fig. 4-1. In t h e discussion of the clyclotron it has b w n pointed out t h a t the axial restoring force on a particle above or Iselow the median plane is caused by a radial conil)onent of ficld (F,= q v K , ) , an11 tlint such a r i d i a l component can occur only when the field lines bulge outward with respect t o the 1oc:~l center of curvature. If they bulge inward, the axial force is defocusing (refer 55

EDGE FOCUSING

56

oxiolly focusing rodiolly defocusing

neutral

oxially defocusing rodiolly focusing

Fig. 4-1. The effect on the orbits sf adjusting the angle of the faces of the magnets in a raae-track synchrotron. to Figs. 2-3 and 2-4). It is therefore apparent that the slanted ends shown in Fig. 4-la produce a radial coqponent which drives a high or a low particle back towards the mid-plane. I n Fig. 4-lb no axial force is generated, while in Fig. 4-lc the radial field component acts to defocus in the axial direction. I n a cyclotron, bilateral focusing forces occur because the radial gradient of field is just big enough (corresponding to a field index between 0 and 1 ) . In the present instance of slanted magnets ends, the greater part of the path through the fringing field lies in a region where the gradient is very large, so if the forces are axially focusing they are also radially defocusing and vice versa. There is a second criterion which may be used to decide qualitatively whether an edge focuses or defocuses, in addition to the preceding remark about the intersection point of the extended magnet faces. If the outward normal to the face of the magnet, a t the point where the orbit meets the magnet, lies outside the orbit in the straight section, then the effect of the magnet's edge is to focus axially and defocus radially. The angle 0 between the orbit and the normal is then taken as positive, as shown in Fig. 4-2a. If the normal lies inside the orbit in the straight section, the edge defocuses axially and focuses radially and the angle 8 is taken as negative. This is illustrated in Fig. 4-2c. As an alternative way of considering the effects of slanted magnet ends, we note that the configuration of Fig. 4-la and Fig. 4-2a produces a field which on the average decreases slightly with rising radius, because large orbits are clearly immersed in the field for a lesser fraction of a turn than are small

positive

8

zero

B

negative

8

T H E AXIAL FOCAL LFSSGTII OF AS EDGE

57

ones, thus Icnding to axial stability by a gcmeralization of the arguments presented earlier in Chaptvr 2 ; such an approach to the analysis of stability will be given in $ 13-5. In the inrmediately following scctions wc will adopt the conccpt of forces engendered by the radial components of field a t the nragnct faces, in order to derive first-order expressions for thc. effects. 4-2. T h e Axial Focal L e n g t h of an E d g e

Consider a magnet with its end cut a,t a slant with a positive value of the angle 6 so that oncoming particles ate focused xial ally, as indicated in Fig. 4-3.

-

vertical seotion, viewed from center of machine

view of magnet from above, at elevation z obove midTplone

Fig. 4-3. Components of the fringing field at slanted magnet ends which focus axially. At a distance z above the median plane, the fring'ing field near the magnet end has a horizontal component B,, perpc.ndicwlar to the face. The radial con>ponent of this is Bnsin 8, where the angle 8 between the outward normal to the face and the path of a particle a p p r o a ~ h i n gthe magnet is taken as positiw if the normal lies outside the orhit. Whcn 0 is positive the radial component of ficld creates a force on the nloving projectile directed towards the median plane, so the equation of motion for a particle with azimuthal velocity v = d L / d t is dLz M7 qv Rh sin 8 = 0; dt-

+

hut

so, on substituting, we find d l z G' - --

dL

M

Bn sin 0.

EDGE FOCUSING

58

On the mid-plane in the straight section choose a point sufficiently far from the magnet that all fringing field components have vanished. Let the azimuthal coordinate of this point be L1. At a second point L2 sufficiently within the magnet that the curvature of the field lines has disappeared, there is only the axial component Ba. Integration of the last equation between these limits yields

9 l

r d v Z = -M sin 0

Bh dL

=

-9 M tan 0

h:^

Bh cos 8 dL.

(4-2)

Now evaluate the integral on the right by carrying a magnetic pole of unit strength through a rectangular path lying in the vertical plane; see Fig. 4-4.

I / / /north "//,I

Fig. 4-4. Vertical section at the end of a magnet and the path of integration. Start a t L1 on the median plane and move up a distance z ; no work is done since the field is zero. T o approach the magnet on path b (the assumed path of a projectile) requires work in amount

When returning to the median plane on path c, work Boz is done by the field. On the last leg, d, no work is done since on the median plane Bh is zero. Because no current has been linked, the total work around the loop must be nil so that there results

lL;BI cos

0 dL = Boz.

(4-3)

Eq. (4-2) therefore becomes

Q (vz)~a- (v.)L, = - M BOZ tan 0. But

Use this in Eq. (4-4) and then divide by v to obtain

(g),, (d),, -& -

=

BG tan

e=

X

--r tan 0.

Here we have applied Eq. (1-2), Mv = qRr, in order to introduce the radius of curvature in the magnet. This expression measures the change in vertical slope of the orbit in passing completely through the fringing field; for vertical

THE RADIAL FOCAL LENGTH O F AN E D G E

59

focusing action the slope down st re an^" a t I,, should he less than the slope "upstream" a t I&, so the right-hand side should bc negative, as indeed i t is. If the angle 0 were negative, the action would be diverging.

Fig. 4-5. Ray paths in thin lenses. We may correlate these results with the propertics of thin lenses as shown in Fig. 4-5 where z/f, = t a n 4, so for small angles

The change of slope calculated in Eq. (4-5) is represented by 4 so t h a t

f"

2

=

- (z,lr) t a n 0

and

the minus sign indicating axial focusing when 0 is positive. It is apparent that a similar axial convergence also occurs when the projectile leaves the magnet, provided t h a t 0 is again positive. This derivation involves a swindle arid hence the result is only approximate, though still exceedingly useful. The trouble lies in the implicit assumption t h a t the focusing action produres only a change of slope and not a change of displacement z. This assumption is true only if the fringing field is a true step-function, producing a so-called "hard edge," and this is possible only if the height of the magnet gap is infinitesimal. With a practical finite gap the fringing field has a finite azimuthal length throughout which the vertical force continues to act, thereby changing ttre disldacement z and hence altering the vertical force. T o obtain a really accurate value for the change in slope a t a practical "soft edge," numerical integration through the fringing field must be carried out. 4-3. The Radial Focal Length of an Edge Consider two ions of equal velocity moving side by side on parallel paths but separated radially by distance x as they approach the slant face of L: magnet, the normal to the face lying outsidc the orbits so t h a t 0 is positive,

60

EDGE FOCUSING

as in Fig. 4-6. Particle A reaches the field first, so that by the time particle B arrives, A has already penetrated a distance d = x tan 8 (if the slight curvature of path is neglected) and therefore A has turned through a n angle 4 given by

This represents the divergence which the slant edge introduces between the orbits. Similarly on leaving the magnet, with positive 8, particle A stays in the field longer than does B, so that divergence again occurs.

When the analogy of thin lenses, as in Eq. (4-6)) is used, we find 2

fz

=

I%/r)2tan e1.

(4-9)

SO

fz =

:-, r

tan 8

the positive sign indicating a defocusing action with 6 positive. All in all, nonnormal incidence on., or emergence from, a magnet a t a positive incidence angle produces axial focusing and radial defocusing, while the reverse occurs for negative angles of incidence and emergence. This is an important phenomenon, for some varieties of accelerators derive a large portion of their focusing properties from such localized "lenses," as will bc secn. (Edge focusing has long been employed in certain varieties of mass spectron~etersand spectrographs to produce axial focusing of ions diverging from a source.*) 4-4. Matrix Representation of a Magnet Edge

When an ion passes through the fringing field of a magnet's edge, the latter acts approximately as a lens of negligible thickness, so the transverse displacement y (axial or radial) remains unchanged. Thus *See, for example, the article by K. T. B,iinbridge "Charged Particle Dynamics and Optics," in E. Segrk, Ed., E'xpc~-irnei~tnl Nuc1t:ar Physics, John Wiley & Sons, Ncw York, 1953, Vol. I, pp. 578-582.-Also W. G. Cross, "Two-Directional Focusing of Charged Particles in a Sector-Shaped, Uniform Magneiic Field," Rev. Sci. Instr. 22:717-722 (1951).

T H E ZERO-GRADIENT SYNC:]-IROTROS Y =

61 (4-1 1)

?/I.

On the other hand the slope 3' -- ( l g / d L of tlw path is :~ltcrctlby an arnouni which is proportional to the transvcrsc displacwncnt. This change of slope has been secn in Eq. (4-6) to be 4 = z/f, for axial lrlotion ancl = x/f, for radial by Eq. ( 4 - 9 ) .If the slope is yl' on reaching the lense, its valuc on lcuving is therefore

+

Now write the equations for y and y' in sylnlnetrical form,

in order to see that the matrix formulation is

The sign of f can be taken care of later on, in any particular instance. 4-5. The Zero-Gradient Synchrotron

If one builds a race-track synchrotron with a uniform field urithin the magnets (no deliberate variation of field strength either azimuthally or radially), then n = 0 in the guide field, so there is relatively powerful radial focusing [v, = (1 - n)x = 11 and no vertical focusing [v, = nx = 01. By eniploying say four magnets with four interposed straight sections, eight magnet faces are obtained which, if properly oriented, will supply an adequate ainount of vertical focusing, the associated radial defocusing introduced by the faces being less than the radial focusing furnished by the bodies of the magnets, 60 that all in all the system is stable in both planes a t once.

Fig. 4-7. Zero-gradient synchrotron magnet.

It may he remarked that, within limits, it is not necessary for all straight sections to be identical in length; one may increase half of them by decreasing the other half, keeping the average length as before.

EDGE FOCUSING

62

A machine of this design (Fig. 4-7) is attractive because it is possible to drive zero-gradient magnets to higher fields than has been found feasible with magnets of finite n, so that the same energy may be obtained in a smaller machine. 4-6. Axial Betatron Frequency in a Zero-Gradient Synchrotron

Since n = 0 in such a machine, the body of a magnet supplies no axial forces and hence acts like a straight section of length m. The cycle of edge, magnet, edge, and straight section alters z and z' according to

where z'= d z / d L a,nd s and m are the straight section and magnet lengths measured a t the radius where the equilibrium particle travels. Multiplying out in pairs leads to

where

The criterion for stability, developed earlier,,iv now invoked, and axial stability will exist if uz is real in the expression of Eq. (3-69);

+ Mzz).'

cos U, = %(Mi1 Substitution for the M's gives

cos u* = 1

sm + -mf z + f-sz + 2fz

It has been seen in Eq. (4-7) that the axial focal length is f, = -r/tan 0. The total curved length of path is given by Nrn = 2xr, where N is the number of magnets each of length m. Therefore

so that COs

Uz =

2?r 27r s 1 - - tan 0 - - - tan 8 N N m

s + 2r2 - - tan2 8. N2 m

(4-19 )

As before, uz is the phase shift of the axial betatron motion perssector (one magnet and one straight section). The value of B, may .be found from this

RADIAL BETATRON FREQTTENCY I N A ZGS

63

equation in terms of known values of the parameters on the right. Then the number of axial betatron oscillations per turn can be found from Eq. (3-70) :

Alternatively, a n approximate value of v," may be obtained by expanding cos cz in Eq. (4-19) into 1 - a,"2 and then making use of Eq. (4-20). This procedure gives N N s. -7 (4-21) ',v = - t a n 8 - - tan 19 - -tan' 0 . . .. a

+am

+

m

This shows t h a t the axial focusing in a zero-gradient synchrotron is supplied entirely by the slanted ends of the magnets, which must be so cut that 8 is positive, as in Fig. 4-2a. The bigger this angle and the greater the number of magnets, the bigger is vZ2.

4-7. Radial Betatron Frequency in a Zero-Gradient Synchrotron The general radial equation of motion for a particle within a magnet is, by analogy with Eq. (3-15) : d'x (1-n) dL'+-r?x=O, where L is the azimuthal distance and r is the constant radius of curvature of the equilibrium orbit. Since in this machine n = 0, :t solution is

x's-=dx dl;

-.

A sin L B L -- + - cos r r r

Solve for A and B by letting z = xl and x' = z,' a t the beginning of a magnet where L = 0. Then a t the magnet's end where L = m, the position and slope are given by rr1

xz = x1 cos --

7

+ rxl' sin -TrLr

Consequently, the effect of the trip througli one cycle of the structure consisting of edge, magnet, edge, and straight section, is given by

EDGE FOCUSING

64

:(=)I(

:)(; :)(-;in: cos -

This gives

where

r 2 m m M2, = - cos - ( lsin fi r fi" r m r m M22 = cos - - sin r fz r The condition for stability as given by Eq. (3-71) is that a, should be real in the expression (4-30) cos uz = +(Mil M22>, where a, is the phase shift of axial motion per sector. Substitution for the M's frorn Eq. (4-29) gives

l)

+

+

-a

+

The radial focal length as given by Eq. (4-10) is

f, = r/tan 0 EO

=

Nm/(2s tan %),

that

If this is solved for a numerical value of a,, the number of radial oscillations per turn may be found from the expression

An approximate value may be obtained by expanding cos u, int,o 1 - u,'/2, while on the right cos 2s/N must be expanded to four terms and sin 2s/N to

thrtc, in ordcr to ohtain all contributions to the coefficimts of inverv powcrs of A* u p to t l ~ cfourth. The result is

Kate that t l ~ cfirst two terms arc indelwndent of B ant1 that the usc of slnntcd ends, in order to produce axial stability, a t the same time acts to decrease v,.

Chapter 5 RESONANCES

5-1. Introduction

I t has been seen that in weak-focusing machines stability can exist only if the field index has a value between 0 and + l . In accelerators in which the particles spiral outward, such as fixed-frequency and frequency-modulated cyclotrons, this range is covered between the center of the machine and the radius of the orbit that represents the maximum momentum. Therefore v, rises and v, falls during the acceleration process, since v, = n x and v, = (1 - n)x as given by Eqs. (2-15) and (2-24). It is illuminating to follow the changing

values of these parameters by plotting one against the other as in Fig. 5-1. The resulting curve is particularly simple for v,2

+ vZ2 = [nfiI2+ [(I - n)jiI2= n + 1 - n = 1.

(5-1) This is the equation of a circle with unit radius. The operating point (sometimes called the tune) therefore moves along a quadrant of this circle, starting a t the point where n = 0 , v, = 0 and v, = 1 and ideally continuing to where n = 1, v, = 1 and v, = 0. For all positions reached, except the extreme limits, bilateral stability should exist. With a circular synchrotron or betatron, n ideally maintains a single value so that v, and v, are each fixed and the operating point remains static a t some chosen point on the circular arc, usually where n = 0.6 so v, = 0.775 and v, = 0.633. Unfortunately, however, the permissible locations of the operating point are 66

COUPLED RESONANCES

G7

somewhat restricted in actual practice by factors which wcre not relevant to the preceding c.h:tptc,r. Thew limitation> an: t l ~ esubject of the following paragraphs. 5-2. Coupled Resonances

+

The equations of betatron motion have been derived in the forms S w2nz = 0 and x u2(1 - n ) x = 0 under t h e assumption that the two motions are entircly unrelated. Actually tlus is not trucx (though for many purposes i t is a close enough approximation if the amplitudes are small). For example, the analysis of radial motion assumed that it occurred wholly on the mid-plane of the magnet gap where the axial component of field B,, which supplies the radial restoring force, is equal to the total field B, the variation of K with radius being describcd through use of the index n . But in actuality a n axial motion occurs simultaneously, and since B, vanes with z the radial force really is tlependcnt on the axial displacenlent as wvll. A rigorous analysis results in the appearance of factors dependent on both z and x in the equations of each mode of oscillation. Although such involved expressions will not be discussed here, it is not difficult to imagine that the two motions will react on each other. If the axial and radial frequencies happen to be related in a manner involving small integers, it is possible for the energy in one inode of oscillation t o be transferred to the other mode, and back again, as in the mechanical case of two coupled pendulums. This in itself would cause no harm, provided the vacuum chamber were large enough in each dimension so as not to intercept the particles; but in practice the vertical clearance is usually less than the radial, as is notably true in machines with dees.

+

Fig. 5-2. Coupled or difference resonances. The values of the field index at these critical points are shown for a circular machine.

This state of affairs is said to be due to coupled resonances between the two modes of oscillation. They are also known as difference resonances since they may be represent,ed in the form Av, - Rv, = C where A , B, and C are integers. T h e v, vs. v, diagram becomes crossed with lines indicating where such conditions occur, some of which are shown in Fig. 5-2, the associated n oaluw for

G8

RESONANCES

a circular machine being indicated. (Such lines really should be drawn as bands with rather indeterminate edges, since the resonant conditions are not infinitely sharp.) Partial or total loss of ions may occur if the energy gained per turn is so low that the resonant condition is maintained long enough for the oscillations to die down in one mode and to build up in the other to disastrous magnitude. This is particularly true in a synchrocyclotron in which only a few kev are gained a t each acceleration and the first serious resonance encountered (at n = 0.2, where v, = 2 4 , causes a cataclysmic "blow up" of the beam in the axial direction. Consequently the particles must be put to use just before this occurs. With cyclotrons, on the other hand, the dee potential may be several hundred kilovolts and the resonant region is traversed before the axial displacements can grow to a troublesome size. One might expect that the other coupled resonances could be rushed through equally well, but in practice the ions fall out of step with the dee voltage, as discussed earlier, and so must be used before deceleration sets in. The radius of utilization for cyclotrons often is near the point where n = 0.3. The condition where v, = 2v, is now often called the Walkinshaw resonance after the investigator who emphasized its importance in the accelerators to be described in Chapter 13. The magnitude of the axial motion arising from such a coupled resonance is easily calculated. Consider the motion z = zm sin w,t. The corresponding velocity is 5 = zmw, cos w,t and its maximum value is b = zmw,. The energy in this mode of vibration may be computed a t the moment when it is wholly kinetic; and since the velocity is small the nonrelativistic expression is adequate:

T,

=

+M&2

=

$Mozm2w:.

Therefore the amplitude is

But

V,

=

w,/w, where w is t,he revolution frequency. Hence

A similar argument for the radial motion shows that

In a coupled resonance the energy of lateral motion is transferred from one mode to the other, so T , = T, and therefore

For example, in a synchrocyclotron the resonance v, - 2v, = 0 occurs a t n = 0.2, where v, = (I - 0.2)" = (0.8)" and v, = (0.2)%, so that z, = (4)55xm = 2 ~ , ~ ;

SUM RESONAKCES

69

any radial oscillation amplitude is multiplied by two when the energy is transferred to the axial mode.

5-3. Imperfection Resonances There is another class of resonances which can lead to an unlimited increase in the amplitude of the radial or vertical modes of oscillation, without transfer of energy from one of these modes to the other. Since the vacuum chanlbcr is finite in extcnt, such conditions, if pcmistent, always lead to complete or partial loss of ions. T l ~ c s eresonances are due to some imperfection in the magnetic field which is cncountercd on every revolution of the particle, either until the ion spirals out to where the iinperfection no longer exists or until a tiinedependent field error may vanish. The most, serious ones occur when the betatron frequmcy is an integral 111u1tiple or submultiple of the revolution frequency, i.e., when v, (or v,) is a n integer, the ratio of two integers, or zero. The particle then meets the disturbance repetitively and in a similar phase of its oscillation, so that any aniplitude prescnt a t the first traversal is amplified t h e after tirw. Such resonances represent true instabilities, if continued long cnough. Typical examples of these in~perfec.tionresonances are as shown in Fig. 5-3, with the corresponding values of n indicated for a circular machine. The

Fig. 5-3. Imperfection resonances. Values of the field index are indicated for a circular machine.

most serious of these are a t V , or V, equal t o 3 or 1. Those at v, = t and (n = 0.06 and 0.11) are barely detectable in synclirocyclotrons, whilc thtl swere loss of beam at v, = Q (n = 0.25) is very hard t o distinguish from that due t o the cwupled resonance a t n = 0.20.

5-4. Sum Resonances Othcr imperfection instabilities occur a t what are called S I O ) L reson(lnCes such as thobe shown in Fig. 5-4. Of thcsc, only the two which cross t l ~ epit11 of the operatii~gpoint are relevant in cyclotrons or synclirocyclotrons.

70

RESONANCES

5-5. Resonances in a Race-Track Synchrotron

It has been pointed out that the presence of straight sections in a synchrotron v2)% has a value greater raises both v, and v,. Therefore the quantity (v,2 than unity and the possible positions of the operating point on the v, vs. v,

+

Fig. 5-4. Sum imperfection resonances. Values of n are shown for circular machines.

diagram (Fig. 5-5) lie on a quadrant of a circle with radius greater than 1. Approximate equations for v,2 and v 2 in terms of machine parameters were given in Eqs. (3-77) and (3-78). A rough value of the sum is found to be

Fig. 5-5. Typical resonance diagram for a synchrotron with straight sections. The locus of the operating point has a radius greater than unity.

T H E E F F E C T O F SPACE ('HA'RGE

71

Although the operating point in a synchrotron docs not necessarily move, as it must in a machine with spiraling orbits, ncvertlleless therc are inadvertent motions, since it is next to in~possihlc~ to hold n rigorously constant over the wide range of f i ~ l dstrengths involved, eyen with the employment of programed currents in pole-face windings. Thwefore a nominal operating point is chosen in t h e middle of a wide safe region on the v , ~ , diagram, so t h a t some accidental n~overnent can occur without trespass into a danger zone. For example, in the 3-Bev proton synchrotron a t Brookhaven National Laboratory (known as the Cosmotron) we have s = 10 ft and m = 47.1 ft, so the radius of the operating path is (1 + 10/47.1)% == i.1 approximately. The index has a nominal value of 0.6 but slides to a s high Ls0.72. The beam loss a t n = 0.63, where v, = %,appears t o be insignificant, as does that .at v, - 2v, = - 1, where n = 0.61. 5-6. The Effect of Space Charge

Up to the present point, the problem of stability has been approached by considering the action of the magnetic field upon a single ion. I n actuality every particle travels in the company of a great Inany similar particles, and these exert electric anti electromagnetic forces on each other. I t will be instructive to examine the situation, to see if our previous conclusions need any revision. We will do so in the simplest approxirnatiori. Assume a synchrotron without straight sections a,nd with an equilibrium orbit of radius r. 1xn:igine the beam of ions to he of circular cross section of r. Let the particles, each of chalrge q, be uniformly disradius a , with a tributed throughout this torus, and let 6 be the number of ions per unit volun~e. Then the total charge in the system is

<

Q = ( 2 r r )(ra2)q6.

From this it follows that

Each particle is subject to an electrostatic force of rc'pulsion which tends to increase its distance y from the center of the beam. Only the charges lying a t a radius equal to or less than y will play any part in this force, and this interior charge per unit length is given by Q'

=

ryQ6;

so on substituting for 6 we find

>

a, we may treat the situation as though a test charge were on the Since r surface of an infinitely long straight column of r:tdius y and charge Q' per unit length; so by Gauss's theorem the electric field a t this surface is

72

RESONANCES

where the permittivity of space is ko = 1 / ( 3 6 ~x lo9) farad per meter. Tlie outward force on the test charge is

so on substituting for Q' from Eq. (5-9) we find

But there is also a force on the test charge due to the magnetic field produced by the interior column of ions moving with velocity v = pc. This column represents a current given by I = ?ry7qSpc; (5-13) so on substituting for 6 from Eq. (5-7) we obtain

Consider this current to flow in an infinitely long straight column of radius y. Then the magnetic field a t y is I B = = PO(5-15) 2ay

where the permeability of space is pO = 4r/107 henry per meter. Such a field produces a force on the test charge which is directed toward the center of the column of ions. This force is

The net outward force on the test charge is therefore

But poke = 1/(9 X 1016) = l/c2. Hence

Thus the electrostatic force of repulsion exceeds the attractive magnetic force until the velocity reaches that of light, when the two forces exactly cancel. If we set the general parameter y equal to the axial displacement z , the equation of axial motion takes into account the above force in addition to that generated by the radial gradient of the field:

TlIE EFFECT O F SPACE (IIIAI1GE

Now divide througli by JI, and in the second term write is the total energy. Rut since w = v/r, then we have

73

,M = E/c2, alierc E

Therefore Eq. ((5-20)becomes d'z dt"

+ &(n - dn)z = 0,

where

Here d n is the cliangc in the effective value of the field index brought about hy space charge. If v, is the axial betatron frequency under these conditions, me inny write Eq. ( 5 - 2 2 ) as d' z - + w"vz?z=

dtJ

01

where vZ2 = n -- dn. Wc sec that tlw space charge reduces thv net restoring force and lowcrs thv betatron frequency. Write V,Ofor the axial betatron frequency in the absencc of space charge, so by the familiar Eq. (2-15) vZ< = n , it follows t h a t

If we set thc general transverse coortlinatr !I in E;q. (5-19) equal to thc. radial deviation coordinate z, the equation of radi:tl motion is

and on going through the s:me procedure a s before we find

d?x -+d(l dt'

-

n

-

dn)x

=

0;

so

d2x - + w'v,!x dt'

=

0

where v,?

=

1-

II

- dn.

B u t if spare charge mere not present we would have v,; v,? =

vd2

- dn.

=

1 - n ; so

74

RESONANCES

Thus space charge also reduces the value of the radial betatron frequency. If we subtract this from Eq. (5-26) we find v,2

- v 3

=

(5-32) This expression permits us to plot the locus of the operating point on the v, vs. v, diagram as the betatron frequencies fall with increase of space charge, provided we know the values appropriate when space charge is negligible. Thus for example, it will be seen later that in the synchrotron a t the University of Birmingham, England, the field index has the value n = 0.67, so the designed betatron frequencies are given by v,o' = n = 0.67 and vXo2 = 1 - n = 0.33; therefore Eq. (5-32) becomes v,2 - vZ2 = 0.67 - 0.33 = 0.34. This curve is plotted in Fig. 5-6. If we assume that any of the resonance lines which cross zo

2-

vzo2.

Fig. 5-6. The locus of the operating point when space charge plays a part, and some of the resonance lines which it crosses in a circular synchrotron with nominal field index 0.67.

this curve will cause destruction of the beam, then the space charge must not be so great as to move the operating point from the designed value (at n = 0.67) past any resonance, for even if the operating point a t injection lies between two resonances, the space charge forces will decrease as the energy rises and the point will necessarily traverse the dangerous region as it slides back to the designed value. The space charge limit of injected charge is therefore reached when the initial load of ions is so great that the operating point is forced to the nearest serious resonance. I n the present example this occurs a t v, = 0.5, a t which value we see from the figure that v, =(0.769hhe necessary change in the effective index which brings this about is given by Eqs. (5-26) and (5-31) :

To find the number N of injected ions which will just produce this change in the effective index, we solve Eq. (5-23) for Q and set N = Q / q . This gives

I n MKS units, the lengths are in metcm, tlie ch:~rge in coulombs, and t11c cwrgy in joule*. If wc asslunc3 singly ionized projrctilcs and express the cncrgy 111 JIev, this rctluces to N

=

0'

d'

2.18 X 10IL- I!: --- dn 1-82 '

I n the Birmingham synchrotron, injection is a t T = 0.46 Mev, so we havc Eo = 938.46 Mev and ,8 = [ ( E L-I';,)"/E:']% = 0.0313. The radius of the orbit is r = 4.5 meters, wliilc that of a cylindrical column of ions whicli will just fit iribicle the narrowest dinlension of the v : ~ c n l ~rh:mbcr n~ is tr = 0.05 meter. If we set d n = 0.08, then we find that N = 13.9 x 1010 ions. T l ~ ccorrcsponding injected current may 1)c found from tho time to complcte ~ I I V revolution a t ~njection:T = 2rr ' ( o r ) == 3 X LO-6 second; so the current I < I = Q / T = A r ( ~ / r= 4.7 milliamperes. (These (*omputations assume that tlhc ions are uniformly distributed :irol~ntlt he circular magnet; it will bc sliomn in ('haptt~r6 that actually only about l d f of the circurnfcrcnrc is filled wit11 particles; so the local charge dmsity is twice the averagc, and consequently thc critical charge and critical currmt are about half of the :hove figures.) I n practice, the injected current a t Birrrlingham is inore like 1 milliampere, so t l ~ c space charge 11mit is not re:~chc~l.Kcvc~rthclcss,in some of the more recent synchrotrons, wherch every effort is being m:de to obtain as large a be:tm of ions as is possible, the space charge effcct may we11 3et the upper value. The analyhis given above is considerahly idealized, for in many synchrotron,i the injected hc~irnis not circular in cross section. Tk~enext approximation is to imagine it to hc elliptical in sh:~pc.. The :inaly~isis then considera~~ly rnore involved and the locus of tlie operating point followe a curve which is marc. or , that a different set of resonances are crossed; but less parallel to the v, a ~ i s SO thc liiniting number of injected ions is not substantizlly different. I n any c a s ~ it, is clear from Eq. ( 5 - 2 3 ) that the Irouble is most pronounced wlien the energy is low, so that injection at a high cncrgy is advisable.

E =T

+

PHASE STABILITY

6-1. Introduction

A little reflection will indicate that the method of operation of a synchrocyolotron or synchrotron, as expounded thus far, gives little promise of accelerating any substantial fraction of the ions up to the final energy. I t has been stated that these machines work because the oscillator frequency is continually adjusted to equal the revolution frequency of the particles, so that the ions stay in phase with the voltage and are therefore always accelerated when they reach the region of the electric field. Such a statement, made for simplicity, puts the cart before the horse; for in practice the oscillator follows an externally irnposed schedule. In a synchrocyclotron this is determined by the motion of a mechanical capacitance, while in a synchrotron the oscillator's frequency is made to track the rising value of the magnetic guide field, generally by electronic methods. If there is to be a synchronous particle, its radian frequency w, and its total energy E, are dictated a t all times by the radian frequency w,, of the oscillator through the relation w, = w,, = q B / M , = qBc2/E,. It is conceivable that the peak voltage y, across the accelerating gap could be so chosen that the energy qV, gained by an ion in crossing a t the peak would be just that needed to raise the energy to the higher value of E , required during the next turn; but the number of such successfully accelerated ions would be vanishingly small. Particles in an accelerator do not advance entirely side by side, but are also spread out into a column of appreciable azimuthal length; so one might expect that if the front end of such a column reached the accelerating gap when the voltage was a t its maximum, all later particles would receive less than the necessary energy gain. Consequently the column would lengthen and ultimately many of the ions would undergo deceleration because of arrival a t the gap when the electric field was directed the wrong way. Douhtless many investigators contemplated the scheme of varying the oscillator frequency as a means of avoiding the difficulties inherent in a cyclotron (i.e., the lowering of the revolution frequency bccause of both the relativistic increase of mass and the wakening magnetic field needed to obtain orbit stability) but abandoned the idea because of the exceedingly few particles which would be expected to be brought to full energy in this manner. 76

T H E PRIISCIP1,E O F PHASIC S T A B I L I T Y

77

6-2. The Principle of Phase Stability

I t was suggested by Veksler in 1944 and independently by McRlillan in 1945 t h a t a natural solution to the difficulty \\-as a t hand, provided certain easy r e q u i r e ~ n m t swere fulfillctl. T h e plicnonienon t h a t bupplies tlic solution is known as t h e principle of phase stability T h e requirements are: First, t h a t the peak voltage across the accelerating gap inust be somewhat greater t h a n would otherwise be needed ; Second, t h a t the particle orbits must he such tliat a change in energy and monlentum must be accompanied by a cliungc in the period of revolution; Third, t h a t the variation of t h e oscillator's frequency be made sufficiently slowly.

Fig. 6-1. Successive phase relations of synchronous and nonsynchronous ions in a weak-focusing ;~ccelerator. T h e way in which phase stability acts is illustrated in Fig. 6-1, drawn for the case of a ~wak-focusingn1:lchine in wl~ich,a s is already known, a n increasc Ire the phase angle of energy causes a n increase in the period of rotation. Let a t which a synchronous ion of energy E , crosses the gap of peak potential V , , so t h a t the cncrgy gain needed to keep the ion syncllronous is AE, = qT7,,,sin +,. Thercfore succwsive crossings vontinue to occur a t +,, separated by the syrlchronous period T, = T,,. It is important t o notice t h a t 4, has been chosen t o lie in t h e second quadrant, between a/2 a n d a. T h e reason for this will becoliit. clcar 1:ltw on. ,4t this stage in the discussion, the argument will be consitlcrably siml)l~ficd and will suffcr no loss in generality if we irimginc. tliat accclcration occurs only once 1)er turn. This actually is the c:kse in many .ynclirotrons. \Yith a synclirocyclotron tllcrc3 really :ire two acccleratior~sper turn, but to carry on the argument on such a basis requires us continually to rcn1cn11)cr t h a t the sinusoidal voltage plot of Fig. 6-1, and of similar plots to follow, ~ i l u >I)(% t "turned over" every half cyclc, hir~ctif a p:trticle is acccllcratctl a t on(. gap, i t nil1 :ilso be :LC~ , though celerated a half cycle latcr a t tlir next gap 180" away in x z i n ~ u t l even the voltage plot a p p c w s to indicate. tlecelcration, for the particle is now tr:~vcling in the opposite direction :Icrosq the gal). n liic~l~ also has energy E, hut wllic.11 rcaclrcs the gal) Xow con4dt.r a 1):rrtlc~l(~ at < +*, a s shown in Fig. 6-1. I t experiences a n energy gain AE > AE, and

+,

78

PHASE STABILITY

hence immediately becomes nonsynchronous, for now E > E,. This makes its period exceed the period 71.f of the oscillator; so the gap is next encountered one turn later a t $2, closer to 9,. The energy is then raised again, though by a lesser amount, and the phase shifts further in the same direction. This process continues on subsequent turns. Meanwhile the synchronous energy is steadily rising, because of the changing oscillator frequency, and if this change occurs slowly enough E , will not catch up with the particle's energy E (including all its accumulations) until the particle's phase exceeds $,. Let $' be the phase a t which E, becomes equal to E. Depending on circumstances this can occur when

Fig. 6-2. Possible excursions in phase of nonsynchronous ions. In ( a ) the particle is always accelerated, while in ( b ) it undergoes deceleration when its phase exceeds n.

4' < ?r or when 4' > T , as shown in Fig. 6-2a and Fig. 6-2b, where, in order to save space, the successive values of the phase are plotted on a single wave. During all this time, the ion's period T has been changing with respect to that of the oscillator, the difference between T and T,/ rising from zero when the phase is reaching a maximum when the energy departs most from the synchronous value, and falling to zero again when the phase becomes 4'. At this point the ion is again synchronous, with E = E, and o = w,j. Immediately w,f is made to change, so E, rises and the particle for the first time finds itself with an energy deficit compared to the synchronous value. Its revolution period is therefore less than T,, and the next gap crossing is a t a phase slightly less than 4'. The energy then gained is below the synchronous value, as in Fig. 6-2a; or it may even be negative, as in Fig. 6-2b, so that a further shift to a smaller angle takes place. This continues until the accumulated energy increments total up to the new value of E,; this occurs just as 9, is again reached. A complete phase oscillation has then transpired and the process repeats. It will have been noted that oscillations in energy also take place, maxima and minima occurring between the synchronous values which are reached a t the extremes of the excursions of 4. Nonsynchronous particles therefore gain energy a t the same average rate as does a synchronous ion but deviate noticeably from the average in so doing. Practically all of the successfully accelerated particles in a synchrocyclotron and synchrotron fall into this nonsynchronous category and the output of high-speed ions necessarily has some sprcad in energy. Oscillations in radius of curvature accompany the fluctuations in energy, so

T H E PRIKCIP1,E OF PHASE STABILITY

79

the vacuum chamber of a synchrotron must be made wide enough to accommodate such radial oscillations, as ~(111as tlic radial botntron motions which art> superimposed. As noted abovc, the revolution frequency rises and falls about the steadily changing synchronous v:ihie. Thus there are four types of so-called synchrotron oscillations associated with phase stability: oscillations of phase, of energy, of radius, and of revolution frequency. The common frequency a t which all occur is known as the synchrotron frequency. This is always less than that of revolution by a factor which ranges from several dozen to several thousand. Since the betatron frequencies are of the same order of magnitude as the turning frequency, it is clear that the betatron and synchrotron motions may be analyzed separately, just as if the other did not exist, except that their amplitudes must be added when considering chamber dimensions.

Fig. 6-3. Energy relations of synchronous (straight line) and nonsynchronous (curved line) ions when the latter are ( a ) always accelerated and ( b ) alternately accelerated and decelerated.

The action of phase stability also can be exemplified as in Fig. 6-3a, where kinetic energy is plotted against the number of turns. For example, if the synchronous ion gains a constant energy per turn, the :graph is a straight line of fixed slope. The wavy line represents the hehavior of a nonsynchronous particlc whose energy oscillates about the synchronous figure. Where the slope of the wavy line exceeds that of the straight one, energy is gained a t more than the ideal rate, and vice versa. The phase slip is in one direction or the other depending on whether the wavy line is above or below the straight one, the phase momentarily passing through 4, whenevw both lines have the same slope. A wavy line with a slope that is everywhere positive, as in Fig. 6-3al indicates that the nonsynchronous ion is always acce1er:tted a t the gap, as was illustrated in Fig. 6-2a. If the phase slips so far that 9 > ?rl as in Fig. 6-2b, then there are regions of negative slope corresponding to intervals of deceleration. This condition is shown in Fig. 6-3b. The fact that the wavy lines have maximum slopes which exceed that of the straight one, is an expression in graphical form of the earlier statement that the available energy at, a gap crossing must exceed that required by the synchronous particle if phase stahility is to exist.

80

PHASE STABILITY

Inasmuch as sin (a - 4,) has the same value as sin +,, it is clear that the synchronous energy will be gained if a particle crosses thc gap a t either of these two phase angles. Nevertheless, oscillations of phase occur about only one. In a weak-focusing accelerator a momentum increase brings about a rise in rotation period, and on this basis the arguments given above showed that phase oscillations occur if 6 lies between a/2 and a. If one attempts to repeat the arguments for the same type of accelerator with 4, located between 0 and a/2 (at ?r - 4,) it is easily seen that oscillations will not take place about the phase angle.

Fig. 6-4. Phase oscillations do not occur, in a weak-focusing magnetically-guided machine, about a synchronous phase angle in the first quadrant. Thus, in Fig. 6-4 a particle a t & receives more than the synchronous energy increment AE, associated with the angle a - 6 ,takes more than the synchronous time to complete a turn, and next reaches the gap a t &', further away from a - h than before. Similarly, an ion a t 4b, which receives less than AE,, next appears a t the gap a t +b', also further from ?r - 4,. Particles retreat from this angle, rather than oscillate about it. (However, i t will be seen later on that special types of accelerators exist in which the reverse is true; that is, for some machines phase oscillations take place only about a synchronous phase angle located in the first quadrant.) A further interesting conclusion may be drawn from the last paragraph. With 4, between a/2 and a , as is required for a weak-focusing machine, the phase a - 4, may be considered as a limiting angle for oscillations about 4,. Any excursion of + will be stable about 4, unless i t is so large as to reach the region to the left of a - 4, in Fig. 6-4, i.e., unless 4 becomes smaller than a - 4,. What then occurs will be described later. To obtain a quantitative understanding of phase stability, it is profitable to start by investigating the circumstances under which a momentum change will cause a change in period, for the situation is somewhat more subtle than appears a t first sight. The period depends on the velocity and on the orbit length, and the orbit is determined by the magnetic field; so the first step is to consider the influence of the field on changes of momentum and of orbit length. The relation between these latter quantities is called the momentum compaction. 6-3. Momentum Compaction

Momentum compaction measures the extent to which equilibrium orbits of different momenta are crowded together in the magnetic field. If two such orbits

are packed closely together, like tightly nrsting wire rings, their circunlferences differ only slightly. Such a condition rey)resents a large niomentuni coinpaction and tlic particle with the grcatcr velocity may have t lie shorter period. On thc. other hand, a srnnll momentum compaction exists if the field is such t h a t tho two momenta correspond to orbits of markedly different circunifcrcnccs. This difference may be of more iri~portancethan the tlifferencc in velocities, so that tlic faster particle may have tlir longer period. Under special circuitistanccs tlic interniediate situation can arise where the incrcascd velocity is just compcnsated by tht. increased path, resulting in the samr period for both orbits. It is apparent, t h a t n quantitative measure of thc compaction is neetled to dctcrmine which of these possibilities is appropriate in a givcn accelcrntor. Furthcrrnor~,bince nonsynchronous ions gain thc proper energy only on tlic avcragc, it is clear that a t every instant particles will exist with more and with less energy than thc avcrage ~ a l u cand hcncc will describe orbits with larger and smaller radii than that of the synchronous ion. 'This nieans that the vacuum chamber and the radial width of the magnet of a synchrotron will have to he wider than if only synchronous particles wrrc being handled. Since it is tlesirable to lose as few ions as possihlc by collision with the inner and outer walls, and since an unncwssarily wide rnagnct can waste very large sums of money, a knowledge of the radial spread Iwr unit of monientuun change is of great economic iniportanc~. Consider two particles, one with monlentuin p and orbit Icngtli 11,the other with momcntu~np d p and orbit length L dIJ. Thc momentum compaction a is defined as

+

+

Thus a measurcs the fractional change In momentum for a given fractional change in path. An alternatiw an11 entircxly cquivalent definition can be obtaincd by iinagining that the real orbits, whatever thcir shapes, :ire rcplaccd by circles with circumferences eq11al to tlic wtual path lengths. This procedure d K for ~)articlcsof monlcntuin p and gives thc cquivalent radii R and R p dp. In this picture, a is defincd as

+

+

For a circle, I, = 2?rR and hence d L / L = dR/R so t h e two definitions are equivalent. It must be emphasiztd that the radius ,R may have no physical counterpart in any actual machine; i t simply dCsc.rit)e:;a circle of circumference equal to the rral path. An instance whcre R differs from the orbit's ratlius of curvature r xilay he found in a synchrotron with straight sections. I n an entirely circular machine R and r are identical. A word of caution on nomenclature is in ordrr for those who study furtlier in tllc field of accelerators. There is a lamentable lack of agreement on the

82

PHASE STABILITY

definition of momentum compaction. Authors are divided into two camps; those who employ the definitions given above, and those who employ just the inverse, so that a = (dL/L)/(dp/p) = (dR/R)/(dp/p). Even worse, the same name and the same symbol a! are used for both. The calculation of a will now be given in a few examples. For a circular weakfocusing machine, an ion of momentum p describes an orbit with radius r and dp, length L = 2sr. If in the same magnet we inject a n ion with momentum p it must seek an orbit with a greater value of the product Br. Whether or not the field falls off with radius, this new orbit will be located a t some larger radr and its path length will be L dL = 27r(r dr). The difference dius r in paths is dL = 27rdr so that

+

+

+

+

Now consider the effect of the field configuration. Since p = qBr, then we have dr/r; and since n = - (dB/B)/(dr/r) then dB/B = -ndr/r. dp/p = dB/B Consequently we have

+

and the momentum compaction is

If n has the customary value 0.6, as in a weak-focusing synchrotron, then a = 0.4.

As another example, consider the impractical cyclotron with uniform field throughout. Then n = 0 and a = 1. Any fractional change in momentum is accompanied by an equally large fractional change in radius. I n an operable cyclotron the field is almost uniform a t the center but gradually drops off so that near the final radius n may be as large as 0.3 so that a = 0.7. I n this latter region an increase in p requires an even larger fractional increase in r, since B is falling. This ever widening space between successive orbits is of great help in the process of getting particles completely out of the magnetic field, as will be described later. I n computing the momentum compaction of a race-track synchrotron (one with field-free straight sections separating portions of the magnet), the stipulation is added that the magnet's ends be cut perpendicularly to the equilibrium orbits in order to exclude the zero-gradient synchrotron, which must be conNs, sidered separately. For the particle with momentum p, we have L = 27rr where the first term is the path length in all the magnets and the second is the path length in N straight sections each of length s. The path for the ion with momentum p dp is L dL = 2a(r dr) Ns, the contribution of the straight sections being the same as before. Then

+

+

+

+

+

MOXIEKTUM COMPACTION

83

where TU is the length of each magnet, so that, 12' = Z1ar/tn.T h e field varies with radius of curvature in the sarne way as in the circular machine, so again

Therefore a = (1 -

71) (1

+ s/w/).

Since the field is uniform in a zero-gradient synchrotron, an ion changing its nlonicntunl from p to p dp niu,st obtain its incre:ised Br cntirclly from n 1argc.r r. Both orbits turn through thc same angle in a magnct in spitc of its slantcd faces, and a geometrical construction as in Fig. 6-5 shows t l i i ~ tthe centers of

+

(centerline of magnet \

Fig. 6-5.

Geometrical relationships in a

magnet with slanted ends.

curvature are separated. Sirice T I N is the half angle through which each of the ,I' magnets defiects the heam and fl is the s h t angle of the magnet face, it i h evident that r -td r - y = r - y so that tlr - d~ cot 0 = -d.j cot (a/il;). Ilence dx = Kdr \\here I< -- l/(cot fl - cot ( a / & V ) )For . all ion with momentum p the N s , where s is the lrirgth of a straight scction at the total path is I, = 2ar position of tlu* particle, while for an ion of n ~ o ~ n c n t u np i rip tllc total 1):~tliis lJ dlJ = 2r(r Or) AT(,5 2dh) since the path i11 each straight section is increased by d.\ a t each end. Then dL = 2adr 2Ndu and we find

+

+

+

Since N

=

+

fLar/rt~,then r

+

+ N s l ( 2 a ) = r(1 f

+

+

s/tti).

Therefore

84

PHASE STABILITY

In general p = qBr, so since B is independent of radius in this machine it follows that d p / p = dr/r. The momentum compaction for a zero-gradient machine is therefore

Substitute for K to find 1

(cot 0 - cot ( s / N ) Here 6 is the slant angle of the magnet face, and the magnet length m and tlic straight section length s are measured a t the position appropriate for particles p. As a numerical example, a somewhat simplified version of thc of n~omentun~ zero-gradient synchrotron under construction in 1961 a t Argonne National Laboratory may be considered. Here m = 52 ft, s = 14 ft, N = 8, and 0 = 12". With these figures the momentum compaction is 0.6. In general it is seen that if axial stability is attained either through the use of magnets with zero gradient and slantcd ends or by the employment of a finite constant gradient and square-cut magnets, the momentum compaction is reduced below the value CY = 1 that would occur in an impractical cyclotron with uniform field where axial stability does not exist. For constant-gradient (weak-focusing) machines the compaction may be classified as small. In comparison with the strong-focusing accelerators to be described later, in which exceeds unity, weak-focusing devices are characterized by orbits which are relatively widely spaced for a given difference in momentum. As will be seen in the following section, the value of the compaction has an important bearing on whether an increase in momentum causes a rise or a fall in the period of rotation. 6-4. The Relation Between Period and Momentum

The problem of phase stability has been studied in detail by a number of authors in various degrees of approximation. The treatment can become exceedingly complicated if one handles the betatron and synchrotron oscillations as occurring simultaneously, as indeed they do. Such a rigorous analysis leads to equations which are difficult to interpret, so it is customary then to imagine that the motions are uncoupled. This leads to a disentanglement of the equations of oscillation, which may then be scrutinized separately. For the present purpose it appears more illuminating to suppose the motions to be independent to begin with, this being not f a r from the truth if their periods differ markedly. The final results are much the same and the argument is easier to follow. Consequently, in this chapter we will pay no attention to the betatron motions, which are assumed to take place a t constant energy, and will concentrate on what happens when the energy does change.

\Ye will not take into account, the loss of energy hy elcrtromagnctic 1xt1i:ition arising fro111the continual cmtr:ll :wct~ler:~tion of chargctl prticleh iiioving in a circuliir oibit. This can be very i111~)ortant for clectron accelerators, b u t is utterly insignificant with hcavic~rions. A few furtl~crrc1n:irks on the subject will be niadc in due course. As an additional simplification, we will neglect the accelerative forces caused by the time-changing m a g ~ ~ c t flux i c within the orbit (the "I~etatron action") which orcurs in any accelerator with a pulscd field. Later it mill bv shown t h a t the mergy change brought :ibout in this ninrincr is usually quite snxtll compared with that produced hy traversal of the osci1l:itorclriven gap. Thc first stcp in a quantitative analysis is to inquire whether an ion with more than the synchronous momonturn will have a period of revolution which is more or less than the period of a synchronous par1,icle or just equal to it. For an orbit of length L, traversed in the period T a t velocity v, me have VT = I,, so that (JT 11 - -(Zfl . - - dl, - - d- dL - (6-8) 7 - L 1) L p This esprcssm thc fractional change in period of rotation as a function of the fractional changes in the orbit length and in the velocity. By use of Eq. (1-33), = dpl(py'2),this hecomes which states that c@/P =

and by Eq. (6-1) this can be writt,en a s

This, the desired relation, is of considt~rablcsignificance. It is pcrmissiblc to identify T and p with the period and the momellturn of the synchronous particle; then d7 and dp arc the changes in these parameters that are associated with the nonsynchronous ion. It has been seen in Eq. (6-3) that in a wmk-focusing synchrocyclotron a = 1 - n, with n rising from zero :LS acceleration proceeds but never cxceecling the value unity even in principle, the practical upper limit being 0.2. Corwquently l / a starts a t unity and rises thereafter. Ions leave the source with :t small hut finite energy, so in~tinllyl / y V s just a sh:~deuntlcr unity and drops subsequently. Tliercfore the 1)rncketcd tcrni in Eq. (6-9) is always positive in a weak-focusing rnagnetic:dly guided accelerator and ~ T / T has the snrt~t?sign as dp/p, an increasccl nlonienturr~bringing about an incrense in the periotl, and vice versa. Otherwise stated, the influence of the longer path resulting from a rise in momcntu~nis more important than t h a t of the associated increased velocity. The same conclusions hold for weak-focusing synchrotrons, for with a value of n fixed between 0 and I and with a large injection energy, l / a always es-

86

PHASE STABILITY

ceeds l/y2 by a big margin. Straight sections, whose influence on a is shown in Eq. (6-6), would have to be inordinately long to reverse the situation. In the case of such weak-focusing machines, phase stability will exist if the synchronous ions are accelerated when the electric field is decreasillg, as was argued earlier in § 6-2. As will be emphasized in Chapter 12, there are classes of accelerators (for example, the alternating-gradient type) in which the momentum compaction exceeds unity; so that if y is not too large, the quantity ( l / a - l/y2) is negative, and consequently an increase in momentum causes a decrease in period of rotation-just the opposite of what occurs in a weak-focusing machine. As a very special case, we may consider the linear accelerator, which also functions by virtue of phase stability. Such a machine may be thought of as a cyclotron rolled out into a straight line, but with the important difference that each gapto-gap distance is fixed. Consequently there can be no question of a possible change in path length between successive gaps; so the term l/a does not appear in Eq. (6-9), which reduces to

More specifically, since the equilibrium orbit is straight, the radius R of the equivalent circle is infinite and hence so is a ; particles of different momenta are not displaced transversely with respect to each other, but lie in orbits which are coincident. The ininus sign appearing in Eq. (6-10) indicates that an excess of momentum brings about a dccrease in transit time from gap to gap, as is also evident from the simplest considerations. It is not hard to see from Fig. 6-6

Fig. 6-6. Synchronous and nonsynchronous phase relations when a momentum increase reduces the period between accelerations, as in a linear accelerator. that in this circumstance, phase stability of nonsynchronous particles will be possible only if the synchronous ion reaches the gap when the electric field is rising; that is, must lie between 0 and 7r/2. Then a particle a t +, receiving more than the synchronous energy increment, next reaches the gap a t +,' closer to +,, so that oscillations occur about +,. Equation (6-9) also shows that if y2 = a , there is no change in period when the momentum alters, and in this situation phase stability does not exist. This

+,

THE PHASE EQUATION

87

cnergy is c:illc,l thc trnnsitlon or criflcnl cnergy. E:xan~plescvhcn this occurs will bc discu~scdlater in c~~tlnection with tllv alternating-gradicl~tsynchrotron and the isochronous cyclotron. 6-5. T h e Phase Equation

I h v i n g been under n11:it contlitions 1)h:~scst:~t)ilitycan exist, we will now sc of u nonsyricl~ronousion as a function tlcrivc an cxprwsion for tlic ~ ) l i : ~ angle of time. The instantancous tiiffcrcnce in cnergy between a nonsynchronous and a synchronous particle is givcn by

AI< = li - E\.

(6-1 1 )

In crossing an accelerating gap of peak voltage I,,,,, these ions gain yT',,, sin 4 and qT',,, sin 4, respectively. The cliangci in the difference of their energies per :~ccelerationA is therefore

X o w dA/dt is the number of accelerations per second, so if there are S gaps distributed around the machine, then

(G-I 3) where CQ,is the radian frequency of revolution of the synchronous particle. Hence dA = 1Vw,dt/(2n) arid E(l. (6-12) becomes

The rate at, which the nonsynchronous particle changes phase is &/at, so the phase ch:mges b y ~,d+/'tlt in one period 7 , of the synchronous ion. This change call alzo be expressed ah the product, of the radian frequeucy wrJ of the oscillator aild the excess time AT that the nonsynchronous particle takes to 1n:ikc thc tril). Hence

I n order to include future pohsibilitics, we now run the oscillator (or the amplifiers driving the S gaps) a t some intcgral illultiple h of the synchronous revolution frequency so t h a t (6-16) wrf = hwp. This causes peak voltage to be c-leveloped across the gaps to no apparent purpose w h m 1,:lrtirlcs arc1 far from tllese regions; a v:dtie of h other than one is not nect>ssary,hut it ran 11:1v(' rert:tin :1dv3rltilgeh which will appear later. It is custonlary to c:Jl h t l ~ harmonic. . ~ t i n ~ b or t r the hrlrrnonic order. With this :~clditionEq. (6-15) can be written

PHASE STABILITY

88

Now by Eq. (1-31) /?2d~l/q = d-y/-y so d p / p

=

d E / ( P 2 E )and Eq. (6-9) becomes

where finite changes have been substituted for differentials to suit the present situation and thc subscript s has been addcd to identify the synchronous value. Substitute in Eq. (6-17) to find

n Eere we have introduced the abbreviation

Rearrangement of Eq. (6-19) yields

When this is substituted in Eq. (6-14) the result is

NqVm (sin 4 - sin 4.). 27r

This is the equation toward which we have been working. It expresses the timedependence of the phase + of the nonsynchronous particle as a function of the phase, frequency, and energy of the synclironous particle and of the contour of the magnetic field, since r depends not only on y, but also on the momentum compaction a. 6-6. The Analogy of the Biased Pendulum

Consider the situation during a time interval so short t h a t the changes in and in r are small. Equation (6-23) may then be written as

E, d'4 hw?r dt2

ws

NqV, sin . 4 = --Nqvm sin +a. 27r

27r

This equation is siniilar to that of a biased pendulum, such as is formed by wrapping a string around tbe axle and hanging a weight from it, as indicated in Fig. 6-7. (The angle 4 is tlic supplement of the usual angle of displace~iient.) Such a pendulum has two particular positions 4 = 4, and 4 = ?r - 4, a t which

Fig. 6-7. Thc two equilibrium positions of

$5

a biased pendulum.

tion about +,, provided the release point, if on the right, is not so large as to carry the hob, one half cycle later, into the uilstable region where 4 < a - 9,. The applic,xl)ility of this analogy to :In accclcrntor should be clear (see Fig. 6-81. Injection of a particle with the synchronous energy (equivalent to release

Fig. 6-8. Values of

+ which are less than a - +,

are unstable.

of the pentlulum from rest) a t any phase between a - 4, and 4 , will be followed by owillations ahout +,, and injection :it + > +, also engentlers stalde oscillation unless the injection ~ h n s cis so large its to carry the phase, one half cycle later. into the unstakde region \\-here 9 < a -- 9,. If the ion ever crosses tlie acrelcrating gap in that region, the phase shifts still further to the left so that the particI(~first recc,ires Icss than the y w h r o n o u s energy and soon ib :~ctuallytlecc1er:~tcti. By the tiine the phase has moved so far to the lcft that nccelcration once again sets in, the enel,gy deficit is so great that it cannot bc made u p hy the acceleration produced in the shift from 4 = a t o 4 = 0, and the phase, rontinucs to slide to tllc lcft ~ntlcfinltelg.This means that this particle altcrnwtc~lygains and lows mergy but n~nintainsa constant average value., irrespective of the cxistcnce of tlie gap ~oltagcb,which, as time goes on, runs at a frequency niorc and morc remote from thr. average revolution frequency of this partirular ion. I n a synchrotron, where tlie magnetic field is rising, such s particle is soon driven to the inner w:~11 and lost,. With a synchrocyclotron, d i e r e the fieltd is static in time, the proress of capture into ph:tse-stable orbits, or of rejection therefrom, occurs soon after the pabrticlcs leave the ion source. The rcjertcd ions circulate about the source until they strike it a t a moment

PHASE STABILITY

90

when their condition of minimum energy coincides with the inward swing of a radial betatron oscillntion. Note tliat the injection of a particle with r i m e than the synchronous energy is siinulatcd by starting the pendululn with s o ~ n einitial velocity. The niechanical motion and the oscillation in phase are both stable unless the point of release and the initial energy are such as t o make 9 < a - +, on the first excurhion to small values of (6. Thus, trapping in phase-stable orbits is possible even if the injection energy is greater than the synchronous value. Ions injected with less than the synchronous energy will shift in phase to the left and will be caught in phase-stable orbits, provided the net result of any dcccleration and acceleration brings their energy to the synchronous value hcfore the phase slips to the unstable region where 4 < a - 9,. Tlie permissible energy error a t injection, of either sign, depends on the valuc of (6, and on the phase a t the moment of injection. (A quantitative expression will hc developed beforc long.) This property of succes~fultrapping of particlt~s of other than the synchronous energy is of vital importance in synchrotrons, for the auxiliary accelerator used as an injector always produces some energy spread in its output ions. Weak-focusing synchrocyclotrons and synchrotrons automatically set the phase-stable angle between ~ / and 2 a, since this is the only quadrant in which nonsynchronous particles are accepted. The designer has merely to choose thc value of sin +,, by making the peak gap voltage such t h a t qV, sin 9, is equal to the average energy to be gained per turn. This latter figure is sct by the final energy, the average velocity of the ions when in the machine, and the duration of the acceleration interval. 6-7. The Phase Diagram

We return t o Eq. (6-24)) valid over short intervals during which w, and r change only slightly, and change the indepc~nd~nt variahlc from the time to the ~)liaseangle. Since

Eq. (6-24) may be expreseed as d 9 d d+ dt d+ dt

AToV,h o,2I'(sin 9 - sin 9,) ~TI?,

=

0.

Multiply by d 4 and integrat,e:

(2)'+ For convenience define

u,?F (cos 4

+ 9 sin 9,)

=

constant.

(6-26)

+,

m d note that tIiis is positivo since cos +, is nrlg:itive .~rhcn lics in the second quadrant, as occurs in weak-focusing ni:~gnetic~lly guitletl nccclerators. I n those clt)vices in whicll +, 11c.i in tlrc first qundr:int, as for c~x:urrple,n linear accelcrator, r is ncgatlx e, as 1s s l i o ~ ~by r i Eqs. (6-10) and (6-21). JTith this substitution, Eq. (6-26) beconles 2Q2

(COS4 f 4 sin 4,)

=

co~lstant.

, Evaluate tht. constant by wing the initial condition t h a t d$~/dt= ( d d ~ l d t )when 4 = +,. H m c e

An interesting specihc casc may he invcstlgated by setti~ig4 , = a - 4,. As mcntioncd carlier, t l ~ ds u e r c p r e w ~ t sa maximum di.iplacemclit from 4 , (i.e., a minimum valuc of +) if stable o>cillations ill phase are to occur. Consequently it folio\\ ti that (d+/dt), = 0. Since in general - c o b (7r - 4,) = cos I$,, Eq. (6-29) t)ecomrs

1;rorn this e~prcssionthe dinwlriiodess quantity dr#~/(Qdt)may 1)e plotted aq a function of + for variou. values of 6,.l h t first, it will bc instructive to convcrt d+/(Qdl) into expressions deicribing the difkrellce ill energy A E and the diffcrcnce in thc equivalent radii All of t h r orhiti of the synchronous and the nonsynchronous ions. Return to Eq. (6-22) and rewrite i t a s

Therefore

T o n u k e thc vonvcrsion to h R , we liavc recourse to the definition of inoillenturn coinpaction, 15q. (6-2) :

Use Eq. (1-31) and replace differentials by finite chunges to obtain

Substitute for A E / E , from Eq. (6-22)

92

PHASE STABILITY

Then

Let F(+,+,) represent the function appearing on the right side of Eq. (6-30) ; that is, define

so that Eqs. (6-30), (6-32), and (6-36) may be written

constant over short intervals, so it is apparent The denominators are pra~t~ically that d4/dt, AE, and AR are proportional to the same function F of and 4,. If eubstitution is made for D and r there results: C$

d+/dt

[

cos 2rEO

-NqV,h

ws

+,

$6

(6-39)

=

The function F(+, 4,) is shown in Fig. 6-9 (often known informally as the fish diagram). The left-hand intercepts with the horizontal axis represent those extreme amplitudes of + (here chosen as the particular value where + = ?r - 4,) where F = 0 and so where d+/dt = A E = AR = 0. The intercepts on the right, also a t F = 0, give the corresponding other extreme amplitudes a t > h. These latter amplitudes are extreme only in the sense that if they are exceeded, instability will result a half cycle later. = 165", so that the synchronous ion reSuppose the choice is made that ceives qV, sin 165" = 0.259qVmin crossing the gap. Then the nonsynchronous particle which will undergo the widest oscillations in phase, in energy, and in radius will reach a minimum phase angle = 15". This condition is given by the intercept a t the left, where F = 0. On successive accelerations the phase increases and F grows (drpldt, AE, and AR all increase) to reach a maximum of 1.62 = 165"; this is followed by a reduction, just as 4 passes through the value = until zero is reached a t = 262". This completes half a synchrotron oscillation and the remainder follows the lower part of the curve. The total excursion in

+

+,

+

+

+ +,

TIIF, PIIAPE DIA\C;IlLI3t1

93

phacr is 262' - 15' = 247"' S o t c that it is not ~ymmetriralabout $,, and tlmt on the portion of the excursion where > 180°, the particle undcrgocs decelera-

+

tion at the gap. As another example, assume 4, = 120". This makes the synchronous ion rcceive 0.866 of the peak energy gain when it traverses the gap. The nonsynclmnous ion which is a t the limit of stability oscillates in phase over a more r c strictcd range, from 4 = 60" to $ = 150°, again aayninlctrically with respcct to 4,. The cwsursions in energy and in radius :Ire sr~inllert1l:in in the prel.lou+ case, and thert. is no deceleration.

Fig. 6-9. The phase diagram for the case where & = x - +,.

One can draw the following general conclusions. 115th a large value of 4, (near a ) , the average energy gained per turn is small, wide oscillations in phase develop, the beam of ions exhibits a relatively great inhomogeneity of energy, and the currcnt is big, since a large fraction of all the ions present becornes trapped in phase-stable orbits, assuming the ions to have been uniformly distributed in :izirnuth to start with. On the other hand, with a small 4, (ncar a/2), there is a large average energy gain per turn, small oscillation in phase, little energy sprcad, and a relatively small current. As extreme examples, if +, = a, all the ions are trapped, they fluctuate widely in energy but gain nothing on the average; while if 6, = a/2, the appropriate curve, not shown in Fig. 6-9, has degenerated to a point and very few ions are caught, but these few have a vanishing energy spread. It must he emphasized that Fig. 6-9 and the remarks made in referencc to i t were based on the supposition that 4, = a - +,, mhich is indicative of the limit of stability. I n practice, many ions will. enter the system when 4, is greater, and

94

PHASE STABILITY

to describe their behavior requires a different set of curves, which may be obtained from Eq. (6-29) by the use of suitable initial conditions. Such curves will be of a similar nature to those shown in Fig. 6-9, but will lie within the curves displayed, for the same values of 4,. 6-8. Permissible Error of Injection Energy

It was mentioned earlier that because of phase stability a synchrotron will accept ions from its injector with a certain spread in energy. This statement can now be put on a quantitative basis. Rewrite Eq. (6-40) as AE/D = F , where all the quantities in the denominator D are assumed known, 7, being the measure of the mean energy of the injected particles. Let AE represent the half-width in energy spread of the injected beam. In this way a numerical value of F is obtained, and in Fig. 6-9 two horizontal lines may be drawn, passing

LE=-~

4 Fig. 6-10. Phase diagram showing a small spread in energy of the injected ions.

through r+F.This is indicated in Fig. 6-10, where only the curve for one chosen value of 4, is shown. A particle a t point a, where 4 = 4,,, = a - 4,, must have F = A E / D = 0, so only a particle with the synchronous energy is accepted in a phase-stable orbit a t this value of +. As the particle drifts in phase to larger values of 4, it develops an energy error, so when the point b is reached, the error is measured by Fb. The synchrotron cannot distinguish between an error developed inside and one developed outside the machine, so i t will equally - well accept a particle injected a t phase + b with an energy error Fa (or less). A larger injection error is permitted for entry a t greater phase angles. Beyond point c, energy deviations are allowed which are bigger than those produced by the injector, so from c to d the machine will accept all the particles presented to it irrespective of their energy errors. At phases between points d and e, the synchrotron again imposes a limit on the maximum error in energy that it will accept. If the energy spread from the injector is so large that the two lines a t F = r+hE/D lie outside the curve for I$,, as shown in Fig. 6-11, the ions with extremes of high and low energy are totally rejected, as is indicated by the ,, particles are accepted only shaded areas. At all values of + from +,in to ,,+ with energy errors from zero up to the value specified by the curve. Thus, in order that as large a fraction as possible of all the injected ions should be accepted, they should have a small spread in energy. If the spread is of some considerable magnitude, a large value of +, should be employed and

PERMISSIBLE ERROR O F INJIXTIOK 14XERGY

95

the gap voltage should be increased, since this lowers F, as indicated by Eq. (6-40). Similar considerations apply to the frequency-modulatcd cyclotron, though a t first sight it may appear that an error of injection energy is impossible, since all ions start from rest. H o w c ~ e r ,zero energy is correct only for the initial value of the oscillator frequency, and this frequency is made to decrease as tlic particles are arcc1crxtc.d. Conseqi~ently,after the start of the modulation cyclc, zcvo energy of tlic cntclring ion rrprescnts a steadily rising error of injection rnergy with respcact to the oscillator frequency then existing. Ions will be accepted into phase-stable orbits only during t h a t interval in which

Fig. 6-11. Phase diagram showing a large spread in energy of the injected ions.

this energy ~ r r o lies r within the permissible range. From this viewpoint, a large is desirablr, since then the numbcr of accepted ions is the greatest. On the other hand, a big +, implies only a little energy gain per turn for the synchronous ion and also, on the average, for the nonsync~hronousones, so the orbits do not grow in size rapidly. Correspondingly, a large implies large energy oscillations accompanied by large excursions in radius a t the slow frequency of the synchrotron oscillation. As a result, :it the first inward swing of this oscilI:~tion,many particles undergoing an inward betatron motion will collide w ~ t h the ion source and be lost. C'onvcrsely, ~f +, is small. the orbits expand rapidly and the radic:tl synchrotron aniplitude IS snlall, so t h a t fewer particles are lost in this manner. Rec:~pitulating,in a synchrocbyclotron a big value of 4, leads to thc greatest initial acceptance of ions, ~xhllewith a small +., the ulti~natc retention of those that have been accq)tetl is the most. It is apparent t h a t the largest number to reach full energy will be obtained with some intermediate value of +,. Detailed calculations arrive at, slightly different optimum values, depending on whether the dee voltage or thc rxtc of frequency modulation is assunled adjustable. The best practical compromise lies in the neighborhood of 150°, ~vllcrcthe synchronous ions pick u p one-half of the available energy. It is notcluorthy that the anil)litutles of the cxcursions both in energy and in raclius are inversely proportional to the, squ:tre root of the harmonic number h , as is setn in Eqs. (6-40) :ind (6-41).I t is therefore profitable to run the oscillator of L: hynchrotron :it :i high nlultil)l(, of the rotation frtyucncy, bince tllis rctlucw tlw Ilecesbary n i d t l ~of tllc ac'uunr c ~ 1 l : t r t i l m and niagnct :i]wrtu~c(nlthough it also lowers the perniishiblc ~mcrgysl~readof the injector). How far

+,

+,

96

PHASE STABILITY

this can be carried depends largely on matters such as the availability of oscillator and amplifier tubes and of suitable ferrites with which to control the frequency, and similar practical considerations. 6-9. Frequency of Synchrotron Oscillations of Small Amplitude

If the excursions in phase are small, it is profitable to measure them with respect to 4,. Let (6-42) f E 4 - 4%. Then d2+/dt2 = d2[/dt2 and sin 4 sin 4

=

sin f cos h f cos 4,

+ cos [ sin +,, so if [ is small

+ sin 6.

(6-43)

Consequently, again assuming that a,,E,, and r are constant over the time interval under consideration, we may write Eq. (6-24) in the form

where, as in Eq. (6-27),we use the abbreviation

Equation (6-44) describes simple harmonic motion provided a2 is positive, in which case a is the radian frequency of these small oscillations. That n2 is positive is indeed the case for weak-focusing magnetic accelerators where r, defined by Eqs. (6-20) and (6-21) and discussed in alternate form in connection with Eq. (6-9), is positive and where cos 4, is negative, since it has been seen that phase oscillations will occur only if 4, lies between s / 2 and 7.For accelerators in which r is negative (for example, a linear accelerator, as shown earlier in Eq. (6-lo), it is clear that Eq. (6-44) can describe an oscillatory situation only if 4, lies between 0 and s/2, so that cos 4, is positive. The necessity for this was argued on physical grounds in connection with Fig. 6-6. 6-10. Adiabatic Damping of Synchrotron Oscillations

It must be emphasized that Eqs. (6-39), (6-40), and (6-41), which lead to the phase diagrams for d$/dt, AE, and A R for a nonsynchronous ion a t the limit of stability, were derived on the assumption that the energy and the frequency changed so slowly that they could be considered constant over the interval of interest. These equations therefore may not be used to predict values separated by long intervals when significant changes in energy and frequency will have occurred. But it is nevertheless desirable to see how the amplitudes of the synchrotron oscillations vary during the course of the entire acceleration process which, in large synchrotrons, may last from one to several seconds. Happily, the time-varying parameters change slowly with respect to the synchrotron

oscill:ttion, so the K K R approxirn:ttc ~ r ~ . t h oofd solution is applicable to the d1ffcrcnti:~lcyu:~tionswith vnriwhlc eoefir.ionts. \Ye consic-lc,r first the long-time beh:~vior of the oscillationh of tlie phase angle + of nonsyncllronous ion, This is governed by Eq. (6-23). For convenience let

Then Eq. (6-23) becomes

K e limit the discussion to snlall arnplitudcs and again set t: = 4 - +,, so t h a t ~ sin +,, as in Eq. (6-43). Vsc this in Eq. (6-47), perform sin ==; ( C O 4, the indicated differentiation, divide by u and substitute Eq. (6-46), where u appears on the right. This gives

+

+

But by Eq. (6-45), the quantity in I~racltetsis -!2', :dthough 0 is no longer interpreted as the synchrotron oscillation frequency Thus we may write

This equation exhibits a damping term proportional to d$/dt, and we know that t h e caeficient u/uchanges dam-ly. T o apply the W K B approximate method of solution, we set

where f ( t ) is some function as yet undeterniined and, be it remembered, depends on time since it contains W, and r. Since Eq. (649) is identical in form with E:q. (2-32), wherein the damping of betatron oscillations was investigated, the solution follows the same pattern with CPreplacing b and u replacing d l , so by analogy with Eq. (2-37) wc haw:

where K is the constant of integration. Substitute for n, u, and Eqs. (6-45), (6-46) and (6-21) and set E, = y , Eo, l,o find t h a t

r

by use of

where all tlw fixed quantities have been absorbed in the constant. This exprcssion hlicnvs that the amplitude of small phase oscillations is reduced as the energy rises. For psan~ple,compare tlie valuc. (f at n final energy of 8.3 Bev

98

PHASE STABIIJITY

10) with gi at an injection energy of 50 Mev ( y = 1.0533) ; we find tr/[i= 0.65/1.72 = 0.38. To discuss the time dependence of the energy oscillations, we return to Eq. (6-14), here repeated: (y =

1 dAE w, dt

-

(sin 4 - sin 6).

This can be altered into a form involving second as well as first derivatives by differentiating both sides with respect to time:

Eliminate d+/dt through Eq. (6-22) and multiply by

w,:

Once again we measure the phase displacement from 9, by setting = - +,T. Then cos 4 = cos (E 9,) = cos E cos 4, - sin E sin 4,; and if E is small, then cos 9 z cos 4, - E sin 4,. Therefore AE cos 9 = A E cos 4 , - AEi sin +,, and since both AE and E are small we drop their product to obtain A E cos 4 m AE cos 4,. This approximation permits the use of the abbreviation Q2 as defined in Eq. (6-45). Therefore

+

Note that the sign of the coefficient of the first derivative is negative, so that antidamping, ie., growth, is to be expected. To apply the WKB method, set A E = AE,f(t) exp [f j J Q dt], form the first and second derivatives, and use t4heseexpressions in Eq. (6-56). Drop terms in 7 and f& and then divide by 2jQf. This gives

and on integrating we find

f

=

Kus%'-$$,

where K is a constant. As a result of substituting for 2! from Eq. (6-45) we find that rS2- 1 A E = constant (6-59) exp [ A j dt].

]

/*

Here, as expected, it is seen that A E increases with rising energy; in fact, comparison of Eq. (6-59) with Eq. (6-52) shows that the dependence of AE on y, and a is just the reciprocal of that exhibited by 6. In terms of the previous example we see that between 50 Mev and 8.3 Bev the energy inhomogeneity is increased by a factor 1/0.38 = 2.63. Most of the energy spread

in the output of a synchrotron or spnchrorpclot~~on has its origin in this :intidamping of thc encrgy osrillations of thc nonsynchronous ions. In order to investigate thc radial synchrotron niotion as a function of time., Eq. (6-34) is v rittcn in the forrii

This is used in Eq. (6-14') in conjunction with the nhhreriation

Differentiate this with rc,,ipect to tirnc, rcinenhc~ingthat time dcpcnclcnt :

p,

R , and AR are

Multiply the last t r r m on the right by w /w,. This tlmrm then hrcomrs equal to ( G , / U $ ) ~A( R~ / K ) / d t hy 14:q. (6-02). Express Eq. (6-63) in this form and carry out the indicated differelltiation, clropping terms in p, if?, L,i, i2?, 8 , and R/R' as negligible. After collection this gives

[2CL--R w.R]

p dl AR -.-+ X dt"

6"p

NqV,w, - cos 4

d -AR =

[

dl

((i-64)

?nl!',,

Kow use Eq. (6-22) to express d 4 l d t as a function of AE and use Eq. (6-60) t o give A E ill terms of AR. Put this rxpression for d+,'tlt in Eq. (6-M), multiply by R I P = R/(a/Yy) and again assume small oscillations meawred from 4, so that AR cos 4 z AR coq 4, by an a r p u n ~ r n tsimilar to that used in converting Eq. (6-55) to F:q. (6-56). Introduce the abbreviation R2 a1 in Eq. (6-45). All this yiclds

IIere there is n damping tcmn in co~npetitionwith an antidamping term. T o use the \17KEi niet1lot-l of solution set h R = A R , , ,(~t ) x exp [ * j .f d t ] . 1-w this and its dtv-iratircs in Eq. ( 6 - 6 5 ) .drop the sm:ill terms and obtain f = w,$?(t$"-l

P

Substitute for R and AR

=

constant

,-l

-

to find

-'&]"

"

ffy,(I/ff -- l/y,-)

e ~ [A p j

R 11

(6-67)

100

PIIASE STABILITY

This shows that A R is damped, its ralue decreasing as the energy rises. This accounts for the fact that although a synchrotron is initially filled wit,h ions performing wide radial synchrotron oscillations with an envelope that fills the vacuum chamber, the full-ecergy beam becomes squeezed down to a very narrow bundle. For such a. rnachine, the momenturn conlpaction does not vary and may be placed in the constant. I n that case the energy dependence of A R is y,/(y," 1) times the energy dependence of A E , as shown by Eq. (6-59). For the previous example of an 8.3-Bev synchrotron with a 50-Mev injector, it is found that the radial width of the beam caused by synchrotron oscillations is damped by a factor 36. A similar damping of the radial osrillations occurs in a synchrocyclotron, but detailed knowledge of the behavior of with energy is needed to compute A R a t specific energies. (Y

(Y

6-11. Over-all Motion of Ions

I f the injector for a synchrotron is a steady-current device such as a Van de Graaff machine, the ions enter in a continuous stream. If a linear accelerator is used with a driving frequency perhaps 200 times the rotation frequency in the synchrotron, the stream of ions is segmented, like beads on a necklace. The process of capture into and rejection from phase-stable orbit goes on, until the surviving ions are left in long bunches, with an empty space ahead and behind each. The number of such groups depends on the harmonic number of the oscillator and on the number of accelerating cavities, the length of each group depending on the range of phase angle representing stability as deter= 150" the range is apmined by Fig. 6-9 for the existing value of +,. If proximately 185O, so if h = 1 and N = 1 there is one group with azimuthal length 185" and no ions elsewhere. If h = 2 there are two bunches, each about 92" long separated by corresponding gaps, and so forth. The bunches are bounded by envelopes which resemble sausages. The more energetic particles (in a weak-focusing machine) lie towards the outer side and gradually drift backward to the rear end of the sausage, since their periods exceed the synchronous value, while the less energetic ions creep forward near the inner boundary. Some hundreds or thousands of trips around the machine may be needed to complete one fore-and-aft cycle. These paths, however, serve merely as reference lines about which the very fast radial and vertical betatron oscillations take place, so if the underlying pattern were not known, the motion of the ions within a sausage would resemble that of a swarm of bees. Only a sophisticated eye can detect any order. The same general pattern occurs in a synchrocyclotron. Here, usually, h = 1. The angular extent of a sausage is not as long as was calculated above because of the loss of particles intercepted by the ion source on the inward swing of the first synchrotron oscillation.

+,

SE’h’CIIROTRON

CXXILLATIOSS IN A CYCLOTROS

101

6-12. Synchrotron Oscillations in a Fixed-Frequency Cyclotron The process of acceleration in a fixed-frcqnrncy cyclotron may be considered as one quarter of a synchrotron oscillation of nonsynchronous ions. Imagine the osrillator frequency to bc ad,justcd to be equal to that of ions just leaving the source (U = @/M,,), with the dee voltage then at its maximum. Only for one brief instant is there synchronization, since any energy gain The synchronous whatever will raise the mass and lower the rotation f:requency. phase angle $a is IfSO”, for only this will produce no mass increase, as is demanded by the oscillator. The actual angle at the start is 90”, with peak voltage about to be gained. As in a synchrocyclotron, continued acceleration causes the period to increase, since all energy increments are in excess of the synchronous inrremcnt (zero) and so the phase shifts closer to +. As is evident from the phase diagralu of Fig. 6-9 for & = 180”, the energy “error” AE becomes a maximum (the full-output energy of the cyclotron) when 4 becomes equal to $+. At the same moment the excursion in radius, AR, reaches its peak (near the edge of thr poles) and the phase changes most rapidly. If no target intcrcepte the particles, further passage through the dee gap will start a decelerativc process, tllta phase angle moving to larger values, with AE and AR decreasing. At 4 = 270” the ions receive peak decclerative dee voltage and come to rest at the ion source, having experienced half a synchrotron oscillation.

Chapter 7 FIXED-FREQUENCY CYCLOTRONS

7-1. Energy and Types of Projectiles

Although for simplicity we have thus far usually considered protons as the projectiles in cyclotrons, this restriction does not necessarily hold; protons, deuterons, and helium ions are all very commonly used, while to a much lesser extent ions of heavier elements have been employed. Bcause the demand for more versatile machines had not yet arisen, most cyclotrons of the past have been designed in such a manner that changing the frequency of the oscillator by any appreciable amount is a rather major project not to be accomplished in an afternoon. One single frequency is chosen in consonance with the field of the magnet and the ion to be accelerated, the circuits being tuned in accord. From the fundamental relation w = qB/M it is seen that the charge to mass ratio is the important ion characteristic. Since the proton has a unique value for this ratio, a cyclotron adjusted for protons cannot accelerate any other particle (unless "harmonic" operation is employed, as will be described later.) If the machine is tuned for deuterium, of charge 1 and mass 2, it will also handle doubly ionized helium and singly ionized light hydrogen molecules, since all these enjoy q / M = 4.In theory sextuply ionized carbon is equally suitable, but is practically unattainable in significant quantities. The very small change in frequency or field needed to compensate for the not quite integral relation between the masses of these particles is well within the tuning range needed for seeking resonance. Since a "deuteron cyclotron" thus offers a choice of three projectiles compared to a "proton cyclotron's" one, it is not surprising that of all the fixed-frequency instruments operating in the world in 1960, with energies ranging from 1 to 25 Mev, forty-four are for deuterons and nine for protons, while one is adjusted for 27-Mev nitrogen ions, triply charged. The more recent plans for cyclotrons capable of handling a wider variety of projectiles will be considered in Chapter 13. 7-2. Operating Frequency

As has been shown in Eq. (1-4), the radian frequency of revolution of the ions is given by w = q B / M . For low energy prot,ons, with q = e = 1.6 X 10-l9 102

MAGNETS

103

coulomb and Mo = 1.67 x 10--'' kilogram, the oscil1:ltor will be in resonance i f its frequency is sct a t

where f is in cycles per second and R is in webers per square meter. I n more customary units, this can be expressed as where f is in megacycles per second and H is in kilogauss. Thus for the typical value of 15 kilogauss, the oscillator rune a t 22.8 Mc/sec. If the cyclotron is tuned for deuterons, with twice the mass and the same charge, the frequency is half this, for the field will be kept a t tlie same value to get the most energy of which the magnet is capable. These are vcry convenient frequcncics, for a number of triodes and tetrodes arc available co~ntnerciallywith adequate power. rating. For a large cyclotron with magnet poles 60 inches in diameter, the R F power needed to drive the dees is in the neighborhood of 100 to 200 kilowatts, of which perhaps 20 kilowatts appears as kinetic energy of the particles a t final energy; most of the rest goes into resistive losses in the resonating circuit, the balance appearing as energy of particles which are lost a t intermediate radii. Ready-made tubes of adequate power were not always on the market and until the late 1930's many cyclotrons were driven by home-made triodes continuously evacuated by diffusion pumps. Peak voltage, dee to dce, varies considerably with energy, as will be discussed in detail further on. For the larger machincs with 20 to 25 Mev o u t l ~ t the, , figures run to 200,400, w e n 500 kilovolts. 7-3. Output Current of Ions

The current of ions, generally called the beam, may be observed on an insulated, water-cooled probe or targct (connected to ground through a current meter) which is interposed in the path of the particles a t the radius corresponding to the energy of interest, generally the maximum. The current is composed of short pulses occurring a t the oscillator's frequency, if the ion source is built to emit towards one dee only. Each pulse lasts several millimicroseconds-a few percent of the oscillator's period. A single pulse may contain something like 10"r loR ions, so the time-averaged current can be a milliampere or more,

7-4. Magnets T o the casual observer, the magnet of a cyclotron has so much bulk that the vacuum chamber between the poles often passes unnoticed. The magnets have been built over a wide range of size, with considerable variation in relative dimensions; the unit with the greatest pole diameter is not necessarily the heaviest. The original experiments were carried out on magnets of 4-inch diameter and later of 11 inches. Effective research instruments now in

104

FIXED-FREQUENCY CYCLOTRONS

operation (1960) w r y in pole diameter from 18 to 90 inches, with the weight of iron running from 6 to 370 tons. (Accelerator magnets are customarily specified by the diameter of the pole tip, irrespective of the size of the biggest closed orbit.) The field strengths generally range from 14 to 18 kilogauss, although there are machines a t the extremes of 6 and 20 kilogauss. The magnets generally are of the double-yoke variety; two vertical side pieces separate top and bottom horizontal members to which are bolted the cylindrical poles. The tips of the latter sometimes are somewhat tapered to a smaller diameter, but not always. When the machine is so large that these six fundamental pieces cannot each be made from a single forging, they are built up as laminated structures from as heavy members as is practical. Mating surfaces are machined to high precision to ensure parallelism and coaxiality of the poles' faces. Usually the poles' axis is vertical, so that the orbits lie in a horizontal plane, although a few machines have been built with the axis horizontal. I n this discussion we assume the former orientation in order to be able to employ the convenient words "top" and iibottom." 7-5. Exciting Coils

For machines with poles up to about 60 inches in diameter, the exciting coils are usually fabricated separately, clamped between end annuli and then jacked into position around the poles in the already assembled magnet and bolted into place. This procedure may involve pole pieces with removable tips in order to afford room for the introduction of the coils. With very large machines a temporary turntable is mounted around the pole and the winding of each coil is carried out in situ. The windings are often made from rectangular or flat copper or aluminum, with an internal water-cooling channel. Interturn insulation is sometimes of paper, with sheets of some phenolic resin between layers. In some instances the conductors are solid, heat being removed by water-cooled discs between layers (or alternate layers) ; sometimes the windings are slightly separated and cooled by circulated oil. If the various layers of the coils are not connected in series, but in some series-parallel combination, the number of ampere-turns in the upper and lower coils may not be identical, because the cross section of the conductor changes if the die wears during manufacture. This can lead to a magnetic median plane which does not coincide with the geometrical median. Such a situation can be corrected by shunts connected across appropriate portions of the windings. There are no hard-and-fast rules on the amount of conductor per pound of iron; the ratio of iron to copper ranges from a high of 29 to a low of 1.5; more uniformity occurs in the ratio of iron to aluminum, which runs close to

Courtesy of Lawrence Radiution Laboratory

PLATE 2 The original cyclotron experiments of Lawrence and Edlefsen in 1930 employed vacuum chambcrs formed from flattened glass flasks with dees made of silver coating.

FIXED-FREQUENCY CYCLOTRONS

106

Both Courtesy of Lawrence Radiation Laboratory

PLATES 3 and 4 The 4-inch vacuum chamber shown at left, with a single dee, was used by M. S. Livingston to demonstrate the validity of the cyclotron principle in 1931. A Faraday current collector lies behind the septum and the deflector. The filament support is at r i0~ h t .The glass insulator of the dee stem is broken. Vacuum chamber and . ~the . single dee, shown i t right, is of the 11-inch Berkeley cyclotron which produced Mev protons in 1931. -

-

Courtesy of Lawrence Radiation Laboratory

PLATE 5 The 27-inch cyclotron at Berkeley, about 1932. The peculiar shape of the magnet's yoke arises from its conversion from a Poulsen arc generator of R F current, formerly used in radio communication. The helix on the right is a guard to keep experimenters away from the deflector voltage.

Courtcxy o f Argonne National Lnboroto~:l

PLATE 6 Assembling the magnet of the 60-inch cyclotron at Argonne National Laborator!..

108

FIXED-FREQIJENCY CYCLOTRONS

Courtesy o f Brookhaven National Laboratory

PLATE 7

Jacking one of the exciting coils into position on a 60-inch cyclotron. 10. Such variations are due to the whim of the designer or to the relative importance of initial cost and operating expense. 7-6. Vacuum Chambers

I n small cyclotrons the vacuum chamber is generally a brass, bronze, aluminum, or stainless steel box, into the top and bottom of which are inserted discs of iron forming the final pieces of the magnetic circuit. These lids must be thick enough not to flex appreciably under the combined action of magnetic force and the pressure of the atmosphere; in 60-inch cyclotrons this means a 5-inch thickness. In large machines, the entire inside surface of the chamber is covered with water-cooled copper sheets (liners)since this surface forms part of the resonating circuit; for low-power units, copper plating of all iron surfaces is generally adequate. The over-all height of the assembled chamber is an inch or two less than

the lleiglit of tlie gap between the poles of' the magnet, so when the chamber is in place sittiug on three snlall hlocks of' brash, tllcrv are shirnmng yaps of equal tliickness between tlle outsldc surfaces of the lids and the magnet's pole tips. Into t11cse gaps may be placed symnetrically d~sposedsheet iron discs of various diameters in order to bring about the most clesirable drop-off of field strength with radius. T o this s:me end, ec~~nlpcrrii:~ncnt sliiins arc often bolted to the inside surfaces of the lids 111 tlie forin of Rose rings. Thebe are rings of iron, perhaps to +-inch thick and an inch or two in radial width, located a t various radii near the periphery. Tlicir functlon is to postpone the inevitable rapid decrease of ficld in this region, so that the useful diameter is increased. dornetimes, if sufficient study with scaled model magnets warrants it, these. contours are machined into the lids perruanently. Nonsymmctrical pieces of ~ within the chamber sheet iron may :dso be inserted in the gaps 111 (lase t h field is not azimuth:rlly uniform or does not have a flat median plane, because of tlic presence of undetected blow holes in the interior of tlie poles, lack of uniformity in the quality of tlie iron, or the failure of structural bolts to seat completely. \Vlien the proper array of such pot shims has been discovered, the j)icces of iron :ire son~etimesriveted to alunlinuin sheets which arc slid into tllc sllim gaps, for easy replacement after the next removal of the vacuum chanibcr. Nowadays, no one ever attempts to get a cyc~lotroninto operation without first mapping t l ~ efield with great precision and removing all inlioinogeneitics to an accuracy of 1 p r t in several thousand. In tlic early days, the desirability of this was not known since thc theory of orhits had not been devclopccl. Shimmzng the field was m expected procedure after every reassembly of thc vacuum chamber; this consisted in placing odd-shaped pieces of iron shect into tlic shim gaps, prarticnlly a t random, until thc hest beam of particles was :rttained. Sinccb the ficld had to be turned off before any of these scraps of iron rould be m o ~ w l this , w : ~ s:t tinlc-consuming proctls%,taking I~oursa t least, ant1 ionictirncs days, coniplicuted by tlle fact that operators did not agree on tlie t m t positions of shinis and location of the ion source; various combination,i often would t)c cqually satisfactory.

:-

7-7. Dees and Drivers Decs (Fig. 7-11 are inatlc of copl)er, preferably the OFHC (Oxygen-Frcc High-Conductivity) variety, since this appears to have the best ability to hold voltage without sparking. Copper water-cooling tubes are hard-soldered to the interior surf:tces. ( T h r use of soft solder in any evacuated system exposed to high voltago or fast particles is asking for trouble, since i t sputters easily.) The inside apc'rture ranges in bright from 1 to 6 7 inches in different inacliines The tlecs :trc supported in cantilever fashion by dee stems; in small machines tllesc are ropl)cbrrod> fastcnctl :it the far end to glass insulators attaclietl to the t:mk wall. In t l ~ cearly days the joints were m:itlc hy rctl sealing wnu, wtlr water cooling tubes in close proxin~ity.Later liiacliiries used flanged glass 1)ipeh

110

FIXED-FREQUENCY CYCLOTRONS

with rubber or soft metal gaskets, bolted into place. The lumped inductance, resonating with the capacity of the dees, was connected to the stems a t the ends of the insulators and coupled to the oscillator or oscillator-amplifier (Fig. 7-2). I n the later 1930's the importance of ohmic losses in resonating circuits became appreciated as dee voltages were pushed to higher values. Dee stems grew in diameter from a 4 inch to 2 or 3 inches, and the lumped inductance was replaced by a resonant parallel-conductor transmission line formed of copper tubes 3 or 4 inches in diameter. These changes were so advantageous that the next step carried the idea even further; glass insulators were abandoned, the dee and dee stems lying entirely within an evacuated metal enclosure so as to form a resonant quarter-wave transmission line, somewhat foreshortened owing to the large capacity of the dees. To cut down losses, the dee stems became cylinders up to 12 inches in diameter. In some instances, each dee stem lies within its own shield, so as to form (a) glass-supported system

shorting bar /

(b) shielded 1/4lines

>dummy

dee

single X/4 coaxial tine w~thdummy dee

Fig. 7-1. Typical types of dees.

South Pole of Magne

lid 'of vocuurn to vocuurn pump

-gloss

cylinder

North Pole of Magnet

Fig. 7-2. Schematic drawing of early cyclotron with dees supported on glass insulators and resonating with a lumped inductance. The ion source is a simple hot filament. The external deflector directs the projectiles into, a re-entrant bombardment chamber. a coaxial line (Fig. 7-lc) ; in otlicr machines both r,tcrns are inside a single tank, thus forming a shielded-pair line t Fig. 7-lb) . A few cyclotrons h a w only one dce, in which case the stcw is loci~tedso as to be perpendicular to t11c open edge of thc dee. This yiclds a syn~rnctricalvoltage pattern across the edge, which is often lacking in the usual two-dcc arrangement, for in spitc of their large capacitance the dees still act somrwhat as part of a transnlission line, with the highest potential a t the "open" end. This has a bad effect, for with unequal energy increments a t alternate passa,ges across the gap, the centers of successive half turns tend to "walk" in the low-voltage direction. I n order to obtain :t vertically symmetrical field pattern, a dummy dee faccs the active electrode in single-dee machincs (Fig. 7 - I d ) . Such s dummy consists of s coppc'r frame bolted to ground and equipped with flanges that mimic the opening of the real dce. Oscillation frequency is grossly controlled by copper shorting bars (Fig. 7-3) or spiders reaching from stem to outer cylinder, or, in the shielded-pair structure, from sten1 to stem and also to shield (Fig. 7-1 b ) . In a t least one instancc the shorting bars are bolted and hard-soldered into position; more generally they are movable, the operator releasing clamps or bands which hold highconductivity spring stock, in the form of narrow fingers, against t11c rl~aincon-

112

FIXED-FREQUENCY CYCLOTRONS

Courtesy of Lawrence Radiation Laboratory

PLATE 8 The vacuum chamber of the 27-inch cyclotron at Berkeley, about 1932, shown on :I dolly and with the iron lid removed. The dees are supported by glass cylinders, the vacuum joints being of sealing wax. Two filaments (one of which is a spare) arts supported on metal bellows above and below the gap between the dees. The de. flector is in two sections, each supported on a glass insulator. The projectiles arc: directed to an internal bombardment chamber, where one of several targets may be: positioned by turning a handle. Products of disintegration are observed with the: protruding ionization counter, after passing through a vacuum-tight mica windov, and one of a variety of stopping foils (chosen by another handle) in order to afford an estimate of their energy. A peculiarly shaped Dewar vessel for liquid air may bc seen at the left, diametrically opposed to a pipe which will be connected to the vacuum pump. ductors. Usually a change in position means a shutdown, but in a few c a m the operation can be carried out by remote control with the system evacuated Fine tuning is often accomplished by motor-driven trimming capacitor platc: which can move toward or away from the sides of the dees. Such control is particularly important with modern high-Q circuits (5,000 to 10,000), since a frequency shift of 1 kc/sec (out of 10 to 20 Mc/sec) can significantly lower the dee voltage. The temperature of the cooling water for the dees sometimes must be maintained steady to less than 1°C, since larger variations can cause sufficient warping of some surfaces t o alter the capacitance by a noticeable amount.

DEE3 AXD DRIVERS

113

PLATE 9 The vacuum chamber of the 42-inch cyclotron at Harvard University, 1939. The dees are supported by glass insulators, which are joined to metal with rubber gaskets and bolted flanges. The deflector is supported by a glass cylinder and metal bellows, so it may be adjusted in position. The target chamber is seen at the rear, with a handle that opelates the air-vacuum gate. 'The ion source is introduced through the hole between the dee insulators, and above it may be seen an "electron catcher," supported from the left. Photograph by Paul Donaldson, from W h y Smash Atoms? by Arthur I<. Solomon. Reproduced by courtesy of Harvard University Press. The copper-lined ~iictal~liicI(1around the dee stcrns, which bolts to the vacuum chanibcr between the poles, often serves also as a vacuum n~anifoltl, tlie diffusion pumps bcing usually coupled in behind the shorting bar so t h a t n large pumping aperture in thc shield :wound the dee stcm will not impcde the flow of current. With such an arrangement the shorting bar must be of an open structun: in order not to cut down the pumping speed. Occasionally an additional vacuum pump is connected directly to tlie acceleration chambcr itself. The drive and feedback lines enter the shicld through glass insulators. Coupling to the dee circuit may be by inductive, capacitative, or conductive means. Dees often are so mounted t h a t their positions in the gap can be adjusted. This is useful in compensating sag of tlie cantilever support or irregularities in shape. I n some cyclotrons the inside surf:tces of the dece tire lined with plates of carbon, so t h t misdirected ions will induce a short-lived radioactivity ratllcr

PLATE 10 The dees of the 60-inch cyclotron at Argonne. When the housing in which they are mounted is moved toward the magnet at the left, the vacuum pump manifold (rear center) may be bolted into place.

trimmer capacitor

South Pole of Magnet

Fig. 7-3. Schematic drawing of late cyclotron. The ldees and their stems form a ~ / 4resonant circuit. The ion source is a hooded arc. The internal deflector (not shown) lies within the far dee. t h a n the long-lived activity of colqwr. F o r siii~ilarrcasons, h:trd soldcrs not containing sil vcr are clcsirahlc. I n the early cyclotrons, nothing was stahillzed and the operator was continually busy readjusting tliv exciting fivld of the magnet's motor generator in order t o Iiold thc I)ca~nc u r ~ w i tso~licwhcwric.:~r its peak, without a n y knowledge of whether t h e oscaillator or i1i:rgnet had drifted, or in wliich ilircction. I n modern cyclotrons, the o1)crator is often attending to otllcr matters in tlit: control room, for servo circuits niaint,aln resonance nutomatic:tlly. Various schcmes of control and feedback h:t\c been uscd. T h e magnetic field is generally hcld constant hy continual nlonitoring, quite oftcn by nuclear magnctic resonance methods, the l m b e w11ic.h contains the sample and tlic exciting and pick-up coils I)eing pl:~cetl on the roof or floor of the vacuuiil chamber near its center. T h e oscillator or oscil1:~tor-ai111)lificr is sonietimes crystal driven, or its frequency niny 1)c continually coinpared with a monitor crystal. I n so~rlcinstances, the csirruits 11:tve bevn designctl t o continually lrlaxiiniee thr: beam of ions, rathcr than to hold the field and frequency a t fixcd values. A plienoincnon known a s multipacting (or multipactoring) is often a source

116

FIXED-FREQUENCY CYCLOTROXS

PLATE 11 The shorting bar of the Argonne 60-inch cyclotron. Note the metal fingers which make electrical contact; the inner ones are clamped to the dee stems by metal bands, while the other press against the outer wall by spring action. The driving loop, grounded at its center through a choke-coil, lies between the dee stems. The gimbals support of the left dee is visible. of trouble in the development of radiofrequency high voltage in particle accelerators. Ionization of the residual gas occurs and the freed electrons are driven back and forth to strike the dees, dee stcms, or surrounding metal walls. This results in secondaiy emission of further electrons, and the process becomes multiplicative when the voltage and distance are such t h a t the time of transit is half a cycle. This puts so great a load on a self-excited oscillator that i t sometimes cannot build up to a sufficiently high amplitude to supply adequate grid excitation. For many common dimensions of the space wherein the electrons dance, this block is often temporary; for if a high voltage can once be established, the electrons are driven to the walls within less than a half cycle and hence do not oscillate continually. The problem is one of getting started. There are several solutions. I n one, the grid of the oscillator is connected

Courtesy of Oak Ridge National Laboratory

PLATE 12

k

Cut-away view of the Oak RidgeH-inch cyclotron. The particles rotate in a vertical plane between the p o l ~ sof a magnet with U-shaped yoke. so as t o afford adcquate excitation until the block is passcd and is thcn reconnected so as not to overdrive when normal opcr:ttion has been established. Sometimes a small separate oscillator with high drive gets the system under way and is completely disconnected when the main oscillator takes over. Alternatively, the electrons may be swept to the walls by application to the decs of a DC bias potential of several thousand volts, the dees being insulated from ground but coupled in the R F circuit by a built-in capacitance of low

118

FIXED-FREQUENCY CYCLOTRONS

impedance. Sometimes this clearing field is supplied by putting a DC potential on a grid of wires strung in the region where multipacting occurs. Finally, an oscillator-amplifier system may be employed, so that adequate drive to the amplifier is available a t all times.

7-8. Ion Sources

In the most primitive cyclotrons, ions are produced a t the center of the machine by a very simple method (Fig. 7-4a). A hot tungsten filament, held in

floor of chamber

Fig. 7-4. Ion sources: (a) simple filament; ( b ) arc, open at top; (c) hooded arc, shown opposite dee with "feelers." In practice the support for the source is usually parallel to the dee gap, rather than passing under a dee. (d) Sketch of dee with "feelers." (e) Sketch of dee with "pullers."

ION SOURCES

119

place by a radial pipe containing the power leads, is mounted near the floor of the vacuum c>l~arnbcr. The fi1:rnient is kept :L few hundred volts negative with respect to ground, and a tungsten plate, just below tlie filament and a t its potential, helps direct most of the ions upw:~rd in tight spirals around the lines of tlie magnetic field. The whole tlec chamber is flooded with the projectile gas a t low prcssurc and sollie of the ions cre:ltcd near the median plane are pulled into wllichcver clee happens to be negative and are thus started on their career as projcctilcs. A later de\dopnient is to surround thc filament structure with a hollow cone of metal a t ground. Gas is fed into this cone through a small pipe. This pcrrnits a n arc to be struck i11 t: region of relatively high pressure while allowing a good vacuum to be niaint:tincd in the rlegion around the dees. The ion source thus resembl~sa volcano wltli tlie tip of its crater slightly below the cyclotron's mid-plane (Fig. 7-41)) ; ions forriled just outside are available for use. More recent ion sources are of the hooded arc variety (Fig. 7-4c). Thc volcano is replaced by a hollow cylindcr or c h i m n e y , closed a t the top but with a hole in its side a t mid-1)lanc height, facing one dee. The electric field between the dee and the chiinncy pulls ions out (of the plasnia a t the hole. Sometimes an insulated n1et:tl disc replaces the usual plug a t the top of the chimney; this collects electrons until its potwtial can repel them, so that they oscillate u p and down the interior of the cliimncy thus enhancing their chance of ionizing a gas atom. T o increase the field strength between dee and chinincy, the upper and 1owc.r rdges of the dce are often extended by metal lips or fcelers which come close to the cliimney (Fig. 7-4d). Although this wheme tends to give an unwmted rertic:~l coinponent to the ion's velocity, it has the advantage of permitting t110 ion source to be moved about, until thc position i$ such as to nmxin~izet l i ~number of particles reaching the dcflectmg system (see $ 7-11) by w l ~ i c lthey ~ are brought out of the vltcuu~u tank for use elsewhere. RZorc rcwntly tlicw has been a. trend t o ~ m r d sreplacing tlie lips by two vertical bars or plillerr (Fig. 7-4e1 across the opening of the (lee, so as to form an elementa~~y grid that hclps extract and launch the ions. This procedure does not producc. as large :t vertical velocity but leaves lpss freedom in t,he position of the ion sourcc. I n some instances a series of four or five vertical grid bars h a w bccn used, placed along the openings of each dee, so that the ions pass through the slots thus formed, on successive turns. This is useful in intercepting, a t an early stnqe, particles which could not succeed in leaving tlie tank :myway, thercby reducing tlie radioactivity of tlie tank and dees. T o fnci1it:ltc the rq)l:iccnwnt of l)l~rnctlout filanicrits and erotlcd cliinineys, iiiotlcrn ion sorlrccs arc tlehigned for r e i n o ~ a land replacement through air lock!: without t l ~ closs of vacuuni in the main clinmber.

120

FIXED-FREQUENCY CYCLOTRONS

7-9. Early Orbits in Cyclotrons

I n the simplest concept of a cyclotron, one considers the gap between the dees to be infinitely narrow so that the electric field may be taken as constant during the vanisliingly small time of transit of the ions through i t ; tlie particles are imagined to spend essentially all of their time inside the dees where there are no electric forces.

Fig. 7-5.

Coordinate system used early orbits.

analysis of

I n actuality the gap between the dees may be as large as perhaps 4 inches; this provides space for the introduction of the ion source and probe targets. As a consequence, upon emergence from the source the ions are sirnultaneously subject to both a constant magnetic field and a time-varying electric field. As acceleration proceeds, a larger and larger percentage of each turn lies within the dees so t h a t the simpler picture ultimately becomes justified. A rigorous analysis of the true state of affairs during the first few turns is extremely complicated, since i t depends largely on detailed knowledge of the shape of thc electric field between ion source and dee. Such calculations have been performed with varying degrees of approximation, often with the aid of specific field plots obtained with an electrolyth tray. For our purposes, we will make the simplest possible assuniptions in order to obtain a bird's-eye view of what occurs. The result is in reasonable agreement with the more exact studies. Briefly stated, the conclusion is as follows. Even though particles may be emitted from the source for a full half cycle (while the field between the dee and the ion source is accelerative), within less than one turn this wide spacing of ions has shrunk considerably and they all cross the gap in a narrow bunch close t o the peak of the voltage wave. This conclusion is of interest, since in subsequent calculations on the ininiinum clce voltage needed to attain a given energy, i t will be necessary to know tlie starting phase of the ions; the present investigation will show t h a t without serious error this phase may be taken to be t h a t a t which the voltage is a t its n~:txin~uni. I n the simplest approximation, the electric field brtwccn the dew is considered uniform, as though it were produced by two parallel plate electrodes,

and tlrc ph+cnl ion source i i r q ~ l n r c d by :I firt~tious point which cnrith p:trtic*lvh.Tl~cbwl):ir:~tion h ~ t x w c ~tlic r ~ tlcw i y t:~lickn'L* b o I:~rgctlrnt the ion5 (lo not pcnc.tr:itc n i t l ~ i nthem, a t least for sc.reral turns, and the magnetic ficltl is niliforin. \\'c cirrploy n nglit-llandcd coc~t~lin:~tc systcm, with tlle origin a t the ,iourcc,; the 2 :rxis i i tlic tlirvction of tlrc ~nagric,t~c f i c k l t l , !I is parallel to the open faces of tlre dces, w l d e the electric field acts along .r, as drowri in Fig. 7-5. The equations of motion of a positively c711ar~cclion on tlic mid-plane are tlicw

r

(Y -

dt-

=

Mn

6, c70s~ , , ( t

+ tl,) + -, nio nZd?4 (It -

(1

1Ic.r~.8,,, is t h r amplitude of tllc clcrtric~fic~hl,w,, is the radian frequency of tlic oscillator, : m i t,, has bcrn introtluccd to take : w o u n t of ions starting : ~ t tiiffcrcnt cl)oc.lls of thc volt:ige cycle. \\'v m:iy not ~tlcntifythe factor yH-/JI0 wit11 tlrc r:tcli:m frcqwncp of revolution of tlre partirlcs, for sucli an itlcntific:ltion inlplic\ motion in n magnetic firld only, which is not now tlre casc. H o \ v c ~ c r ,wc :ire a t liberty to specify R., :ml we so choose it that qRI/JIo cquals w r j . The rotation frequency of tho particles i ~yet , to be found. ITence we n-rite x q d!4 - = - F;,, cos w,,(t to) wrj dt' M(, flt

+ +

and set the boundary conditions that at t = 0 all position and velocity cotnponcnts are zcro. Tlre rrlctl~odof solution of thrse equations is w r y tedious ant1 nil1 not I)e repeated here. The results art., with 0 = w,,t and Oo = wrjtofor convenience, 98, X = -[-sin

2Mowr/"

( A n

?/=2blow,- [ -

cos 0" sir) 6

Oo sin 0

+ 2 sin el, ( I

+ 0 sin (0 + O,)] - ros 0 )

+ 0 cos (0 + Oo)].

(7-8)

T h a t thew arc. indcctl solutions is readily vchrificd hy differentiation and inp:~tlls 11l:iy be sertion in the, cquations of motion. Fro111 tllcsc. cq)rc.\ions t h ~ plotted, for c+osen values Bo of tlie p11:~sc of the. electric field a t the moment of emission (i.e., a t w T j t = 0 = 0). Typical paths are shown in Fig. 7-6, where the dots indicate the positions of the particles a t intervals of ?r/4 radians of the oscillator's cycle. Thus in Fig. 7-(ia, t& = -a/2 and the field is becoming accelcratiw In the direction of 7 . r ~ 1 ~ a1t . the moment of emission from tlre iource, so the ion undergoes n nr:tui~num acceleration in this direction. Tn Fig. 7-C,d, where we have 0,) = T 1,the field is weak at the start and about to

122

FIXED-FREQUENCY CYCLOTRONS

Fig. 7-6. Paths of ions emitted at different phases of the dee voltage, on the assumption that the particles do not penetrate within the dees. The dots show the positions of the particles at the indicated number of eighths of a cycle since the first ion was emitted at 8, = -a/2. reverse in direction, so the orbit initially describes only a small loop. Obviously a plot drawn for 00 = 7r/2 would be identical with that for 00 = -7r/2 except for rotation of the whole pattern by 180'; this implies a n initial motion in the -x direction, which is impossible with the usual hooded arc which is open on one side only. Since the assumed boundary condition sets wTrt = 0 = 0 irrespective of the value of wrfto (i.e., of Oo), the curves give no immediate informat,ion of the

TILIXSIT TIAIE

123

relative positions of the ions, for different values of Bo, on :x single tinle sc:tle. This information can be obt:tincd by realizing t h a t whcm the ion of Fig. 7-6a has been in orbit for ~ / 4radians of t h e oscillator's cycle, the ion of Fig. 7-6b is just enierging from the source, while the plrticle of Fig. 7-6c has still t o wait for one eighth of :i period before it ~ 1 1 1sltnrt. Conscqlwntly tlie dots niay be nurnlr~cred :is s h o ~ v n ,corresponding to the numkjcr of eighths of an oscillator period which h a w ela1)scd since the first ion was emitted. T l i c r ~ f o r e t h e dots with identical numbers represent pos~tionsa t the sanw instant of time. An interesting observation m a y thtm be in:~c-lc.At dot G (after $ of n cycle has passed, rounting from the voltage null a t - ~ / 2 ) , all the ions are closc to the mid-line betwccm tho clcc,s, whcrc .c = 0, quite irrespcctivc of thc voltage phase existing when they Icft the s o u r c ~T1icre:~ftcr all ions again return approsinlately to the mid-11nc a t intervals of one 11alf cycle (dots 10, 14, 18, e t c . ) . As far a s further accclerntio~iis conccrnccl, t h c p:trticlcs hehave us though they all were emitted very near to the monwnt w1ic11 tlie tlec voltage was a t its peak value. This fact will be of considerable use later on, when t h e minimum required dec roltage is calculated. T h e curves are plotted in units q~,,,/(2Ml,o,,,'),one unit representing 0.5 cm if t h e particlcs are protons (for whicli q/M0 = 9.51 x lo7 coulomh/kg), if the oscillator frequency is 22 RIc/,iec (c>t, = 1.37 x lOQadians/ses) and if the peak electric field is 2 x 10" volts/m, corresponding to 100 kilovolts hctwecn dees separated by 5 cm. T h e initial :issumption t h : ~ t the pxrticlcs do not penetrate into the region free of electric field within the dees is therefore satisfied for the first half-turn to the right, a s Fig. 7-6 shows, and by the t i ~ i i c this first half-turn is complcted a11 the ions are bunched so as t o cross t h e mid-line almoit together. Thereafter the dee-to-dce gap becomes a progressively smaller fraction of a turn, so t h a t we are soon justified in assuming t h e gap length to bc ncgligible. 7-10. Transit Time Except in 7-9, i t has hccn assumed throughout t h a t the accclerating gap in a cyclotron is of negligible lcngth, so t h a t no change of electric field occurs w111lc t h e particle is in transit across t h e gap. V'e will now sre to what extent this assumption is justified, for in practice thc gap m a y be several inches long a n d for a n orbit of small radius t h e transit timc i:j an appreciable fraction of the period of the oscillator, so the particle re:~ily is subject t o a time-varying force. Wlicn calculating the gain of energy under these circumstances, the relocity appears in the :tnalysis, and i t is convenient t o treat i t as the average velocity of tlie particlc, when within the gal). T h ~ is s intlccd a very good approximation if the energy increment is only s small fraction of the energy already attained. T h u s by Eq. (1-28) we have

124

FIXED-FREQUENCY CYCLOTRONS

P

-

1

1

h

-

EJ dB; Eo' -. dE E L - E '0 LE T+22T&T+Eo

For protons, Eo = 938 Mev. Assume the energy dE gained a t the gap to bc as great as 0.1 Mev. Then even if T is as small as 1 Mev we find that dp/p = 0.05. Thus the velocity changes by only 5% and the assumption of a constant average value is justified. Let us measure time and the phase of the voltage from the moment when the electric field is zero and changing from decelerative to accelerative. Let 4 bc the phase a t the instant t , when the ion reaches the center of the gap, so that

4

~rjtc. Then the distance traveled from the center of the gap in the interval between t, and any other time t is given by =

z = v(t - t,).

On substituting for t , from the expression above, we have

Solve this for t to find

The time dependence of the electric field is & = Em sin wrjt,

where for the sake of simplicity wc assume &, to be constant along the path. Then the field a t a distance x from the center of the gap is given by & = 6, sin

(

%f++.

)

But w r j = v/r because the synchronous ion has the same frequency as the oscillator. Therefore E = E. sin

(4 +

4).

The energy gained in crossing the gap of length g is then

AT

=

/

n/2

q~...

-u/2

sin ( 4 r

+ 4) dz.

Use the trigonometric expression for the sine of a sum and thcn integrate, remembering that 4 is a constant while the particle crosses the gap. Then

AT

=

2qr8, sin

(g)sin 4.

Multiply hy g / g anti recall that g t ; , = I.,,, the p w k voltage. This gives

where, i t will be recalled, + is the. pliasc when the ion is a t the center of tlic gal). T11u.s tlic cffccti\,e pcak volt:lge k)o(~niicsthe actual 1)enk v:duc n i u l t i l ) l ~ r ~ l hy the constant br:~cltetedtcrni (known as tlic transit-tinic factor) wliicli (lcpends on the ratio of gap lcngtli to orbit radius. I t is alq)arclnt that a vanishingly sin:dl gap length is desirable in principle, since a function of tlie form (sin y ) / y falls froui unity as y increases from zero. But in practice the reduction of energy gain because of a finite v d u e of g is of little consequence in a cyclotron. For example, in many machines tlic, g:~plength is a constant : m l evcn a t s11ch small radii that g = r , thc transittime factor is (sin & ) / $ = 0.96; so only 45% of the available voltagc is lost. I n some machines the open cdgcs of the dees are flared back so t h a t the g a p :ire wcclgc-sl~aped,with tlie apexes a t tlie center of the machine; this f>icilitates the introduction of probe targets and reduces the capacitance of each tlcc to ground. For a flare anglc I9 we Ii:~vc.rI9 g antl the transit tinic factor I ~ c o i i ~ c s sin (I9/2\ /(I9/21. Even i f I9 is us large :IS 30°, the factor is still 0.99. It must bc mentioned that tho abovtk :tn:dysis of the effects of transit time is considerably idealized, since it has heen assumed that tlie electric ficld i h constant tllroughout the entire volume of the gap in which the particles rnovc~ A more complicated analysis n ~ u s tbe made if a nonuniform ficld is assumed ;

7-1 1. External Beams

The ions in a cyclotron arc somctim~xused by mounting the targct on t l ~ c end of a probe located in th(3 gap between the dees, so as to intercept ions at whatever radius corresponds to the dcsired energy. This is :idequate if the purpose is to produce neutrons or to make the target radioactive (provided tlic material docs not sputtcr wlieri bombarded, cvcn though water-cooled, :mtl i. not inconipatible with a vacuum). Such a method is not only inconvenient to some clcgrce but for some purposes is entirely inadequate, as when the projertileq are to he used in a ctrvice such as a scattering chamber which must be free of cyclotron hackground radiation and hence is 1)eliind a thick wall of concrete. For such purposes, the projectiles must be frccd from the grip of the ~n:ignct :inti directed outside the vacuunl chamber antl beyond. The equipment that acconlplishes this is called the deflector system. I n fixedfrequency cyclotrons, where tile energy is at most :t few tens of Mev, the particles that have reached final energy can be deflected by an electric ficld dircctcd radi:~lly outwarcl. This partially ovcrcoiucs the inward force due to

126

FIXED-FREQUENCY CYCLOTRONS

the magnetic field, with the result thc radius of curvature is augmented during a portion of the last turn, and the particles curve directly out to the chamber wall, either to be used on a target clamped thereat, or to be transmitted through an evacuatcd pipe to some distant area. All this is much easier said than done, and if 30% to 4070 of the ions which might have struck an internal probe are successfully extracted, the system is working very well. Cyclotron deflector systems may be divided into three general classifications: the DC, RF, and DC-plus-RF types, depending on what combinations of electrostatic and radiofrequency electric fields are employed.

I\target

L

deflector

Ltarget

Fig. 7-7. ( a ) DC deflector. (b) DC-plus-RF deflector. In the DC variety shown in Fig. 7-7a, a DC electric field is applied between two curved plates forming a cylindrical condenser. The inner one is called the septum and the outer the deflector. In single-dee cyclotrons, both plates may be mounted behind the dummy dee and hence are free of all R F fields. In twodee machines, both plates lies inside one dee; the septum is conductively connected to it, while the deflector is held on the end of a long metal arm extending down the interior of the dee stem, this arm being supported on one or two

13XrI'ERSAL BEAMS

127

insulators. With the single dce, adjustments in the positions of the deflector and septum m e readily madc, wllile in the two-dee systcrn, control of this sort is much more limited. I n the DC-plus-RF deflector of Fig. 7-7b, the outside wall of one dee is cam-shaped, full-speed ions passing tllrough a slot in tlie step. The outside

PLAT'E 13 The septu Oak Ridge

rd

of the %-inch cyc formed of an array cooled tubes.

tron a t waiter-

Courtesy of Oak R ~ d g eNntzonnl Loboratmv

surface of the dec wall then :wts :is the septum while the deflector plate is held nearby, on a stem passing tlirougli an insulating cy1intlric:~lsup1)ort on the w:~11of the v:~cuum tank. 'l'l~c total elcctric field acting on tlic particle as i t pnsscs down tlie dcflcrting cllni~nc~l is thus coinposeti of tlic DC: potcnti:ll applied to the dtsflec-tor plate l)lus the IIF ficltl \ ) c t n w n ttrc p h t c ni~tltlc~,.11-it11 p u ~ c RF deflection, tlic dcflertor pl:~tcis g~ountlcvi.In I)otll thew systeill~t l ~ e(I(,flccting fic.ld varics in ningnitutle with time and l~ellccwith distance tluwn the cl~annel,and dclmids on thc voltagc 1)llasc (luring this transit. Co111j)ututions

128

FIXED-FREQUENCY CYCLOTRONS

on deflector design are hence more complicated, as well as being dependent on assumed operating characteristics. If only R F deflection is used, one particular dee voltage is required to obtain a n external beam, although some slight latitude is possible if multipacting is controlled by a D C bias voltage on the dees, for the potential may be varied, within limits. At first sight, one might hope to so position the deflector and septum that the next-to-last orbit would clear the surface of the septum nearest t o the center of the cyclotron, while the final orbit would pass neatly down the channel between deflector and septum without loss of any ions. But this does not occur in practice. The opening of the ion source has a finite width and hence so has the beam of ions as it starts out. Since the magnetic field has a gradient, particles on the inner and outer borders of the beam do not experience quite the same values of field over their total paths. In addition, the radial betatron oscillation frequency does not have an integral value, so a t the azimuth where the deflecting channel begins the last and next to last orbits are a t different phases of this oscillation. As a result of all this, the particle density does not drop to zero in the region between the last few turns, and consequently the septum of finite thickness intercepts many ions. The initial portion of it is sometimes called the beam splitter for obvious reasons; this part is particularly subject to bombardment and thus to destruction by melting, and hence is of special design. I n low-power machines a thin sheet of tungsten or molybdenum is often used successfully, but with more energy and current water-cooled structures are necessary (Plate 13). These take the form of a "picket fence," composed of needlelike water-cooled tubes, or of a copper sheet with water-cooling above and below a long tapered slot in the mid-plane. This latter expedient increases the area of surface which intercepts ions and so reduces the energy density. (The axial height of the beam of particles may be only a few millimeters a t the final radius.) There is recent speculation and some experimental evidence, as mentioned earlier, that it is worth while to insert slotted diaphragms inside the dees a t a small radius, so positioned as to let pass only those ions which ultimately will reach the deflecting channel under optimum conditions of position and direction. Wear and tear on the septum are reduced and the homogeneity in energy of the deflected beam is improved. If these sanguine hopes are fully justified, the cyclotron may approach the Van de Graaff machine as an instrument of precision, but a t higher energy. (At the present time, the external beam from a cyclotron has a n energy spread of about 1% of the nominal value.) Although there is little scientific value in allowing the deflected beam of ions to penetrate a thin vacuum-tight window of metal foil and to emerge into the outsitlc air, few cyclotron builders can resist the temptation to do so after a machine first gets into operation. The flame of ionization is dramatic proof that the apparatus is working. Measurement of the range in air, with

PLATE 14 The beam of 21.6-Mev deuterons emerging th~otighan aluminum-foil window in the 60-inch cyclotron at Argonne National Laboratory, 1952. The straplike object leading down from above is a transmission line of low characteristic impedancc which supplies current at 220 kc/sec to the filament of the hooded arc. On the right side of the vacuum tank is the housing for one of the trimming capacitor plates, used as a vernier control on the frequency of the dee system.

due allowance for the stopping power of the window, permits a rather inaccurate calculation of the particle e n u g y , which generally comes out on the high side of the true value. since heating the air lowers its density and increases the range. After such a display of fireworks the room should not he enterell until remote measurement intlicntc~s a sufficiently low lerel of air-borncl radioactivity.

7-12. Deflector Calculations Although the qualitative remarks a l ~ o v emay Suggest t h a t deflector design is a simple straightforward process, this is far from being the case, :is is illustrated by the following paragraphs on a 1)C deflector in which the electric forces are not time dependent, a s opposed to the more complicated case wllerc R F fields are employed.

130

FIXED-FREQUENCY CYCLOTROKS

The stream of ions moving down the deflecting channel has a finite radial width, and since the magnetic field gets progressively weaker as the ions move towards the tank wall, the outermost particles are always in a smaller field than their inner neighbors. This causes the width of the ion beam to increase, and generally necessitates a continual widening of the channel. When the ion is at a point where the channel width is w, with voltage 17 across it, and where the magnetic field is B and the radius of curvature of the path is p, the equation of forces is

whence

Mu =

2. B--

v

(7-12)

wu

Immediately before entry into the channel, at the point where the field is Bo so and the radius of curvature is po, we have quBo =

and Eq. (7-12) becomes

The deflector channel is tipped somewhat with respect to the final circular orbit of radius po, and the initial radius of curvature pl in the channel is slightly greater than po. Hence just after entry a t a point where the field is still Bo and where the initial width is wl, we have

Use Eq. (7-13) to eliminate M from t'his and then solve for V / v :

Put this in Eq. (7-14) to find that

This is in a form convenient for later computation: p = K 1 / ( B - Kz/w) where Kl and K 2are constants. The permissible magnitude of pl depends on practical considerations. By eliminating Bo from Eqs. (7-13) and (7-15) we find

But M v V s twice the nonrclativihtic kinotir cnc>rgy 1'. If this is c y m s s c d electron-volts, then 1- is in volts and n-e gct

111

\\here S is the degree of ionization of tht) projectile. For a given T and PO, this equation determines the maximum permissit)le p,, ;since the voltage and the voltage gradimt must be smnll enough not to cauqe spark-over. (The error in using nonrc~lativistictlynanlics is e:tsily conll~t~nsatetl hy an empirical adjustnicrit of T 7 whcn the machine ih running.) For csaniple, if for proton5 7' = 20 M e v ant1 po = 26 inches and if we choose = 1.20 then \'/w is 257 kilovolts per inch, i.e., 64 kilovolts across inch which is reasonable. But we are now in a dilemma. Ilvcn ~f we ahsign L: fixed value to ul or let it increase in some arbitrary way, we cannot fmd suc.cessive values of p from 131. (7-17) (in order to calculate the poeltion and s l ~ a p cof the orbit) until we linow B a t all points along the path, and we cannot assign vttlucs to B (frorn calculated or measured values of the ficlld as a function of radius) until we know where the orbit lies. Furtlirwnore, how long is the channel and where i1oc.s it start? The procedure of design is one of trial zind error, though there are a few tricks which c:ln shorten the l : ~ l ~ o rOne . knows where the particles are to emerge from the tank and in what dircctron the central ray should be traveling a t that point. 'rhc magnetic field is nw;i+urcd as a function of distance frorn the niacliine's center, and contour circles, in steps of 50 or 100 gauss, arc made on a large-sc:lle drawing, Fig. 7-8. Tlie 1)nrticles arr assumed to move in a magnetic field only, and the orbit is traccd backwards from its point of enicrgence in arcs so short t h a t the field may be conbiderod constant over each. Tlie path is found through the relation t h a t each successive product B p equals BoPo, \\here po is the radius of the final circular orbit a t full energy and Ro is the field at po. Since the arcs must join smoothly, t h e various centers of curvature 1w on the last r : ~ d ~ uof s the preceding arc, as indicated in Fig. 7-8. This rewrecd-direction path will either continuously approach the cyclotron's centcr or rlsc it will attain a minimum distance from it, and then recede, depending on tlic cllosm angle of emergence from the vacuum tank. Let this be Drawing A. ?;ow to the same scale on tracing paper we make Drawing B by laying off a set of scmicirclcs that are tangent a t one point so their centers lie along a conrmon diameter. Let their radii be PO, pl', pll', P1l", . . . , the primed quantities being trial values of the initial radius of curvature of the orbit when in thc deflecting channel. The largest ratio of any of these t o po must not imply too high a value of T'/U'~ as determined hy Eq. (7-19). The center point of the cllrvc with rndlua po on Drawing B is then pivoted on the center of the cpclo-

+

F I X E D FREQUEXCY CYCLOTROKS

I Drawing B

1

Tend of deflector

\

Drawing A with Drawing 8 imposed

Fig. 7-8. A method of determining the starting and ending point of an electrostatic deflector. tron on Drawing A with a thumb tack, and Drawing B is rotated until one of the semicircles (or an interpolated one) becomes tangent a t some point to the backward-traced orbits on Drawing A. This point locates the exit end of the deflector channel, while its beginning lies beneath the common point of all the semicircles, i.e., at radius po. If both ends of the channel do not lie within the bounds of one dee (for an internal deflector in a two-dee system) or within the free space in a single-dee machine, one must try again, with another point and direction of emergence. This technique assumes that the radius of curvature of the path remains constant over the full length of the channel, and it must now be seen to what the septum and deflector. For example, it may be assumed that each of these

THE SEPARATION OF EQCILIBRIlJM ORBITS

133

has a constant, but different radius, so that the chainnel widens in proportion to its lmgth. Trial positions o f such a scptum and dcflcctor are inscribcd on Dr:twing A, and tlie path is traced in thc proper, forward direction, again in small arcs, each with a new value of p calculated froin Eq. (7-17), a new value of B as rc:rd from the contour map, antl a new value of w as determined from tlie drawing of the assumed sl~alwof deflector and septum. In general i t will be found t h a t p will not remain constant,, but will rise and fall in magnitude. ( I t can c w n bccome negative, if B falls away faster than w grows.) If tlic path runs into the assurricd positions of the channel boundaries, we are in trouble and must start again with some parameter :dtered. Such a forward tracing of the orbit should he cxrried out not only for particles which enter tlie channel midway across its width, but also for two particles which, when entering, just graze the septum antl the deflector, rcspcctively. 7-13. T h e Separation of Equilibrium Orbits

Even though complete separation of turns of the ion beam rarely occurs a t the final radius for the reasons given earlier, it is nercrthcless interesting to compute the th(~orcticals~p:tration of thc successive turns of the equilibriurli orbit. Il'e start with Eq. (1-36) :

Bgrc

=

( T q - 2TEo)".

Take the logarithmic derivative, to find

From the definition of the field index, Eq. (2-11, we have d B / B = - n d r / r ;

so

\Ye now interpret dr as the increase in radius per turn, that is, the separation hetween succcbsive turns. Then dT is the energy gaiined per turn, 2qV,,, sin 4, d i e r e T',,, is the. peak voltage across thc dee gap and 4 is the phase anglr of thc voltage (measured from the voltage null) a t the moment when the ions iiiake the traverse. The factor 2 occurs because there are two crossings per turn. Since T is very much smallcr than Eo, we may drop the terms T and T" to obtain dr -. - qV, sin 4- . r T(1 - n) In a cyclotron n may be about 0.3 at tlie radius of the entrance to the deflector, say 25 inches. For T = 20 Mev :rnd for qV,,, = 200 kev, then dr = 0.35 sin cp inch. If sever(. loss of beam on the septum is to be a ~ o l t l e d advantage , must btl takcn of any d i ~ ~ ~ i n u t in i o nparticle density in thc space Ixtween the final turnh of tlie ideal equilibrium orbits, by niaking the radial separation of such path.

134

FIXED-FREQUENCY CYCLOTRONS

sufficiently large that the last two turns can straddle the septum. This condition is optimized when sin + = 1,that is, when the last acceleration takes place a t the voltage peak. The consequence of this will be explained in § 7-14. 7-14 The Minimum Dee Voltage in a Cyclotron

If the particles in a cyclotron always stayed in exact synchronism with the accelerating voltage, there would be no need to hurry in bringing them up to full energy and the dee voltage could be as small as desired (disregarding loss of ions from scattering by residual gas, which of course increases as the path gets longer). Such is not the case, and if the revolution frequency initially equals that of the oscillator, it will soon drop below it, for as soon as any energy is acquired, the rotation frequency becomes less, and this reduction is accentuated by the smaller magnetic field encountered a t greater radii. This results in a shift to a greater phase angle, which will bring about deceleration if continued for very long. Obviously the phase shift would be of no consequence if the dee voltage were so large that full energy were attained in one acceleration; so i t can be seen qualitatively that the smaller the voltage, the sooner trouble will develop. An elementary analysis of the situation will now be presented; more sophisticated treatments can be found in the literature. Let the oscillator's frequency w r j be set equal to that of ions with very small kinetic energy, so that

where Bo is the field appropriate to such resonance. At a larger radius the field is weaker and the mass has grown, so that the radian frequency of revolution, given by

is smaller than w,,. Let the phase of the voltage be 4 when the particle crosses the accelerating gap, being measured from the voltage null where the field changes from being decelerative to accelerative. Then d+/dN, the change of phase per turn, is

+

hut M/Mo

=7 =

+ T ) / E o= 1 + T/Eo so it follows that

EIEo = (Eo

In this expression for the change of phase per turn, the first term represents the contribution arising from the drop-off of field and the second that caused I)y tlic relativistic increase of mass. Both terms are positive and both increase as the particle gains energy, so the phase advances faster and faster. For a

numerical exnn~ple,it is convenient to return to 1
Fig. 7-9. ( a ) The relations between B, Bil, B,, b. and K . (b) Phase angle with relation to gap voltage.

It is convenient to introduce three new paranletem: first, B,., the field a t the center, second

136

FIXED-FREQUENCY CYCLOTRONS

the amount by which the central field exceeds the resonant field Bo for slow ions, and third

the amount by which the central field exceeds the field a t any radius. This quantity b must be measured experimentally, point by point, for any given magnet a t any particular exciting current. Within limits, b can be altered by means of shims and Rose rings. Now multiply Eq. (7-25) by BJB, and arrange i t in the form

By Eq. (7-26) set Bo = B, - K and then by Eq. (7-27) set B, - B = b. This gives

Now b and K will be about a percent or so of B,, and IT will be a few percent of the rest-mass energy, so the first three terms in the bracket are comparable in magnitude. The last term is much smaller and may be dropped. T o a good approximation, therefore, the relation is

The first term represents the effect of the radially falling field, the second that of relativity, while the last gives the compensating effect of the field which deliberately has been made over-strong a t the center We now change the independent variable from the number of turns to the kinetic energy, and introduce the peak dee-to-dee voltage V,. The energy gained per turn, during which there are two accelerations, is

E' = Z ~ Vsin , Q; dN

(7-31)

so if we multiply by d+/dT we get

d4 dN

-=

dQ = - 2qV,d cos Q 2qV, sin Q dT dT

On comparison with Eq. (7-30) this yields d cos Q

-

T

dT The independent variable T will now be expressed as a function of the radius, and since protons, deuterons or helium ions cannot be accelerated in a fixed-

THE ~ I I X l M U ~ DEE I VOLTAGE: IS A CYCLOTRON

137

frequency cyclotron to more than a few tcms of Rlev, we may use the nonrelativistic expression for the kinetic energy, 13q. (1-37) :

rl T BZf Hence - = and Eq. (7-33) becomes 2Mu d(r3 ---J

Since B .=; B, over most of the useful radius arid drops only a few percent near tlie largest r:tluc of 1-, further calculations will be only slightly in error if we substitutc B, for R. On integrating we find COE

4 - COS 40 =

(7-35)

where +o is the. phasc when r = 0. This is the desired relation. The first t c m in tlic integr:il must be h:~ndlcd graphically and it will be convenient to treat the others in like nianner. From the measured values of B as a function of r, we plot b/l3,. against r', as shown in Fig. 7-10a, out to

G

2

rd'

Fig. 7-10. Graphical representations of the integrals in Eq. (7-35). the value rd%wliere the entrance to the deflector is located. The second term within the intogral plots as a rising straight line, except for a slight reduction in slope a t large radii because of the> decrease in ri~agnleticfield, as in Fig. 7-lob. The ordinates of thc two curves arc added to produce a third curve, Fig. 7-10c. The area X utitler it out t o r d 5 s the surn of t l ~ cfirst two integrals and is a measure of tlic increase in phase caused by field contour and the rising mass. A plot of the third term within tlie integral is a horizontal straight line and

138

FIXED-FREQUENCY CYCLOTRONS

the area Y under i t measures the corrective phase shift introduced by making B, greater than Bo by the amount K ; see Fig. 7-10d. At the moment, we do not know the value of K and therefore are ignorant of the size of the area Y. If we are interested in finding the minimum dee voltage for a deflected beam and a t the same time desire to intercept as few particles as possible a t the septum, i t will be well to have the equilibrium orbits well separated a t the last two turns, as explained a t the end of 8 7-13. We therefore arrange to have the final phase a t a/2, so the particles cross the gap a t the voltage peak. (Thereby we sacrifice the opportunity of letting the phase slip forward the maximum amount to T.) It has been seen that almost immediately after leaving the source, all the ions cross the gap close to the voltage crest, so we also set the starting phase $0 a t a/2. Hence in Eq. (7-35) we set $0 = a/2 and $ = + d = a/2, and for convenience introduce the abbreviations

Then Eq. (7-35) becomes cos a/2 - C

O a/2 ~ =

A Vm

-- (X - Y ) ,

(7-39)

whence

X = Y This means that the total phase advance X equals the total phase retardation Y. But X has been found by graphical integration, so the area Y is now known. Since the base of Y is ri2, as shown in Fig. 7-10d, its altitude K / B , is given by

The rectangle Y is now constructed and superimposed on the drawing of X in

Fig. 7-11. The location of the synchronous radius

T,.

Fig. 7-10a; this is shown in Fig. 7-11. The crossing point determines the square of some particular radius r,. From the center out to r, the ions have undergone

a n artificially irduccd phasc rctartlation ineasurcd by Y', a i sliown in Fig. 7 - l l c , :mtl a siri:illcr nat11r:il plr:isc, at1vanc-c inc~:i~ru.otl 11yS';on tho whole tlioy liavc lost I)II~L'c' in the : m o u n t Y' - S' in ~ i l o ~ ~fro111 n g r = 0 to r = r,. Fro111 r, to r d t f i ~pl1:lse atlvuncc X" is greater 1,fiari tlicl ph:tsc rctartlation Y", so tlic net gain is A"' - Y". F o r the whole trip t h e ph:isc ch:mgc is zero. Thcsc rel:itions, scparatcd for greater clarity, are sliown in Fig. 7-1 1. T h e radius r , is tliat a t which the nct loss of phasc ceases and the net gain begins and i t is thercforc the radius :it which t h e ions niornentarily turn in exact synchronization witli the oscillator. Although tlic r a l u c of I-, it; now known, the p l ~ n s e:it r, is still undetermined Wc may find i t hy integrating from r = 0 to I- = r,, using the :issoriated anglv. 4,) = a ' 2 :~11(1 4 a ~ i dI he appropriate areai X' a n d I"' which rcprcstvlt t h e in1 egr:A.:

This shorn that IT,, varies inversely with cos +,, a s ( ~ q x ~ t cTdh. a t is, less tlcc voltage is ncctlctl if more revolutions art. nllo~wcl,which occurs when the induced phase shift itowards sinall valucs of +) is iiicrc:isccl. T h c r n i n i ~ n l ~ n l ~ : i l n cof T',, is ohtnined when = 0, whence i t follows that

+,

hy uhc of 131. ( 7 - 3 6 ) . This cxprc+eion g ~ v wthe least valuc of tlrc pcnk gap voltage wllich will produce particles :it the, f i n d m m y y with orbit5 v p a r a t c d a.: much a.: possihlc. T h e ion4 lea1.c t h e source a t I$ = 4,) = T '2, have \)eel1 cal~sedt o slip hack in phase t o 4 = 4, =- 0, :uid h a w then moved forn-ard t o 4 = 4,l = T / O . If T 7 , is raised ahove this ~ t i i ~ i i r n u t11c ~ n , i~iitinlphnie retardatiol~ will not he a s great (i.e., 4, will not he a s small a i zero) hut t h e ions TI ill still reach t h e final radius when crossing t h gap ~ a t t h e ~ o l t a g ccrest. A typical n~ininluuldee voltagc for :I tltflcrtcd I~crnilof 21-?tIw deuterons, in a 60-inch cyclotron with a finnl ratlius rd of 25 7 5 iliclics, is 260 l i i l o ~ o l t s l)c:~k,dce-to-rlec. Calculations such ah tlrc a h \ (, c:m predict q11c.h a v:tluc to within a fcn- lwrccnt. A somcwh:tt lower threshold decl potential is required if the projectiles arc used t o bo1ni):ird a n internal probe targct, so t h a t extraction try the deflector is not nccded. Then there is no necessity for the. last acceleration to occur a t the voltage l ) c ~ i kand , the minimum potcmtial is t h a t which prlmits a s inanv turns as possihle for the ions to attain finnl energy without nlloming them ever to get into a rltwlrrating electric field. Since they start a t the voltage crest whcn $1, = T / % , the greatest pcmissi1)lr p1i:lsc~ retardation will carry then)

140

F1SEI)-FREQUENCY CYCLOTRONS

hack to $, = 0, just as before, hut the greatest subsequent phase advance may now carry them forward to a final phase #Jf = T. They reach the final radius with orbits closely spaced, since the last accclerations occur when the gap voltagc is approaching zero. The total phase rctardation and advance are no longer equal, and the simplicity of the previous analysis disappears. The initial phase retardation is found by integrating Eq. (7-35) from 40 = n/2 t o 4, = 0 while r changes from zero t o r,. This gives

wherc Y f and X', the areas out to r,, arc as yet unknown. The subsequent phase advance is obtained by integrating from = 0 to +/ = n, while r changes from r , to rf. A cos n - COS 0 = - - (X" - Y"), (7-47)

+,

v m

whence

Equate these two expressions for T7,n/A and eliminate Xu and Y" through the Y". This gives use of the relations X = X' X" and Y = Y'

+

Y

+

=

X

+ X'

- Y'.

(7-49)

This must be solved by trial and error, since only X is known (by graphical integration). Guess a t a value of r.2, so that X", Y', and Y can be found. Repeat this until r,"as been so chosen that Eq. (7-49) is satisficd. Then the associated values of X" and Y' will determine T',, through Eq. (7-44) ; that is, V,/A = Y' - X'. 7-15. Electric Focusing and Defocusing Forces at Dee Gap

The shapes of representative electric lines of force between the dees of a cyclotron are somewhat as shown in Fig. 7-12. An ion traveling from left to

JC-

-6 ,

Fig. 7-12. Electric field lines and their components at the dee-to-dee gap.

----

right on the plane of symmetry will experience only a horizontal forcc, hut ions above or brlow this plane will he acted on by a rcrtically focusing forcc

during the first half of the gap transit, a n d by a. defocusing force during the second half. Assuinc t11:it the particle's velocity ik so high t h a t the gain in velocity in crossing t h e gap is negligible in coi1lp:irison: the velocity is taken to be constant. Furtlier, also assume tlie clectric ficld to be constant during thc time of trarers:il. 'T1ic.n a n ion approacliing tlie gap on :In orbit parallcl to the iiiidplane b u t somewhat above it will acquire a sruall downward velocity component as i t enters the gap, wl~ichwill be exactly neutralized as it leaves, resulting in a not shift of the orbit t o a plane closer to thtl p1:me of symmetry. h-ow remove the static-field restriction, and consider wlmt happens if the ion (still nioving with constant velocity traverses the gap wl~ilethe clcctric ficld is rising in time.. Tlie dcfocusing f i ~ l dconrponcnt during the second half of the trip is now stronger t h a n the focusing coinponent in the first half, wliiclr results in the acquisition of a net velocity component away from the plane of syninletry. On the o t h c ~hand, if the crossing is m : d e wllile the electric ficld is tiecrcusing, the situation is reversed and the ion acquires a net transverse velocity directed tow:irds the median p1:tnc. I t is therefore highly desirable t h a t all traversals occur after the gap voltage has reached its peak a n d has begun t o dcclinc. Unfortunately, this is not always t h e caw. I n d~scussingt h e required dce voltage i t has kcen shown that it is profit:~blc to havc the magnetic ficld near the center of the nlaclline soincwliat stronger than t111eresonant value; this has t h e effcrt of inalting the ions initially slil) in p l m e so t h a t they cross the gap whcn the field is rising. It is not until a considerable fraction of the final energy has been rcachcd t h a t the phase changes so t h a t acceleration occurs when the clectric ficld is dccreasing. Therefore, early orhit:3 undergo electric defocusing; :inti since Itcar tllc ccntc,r of the nlnclrine tllc 1n:~gnctic field is almost unifonii, thcrc is T cxry little compcns:tting focusing of the nmgnetic variety. Tlie early orbitb tllercfore ( h i a t e from tlic mrdian plane in a n exponential manner and a l:+rgc. fraction of the pro~ec.tilcs :Lrc lost on the dces. Only those ions wl~ichs t a r t off with little o r no vertical velocity ever survive to reach full energy. (Strictly spe:~king, there arc atltiitionnl forccls on the particles k~ecausr~ of their inotion through the R F niagnetic field associated with t h e changing elcctric field 1)ctwecn the dces; hut thcscl forces are negligible for thc initial orbits bccnusc of the. low vclloc3y. Even when greatcr speed is nttninetl, such forcos a r e outwl.cigl~otl197 thosc t l l l c to tlre ri*inp; field index. Calcul:rtiorls involving thcx R E inagnctic ficltl will 1)e presented in # 14-5 in conncctio~l1~1th linear nrcclcr:~tors,where thcrc :we no stubilizlng inflwnces of :t guidc ficltl I I t shoultl I N , ~nentioncdt h t l l ~ o r ri i a n effcrt (lrnown :is vr.lortty f o c ! ~ , s ! n ~ ~ or second-orilcr f o r u a ! n q ) wllic*l~can allm late the i n i t i d loss of ions in a cyclotron, thougli b u t to a small degree. If we realize t h a t t h e ions actually do gain

142

FIXED-FREQUENCY CYCLOTRONS

velocity in going from dee to dee, we see t h a t they are in the defocusing field of the gap's second half for a shorter time than they are in the focusing field of the first half. This helpful action can predominate only if traversal on the rising voltage wave occurs very near t o the voltage peak, when the change in field between the first and second halves of the crossing is very small. Several schemes exist t o counteract the loss of ions a t small radii. ( a ) The use of a few carefully located vertical grid bars across the faces of the dces are coming into vogue, a s has been mentioned earlier. These act somewhat like cylindrical lenses and do not induce such strong vertical velocity components as the "feelers" which are often attached t o the top and bottom of the dees to help extract ions from the source. ( b ) Higher dee voltage permits a less extensive phase shift into the rising side of the voltage wave. (c) By the expedient of slightly reducing the magnet gap a t the center of the machine, an initial drop-off of field a t small radii is created, so that magnetic focusing forces are brought into play. This is particularly useful in synchrocyclotrons where the total path length is very long, and a larger total decrease in field with radius can be tolerated because of the continual reduction in the oscillator's frequency. If the entire opening of a dee which the ions are approaching were covered with a grid, while the opening of the dee being left remained unobstructed, the electric lines of force would not diverge in the second half of the dee-to-dee gap, and the convergence of the lines from the open dee t o the gridded dee would bring about axial focusing. Unfortunately, such a scheme is not practical except for the first few turns, as noted above, where the ion paths are well separated. At larger radii the distribution of ions is fairly continuous, so if grid wires arc not to intercept more ions than they save, they must be extremely thin, in which case they cannot withstand the impact of the ions which strike them or the occasional sparks to which they are subjected. (This same situation exists in a very dominant form in linear accelerators. With such instruments, as has been briefly noted, phase stability is present only if the synchronous ions cross the gap when the electric field is rising and the discussion above has shown in that situation the electric forces are defocusing. Consequently, linacs have required relatively massive grids across the entrance ends of their drift tubes, the exhaust ends bcing left open. For a long time such an expedient was mandatory if any appreciable number of particles were to survive, even though the grids intercepted a large fraction of the ions. Quite recently magnetic focusing devices have been incorporated within the drift tubes so t h a t the grids may bc dispensed with.) 7-16. Variable-Energy Fixed-Frequency Cyclotrons

The magnetic field in a conr-entional cyclotron is usually adjusted for the highest possihlc energy by the use of iron s h i m to produce the proper contour for focusing. Because of diffcrcnt degrees of saturation in various parts of tlle iron path, a new arrangcmcnt of shims is required if a lower energy is wanted,

HARMONIC ACCELERATION OF HEAVY IONS

143

antl since i t takes much time to 1n:ike such a rearrangement, this is s t ~ l t l o ~ r ~ done. N o s t cyclotron\, tlicreforc~,arc I I ~ I Y:IS~ singl(1-cncrgy dcrircs. Slwlr :I 1ilnit:rtion is of I ~ t t l (in~portance > if the S11nc6on of tlif nmchinc is s i ~ r ~ p ltoy 1)rooduce neutroi~sor r:idioactivc ~hotopcs,hut there : ~ r cother occ:~sions wlieii variability of energy of the dr+lecttcd t)ealu is extrcmcly useful. A simple way to attain i t is by the insertion of nlctallicl foils, of difftrcnt thic,knesses, in t l i ~ h are of lowpath of the cxtcrnal twarn, so that the projectiles ~ v l ~ i rpenetrate ered energy. This technique, t h o ~ ~ guscful, h has seritous disadvantages: if too much energy is removed, the foils melt, and in any case they introduce an angular divergence of the bcam which can only be partially compensntcd by subsequent focuing inagnets. Worst of all, the foils increase the inhomogeneity already present in thc cncrgy. Variable energy has been obtnined in a few cyclotrons I)y operation a t :I niodest field strength, where saturation is all but negligible. The field niay then be changed over a limited range and still maintain a suitable contour with a single arrangement of s h i m . It is newssary, of course, to retune thc~ oscillator and dee system for each energy, and if the changes are to be made with acceptable rapidity, this must be done without losing vacuum, so tlic shorting bars on the dee stems must be mnnipuhted Croni outside by ingenious mechanisms. I n this manner, a change in energy by ,about a factor 3 has been obtained in a few low-energy cyclotrons, in response to a change of field and frequency by fl.A considerably greater flexibility is anticipated from the recent developn~entsin "azimuthally varying field" cyclotrons which will bc described later.

7-17. Harmonic Operation of a Cyclotron: Acceleration of Heavy Ions Under the usual operating conditions with 180" decs, the frequency of thcl oscillator is al~proximatelyequal to t h a t of particle revolution, and aithough the electric field changes direction every half cycle, so also does the projectile, with the result t h a t every traversal of the gap between the dees is accelerative If the electric frequency is increased by the factor 3, so t h a t the oscillator goes through 1.5 cycles while the particle makes half a turn, it is clear t h a t tllc electric field is still in the proper direction to cause acceleration every time the particle rtaaches the gap (although peak 1)otenti:tl is also developed at moments when ions are not in a position to experience i t ) . Operation of this sort is called htrrmonic acceleration, antl the harmonic number or harmonic. order, given by h = a r f / a ,can be any odd integer, the value unity corresponding to the usual "fundamental" mode. (IVith even ~ a l u c sof h it is apparent, that the ions are alternately :tccclerated and dccelerated a t the gap between 180" dees, with no net gain in energy.) T h e basic expression a = qB/M may therefore be re-expressed in the form

144

FIXED-FREQUENCY CYCLOTROIVS

Thus if I3 and w , are ~ so chosen that protons will be accelerated when h = I , then the same values of B and w,l mill be appropriate for acceleration of a singly charged ion with triple the proton's mass, for then h may have the value 3:

Harmonic operation therefore permits a cyclotron which is adjusted for protons (with h = 1) also to accelerate 3He+ or singly charged molecular ions such as (IH 2D)+ on the third harmonic. This type of operation plays a crucial role in the acceleration of highly ionized ions of heavy atoms. The cathode of a cyclotron's ion source generally is not more than 100 or 200 volts below ground potential, so the emitted electrons do not have enough energy to remove, say, the first 6 electrons from a projectile gas such as nitrogen, the ionization potentials of these electrons being 14, 29, 47, 77, 97, and 550 volts respectively. Reasonably large numbers of 14N" are formed in the source, however, and if thc field and frequency are chosen to be in resonance for 14N6+ with h = 1, then the system is also in resonance for 14N2+with h = 3. That is, if M is the mass of a nucleon, we have

Thus doubly charged ions which are withdrawn from the source are accelerated on the third harmonic and soon acquire enough energy to become further ionized a t a collision with a residual gas atom. This throws them out of resonance and for some time they circulate without any further average gain in energy, but before long additional collisions bring them to the 6+ state of ionization a t which they are resonant with h = 1. It is obvious that the final orbits are not centered on the axis of the cyclotron and that many particles are driven into the side walls of the dees. Those which reach an internal target are few in number (perhaps one thousandth of those leaving the source) and they strike the target over a range of angle and with a considerable spread in energy. If the gas pressure in the dee chamber is increased, there is a larger chance for the state of high ionization to be reached closer to the center, so that the orbits are less eccentric and the particles have a greater opportunity to approach the theoretical limit of momentum given by p = qBr. Thus it is seen that early acceleration a t h = 3 can be followed by later acceleration a t h = 1 for particles in which the state of ionization changes by a factor3: 12C2+ to 12CW 14N2+ to 14N6+, 1602+ to 16OG+ 20hTe3+ to 20Ne9+ etc. It has been observed experimentally that pre-stripping acceleration can also occur with h = 5, to be followed by final energy gain a t h = 1: 12C+ to 12C" etc. Heavy ions afford interesting alternative projectiles for the study of nuclear disintegration, and may have great use in the production of transuranic elements, since the product nucleus can have a charge and mass considerably in excess of the target. There is experimental evidence that ion sources

for cyclotrons can be c l e v c l o p ~ ~tol yicsltl 111uchlarger currents of ~iiultiplyionized atoms than is now ~)ossihlc,for a e c ~ ~ l e r a t i von n tlw inore efficient funtiariiental illode. Linear accelerators already Iiltvr been built specifically for heavy ions. It has :dso been e ~ t a b l i s l ~ et11:it d a c c c l c ~ ~ a t i or ai n~ occur for h = 2, tliougli in a n unexpcclcd nrid inefficient n1:mrier. Second h a r ~ u o n i coperation iiilplies t h t one, re\ olution occurs in two cyclc~sof tlic oscilliltor, :tnd this can 11:~p1)cmif t h ~ ' center of the o r l ~ i tis displaced fro111the gal) between the dees to such a n extent t h a t the partivlv sl)cnds 5 of a n el(1ctric c~yclcnitliiri on(' (lee anti of a cyclc n-ithin the otlicr. Cridcr tliesc, circuiiistanccs acccleration car1 occur a t both travcmals. This situation has 1)ccn ok)ocrvctl in a cyclotron tuned t o wccelrr:ttc "He" on the funtlnli~cntaln i o ~ l c with ; tliv o : u ~valut~sof I3 antl w,,, it h i s been fountl t h a t a detect:tblc nuniber of "IHD) + ions can also be in resonance, witli h = 2. T h u s

+

7-18. Multiple Dees and Mode Operation

It is interesting to consider proposals for (lee s l i a l m other than the seiiiicircular 180" t y l ~ d~scussctl s tliuh far. Four dew, c:acl~of 90°, offcr interesting cllaractcristlcs. Since it is inconvcnicmt to excite :ill of t l ~ c n l only , two are drlrcn by t11e oscillator, the otlicr pair being groundell d u r ~ ~ n r i cinscrted ~s t o produce sinlihrly s l ~ a p c delectric fields a t all gaps. Consitlcr the uoual case of operation on the out-of-p11:tse ("pusllpull") modc, wltli tlw A p o d i v e ~ l ~ tlec. c n B is negative, a n d with h = 1 so there is one rerolution per oscillator cyrle AS in Fig. 7-13a. Accclcration occurs a t inotants 90 clectrical degrees np:trt. \\'lrcn t l ~ cproton is in g:ip I , arjljroac11ing dcc B, t h a t tlcc iiiust be ncg:ttivc; 90 clert~ic;iltlc~gl-eels Inttlr, B rllliht I)(' positive antl ~lrcferahly slioultl 1i:~ve tlio s:amc :lIjsoIutc value of potential.

I/

+ -

/

- n/-),A/-\,, (b) h=3 \

/

push-

Fig. 7-13. Harmonic oper:ition with 4 dces csach of 90".

146

FIXED-FREQUEXCY CYCLOTRONS

Therefore gap crossings should always occur 45' ahead and behind the voltage nulls. The energy gained per turn is AE = 4qV,/d5 where I.',, now is the peak dee-to-ground voltage. This number is also the greatest possible gain, for if the rotation frcquency w decreases, so that gap crossings occur later and later, the voltage :it gap 2 increases by an amount which is smaller than the decrease in voltage a t gap 3, because of the shapc of a sine wave. A siniilar argunicnt holds if OJ should increase, so t h a t crossings occur earlier; the voltage a t gap 2 decreases niore than it rises a t gap 3. If the oscillator frcquency is tripled, so h = 3, the impulses arc separated by 3 X 90" = 270 electrical degrees (see Fig. 7-13b). For equal energy increments on entering and leaving a dee, these instants must occur 270°/2 = 135" on either side of a voltage null, so again AE = 4 q ~ 7 , / d % I n general, out-of-phase operation with four 90" decs is possible when h is any odd integer. In principle, i t is possible to make the two dees oscillate in phase ("pushpush"), both going positive or negative together, by an appropriate change in tlie connections of the feedback circuit. Acceleration can then occur for h = 2, with AE = 4qT',,, as shown in Fig. 7-13c. I n general, "push-push" operation is possible with h = 2, 6, 10, 14, . . . , but no net acceleration can occur with h odd or of value 4 , 8 , 1 2 , . . . . The potential usefulness of such a system with four 90" decs is apparent when i t is realized that a single oscillator frequency is appropriate for protons using the push-pull mode with h = 1, and also for deuterons with h = 2 and the dees driven in phase. Another alternative is the use of two 120" dees separated by two 60" duinniies. Arguments similar to those above show that acceleration will occur for out-of-phase operation with h = 1, 5 , 7, 11 . . . and t h a t h E = 4qV,,, d 3 / 2 . The same energy is gained with the in-phase mode for h even, except for the values 6, 12, 18 . . . . Here again, by shifting modes and harmonic number, one can accelerate protons and dcutcrons a t the same oscillator frequency. Similarly, 60" dees with 120" dummies yield AE = 4qV,,,/2 with the pushpull mode and h = 1, 5 . . . , while for h = 3, 7 . . . the energy gained is AE = 4qT7,. In-phase operation with h even (except for 6, 12, 18 . . .) gives AE = 4 q T 7 f~l~/ 2~. An experimental proton cyclotron with three dees, each of 60" and symmetrically disposed, has been operated successfully. The dees are identified as ABC, in the order in which the ions pass through them. With the electrical phasing circuits so arranged t h a t peak voltage appears on the dees in the sequence ABC, the so-called fortcard modes are obtained. These produce acceleration for h = 3n 1, wliere n is an integer. JVith n = 0, so h = 1, there is one cycle of the three-phase s y s t e n ~per turn. Each 60" dee is 60 electrical degrees wide, so symmetrical acceleration occurs if an ion enters a negative dee 30" before a voltage null, so that it gains energy qV,,, sin 30" = qVrn/2, ant1 leaves with the dcc positive, 30" after tlie null. This occurs for each dee, so the energy gain per turn is 6qVw,/2 = 3qV,,,. The same energy gain occurs for a11 even values of n, i.e., h = 1, 7, 13 . . . . For odd values of n, we have

+

h = 4, 10, 16 . . . . For the first of these ( h = 41, each dee is 240 electrical dcgrceb wide, so ioris enter and lcave 120" on either side of a null and the energy gained is qVnIb/$/2; the increment per turn is hE = 3 d 3 qV,. This holds also for the above higlicr values of h. The bacJiw(lrd m o d e occurs when the circuitry is arranged so t h a t the deos rcnch peak potc,nti:~lin the order ACB, tl~ouglltllc ions still pass through them in the order AI3C. For tliese iiiodcb acceleration occurs when h = 3n 2. For even n, h equals 2, 8, 14 . . . :tnd i t is not hard to diow t h a t AE = 3 ~ 3 q V l , , , wliilc for odd n , h nlust be 5 , 11, 17 . . . and hE is 3 gV ,,,. \\?t11 the nmtral mode, all three decs are made to oscillate in phase. Here acceleration is g i w n if h = 3n + 3 = 3, 6, 9, 12 . . . . Ions enter and leave a t peak potential, so AE = 6 ql',,,. Since tlic. three-dcc cyclotron can operate with a variety of values for h , i t can accelerate a numbcr of different projectiles, even with a fixed oscillator frcqucncy. Thc circuitry is cwlrl)lcx, I~owcver,and it sccii~slikely that the w r i a h l e - e n c r g :~ziluutlinlly-varying fiel(1 cyclotron with continuously wdjustable field and frequency, to be described later, will Le a more useful tool.

+

7-19. Shielding

The output, of neutrons and gwii-nixL radiation from modern cyclotrons, wllether deliberate or caused by niisclirccted ions corning to rest insidc the inacl~inc,is so largc that a considerable investment in shielding rnust be inadc, not only to reduce background radiation in experinicntal set-ups, but to protect personnel from dangerous exposure. Initially, no protection was supplied a t all. Then 1:irger currents and higher energies raised the radiation level, and as the 1)iological hazard become increasingly recognized, neutron shielding bc.gan to appe:w. At first, this took the form of 5-gdlon cans of water or blocks of paraffin piled about the vac~uuinrhaln1)cr. This inconvenient makeshift gal-? way to water tanks several fect thick foriiiing a wall around the entire cyclotron, substqumtly augmented by more water tanks d d c d as a ceiling over the enclosure. Nowadays, cyclotronb are built in large vaults with concrete walls 5 to 11 fect thick and with ceilings 4 to 5 fect thick. Access doors, of equ:tl thickness, move horizontally or ~ e r t i c a l l y .Contluits and piping from the vault go to cellar are:ts normally uninhabited. Vision into the vault may be by closeclcircuit television, by periscopes, or by the use of glass windows laminated to a tliicltness as great as the concrete walls. Sometimes the windows consist of glass-walled tanks filled with water, or they may be of two tanks, the inner one containing water to slow d o n n and capture neutrons, the outer one being filled with a saturated solution of zinc bromide to a1)sorb the gamma rays t h a t are produced when the neutrons are ci~pturetl. Rooks and articles on fixed-frequency cyclotrons are listed on 1). 344. Paperb tlo~cribingparticular machines of this t y may ~ be found on 111). 345-348. l
SYNCHROCYCLOTKONS

8-1. Introduction

Synchrocyclotrons, it will be recalled, look and operate very much like cyclotrons, except t h a t the frequency of the voltage applied t o the dees is made to decrease as the particles gain energy in order always to match the decreasing frequency of revolution of the ideal, synchronous projectile. Most of the ions are accelerated by benefit of the principle of phase stability, their energies oscillating about the steadily rising value determined by the frequency of the oscillator. Geometrical stability of the orbits is obtained by the weak-focusing technique, with the field index rising gradually from zero as the particles spiral outward. As will be demonstrated before long, the dee potential is only a few kilovolts, so the pitch of the spiral is very small; consequently the particles spend a relatively long time a t every radius, and resonances can be very troublesome. Mention has already been made of the particularly severe one a t n = 0.2, where the radial betatron oscillation frequency V~ is twice the axial betatron frequency v,; this resonance is practically impassable, so the target or the deflecting mechanism is so placed as to be reached by the ions just before this harmful condition is attained. The magnets for these machines differ from those of cyclotrons chiefly in being larger, since it is possible to attain much greater energies by taking advantage of the existence of phase stability and hence avoiding the energy limitations of the simple cyclotron. Since the oscillator frequency is modulated to take care of the increase in mass, i t is easy to increase the degree of modulation slightly more to compensate for an increased drop-off of field with radius, which often runs t o as high as 3% to 5%. It has been shown in Chapter 2 that the amplitude of the axial betatron oscillations can be reduced if the field does so weaken; this permits a smaller gap height in the magnet, with consequent reduction in the amount of copper in the exciting windings and in the size of the power bill. The appropriate contour is generally machined into the pole tips after a careful theoretical study backed up by field measurements on a small-scale model. The field strengths a t the center range from 16.3 to 23.3 kilogauss in different machines. There are 18 synchrocyclotrons operating in the world today (1960). The 148

sizes range from the 35-inch tliamctc>r magnet a t Princeton, for 20-XIcv protons, up to the 189-inch, 730-Nev niachine at Berkeley, ('alifornia (I.(,ccntly enlargrd fro111 184 inchch), the 197-inch 600-l4ev instrument at Geneva, Switzerland, arid t l ~ cm o n s t ~ ~ o ~236-inch is 680-RIcv "phxsotron" (a* the Russians c3:rll i t ) a t Duhnn. The u.13ght uf iron in these cx:tmples is 40, 4000, 2500, :inti 7200 tons, rcspcctively. Threr synchrocyclotrons operate exclusively with deuterons a t about 30 Mev. Separate and easily removablc~vacuum chambers are not practical in the larger macliinr~s;instead, thc pole faces of the rnagnct serve as the "lids" of the chaml)er, the joints being made tight with gaskets. 8-2. Frequency Range

At every instant the synchronous ion and the oscillator have the same radian frequency = qB/M = qBc2/E == qRc2/(En T ) . Consequently the frequency in cycles per swond is given by (J

+

In the usual units with f expressed in RIc/sec arid B in kilogauss, this hecoincs, for protons,

Note t h a t this differs from Eq. (7-2), a p ~ ~ l i c a bto l c a, fixed-frequency cyclotron, only in the appearance of thc term T/ISo.The frcqucmcy range of the operatirig cycle thus dCpcnds both on t h t final k ~ n c t i cenergy and on the drop-off In field from the central value to that a t the final radius. As an example, the machine a t Berkeley operates between 23.3 and 22 275 kilogauss when bringing protons to 730 Mev, so the oscillator must be modulated from 35.4 to 19.1 RIc/sec. When deuterons are the projertiles, Eq. (8-1) becomes

where now Eo is the rest energy of a deuteron, 1876 Mev. When producing 460-Mev deuterons, using the s m e magnetic field as before, the Berkeley machine requires the oscillator to run from 17.7 to 13.6 Rlc/sec. Doubly charged H e q o n s (alphas) are accc1er:itcd to twice the deuteron energy undcr tlic s:ime operating conditions as for heavy hydrogen. The change in frequency ordinarily is made with :r rotating multibladed capacitor, the repetition rate varying from as little as 60 to as high as 2000 cycles per second, in different installations. I n one machine, t h a t belonging to Columbia IJniversity, the rotor is mounted s t the electrically ideal position -directly between the working edge of the dce and ground-though turning such a capacitor in the strong magnetic firld offus many engineering dif-

150

SYNCHROCYCLOTRONS

ficulties. Usually the rotor is as far from the field as the designer can put it. In the synchrocyclotrons a t Berkeley and a t Gcneva a large multibladed tuning fork supplies the alteration in capacitance, while a t Bonn ferroelectric units are caused to change capacitance by variation of a DC bias voltage. 8-3. Dee Voltage

The accumulation of energy, from zero to the final value, must take place in the time of modulation of the oscillator from the initial to the final frequency. A typical value is, say, 0.01 second. If the final energy is 500 Mev, the ions ~ = 5 x 101° ev/sec. Supmust gain energy a t the rate of 5 x los e ~ / 1 0 - sec pose the average frequency is 20 Mc/sec, so that the average period is 5x sec. Then the average energy gained per revolution must be 5 x 101° ev/sec x 5 x 10-8 sec/rev = 2500 ev/rev. If the synchronous phase angle is +, = 150°, so that sin +, = 0.5, and because there are two accelerations per turn, a single dee with a peak potential of 2500 volts is adequate (rather than the 100 kilovolts or more required with cyclotrons). Such a low voltage is very convenient, for a comparable potential occurs across the tuning capacitor where the distance between electrodes is small and sparking must be avoided. 8-4 Rate of Frequency Modulation

To investigate the influence of the rate of change of oscillator frequcncy on the energy gained, we repeat Eq. (6-9) :

where a is the momentum compaction and 7 = E/EU.This expresses the general relation between change of momentum and change of period. Since T = 27r/w, it follows that ~ T / T= -dw/w, and when Eq. (1-31) is used to convert momentum to energy we find that

As in Eq. (6-20), we define

Hence Eq. (8-5) may be written

Let the energy gained per turn by the synchronous ion be 2qV, sin are two gaps, with a peak potential V, across each.) Then dl3 dE 2?r 2qVmsin 4, = - T = - -' dt

dt w

+,.+ (There

Solve this for d E / d t and substitute in I 3 l . (8-7) to find

Before this rclation is discussed, i t will be iwll to d i s c o ~ e ra niore explicit depcnticnce of r on the cncrgy than is given by +;q. ( 8 - 6 ) . By rearranging the t e r m in its clrlfinition and by using Eq. (1-21) wc find:

But in a synclirocyclot~.ona = 1 .- n , as s1ion11hy Eq. (6-3). Thcreforc

It is apparent, that the dependence of r on the energy (or velocity) cannot be g i w n in general terms for :L ,iyncllrocyclotron, hut must await t1et:tilecl knowlctlgc of the s1)ccific field contour as n function of radius, and hence of energy, for any g i w n magnet. For the present discussion we must therefore be content viith the approximation t h a t the field is unifol-111, so that n = 0 and = 1. We return to IQ. (8-9) and note t h a t the value of dw/dt is controlled by the shape of the plates of the tuning capacitor and by its rate of turning. If we assume that dwldt is constant, as well as that r = 1, it is clear t h a t qF7, sin 4, must increase throughout the accderation process, since w falls and 8; rises. If ive also illblbt that 4, remiin fiucd, tliis requircs IT,,, to undergo a large rise, nliicli ni:iy bc inconvenient or i111j)ractic~al.But it is only in the early stages that 4, must rcn~:lin near 150" (in order t h a t the largest number of particlcs I)c caught :mI 1,ctaincd in ph:isc-stable orbits, as discussed in 6-8), so subscclumtly +, may bc allowed to increase, as will occur automatically if I;,,, is held constant. I n practice it is customary to reduce the value of dw/dt throughout the acceleration interval and thereby to permit a smaller rise in qV, sin +,. 8-5. Frequency of the Synchrotron Oscillations

Tlic frcclucncy of iynchrotron osci1l:ltions of small amplitude has been sllown in Eq. ( 6-45 I to be =

hy4'&qr 2aF,

cos

+,,

is tlic j)e:di potential :icross e:wh ncccltmt~nggap of which there :u.c wliere IT,,, .V pvr turn, atid where h = w r f / w . Assunle, as in a synchrocyclotron, that S = 2, h = 1, 4, = 150°, and, as in S 8-3, that the energy gainod per turn is

152

SYNCHROCYCLOTRONS

AE = 2.5 kev. Then h E = NqV, sin 4, so that qV, = 2.5 kev = 0.0025 Mev. As the protons start out and for some little time thereafter, their total energy is E, .-:938 Mev, and near the center of the machine the field is practically uniform; so n- 0 and therefore r = 1, by Eq. (8-11). Then from Eq. (8-12) we find that w = 1170Q, so that each synchrotron oscillation lasts for 1170 revolutions of the particles. The final energy, say 500 Mev, is reached where n = 0.2; E , = 1438 Mev, P2 = 0.574 and r = 1.436. If 4, and AE, remain as before, it is seen that w = 12050. Thus throughout the entire acceleration process, the synchrotron frequency is very much less than the revolution frequency.

8-6. External Beams

Electrostatic deflection techniques of a simple sort can be employed only when the energy is low, as is the case in two small machines which have been converted from cyclotrons. For the larger devices the energy gained per turn is so small a fraction of the final kinetic energy that the separation of turns is but a few tenths of a millimeter and all the beam would be intercepted by a septum thick enough to withstand the occasional sparking between it and the deflector plate. The earliest method of obtaining external projectiles of some hundreds of Mev employed scattering from an internal target; an exceedingly small fraction of the beam is diverted into the desired direction and passes (sometimes through a magnetically shielded channel) out of the vacuum tank for use elsewhere. A more efficient procedure is an elaboration of the slotted beam-splitter used (for other reasons) in cyclotrons. The septum and deflector are each replaced by two rectangular bars, curved to the proper radius, extending for something over 90' in the region opposite the single dee. The structure amounts to a cylindrical capacitor with a slot, perhaps an inch high, a t the median plane of each plate; the useful volume is that between the plates a t slot height. Since the electric fringing field must not affect orbits a t smaller radii and since the necessary field strength is above that which can be held without breakdown if kept on continuously, the voltage between inner and outer plates is applied in a pulse, once per modulation cycle of the oscillator. The rise time is about equal to the rotation period of the particles, full voltage is maintained for about twice this time, while the decay is short compared to the repetition period. Rather surprisingly, the electric ficld is applied in a direction to drive tlie ions inward. If we picture the particles as rotating clockwise, with the electric deflector extending 120" from 10 o'clock to 2 o'clock, the result of traversal down the channel is to displace the center of curvature in the general direction of 7 or 8 o'clock. Consequently when tlie ion has made a full turn and is back a t the azimuth of the start of the deflector, its orbit passes on the outside of

t h a t structure; i t then enters a final channel in which a inngnetic shield r e d u c c ~ the field to ahout one third its nornlnl value and the particle is thus let1 out o f tllc vacuuril c~h:rn~hcr in somctlling less tli:an :t quarter turn. Tlie mngnelic. shirltl in this final ch:inrwl consists of two I ~ r of s iron, of rcct:mgul:tr rrois scction, supported siclc by sitlc so :I.: to form the inside and outside walls of the channcl. consitierablc portion of the flux from the main poles of t h e n1:qpd is thus divcrtct-i through t l i t w bars, so the field is lowcred in the region h c t u w n them. I'nfortunatcly the field is also depleted outside tllc bars; t h e pffcct is quite noticeal~le soinc3 inchc.: ton-nrd the ccntcr of thc ~ ~ m c l l i nn-llerc c supposedly ~lorinnl orhrt.: arc. 1oc:~tctl. Tllc field illust 1 ) ~ returned t o its full value in this rsgion, :lnd tliia is :~ccon~plishcd by ~ n o u n t i n g w r t i c a l shccts of iron a h o r e and below the nicdian p1:tne. T h e size anti disposition of these compensators is d(tcmnincd hy calculation follon-ed t)y oinpiricnl : d j u i t ~ n e n t . I n the largc. Rcrkclcy syncl~rocyclotronin which this scheme was first used to deflect 190-Rlev deuterons, a potential of 200 kilovolts was pulhed across a 1-inch gap. A t ~ o u t1% of the internal henm was sucsessfully deflected. Details of the system and of the pulsing circuit can I)c found in the 1iter:tture. A more r e c m t d e ~ e l o p l n e n tin the cxtr:~ction of particles from synchrocyclotrons is known as ~ e g e n c r a t z ~dt7flection. ~e I n hricf outline the s c h e n ~ eis the following. 4 s the ions sl)iral outward, t l ~ c ylittle by little fall under the influence of a "buinp" in the guide field--:~ region where the field hiis been made overstrong. T h e radius of survatnre therein is reduced, hut i t returns to full value when the particle leaves the h m p , the new renter of curvature bcing sliiftcd In :i direction roughly 90" "upstrcwn" from the hump's a z i m u t l ~ . As a result of repeated traversals, this n ~ o t i o n~)rotiucesa n ~ u c hlarger separation between turns than WE brought at)out by a n increase in energy. Ultim:ttcly the sep:mtion becomes large enough to clear thc thickness of t h e inncr wall of a magnetically shielded tunnel a n d t h e particles leave t h e machin(.. Extraction efficiencies of :t few pcrsent 21al;c been obtained. This ejection systein also m a y he looked on a s one in which a deliberately produced error in the guide field causes the operating point to move close to a resonance which causes a rapid growth in the radial betatron oscillation amplitude. 8-7. Shielding

Shielding problems are more severe t h a n with cyclotrons, in spite of t h e much s n ~ a l l e rtin~e-averagedcurrents. \Yith the. higher energy of the primary projectiles, the output of neutrons and nwsons is concentrated inorc in the forward direction so t h a t shielding a t the median plane must he augmented. A thickness of 20 feet of concrete is required for ~ n a c h i n e sin t h e 500-Mev region. References to papers on synchrocyclotrons and on regenerative deflection m a y he found on pp. 351-354.

Chapter 9 SYNCHROTRONS

9-1. Proton Synchrotrons, Operating and Under Construction The highest particle energies yet attained by artificial means have been produced in proton synchrotrons, in which a relatively modest amount of iron is distributed into a ring-shaped magnet of large radius. Particles are injected into this from an auxiliary accelerator and gain energy by passing repetitively through one or more accelerating units whose frequency rises to keep in step with that of the rotating ions, the magnetic field increasing all the while to maintain the orbits a t constant radius. There are fire weak-focusing proton synchrotrons now operating (1960), the first coming into operation in 1952. In the order of completion these are: "Cosmotron" a t Brookhaven National Laboratory, Long Island (Plate 15) Synchrotron a t The University, Birmingham, England "Bevatron" a t Lawrence Radiation Laboratory, Berkeley, California (Plate 16) "Synchrophasotron" a t Dubna, USSR "Saturne" a t Saclay, France (Plate 17) All of these employ a finite gradient with the field index close to 0.6, their peak fields ranging from 12.6 to 15.4 kilogauss. Five to eighteen pulses of protons are produced per minute a t the various installations, each pulse containing 1010 to 1011 particles. Five other weak-focusing proton synchrotrons are under construction in different parts of the world with energies distributed from 1.0 to 12.5 Bev. Two of these are zero-gradient machines. 9-2. Magnets Multi-Bev synchrotrons are impressive pieces of equipment and represent large engineering undertakings. The 3-Bev Cosmotron magnet weighs close to 1700 tons and is built in four quadrants, with an orbit radius of 30 feet separated by straight sections 10 feet long. The 6-Bev Bevatron magnet weighs 9700 tons, has 20-foot straight sections and has quadrants of 50-foot radius. 154

MAGNETS

155

Tlie ltussian n~achine11as a n orbit radius of 92 feet, four straight sections of 26 feet and w i g l i t of 36,000 tons There arc two basic alternative cross scctions for ~ynclirotronmagnets: the C shape and t l ~ cp~cture-frameshape, also called the H shape (Fig. 9-1). The injector

accelerator

Fig. 9-1. Proton synchrotron of four sectors, and schematic enlarged cross sections of "C" and "picture-frame" magnets, with and without poles that protrude beyond the exciting windings.

C, u s ~ da t Birmingham, a t Saclay, and in the Cosniotron, has many advantagc.~ of convenicncc., since the coils and vacuum chnrnber can be installed or re~ ~ i o v cfor d scp:lir without disturbing the iron, and one whole side of thc ch:wubcr is availablc for installation of viicuum pumps and other components. On the otlicr hand i t is less cficient magnetically than tlic symmetrical picture-framc type-whicli, however, suffers from inaccessibility of coils and cliani\~cr.Picture-frame magnets arc uscd only in synchrotrons of the race-track variety (such as the Bevatron and tlie ~ n a c l ~ i naet I h b n a ) wherc field-fret straight sections connect the curved rrmgnets. (This arrangement has been used also with tlie C cross section.) The straight sections are used for vacuulnpurnp connections, internal targets, deflecting equipment for injection and ejection, accelerating units, beam obscrvation ports and similar auxiliarics. The magnetic gap, whether in the C or in the H type, sometimes has poles \vliich protrude beyond the exciting coils and sometimes not. This choice and tlic dctuils of the shaping of tlic iron surfaccs involve a long series of com-

SYNCHROTRONS TABLE 9-1 WEAK-FOCUSING PROTON SYNCHROTRONS OPERATING I N 1960

Maximum energy Injection energy Maximum field Injection field Field index Orbit radius No. straight sections Length str. section Rise time Rep. rate, per min. No. accel. stations Harmonic order Oscil. freq. min. max. Vac. chamber width height Weight magnet copper Injector Energy gain per turn Protons per pulse Completed

Lawrence Radiation Lab. Berkelel~ Calif. "Bevatron"

Joint Instilute for Nuclear Research Dubna USSR

Centre d'h'tudes Nucldaires Saclay France "S'alurne"

Units

Brookhaven Nat. Lab. Uplon L.I. N. Y. "Cosmolron"

Bev

3

1

10

2.94

Mev

3.6

0.46

9

3.6

kgauss

13.8

12.6

13

14.9

gauss

295

217

150

328

0.6

0.67

0.65

0.6

30

14.75

91.6

27.6

4

0

4

4

ft

10

0

26.2

13.1

aec

1

1

3.3

0.8

12

6

5

18

1

1

2

1

1

1

1

2

Mc/sec Mc/sec

0.360 4.18

0.330 9.3

0.182 1.45

0.76 8.45

inches inches

25.6 6.3

13.8 3.9

59 14.2

20 6

tons tons

1650 70

9700 350

36,000 460

1080 55

Van de Graaff

800 10 CockcroftWalton

linac

linac

Van de Graaff

1

0.2

2.5

1.16

10" 1952

5 X 109 1953

ft

kcv

Birmingham Unzverstly Bfrmzngham h'ngland

putations and of experiments with small-scale magnets, the goal being the attainment of a broad radial region having the desired constant value of field index over the full range of field strength to be employed. Generally speaking, pole-face windings or auxiliary magnets in the straight sections must be employed, as has been mentioned in the earlier discussion of the control of t h e field index. The Birmingham machine has no straight sections, the C magnet forming a

TABLE 9-2 WELIT<-FO('ITSISG PROTON SYNCHROTRONS UNDljClt C'OSSTKCCTION I N I N 0

Feature Maximum energy Injection energy Maximurn field Injection field Field index Orbit radius

C'nila

!l'ecIut~(al C'flt~. lleljt Netherlands

l'n?rc~tunI'enn. Arc d e , alor l'rificetun N.J.

Nutherford Hzuh Enrrei, 1,d. Hui cell

Brigland "Ntmrod"

Argunne Nat. Lab. Lemont, 111. "ZGS"

Australtan Nat. Cnau. Canberra

Rev

3

7

12.5

10.6

MPV

3

15

50

8

kyauss

13.!15

14

21.5

80

EnllRS

270

290

477

850

0.58

0.6

0

0.55

30

61.6

71.2

15.7

16

8

8

4

ft

No. straight sections Length str. sections

ft

4 Rs 6

14 & 11

20 & 14

8.2

Rise time

sec

1/40

0.72

1

0.8

1140

28

15

1/10

4

1 (double)

1 (triple)

?

8

4

8

1

1.43 8.04

4.4

1.9 9

Rep. rate per min.

No. accel. stations Iiarnronic order Oscil freq. min. max. Vac. chamber width height Weight magnet copper

Mc/sec Mc/src inches inches

11.3 3.4

7.0 2.75

36 9

32.50 5.25

8.5 diarn.

tons tons

? ?

350 27

7000 250

3500 70

air core 77

Van de Chnatf

hnac

linac

cyclotron

ti0

5.5

12 thrn 7

2

10'2

?

C-W or

Injector

cyclotron Energy pain per turn

kev

Protons per pnlse A\e. current

14

Fa

7

0.04

Completion date expected

1962

?

circle with an orbit radius of 14.75 feet. There is but one exciting coil, which is wound between the vacuum chamber and the ve.rtical leg of the C. Though relatively inexpensive, this winding procedure produces a troublesome stray field a t some distance. I n other C magnet installations the exciting winding for each magnet quadrant forms a coil which goes down one side of the gap next

Courtesy of Brookhaven National Laboratory

PLATE 15

The 3-Bev Cosmotron. The pressure tank of the 3-Mev Van de Graaff injector is seen at the left. A small analyzing magnet in the foreground separates the HH+ ions from H + and directs the latter towards the electrostatic inflector located in a straight section of the synchrotron. The smooth appearance of the laminated magnet is caused by a plastic dust cover. to the vertical yoke and returns on the other side of the gap and outside the iron, these return conductors being placed half above the gap and half below, to afford access to the vacuum tank. With picture-frame magnets, the conductors lie entirely within the yoke, forming loops around the vacuum tank (and around the poles if they protrude). At the ends of a magnet with either type of cross section, the windings are bent so that half pass u p and over the vacuum chamber and half pass down and under, t o leave a clear space for passage of the particles. The windings of the Bevatron and the Birmingham machine are cooled by forced air; in the other installations the conductors contain a channel for water cooling. Because the magnetic field changes with time, eddy currents are induced in the iron and these currents produce distorting fields in the gap. T o reduce

MAGNETS

159

Courtesy of Lawrence Radiation Laboratory

PLATE 16 The 6-Bev Bevatron. The ion source ancl 600-kilovolt Cockcroft-Walton unit lie within the cubical enclosure at the right. In the center is the 10-Mev linac which introduces protons into a straight section between two of the four magnets which form the synchrotron. these fields to tolerable values, particularly a t injection time when a few gauss due to eddy currents may he several percent of the nominal field of 150 to 300 gauss, the magnet is fabricated from flat laminations, after tlic manner of an alternating current transformer. Chonce of the lamination thickncss is a complicated matter, depmding on the ratc of rise of the field, the characteristics of the chosen type of steel and the extent to which the eddy current field can be used to coinpensate the residual field left over from tlic previous pulse. All of this is con~pouncledwith practical matters such as tllc flatness, the dimensional tolerances to be obtained in quantity production of sheets or plates, the method to be used in cutting to shape, techniques of insulating and of bonding t,ogether, and the cost of assembly. For machines in ~ ~ h i rthe l i rihe time of the field is about 1 second or more, a thickness of 4 or 4 inch is adequate. Much thinner laminations, such as 0.03 to 0.06 inch, : ~ r cnc~clctli f tlrc pulse ratc is, say, 20 per sccond, as in the 3-Brv proton sy~iclrrotron untler construction hy Princeton and tlrc IJniwraity of Pennsylrttnia, and in the 6-Bev electron synchrotron being built by Harvard and t h e Uassacliusetts Institute of Technology.

160

SYNCHROTRONS

Courtesy of Centre &Etudes NuclBaires d e Saclay, Commissariat d Z'Energie Atomique

PLATE 17

The 3-Bev proton synchrotron "Saturne" at Saclay. The Van de Graaff injector lies to the riglit, out of the field of view. Usually the laminations are stacked together into units which are assembled fanwise to form the curved magnet. This leaves wedge-shaped air gaps, which reduce the effective quantity of steel and lower the field for a given number of amper-turns. Sometimes this situation is partially alleviated by inserting into the wedges further laminations of reduced radial width, to produce a set of smaller air wedges. For the 12-Bev proton synchrotron under construction a t the Argonne National Laboratory near Chicago, all laminations of nominal &-inch thickness will have a wedge-shaped cross section in order to attain the highest possible field. The stored energy in the magnet of a big proton synchrotron can run into tens of millions of joules, which is equivalent to the energy involved in lifting the magnet several feet off the ground. This energy must be delivered in perhaps a second, so the peak power demand, when copper losses are included, may be as high as 100,000 kilowatts, though the time average is perhaps one tenth as great. It is usually uneconomical to pay the "demand charge" for such a large burst of power, even if the distribution and generating systems are such that it can be delivered without causing a serious dip in the voltage.

MAGNETS

161

Coultesy o f the Institute of Physics, Rome, and High Voltage Engineering Corp.

PLATE 18

The 1-Bev electron synchrotron at the University of Rome. In the left background may be seen the 3-Mev Van de Graaff injector with its pressure tank withdrawn. As a result, thc energy is accumulated slowly, stored in massive flywheels, and then delirercd to the magnet. A cotnmon system employs one or more alternators coupled to motor and flywheel, ~ ~ i mercury-arc t h rectifiers to convert to direct currcnt for use in the magnet. I n such systems the rise time of the field ranges from $ to 3 seconds. By suitable electronic switchgear, the stored energy is then drained out of the magnet, in a co~nparabletime and is fed back to the alternators, which now act as motors, to help re-cnergize the flywheel, which will Imve dropped in speed a few percent. H e a t is developed in the windings during both directions of energy flow, and the continuously energized motor makes up this loss, as well as that from hysteresis and eddy currents, during a recovery period of a second or so. The complete cycle repeats a t a rate whicll varies from 5 to 18 times a minute in the different machines. A t Birmingham s direct currcnt generator is used rather than an alternator with rcctificrs, while in the synchrotrons k i n g h i l t a t Princeton and at

SYNCHROTRONS

162 ,..q

r.. \ \,, ri-t-

HIGH-VOLTAGE GENERATOR

-

DC MAGNET INFLECTOR

GUIDE MAGNET

BENDING MAGNET

PlCClONl MAGNET "2 SWITCH1 NG MAGNET DEFLECTING MAGNET

SWITCHING MAGNET

BENDING MAGNETS

MESON AREA

PLATE 19 Sketch of the 12-Bev zero-gradient proton synchrotron under construction at Argonne National Laboratory in 1960. Harvard the stored energy is not so great but that the flywheel can be replaced by capacitors and auxiliary inductances which form a resonant circuit with the magnet, the repetition rate being 20 to 30 per second. The 10-Bev synchrotron under construction in Australia has unusual characteristics. No iron will be used, the peak field of 80,000 gauss being obtained wholly from copper windings. The peak power will be 500,000 kilowatts supplied by a homopolar generator which also serves a s a flywheel. The rise time of the field will be 0.7 second, with one pulse every 10 minutes. The zero-gradient synchrotron, known as the ZGS, now being built a t

Argonne National 1,aboratory to produce 12.6-Rev protons, affords some intcrcsting fc.at,ures. T h e guide magnet 1s fornltd of 8 arcs, each about 58 fcct 14 and 20 feet long, to gi\.ca in length, scp:tr:ttcd by straight s ( ~ ~ t ~ alternately ons :Ltotal circuniftwncc of 563 feet. Tllc cross sccticm of the magnet is of tlic 1)icture-frnnlt. typcl without protruding poles, tlic. opposing inncr fxccs of thc stccl la~ninationsbeing strictly p:1rallcl to producr a uniform field. Vcrticnl focusing is supplied entirely by the 16 slanted ends of t l ~ emagnets. Accurately positioned water-cooled conductors run down one side of the gap and return on the other, being bent at the ends so as to pass over or under the vacuum channel, which is 32i inches wide and 5 : inclics high. Such a magnet design produccs an adequately uniform field up to about 17,000 gauss, since the majority of the reluctance in the magnetic circuit is supplied by the air gap. Above tllis fiqure 5:lturation of the steel Iwgins to be noticenl)ltb; the pernieahility of the steel drops : ~ n dtlie metal plcays :in increasing role in determining the re1uct:mcc. Tliu3, a certain field a t tlie cdge of the vacuum cha111t)c.r is producetl by a loop of flux which harcly clears the conductors, but a larger loop wli~cli passcs through the center of the c h a ~ n h e rlias a longer path in the iron, so that tlic clentral field is weaker th:rn t h a t at the edge of the chamber. This un1iapl)y state of affairs can be corrected by boring holes in the steel laminations just above and below the conductors and parallrl to them, for the full length of the magnet. Because of their position, these holes increase the reluctttnce more for short loops than for long ones, so that the edge field is brought down to acceptable equality with tlic central v d u ( > . (At low fields, the holes havc little disturbing effect, since tile reluctance of the main gap is then the controlling f:tctor.) The choice of t l ~ c position and size of the holes lias involved a long study of a series of model magnets, but the results are worth the effort, since a field adequately unifonn up to 21.5 kilogauss is attainable. Inasmuch as the fringing fields a t the ends of the magnet play a crurial role in the ZGS, special attention lias been given tlo their control. T o prevent a change in the effective length of a magnet because of saturation a t the open ends of the gap, the gap height is gradually i n c w a w l as the end is allproached (vertical flaring). and a t the same time t l r ~vertical spacing of the conductors is increased so as to follow tlie separating magnetic equipotentials across the gap. Widening of thc magnetic gap in the r:dixl direction also occurs a t the endq of the magnets (radial flaring). This not only prorides more room for extraction of particles but removes the conductors from close proximity to the side borders of the large region in which the. particles travel a t injection. Fabrication wrors in the placement of the exciting conductors will generate errors in the field which die out rapidly with radial distance, and radial flaring of this nature diminishes the deleteriouq effects of such errors in the re,'~ l o n s where the orbits are most sensitive to them, that is, :it the magnet ends where all the vertical focusing forces are generated.

SYNCHROTRONS

.

Another feature of the Argonne magnet design is of interest, since it improves the effective uniformity of the field. Four of the eight straight sections contain small magnets driven hy a steady current. With a field of about 800 gauss, each of these magnets produces a 4' deflection of the 50-Mev injected protons, so only 344' is left for the main, pulsed, magnets. The centerline of the envelope of widely oscillating ions a t injection time coincides with the centerline of the gap in the main magnets, but a t high energy, when the effect of 800 gauss is inappreciable, the centerline of the orbits' envelope (now greatly reduced in size by adiabatic damping) passes diagonally through each main magnet, being near the inner or outer edge of the chamber at alternate straight sections. Such a path effects an averaging of undesirable residual gradients a t high fields, for, from the symmetrical construction, any gradients will be of opposite sign on opposite sides of the center line. 9-3. Vacuum Chambers

The design of a vacuum chamber for an accelerator with a pulsed magnet offers many knotty problems. The structure must resist atmospheric pressure, but a thick roof and floor will decrease the inner space allotted to particle oscillation. To maintain this space by increasing the over-all height of the magnet gap is extremely expensive in terms of copper, power, and stored energy. Further, the generation of eddy currents must be discouraged in view of the perturbing fields they set up. This last requirement argues for metal parts that are thin, have small horizontal area, and are made of a material with high resistivity, such as stainless steel. I n the Cosmotron, the vacuum chamber within each magnet quadrant is 47 feet long, 26 inches wide, and 6 inches high, inside. The roof and floor are each composed of about 300 ('joists" of stainless steel 8 inch high and 2 inches wide, separated slightly from each other and screwed to and insulated from side walls of stainless steel 1 inch thick. This framework is made vacuum tight by an outer covering of synthetic rubber shceting. The chamber for each quadrant is installed by slipping it between the poles of the C-shaped magnet. The Bevatron's magnet is of picture-frame cross section. The upper and lower pole tips are separated by massive side members of stainless steel, and this entire substructure is surrounded by a vacuum-tight skin of stainless sheet, so that the major portion of the poles is within the vacuum, the pole faces forming the roof and floor of the chamber 48 inches wide and 12 inches high. All this is assembled inside the yoke and the remaining parts of the poles. A rather similar scheme is employed in the Russian magnet, also of the picture-frame type; but the chamber in which the poles find themselves need be maintained only a t a rough vacuum in order to prevent collapse of an internal very thin-walled high-vacuum chamber 6 feet wide and 1.3 feet high. Somewhat the same method will he used in the Argonne 12-Bev proton synchrotron now being built; the magnet laminations, with an interior lining

FREQUENCY RANGE:

165

of plastic, will form a rough vacuum tank, relieving the pressure on a thin stainless steel high-vacuum chamber with internal dimensions 32.5 x 5.25 inches. The 1-Rev Birmingham synclirotron enlploys a porcelain chanlber of 60 sections, gask(>tedtogether. It is of oval cross section, 10 cm high and 35 cm wide inside, with 2.8-cm mi11 thickness At Delft, a thin stainless steel skin is clamped over elliptical truss punchings; this chamber fits in the 30 x 10 cm gap between thc poles. The synclirotron a t Saclay cniploys a glass fiber and epoxy resin cllamber reinforced with stainless s t e d ribs and lined with thin stainless steel sheeting. 9-4. Frequency Range

The scheduling of the frequency of the oscillator in a proton synchrotron is considerably more complicated than in a synchroeyclotron. The magnet has such a large inductance that the rate of rise of field is not subject to detailed control, and the ficld, therefore, must he taken as the independent variable, changing a t a rate which may not he unifornl and which may vary from pulse to pulse. I t is uneconomical to make the vacuum chamber much wider than is required to contain the radial betatron and synelhrotron oscillations, so the equilibrium orbit should be constrained always to lie close to the mid-line of the chamber. This means that tllc synchronous energy is that demanded by the changing field. But the synclirorious energy clctcrinines the synchronous frequency. Hence the oscillator's frequency and the synchronous energy must both follow the magnetic ficld. For reasons largely of convenience in mechanical design and experimental use, most proton synchrotrons arc. h i l t in "race track" form, the magnet being divided, typie:tlly, into four quadrants which are :separated by field-free sections of \acuunl chamber. The lengtli s of each of these N straight sections must be added to the length of curved path, L'rr, when calculating t h e radian frequency of tlle synchronous ions.

(Here Eq. 1-14 h:th h e n used to c1iinin:~te P. I T l k is also the required frequency given :is a function of the energy. T l ~ eexpression is useful in tlctclrmining the frcqucncies associ:~tcd with the f i n d and the injection energies. It will bc. rwnlled t h a t a finite starting velocity is required to produce a nloirientuin consistent with the synchrotron's radius and the inininlurn guidc field that can X)e produced with adequate precision. Writing the equation above in the form w = constant X (1 - l / y 2 ) % , where = E/Eo, we see that w is more sensitivc to when is small th:ln when i t is large; for a machine of given ratliui the final f r q u c n c y dcpcnds very little on the final energy, provided it 1,e 1 or 2 Bev or nlore, wliereas the initial frequency drops rapidly

166

SYNCHROTRONS

with lowered injection energy. A high injection energy therefore markedly decreases the required band width of the oscillator, the ratio of the extreme frequencies being roughly 11 to 1 a t 3 Mev but only 3 to 1 a t 50 Mev. It is necessary to know, as described above, how the frequency will vary as the magnetic field changes. To find this, we use the expression derived when considering the necessary size of the magnet, Eq. (1-35) : whence When these two expressions are used in the above equation for

w

i t follows that

If it could be guaranteed that B would change with time in a known and reproducible manner, then appropriately shaped capacitor plates on a rotating shaft driven by a clock could adjust the oscillator to such a predetermined schedule of frequency change. This is equivalent to saying that if the speed of an automobile rigorously followed some known pattern as time advanced, an independent clock connected to the steering gear could be relied on to drive the vehicle along the center of a tortuous but well-surveyed road, without reference to any landmarks en route. For a wide variety of practical reasons, the field does not follow any predetermined schedule with sufficient accuracy to make such a tracking scheme successful. Landmarks must be observed, either continuously or from time to time. Various ingenious schemes have been used. For example, a coil of wire in the guide field develops a voltage proportional to dB/& and an electronic integrator converts this to B; this information goes to an electric analog computer which solves the above equation and sends the resulting value of w to the frequency-controlling circuit. Sometinies the Hall effect is used to obtain B directly. Alternatively, peaking strips are used; these are needles of a magnetic alloy which produce a signal voltage when the rising guide field overpowers a known fixed biasing field applied to the needle. A set of such strips will give a series of signals as the field rises through successive known values and thereby permits corrections to be applied to the frequency of the oscillator, which runs by dead reckoning in the intervals between check points. In a similar manner, nuclear magnetic resonance equipment can be used to monitor the field a t some chosen set of values. Such control of the oscillator frequency is particularly important right after injection. For a constant value of dB/dt, w rises very rapidly a t first, then levels off to a modest rate of rise as the ions' speed asymptotically approaches the limiting velocity of light. The frequency must be controlled to within O . O 1 ~ l or better during the initial moments. After a number of synchrotron phase

ACCELERATION BY CAVITIES

167

oscillations have occurred, thc original load of ions lwcornes separated into tlrose that :\rv drivcn to thc wnllh and those that :we acrq)tct-l in phasc-stahlc orhits. The latter 1)c.rouit~bunchtatl into groups wparatcd in aziniuth and hcncr are capable of inducing w vo1t:tgc pulse in p:rsc,ing tlirough an insulated electrode (for cxsrnple, a cylindrical or rectangular \)ox) supported in a straight section. Such a pulse announces the time of pmsage of a bunch and can be used either to generate the frequency directly or else to correct it. The energy that must bc gained a t every turn is easily found. In general Mu = qBr, so with r constant, a steady rise of R requires a steady rise of Mu.This does not actually occur in synchrotrons, where acceleration is produced a t a finite number of stations; so not cven the best behaved particlcs follow a truly circular path. Nevertheless with very little error it may be considered that M u does rise continually and thc time rate of change of inonientum may be equated to a, hypoth~ticaltangential force acting over tlic entire path. Thus F = d ( M v ) / d t = q d R / d t . Multiplying by the circunlfercnce C gives the work done or the energy gained per turn, .hE. Hence

In most large synchrotrons the full change in B occurs in about 1 sccond and AE has a value ranging from 1 kev to 100 kev per turn, depending on machine size. These rather modest figures emphasize t,he fact, that it is the difficulty or expense of raising the magnetic field quickly which determines the rate a t which energy is gained, rather than problems of generating a large accelerating voltage. If dB/dt is not constant, A E must be tailored to fit, within rather close limits. Information to effect this is sometimes gained from sensing electrodes, somewhat similar to those which indicate the time of passage of a bunch of ions. I n the present instance the electrodes must report the radial position of a bunch. If i t lies inside the centerline, more energy must be delivered; if outside, less. Servo systems make the correction by changing the frequency or the amplitude of the accelerating voltage. 9-5. Acceleration by Cavities

A coaxial transmission line, one-quarter wavelength long, shorted a t one end and open a t the other, develops a large voltage across the gap a t the open cnd when the line is driven a t its resonant frequency, as is shown in Fig. 9-2. A somewhat similar unit, with both conductors having holes in their ends so as to form part of the vacuum chanlber, is often used as the accelerating device in an electron synchrotron. An appropriate length is cut out of the glass or ceramic doughnut-shaped vacuum chamber, and the inner and outer surfaces are coated with a metallic film, except for a small gap (across which the accelerating voltage appears) in the inner coating at one end (see Fig. 9-3). A driving loop is inserted, perhaps in the form of a coaxial

SYNCHROTRONS

168

cable of which the outer conductor is connected to the outer coating of the cavity, while the inner conductor passes through the insulating wall and contacts the inner coating. Because of the dielectric constant of the glass or ceramic, the physical length of the unit is less than h/4. T o reduce eddy currents, longitudinal scratches are made in the coating, and the remaining

u

(7

the inner and outer conductors of a h/4 coaxial resonator. The lines of electric field become longitudinal at the "open" end.

strips of conductor are connected together only a t the gap. The doughnut is then reassembled, the joints being made vacuum tight with gaskets or cement. A somewhat similar accelerating cavity is often used in a proton synchrotron, though located in a straight section. It takes the form of two opposed coaxial

-

F P particles

P

--+

Fig. 9-3. Coaxial cavity used in electron synchrotrons.

h / 4 lines, with the voltages out of phase in order t o produce a large potential

difference across the gap which is bridged by a vacuum-tight insulator as shown in Fig. 9-4. Rings or rectangular frames of ferrite surrounding the inner conductor add to the inductance of the cavity and so reduce its physical length. The system is tuned by altering a direct current flowing in a bias loop which links the ferrite, since this material enjoys the property of changing its permeability when a steady magnetic field, in which i t is immersed, is altered. Sometimes the cavity is untuned, as in the Cosmotron. I n this case conductors from the variable-frequency oscillator-amplifier are made to link the ferrite rings, which then serve as the core of a transformer, the stream of protons acting as the secondary circuit. 9-6. Acceleration by Drift Tubes

An alternative acceleration scheme for a synchrotron is the use of a drift tube fitting snugly inside the vacuum chamber. If the tube is 180" long in azimuth and is driven a t the particle's frequency, the action is like that of a

a4CCELEKATION BY DRIFT '1'UBES

Copper- lined steel cavity

Cooling air

particles

-

~HH

-

r r

RF driver

Fig. 9-4. Acct:leration cavity tuncd with ferrite and the voltage distribution within it. cyclotron with a single dcc. [oris are accelerated t ~ ytlie fringing clcctric field t~ctweentlie ends of the tube and ground, the force being attractive as tlic ions enter tho tube a n d repulsive as they 1e:ive it, since the clial-gc on tllc tube changes sign while the partirles arcbwitliin it. T h e frequency is controlled hy resonating the tube's fixed c:ipacitance with a variahle inductance, which often is mack with a ferrite core supplietl with a variablt. bias field. Tlir Birmingham synchrotron operatths in this manner, t h e drift tube being formed by a metallic deposit on the wall of the ceramic vacuum chamber. T h c tube is 102" long. T h e Bevatron employs a drift tube with a length of 10"; in this case t h e tube is supported in a straight scction. It is interesting to consider tlie situation when such short drift tubes are used. As a preliminary, we assume that the azimuthal extent of tlie electric

170

SYNCHROTRONS

field between the ends of the tube and ground is negligible, so that transit-time effects in crossing the "gap" may be neglected. It has been seen earlier that a weak-focusing machine exhibits phase stability if energy increments are received a t phases when the accelerative field is decreasing, so that an ion arriving earlier than the synchronous particle receives more than the synchronous energy gain AE,. When the drift tube subtends 180°, the phase a t entry can lie anywhere between C$in min = a/2 and $I, ,, = a , and therefore ,,, = 2a, the phase a t exit lies correspondingly between C$,ut ,, = 3a/2 and as shown in Fig. 9-5a. The energy gain, both a t entry and exit, occurs when

Fig. 9-5. The range of entry and exit phase angles for a drift tube of length (a) 180" and (b) 135". the accelerative force is decreasing, for the potential of the electrode changes sign while the particle is within it, and phase stability is produced. If the dee or drift tube is >90° but <180° in extent, both energy increments can still occur when the accelerative field is decreasing, but the possible values of the entry and exit phase angles are more restricted. Fig. 9-5b is illustrative of the situation for a tube 135" long. But if the tube is <90° long and if the voltage changes sign while the particle is inside it, then one acceleration occurs when the field is falling and the other when it rises, with the result that an early ion does not receive more than the synchronous energy increment and therefore phase stability is lacking. On the other hand, if the particle is not inside the tube a t a voltage null, phase stability can be attained, as is shown in Fig. 9-6, where repre-

Fig. 9-6. For a drift tube <90° long in a weak-focusing proton synchrotron, the voltages phases at entry and exit must lie in the first quadrant.

are indicated. I t sentatire positions of thc entry and exit 1)liascs +,, find is clear that :I net gain of energy occurs in the first quadrant (0 to 7~/2),since with the tuhc positively chargcd, a po\itivc ion l o s ~ scncrgy when it cntors a t +,,, but gain$ a greater amount whcn it Icavc1s :it + , , r ,. A s ~ n ~ i l anet r gain occurs in the. fourth q w d r a n t , when thc t u h i\ n e g a t i ~c and tlic energy gain a t cntry exccctis t l ~ closs a t exit. In 111~second and third qu:idrants the next energy change is a loss. Kom suppose the nct gain, :it the indicated angles +,,, and +,,,,, in tlic first qut~drunt.is the requircd v d u e hE,. Then it is clcar that phase stability will exist, since an early ion, whicli enters the tube a t +',, and leave. a t +',,,+ ,, gains energy in excess of AE,. It will therefore take morc than n ptbriod of the osci1l:~torto ninke a turn (if the machine is weakfocusing) and so will next traverse tlie tube a t phase angles closer to 1 and On the other hand, if operation is ntteniptcd with the synchronous phnhc angIes lying in the fourth quadrant, it is easy to see t h t an early ion will gain less th:m AE, and hence will retreat from thcse angles rather than oscillate ahout them. We conclutic~that I)aswgc>through a drift tuhe less than 90" long in tz n-eak-focusing positivc-ion nccelcrator must occur when the tul)els potential is positive and rising. Thc nct gain of energy in passing; through a drift tube is

+,,

A E = qTr,,,(sin4,,, - sin +,,,,),

(9-5) wherc T',, is tlie peak potential of the tuhe. But the angular length 6 of tlie tuhc is 6 = ,,,,+ - ,,, SO it follom-s that

+

+

AE

=

yV,,[sin

+in

- sin (0

+ +,,)I.

(9-6)

This approachw zero as 6 goes to zero and reaclies L: maximum of 2gT-,, sin +,, when 6 beconic~s180°, as in the usual cyclotron. A short tube is therefore inefficient in the sense t h a t a large peak potential must 1)e generated to effect a s~iiallenergy gain. On the other hand, the inechanical structure and the capacitance are small, so the system has certain important advantages for engineering re:tsons. 9-7 Frequency of Synchrotron Oscillations

It lias heen hliow~in Eq. (6-45) that tlic frequency of synchrotron oscillation:: of sn~nll:unplitude in general is given 1)y

where I', originally defined in Eq. (6-20) a s ( l / a -- 1/y2)//??,has been shown in FJq. (8-10) to be expressible in the alt ernative form r = 1 (1 - a)/(ar P ' ) . For a race-track synchrotron the momentun1 compaction was given as = (1 - n ) ( l s / m ) in Eq. (6-6); so for such a machine we have

+

+

172

SYNCHROTRONS

For a numerical example, consider the Cosmotron, where the length of a magnet is m = 47.1 feet, the length of a straight section is s = 10 feet, and the field index has the nominal value n = 0.6. The energy gained per turn is AE = NqV, sin 9, = 1 kev; and since there is one accelerating cavity which operates a t the same frequency as that of rotation, we have N = 1 and h = 1. If we take 4, = 150°, then qV, = 2 kev. At the injection energy of 3.6 hlev, we find P2 = 0.0076 SO = 140 and hence w , = 15512. ,4t the peak energy of 3 Bev, where we have ,B2 = 0.943, which makes r become 2.13, the result is that w , = 1265Q. Thus the synchrotron oscillation frequency becomes less and less, the particles initially taking 155 turns to complete one phase oscillation and ultimately requiring 1265 turns. From the form of Eq. (9-7) it can be seen that the existence of straight sections raises r and hence increases the synchrotron frequency a. 9-8. Betatron Action in Synchrotrons

In discussing phase stability, the effcct of the time-changing flux on the energy of the ions was neglected. The justification will now be given. The emf generated around the orbit depends on the time rate of change of flux inside the orbit. If C-shaped magnets are used, with the vertical leg inside the ions' paths, (Fig. 9-7a), all the flux lines in the gap that lie inside the orbit are

Fig. 9-7. Flux lines and orbit position for different magnet configurations. The proton is receding and the center of the machine is to the left in all cases. counterbalanced by their continuation, in the opposite direction, in the vertical leg. The only uncompensated lines inside the orbit are those which, when in the gap, lie outside the orbit, and these are so oriented as to decelerate the particles when the flux increases. If the vertical leg is outside the orbit, as in Fig. 9-7b, only the lines in the gap which lie closer to the machine center than the orbit produce an emf, and this is accelerating. Even when the ions are misplaced from the ideal central path, so they receive the maximum effect, the energy gained or lost in a typical machine is of the order of a few dozen electron volts, compared to the hundreds or thousands received from the oscillator. T i t h magnets of picture-frame sections, Fig. 9-7c, a particle traveling down the center of the chamber has almost zero net flux inside its orbit, the only contribution being due to a lack of symmetry in the field if the design produces a gradient. Particles near the inner wall receive the maximum deceleration,

IS,JE(:TION OF PARTICLES INTO S Y N C H R O T I I O S S

173

because of tlie flux in the inside Icg, wIlile thosc a t tlie outer boundary get the iilost acceleration. Oncc again, the effect is small in comparison with t h e oscillator's voltage. 9-9. Influence of Injection Energy

Tlicre arc a nunll)er of reasom why a high injection cncrgy is tlesirwhle. First, i t iniplics a h~gliervnlue of field sn the synclirotron magnet a t the m o n ~ e n tof injection. This means t h a t residual fieltls and fields due t o eddy currents in tlic poles, in the vacuum c1iunlt)er and in tlie exciting c~onductors,a r e relatively snl:rller arid Icss tllsturlnng. Second, tlic required range of frequency over which tlie oscillator must operate is reduced. This range is determined almost solely by tho injcction energy, for the particle velocity ( a n d hence the frequency) :$pin-o:~chesa constant vnlue as the energy rises. Third, thc rate of r h m g c of f r c q w n r y dccre:ws with rising mergy, so t h a t a large injection cncrgy soniewliat cases the prohl(m of making the oscillator frequency track the magnetic field. Fourth, large initial energy retluccs t h e loss of particles due to scattering hy the gas. This loss is most serious when the ions are traveling slowly. And f i n d l y tlw troul)lesoine effects of space charge are di~ninidied T h e mutual rrpulsion of the particles in t h e beam tends to expand it, thereby counteracting to sonw extent t h r focusmg forces of t h e magnetic field; but this effect appro:dles zero w ~ t hrising energy. (This was discussrd in 5-6.) Against tlicw arguments must be sct a nurnbcr of practical considerations. Cockcroft-\V:xlton generators ul) to about 1 Rlev a r e available commercially, a s a r e Van dc Grnaff accclcrators to about 5 hlev, whereas t h e very few proton lincw accelerators t h a t now exist have been individually designed a n d h i l t essentially by hand a t a higher cost, per volt. And i t is only recently t h a t oscillator ant1 un1)lificr tubes liavc colncl into I~eingof sufficient power rating t h a t only a few arc required to drivc a linac, as opposed t o the dozens of wi:allcr tubcs ~ w p i r e da fen- years ago. 9-10. Injection of Particles into Synchrotrons

Injecting particles into a synchrotron 1s not as easy :IS i t might seem a t first sight. It is not possible to shoot them in tangentially, a s in Fig. 9-8, for a t

the 1)oint of t;rnpcwcay t l ~ cfivld nus st I)(, simultuncously zero a n d finite, w h ~ l e tlic radius of cuivatule must be both finite and infinite.

174

SYNCHROTRONS

Nor is it possible to direct the ions diagonally across the vacuum chamber as in Fig. 9-9; whatever the value of magnetic field, there is an ideal circular orbit appropriate for the ions of the injection energy, and radial betatron oscillations will take place about it. Since the particles come from outside the

\

Fig. 9-9. Diagonal injection also will not work.

chamber, they behave as though their entering path were part of an oscillation with amplitude of such size as to lie outside the chamber and the next outward swing will drive them to the wall. Nevertheless there are successful nietliods of injection, but as the detailed analysis is extremely complicated we will give but a brief outline of how it can be accomplished. One obvious method, indicated in Fig. 9-10, is to mount electrostatic in-

Fig. 9-10. A satisfactory arrangement for single-turn injection.

Fig. 9-11. An impractical injection scheme.

flector plates on the inner and outer walls of the vacuum chamber in a straight section and to introduce the particles froin the injector a t a slight angle. An electric field of proper strength will bend the orbit and launch the ions a t the correct radius to follow a n equilibrium orbit. There are a t least two objections to this sclienie for weak-focusing synchrotrons. First, the electric field must be turned off before the first ions make a complete revolution; otherwise they will be deflected outward and lost; so the technique limits the number of particles injected to those contained in one circumference (or somewhat less, since it takes time to discharge the deflecting plates). Second, with the large radial width of chamber in weak-focusing accelerators, the required voltage on the plates bcco~ncsimpractically high, unless the injection energy is low, and there are overriding reasons why it should be great, as already seen. I n strong-focusing synchrotrons, on the other hand, the chamber width is much reduced and thc necessary voltage is within reason; furthermore,

particles.

3Iultiple-turn injection into weak-focl~singsynchrotrons is possible undcr w n c . circun~stanccs.This leads to :t 1:1rger numt)er ol; :tccclcratetl particles 1 ~ ' 1)ulsc. A4swill appcar a littlc later, tliib 1, the caw when tlie nlomcntuln coniluc(Fig. 9-1 1 ) R pair of c u r v ~ d11letiLl inflector platea closdy tion ih ,1111111. 1ru:~gin~ sl)acctl so t l ~ a at rcasonablc voltngc hetween them will launch the ions on their proper equilibrium orbit if the exit opening between i,he plates is a t the proper radius for the magnetic ficld then exist~ng.(For s~mplicity,we assume tht' inflector t o bc. in a straight section, but this is not necessary, if magnetic shields are sup1,lied.) Unfortunately, after one turn the ions will collide with the back sidc of the inner electrode. This difficulty can be avoided by injecting a t a moment when the equilibrium orbit is sn1:tller in radius than the radial position of thc moutll of the inflector. When the particles leave this aperture they are then not "on course" but a t the extreme outward displacement of :t r:xdi:tl betatron oscil1:~tionwhich will develop inlmed~ately.By properly choosing the value of v, (the number of radial betatron oscillatioiis per turn) i t can be arranged that a t the end of one turn the owillation is a t the extreme amplitude of an inward swing, in this manner avoiding a collision as is

Fig. 9-12. A practical method for rn~iltiturn injection.

indicated in Fig. 9-12. The most obvious way to bring this ahout is to design the machine so that v, = t . LJnfortunately, after two turns the betatron oscillation will be a t its maximum value outward, so what was saved after one turn mill he lost after two. If we make v, = i,it will take three revolutions for the ions to conlpl~tetwo radial oscillations, at which monicnt they will strike the inflector. Such an enforced delay in the moment of theoretical collision with the back sidc of the inflector is of considcrable value, for the guiding magnetic field is rising a11 the time, and if the accelerating voltage has not yet been turned on, the radius of the equilibrium orbit is (,ontinually shrinking, and the disaster may be avoided, after all. (Much higher values of v, are inconsistent with thc index n heing in the neighborhood of 0.6.) So far, the ions I1:~r.c been assumed to be injected exactly parallel t o (but displaced from) the equilibrium orbit. But not a11 the injectcd ions fit this description. .Just as i t is practir:dly impossible to obtain a parallel beam of light from s searchlight, so is i t practically impossible to get a parallel beam of ions to coin(. out of the inflector; they will emerge with some divergence ( a spread in angle). What happens is best seen if we imaginatively "unroll" the orbits out into a straight line, as in Fig. 9-13 (which is grossly exaggerated in scale). Here the solid line represents tlie course of the ions launched parallel to the equilibrium orbit; the dotted linc shows those directed a t the extreme

SYNCHROTRONS

176

\equilibrium orbit

I

to center of machine

Fig. 9-13. Paths of ions in an "unrolled synchrotron, seen from above. inward angle of the divergent beam, while the dashed line indicates those emitted a t the extreme outward angle. The figure is drawn as though all the ions originated from a point; this point "source" then forms point "images" of itself spaced around the machine a t distances equal to half a radial betatron wavelength. Actually, the ions emerge from an area (or from a line source, when seen from above) located a t the nlouth of the inflector (or somewhere further back towards the injector, depending on other conditions), and from each point of this finite source the particles come out with angular divergence. Three representative points on the source are indicated in Fig. 9-14, and from each

Fig. 9-14. The divergent paths of ions from all points on the surface of a "source."

Fig. 9-15. The formation of an image by ions emitted from a finite source.

point the mean beam and the two limiting beams are indicated. Therefore, each of the "images" formed later, by the crossing of the beams, Fig. 9-15, really has a radial width about equal to that of the source. The entire beam becomes alternately diffuse and concentrated, "images" of the "source" occurring a t the azimuths of greatest density, one half wavelength apart (Fig. 9-16). If v, = the inflector appears a t locations A, and in a static field all ions would be lost after two revolutions; while if v, = $ the trouble comes a t locations B and loss would occur after three turns. Since the images have a finite radial width, any rise a t all in field will permit some of the ions on the inner edge of the image to escape collision; but unless the power supplied to the magnet is increased inordinately, this may be a disappointingly low fraction,

+

I K J li:CTI( I S OF P A R T I C L E S IKTO S'YXCHZZOTKONS

,' equ~l~brturn . . orbit

177

to center of rnochine

Fig. 9-16. Thc envelope of the beam. If v , = 4, the circumference of the machinc is the distance AA; if v , =- z, it is the distance B E

because of thc smallness of tlie cliangc t h a t r a n be made in t h e field in tlic short time t h a t has elapsed. Help m a y then be obtained by making use of energy transfer from the radial to the vcrtical mode of oscillation, so t h a t tlic outward swings arc reduced in magnitutle, with thc result t h a t more ions miss the inflector. B y the time the m c r g y f o w s back t o the radial mode, a pronounced shrinkage of the c c ~ u i l ~ b r i u orbits m will l ~ a v eoccurred. Deliberately induccd and prograined "errors" in the riiagnetic field m a y be eniploycd wit11 benefit to effcct such energy transfer. Experiri~entalfigures are scanty, but i t appears t h a t losses t o the inflector in one synchrotron range from 10% to 70y, depending on adjustments. Injection is continued without applicrttion of the accelerating voltage until tlle cquilihriurn orbit of t h e earliest particles has moved t o the center of the vacuum charnl)er, a t which time acceleration is started. T h e radial oscillations about this prrrticular orhit are miall, for they were engendered when the orhit was closc to tlie inflector. T l ~ clast particlles injected have radial oscillations of amplitutlr equal t o t h e distance bctwcen the inflector irioutli and the chanil)erls centcrline. O n the whole, when acceleration st:trts t h e ions present have radial oscillation amplitudes ranging from almost zero ~ , oabout half tlie chamber width. Vertical dirergence of the injected beam produces vertical motion, so the cnvelopc of tlie particlc trajectories fills the entire charnbcr in m~idtli, liciglit, and circurnfcrence, ions having been injected for perhaps something over a d o e m turns. Phase oscillations s t a r t when the R F ro1t:~ge is turned on, with conscqwnt loss of particles, the survivors bccorning grouped in arcs with empty spaces ahead and behind. T h c number of such groups depends on tlic numhcr of accelerating cavities and on the integral ratio h of the oscillator's frequency to t h a t of revolution. A variant of tlie above method of niiszing t h c inflcctor (careful control of t h e amplitude of radi:d oscillation) will be employed in the Argonne zero-gradient proton synchrotron. By proper choice of the relative lengths of straight sections a n d magnets and of the slant angles of tlic nlagnet ends, the frcquencies of tlie betatron oscillations will be made closc t o v, = p and v, = 2. As usual, injec-

178

SYNCHROTRONS

tion will take place outside the equilibrium orbit so as to induce radial motion, but in addition the inflector will be above the median plane in order to create axial oscillation. After four turns the beam will return to the radial position of the inflector mouth but will lie as much below the mid-plane as it lay above initially, so that complete clearance will be attained. Only after 8 turns have elapsed in a static guide field would the image be formed directly a t the inflector plates but as the field is actually rising, a high fraction of all the protons in 110 turns is expected to be launched successfully. Some slight deviation from the exact values of v, and v, quoted above may be necessary to avoid the resonance a t v, - 2v, = - 1, if this should turn out to be harmful. Under some circumstances, it may be desirable to eliminate deliberately, and a t an early stage, those ions which will be lost in the long run. This can be done by inserting a chopper shortly after the ion source, under adequate time control; this might consist of an electrostatic or magnetic system which sweeps the beam periodically across a hole in a barrier. Such a procedure (which adds complexity to the entire system) may be desirable if the unwanted ions produce too much radioactivity in the synchrotron walls, or if early bunching in the synchrotron (such as was described in 6-11) is necessary so that the frcquency of the accelerating voltage may be controlled by linking it with the revolution frequency directly. This can be accomplished on a bunched beam since the passage of each bunch will induce a voltage pulse on an insulated electrode placed close to the inside of the vacuum chamber wall, as mentioned in $ 9-4. I t was stated earlier that multiple-turn injection is best favored by a machine with small momentum compaction, a. Since the momentum is given by p = qBR, where R is the radius of a circle with perimeter e'qual to the orbit length, then d p l p = dB/B dR/R, so that Eq. (6-2), given as a! = Rdp/(pdR) can be expressed as a = 1 RdB/(BdR). From this it follows that

+ +

Therefore, dR/dB is larger when a is smaller and more ions will miss the back side of the inflector for a given change of field. 9-11. Synchrotron Targets

I n machines with spiral orbits, a target fixed in position may be cinployed, for it intercepts only ions of full energy. This convenient state of affairs does not hold with synchrotrons in which the radius of curvature is fixed for all energies. Several ways out of the difficulty have been employed. It will be remembered that the oscillating ions fill the entire cross section of the chamber just after injection, but as acceleration proceeds the amplitudes die down; in the final stages the beam occupies but a fraction of the available space. Since in large proton synchrotrons acceleration takes place over an interval of from

EXTERNAL BEAMS

179

1 to 3 seconds, there is opportunity to Insert target,^ mechanically, either by fast-moving pibtons or by magnetically operated structures mounted on a li1ngc~-fl7p-~p targets. A l t e r n a t i ~ely, targets nlountetl on a n outer wall ~ 1 1 1be struck by full-cmrgy ions when the magnetic field begins t o decay, or targets on either wall will be bombarded if the energy gain is controlled so a s to be greater or lesh t h a n the synchronous value. \\'it11 open-sided C-s1lal)ed ~nagiieta,target+ can b~ nlountcd a t a n y azimuth. Products of dibintegration t h a t are elect,rically ncuirnl will leave the magnct a t a tangrnt, while negative n ~ c s o n swill be deflcc+etl outward, leaving the nagn net a t pwitions and a t angles t h a t dcycn(1 on their moinenta. If the targct is judiciously located u p t r c a i n from a straight scction, positively charged particles of appropriate nlornentum \fill leave the machine a t t h e field-free region, heading for the area inside the inagnct. T h c position of targets is more restricted with picture-frame n ~ a g n c t s .T a r gets ordinarily must either 1)c in straight swtions or be introduced slightly upstream by some mechanic:d device if the bending action of the guide field i\ to a c t on the nucle;tr debrib. However, if t l ~ ccxciting co~itluctoraof the nlagnct do not lie on the. mct-lian plane, c1i:~nnels 111:iy h horcd tliro\~gllthe vertical parts of the yokc for the insertion of targets and for the passage of tlie products of disintegration. 9-12. External Beams

Multibillion volt proton b e i ~ m sare much too "stiff" for electrostatic dcflertion tecl~niqucs.Tllcoretical studies havc indicated t h a t succcss should be attained with thv regenerative mctliod of ejection, a s was described briefly in 8-6, b u t i t has not yet been attcw~pted I n the usual scnec, inagnctic tloflcction is almost out of the question, since the fieltl m u ~ be t built ul) extremely rapidly if ions still untlcrgoing accc~lcrationarc not to lw :rfl'cctctl ndvcrbcly (though huch a technicluc has b c m applicd to ohtnin h u r ~ t sof :1 few niicrosecon(lsl tluration, by discharging a large hank of c:il)aritors through a n air-cored inagnet, con~posedof a few turns of roppt3r bus bar losated in n straight section; the inductance must hc kept low to o1)tain :t rapid rise of field). At Birmingham, a "flat-topped" 1)ulsc of 4 milliseconcls is produced in :t magnet which is pneumatically inserted a t the corrsct time. T h e Picciorii ejection schsme is x practical alternative already in use a t the Cosinotron. It is as fol10~1-s.\I'llen the full energy has been attained in the synchrotron, t l ~ caccelerating voltage is turned off, but t h e magnetic ficld is permitted t o continue to rise; this makes the ions spiral inward towards tlrc inner boundary of the v a c u u n ~clrariiber. T h e inward swing of their radial betatron oscillation makes them collidc with a very thin targct (the lip t a r g e t ) which has been plunged into position after t h e wider oscillations existing a t injection tiinc have rlaniprd down (scc Fig. 9-17). Passagc through this targct on the inward bwing of a radial oscillation of alnplitutle A l does not noticeably

s

180

SYNCHROTRONS

deflect an ion, but it does remove a little energy, so that the particle a t once finds itself a t the peak inward swing of smaller amplitude A2 centered about a new equilibrium orbit of slightly reduced size. Continued traversals of the thin target continue, with the amplitude of oscillations dying down and the orbit moving farther in, until finally when the amplitude is down to A,, a thick portion of the target is encountered (the jump target). This removes a lot of energy in one passage (several Me\.)--so much that the new equilibrium orbit lies well beyond the inward swing of the last oscillation. Hence, on emerging from the thick target the ion finds itself a t the peak outward amplitude

Fig. 9-17. The changes in radial oscillation amplitudes on passage of the particles through the Piccioni target.

( A , + 1 ) of a large betatron oscillation. One half betatron wavelength further around the machine the ion is a t the extreme inward swing of this large motion and is now closer to the machine center than ever before-i.e., an inch or two inside even the inner boundary of the wide bean1 t h a t existed a t injection time. At this azimuth the ions traverse the gap of a so-called Piccioni magnct whose field is antiparallel to that in the main magnet; the Piccioni magnet therefore puts a reverse curvature into the orbit for a short distance, so that the particle traverses the next main magnet sector diagonally. At some point further around it reaches the outer wall of the vacuum chamber and escapes through an appropriate thin window. (This last remark applies to machines with C magnets, where the chamber's outer wall is unobstructed; in machines with picture-frame cross section, escape of the ions occurs only a t a straight section.) It may be necessary to employ a second magnet to deflect the ion farther away from the synchrotron and out t o the experimental area (see Fig. 9-18). The purpose of the jump target is to allow the Piccioni magnet to be positioned where it can be turned on (or shoved into place) without its increasing field (or fringing field) causing intolerable distortion to the circulating ions during acceleration time. The extra inch or two by which the ions are driven inwards enables the Piccioni magnct to be displaced from the accelerating region just barely enough to permit this to happen with some degree of success.

ET,FXyTR0 N SYNCHROTR O S S

181

Two complete sets of Piccioni ejection components will be employed a t the Argonnc ZGS, so that external lwmis of protons will lw nvwilal)le in two arcas. By thc u w of bwitcliing niagncth, tllcsc rvgions can 1t)c increased to four. 9-13. Shielding

The magnit,ude of the shielding prol)lcni 1iat-l not bccn appreciated when the first proton syncllrotrons were built and protection has been obtained by putting more and more blocks of concrete around tlie machines as improved operating efficiencay has raised the output of ions per pulse. This has reduced the

Fig. 9-18. Ejection by the Piccioni magnet. deflection

area available for experimentation with the high-speed particles and thereby necessitated much crowding of cquil)n~cntand loss of niachinc~time while apparatus for one experiment is rcmovcd and that for another is installed. The designers of the more recent macliincs have endeavored to profit from this experience by placing the acceltmtors conipletely underground or burying them under mounds of dirt, and a nlore substantial fraction of the budget now goes to providing large experimental areas wherein several arrays of detecting and analyzing equipment may be in process of installation or removal while a t the same time observations are being made with another set-up. The capital investment of big accelerators is so large that two-shift operation is customary and round-the-clock use is contemplated for some of tlie machines now under construction, with a minimal weekly shutdown scheduled for routine maintenance. 9-14. Electron Synchrotrons

The synchrotrons described thus far have all been devoted to the acceleration of protons, but weak-focusing ~ynchrot~rons for the production of higlienergy electrons are also in use, having predated those for protons by several years. Since electrons are much lighter, these machines are comparatively small, a 70-Mev affair having an orbit diameter of :t little less than 2 feet in a magnet weighing 8 tons with a field of 8.1 kilogauss. Several have been built in the 300- to 400-Mev range. N'ith magnets of nlodest size i t is possible t o

182

SYNCHROTRONS

form a resonating circuit with a bank of capacitors, and the rise of field repeats a t 6 to 60 times a second, depending on the installation. Acceleration is accomplished by means of an RF cavity, such as is described in $ 9-5. Since electron velocities closely approximate the velocity of light a t energies in excess of 1 or 2 Mev, the required band width of frequency modulation is small. For low-energy machines, a separate injector is often omitted and betatron action brings the electrons up to about 2 Mev, by means of a small central core which saturates and thereafter maintains a constant flux. The R F cavity is turned on just as betatron acceleration ceases. Two weak-focusing electron synchrotrons are now operating (1960) a t the 1 Bev level. The machine a t the California Institute of Technology was originally constructed as a nominally quarter-scale prototype of the Bevatron and has a four-sectored orbit with a radius of 12.33 feet. The field index is 0.6. Injection is a t 1 Mev from a pulse transformer. The guide field rises in 1 second from 12 gauss to 13.6 kilogauss, the energy being stored in a flywheel. The single accelerating cavity is frequency modulated between 37.6 and 40 Mc/sec, this range representing the fourth harmonic of the orbital frequency. Each pulse, occurring once a second, contains about 1010 electrons a t an energy close to 1.4 Bev. The weak-focusing electron synchrotron a t the National Institute of Nuclear Physics in Rome, Italy, is also four-sectored, but with an orbit radius of 14.2 feet. Injection is from a 3-Mev Van de Graaff machine. An energy of 1.2 Bev is reached a t a peak field of 11.1 kilogauss, the repetition rate being 20 per second. Capacitors are used to store the energy for the magnet. The accelerating voltage across two cavities is driven a t the fourth harmonic of the revolution frequency; that is, the R F runs between 42.6 and 43.7 Mc/sec. 9-15. Radiation by Electrons in Circular Accelerators

Classical electromagnetic theory still rules in macroscopic matters, and a charged particle rotating in an orbit undergoes acceleration even if its speed remains constant. It may therefore be expected to produce electromagnetic radiation a t the expense of its kinetic energy. Such radiation is observed with electron synchrotrons and betatrons, being emitted in a narrow cone directed forward, tangent to the path of the particles. Some of the energy lies in the visible spectrum and if the vacuum chamber is of quartz or glass, a brilliant source of light can be seen if one looks "upstream," tangentially to the orbit, from any azimuth. It is shown in several of the articles listed in the references a t the end of this book that the rate of emission of radiation varies with the fourth power of the particle's energy, inversely with the radius of curvature, and inversely with the fourth power of the rest mass. For the latter reason, although the energy loss is utterly negligible for proton machines, it is extremely important in circular electron accelerators. The energy lost by radiation must be made

EKERGY 1,OSS RY RADIATION

183

up by the accelerating mechanism before further gain in kinetic energy can occur: onc is, as it were, trying to fill a badly lcaking hucket. I n the HarvardMIT alternating-gradient machine (to bc described in 3 12-8), this loss will amount to 4.5 Mev per turn when the electrons an. close to the top energy of 6 Bev, even though advantage is taken of the l / r facxtor by using a large orbital radius of curvature (86 feet) and a correspondingly low peak field (8 kilogauss). Tlic sixteen pairs of accelcratirig cavities will consume 350 kilowatts of power, operating a t 475 Mc/sec ( t h ~360th harmonic of the rotation frequency). The generation and distribution of this power involves considerahlc ingenuity. Furthermore, the radiation damage to be suffered by any plastic con~ponentsof the vacuum chamber has bern a source of considerable worry. More important than either of these factors, in llmiting the ultimate energy of such a circular machine, is the recoil suffered by each electron as it emits a photon of radiation; this produces an additional component of radial oscillation, thereby demanding a wider vacuum chamber and a more expensive magnet and power supply. I t is therefore believed that 6 to 8 Bev is probably the econonlic limit for electron synchrotrons. If greater energy proves desirable, it will doubtless be reached with electron lintvw accelerators of fantastic length ( a mile or more; see 5 14-13), sincc such machines do not suffer from radiation losses. Review articles on proton synchrotrons are givcn on p. 354-355. Papers describing particular machincs will be found on pp. 355-358. Reviews and detailed papcrs on clectron synchrotrons are listed on pp. 358-359.

BETATRONS

10-1. Introduction

The magnetic field in a betatron resembles that of a cyclotron to the degree that it extends from the axis of the cylindrical poles all the way out to their periphery. But, as in a synchrotron, the field changes with time from zero to a maximum and back to zero again. Also as in a synchrotron, the electron projectiles rotate a t a nonlinally constant radius in a toroidal vacuum chainbcr near the edges of the poles. While the flux within the orbits is increasing, an electromotive force is developed around the path so that a tangential force is applied to the particles, thus accelerating them to high energy. (One may think of an ordinary transformer as operating in much the same manner, except that the electrons in the secondary circuit suffer so many atomic collisions that they acquire only a slow average velocity, and the path is deterrnincd by the boundaries of the copper wire rather than by the magnetic field.) 10-2. T h e Two-to-One Rule

The essential trick that makes a betatron work is the mechanism which forces the electrons always to have the proper energy so as to remain a t a fixed radius in a field which increases with time. The problem is similar to that encountered in a synchrotron, but the method of solution is quite different. If the magnetic flux within a circuit of radius r changes with time, the electromotive force developed around the circuit is d @ / d t . This may be expressed as the product of a circumferential electric field strength & and the circumference:

As a result, the force acting on a charge q is

and therefore the time rate of change of momentum is

This expression governs the acceleration process. 184

T o mnint:tin circulnr orhits, we must Irnrc, as in Eq. (1-21, if the rattills rcniains fixed then

Mv

= q R r ; so

The last two ecluations must be satisfied simultaneously if the machine is to work. This means t h a t

T h a t is, the rate of change of the. space-nvcragecl flux drnsity in the central iron core within the orhit must equal twice thc rate of change of the guide field a t the orbit. But "flux density" is just another way of saying "field," so the criterion for operation also can hc given :IS

dB, - = 2 - 1 1dn dt

dt

nhcrc '75, is the space-averaged ficltl strength in the core and B,, is the guiding field a t the o h i t . Both of Eqs. (10-51 and (10-6) express the well-known "2-to-1" rule of thc I)ctatron, which permits simultaneous acceleration ant1

Fig. 10-1. The relation of the guide field B,, and the space-averaged field B, in the core of a simple betatron. Injection occurs shortly after both fields pass through zero and final energy is reached about one quarter-cycle later. maintenance of the orbit a t a constant radius. The r111c also can be expressed without reference to time, for if wc integrate hctwccn initial and final values wc find (10-7) B, = 2 ( B , - B,,).

n,,

Thus the change in the space-arrragcd ficld in the core must equal twice the change in the ficid a t the orhit. If the initial r.:~lues happen to be zero (as is often the case, though not ncc.essarily1 we find

An electron gun, con~posedof a hot filament and a grid a t a positive potential so as to inject electrons with a finite initial velocity, is mounted close to

186

BETATRONS

the wall of the chamber. Injection occurs when the guide field is slightly abovc zero, at a value consistent with the injection energy. Hence acceleration takes place during almost a quarter cycle of the alternating current that drives the magnet, as is indicated by the heavy lines in Fig. 10-1, the final energy being reached a t the peak of the ficld. 10-3. Flux Change and Energy Gained The relation between the energy gained and the flux change in a betatron is easily found. The energy increase per turn is

hut also by Eq. (10-1)

Therefore on equating these two expressions we get

On canceling dt and then integrating, we obtain

Since electrons very quickly approach the velocity of light, the radian frequency of rotation w may be taken as constant without sensible error, so that there results A+ = K AE. (10-13) Therefore a chosen energy gain hE, produced by betatron action, requires a specific change in flux A@ within the orbits. 10-4. General Description Orbit stability is obtained just as in a weak-focusing synchrotron, the guide field being designed to maintain a constant value of field index, generally in the neighborhood of O.G. At first sight, betatron magnets look much like those of cyclotrons, but there are important differences. Since betatrons usually operate a t 30 or 60 cycles per second, often being resonated with a large capacitance, the whole structure is made of laminated iron sheets to minimize eddy currents. The doughnutshaped vacuum chamber surrounds the central core which, like a cyclotron, has an air gap, though its function is only to control the reluctance of this part of the magnetic circuit so that the 2-to-1 rule will be obeyed. The vacuum chamber is of glass, quartz, or ceramic, lightly silvered on the inside to prevent the accumulation of static charge from misdirected electrons. If this film is of low

T H E BIASED BEI'ATROIN

187

resistance, i t niust be interrupted a t one azimuth, otherwise the circumferential emf will generate a strong current in it, instead of accelerating the electrons. ll'ith such an interrupted heavy coating, :tcceleration takes place only when the particles are crossing the gap in the coating, the entire cnlf then bcing concentrated a t t h a t region. I n some m:whines several gaps are used to lessen the amplitude of the induced radial oscillation. Betatrons have been used extensively as x-ray generators. As the final energy is approached, the orbits are expanded or contracted by altering the flux in the central core so t h a t the electrons strike a target on the outer or inner wall. The change in flux is accomplislid hy placing in the core's air gap several small pieces of metal which saturate at, the dcsircd flux density, or clsc by pulsing currents through special windings a t the proper nioment. The basic relation of Eq. (10-8) can be written as @ j ( r r 2 )= 2B, and this shows t h a t for constant K g a reduction in @ is acconipnnietl by acceleration a t a snlaller radius, and vice versa. Consequently the electron beam, which is of finite cross section, is swept onto the target a t a rate depending on the abruptness of the flux perturbation. I n this way the x-ray output can bt. strctchcd to several hundred niicroseconds. I n some instruments the beam of electrons is brought out of the machine through a magnetic shielding channel, entry into this being brought about 11y pulsed magnetic perturbations. Sonietirnes these are so designed as to put the orbit into a condition of radial instability, in :t manner similar to tlie regenerative technique used in extracting the bean1 from a synchrocyclotron. The names of Slepian, Breit and Tuve, lVideroe, lJ7alton, Jassinsky, and Steenbeck are associated with the 18-year period during which partially surcessful attempts were made to accelerate electrons by the means embodied in the modern betatron. Little by little there were discovered the 2-to-1 rule, the necessity for orbit stability and of damping of the oscill:ttions, the shape of the magnetic field t h a t would bring about these ends, tile importance of gas scattering, and the technique of launching c.lectrons properly. The first fully successful betatron (producing electrons a t 2.2-&lev) was built by Kerst in 1940, and in 1941 there appeared tlie fundanlental studics on betatron oscillations, by Kerst and Serbcr, that p1:icetl orbit theory on a sound basis. The largest betatron yet built is the 300-Mev machine a t the University of Illinois. 10-5. The Biased Betatron

Because the average flux density in thc core inust be twice the flux dcnsity in the pole tips a t the guide field, the core will saturate before the pole tips do if the driving nlagnetomotive force is increased too much, so that the 2-to-1 relation becomes destroyed. Consequently the energy of a simple betatron of a given radius is limited by that flux density in the guide field a t which the douhlcd density in the core has not yet approached tlic saturation point. This illcans that the guide field ordinarily does not excced a few thousand gauss.

188

BETATRONS

A means of approximately doubling the energy for a given radius is supplied by the so-called biasing technique. In principle this consists in supplying independent control over both the core field E, and the guide field B,, in such a way that when the latter changes from zero to peak value the core field changes from -B, , ,,to +B, ,,, (instead of changing from zero to +B,),, , thus supplying twice as much change in flux through the core yet without raising thc flux density to the saturation point. In conformity with Eq. (10-13), i.e., A@ = KAE, this permits doubling the final energy of the electrons, which are contained in an orbit of the original size by the guide field being driven to times its former peak value. In the field-biased betatron, a steady positive biasing field a t the orbit is superimposed on the usual alternating guide field, so that the net guide field changes from zero (or slightly less) to peak value during the half cycle that the purely alternating field in the core changes from its negative to its positive peak. Flux biasing is an alternative, pulsed, technique wherein the core is given a permanent negative bias and is pulsed to an approximately equally large positive value during the time in which the guide field is driven from zero to its crest. A number of circuits have been devised following these general lines, with auxiliary windings being used around various portions of the betatron yoke and poles. I n some instances the air gap in the core is eliminated so that the stored energy is lowered and the cost and complexity of the power supply is reduced. I n general, biasing reduces the weight of iron for a given energy. Even so, the betatron is a t a disadvantage when compared with an electron synchrotron, for the large amount of iron needed in the core can be used to greater profit when distributed in the annular form of a synchrotron magnet of greater radius. Furthermore, radiation losses vary inversely as the radius, so a machine of larger diameter is to be preferred. Most important of all, as far as the attainment of very high energy is concerned, such losses can be made up in the synchrotron simply by delivering more power to the accelerating cavity, whereas the betatron requires a still larger core for such compensation. References to articles on betatrons, on the energy lost by radiation and on gas-scattering are given on pp. 359-360.

11-1. Introduction

The microtron, sometimes called an electron cyclotron, employs an ingenious idea due to V&sler to circumvent tlie loss of synchronization that occurs in a fixed-frequency cyclotron because of tlic relativistic increase of mass with energy. This cff'ect increases the period of revolution so that particles lag in pl~asc with respect to the accelerating voltage. In the microtron, i t is so arranged that the energy gained per turn is a constatit. The result is t h a t the time lag per turn is fixed a t a single value and this is set equal to an integral multiple of the period of the oscillator which runs a t constant frequency. The time required for successive orbits therefore increases by one or more complete periods of the oscillator and the particles always reach the gap a t the same phase of thc voltage, irrespective of their total energy. Although the necessary cncrgy gain is acquired only by those electrons which reach the gap a t a particular voltage phase, the principle of phase stability is operative, and electrons somcwliat ahead or behind the synchronous phase (which 1ic.s on the falling-with-time portion of the electric field since the dcvice is weak-focusing) acquire the correct energy gain on the average, as in a synchrocyclotron or synchrotron. The oscilla1,or frequency for an electron microtron turns out to lie in the microwave region-that is, a t a wavelength of a few centimeters. Thus far, most such machines have operated a t only a few Mev, though one has heen h i l t for 29 Rlev. I n its cxisting form, tlie dwice is unsuitable for heavier particles, hut a hint of what future developments may bring will be presented later on. A microtron is constructed by mounting a resonant accelerating cavity ntlar tlie periphery of a uniform magnetic field. The cavity, pierced by two holes to permit passage of the electrons, is driven by an external fixed-frequency oscillator. I n what follows we imagine the accelwating gap to be of negligible length, to avoid the complications of transit-time effects. The error thus committed is quite small. The source of electrons is usually the edge of one of the holes in the cavity, electrons being drawn from i t by the intense electric field, though sometimes a hot filament in the "entrance" hole is used. Successive 189

190

MICROTRONS

orbits are circles of greater and greater radii, all with a common tangent a t the point where they thread the cavity. See Fig. 11-1.

Fig. 11-2. Dots indicate the positions of electrons at intervals of the veriod of the oscillator, for the particular case where p=2andv=1.

Fig. 11-1. Orbits in a microtron.

The fundamental cyclotron relation, re-expressed in terms of the rotation period T and the total energy E, is

If the magnetic field is constant in space and time, then 2?r AT = -AE,

Bee" that is, the change in period is directly proportional to the change in energy, quite independent of what may be the total energy. This is the basic fact on which the microtron operates. 11-2. Conditions for Resonance

Express t,he energy gained per turn as a fraction AE

=

CEO.

e

of the rest mass energy Eo: (11-3)

The first turn is made by an electron which started from rest and gained eEOin crossing the gap, so its total energy is El = Eo rEo and Eq. (11-1) becomes

+

I n order that this particle may return a t the same voltage phase that it first experienced, the rotation period must equal some integral multiple I*, of the electric period T,! of the oscillator; that is

COSDITIONS FOR RE;dO&'Ah'CE SO

191

that

(At the moment, it appears that p may be any positive integer, hut with the subsequent conditions it will turn out that the ~niniii~lun value nlust he 2.) The same energy increment occurs a t the second acceleration, so the second rotation period is

Subtract Eq. ( 11-4) from Eq. (11-7) to find

wliicli re-ex~)rc.ssesEq. (11-2) ; t11e change in 1)cnotl of successive orbits is proportional to the energy gained per turn. \\-c now make this difference in periods equal to an integral multiple v of the oscillator per~od.Then 72

-

71

=

VT,.f;

so from Eq. (1 1-8)

Aftcr the scrond turn the particle again rtachcls the gap a t the original pliasc and receives the same energy increase. Obviously I4:qs. (11-6) and (11-10) determine the oprration, but they can be put in more useful form. By eliminating 7 , f between them we find the important rclation

where, it will be recalled, e or Eq. (11 -10) yields

=

Al?'/lSo. Putting Eq. (1 1-12) in either Ey. (11-6)

This can be Iretter expressed in terms of the n-n\elength X of the oscillator. Since 7 , f = l/f = X/C, it follons that

1~:c~uations( 1 1 -12) and ( 1 1-14) tlctcrliiirlc the elm-ating paranieters. Note that tlic energy gain cannot be arljitr:wy, :LS in a conventional cyclotron, but niusi,

192

MICROTRONS

have one of a set of specific values determined by the choice of the integers p and v. A smaller value of e results from increasing p if v is fixed, or from decreasing v if p is already chosen. It is also seen that p must exceed v in order that E may be positive and finite, and since the minimum value of v is 1, the least value of p is 2. If E is as small as 1, the energy gain per turn is 0.511 Mev for electrons. This is hard enough to develop in a single cavity, so we need be concerned only with possible choices of p and v which cause E t o be 1 or less. Furthermore, BX should be large in order that a practical value of h may correspond to a large enough value of B to hold electrons of an interestingly high energy in an orbit of modest radius. The quantity 2?rEo/(ec) equals 10,700 gauss-cm for electrons and 1.97 x lo7 gauss-cm for protons. Sample values of the parameters for electrons are shown in the Table 11-1, from which it is apparent that Bh rapidly reaches TABLE11-1 ILLUSTRATIVE PARAMETERS FOR AN ELECTRON MICROTRON WITH e = E/Eo 5 1 e = - v P-v

BX

=

10 700 gauss-cm P-v

impractically low levels if p is raised much above its minimum value, even though smaller energy gain would then be required. The largest Bh for electrons is reached when p = 2 and v = 1, so that E = 1, whence it is clear that microwave oscillators are involved. If h = 10 cm (f = 3000 Mc/s), then B = 1070 gauss. For 29-Mev electrons, Br = 98,100 gauss-cm, so r must be 91.7 cm. A machine of this character is practical and has been built, although pulsed operation is necessary since available microwave oscillators cannot develop continuously the necessary power of about 2 Mw. The time average output of current a t full energy is about lop9 ampere. A smaller machine has produced 1 microampere of 5-Mev electrons a t a duty cycle of 0.07%. Experimentd ~nicrotronsoperating a t shorter wavelengths have been constructed, but a t sonlewhat lower energy.

I t is intcrwting to calculate thc scpnration of turns a t the azimuth opposite. the cavity. From Eq. (1 1-9) we have r,,,~- 7, = V T , / and in terms of t>heorbits' radii and velorities this can I)e written 2~r,+ -~2 ['n+1

=

u,,

v X-' C

Electrons very rapidly approach the rcXocity of light: for example, /3 = 0.866 a t T = E,, and p = 0.945 a t T = 2Eo. Hence with little error this equation may be givcn as

where the D's are the diameters of the orbits. Since the circles are tangent a t the cavity, the orhit separation half way around is v X / s . This is sufficiently large that i t is easy to install a. pipe of nlagnctically shielding material to extract the clcctrons from the finti1 orbit. Perhaps lt)ecause of this. a 5.6-Mer microtron is l~lannedas the injcctor for the 1.2-Bcv electron synchrotron a t Lund, Sweden. If t l ~ cmagnetic field is strictly ~ ~ n i f c ~ then r m , tlhe field index n is zero and thcre are no niagnetic axial focusing forces, stability in this direction depending solely on the small focusing that occLurs:is the particles cross the gap when the electric field is falling with time, as was hriefly discussed in 7-15 in the description of cyclotrons. I n practice, the magnetic. field is not truly uniform, since the resonant cavity is placed near thc poles' ecdges where the field may be down a percrnt or two. This supplies a I-estoring force once per turn, but since the majority of the path is in a unifor~nfield, the synchronous behavior occurs much as described for the ideal cab?, though there dout)tless is n smnll amount of phase shift. The turns are so few in number that no serious phase shift occurs. 11-3. Variability of Energy

Flexibility in output energy of a microtron can he obtained hy employing injection a t variable energy. Let tlre injwtion energy hc K E 0 , where A is some fraction. Thcn Eq. (11-4) is replaced by

and Eq. (11-6) bccon?es

The diffcrenw in periods of succc~esircorbits rc.mains unchangcd, so no alteration is needed in Eqs. (11-7) through (11-11). On eliminating ~~f from Eqs. (11-17) and ( 1 1 - l o ) , n-c3 find

and this used in Eq. (11-10) gives the analogue of Eq. (11-14) :

BX

+ K) ------. p - v

2a& (1

= -

ec

(11-19)

+

Since the output kinetic energy is T = K& NNE&where N is the number of turns, it is clear that the final energy can he altered by changing the injection value, and by making a suitable adjustment in B. 11-4. Speculative Elaborations

Several investigators have studied the feasibility of adding axial focusing to a microtron by cutting slots in the magnet (Fig. 11-3); the effect of such an

Fig. 11-3. A suggested configuration for a proton microtron.

linoc

arrangement of edges gives rise to what is called sector-focusing, as will be described in Chapter 13. The possibility of a high current microtron for protons in the Bev region has been examined by Roberts. This involves not only sectorfocusing, but the replacement of the single accelerating resonator by a linac of 10 to 20 Mev, in order to increase the energy gained per turn as is necessary for practical values of B and A. Thus if NE = 0.02 (AE = 18.76 Mev) and if p = 51 and v = 1, then B = 16,000 gauss and X = 24.6 cm. Since the efficacy of a linac drops rapidly as the velocity of the entering particle departs from a single value, it would be necessary to inject the protons into the microtron a t a high energy, perhaps 500 Mev. See pp. 360-361 for references to papers on inicrotrons.

-

-

-

ALTERNATING-GKAL)IEN?' SYNCHROTRONS -- --

12-1. Introduction

I n the constant-gradient machines so far considered, radial and axial stability is obtained by causing the average guiding magnetic field to drop off sligl~tly towarcls it outer edge. I t has been shown that if this fall-off were consitlerablr, so that the field index would excecd unity stronger rwtoring forces would come into play for axial motion, but unfortunately accompanied by the development of defocusing forces in the radial direction. Conversely, if the magnetic field were to increase as one moves away from the center, so that the field index mould be negative, radial stability would be greatly improved but axial stability destroyed. Fortunately, there is a srnall regLon of overlap, associated with an index lying between 0 and f l ,in which stability in both directions can bc obtained, and on this fact depends the successful operation of all weakfocusing accelerators. The focusing forces in this critical region are not largc, however, so t l ~ a the t betatron oscillations have considerable amplitude, thereby requiring vacuum ch:tinbers of large cross section and consequently very heavy and expensirc inagnetb. A new concept was advanced in 1952 1)y Courant, Thingston, and 8nydt.r (it had been suggested two years earlier by (histofilos, but was not publishcd). This was simply the realization t h a t a long-cstablished fact of geonlctrical 0ptit.s was applicable to synchrotrons. If :t pair of lenses of focal length f l and f, are separated by the distance d , the net focal length F of the combination is given by l

F

l + --l - -.d

=

f,

f,

fif,

Now if the lenses have equal focal lengtl~sbut onc is converging and the othcr diverging, so t h a t f a = -fl, then i t follows that

The focal length of the pair is therefore always positive (converging). The physical explanation of this, of course, is t h a t the angular deviation in each 195

196

ALTERNATING-GRADIENT SYNCHROTRONS

lens is proportional to the distance from the axis a t which traversal occurs, so that the convergence angle 8, always exceeds the divergence angle Od, no matter which lens comes first, as is indicated in Fig. 12-1.

Fig. 12-1. The angle of deviation increases with distance from the axis of an optical lens, so the net action of separated and equally strong converging and diverging lenses is converging. The application of this idea to a synchrotron is a s follows. For a certain azimuthal length, the magnet is built with a large radial gradient, the field index being much greater than unity so that powerful forces, proportional to the displacement, are produced which focus axially and defocus radially. This is followed by an equal azimuthal length of magnet in which the gradient is reversed, so it focuses radially and defocuses axially. Since the orbit is always further from the axis in a focusing magnet than in a defocusing one, the forces of convergence are stronger than those of divergence and the net result is to steer the particle closer to the axis (see Fig. 12-2). Consequently the system is

focusing

defocusing

Fig. 12-2. The focusing and defocusing forces increase with distance from the axis of alternating-gradient magnets in a synchrotron, so the net action is focusing. stable in both the axial and the radial directions. An idealized array of such magnets forming a synchrotron is indicated in Fig. 12-3. If the field index is sufficiently large, the nct restoring force is much greater than in the weak-focusing (constant-gradient) accelerators considered previously, so the amplitudes of the betatron oscillations are much reduced, for a given angle of deviation. T h a t this is so may be seen by referring to Eqs. (2-29) and (2-30). Therefore, magnets of considerably smaller cross section are sufficient. This technique is known as alternating-gradient focusing, or as strongfocusing. I n common parlance, the former modifier is simply the letters AG, and a synchrotron built on this principle is a n AGS.

THE STABILITY DIAGRA A l

197

It nlust be miphasized that t h r high-frequency wiggles, formed in the orbit 11y passage through the individual magnets, do not constitute the betatron motion. This 1attc.r expression refers to the oscillation of the orbit as a whole about the mean azinlutllal path. The l~etatronoscillations are no longer sine waves, but rather sine waves on which a high-frequency component is superimposed.

Fig. 12-3. Schematic representation of an alternating-gradient synchrotron and enlarged cross sections of successive magnets.

-

to center of machine

It is clear that the magnets cannot be too long, or the defocusing force will drive the ion$ into the walls before the corrective ,action of the next magnct can come into play. It should not be surprising if stability depended in some way on a critical relation between the strength of the magnetic gradient and the length of the n1:ignets-in other worcls, on their number-and indeed thls is found to be the case. Since the focusing (or defocusing) force per unit of displaccnlcnt is constant throughout tho length of each magnct, the matrix method of Chapter 3 may be employed to find the conditions of stability and to determine the frequencies of the betatron oscillations. The details of thew calculations will bc presented t~eforelor~g,but for the nloinent i t will be illuminating to quote the results and to cuainint~t l l e ~ rconsequences. 12-2. The Stability Diagram

Consider :t circular synclirotron ( n o st] aight sections) coinposed of 2N rnagnets which have the saine streilgtli of fjeld a t the mid-points of their radial apertures; thc radial gradients, honevcl, are :tltcrnatc ill direction and inay

198

ALTERNATING-GRADIENT SYNCHROTRONS

perhaps not have the same values. This situation is described by writing the field indices as nl and -n2, so that the symbols nl and n2 each represent a large positive number. For convenience set PI

= nl"/N

p2

= n2H/N.

(12-1)

It will be shown that if the system is stable the shift in phase a, of the axial betatron oscillation in a length of one sector (that is, in one pair of magnets) is determined by the expression (12-2) cos U r = cos rpl cosh w2- ('12 - 'z') sin r p l sinh rp2, 2~1~2 and if the indices are so large that unity may be neglected in comparison, the corresponding expression for radial motion is given by cos U,

=

-0.4

1-

cosh rpl cos rp2 -

sinh r p l sin rm. (12-3) 2~1~2 If we call the right-hand sides of Eqs. (12-2) and (12-3) the functions F, and F,, then it is clear that F , and F , must each lie between +1 and -1 for these equations to be valid. It is impossible to see by inspection what are the conditions on p l and p2 such that F , and F , will lie between such limits. We can, Iiowever, assign a value to ~1~and then choose a series of values for pl, in each case computing F,. It will be found that F , sometimes exceeds +1 and some-

\

--"2

A' 2

Fig. 12-4. The stability diagram for a circular alternating-gradient synchrotron, shown for the case where N is a large number. If N is made smaller, say 10 or less, the pattern becomes progressively less symmetrical about the diagonal.

T H E STA131LIT'ITDIAGRA 1\I

199

times is less than -1. By repeating these calcl~lwtionsfor different vaIucs of p2, tl~cbarea rc~)resentrtlby a plot of p I 2 vs. p-%il;ty I)c bwnnrd, much as is a tcle~i,sionscreen, so that tlic associatrtl value. of p12 arid of p2') can be found a t wliich F , p a s s ~ sthrough tlie valucs 1 and - 1. A similar procedure may then be carried out for F,.The relevant results arc shown in Fig. 12-4, where on a graph of p12 -- n1/N2 against = n2/N2 are shown the locii where F, and F , each have the values $1 and -1. In the areas between tliese liniits we may set F, = cos a, and F, = cos a,;outside these regions we may not do so. Where cos a, and cos a, exist, we have axial and radial stability, respectively. Bilateral stability occurs only in the common region of overlap. Because of its shape this area is often called tlie "necktie." If the figure were drawn to include mush larger values of nl/N2 and n2/N2, it would be seen that other locii exist where cos a, and cos o, have the values of +1 and -1, with additional regions of overlap where stability in both directions can be obtained. Such areas are small, however, and occur a t such high values of nl a11cL n2 that it is doubtful if the gradients they represent could be realized in practice. Consequently these "islands" of stability have never been used in accelerator design. If the common choice is made that the two field indices have the same absolute value n (PO that identical magnets may be used if desired, alternately oriented), the second terms in Eqs. (12-2) and (12-3) become zero and we find

+

This is represented in Fig. 12-4 by the diagonal line down the center of the necktie. At the "knot" of the tie (the origin), both cos a, and cos a, are + I , while a t the tie's tip both are -1. Somewhere in hrtween they must both be zero. To find where this occurs, we recognize that cosh (rn:/N) never vanishes, so that the burden must fall on cos (?rnt,lN). This becomes zero when

This point is marked with a circle, and is seen to be not far from the widcst part of the necktie. Excitement ran high during the early discussions of the alternating gradient principle. Large field indices mean powerful forces and reduced amplitudes of betatron oscillations, so that magnets with apertures only 1 and 2 inches on a side appeared possible, about one tenth as large as are needed for constantgradient machines. Values of n running up into the thousands were contemplated. Even with the realization that a truly constant value of n probably could not be maintained over the wide range of field strength needed from injection to final energy (largely because of different degrees of saturation in different parts of the magnet yoke and pole tips), nevertheless a considerable

200

ALTERNATING-GRADIENT SYKCHROTRONS

wandering of n could be tolerated without traversing the boundaries of the necktie region of stability, particularly if the nominal value was chosen so as to put the operating point close to the widest part. These initial hopes were soon sobered by consideration of the effects of resonances on the orbits. It turns out that not the whole area within the necktie is stable when all deleterious influences are taken into account, for it becomes crosshatched with a pattern of resonance lines or bands which the operating point must not approach too closely. ( I t is perhaps apparent that the necktie diagram is the equivalent of the earlier plot of v, vs. v, but with different coordinates.) These danger bands arise from resonances between the radial and axial oscillation frequencies or between either of these and the rotation frequency, the latter effect being emphasized by any errors in construction. These include such things as nonidentical values of n in all sectors, misalignments of magnets as a whole either radially or axially, the warping or twisting of any of them, the failure of the median planes to coincide from unit to unit, nonuniformity of length, and in fact every conceivable error in construction and assembly that can possibly occur. Just because the focusing forces are so strong, the precision with which the magnets must be built is correspondingly augmented, if the operating point is to remain a t all times within a very small area of the necktie. Such severe restrictions do not, however, place the alternating-gradient machine outside the bounds of practical construction. Extremely high values of n have had to be toned down t o figures like 200 or 300 (still a notable increase over the value 0.6 in constant-gradient accelerators!) and extraordinary care must be employed in manufacture of the magnets. Moreover, the units from which the full magnets are assembled must be randomized so as to distribute and not to concentrate any errors in fabrication or quality. Particular attention must be given to the correct placement of the units and means must be devised for restoring this condition in case of subsequent movement of the underlying earth. Foundations are required to be stable to a degree previously unheard of in the architectural profession. This is complicated by the virtual certainty that, from time to time, experimental rearrangements will require moving extremely heavy blocks of radiation shielding in a region very close to a portion of the foundations of the magnet. Root-mean-square errors in position may not exceed about 0.5 mm in very large machines, such as for 25 Bev. On the optimistic side, if the best of care proves inadequate, palliative measures can be taken in the form of specially designed auxiliary magnets located in straight sections. These can supply corrective fields and gradients which are introduced a t appropriate times throughout the acceleration cycle by circuits programed with sufficient elaboration. Having completed this general discussion, we will now derive the expressions governing orbital stability.

BETATRON FKEXJUESC1E;S

20 1

12-3 Betatron Frequencies

I n order t o dctcrininc tlie critcn:t for stability of orbits in :in alternatinggradient synchrotron a n d t o calculate the frequoncics of t h c betatron oscillations, we will apply the matrix metliocl t1cvelol)etl in Chapter 3, arid t o he general and practical we will consider :L racetrack synchrotron, since there must be some space bc.twcen magnets t o :illow t l ~ cwindings to be foldctl u p and down out of the way of t h e particles. First wc nil1 treat the axial motion, b y tracing a particle in succession through an axially focusing magnet, a straight section, a n axially defocusing magnet and another straight section-these four units constituting one sector, of which there arc S in the entire rnachinc.. IT'hcn radial niotion is studicd, the units of thc sector will occur in t h e order dcfocusing, straight, focusing, straight. Whcn a particle is within a nlxgnct which focuses axially, the equation of motion is identical with t h a t of Eq. (2-12) :

although n l is now a large positive number instcad of lying between 0 and + l . Since thc vclocity is ronsidercd cor1st:int In discussing hetatron motion, wc m a y sct d t = dL/v, where d L is the azimuthal distance traveled in tinie dt. W e also have w = v/r, so t h e equation m a y be written as

A satisfactory solution is nl 5 5 L

z = '4 cos ----

r

-+ B sin nlsL r --1

(12-8)

A t t h e beginning of t h e focusing magnet, whcrc 1, = Ll = 0, let z = zl a n d -c,. From these initial conditions we dctc:rniinc t h a t A = z l and I{ = rz,'/nllh. Consequently a t thc end of thc focusing magnct of length m, n h e r e L = L:: = rn, we have

-

;I

- I

f

where for brevity we set /I l

l~n~

= -7-

(12-1 2)

Passage of the ion through the axially focusing magnet and then through a straight section of length s results in a displacement z3 and a slope 23' given by

(l:,) (1 :) (yg

.),:l(

r

sin . $I cos sin $ 1 1

=

(12-13)

In this expression the matrix for the straight section is from Eq. (3-24) ; that for the magnet follows from Eqs. (12-10) and (12-11). On carrying out the indicated multiplication we find

snl)$ cos $1 - 7 sin r

sin

+I

r

- sin

nls

+ s cos

(;:,)

m s $1

(12-14)

We now consider axial motion through the second magnet and the straight section which follows it. Since this magnet defocuses axially, its field increases rapidly with radius, so the index is negative. We express this as -n2, where n2 is a large positive number. The equation of motion applicable within this magnet is then

This represents a growing displacement, and an appropriate solution is

n$L

+ D sinh n245L -. r

z = C cosh r

(12-16)

Therefore zf

E

n254 n255L M L2n\ 3 A? = C - sinh -+ D cosh dL 7' 7' T

--a

r

(12-17)

The constants may be determined from the initial conditions. At the beginning of the magnet, let L = 0, z = 23 and z' = z3', these values of the displacement and slope being the same as when the particle leaves the preceding straight section. Then, since cosh 0 = 1 and sinh 0 = 0, we find that C = z3 and D = rz,'/n2x. Hence a t the end of the defocusing magnet where L = m, the parameters are r ~4 = 2 3 C O S ~ $2 - zZ' sinh (12-18) n24r

+

zql =

where

23

n2+" sinh $Z r

+ zapcosh $2,

(12-19)

After tral-crhing the serond straight sect,lon, t h ~ion's position and slopr arc. given hy

r (12-21) - sinh

cosh

G2

T o obtain zr, and 2;' in terms of zl arid zl', we subqtitute in Eq. (12-22) thr. value of 2.; ant1 2:;' given by Eq. (12-141: that is, we niultil)ly togetlicr t l i ~ matrices of t h ( w equations. The result is

where

+

72.2'5

cosh $'L s - sinh r

cosh #z

n2'4 r

M21

= - sinh

A422

= - sinh

n215

r

+ s nZ% r sinh -

+?

T

+.?

+ s cos

sin

II/:!

(12-26)

r

h ( 2sin $1

+ s eos

I)

t

eos #I eosh

$2.

(12-27)

\I-c now apply the criterion dewloped in Chaptc>r 3. The system will be sta1)Ie in the z direction if (Ji,, $- !112,1 lies between -1 and f l , and if this is so we may set (1 2-28 I ros u, = +(MIL M 2 2 ) .

+

Here u, is the phase shift of the axial l ~ t a t r o nmotion per sector. Suhstitutc for Ad11 and from Eqs. (12-24) and (12-27) and recall that 2ar = 2Ntt1, where N is the number of pairs of magnets, so t h a t r = N m / a . Consequently tile quantities and q2 given in Eqs. (12-12) and (12-20) bccoine

204

ALTETINATING-GRADIENT SYNCHROTRONS

and Eq. (12-281 may be written as cos u,

=

cos #I cosh

$2

---

+ sa

- (nl - n2) sin

sinh

2 (nln2)

(n2%cos #I sinh

$9

#, - nl'%in

cosh

- -S"T? n l W $ sin 2m'NL

sinh #2,

(12-30)

where, it will be recalled, both nl and n, are positive numbers. The phase shift of axial oscillation per sector is thus given in tcrms of machine parameters. Radial stability may be investigated by writing the appropriate equation of motion for a particle within a magnet:

d'x z

+

(I - n) ~

X

=

o

.

(12-31)

Since in alternating-gradient machines the field indices are very large, unity may be neglected in comparison. Consequently since the first magnet is axially focusing, it is radially defocusing, so we h a r e

while for the second magnet, which focuses radially, the expression is

I n both these equations n, and n, are positive numbers. By comparing Eq. (12-32) with Eq. (12-7) and Eq. (12-33) with Eq. (12-15), it is clear that the analysis of radial motion will follow that for axial, provided the field indices (and hence the subscripts) are interchanged. The phase shift of radial motion per sector is hence given by

cos

uz =

cos $3 cosh

- (nz - nd sin t+b2 sinh $1 2(n2nd5$

s"? -2m!h" n2%n14$sin G2 sinh #I, where and $, are defined in Eq. (12-29). If there are no straight sections, so that s = 0, Eqs. (12-30) and (12-34) reduce to Eqs. (12-2) and (12-3), for each n in the second terms of Eqs. (12-30) and (12-34) may be divided by N-o form p1 and p2, and by Eqs. (12-1) and (12-29) it is seen that apl = nn15"N = and similarly apz = $9. Several alternating-gradient synchrotrons have been built with different absolute values of the field indices in alternate magnets, but in most machines

PHASE STABILITY

205

the absolutc v:tlues are alike. T o describe such a symmetric accelerator \vc have but to put nz = nl = n. Then with the dcfinition

\vc find identical expressions for cos a, and cos a,. (Note t h a t the second terms of Eqs. (12-30) and (12-34) bccomc zcro.1 Thus:

+ sar~J(' (COS$ sinh $ - siri J/ cosh $1 mN -

,s'a!n 2wl-N-

- 7; siri $ sinh

9.

(12-36)

Again, herc n is positive. If s = 0, this takes the simple form quoted in Eq. (12-4). Accurate v:tlues of the betatron oscillation frequencies v, and v, may be found by first obtaining values of the a's from Eqs. (12-30) a i d (12-34), or from Eq. (12-36) if the riiacliine is synimetric>:il, using known values of the machine parameters S,s, m, n,, and n2 and then apldying the general relation

which was derived in obtaining Eqs. (3-70) arid (3-72). T o obtain approximate values, the trigonometric and Iiyperhol~c fiinctions may he expanded; set cos a = 1 - &/2; sin = - k3/:3!$- p/5!- . . . ; cos = 1 - $'/2! fi4/4!s i n h $ = J / + + 3 / 3 ! + + 5 / 5 ! + . . . ; c o s h J / = 1 +$2/2!+$4/4!+ For example, the result for thc symmetrical machine is

+ +

e m . ;

+

+

. . a .

This inversc dcpendrnce on ,\i is in :~ccortlwith the implications of the necktie tliagra~nFig. 12-4 (which strictly is apl~licableonly if there are no straight scctionsl, for it is seen that a large ' \ L puts the operating point near the tie's knot \\here a is small and so also is v. Considering only the most important term in the expression above, i t is evident that t l introduction ~ of straight sections raises tlic 1)etatron frequcncies, as it docs in weak-focusing machines. 12-4. Phase Stability

The projectiles acquire energy in an alternating-gradient synchrotron by the same method as in a const:~iit-gradientmachine, that is, by passing through onc or more gaps ncross which an :~lternatirig ~ o l t a g eis impressed, tllc frequency of this voltage being raised continu:tlly to keep in step with thtl rising frequency of revolution of tlre ions ,Just as before, the peak voltagcl across the gap (on the simplifymg assurnptioi~that there is hut one) must exceed t h a t rccluirecl by the synchronous ion, so tlr:tt, this particle can acquire the synchronous energy increnicnt by arriving a t t l ~ cgap a t the appropriatc

206

ALTERNATING-GRADIENT SYNCHROTRONS

phase. But those ions which do not reach the gap a t exactly the correct moment will gain too much or too little energy, so their momenta will differ from the ideal value. We must again inquire whether their subsequent revolution periods will be greater than, equal to, or less than that of the synchronous particle. The answer, as previously, depends on the value of the momentum compaction a of the machine. By a fairly lengthy argument to be presented shortly, an expression for a can be obtained in terms of machine parameters. (For simplicity, we consider a t the moment only the symmetric accelerator in which nl = n2 = n . ) It will be shown that

Here n is the common absolute value of the field index, while N is the number of magnet pairs; all straight sections have length s and all magnets have length m. Since this expression gives values of a which may be considerably greater than unity, it indicates that orbits with a wide spread in momenta are squeezed together in a narrow radial region. Note that a high value of nlomentum compaction is a characteristic of alternating-gradient machines. This is in sharp contrast with the low value found in constant-gradient accelerators. To understand the relevance of this, careful attention must be given to the familiar expression, Eq. (6-9) :

+

Recall that y E/Eo = ('T Eo)/Eo, where T is the kinetic and E o the rest energy. As T rises from zero, y also rises, starting at unity, and consequently l/r2decreases, from an initial value of 1. With a greater than unity in a strong-focusing device, l / a is less than 1. The quantity (l/a - l / y 2 ) is therefore negative a t those low energies for which y2 < a, it becomes zero a t a critical "transition energy" y, (Fig. 12-5) given by y: = a, and is positive a t

Fig. 12-5. The transition energy

y,

in an

AG synchrotron, in which the momentum compaction a is constant.

higher energies when y2 > a. Hence at energies below the transition value an increase i n momentzrm causes a derrense i n period of revolution and at energies above the transition energy an i n c r e a s ~i n momentum causes an increase i n revolution period, while at the transition energy a change in momentum has no effect o n the period.

PHAPF: STABILITY

207

T h u s a t high enough energy an AG mac,lline acts in t h e same way a s a CG tlevice bcliavcs a t all c n e r g i c ~ ,and pliahc stability occurs if tlw synclironouh articles cross tlie accclcratirig gap whcn tlic c~lcctric ficltl is falling \\it11 time. T h e particles of high energy :ire w r y relativistxc, a n y energy incrcment being associated more with a rnass i n c r e : w t h a n with :t change in velocity. I n spite of t h e close crowding together of orbits of different momenta a n d consequent sniall change in path, the change in vclocity is even smaller. At low energy in a n AG accelerator, the. change of period has t h e opposite sign of a change in mornenturn, and i t is not hard to see t h a t phase stability is obtained only if t h e synchronous particles reach t h e accelerating gap when t h e electric field is rising ( a s with a linac). Because of the large radial gradients in AG magnets, only a small radial displacement is necded (alternately inward and outward) for t h e ion to find a n appropriate Br for its greater ~nornentum.T h e increased path length is less import:tnt t h a n t h e increased velocity a n d a shorter revolution pcriod results. With a n AG synchrotron, positive actioi~must he taken t o make the voltage a t t h e accelerating gap shift in phase whcn the transition energy is reachctl. It will he noted t h a t a s this energy is approachd and passed, phase stability becomes weaker, vanishes entirely, and then gains in strength again. C'onsitlerable difficulty in avoiding a heavy loss of particles was anticipated originally, b u t experiments a t Brookhaven National Laboratory with a small electron machine built t o test t h e point have indicated t h a t the troubles are not as serious a s had been feared. If the momentum compaction is not too large and the injection energy not too small, the transition energy can lie below t h e injection value, so the difficulty is sidestepped entirely. This is t h e case with the 1-Bev electron AG synchrotron operating a t Cornell University.

TABLE 12-1 COMPARISON BETWEISN WEAK-FOCUSING (CONSTANT-GRADIENT) ,4ND STRONG-FOCUSING (ALTERNATING-GRADIEKT) ACCELERATORS

Field index n Momentum compaction a Change in period of rotation associated with an increase in momentum At low enerav At hrgh enc& Contiition of awelerative electric field appropriate for phase stability At low energy At high energy Existenw of transition energy

CG

AG

<1 <1

>1

increase incrcaqe

decrease increase

falling falling no

rising falling ves

>I

It must be emphasized t h a t these considerations on changes in momentum and in period apply t o time intervals so short t h a t t h e guide field m a y be considered a s constant. Over long intervals the guide field rises and tlie orbits

208

ALTERNATING-GRADIENT SYNCHROTRONS

remain of essentially constant length since the energy is adjusted to keep them centered in the vacuum chamber. Therefore as the energy increases, thc rotation (and oscillator) frequency must continually rise, irrespective of the existence of a transition energy. After these general observations on phase stability in an alternating-gradient synchrotron, attention will now be given to the derivation of the expression for the momentum compaction in such a machine. Since a depends on d L / L , the first step is to discover the shape and length of orbits of momenta p and P + dp. 12-5. The Shape of Equilibrium Orbits

Imagine that the slowly rising magnetic guide field is held fixed a t some intermediate value, so the particle orbits within it can be examined a t leisure. The guide field B appropriate to a particle of ideal momentum p is found a t radius r, roughly midway across the radial width of the magnet gap, across which a wide range of field strengths exists because of the large built-in gradient. If alternate magnets are properly placed, the same field B will be found in the next magnet a t the same distance from the center of the machine, even though the gradient is reversed. Hence the orbit for momentum p lies on a smooth circle if there are no spaces between magnets; if there are such straight sections, the orbit is a series of circular arcs smoothly connected by tangential lines. This path is called the p orbit and is appropriate for a synchronous ion. But most of the particles are nonsynchronous and acquire the proper energy only on the average through the operation of phase stability. Consider such an ion with momentum p d p ; this requires a larger value of the product Br. In a radially focusing magnet this can be found a t a radius somewhat greater than r, since the field out there is stronger (see Fig. 12-6a).

+

rodiolly focusing rnognet

rodiolly defocusing rnognet

Fig. 12-6. The relative positions of orbits with momenta p and p sive AG magnets.

+ d p in succes-

I n a radially defocusing magnet the appropriate Br is located a t a radius smaller than r, as shown in Fig. 12-6b, because the large field gradient makes B rise more than r falls. If we were to launch ions with momentum p d p a t the beginning of each magnet, appropriate orbits in each would form a series of disconnected cir-

+

TI-I 1' SHAPE O F EQIJI1,IBR IUhI OIIBITS

209

cular arcs lying outside r in radlally focusing niagnets a n d inside r in defocusing rcgions. I t is obviously impossil)lc for a n ion launched only once to take such :r l)ath, but i t does the best it can, following a curve which overshoots this c o u ~ win focusing regions and u n d t n h o o t s i t in dcfocusing areas, a s s l ~ o w nin Fig. 12-7. ( J u s t why this is so, will be explained before long.) T h e focus

-p

orbit

Fig. 12-7. The short arrows indicate where the equilibrium orbits of particles with momentum p $- dp should lie; such a discontinuous path is impossible and the actual path lics entirely outside of thc orbit of a particle with momentum p. The apparent reversed curvature is not real and results from an exaggerated scale. outermost c~xcursion,iare portions of c o m e waves .rllicli, if there are spaces between magnets, a r c connected by straight patlis to portions of hyperbolic cosine orbits a t the innernlost regions. This curved path is the equilibrium orbit d p particles. for t h e p r bear in niind t h a t a11 magnetic K h e n looking :it Fig. 12-7, the ~ w d e should fields point in thc same direction, so a11 curved portions of the orbits really arc concavc toward the n~acllinecenter, the c?os and cod1 portions being with reference to the curved p orbit. This is next to iirlpossible to depict nmmingfully in a sm:ill drawing; t h e displayctl rwersed curvature results from a sc:tl~ wllich is grossly cornpressed in azntiuth :mtl exy):indecl in tho r:icli:tl direction. \Ve will now exanline, to first order, the e c p t i o n s of radial nlotion for an ion with momcntuni p clp, using :is a refcrenw t21e orbit of a particle with moriientum p. It has been seen eurlier in Eq. (2-26) t h a t with respect to such an orhit, tlic trajectory of the particle with inoiuentum p tip follows thc expression

+

+

+

for a n y portion of the path for which the field index is n. Herc all quantities except x a n d t a r e constant, w, r , a n d p l)&lg associated wit,h t h e reference ion, wllilc tip is tlie finite extra niorncmturn of the particle under investigation. TVe have dropped tlic subscript e previously appentlcd t o r. Assume tlie velocity t o be constant so t h a t dt = dL/o, a n d xvith w = v/r we obtain

210

ALTERNATING-GRADIENT SYSCHROTRONS

The field index is a large number in alternating-gradient machines, so we drop unity in coinparision with it. The subscript 1 has previously been assigned to a magnet which focuses axially and defocuses radially, so for this magnet we write

where nl is a positive number. Correspondingly, for the radially focusing magnet we have

and here n2 is also positive. For zero values of the second derivatives, appropriate solutions of Eqs. (12-43) and (12-44) are

These correspond to the disconnected paths of ions launched on-course a t the beginning of each magnet, as mentioned earlier. Since a particle cannot follow such a discontinuous orbit, we choose less restricted solutions: r dp nl P

XI = ---

nlt5L + B cosh r

T h a t these are indeed solutions may be proved by forming the second derivatives and substituting in Eqs. (12-43) and (12-44). In radially focusing magnets the orbit is a portion of a cosine wave about a circular arc lying a distance r dp/(n,p) outside the orbit of the ion with nionientum p, while in defocusing regions it is a portion of a cosh wave referred to an arc lying a distance rdp/(nlp) inside the path of the reference particlc. If successive magnets are in contact, the constants A and B are so choscn that the cos and cosh curves join smoothly, while if there are straight sections between the magnets, the curved parts of the orbits are joined by straight lines. This latter situation is indicated in Fig. 12-8, where for clarity the p orbit has been drawn straight. (Note again that the reversed curvature in defocusing regions does not exist in reality, but results from an exaggerated scale and the straight-line representation of the p orbit.) It will be convenient to measure distance along each straight section from a zero taken a t its left end where a particle enters. On the other hand, the forms of Eqs. (12-47) and (12-48) show that distancc within the magnets is measured from their midpoints. Two relations arc needed to dctermine the

focus

f~rst stra~ght

defocus

second straight

Fig. 12-8. The junctions of thc e h n c n t s of an orbit in an alternati~lg-gradicntmachine. Exaggeration of the scale makes the path appear to have rcversed curvature. constants A :11111 H which :11)1)0:w in Eqs (12-47) ant1 (12-48). F o r t l ~ efir5t, we riotc t h t t110 slol~c:it thc r i c , l ~ t - l ~ a ~end i ( l of t h r fucuslng nlagnet, ~v11c.r~ I, = m/2, cqu:~l- tllc slol~c:it, t 1 1 ~left c~aelof the tlofocnbing ~ n a g n c twhcrc I, = -1u/2. \\ 11ci1this c o r ~ d a t i o i iis iil;~d(,by (liff~mmtiatingE ( p . (12-47) and (12-48) wit11 ic,sl)c~tto I; and e w l u a t i n g a t tllcsc pcririts, i t should bc rec:tlletl t h t (1 cosl~ 0 51nl1 808, anti tllat bc>csuac bin11 8 = (eH- e - 0 ) /2, then sinh (-8) = - .in21 0 Tlw rcsult is

-

+

Usc Ecp. (12-47) and (12-48) and remc.~nbert l ~ cvsli t 8 = (e@ e-")/2 so I= cosl~ 8. \\.e uhtain

t h t cod1 (-8)

\Vhcn Eqs. (12-49) ant1 (12-50)

:IIY

solved for A :inti 1.: tho rcsults arc

ALTERNATING-GRADIENT SYNCHROTRONS

212

Tlicse rcsults will be of value later. 12-6. Momentum Compaction

Momentum compaction has been defined in Eq. (6-1) as a = (dp/p)/(dL/L); so to find ac we need a comparison between the path lengths of particles with dp. An element of the curved orbit of the p particle momentum p and with p is rd0, SO its total path is Lp = Jrd0 2Ns, since there are N pairs of straight sections each of length s. The integral is from 0 t o 2n. Similarly for the more energetic ion we have Lp+dp= J(r x) d0 2Ns. Therefore the path difference is dL = Lp+dp- Lp = Jxd0 and we find that

+

+

+

d I, Lp

1

rd0

+

+ 2Ns

(r/N) (27rr

(12-54)

+ 2Ns)

The numerator of this expression is the value of x integrated over the two magnets in one sector, while thc denominator is r multiplied by the length of one sector. Therefore the entire expression on the right is the average value of x in a pair of magnets, divided by r. Hence we have

The subscript p, having served its purpose, is now dropped. An explicit evaluation of z / r is required. This is obtained by integrating, over a sector, the expressions for xl and x2 given in Eqs. (12-47) and (12-48), adding the results, and dividing by r timcs the sector length 2 ( m s ) . We find

+

-

Er

m dp 2(m s)

+

A sin (&) +

np(m

+ s)

dp + s) nlp

m

2(m

+

B sinh (&). n1'4((m s)

+

(12-56)

Now by Eqs. (6-1) and (12-55) we have

When Eq. (12-56) is used in this, and substitution for A and B is made, according to Eqs. (12-51), (12-52), and (12-531, and in addition r is eliminated through the relation r = Nm/a, the result is

I

+

-1.

(12-is)

roth (;+,I - all' rot (;fi2) ~ m , ~ ~ ' ' t l(m.2.) ~'" From tliih ~'xlwvssion,the m o i n ~ n t ~ ~ co~t~paction in N 11i:ly be found in tcrins of the machine parameters, the quantities I$, being defincd in Eq. (12-291 as fi215

+, = n,%/N. The numbers nl and n2 are bot,h positive. Most, but not all, alternating-gradient synchrotrons are synmctric; t h a t is, the ficld index has thc same nt)solutc xalue in all magncts. T o exprcss this, we set n, = n2 I= n, and note that the f i l ~ tterm in Eq. (12-58) bcconic zero. For such a ninc.hine we have

This is the expression disl)l:~yet-icarlier in ICq. (12-39) As a special case, consider tlit sinlplificd marlline with no straight s c d o n s , so s = 0; and let us make the further stipulation that cos a, = cos a, = 0. This condition is rq~resentedon the "necktlc" of Fig. 12-4 by the point on tlw diagonal, closc to the widest part of thc Lie. It was shown in Eq. (12-5) that a t this point we have n M / N = 4. Under these conditions the last expression reduces to

This simplified expression makes it clear t h a t when n is large, say several hundred, tlw illomenturn compaction h:1s a sut)st:mtial value, very much gre:iter than thc figure (less than unity I whic,l~occurs in weak-focusing accclcrators. I t is not difficult to show that the general expression of Eq. (12-58) also applies to a constant-gradient nlachine, when wcl rccall that unity was dropped in cornparison with nl and n2. Thcl coefficient of x l / r in Ey. (12-43) must bc changed from -nl to (1 - n ) , so we sct nl = n - 1, while the coefficient of r 2 / r in Eq. (12-44) must be altered from n, to ( 1 - n ) , so we set n2 = 1 - n . With thcsc sul)stitutions, the second term in Eq. (12-58) vanishes :tnd tllc first term yicltls a = (1 - n)(l s/m). This result was derived earlier as Eq. (6-6) by much sinipler arguments.

+

+

12-7. The Use of "Half" Magnets

It is apparent that in the center of focusing and dcfocusing magnets thch ions are traveling par:tllel to the centerline of the c.hambcr. C:onsequently if a straight (field-free) section were introduced in thc middle of a m a g n e t a s

214

ALTERNATING-GRADIENT SYNCT-IROTROSS

by cutting it in two and drawing the halves apart, aziniuthally-no additional transverse displacement in the equilibrium p dp orbit would result. For this reason some AG synchrotrons are built in this manner, half of a focusing magnet being in immediate contact with half a defocusing unit; this is followed by a gap and then anothcr pair of half magnets in rcvcrsed order, and then another straight section. This pattern is represented by the symbols . . . FOFDODFOFDOD . . . or FOFDOD for short, where each magnet is a "half length." The situation considered earlier was . . . FODOFODO . . . called FODO, each magnet being of full length (Fig. 12-9).

+

Fig. 12-9. Two commonly used arrangements of focusing and defocusing magnets and straight sections.

It is obvious that the calculation of cos a for FOFDOD is morc tedious than for FODO; six matrices must be multiplied together to traverse a complete sector, rather than four. Since general principles are of greater interest than detailed calculations, the FOFDOD computations will not be carried out here. 12-8. Existing and Future AG Synchrotrons

The first alternating-gradient synchrotron ever t o be put into operation is the 1.5-Bev electron machine a t Cornell University. Its 20-ton magnet has 16 sectors, and the field index has values +14.15 and -16.25. Injection is from a 2-Mev Van de Graaff accelerator into a field of 20 gauss, which rises in 0.01 second to 13.5 kilogauss, the energy being stored in a resonant circuit. The radius of the orbit is 12.5 feet, the vacuum channel is 3 inches wide and 1 inch high and the output of electrons is about lo9 per pulse. The transition energy lies below the injection value, so phase stability is never lost. The Harvard-MIT 6-Bev alternating-gradient electron synchrotron has been mentioned briefly in § 9-15. Completion is expected in 1961. The field index has the value of *91 and the 48 sectors and associated straight sections form a circle of radius 118 feet, the actual radius of curvature of the orbit being 86 feet. Injection will be from a 20-Mev linear accelerator and the field will rise from 20 to 7600 gauss in 1/120 second, energy storage being by capacitors and auxiliary inductances. The electrons will travel in a chamber 5.2 x 1.5 inches. A somewhat similar clectron machine is being built in Hamburg, West Germany, with an energy of 6 Bev, an orbit radius of 104 feet, and a slightly higher peak field. Alternating-gradient proton synchrotrons of impressive energy and physical size have been constructed a t Brookhaven National Laboratory on Long Island (Plates 20,21) and a t Geneva, Switzerland, the latter device under the aegis of a group of west European nations known as CERN (Conseil Europkenne pour la Recherche NuclBaire). Both machines are designed for the 25- to 30-Bev

I i X I S T I E G AND FUTURE XG S Y S C H R O T R O N S

215

Courtesy of Brookhaven National Laboratory

PLATE 20 The 33-Bev alternating-gradient roton synchrotron at Brookhaven National Laboratory, shown before shielding &rt was piled on top of the ring building which houses the magnet. The 50-Mev linac injector is in the shelter at lower right. The experimental-area structure straddles the synchrotron, and nearby is the building for controls, power supplies, and laboratories. region, the a s s m ~ b l i c sof magnetb, str:~iglit sertions, accelerating units, a n d vacuum p u n ~ p sbeing placed in circular undcrgrouixl tunnels about 30 feet widc, 30 fcet high and half a mile in ci~cumference.Initial operation of the CERX device took place in late 1959, and early tests of the 0 t h machine occurred in mid-1960. Energy storage for the magnets is b y flywheel, a n d acceleration is produceil by ferrite-tuned coaxial cavities, 12 double cavities a t Brookhaven and 16 a t C E R X , cac~lh entire assembly developing twice the synchronous voltage. Injection is from 50-AIev Alvarez-type linear accelerators, with pulsed clectrostatic inflection for one turn. T h e Brookhaven m:~chine passes tllrougli its tr:~nsitiouenclgy a t tll)out 7 Hev, the 1Surol)c:m liiaclii~iea t :tbout 5 B c r . T h r yield of ion5 i~ 10"' to 10" per pulhc, wliich rel)eats perhaps once every 4 secontls. I n c w l l instnllation the 111:igu~tring p s x s t l ~ r o u g la~ large I)uilding where tllc c x l w i ~ n e n t c r sand tlicir ccluipiilent, : ~ r csliicl(lctl from thc n~:trhinta r:diwtions k)y t l ~ i c kconcrcte walls. T l ~ c s o:wc ~)icrcctlwith ch:mricls to p e r n ~ i t passage of subnuclear particles procluccd at targets driven into t h e final orhits. It is belicred t1i:~t external beanis of protons also m a y be extracted.

216

ALTERNATING-GRADIEKT SYNCHROTRONS

TABLE12-1 ALTERNATING-GRADIENT ELECTRON SYNCHROTRONS

Feature Maximum energy Injection energy .\larimurn field Injection field Field index Betatron freq.

Unit

Tokyo Uniu.. Tokyo Japan

HaraardMIT, Cambrtdge Mass.

Germqn EleclronSynchrotron, Hamburg Germany

Lund Uniu., Lund Sweden

Bev

1.5

1.3

6

1.2

Mcv

2

6

20

5.6

kgauss

13.5

10.8

7.6

11

gauss

20

54

25.4

55

{"L53

15

91

11.2

1.25 2.25

18 18

6.4 6.4

ft

12.50

?

86.5

? ? 11.96

16

8

48

16

aec

0.01

vi a's

Orbit radius No. straight sections Rise time Rep. rate per niin No. acccl. stations Harmonic order Oscil. freq. T'ac. chamber

Cornell Cniu., Ithaca N.Y.

0.008

3600

1290

3600

7

1

16

16

1

360

45

Mc/sec

85-87

138.1

475.8

402

width

inches

3

5.9

5.2

2.3

height

inch~s

1

2.1

1.5

focus 2 , defocus 3 45

tons tons

20 3.5

53 7.9

298 49

570 80

32 6.5

Van de Graaff

linac

linac

linac

microtron

10-120

6

800

?

109

1010

10"

t

1961

1961

?

Weight magnet copper Injector Energy gain per turn Particles per pulse Completed Completion expected

8

kev

j

1,4

1954

Three alternating-gradient proton synchrotrons have becn reported from the USSR, where they are known as synchrophasotrons. A 650-Mcv ~uachine has been completed. Its orbit radius is 5 meters and there are 17 pairs of magnets in which the field index alternately changes sign with the values 27.5 and 28.5, with v, = 3.2 and v, = 3.3. This device has served as a model for a 7-Bev accelerator scheduled for operation in 1961. A novel characteristic of this latter machine is t h a t the transition energy, yl, has becn raised above the final operating energy, so no jump in phase of the oscillator is required. Since

Hrooktrnaw~ h'nt. l , , ~ h . . C p l n n . L.I.

Maximrun rnrrgy Injrrtion energy hlauiniurn firld Injrrtion field Firld index Retatron frrq. v, =

33

SO

Mrv

50

100

kg:m.;n

10-13

11

121

!I0

357

410

5.75

12.75

ft

280

517

240

120

see

1

3.8

v,

No. straight srrtions Ria. tinw

1'SSK

Rev

g:lllsc

Orhit radius

Scrpukw,

v..'J

C'n ;t

Frafurc

R P rate ~ prr rrtin

20

5

No. nrrel. stations

12 (doublr)

?

Harmonir ordrr

12

30

Xlr/s~r Mr/ser

1.4 4.5

6.1

Var. rharnh~r u idt 11 height

inrtrps inrl~es

(i

\\-rieht rnaxnet roil

tons tons

Osril. freq. min Illax

Injector Energy gain per t u r n

2.0

2.7

7.80 4.72

3400 130 (Al)

4000 400 (('11)

2100 Y

linar

linar

linac

80

100

Protons per l)lllse (prelin~irlnry)

10''

?

Corr~pletrrl

1900

('o~npletion exg~wtcd

krv

>

application of the usual techniques would requirc a very high value of coinpartion in order that orbits of diffcrcmt ~nomentamight bc very closely packed together with essentially no difl'erence in path lengths. The change of period associated with a change in mon1cmtun1, essential for phase stability, then conlcs alinost wholly froin the change in velocity, small as that may be. Hut since a is proportionnl to n, a n extremely large value of the field index is required, which in turn demands ilnpractically close tolerances in manufacture and plncenlent of thc magnet, if resonancce are to be avoidctl. Tllc alternative sol~ltionproposed l)y t l ~ eR~lshiansconsists in making 14 inngnc~s

-yt2 = a,

218

ALTERNATING-GRADIENT SYNCIIROTKONS

Courtesy of Brookhaven National Labo~atory

PLATE 21 Magnets and exciting coils of the Brookhaven AGS.

out of the total of 112, with reversed fields of half strength, though their gradients and those of the normal magnets alternate in direction in the usual way. This results in a reversal of curvature of the trajectories near these regions, so the equilibrium orbit becomes distorted from t,he usual circle of a normal AGS. In consequence, though it is not easy to see, a particle with more than the equilibrium momentum follows a path distorted in a different manner but yet of a length very close to that of the equilibrium ion, thus producing the same effect as would a very high momentum compaction. In practice, the effective compaction has been raised to the value a = 500, which puts the transition energy a t 20 Bev. A disadvantage of the scheme is that the total circumference of the magnet assemhly is increased by about 25%. Construction near hioscow of a 50-Bev proton machine, based on experience gained from the 7-Bev instrument, was started in 1959. Here the reversed-field magnets number 15 out of a total of 120. If the scheme to raise the transition energy beyond the output energy is not satisfactory, these 15 magnets can be connected in the normal AG manner; the final energy will then be 70 Bev and the transition energy will be a t 9 Bev. A list of the important papers on the theory of alternating-gradient focusing is given on pp. 361-362; references to particular installations appear on pp. 362-364.

FIXED-FIELD AL'I'ERNATI NG-GRADIENT ACCELERATORS

13-1. Introduction

T h e s u b j w t of the present cliaptcr is thc c1evelol)mcnt of nlagnctically guided a n d focwscd accelerators in wliic~liit, is possible t o eniploy niagnetic fieltls wl~icliribc mith increasing r:~dius but y t t perinit the orbits t o exhibit axial as well a s radial stability. T h e history of this is interesting. T h e original l'apers on the subject were pnblishccl by L. 13. Thomas in 1938, b u t t h e reasonableness of the suggestion was not apprcciated :it t h a t time and i t lay dorniant for t r v e l ~ cyears. X o t until 1950 w:is this early theoretical work resurrcctctl hy the group a t Berkeley, when two small experiinental electron cyclotrons, following the suggestions of ' ~ h o m a s ,were built and studied under strict security classification, publicatiori in the open journals being withheld u n t ~ l1956. LI(~m\vhilc,stirilul:~ttd hy the announcwnent of the alternatinggradient princ~plein 1952, the itle:~s of Tiionins were independently revived and Rabinovicli :ind extended by 0lik:~wain 1953, hy K o l o ~ ~ n s k iPetukhov, j, in 1955, :ind by Syrnon in 1955. S i n w them t l ~ vtlir.ory has dctvelopctl rapidly in tllc hands of a nuniber of people, a l ~ dthc essential unity of all types of r ~ ~ g n c t i c a l controlled ly accelerators has becoinca iiiore :iplwent. Pl'o single expression has Ixen universally adopted to describe the modern elaborntioris of tlie T h o n ~ a sprincil)le. Some authors employ the adjective "fixed-field :ilternating-gradlent" (FE'AG) , others prefer "aziniuthally-rarying field" ( A V F ) , while soinc3 use the words, "sectorial" or "sector focused." Katlial a n d spiral ridge (or scctor) arc also cormion expressions for two varieties of tlie underlying itie:~. T h e important cnginecririg characteristic of all devices of this class 1s tli:it thc guiding magnetic field is static in time. T o consider Tliolnus focusing t ~ u a l i t ; ~ t i v c l yimagine , n ryrlotron in which t l ~ epole faces are not flat but arc sinusoidnlly coritourcd in aziniutli so t h a t the gap between the poles b t v m ~ ~ e:tlteln:itely s larger :md smaller :is one goes around t h e mac2iine. Therefore tllv inagnetic f i ~ l t ivaries with azimuth ; regions \\here i t is high arc collocp~iallpknown :tb hills and w l ~ c r eit is low a5 valleys. I n tlic origiixd f o r n ~the , loci of the, hill inuxiii~aan(l valley rniniinn arc radial, 219

220 FIXED-FIELD ALTERNATING-GRADIENT ACCELERATORS so the device may be called a radial ridge cyclotron. Because of this azimuthal change in field strength, there is an azimuthal curving of the field lines so that they are convex toward the centers of the valleys. If the magnet is imagined to be "unrolled,~'a side view of the gap and of the field lines is as indicated in Fig. 13-1, the path of the particle lying in the plane of the paper. At the South Pole

N o r t h Pole

Fig. 13-1. Side view of sinusoidal pole tips and of field lines in a Thomas cyclotron which has been "unrolled for clarity of illustration. The observer is at the center looking outward. median plane of the gap, the field is vertical, but a t points above or below there is an azimuthal component Be which changes sign as the axial coordinate z passes through zero. The equilibrium orbit of a particle with given momentum is not a circle in a field of this sort. The radius of curvature p is small where the field is large and vice versa, for a t every point the relation must hold that Mu = qBp, as in Eq. (1-2). The orbit shape is easily seen (and most easily drawn) if the sinusoidal pole tips are replaced by a square wave pattern, shown in Fig. 13-2. South Pole

N o r t h Pole

Fig. 13-2. Side view of an "unrolled Thomas cyclotron with square-wave pole tips. (-4 Fourier analysis of such a square wave includes a sine wave of similar 1)eriod as its dominant term, so the substitution is well justified. In fact, most sector-focused accelerators are built with such square-wave or step-function pole tips.) Then in the idealized case of infinitely abrupt boundaries betwcen hills and valleys, with uniform fields in each, the radius of curvature has only two values. The orbit weaves back and forth about a circle, forming a "scalloped" path, as shown in Fig. 13-3, and it is clear that the particle develops a radial component of velocity, alternately directed inward and outward. This radial velocity v, interacts with the azimuthal field Be to produce an axial force which turns out to be directed always toward the mid-plane. This Thornas force is quite weak, for with practical variations in

ficld, Imth B0 and z), arc snl:~ll.The point of great importance is that this force exists q111te inilcl)cndcntly of :my radial gratlicnt such as supplies the asml restor~ngf u r w in a nc:tk-focusing accvlcrator. If Thomas pole t l ~ ) sarc added to :i cor~ventionalcyclotron, the axial htabllity is augmented and the betatron frcqucncy is raised. On the c~tllerlimd, the average z-component of the ficltl can be made to increase wit11 radiuh a t the same rate a t which the ion's mass incw.xses, so that thc revolution frrquency remains constant; this means t h a t there is no slip in p1i:tse bctween the particle and a constant-fr+ q w n c y oscillalor which drives the dew. In an ordinary cyclotron, sucll a rntlially rising ficld would cause axial instability, but the Thomas force can ovcrpowcr the. usual force of tlir-crgence and cause the orbit to remain stable. Tllc original c~lcctroncyclotrons a t Berkeley wcre of this type and brought the particles up to lialf the speed of light, that is, to an energy of 79 key, which represents a 15% increase in mass. The same machines, if scaled u p for protons, would produce 145 Mev. valley

valley

--=I

valley

valley

valley

Fig. 13-3. Four-sector Thomas cyclotron with idealized square-wave variation of field. The orbit has two radii of curvature and forms scallops about a circle.

Fig. 13-4. When the spiral angle is large enough, axial focusing and defocusing forces occur at the boundaries between high and low fields, giving rise to Kerst alternating-gradient stabilization.

Two additional and interconnected axial forces wcre discovered later, by Kerst and by 1,aslett. Their existence depends on making the Thomas field h a w a spiral characteristic. This means that the locus of any particular phase of the aziinuth:~lly v:trying ficltl (such as the locus of a hill maximum or of a valley minimum) follows a spiral, rather than a, radius. For simplicity of illustration, again consider a field which varies azimuthally in a square-wave manner, but Ict the abrupt boundaries bctween hills and valleys be spiraled, as indicated in Fig. 13-4. Tlwn a t every boundary the fringing field has a radial component R , , with a gradient t h a t points in at one edge and out at the

222 FIXED-FIELD ALTERNATING-GRADIENT ACCELERATORS other. In passing through these radial fields, a particle with azimuthal velocity zje will experience an axial force, focusing a t one boundary and defocusing a t the other. This gives rise to a net focusing force, thanks to the alternatinggradient principle. Since this force depends not only on the variation of field strength between hill and valley but also on the pitch of the spiral, it can be made much larger than the Thomas force. The second (Laslett) force which arises from spiraled pole tips depends on the fact that the effects of the focusing and defocusing forces encountered a t the boundaries of a hill are not exactly equal and opposite, as assumed above. An enlarged and somewhat exaggerated view of an orbit crossing a hill is shown in Fig. 13-5. The axial force is focusing as the ion, approaching a valley

+

hill valley

Fig. 13-5. A particle has a longer path in the focusing field than it does in the defocusing field. This is origin of the Laslett force.

hill, traverses the fringing field, and is defocusing as it enters a valley. (The situation is reversed if the spiral winds the other way.) But the path through the defocusing field is more nearly perpendicular to the boundary than is the path in the focusing field, so the defocusing force acts over a shorter distance. There is therefore a net residual focusing force, over and above that associated with the alternating gradients. It so happens that the Laslett and the AG forces (which necessarily occur together) have the same values. Parenthetically, it may be noted that the AG force due to spiraling is akin to the edge focusing examined in 3 4-2, since both depend on the azimuthal velocity ve and the radial field B,. I n the zero-gradient synchrotron, the axial force is focusing a t both boundaries of a magnet, since the slant angle of the edge (equivalent to the spiral angle) changes sign, and as a result B, in the fringing field is always directed outward, as was shown in Fig. 4-3. Such a design entails an average field which decreases with radius, since the ratio of the azimuthal width of a hill to that of a valley is steadily reduced. This would offer no advantage in a cyclotron, for it would not produce a constant frequency of revolution. A rigorous analysis of all these effects is beyond the scope of this book, for the problem is exceedingly complicated. We will be content with a rather crude approximate presentation, which nevertheless exhibits the main features of accelerators of this advanced type. But before the task is undertaken, it is necessary t o introduce the term flutter as a preliminary step.

13-2. Flutter Flutter measures the change in field strength between the hills and valleys. Numerous particular definitions exist, according to the tastes of the authors.

Here the choice is made in accord with a field assumed to vary sinusoidally in azimuth a t :my given radius. A field of this sort may be considered as the sum of an average field < B > and a sinusoidal field of amplitude A = f, where the flutter amplitude f is some nunlher less than 1, and the <> means that an average is taken azimuthally (Fig. 13-13). Fig. 13-6. The relation between the average field < B >, the maximum hill field B,,, the minimum \.alley field B,,, and the flutter amplitude f .

The field a t any azimuth anglc 0 is given by

B

=

< B > ( l - f sin@).

(13-1)

The peak hill field and the minimum vallry firltl are lhen expressed by

B,, = < B > ( l f f )

(13-2)

and

B,

=

( 1

-f).

Thc flutter funrtion P is defined as the quotient obtained when the difference betwecn the mean squared ficld and the square of the mean field is divided by the latter:

T o cvnluatc this, we must find
Froin Eli. (13-1)

and sincc the average of sin 0 i:: zero :iiltl t h a t of sin" 0 is {, then

Con~cqucritlyEqs. (1 3-2) a i d ( 1 3-3) become B,, = jl - ( 2 b ' ) y

224 FIXED-FIELD ALTERXATING-GRADIENT ACCELERATORS

F = (Bh - B d P , 82

13-3. Thomas Focusing

Let the z component of the magnetic field vary with azinluth according to the expression B, = (I - f sin 8 ) . (13-11) To a good approximation, is the average field along the circle of radius ro about which the orbit weaves. For an assumed constant momentum of a particle, we then have Mu = qB,p = q ro. (13-12) Here B, is the field a t any point on the orbit where the radius of curvature is p. I t follows that

so by use of Eq. (13-I]),

A theorem of analytical geometry relates the radius of curvature to the radial coordinate and its derivatives with respect to the azimuth angle:

where r' = dr/dO and r" d2r/d02. In the present situation, r' is exceedingly small compared with r, and r never departs very much from ro. Hence we may say that

When this is set equal to Eq. (13-14) we find T"

=

rof sin 8.

(13-17)

, then multiply Integrate this with the condition that rr = 0 when 0 = ~ / 2 and both sides by the radian frequency of revolution w , to obtain

dr = ',,cle --

-wrof cos 8.

S o w the relative orientation of the cylindrical coordinates r, 8, z , is shorvn in Fig. 13-7, with z chosen as vertical. The guide field 13, re:~cts with the azimuthal velocity ve of a positively charged particle to give the central force

THONBS FOCUSING

223

F,. which acts along the direction of dccwasing r , so the projectile's velocity is in the dirtxtion of negative 8. Consequently the average radian frequency Fig. 13-7. Orientation of coordinates and parrrneters.

B&r

6

(angular velocify) of the particle, given hy w = q/M, is positive in the direction of minus 6, SO t h a t w = -dO/dt. Therefore the radial compor~elltof vclocity 71, is found to be dr -

d6 d r --

dr de

w-=

tit

do

tlt

-up.

C'oml)ining this with Eq. (13-18) yields 11,

=

W T ~cos S 6.

(1 3-20)

TI'(, tnnst now obtain an expression for the azimuthal field Bo. The weakening of t l ~ rfield I:: i n the valleys is :-tccompanied hy a bulging of the field 1inc.s towards thew regions, as shown in Fig. 13-1, and it is clear that the azimuthnl conlponent increases with x. \Ye sssumc. this incl-rase to he described adequately by the first term in a Taylor's exptmsion:

Since curl B = 0, then d B ~ / a z= ilR1/ ir,,r;lO),so that

The value of the derivative is obtained from Eq. (13-11) ; so on setting < B,> = A l , /c/,\ye find

BB -- -

xA!!ilo f ~ o 6.s r,,q

-

(13-23)

The Thomas force is found from Eqs. (13-201 and (13-23) :

F,

=

(~tl,n,

= -z M 1 ~ f'

coS 0.

(13-21)

This is always directed toward the mid-plane and hence is focusing. I t fluctuntrs in strength, but thtl average value is



=

--$zMwZf'.

If this is the only axial force, t h r equation of motion is

226 FIXED-FIELD ALTERNATING-GRADIENT ACCELERATORS and the Thomas radian frequency is given by

Since in practice it is difficult to make f even as large as 4, the Thomas frequency is much smaller than that of rotation. The betatron frequency for this maximum practical value of f is Y T ~= W T ~ / W = f/d\/Z= 0.17. This is comparable to the betatron frequency in an ordinary cyclotron, so it is seen that the Thomas force can be of the same order of magnitude as the familiar weakfocusing force produced by a field which falls gradually with increasing radius. 13-4. Spiral Focusing

Consider a cyclotron with spiral Thomas poles. In the region where the field lines are bowed, there is a horizontal component B7,a t points off the mid-plane, and this componcnt is perpendicular to the spiral locus of constant phase of

Fig. 13-8. With spiral pole tips and with vo in the plane of the paper, the field component B , must be imagined to come out of the page at an angle. The view is from the center of the machine.

field; in other words, the "fringing field" is normal to the "edge." A side view is shown in Fig. 13-8 and a plan in Fig. 13-9. It is seen that B7,has com,

,tongent

t o spirol orbit

Fig. 13-9. Field components and the relation of the spiral angle ( to the radius of the orbit and to the tangent to the spiral.

4

ponents Bg and B,. The angle between the tangent to the spiral, where it crosses the orbit, and the radius drawn to the point of tangency is called thc spiral angle 5. It is clear that B, = -Be tan j. (13-28) When we substitute for Be from Eq. (13-23), this becomes

SPIRAL F( )C'UPI-\TG

227

The particle's path is now :issurncd to hc circular ( t h a t is, scalloping is nrglec.tc~tl)so it, vclocity i b 7'0 givm by

the minus sign being uscd for the rcasor~sgiven in establishing Eq. (13-19). Tile axial forw is Eh = - ( / v J ~ ~ =

~ M u l fr o s B t a n

r.

(13-31)

This force is alternately focusing ant1 dcfocusing, tlepending on the fixcd sign of tan [ and tlrc changing sign of cos 0. At first sight it odd seem that t,he average for,-( should be zcro, but :rcrount llns not yct been taken of the diffcrcnt p a t l i lengths in the focusing : ~ n d ticfocuiing ficlds. Fig. 13-10 in.,,

/

-/*\

,t a y g e n t to spiral

.

volley

/ / frmgmg/

r '51'

orbit

orbit circle

circle

Y

# t o center Fig. 13-10. Path of a particle in a spiral triugirlg field of thickness t.

Fig. 13-11. The relatiol~sbetween dr, and dB.

E,

r,

dicatcs that in the region of a hill the orhit lies outside the mean circle am! that the radius of curvature p is smaller than To. The dotted lines indicate the space of thickness t in which the spiral fringing field may he considerc.tl to csist. I,ct E be the angle between the tangents to the circle and to the orhit. I t is :~l'purcnt that the length I, of the orhit in the fringing field is ap~)roxinmteIjr

L

--

t ---------

m s ({

+

t)

and that c is a function of 0. The ef'fectivcncss of the force of Eq. (13-31) tvill be t:iken into account if the force is ~l~ultiplicd 197 L / t . But since wc arc going to colnputc thc averagc force,, tllcs vcighting factor must hc n-rittcn as

so its :i\cmgr value is unity. Kon. t depends on 0, hut { does not and E is small, so wc write

228 FIXED-FIELD ALTER NATING-GRADIENT ACCE1,ERATORS

because e is positive in hills and negative in valleys. The weighting factor is therefore

+

cos ( cos ( (13-33) 1 e tan (. 1 - c tan jcos (j6) cos - c sin c From Fig. 13-11 we see that tan E = dr/(rcl0), and because e is small and r does not differ much from ro, it follows that e % dr/(rodO). When Eq. (13-18) is used for &/do, we obtain d z -f COS8. Then Eq. (13-33) becomes

w.f.

=L

+ "

w.f. = 1 - f cos 0 tan (. Therefore the weighted force of Eq. (13-31) is

F, = z M d f cos 0 tan ( - zMu2f"os2 0 tan". If this is the only force acting, the equation of motion is

d2z - = zf cos 0 tan 5do2

- qf2 cosZ0 tan2 c.

(13-36)

(13-37)

This equation cannot be integrated by direct methods, but a solution can be reached by successive approximations. We note that on the average the first term on the right vanishes, whereas in the second term the factor cos2B becomes +. Therefore the zero-th order approximation gives the equation

For constant values of f and (, this represents simple harmonic motion; designate this approximate solution as Z = 2% sin

(a),

where the radian frequency is

We see by reference to Eq. (13-27) that 6 is comparable with the Thomas frequency, and therefore 2 is a slowly varying function of 0. Hence in this rough approximation, the application of the weighting factor to the fringing fields of the spiral causes the orbit to travel a long distance above the median plane before it passes through to spend a long time below it. The larger the value of 2, the stronger are the in-and-out radial components of field which are encountered a t every traversal of hill and valley, so the ion is forced into rapid axial oscillations, of small and varying amplitude, about the path 2. If we represent these deviations from z as Ax, the displacement from the mid-plane is given by

SPIRAL FOCUPIKG % =

< :.

wit11

5 -+ ai.

229

(13-39)

\Ye now makc t h c ncxt approxiln:ttion by insvrting Eq. (13-39) into the basic Eq. (13-37), but on the right side we drop A2 in comparison with Z. This gives

T h e second term on the right is inucli smaller than the first, so we t a k e its average value and obtain

d z (1' Az - $ ----do' do'

=

2f cos t) t a n

{

-

1' ! 2'-- tan? {.

2

Kow if Eq. (13-38) is a valid rough nppl'oxiniation for the variable 2 , i t is also valid for the ~ a r i a b l e2 . Therefore t h e first and fourth t e r m in Eq. (13-41) canccl, leaving (1' Az - xf cos 0 tali <. do' Since f a n d tan 5 are constzmt and 2 nearly so, because of its long period, this equation m a y be integrated twice t o yicltl Az

-2f cos 6 tan

=

{,

( 13-43)

the constant:: of integration being rn:dc, zero hy proper choice of tlic initial conditions. This value of As is usctl in Etb (13-39) t o give

z

2(1 - j cos 0 tan {).

( 1 3-4 4) hTow use this value of z in Eq. (13-373 ; :tnd drop the term in I". Further, sinre 2 =I Az, Az is inconsequenti:d compared with x . so we m a y set 2 = z. \Ye obtain =

+

d 'z de, -

fY

- 2'- tan"

2

f 2tan2 <. - 2'--2

This is t h c final approxiniation wc sl1:~11USC. If tlie origins of tliesc two eclu:~l tcrins a r e traced, i t will be found t h a t tlie second arises from the weighting factor appliecd to the path through the fringing ficlds, while the first colilcXs from the alternating gradient effect. Now recall t h a t d % / d P = ( l / U 2 ) d2z/dtx, a n d add t h e t u o terms, to find d2z rlt'

-

+ (w'f'

t:111'

{)% =

0.

(13-40)

But the T21oin:ls force of Eq. (13-25) is ncwss:~rilypreberit if t l ~ e r eis a n y azi~ n u t l i a lvari:*r ion of field, so if the machine has no over-all radial gradient tllc proper ccpation of motion is

230 FIXED-FIELD ALTERNATING-GRADIENT ACCELERATORS Since by Eq. (13-5) we have F = f2/2, this can also be given as

so that the axial betatron frequency is

This approximate expression shows the influence of the flutter and of the spiral angle on the axial stability. In most cases the accelerator will also be subject to the effect of a radial gradient in its field, but before this can be taken into account it is first necessary to introduce the concept of the average field index. 13-5. The Average Field Index k

Up to the present chapter in this discussion of magnetically guided and focused accelerators, the orbit's radius of curvature p has coincided with its radial coordinate r measured from the center of the magnet, or if straight sections are involved, with the radial coordinate of the orbit with respect to the center of the portion of the magnet between such field-free regions. The symbol r has been used for both these identical radii p and r; and in particular, the field :?dex has been given, as in Eq. (2-1), in the form

I n many of the FFAG machines the orbit's radius of curvature p and its radial coordinate r do not have the same value, as may be seen by looking again a t the sketch of a Thomas cyclotron, Fig. 13-3. The forces due to motion in a magnetic field have directions determined by the particle's velocity and hence arc associated with its local radius of curvature. Strictly speaking, whenever there is any distinction between p and r the field index should appear in the Eorrn

where dx is perpendicular to the orbit. If the field varies with azimuth, then the index is no longer constant along the orbit. Such a situation, it is true, has been encountered earlier when straight sections were considered in synchrotrons, where after traversal of a region of constant field and constant index, the field abruptly goes to zero; but the analytical methods employed in studying the orbit properties sidestepped the fact that the index becomes indeterminate. The circuinstances are different in the present situation, for the field may vary continually with azimuth. Some new concept is plainly needed to supplant the simple index n. The new paramettlr is the average field index k. The path length L of any closed orbit may be associated with the radius 1L)

THE AVERAGE FIELD INDEX k

23 1

of an equivalent circle through the relation I, = 27rR, as discussed i11 § G:3 al)ropos of the moincntuni cornp:~ction.At evcry point on tlic pit11 t l ~ elocal field R is rclatcd to the local radius of curvnturc p by Eq. (1-2), no\\ vrittcn as B = A l o / ( q p ) ; so since‘ tlw velocaity and mass are constant, the average field is given by

where dB is the element of angle about the center of curvature. This result ~ ~ O W t h Sa t a unique value of is associated with each particular momentum, orbit length, and equivalent radius. Thus not only do me have p = qBp applicable a t every particular point, hut we also have, over the entire path, Now if the azimuthally varying field changes witli distance from the c c n t c ~ of the ninchine, i t is obvious that the avorage field also will exhibit a variation with changes in the equivalent radius. This dependence may be expressed by an equation of the form

analogous to Eq. (2-3). Here < R I > is the average field a t some reference radius R,, and is the average field at a slightly different radius R , while k is some numbrr which describes how varies witli radius in the ncighborhood of R I , k being considered constant in this region. Thus the average field index k plays much the same role as docs the field index n , although k refers to the field averaged in azimuth. Note t h a t a ficld which increases with radius is described by a positive k and by a negative n. This reversal of sign is simply a convention, with no dcep-lying significance. When the derivative of < B > is taken with respect to R , i t follows that,

and alt~rnat~ively

The simplest and best way to find k often is through its relationship with the momentum compaction a. Thus from p = q < B:>R, we have

But from Eq. (13-55) d < R > / < B > = ktlR/R, wlieiice

232 FIXED-FIELD ALTERNATING-GRADIENT ACCELERATORS But by Eq. (6-2) we always have a

=

(dp/p)/(dR/R). Therefore

a = k + l

(13-57)

k=a-1.

(13-58)

and For a simple cyclotron it was seen in Eq. (6-3) that a = 1 - n ; so for such a machine k = - n, and consequently k lies between 0 and -1. For a race-track weak-focusing synchrotron, Eq. (8-6) gave a = (1 - n ) ( l s/m), where s m are the lengths of the straight sections and of the magnets. From this it and follows that k = [ ( I - n)s/m] - n, and since s/m is less than unity in practical machines, k again lies between 0 and -1. The momentum compaction of a typical zero-gradient synchrotron has been shown to have the value a = 0.6, by application of Eq. (6-7) ; hence the k is -0.4, indicating that the average field falls with rising radius, so that vertical stability exists. This way of looking a t the genesis of axial focusing forces in a ZGS was mentioned qualitatively on pages 56-57. I n the FFAG machines that are the subject of this chapter, the average field increases with radius, so that k is positive.

+

13-6. Betatron Frequencies The accurate computation of the betatron frequencies in a general AVF machine is extremely complicated. The field a t any radius goes through N cycles of sinusoidal variation with flutter F, the maxima and minima lie on curves of spiral angle 5, and the dependence of the average field on the radius is described by the quantity k. I n the most general case, F, 5, and k are all functions of the radius. The problem is difficult, since the equilibrium orbit has a continually varying radius of curvature p which is unequal to the radial coordinate r, and the gradients dB/dp and dB/& have values and orientations that are markedly different. Further, the gradients are large and change in short distances, so that use of only the linear term in a Taylor's expansion for the force and motion is far from adequate. The equations of motion do not have constant coefficients but are of the Hill or Mathieu type. All of the mathematical techniques used by different authors to obtain solutions involve some approximations and derive the answers in the form of infinite series. I t would be beyond the scope of this book to expound and expand in detail the papers which treat the subject thoroughly, and it is beyond the abilities of the present author to compress the arguments in any illuminating and satisfactory manner. We will be content with quoting the leading terms in the resulting expressions. These are: v,2 = 1

+k +

e

m

.

(13-59)

It will be observed that the second of these tlifTers froin the w r y approximxte Eq. (13-49) by the inclusion of the f:tctor involving the number of sectors. Xote also that for a s i r n p l ~iliagiiet without flutter and spiral and wherein the ficld falls slightly with radius, these expressions rcducc to the fainiliar equutioris for a weak-focusing cyclotron: v,:' = 1 - n and v,L = n , for in this cast: X: = -n, as shown in 5 13-5. F o r the AVF imtgnct, i t is apparent that thc flutter and the spirals have no effect on v, (to the approximation sho~vn),but can have a profound iriflucnce on v,. The factor N L / ( N L- 1) increases with decreasing S,w l l ~ c lniay ~ be undcrstood qua1it:~tl~elyfrom the optical analogy: a small Y means widely spaced lenses and hence :t stronger net convergence than if the spicing had been closer. Therefore, as far a s obtaining a large v, is concerned, the fewer the sectors the better. But i t will appcar later that radial stability may then be iinpaircd. The approxmlate relations shown above are not sufficiently accurate to br used as the sole criteria in designing a ~nachine.B u t neither are the conlplttc cxprcssions, since an actual magnet seltlorn produces cxactly the field desired because of the variations in pcrrncability of iron t1i.tt accompany differences in flux density throughout the structure. In order to ensure that stability actually will be attained, measurement of the field in a model magnct usually is required, follocved with a point-by-point tracing of the orbit with the aid of a digital cornpuler. On the other hand, Eqs (13-59) and (13-60) are sufficiently reliable to serve as gcncral guides, and me will use tliein for that purpose. The machine first t,o be consicleretl is the isochronous cyclotron. 13-7. Isochronous Cyclotrons

Since an azirnutlially varying field, possibly aided by a spiral characteristic, supplies an axial focusing force which is independent of any radial gradient, i t is possiblc to exploit such AVF forces to overcome the tlefocusing effects caused by an average field which rises In proportion to the projectile's mass, thus prrrnitting axial st:tbility to exist in a machine with constant revolution frcqucncy which would otherwise be unst:~tde.Such a rising field means a positive value of k, and inspection of the last two equations shows t h a t this will increase the value of v,', h u t if vEL is to remain positive, the terms involving flutter and spiral angle must be large rnougli to overpower - k a t all times The ac1v:tnt:lges of such an isochronous cyclotron are many. In the first place, a constant-frequency oscillator is adequate, and the complexity and cost of a frcclucncy-ii1o(lul:itct1 systcm is avoided. Since there is no shift in the phase of the voltage when p:trticles successively cross the dee gap, there is no ultimate deceleration to set a limit to the final c3nergy, as occurs in an ordinary cyclotron. A substantially greater energy bccornt~spractical, although the figure is not unboundctl because of reasons to bc>given later. And finally, the output of particles retuins to tlic high valut, c1i:~racterislicof thc simple cyclotron, since a new batch of ions is withdrann froill the source and an old batch is delivered

234

FIXED-FIELD ALTERNATING-GRADIENT ACCE1,ERATORS

with the output energy a t every radiofrequency cycle. The price paid for these advantages is a vastly more complicated theory of orbits and the necessity of constructing magnetic fields of great complexity and high precision. There are two distinct goals in the design of an isochronous cyclotron; first, to obtain an average field which rises appropriately with radius, and second to maintain orbit stability. There are three basic factors in the solutions of each of these problems, and these may bc applied in various combinations. Consider first the means of obtaining a rising average field: (1) The distance between pole tips may be decreased a t greater radii. The main exciting coils around the poles thereby produce a stronger field near the edges than a t the axis of the poles, because of the reduced reluctance of the air gap. If this procedure is carried too far the peripheral spacing may become too small to allow room for the dees and their necessary clearance to ground. Although in principle an extremely low dee voltage is adequate in an isochronous cyclotron, so that a large number of revolutions may be used to reach the final energy, this is not the case in practice. Magnetic fields cannot be built with infinite precision and some departure from isochronism must be expected, so the accumulated phase shift after a large number of turns may well exceed 90" and thus lead to deceleration. Greater certainty of successful operation follows the use of as large a dee voltage as space limitations will permit. (2) Concentric circles of conductors may be attached to both pole faces, and supported in the planes of the hills, with individual control of the current in each. By proper choice of these currents and of their direction, a significant contribution can be obtained to the desired rise of field with radius. Such conductors are often called t r i m m i n g coils or k coils. (3) The azimuthal width of the hills can be increased a t the expense of the azimuthal width of the valleys, since it is only the azimuthally averaged field which must rise toward the periphery. This can be brought about, for example, by making one boundary of a hill radial, while the other curves away, or both boundaries may curve. To maintain stability, there are three variables a t our disposal: (1) The flutter. This can be produced by employing small spacing between the upper and lower poles of the magnet in the region of a hill and large spacing a t valleys, care being taken to ensure that the minimum space is adequate for the dees. As an aid, "flutter coils" can be placed in the valleys and activated so as to oppose the field produced by the main coils; or pole-face windings on the hills can raise the field in these regions. Both types of coils add expense and complexity, but they introduce an element of flexibility which is not present if the flutter is obtained entirely by shaping the iron. Such flexibility is highly desirable if the cyclotron is to be of variable energy or is to accelerate different particles (see $ 1 3 - l l ) , for the flutter caused by the iron will change with the average ficld strength because of saturation effects. It is sometimes desirable to change the relative amounts of flutter a t different radii; in this

case the coils in valleys and on hills ruubt tw nlultiplc, so that those a t snlall and a t large radii can he powcrcd separately. (The problcrn of finding space for the necessary input contluctors ih not trivial ) (2) Tlic spir:il angle. Once built, t h s cannot hc. c hanged. When the liills arc nidc and with onp houritlary r:ittlal w111lethe other is twrvcd, the precise rue:ming of the spiral angle beco~nesso~newliatnelmlous arid some average value must be used when Eq. (13-601 is employed. In pmctice, a hill so shaped requires model work and orbit tracing. (3) The nu~nberof hills. The li~nitntionson this number will be discussed in 13-9. All in all, the possible field shapes and thc means of attaining them are almost endless--and so arc the difficulties, for many parameters are interdepentlcnt. For example, it is harder to obtain :I strong average field with a large fluttcr than with a srnall one; an increase in t t ~ enunibc~ro f hills nlakes it h:rrdcr to create fluttcr, particularly if the axial : ~ i rgal) is 1:trge; wide valleys supply more room for valley coils hut necessit:itc narrow liills and hence a srnaller average field; pole-face windings on the. hills take up precious and exl?ensivc space; all windings generate lieat whicli nus st be ren~ovtd,even if the profligate use of p o m r is not objectionable. Many t~oinpromiws,tempered by judgment and engineering necessity, must be made. Special problems arise a t the cyclotron's center, since i t is apparent t h a t no significant flutter can he obtained as the hills and valleys converge to a point. Axial focusing in the central region must be foregone or else obtained by the aid of an initial drop-off in field brougl~tabout by supplanting the llills and valleys by :i solid disc of iron. possibly slightly conical. This condition is ~ n n i n tained up to a r:ictius a t which the influence of flut tcr can bc appreciable 1 ) ~ cause of with. cwough valleys. This m m l s , of course, that isochronisrn does not exist a t the st:~rtand some shift of p11:w of the particles will occur. In the discuhsions to follow, this situation will h(, neglected for si~nplicity'ssake and it will be :1sbun1e(lthat the isocllronous ficld re:lcllcs all the way to the center.

a

13-8. T h e Conditions for Isochronism

I n tcir~n.;of the azimutllnlly :I\-cragetl field, the h s i c cyclotron equation is

since y -- E/E,,. The ~nachinewill 1)e i~or11ronoub(that is, the ions will rcvolvc a t coriht:~ntfrequency) if = yBc, (13-62) where B, is thc field a t the center of t11e machine, for then

236 FIXED-FIELD ALTERNATING-GRADIENT ACCELERATORS which is a constant. Equation (13-62) is useful in calculating the increase in average field needed to reach a given energy with a particular particle. Thus for Eo = 75 938 protons with final kinetic energy T = 75 Mev, we have E = T i= 1013 Mev, so y = 1013/938 = 1.08. The average field must rise 8% from the center out to the final radius. Note that the percent increase is less for heavier particles, such as deuterons or helium ions. To describe the isochronous condition in terms of the average field index k, we proceed as follows. Csing Eq. (13-62) and its derivative d = R,dy in the expression for k given by Eq. (13-551, we obtain

+

Since y increases with R , this expression can be put in more useful form if we can find a relation between y and R. To obtain it, we write Eq. (1-35) in terms of and R , rather than B and r :

Rqc = (E2 - E02)M= Eo(y2 - I)$$. (13-65) Also, Eq. (13-61) can be arranged as = wyE,,/(qc2), so Eq. (13-65) becomes

and therefore 02R2 -=I---. c2

1 -t2

Taking the logarithmic derivative of this yields

When this is used in Eq. (13-64) the result is

-

As an alternative expression for the isochronous condition, we recall that r12 = Y2 - 1 by Eq. (1-22), where 7 Mu/ (Mot). Hence k = r2.

(13-68)

Also, Eq. (1-20) shows that y2 = 1/(1 - P2) always, so Eq. (13-67) becomes

Eqs. (13-67), (13-68) and (13-69) show that if a cyclotron is to be isochronous, then Ic must increase throughout the acceleration, starting with the value 1, as shown zero. Since the momentum compaction is always given by a = k hy Eq. (13-57), we obtain from Eq. (13-67) the correlated conclusion that

+

Consequently, in an isochronous cyclotron, succt.ssive turns are more and more tightly packc~ltogctlicr. (This gi1,cs an additional reason for a large tlcc voltage, so t h a t t h t w may be sufficient separation bctwcen the final orbits for t l ~ e introduction of the septum of a dcflcctor s y s t ( w ; scc Eq. (7-20) or (7-21), wherein n is now rcpl:md by -k.) A sopllisticnted w:iy of arriving :lt this last expression is to recall the general expression Eq. (6-9) derived in the chaptcr on 1)liase stability, namely,

For an incrcasc in momentulll t i p to cause no change in thc period of rotation, as is implied hy isochronism (that is, to rnake (17 = O), it must f o l l o ~that (Y = y2. Fronl this point of view, an isochroiioui: cyclotron is permanently a t the transition c u c q y , n h c w 1)hnsc stnl~ilityis lacking. T h e dcpendencc of X on the radius can be determined by the use of /3 = V / C = d i / c in Eq. (13-60) t o obtain

T o find how varies with radius, we return to Eq. (13-68), which says that k = 7 ) ' - p2/(M,,c)' for isochronis~n. But always p = < B > q R , so k = (qR)"(*II,c)? But also k = R d < B > / ( < B > d R ) , so on equating these last two expressions for k we find

Integration bc1,ween R = 0 where yields



:= R, and R = R where



=

,

This gives < B > as a function of R , of the central ficld and of the ratio of the cl~argcto the rest mass energy of the particle. I t is al)parent that a different contour of t l ~ efield is needed for each particu1:tr ion, and if the cyclotron is to be of v:trii~t)lc energy a different contour is :dso rcquircd for each f i n d energy to 1)c re;tcl~c.tl:it :t fixed output r:~tlius. I t nlust I)c en~plinsizedthat th13 cqui~.,tlentr:~tliuiR appearing in the ab0r.c exprcssion is not the same as t11e radial coordinate r of the particle, altliough whcn both r a d ~ i:we large their diffwcnce b c c o m ~ ~ws r y small. 1,ow-energy p:irticlcs dmcmt)e p i t h s wit11 consider:~bleweaving back and forth (scalloping) bout an awr.:rgc c.irc.lc~of r:uhus r , and because of tho rctluced ratlius of curvature p 1x1 the hills, the actual path in a hill is noticeably louger than tlu

arc of the mean circle, so that the average field traversed is larger than if scalloping were not present. Consequently, the required rise of < B > as a function of r is not as great as its rise as a function of R. This must be taken into account in the design of the pole tips and auxiliary windings, by analyses and techniques that are too involved to be described here. I n many cases it turns out that the isochronous average field < B > must actually decrease before it starts to rise, when plotted as a function of the distance from the center of the magnet. It may be of interest to consider the analytical expression for a relatively simple field suitable for an isochronous cyclotron. If the field a t any radius varies sinusoidally with azimuth and goes through N cycles in one turn, an appropriate expression is (1 3-73) B = < B > ( l - f sin NO). Here must increase with radius in a manner suitable for isochronisin, as shown in Eq. (13-72). The value of B given in Eq. (13-73) implies that the sector boundaries are radial. To extend the description to include spiraling, it is necessary to introduce a phase shift, which depends on IZ, into the argument of the sine function. Write

[l - f sin N(B - + ( R ) ) ] , (13-74) where the function @ ( R )is still to be determined, its form and value depending on the shape of the chosen spiral. The spiral fixes the loci along which

B

=

t a n g e n t to spiral

Fig. 13-12. The spiral angle

c.

the field has the same phase, such as its maxirnuin or miniillurn value. Constancy of phase means constancy of the argument of the sine function in Eq. (13-74), so we set 0 - +(A') = conhbnt, whence d@ - -.do dl2 - dR Now as shown in Fig. 13-12, the spiral angle is given by

Therefore Eq. (13-75) becomes

SO

that,

T o find tlie sptcific value of the f~uiction ~t is newhsary first to know the value of t a n 5, :md this will depend on the sli:tpe of the particular spiral in use. Consider the consequences of choosing an Archimedean spiral, for which the polar angle 0 is proportional to the radius: Here p is a constant with dimensions of radians per unit length. Then d 0 / d R and if this is used in Eq. (13-76), we find tan

< = pR.

= p;

(13-79)

The tangcnt of the spiral angle tl~ercforcincrc:tscs with thc radius. Tlic orbits are i ~ l ~ i ~c-ircular o ~ t and so arc approxin1:~tcly ]~erl)endic~~liir to R ; conscyuently, with the sl)iral angle increasing, tllc hills m t l vullcys become rnorc and more tangential to the orbits. The function @ ( R )is, by Eqs. (13-77) arid (13-79),

e(R) =

/gR (

1 =~ pR.

(13-80)

The lower limit of intcgration i~ zcro, since K goes to zero with 0. The expression for the field bcconics

B

=

< B > [I - f sin N(O - pR)].

(13-81)

Archinledcan spiral ridges have been chosen for several of the isochronous cyclotrons alrmdy built or under construction, since the increasing value of the spiral angle relieves thc flutter of s o ~ n cof the burden of maintaining axial stability. It is worth pointing out that as far as orbital stability is concerncd, it is irrelevant whcther the spiral field unwinds In the direction of the particle's motion or in tlie opposite sense. But with regard to tbxtrnction of the full-energy beam, either 11y an electrostatic deflector or by the regenerative method, one or the other direction of spiraling may be prcfcrablc, depending on particular circumstances. 13-9. Radial Stability and Energy Limits in Isochronous Cyclotrons

It has been seen in Eq. (13-67) t h a t the condition for isochronism is h- = r2 - I. The approximate exl)rcssion of Eq. (13-59) for the radial betatron frequency becomes vZ = 7 - +

-...

(13-82)

Ions leave the source with negligible energy; so y = 1 initially, and therefore V, starts at, unity and increases thereafter. (Kate this difference in hehavior

240

FIXED-FIELD ALTERNATING-GRADIENT ACCELERATORS

compared with a weak-focusing cyclotron, where v, starts a t unity and then decreases.) Just as before, care must be taken that v, does not reach any dangerous resonant values and, in particular, any values which represent complete instability. The most important example of this latter group can be discovered by rccalling the matrix method of investigating stability, discussed in Chapter 3. There it was shown that the geometric and magnetic characteristics of the accelerator must be such that the quantity $(Mil M Z 2 ) must lie between +1 and -1 if radial stability is to exist, where the quantity in parentheses is the trace of the transfer-matrix across a sector. I t was also shown that if this limitation on the matrix elements is obeyed, then we may set $(MU M22) = cos a,, where a, is the radial betatron phase shift per sector. Since cos a, equals -1 when a, equals T, this value of a, represents the brink of a basic instability which must be avoided a t all costs. Now it has been seen in Eq. (3-72) that a, = 2~v,/N, where N is the number of sectors; and therefore as v, rises with increasing energy it must not be allowed to become so great that a, becomes equal to T. This necessary termination to the growth of v, is sometimes called the radial s-mode stop-band. We see that v, must always be less than N/2; and since the least value of v, is unity (just as the ions leave the source), it follows that N must exceed 2, which means in practice that N must be at least 3. Two hills and two valleys are not enough for an isochronous cyclotron. It can now be seen specifically how the T-mode stop-band imposes an energy limit. Suppose the minimum value N = 3 is chosen. Then the stop-band is reached a t v, = Na,/(2a) = 3 ~ / ( 2 ~ = )3/2. Now y = 1 T/&, so T = (y - I)&; but v, w y, so T = (8 - 1)Eo = Eo/2 and for protons this is 469 Mev. If N is made 4, the stop-band occurs a t v, = 2, so the energy limit for protons becomes 938 Mev. (In practice, both of these energy limits are somewhat lower, since the neglected terms on the right-hand side of Eq. (13-82) are not negligible; that is, v, is really greater than 7.) Thus the maximum energy can be pushed higher and higher by increasing the number of sectors, although this reduces the value of v,, as shown by Eq. (13-60)) and it also makes more difficult the attainment of a substantial amount of flutter. But even if N approaches infinity, the integral and half-integral resonances of v, offer dangerous pitfalls which, though not absolutely destructive in the same sense as the radial T-mode stop-band, may well cause very heavy loss of beam unless they are passed through very rapidly by means of a large energy gain per turn. Since v, starts a t 1, the first resonance of this sort is encountered a t V, = 3/2, corresponding to T = Eo/2.

+

+

+

13-10. Axial Stability in Isochronous Cyclotrons

By Eqs. (13-60) and (13-67) the expression for v t in an isochronous machine can be written as NZ F(1 + 2 t a n 2 { ) ..., (13-83) v,2=l-y2+NL- 1

+

A t t h e c e n t w -y = 1, a n d it is in1pc)ssiblr t o crcvtte amy flutter. Consequently v, starts off at zero and, a s -y increases while the particles gain energy, v, hecomes i1li:~gin:~r.yout to t h a t radius where t l ~ cflutter begins t o he :tpl)reciable. T o avoid this situation, a weak-focusing, slightly falling field is nccdcd near t h e ccnter of tlle cyclotron, t l ~ vi s o c h r o n o ~ ~coii~htion i h ~ i n gabantloncd for a few turns. Thereafter, a positive arid possit)ly growing value of v, can be 01)tained, provided t h e ternis i n v o l ~ i n gflutter and spir:iling are inadc t o exceed 1 - r'. I t is difficult t o increase F contiri~~ally as the cmergy riscs, so in these cnhes t h e spiral angle is progressively enlarged. A value of v, in the neighborl~ootlof 0.1 to 0.3 would appear to bv adequate, in tlic light of ordinary cyclotron experience. I f the figure slioultl rise to tlic resonant value of 0.5, bcverc loss of 1)enin m a y bc~expected, bccaubc of the liniited axial aperture of t h e dces. T h e use of a falling fieltl near t h c ccmter, to o k t i n axial stability hefore flutter can slipply it, gives ribe t o a difficulty with rcgard to radial s t a h l i t y : when the field stops fallmg :mtl hegin:, to riw according t o the isoclironous wlicdulc, i t ni11st go through :I condition of zelo gradient ho t h a t lz = 0. Equation (13-59) shows t h a t then v, will pass through t h e resonant value of unity. This can bc scrious unless traversed quickly. 'I'l~erc~forca large dec loltnge is desirable. 13-11. Variable-Energy Multiple-Projectile Isochronous Cyclotrons \\'it11 thc advent of isoclironous cyclotron\, the feasibility of variableenergy operation o w r a wide range has greatly incrc:lscd. Since such macliines obtain their p~ol~crlyshnpctl ficltls largc Py by tllr influence of current-carrying coils and polc-f:tcc windings, it ~h c~xpectetlthat the field contours appropriate for liigli or low cncXrgy a t i~ single final r:dius can be ohtninecl simply I)y :djusting tlicw currenth, without the ncwssity of rclocating a n y pieces of iron. t h a t the frcqumcy of the tlce voltage can O p c n t i o n of tliih sort prtwip~)oxcs be clianged in proportion to thc ficltl. If such :t tuning r:tnge is not practical, 1 ,will he described in 3 13-12. 11:trinonic operation can bc u ~ ( ~as A niwcliine with such flcxihility i11 field sliape :tnd oscillator frequency is ideally suited to liandlc a wider m r i e t y of p:trticles t h a n the custoinary trio of deuterons, a l p l ~ a s ,and ~ ) r o t o n s(in the f o ~ l nof singly chargcd nioleculur ions). T h e critical propertie. arcb the cllnrgc: a n d mass. Various plots can 1)e n u d e t o exhibit tlic requirrtl v:iliw- of fic,l(l, r:diuc, and frcqumcy :it :z. rnngc of energies for ions of diffcrcnt c~llarge-to-niass ratio. One of tlir most useful is described in the following p:~r:tgr:ll)hs :tntl the assoc~iwtctlfigul-c. I n tcrnis of t h e average ficld < B > and the ccjuix alent radius R, Eq. (1-36) becomes

242 FIXED-FIELD ALTERNATING-GRADIENT ACCELERATORS where Z = q / e is the degree of ionization of the particle, T is its kinetic energy, and E, is its rest energy. If the ion has mass number A (that is, if i t is formed of A nucleons) this can be written as

.

=

& [((f)'+ 2 f 21".

Here T / A is the kinetic energy per nucleon and Eo/A is thc rest energy pcr nucleon. Hence if R for a proton is plotted against T, corresponding curves of R for ions of different mass and charge can be obtained simply by multiplying the proton's R by the ratio A / Z and by interpreting thc T scale as the kinetic energy per nucleon. Such a plot has an enhanced usefulness if a scale of fR is added, parallel to that of kinetic energy, f being the frequency and R the final radius. The rclatioii between fR (which is equal to w R / 2 ~ and ) T may be derived in a form convenient for calculation in the following manner:

where Eo = 938 Mev. From this, fR is computed for clioscn values of T and recorded on a n auxiliary plot from which the values of T for integral values of fR may be read. f R (Mc sec-' inch)

T / A , Kinetic Energy per Nucleon (Mev)

Fig. 13-13. Relations between kinetic energy per nucleon, magnetic field, radius, and frequency. The number on each line is A/y, the ratio of the number of nucleons to the degree of ionization.

VARIABLE E S E R G Y BY I LXRAIOSIC UPERATION

243

Such 2% univwsal chart as in Fig. 13-13 is of considerable use. For :L machine with assumed values of final radius and limits of oscillator frequency, two vertical lincs arc drawn correspondmg to the niaxiniuni and rninirnurn values of f R , :md two horizontal lines at, the greatest and least values of the average field, each nlultiplied hy the final r:idius The resulting rectangle bouncls thc ions ~ v l ~ i ccan h used and th~1rangc of cncrgy that each will Iiave. (For tlic purpobe of tliis chart, the clistinction 1)etwcen R , r , and p is negligible.) I t is assumed, in the use of the chart of Fig. 13-13, that the nlulticharged ions arc formcd in the ion sourw ratlier than hy :t stripping process which occurs later, as in tlie manner describctl in # 7-17. ITscful quantities of C", S 5 + ,and 05+already have been o b t a i n d from cyclotron sources and it may be prcsuined t h a t higher charge states will be produced as experience grows. 13-12. Variable Energy by Means of Harmonic Operation

Let us as,utncBthat the nlagnetic field in an AVF cyclotron can be varied over :is wide a range as desired and that the limit on energy range is set by the possible variation is oscillation frcquency. Suppose this parameter can cliangcl by a factor 3, from wrjO to w,/"/3. Since an isochronous AVF cyclotron is designed 50 t h t thc turning and t.lcctric frcqucmies always coincide, then a t all times w = a,/ (if the harmonic order is h = I I arid consequently, since w = q l M , < B > will be made t o vary from

to < B O >/3. (For the purposes of tliis discussion, we neglect relativity.) Then the kinetic energy, given by T = (R)"g"/(2M) will cxhange from TO to T0/9. Assuming t h a t the field can be lowered further hut the o s d l a t o r frequency cannot, another reduct ion in energy can he attained by opera1,itig on tlie third harmonic3. (\Ire assume t l ~ cusu:rl structure of 180" does; then as shown in 7-17, acceleration is possil,le only if h -- wrf/w is odd.) Wc start with T = T0/9 and < B > = < H ( ' > l/3, so that again w = w0/3, but i\vh reset the oscillator a t w , ~= wrJO MI that h = 5 . I3y lowering < U > hy a factor 3 t o < B 0 > / 9 and dropping uri ti1 c o l j 0 / 3 , the energy falls by the factor 9 to T0/81. If one en~ploysthc fifth harmonic to reach even lower energy, i t turns out that there is an overlap with thc range covered with h = 3. Both w and < B > will now range from 1/5 to 1/15 of their initial values, so the energy varies from T0/25 clown to T0/225. If it proves inconvenient or irnl~ractical to change the frequency of the oscillator by a factor of 3, a soniewhat more restricted energy band can be covcrcd by a reduced frequclncy range, say of only \/Q to 1. On the first harnlonic the ficld and frequency may be lowercd by this factor and the energy drops from T o to T0/2. Lower energy can I)(. reached hy harinonic operation, hut there is a gap in the spectrum; if the oscillator is returned t o wrj0, the highest rotation frequency a t h = 3 is w0/3, the associated resonant field is < B 0 > / 3 and the cncrgy is T0/9. Rcducing the frequency and ficld by d2 gives coverage down to TU/18.

244 FIXED-FIELD ALTERNATING-GRADIENT ACCELERATORS Still another hiatus occurs, between the lowest energy for h = 3 and the highest for h = 5. With the oscillator reset a t w r f O ,the greatest turning frequency is w0/5, the field is /5 and the energy is T0/25. By running the oscillator and field down by d ? , the lowest field is--/(5 .\/2) and the minimum energy becomes T0/50. 13-13. Existing and Planned Isochronous Cyclotrons

Interest in isochronous cyclotrons of relatively modest energy is a t a high pitch a t prcsent (1960) because of the advantages they offer and in spite of the difficulty of design and construction. It is most alluring to contemplate extending the energy above the limit of a fixed-frequency cyclotron (about 25 Mev for protons), still keeping a very large output of projectiles. Many problems of nuclear physics await solution in the unexplored energy region between 25 Mev and the several hundred Mev figure a t which most synchrocyclotrons operate. I n addition, isochronous cyclotrons appear ideally suited for variableenergy operation, to the great delight of the experimentalist, who is often interested in the energy dependence of some phenomenon. Consequently it is not surprising that many such machines are in the planning or design stage, with a t least six under construction, in addition to a few already in use. Although two are planned for protons of several hundred Mev, most are being built for protons or deuterons up to about 60 or 75 Mev, and several will be capable of accelerating multicharged heavier ions. Mention has been made of the two original Thomas electron cyclotrons that were built in 1950 to prove the feasibility of the idea. These employed steel magnets similar to those of an ordinary cyclotron, with poles 33 and 40.5 inches in diameter, respectively, the pole tips being contoured into three hills and valleys of approximately sinusoidal shape. A completely nonferrous electron model known a s the Cyclotron Analogue was built a t Oak Ridge in 1957. Thc main field, which extends over a useful area of 24 inchcs diameter, is supplied by a pair of Helmholtz coils, while the increase of field with radius is produced by concentric circular windings. The flutter is created by eight pairs of flat, wcdge-shaped coils which generate four hills and valleys with radial boundaries. These coils have more turns a t large radii, so the flutter increases outward. The final energy, a t ,f3 = 0.688, is 193 kev. T h e radial betatron frequency v, rises from unity to 2, a t which point the beam blows up, since the a-mode stop-band has been reached. The axial betat,ron frequency rises from zero and then fluctuates around the values 0.16 t o 0.19. A second nonferrous electron machine is under construction (1960) a t Oak Ridge. This employs a spiral structure. It is designed to produce 450-kev electrons and will be used primarily to study the problerns involved in extracting protons from a future cyclotron (using iron) of some hundreds of Mev. The eight-sector flutter field is produced by single-layered windings shaped much like the arms of a spiral nebula, with the number of turns in

cnch increasing a t the outer ends of the arnms, so tlmt the flutter, as well as the spiral angle, increases witli rising radius. Part of the rcquirctl rise in average field 1s obtained from tlic cxtrclne spiral anglc at large radii. which increases the rclative width of the hills; 1)ut the larger part is furnid~cdby concentric circular windings. I n orclcr that tlw flutter may 1)c :tpl)rc~ciablenear the center, thcrc are only four hills in that r q i o n . Thc axial betatron frequency rises rapidly from zero and then holds a fairly constant value close to 0.25. Although the exclusion of iron would cntail an unduly large. power hill for a proton accelerator, there is a decided atlvantngc in this type of construction for an electron model; the n~agnetic field a t all points can he calculated rigorously and in complete detail. This allows an accurate study of the effects of rcsonanccs and p m n i t s the discovery of those which mill he most serious in a full-scale proton ~n:whine 'I'luls, the four-bector radial model a t Oak Ridge has demonstrated that no disturbance of the btmn occurs a t V, = 3/2, although there is a severe one at v, = 413. Thomas focusing, hut without the isoc~hronousproperty, has heen added to the 42-inch cyclotrons a t Los A1:tmos and a t the Jlassachusetts Institute of Technology, with a gratifying diminution in the axial height of the beams. The 60-inch cyclotron a t Mosrow has bcen converted into a three-sector radialridge niachinc, with the result that the required dce-to-dee volt:xge, for maximum current, has bcen reduced from 160 to 70 kilovolts. The first isochronous proton cyclotron w:m completed a t Delft, Netherlands, in 1958. It employs four radial sectors, thc air gap diminishing with radius, and it accclerates protons to 12 Mev with only 20 kilovolts on the single dce. At the output radius of 14.2 inches the field flurtuatcs between 10.7 and 17.5 kilogauss, with v, = 1.015 and v, == 0.055. T h e centr:tl field is 14 kilogauss, and weak-focusing is employed near the center out to the radius where flutter makes its presence felt. The 43-inch cyclotron a t the University of Illinois has becn rcbuilt as a six-sector, Archimedean-spiral, variable-energy machine and was put into operation in 1959. Protons can bc accelerated from 3.5 to 15 Mev, deuterons from 7 to 14 Mev, "c++ from 11 to 37 Mev, and 4He++ from 15 to 27 XlclAlso in 1959, the Russians installed a t Dubna a six-sectorcd spiral ridgc machine for 12-Mev deuterons. I t requires only 12 kilovolts on its single dec. The betatron frequency v, varies from I to 1.01, while v, ranges from 0 to 0.2. The ficld fo1lou.s an Archimedean spiral with 6' = 0.06 R. A very compact isochronous cyclotron will hc complctcd in 1961 a t thc University of California a t Los Angclcs. Protons with a fixed energy of 50 Mev will be produced in a magnet with poles only 40 inches in diameter. This requires hill and valley fields with peripheral values of 25 and 16 kilogauss respectively. Thc 5.37% rise in field is obt,aincd by reducing the air gap, as the radius incrcases, down to a miniinurn v d u e of 1 inch a t the hills. The magnetic

246 FIXED-FIEIJD ALTERNATING-GRADIENT ACCELERATORS configuration is a four-armed spiral and the two dces, each of 48" and of 1-inch aperture, arc located in two of the four valleys. Dce-to-ground potential will be 50 kilovolts.

Oak Ridge

Colorodo

Illinois

Harwell

Fig. 13-14. Typical shapes of steel shims attached to pole faces to create azimuthally varying fields in isochronous cyclotrons. A three-sector cyclotron is scheduled for operation a t the University of Birmingham, England, in 1961. At a final radius of 18 inches, where the average field is 16 kilogauss, 12-Mev deuterons or 24-Mev alphas will be produced when the oscillator is tuned to 12.2 Mc/sec, while if the frequency is shifted to 16.2 Mc/sec, the same field will yield 32-Mev helium-3 ions. At Lawrence Radiation Laboratory, Berkeley, California, a variable-energy machine for heavy ions and for deuterons is under construction, with completion scheduled for 1961. The top deuteron energy will be 60 Mev. The magnet, of 88-inch diameter, has three spiral hills and valleys with gaps of 7.5 and

Courtesy of Oak Ridge National Laboratory

PLATE 22 Scaled model of' the lower pole of the Oak Ridge isochronous cyclotron. One valley is shown empty, one with a valley coil in place, and one with a valley coil clamped under a retaining plate.

C o v r t ~ s of ! ~O a k Ridge National L a b o r a t o ~ y

PLATE 23 Lower pole of the model magnet of the Oak Hidge isoclmmons cyclotro~l.All three vallcv coils arc in place a d two trimming coils lia\e bee11 acldetl.

248 FIXED-FIELD ALTERNATING-GRADIENT ACCELERATORS 11.8 inches, the maximum peripheral fields being 20 and 14 kilogauss. The changing ratio of hill width to valley width supplies part of the required increase of field with radius, the remainder being obtained from circular trimming coils. A single 180" dee will be used, the frequency being variable over a 3-to-1 range. A 76-inch, variable-energy cyclotron for protons with a maximum energy of 75 Mev and for heavy ions will be put in operation a t Oak Ridge National Laboratory during 1961 (Plates 22 and 23). Three hills are employed, each with one edge radial and the other circular, so that the average field rises with radius; additional control is given by circular coils. Windings in the valleys afford means of changing the flutter to suit different operating conditions. The oscillator and dee system will be variable in frequency by a factor of 3. The University of Colorado expects to complete its variable-energy multipleparticle cyclotron in 1961. The highest energy will be 30 Mev, obtained with protons. The rnagnet has poles of 52-inch diameter and the AVF configuration is a four-sector spiral. This supplies half of the required rise in field, the remainder being furnished by single-layer coils wound around the hills and over their surfaces. Plans are under way to convert the 110-inch synchrocyclotron magnet a t Hnrwell, England, into a four-poled Archimedean spiral structure with 0 = 0.1 R . This represents a vely tightly wound structure; a t a radius of 50 inches the spiral angle is 5 = 78.6") so the angle between the orbit and the ridges is only 11.4". Isochronous operation with protons is expected to extend to 40-inch radius, where the energy will be 140 Mev. Frequency modulation of 2% will be required to carry the energy to 240 Mev, since studies of resonances arising from nonlinear forces indicate that the rise in v, associated with constant frequency operation would bring about a severe instability when u, reaches the value of 2n/3. A 60-Mev deuteron machine is planned for completion in 1963 a t the Swiss Federal Institute of Technology in Zurich. This cyclotron will probably employ three hills with a rather small spiral angle. The University of Florida is planning a machine for 400-Mev protons. As opposed to all the other proton or deuteron cyclotrons mentioned above, which have square-wave pole tips, this device is designed with sinusoidal poles, six in number. There is discussion a t Oak Ridge of an 850-Mev cyclotron with eight spiral sectors to be used as an injector for a 12-Bev synchrotron. 13-14. Spiral-Sector Ring Accelerators

Even if the radial n-mode stop-band did not put a limit on the energy of an isochronous cyclotron, the cnormous mass of steel needed for a multi-Bev marl~inewould certainly make it impractical, just as in the case of a synclirocyclotron. Consequently much thought has been given to the possibility of renioving

most of the. etct.1 from tllc ccmtcr of :I lnryr FF.ZG sy)irnl-ridge cyclotron, Icaving it6 11lngnt.t in the sliapc of :in :tnnulus or ring into which particles arc1 irijerttd at :t rc,lativcxly low cncrgy froill :m auxili:~rymac.hinc. Son~ctirnessuch p r o p o s d tlc.viccv arc called E'E'AG or AVF' syncl~rotrons,h t the nanw is i~mppropriateiincc L'spnchrotron"irnpl~csw pu1hc.d ficdd, whercas FFAG rings criiploy static. ficlds and :ire nothing marc than tlic. outer portions of FFAG cyclotrons. It is wort11 re-erilpllasizing that L: synchrotron keeps ions a t a constant r:~dir~s, in spite of rising inomenturn, 113' the usc of a field which grows with tirile, nhcrcws in an FFAC; ring tlic p:~rtic.les111owslightly in space into a progrcssiwly stronger static field. It has hcen shown that the operating point of an isochronous cyclotron must cross nxmy serious resonances if the energy is great, even if the number of sectors is large. This raises the ql~cstionof how a large or total loss of ions is to 1)c :~roidetlin an FFAG ring of high energy I3orrowing from the technique of the synchrotron, this can 1)o :rccomplishctl by designing thc machine in such a v a y that v, and v, nominally remain fixed, the operating point on the v,v, plot heing chosen to lie as far as possihle from surrounding resonances, so that a little w:~ndcring d w to errors in construction can bc. tolcratrd. This me:ms, howcvcr, that iio~hronitnimutt he al)andont.cl, as inay 1)e seen from the approximate Eqs. (13-59) and (13-601 here repeated:

For isochronous operation, it has heen soen in Eq. 1'13-67) t h a t k = y' - 1, so rises, so niust k . But if is to remain fixed, then k must be constant and we cannot h a r e isochronism. This statement is perhaps unduly pwsimistic, though a t the present time i t appears to he correct from a practical st,andpoint. Actually, the complete equations for hoth v,? and v,? include not only mall terms in F and hut also terms involving their rates of change with the radius. I t has been shown by Teng that when all thcsc terms are takcn into account, i t is possible-at least on paper-to design :t magnetic field which pr3rmits k to vary in the manncr appropriate for isochronism and nl~iclialso keeps the betatron frcquencicb const:mt. Whether such rxtrcmcly coniplcx fields are within the bounds of practical renlixation is yet to l a - r i c d . We will 1c:trc this possible development as a task for future ycnrs anti will continue the discussion on the basis of the approsinlate expressions g i w n ahove. As far as kc.eping v, at a fixed value is concerned, it is apparent that if 1; is static, then either the flutter and the spiral angle must both he held constant. or else if one of these parameters rises, then the othcr must fall by a compensating amount. The first alternative is much simpler. TInis if tan [ = I / n :IS

$1,

r,

-

250 FIXED-FIEIJD ALTERNATING-GRADIENT ACCELERATORS constant, then by Eq. (13-76) we have Rd0/dR = tan [ = l/a, so that d R / R = ado, and In R = a0 In Ro,whence

+

R

=

Ro ea8.

(13-86)

This shows that the hills and valleys follow a logarithmic spiral, the orbits crossing the ridges a t a fixed angle a t all encrgies. To find the resulting expression for the field, we return to the function @ ( R )discussed in Eqs. (13-74) and (13-77) and find that it is now

The lower limit in the integral is Ro since R has this ralue when 8 = 0 in a logarithmic spiral. Hence by Eq. (13-74) the field is given by

wherc, as in Eq. (13-54\, is proportional to RQnd now k is a constant. /

hill max

hill max

orbit

Fig. 13-15. The radial separation h of ridges.

r t a center

If both the spiral angle and the number of hills are large, an approximate expression may be obtained for the radial distance h between corresponding points on adjaccnt ridges. From Fig. 13-15 it is clear that tan {

=

27rR

---, NA

(13-80)

since 2aR/N is the approximate distance between neighboring hills a t radius R. 13-15. Phase Stability in Spiral-Sector Ring Accelerators

+

By Eq. (13-57) we have a = k 1 always, and since in an FFAG ring accelerator k is a constant considerably greater than unity (perhaps several dozen), the momentum con~pactiona is correspondingly large. As far as phase stability is concerned, the situation is the same as in an alternating-gradient synchrotron, as described in 5 12-4. At low energy, the quantity ( l / a - 1/y2) is negative, and a momentum increase undergone hy a nonsynchronous ion causes a decrease in its period ( a rise in frequency), so that the phase-stable angle 4, must lie in the first quadrant, where the electric field of the accelerating

ROTATIOX F'REQUESCY I N SPIRAL-SECTOR RISGS

251

unit is rising n i t h time. But at a sufficiently high energy, ( l / a - 1/y2) becomes positive and +, must be in the wcontl quadrant. Phase stability temporarily disappears as the particles pass through the imnsition energy given b y y" a. 13-16. Rotation Frequency in a Spiral-Sector King Accelerator

Thcrc is :in important difference between an AG synchrotron and an FFAG ring accelcr:ttor, as far as the rotation frequency of the synchronous 1):trticles is concerned. I n the AGS, such ions arc constrained to revolve a t a constant radius, thanks to the action of :In external agency: the servo circuit which forces the energy to be that drmanded by a constant, radius and by a niagnetic field which increases with time. Since the velocity always rises, the frequency of revolution :wound the path of fixed length always increases. But in an FFAG ring there is no external constraint on the size of the equilibrium orbit and the particle is free to scck the radius a t which the average field is correct for its momenturn. The long-term behavior of synchronous ions is therefore the same as thc short-term beliavior of nonsynchronous particles, both obeying Eq. (13-90). The revolution frtquency of the synchronous ion starts off hy rising, because ( l / a - I / y L ) initially is negative, and continues t o do so until the transition energy is reached a t y2 := a. Thereafter the frequency of the synchronous particle is rcduccd, since ( ] / a - l/y') becomes positive. The reason for this also can be seen on physical grounds. The nionientunl coinpaction in the FFAG machine is large, so orbits of different moinentunl are tightly packed together, with only a sniall difference in their lengths. A t low energy, an inclremcnt of energy is associated largely with an increase in velocity, so u i t h an almost constant path, the frequency of rotation rises. But a t high energy, when the velocity changes approach zero, the sinall increiiicnts in path become the donlinating fact,or and the frequency dccreascs. This effect 01)viously depcntls not only on the, gradual npl)roacli of thc velocity to its limiting value, hut also on the ratlii~lrate of increase of the average field, that is, on the value of I;. The largcr this paraineter, the sliortrr is the distance the orbit must expand to find a suitablc field and consequently the increased length of path is less of a dominant factor. Therefore the decre:tse in frequency, a t energies above the transition point, is less pronounced the larger the value of k. The detail5 of the way in which the frequency changes with energy niay be found in the folloming manller. Since T = 2a/w, then &/T = - ~ w / o . Also ~ 7 = Mz)/ (Mot), so that Rq. (1-26) gives drllrl = dp/p = dp/p = ~ l r , / nlicrc ydy/(y' - I ) , Hence Eq. (13-90) limy be written

+

Here a is a constant, sincse hy Eq. (13-57) a = 1; 1 and by Eq. (13-59) = 1 I<, so that ',v = a and v, is to be kvpt fixed to avoid resonances. S o ~ viiitcg~atc1 ct\\eeii the iiiitial and filial frequencies CC, aiid u p arid t h

v,'

+

252 FIXED-FIELD ALTERNATING-GRADIEKT ACCELERATORS corresponding energies yl and y2. (Consultation of a Table of Integrals may be helpful.)

Take the antilogarithm, drop the subscript 2, and obtain

+

The shape of the graph of w vs. y depends on the value of a. Since a = k 1, the exponent is k/(2k 2 ) . For positive values of k , which represents a field increasing with radius, the expression for w plotted against y gives a curve which first rises from the injection value wl, reaches a maximum, and then falls away. As k is made larger, the peak becomes broader and occurs a t greater energy; a plateau with no descent is reached with k = co. The energy at which the peak occurs may be found by taking the derivative of Eq. (13-92) with respect to y and equating it to zero. This yields the value y2 = a , as expected.

+

13-17. Existing and Potential Spiral-Sector Ring Accelerators

The potential advantages of such a fixed-field machine over a pulsed synchrotron are very great. The repetition rate is set by the modulation period of the accelerating voltage and hence by the rate a t which energy is given to the particles, rather than by the very long period a t which it is econon~ical and practical to cycle a pulsed magnet, so that a very much higher time-averaged current of energetic particles should be produced. The complexity and expense of a pulsed power supply for the magnet disappears, to bc replaced by the simplicity of DC generators. The violent mechanical shocks associated with pulsed operation are no longer applied to magnet coils and rotating machinery, which leads to simpler design and reduced maintenance problems. A laminated magnet is not required and eddy currents no longer exist to produce distorting fields. Any needed local corrections to the field are static rather than time-dependent. The range of frequency modulation is somewhat less than that required in a synchrotron, since the frequency first rises and then falls. Of greater importance, the electric frequency does not have to track changing values of the magnetic field, so that mechanical tuning

SPECIFIC SPIKA41,-SECTORACCEI,ERAI'TOIIS on almost any frequency s c h c d ~ ~ lappcS:trs e possibl~.This vastly r c d ~ ~ c ct -l ~ c cw~r~plcxity of the equipment. On the other hand, thc. dmign and construction of t l ~ cn~ngnctarid vacuunl c~l~anlbcr for hurl1 a fixed-field annular wrcc~lcrato~ of'f(xr In:iny scrious problenis for :t proton ~nachinein the multi-Bev rang(.. I n huch :t n~nisochronousring, where V , and v , are t o remain constant, IG is fiwd and hence it, describes the radial dependence of the azimuthally averaged field over the entire radial width of the magnet (rather than the dependence in thc neighborhood of some particular radius, as it does if its v a l w changes). Consecluently the average field, given by ISq. (13-54), may 1.w written as

where the subscripts 1 and 2 refer respectively to the inner and outer radii of the equiv:il(mt circles (:ilmost exactly the radial coordinntes of thc real machine, if i t is large). From this it follows that

If the two radii arc chosen arbitrarily, as well as the corresponding average fields, then k may be found from this expression and the value of v, then follows by Eq. (13-50): v,' z 1 k. Judgment is required in the selection of the average fields, for they will he significantly lower than the peak values appearing in the high-field regions, so t h a t the attainable flutter (nceded to give a reasonable value t o v,) must he taken into account, even though t h a t parameter does not appear explicitly in the expression for v,. T h e product R1 depends on thc injection energy, while < B 2 > R 2 is fixed by the final energy. If R z is made only slightly greater than R1, a saving in weight and cost of magnet is obtained, but such a choice entails a large change of field over a short distance (LC., a large k ) , which may be difficult to achieve with the desired accuracy. Conrersely a small k , though easier to produce magnetically, involves a bigger investment in stcel and a widcr vacuum chamber, with attendant construction difficulties. The magnitude of the task may become apparent by quoting some of the suggested parmieters for a 20-Bev proton accclcrator employing 5-Mev injection: = 69 gauss, R l ==4688 cm = 153.8 ft, <&> = 14000 gauss, R2 = 5000 c ~ n= 164 ft. The vacuum ch:tmber is therefore 10.2 feet wide with a mean circumference of 1067 feet. Thc value of k, c:ilculated from the last equation, is 83 so t h a t v, c (1 li)": 9. T o obtain radial stability, a, must he less than T and since v, = Na,/(27r) always, then v, must be less than N / 2 , as noted earlier. I n the present instance this means ,V must equal 19 a t least. T o be on the safe side the value N = 31 is suggested, so that a, = 0 . h . Suppose the maximum hill and minirnum valley fieltls, R,, and B,, differ by

+

+

254 FIXED-FIELD ALTERNATING-GRADIENT ACCELERATORS 25% from the mean value a t any radius; that is, f = 0.25 in Eqs. (13-2) and (13-3) : B , = < I ] > (1 f ) and B,, = (1 - f ) . Then a t the inner edge where = 69 gauss, we find Blh = 69 X 1.25 = 86.2 gauss and B1, = 69 X 0.75 = 51.8 gauss, while a t the outer riin where = 14 kilogauss a

+

Courtesr~of MURA

PLATE 24

The spiral ridge FFAG electron accelerator of MURA. The accelerating betatron core is at the left. similar calculation gives Bzh = 17.5 kilogauss and Bgv = 10.5 kilogauss. The flutter is given by F = f2/2 = 0.03125. Choose V, = 4.15 so that the operating point is far from a resonance and use the approximate Eq. (13-60) :

v2

=

to determine that tan

-k+?F(l N"

N2 1

+2tan2{)

+ ...,

6 = 40 so 4 = 88". The spiral is therefore very tight and

RADIAL-SECTOR RING ACC'E1,ERATORS

255

the orbits cut in a t only 2". Thc radial distancc from hill-center to hill-center is, by Eq. (13-89), 2rR - ------= -----2r5000 - 25.3 cm. AT tan { 31 X 40 This is also the radial width of :I hill plus a valley; :md since the fields are averaged linearly to obtain the mean field, the hills and valleys are equally wide in azimuth and hence also in radius. The radial width of a hill or a valley is therefore close to 12.6 cm. As yet no spiral FFAG ring accelerator for protons has been constructed. As proof that the general conccpt is correct, a working model for electrons has been built by the Midwestern Universities Research Association (MURA) a t RIadison, Wisconsin. It employs six hills of 57" s1)iral angle a t all radii, the valleys being of zero field for siniplicity in constru17tion. The average field index lz has the constant value 0.7 and the observed values of v, and v, lie satisfactorily close to the calculated figures. Injection is a t 30 kev and the final energy is 120 kev, acceleration being produced through the betatron action of a laminated core linking the orbit, which increaws in radius from 30.5 to 55 cm. 13-18. Radial-Sector Ring Accelerators

An intcwsting variety of annular acwlerator is known as tlie Mark I or Radial-Sector FFAG. This ]nay be considered to ernploy an extreme type of Thomas focusmg, for tlie change of field from hill to valley is increasd to the point where the valley field is the negative of the hill field. That is, in alternate magnets the fields have opposite directions, though in each the absolute strengths arc the same and increase radially :it the same rates. Following injection from an auxiliary device, the particles n-love outward into stronger fields as tlicy are given energy. Thc uniclue characteristic is that in altcrnatc inagnets the orbits curve first one way and then the other, as shown in Fig. 13-16. An cquilibriuin orbit closcs brcause the reversed-field (or "negative")

Fig. 13-16.

Mark I radial sector FFAC uccel crator.

256 FIXED-FIELD ALTERNATING-GRADIENT ACCELERATORS magnets have less azimuthal length than the "positive" ones. Since the fields in both sets increase outward, the gradient always points away from the machine center, but with respect to the local centers of curvature, which lie alternately on either side of the orbit, the gradient is alternating. Thus in the positive magnets there is strong axial defocusing and radial focusing, while in the negative magnets the forces focus axially and defocus radially. Under proper circumstances stability in both directions is obtained. In order that the orbit a t one energy be continuous, one can see from a geometrical construction that the interfaces between magnets must radiate from the machine center. Consider a simplified machine with no straight sections and assume that the variation of the magnetic field along the path in each magnet is so small as to be negligible, so that the radius of curvature may be taken as constant. The net change of direction in passing through a pair of magnets is $1 - 42, so if ,V pairs make a closed path of length L then N(41 - 42) = 2n and

The quantity C is called the circumference factor since i t gives the ratio of the path length to that in a machine of the same field strength but without reversed-field magnets. It will appear later that for axial stability to exist +* must be a t least 241/3; so the minimum value of C is 5. As a result, the equilibrium orbit does not differ markedly in length from the circle of radius r about which it weaves; so to a first approximation and The much greater weight of magnet necessitated by the longer path is the chief disadvantage of the device. The conditions for orbital stability can be determined approximately by the usual matrix methods if the fields are assumed to be of the square-wave type; in fact, the resulting equations are identical with those of an ordinary alternating-gradient accelerator. For a machine without straight sections the result is cos u, = C O S ~ cos #q, where

and

t t l l iind I ~ L bring ? tllr azimutl~allcngths of the, positivt and negative nlagncts a t the inc:rn radius :ipj)ropr~atet o tllc cncrgy in question and n Iwing the conimon value of the index. If m2 is rnade shorter than 2tn1/3, then cos o, exceeds unity and axial instability rcwlts. This is the origin of the ininin~un?value of 5 for tlie circumference factor. Thc description just given has ncglccted all edge effects and the influence of straight sections, but in actuality the outmrt-l norinnls to the edges of the positivc magnets he outside thc o r h t s in the str:ligl~tscdions :md thereby proctucc axial focusing and r:tdial defocw4ng forccs (Fiq. 13-17). Simi1:~r forcr,s are

Fig. 13-17. Mark I FFAG accelerator with straight sections. gcnerated a t the faccs of the negative magnets, smcc "outside" refcrs to the local center of curvature. The effccts of edges and straight sections arc most pronounced in rnacl~ineswith few sectors and sin:dl radius, for tlien the weaving Ixtck and forth of the orbits about a circle (scalloping) is of more significance. As a result, tlie orbits do not follow contours of constant field :mi the analysis beco~ncsmore complicated. X O accelerator of this type has yet heen t ~ u i l tfor protons but n working modcl for electrons has beer1 constructed by RIr'RA and operates as expc.cted. The guide field is produced by a nlultil~licityof polr-face windings on each of the 8 pairs of ~nagncts,the field increasing from 40 to 150 gauss over the radial range of 36 to 52 cm. The average ficld index k has the fixed value 3.36, and v, and v, an1 cxorrespondingly constant. Tnjection occurs at 25 kev and acceleration by mean.; of n betatron core takes the cllrctrons up to 400 key. I t is possi1)le t o vary v, and v, b y changing the fields in the positive and negativc magnets and to alter k with auxiliary windings. In this way stability conditions have been investigated over thc region betwem v, = 1.5 t o 2.3 and v, = 2.5 to 3.0, s c w r e loss of particles being found a t l,hc integral, half-integral, and onethird-integral resonances, as predicted

13-19. FFAG Betatrons The useful injection time in a conventional pulsed-magnet betatron is excccdingly ihort, of ill(. ortler of a microsecond or I ~ s ysincc the clcrtrons nre

258 FIXED-FIELI) AIJTERNATING-GRADIENT ACCEI,E:l
Cotirtesy of M U R A

PLATE 25 The radial sector FFAG accelerator for 400-kev electrons built by MURA. Magnets with reversed fields are of smaller azimuthal width than those with normal fields. Linking the orbit is the core of the transformer which accelerates the electrons by betatron action.

An enormous increase in duty cycle, and hence in output of x-rays, is certain when FFAG betatrons are built. Such a machine may consist of an annular magnet with a fixed field that rises with radius, stability being obtained by radial or spiral ridges, as in the proton machines considered earlier, but acccleration is produced by a changing flux which links the orbits, rather than by an RF cavity. The important distinction from a conventional betatron is that the guide field is always suitable for particles of all energies from the injection to the final values. The usual requirement that the average field within the orbit must be twice the guide field does not apply, since the electrons are free to choose an orbit with a radius appropriate to their momentum. The only function of the betatron core is to supply the flux change A@ that is necessary for

CENTER-OF-AIASS E S E R G Y

259

the desired encrgy gain AE, as was seen in Eq. (10-13) in the description of the weak-focusing lwtatron. If the corc in a FFAG rnachirie supports a sinusoidally varying flux of amplitude @, the available change, of flux, from -@,, to $-@, is 2@,r,.Let injection start a t the negative pcak and continuc for an angle 8, so chosen that the last electrons accepted reach thc dcsircd cncrgy just a t the positive peak of the flux.

I

4

f ~ r s tin

Fig. 13-18. Flux relations in a FFAC: betatron.

Then the first electrons to start must rc:~cll full energy a t angle 0 before the peak, as shown in Fig. 13-18. The required flux change h@is related to a,,, by whence

and the duty cycle is

These last two equations show that if the chosen energy is so great that the entire available flux change 2*,, is required to develop it (i.e., A@ = 2 @ , ) the duty cycle is zero; if the encrgy is lower (or the flux is greater) so that, for example, A@ = a,,, then LI = 0.25. Thus supplying a big enough corc, so that the ar:tilahle flux considerably exceeds that required for the given energy, makes possible a large duty factor and leads to an output thousands of times a s great as from a pulsed betatron. I t niay 1)c noted that the radial-sector and spiral-ridge model accelerators constructed by MURA and described earlier actually are FFAG betatrons, although the bttatron method of acceleration was chosen for these n~achines merely because it was simpler than building a n RI' cavity. I n these models a duty cycle of 30% has been observed. 13-20. Center-of-Mass Energy

I n many nuclear reactions the proton projectile forms a temporary union with a nucleon of a target nuclcus (that is. with a proton or a neutron) after

260 FIXED-FIELD ALTERNATING-GRADIENT ACCELERATORS expending a small part of its kinetic energy in overcoming the repulsive force that exists between it and the nucleus. The rest of its kinetic energy is then used in two ways. Part is cxpended in giving the compound body (or the products formed from it) a forward momentum equal to the original momentum of the projectile, while tlie balance, sometimes called the "available" energy, is used in deforming or exciting the compound or even in creating new particles. This available energy To is therefore never as great as the original kinetic energy T. As a simple example a t nonrelativistic speeds, consider a ball of putty moving with velocity v, on a frictionless table and striking a similar ball a t rest. The resulting shapeless mass moves off with velocity v2. The initial momentum is p1 = Movl, the final value is p2 = 2Movz, so v2 = v1/2 since pl = p~ The initial kinetic energy is T I = M0v172 and the final amount is Tz = 2MovZ2/2 = M o ( ~ ~ / = 2 )T1/2. ~ Therefore the energy To used for deformation (here dissipated as heat) is but one half the projectile's energy sincc T, = T I- T2 = T1/2. Note that this is the maximum value and that it is reached only for bodies with no elasticity whatever; if billiard balls are used, less energy goes into heat and more into kinetic energy. T o compute the energy available for deformation at relativistic velocities, it is convenient to employ a trick. If two perfectly inelastic bodies of equal mass approach each other with equal and opposite velocities v', the initial net momentum of the pair is zero and so is tlie final momentum after the collision. The entire kinetic energy of both particles becomes available as deformation energy, since the whole mass is now a t rest. By use of the earlier expression, Eq. (1-lo), for the relativistic kinetic energy, the available energy is found to be

This expression, valid when both particles move with respect to the laboratory, is also valid when the velocities are measured with respect to the center of mass (CM) coordinate system which moves (if at all) with the center of mass projectile (a)

I C-.v1 (b)

I

CM X

target V

O , X +

'

4

l

-

-+u=v'

I

Fig. 13-19. Center of mass system moving in the laboratorv.

lab

of the particles. Therefore Eq. (13-100) can be converted into an equation giving the available energy when only one particle moves, with respect to a fixed target, by transporting the entire CM system a t such a velocity that the

CENTER-OF-XIASS E S E R G Y

26 1

target remains : ~ rest t in the laboratory. Imtlgine the CAI systern to be a board h two partirlcs move, each with velocity v' with from the ends of ~ ~ h i cthe rcspect t o the ccntcr of mass. The collision will occur a t the mid-point of the board. If the target hody is to remain fixed in the laboratory, the CM system must move with velocity 7( = 1)' with respect to the lab in a direction opposite to the CM velocity 21' of the target (secl Fig. 13-19). Let the yelocity of the projectile be nleasured in thc lab system. If 21,I)', and u were very small rclative to thc speed of light, it would he justifiable to say t h a t v = v' U ; that is, the projectile's velocity in the lab syettm would equal its velocity in the C h l system plus the velocity of the ( 3 1 system in the lab. But since :dl thc velocitics are close to that of light, it is necessary to use the standard rclati~isticexpression for the addition of velocitics: I )

(Kote that this reduccs to thc nonrelati\%tic forin if ti'u v = c if v' = 14 -= c.) I n the p w e n t inst:ince v' = zc so that

+

< c2 and also that

After division by c , this can be rcnrr:tngcd as

where p -- v/c is the projectile's laboratory velocity in units of the velocity of light. I t is nt1ce.isary to use the nt~gativcsign since v'/c and v/c must approach zero together. Hcncc. 1)' 1 - (1 -/3?))$ - - ----. (13-104) C P Square this, changc signs and add unity to c : ~ hsidc to o1)tnin

262 FIXED-FIELD ALTERNATING-GRADIENT ACCELERATORS which, after a little rearrangement, becomes

+

since y = E / E o and E = Eo T, where T is the projectile's energy in the lab system. P u t this expression into Eq. (13-100) ; the latter then can be rearranged, finally, as

or, on solving for T,

Here T, is the energy available to effect some nuclear change, T is the projectile's kinetic energy measured in the lab, the target is a t rest in the lab, and both particles have the same mass. These expressions exhibit the inherent inefficiency of high-speed projectiles, directed against stationary targets, as a means of releasing energy; a large percentage is wasted in giving momentum to the target or to the debris formed from it, as shown in Fig. 13-20. Thus the 6-Bev Bevatron releases only 2 Bev of available energy and

L a b Kinetic Energy T ( B e v )

Fig. 13-20. The relation between available energy T , and laboratory kinetic energy T. the 25-Bev alternating-gradient synchrotron a t Brookhaven makes available only 5.2 Bev. The influence of relativity is again exhibited in the second curve, which shows that the ratio of available energy to projectile energy decreases asymptotically towards zero, from an initial value of 4,as the projectile's energy rises towards infinity. The relation between T , and T is useful in determining the size of accelerator needed to create matter out of energy, such as, for example, an antiproton. Since such a particle bears a negative charge, its creation must be associated with the creation of a normal (positive) proton, in order to keep the charge of the universe constant. Therefore enough energy must be released in the collision of a proton with a nucleon a t rest to supply the rest mass energy of a

INTERSECTING BEANIS OF PAR'ITCLES

263

proton and of an antiproton, so that T,, must havc a minimum value 2Eo = 1.876 Bev. Eq. (13-109) shows that the incidcnt proton must have st laboratory energy T = 6E,, = 5.63 Bev. I t was with such a reaction in mind that thc designers of the Bevatron set its energy a t a little over 6 Bev, in order that the projectiles might be somcwhat above this threshold; and i t is one of the triumphs of modern physics that such a conversion of energy into matter occurred exactly as predicted. 13-21. Intersecting Beams of Particles

The recognition of the low efficiency of particle accelerators in converting kinetic energy into energy available for nuclear transformation when the target is a t rest, coupled with the realization that the efficiency is 100% if both target and projectile have the same mass and are in motion a t the same speed, has raised the question as to the practicability of firing two beams of protons directly a t each other in order to obtain an enormous amount of available energy with modest accelerators. Thus, two 3-Bev Cosmotrons operated head-on would release 6 Bev, whereas to obtain this available energy cn a fixcd target would require a single 33-Bev accelerator. As another example, the collision of two 10-Bev beams of protons is the equivalent of a single n~achineoperating a t 173 Bev. Unfortunately, the extremely low density of particles in both beams and the small cross section of each ion rule out the practicability of extracting the particles from two accelerators and of directing them in straight paths into head-on collisions. It is as though one wcre firing two machine guns a t one another, hoping for an interesting number of impacts between bullets. The only hope lies in using again and again t,hose particles which failed to collide a t the first opportunity, and this implies some sort of closed path. The idea. of storage rings has received some attention. These consist of two vacuum chambcrs embedded in annular magnets with focusing properties, the magnets being tangent a t one point. Both magnets are steadily excited and in the same sense, so if high-energy ions are introduced into both from accelerators, the particles revolve, say clockwise, in both rings, and have repeated opportunity to make collisions once every turn; for if the vacuum is perfect, the particles will rotate forever or until an impact occurs with another projectile. Average lifetimes measured in minutes or hours can be obtained by applying on a grand scale the "supervacuum" techniques which have already been developed for small systems. The problem of injection into the rings has all the difficulties that have been discussed apropos of injection into accelerators. Unless some change is made in the system, the ions will attempt to escape, sooner or later. The change may be an alteration of the particle energy after it enters the ring, either by acceleration or deceleration. An obvious alternative is to use two tangent accelerators also as the storage devices. This implies a magnetic field that is constant in time, so that when

264 FIXED-FIELD ALTERNATING-GRADIENT ACCELERATORS

particles reach full energy the accelerating units can be turned off and the particles may be allowed to coast a t constant radius and constant energy. Geometrical difficulties rule out this scheme for cyclotrons, but a serious proposal has been made to employ two 15-Bev FFAG annular accelerators with a common straight section where interactions could occur. (To duplicate the 30-Bev of available energy would require a single machine of 540-Bev working on a fixed target.) The great cost of such a pair and the present lack of experience with proton FFAG accelerators appears to have postponed this development, a t least for the immediate future. 13-22. Two-Beam Accelerator

A novel means of obtaining intersecting beams has been proposed in a machine in which identical particles are simultaneously accelerated in opposite directions in the same magnetic field. Essentially it is a variation of a radial-sector FFAG annulus, the fields increasing outward and with alternating sign, the only difference being that the negative magnets are identical with the positive ones, rather than being of lesser azimuthal width. Equilibrium orbits close in one turn because of their scalloped shape, the turning angle in a negative magnet being less than in a positive magnet because the orbit lies closer t o the machine center and therefore traverses a region where the wedge\ \'\

Fig. 13-21. Orbits in a two-beam accelerator.

shaped magnet is narrower and the field is weaker (see Fig. 13-21). The radius of curvature alternates between a small value pl in the strong field of the positive regions and a larger value p2 in the weaker field encountered in the negative magnets. Particles are injected a t the inner border and work their

TIYO-BEAhl ACCELERATOR

265

way outward into stronger fields as energy is sup~bliedfrom an accelerating cavity located in a straight section, stability being maintained by the alternating character of the gradient and by the focusing action a t the edges of the magnets. From thc complete symmetry of the situation i t is evident t h a t a magnet which is positive for a particle moving clockwise is a t the same time negative for a particle rotating counterclockwise, 30 if there is a second injector aimed in the opposite direction, a second beam of ions can be accelerated in a direction opposed to that of the first. A single accelerating cavity in a straight section is adequate, since i t oper:ttcs on the two beams in alternate half cycles. Although the orbits intersect a t every straight section, this does not mean t h a t collisions bctmwn oppositely moving particles will occur as often. As in any azin~uth:tlly varying field annular accelerator, resonances must be avoided, so the betatron frequencies inust be held approximately constant. Consequently the averagc. field index X; is fixed, and hence the machine cannot be isochronous. This means that the rotation and oscillator frequencies must alter, so t h a t pl~ases t a b i l i t , ~comes into play, with attencl:int synchrotron oscillations and the grouping of ions into bunches. C'ollisions will occur only when two

Courtesy of MURA

PLATE: 26 The MURA two-beam accelerator for electrons, during construction. Note the cornplete symmetry of alternate magnets, in which the direction of the field is reversed.

266 FIXED-FIELD ALTERNATING-GRADIEXT ACCELERATORS oppositely directed bunches meet in a straight section. The number of bunches moving in any one direction equals the harmonic order h of the oscillator ( h being the ratio of the oscillator's frequency to that of rotation), and correspondingly there are the same number of azimuthal regions, each as long as a bunch, which are free of particles. Therefore if h i s chosen as 2N, where N is the number of positive and negative magnet pairs, each bunch and each empty region is half as long as the azimuthal width of a magnet. The oppositely moving bunches take turns in occupying the straight sections and no collisions take place during the acceleration process. When full energy has been reached, the accelerating voltage is turned off, the bunches spread in azimuthal length, and collisions begin to occur. A model to produce two beams of electrons, each of 40 Mev, was completed by MURA in 1960, and, in early tests, appears to behave as predicted. Articles on the theory of sector-focused accelerators are given on pp. 364365; those describing particular machines appear on pp. 365-366. Papers on intersecting beams are listed on pp. 366-367.

14-1. T h e Widerije Linear Accelerator

The 0rigin:tl linear accelerator may be considered as a "rollctl out" cyclotron. I t ib n devicc in wliich an ion is :~llon-cdto fall m r n y times over a 5 1 1 ~ 1 1 1)otenti:d hill of constant niagnitude, so that the ~xirticlc attains the saiiic cncrgy as if it had fallen once over a larger hill. Instead of a magnctic field in m-l~icliare place11two dees between which the particlc is acccleratcd and within which it coasts until the voltagc changes sign, we now have a straight-line array of hollow metal tubes of increasing length, alternate rncmhers being connected to olq>ositc terminals of an A(: generator (see Fig. 14-1). For the inon-~ent,imagine the gaps between tubes to he of negligible length. -4 partirle will be accelerated if i t reaches the gap at a time when the electric field is in thc desired direction. Tlie ion rcinains in the field-fret. region inside the tube, of appropriate length, for half a cycle, just as in :I cyclotron; and it rearlies the next gap when the potentials of all tubes have reversed hut arc otherwise a t the same phase as before, so that a second accelrration takes place. Since therc, is the same voltage difference between all pairs of tubes, the energy gained hy such a synchronous particle is a constant :it all g:ips. There is, I~owcver, an important dis1,inction betwecn this device and a cyclotron, other than the lack of a magnet. I n the cyclotron, it nmkcs no diffcrcnrc how lnuch energy is acquired in crossing the dee-to-dce gap; as h:ts been seen, the radian frequency is w = yB/M, so the time for half a turn is independent of the radius and hcnce of the energy, provided R : m i M reinnin fixed. Stated othcmise, the distance travcled kwtween accelcrntions is automatically adjusted to the s p e d , so the elapsed time is constant. This docs not hold in a linear accelerator (often cnlltd a l i n n c ) . Each successivc gap-to-gap distance is a predetermined quantity, so the velocity of the particle inust be of just the correct value as each tube is reached, if exactly one half cycle is to elapse while the ion is within it. The length L, of the nth "cell" (composed of a tube and n gap) must be proportional to the velocity. If r is the period of the oscillator, f its frequency, and X its wavelength, then we h a w I+, = I),, T/% = t l n / ( ( t f ) = 1 ' ~ X / ( 2 cso ) , that 267

This equation describes an important characteristic of this variety of linear accelerator, which is sometimes referred to as a "half beta lambda" linac.

Fig. 14-1. The Wideroe linear accelerator. Ions travel from gap to gap in a half. period of the oscillator. Arrows indicate the flow of charging currents.

Since proton linear accelerators of this type have not been built for extremely relativistic energies, we may use the nonrelativistic expression for the kinetic energy: T , = M0v,'/2. Then v, = (2T,Mo)% and P,, = ( 2 T n / M 0 ) % / cCon. sequently tlic length of thc nth ccll is

Suppose the particle starts from rest. Since a constant energy AT is gained a t each gap, we have T 1 = AT; T 2 = T 1 AT = 2 a T ; T , = T 2 AT = 3 a T ; T , = T n - I AT = nAT, where n is an integer. Therefore

+

+

+

and it is apparent that tlic lcngtlis of the cells i n r r as ~ tlie ~ ~square roots of the natural swics of integers. The initial cclls would be iinpractically short if the ions really did start from rcsi,, so h f o r e entcririg the linac they are giwn an initial envrgy of some convenient magnitude n'hT, where n' is an integer somewhat greater than unity. Mien Eq. (14-2) is solvccl for T,,, t h c v r w ~ l t s

If n has thc particular value representing the last cell in the accelerator, thrn T , is the final kinetic energy and it is clvnr that this energy is fixed, being dctcrmined solely by tlie mass of the projectile, the RF wavelength, and t l ~ c geometry of the machine. Aside from the rel:~tivc.ly small energy spread associated \ ~ i t I inonsynchronous ions (for the principle of phase stability is operative, as will be discussed later), the output cncrgy is predetermined whcn the device is built. Consequently. the voltage :lpplieti to the drift tubes must have just the proper value, if particles are to hc. :~creler:tted. Thus far, t h r gaps have been assumed to he infinxtc.1~short. T o consider tlie effect of finite gaps, we may return to the cxpreqsion for the electric field i~ctwccnthc (Ices of a cyclotron as given in E:q. (7-9), for i t is equally valid in tlie present instance: E

=

6. sin

(,? + 4).

Here the amplitude &, of the electric field is assumcd constant throughout the volume of the gap, 4 is the phase of the rc11t:tge when the particle is a t tl~c. center of the. gap, 4 being measured froin the voltage null \vherc the field starts hecoming accelerative, while x is thc distance along the gap from its midpoint. (This distance n a s given as .r in Eq. (7-9), hut in linac theory it is custonlary to use z for thc longitudinal coordinate.) The velocity v is assunlcd constant across :my intlivitlual g : ~ p ;the validity of this was shown in § 7-10. Since in a PX/2 linac we have L PA12 = vnlw,, (we let the subscript n he understood), it follorvs that w,, = 11a/L, and the field may be described 1)y

-

E = 6 , sin

(y + 4).

The energy gained in crossing a gap of Icirgtli g is

270

LINEAR ACCELERATORS

AT

=

qV, [sin (*gn/2W] sin 4. *gn/2Ln

Here the subscript n has been reinserted to emphasize that there is a different value of g and of L a t each cell. The bracketed term is the transit time factor and is seen to be identical with that in Eq. (7-lo), since the distance between gaps in a cyclotron is a half-turn: L = r r , so that ?rg/2L = g/2r. The gain in energy is still close to the maximum in spite of a large gap; even if g = L/2, the factor is still 0.90. The feasibility of this type of resonant acceleration was originally demonstrated by Wideroe in 1928 with ions of the alkalies, while the technique was greatly improved by Lawrence and Sloan in 1930, who used mercury ions, heavy particles being chosen simply to keep the tubes short, as required by Eq. (14-2). In a 1934 model, 36 cells were employed, the last being 16.5 cm long, of which 20% was gap. Singly charged mercury ions with energies of 2.85 Mev were obtained, with only 79 kilovolts across the gaps, so that a voltage multiplication of 34.8 was reached. The frequency was 10 Mc/sec. The virtue, and a t the same time, the chief defect of this type of linear accelerator is that a truly small voltage is used over and over. Hence if high energies (by modern standards) are desired, the system becomes prodigiously long, particularly for a light ion like a proton; it lacks the space-saving attribute of the spiral orbits of a cyclotron. Of course, the length can be reduced by increasing the voltage on the tubes, but practical considerations prevent much improvement in this direction, if the original method of power distribution is adhered to: that is, the use of a lumped LC circuit delivering power through a transmission line to the tubes by means of glass-to-metal seals in the cylindrical glass vacuum envelope. The use of such an insulating container reduces the tube-to-tube capacitance and lowers the power requirement, but serious losses still occur and dielectric hysteresis in the glass limits the potential that can be used. Early developmental work on this device took place when the cyclotron was coming into being, and because of the phenomenal success of that accelerator, the PA/^ linac was dropped as a research tool. Not until after World War 11, when enormous strides were made in the techniques of generating high power a t high frequencies, did the linac again receive serious attention, and this time in a somewhat different form, now to be described. 14-2. The Alvarez Linear Accelerator

In spite of superficial reseinblxnccs to the Wideroe machine, the resonantcavity linac of Alvxrez operates in a somewhat different manner. I t depends on the generation of an AC voltage with peak value that equals or exceeds the energy, rneasurcd in electron-volts, wliicli is imparted to a singly charged ion in passing through thc device. I t is shown in many books on radiofrequency

techniques* that a cy1indric:tl rnetal cavity, of tlianleter = 2b am1 of any lcngth, can be made to osci1l:~te a t a w:tvclcngth h g i w n by

in a manner known as the TMolomode I Fig. 14-2). Here the magnetic field is entirely azimuthal, being strongest a t the cylindric:~lw:tlls and zero on tllc

Fig. 14-2. The TM,,,, mode of oscillation in a cylindrical cavity.

axis, while the electric field is wholly longitudinal, strongest on the axis anti zero a t the walls. The system may be driven by an oscillator coupled to the cavity by a small loop near the cylindrical wall, the loop being oriented so :ts to link the lines of magnetic field. No insulation is required other than t h a t which isolates the feed line from the v:lcuuin in the cavity. Such a resonant system can have a quality factor Q (Q = 27r energy stored/energy lost per cycle) n~easuredin tens of thousands, and it has an enormous shunt resistance. (This is the fictitious resistance which may be considered to be in parallel with tlie equivalent lossless inductance and caapacitance of the circuit and which appears to absorb as much power as do the actual conductors in the system.) Axial electric fields of 3 to 1 &lev per foot can be attained a t power levels which, though high, are not impossible with available oscillators and amplifiers, most usually on a pulsed basis with low duty cycle. Such largc voltage gradients are enticing to accelerator designers, hut there are limits to the fields that can be maintained without breakdown in tlie rclsidual gas, so a high-energy device of this sort requires considerable length. Therefore the period (and hence the wavelength) of the oscillations must be correspondingly large if protons introduced a t one end are to reach the far end before the voltage reverses. The wavelength determmes the radius of the cavity, as shown by Eq. (14-101 ; and it turns out t h a t to produce protons of some tens of Mev would required a radius of some tens of meters. If this unattracltive dimension is reduced to a few feet, the frequency rises to such a high figure that protons advance only a small fraction of the total length during a half cycle; but by mounting drift tubes along the axis, the projectiles are sheltered from the field when it is directed the wrong way. This means t h a t a whole period must elapse between entry into one gap and entry into the next. \;Ye define a cell as a drift tube plus it gap or, equivalently, as half a tube plus a gap plus half the next tube. Then the length of the n t h cell is given b y L, = u , ~= v7,/f = u,X/c and consequently *For cxamplrl, see F. E. T ~ r r n a n Rndio , Engzncr~ing,3rd ed., 1945, McCrnm-Hill, p. 145.

272

LINEAR ACCELERATORS

Hence such a inaclline is sometimes called a "beta lambda" linac. One can think of t h e electric field as polarizing the drift tubes so t h a t periodically the two ends of each are charged to opposite sign, all tubes being polarized simultaneously in the same direction, as indicated in Fig. 14-3. \\'lien

t

=

~

Fig. 14-3. The Alvarez linear accelerator. Ions travel from gap to gap in a full period of the oscillator. Arrows indicate the flow of charging currents. the charging currents flow to the right along the tubes, an equal current flows to the left along the inside wall of the outer cylinder. The tubes may be supported a t their centers by one or two radial rods along which no net radial current flows. The energy a t any cell in the Alvarez linac is easily found; we shall find i t in relativistic terms. B y the use of Eqs. (14-11) and (1-14) we have Lfl2/h2= Pf12= (En2- EoL')/ET,2= 1 - EO2/ ( T , EO) 2, SO it follows that

+

T h e final cXncrgyis thcrcfore l)rcdc.tcrl~l~rlc.tl by the gc~on~ctry of the ~nacllinc~, aside from tlw cmergy s1)rentl tliat rcsults from t h t opcmtion of phaw stability. T o introtlure sonith ~ : m c t y ,wc will trv:tt the ~)rohleniof t r a n < ~time t in the Alvarez linac in more gcner:d ttwns th:m W I Y Y ~c~nll)loyctlin connect~onwith tllc cyclotron and thc Widcroc :tccclcr:ttor. \Vc, assert t l ~ n talong tllc axis of :t ccll the aniplitutle of the time-varying field is a function of tlistancc. from some origin w i t h the cell. T o this end wc nl:ty ( ~ i i p l o yan exl)ression of the form of Eq. (7-9) by making t h e amplitude dcpcndent on z:

=

~,ix) sin

'2T2 (,t +

4)'

since w,f = 2 a c / X a n d t h e cell length is I> = P A . Wc locate t h e origin of x b y introducing the concept of the electric center of tlie gap, this k i n g defined by t h c cxprcssion

J,E~,(Z)sin ( 1

1

2 ~ (12)

=

0,

Tliercforc the origin lies at the gc\onictrical center of tllc gap only if the ficltl is syininctrical. In the general case, where the t)eh:tvior of tlie field is unknown, tllc energy gaincd in traversing n cell is found hy integrating Eq. (14-14) o w r the full lcngth of the cell. ( F o r simplicity we consider only particles rnoving on t h c axis, tllcrc3l)y avoiding the con~plicaticmsof a longitudinal field t h a t varies with radius.) \Ire have g.(z)

sin

4

l,

=

q

/L

=

g

(WS

( 2 +~ 4)

(iz

+

8,(x) sin ( 2 ~ dz ) g sill 4

g r n ( ~cos )

(F)

&.

T h e first term vanislies because of Eq. (14-15), so we ol)t:~in

T h e fraction in 1)raclirts is thc t r m s i t - t i m e f w t o r F , wllich in gcncral iiiust 1)o e v a l u a t d by numerical intcgratiori, using d a t a fro111 cxperinicntal measurenlcnts. T h e last expression m a y then he written as =

g.

sin 4

["/,;-I

EWL(2)(iz

274

LINEAR ACCELERATORS

Here the bracketed term is simply the mean value of the z-dependent amplitude of the field, averaged over the entire cell length. Calling this field 8, we have A E = qFE, .L sin 4. (14-18)

..,

..,

from end A properly adjusted Alvarez linac maintains a constant value of &, to end of the entire structure, and in general the transit-time factor is constant, or almost so (see beyond). Hence if 4 represents the synchronous phase anglc and if it is constant, and since the cell length L increases along the machine because L = PA, then it is apparent that the energy gained by the synchronous ion increases from gap to gap (as opposed to remaining constant in the Widerije machine). It will now be shown that the increments of momentum are constant. The total energy is y = E/Eo and thc momentum is 7 = p / ( M o c ) and by Eq. (1-25) we have -y2 = 1 72, SO that E"Eo2 = 1 p"(Moc" from which it follows that E 2 = EO2 p2c2. (14-1 9 )

+

+

+

On differentiating this, we find

where finite changes have been substituted for differentials. By rearranging Eq. (14-19) we get

But the left side of this equals Therefore Eq. (14-20) becomes

P2 by

Eq. (1-14), so we find that

P = pc/E.

When we equate this to Eq. (14-18) and use L = PA, we obtain ~p = (IFG,

..5 C

sin 4.

(14-23)

This is a constant, or nearly so, since F varies only slightly if a t all, as will be discussed later. Therefore the momentum is a linear function of the number of cells, so a t the nth cell we have pn = nqFE,

.,

X

c sin

4.

I t follows that

wherc for convenience we introduce the abbreviation

THE ALVAREZ LINEAR ACCELERATOR C

-- (IF&,.A

275

sin 9

flo

B y ~ q (1-23) . wc I m w 1 solvirig for T,, we find

+ rl," T,

=

=+,',so ( I &[(I

+v,,~ )"

+ vn2)t' - 11

= yn = 1

+ Tn/Eo, and on (14-27)

We know from Eq (14-12) t h a t T,, d ( y n t 1 s solely on Eo and incchanical p:irametcrs of t l ~ c:~ccc~ler:rtor, so thc final energy fixed. T h e last equation thercfore tells us tliat (' is siini1:~rly determintd wlicn thc. ~ u n c h i n cis built, so by Eq 114 26) it l i secn t h a t tllc a \ eragc pc~alifiel(1 I , ,L, is ~)rtwril)etl,once 4 has been clioscn. Since the. encbrgyproduced by a linac i> comparatively low, tlir change of the projectilc's mass is small :inti the const a n t incrrlment of n ~ o n l c ~ l t ua~t neach gap m a y be rcylxcctl by :L constant incrcn~cntin velocity This ~wrinitsdetcrininntion of thc lengths of succcssiw cells in :I silnldc inanncr. If tlie particles start from rest, t h e length of t h e first cell is L1 = 7 1 )=~ TAU. Therefore L2 = TV2 = ~ ( 2 ' ~ All) = ~ T A and I ) I,, = n ~ A v the ; l m g i h s of t h e cells increaw in proportion to the, first poner of thv series of integers. As discussed carlicr, this 1e:tds to ~ n ~ p o s s i h lshort y initial cellj, so i t is cilotornnry t o inject with ttrl energy of 500 to 800 kcv from w C'ockcl-oft-b'alton accelerator, the initial integer in t h e scrics k i n g chosen in :wcord. T h e effect of transit time can be seen explicitly if tht, amplitude of the electric field is constant along the gap ant3 zero i n e ~ d etlie tubes. T h e transit-time factor in Eq. (14-16) then hecomes

+

where t h e cell number n has hwri :iddctl :is a subscript to rec:tll to nund t h a t the lengths of' the gal's and of tllc cells change :tlong, thc machine. This transittime factor falls more rapidly with increasing values of Q,~/I,,than docs the cm-re,iporitling factor for thc. VTitleroe rnacllir~e as, givcn in Eq. (14-9). At q,,/L,, = j , the v:lluc low is 0.9, mliilr a t g,,/TJ,I = 3,i t has dropped to 0.63. As short :t gap as is podsihle without volt:tgc brc:iktlown is thcrcfore t1csir:ihlc in the Alvarcz linac. T h e ratio of gap to c d l Itwgtl~sIS usually chosen to lie bctween j a r d \ Tt is wort 11 noting that thc ])rocluct 6, T, owurring ill E q (14-18) is thc potential tliffcrtvce across a cell. Since them ficld is zero within a tube and has a finite average> value &, ,", in a gap, the same potential difference appears across a gap, so t h a t

.,

E m 0 avg

=

Ern a&.

(14-30)

276

LINEAR ACCELERATORS

Thus the field in a gap is increased above the average value along a cell by the ratio L/g. Voltage breakdown from tube to tube therefore sets the maximum electric gradient, rather than breakdown along the main cavity. Since the drift tubes decrease the volume available to the electric field a t a region where 8 is strongest and also decrease the volume available to the magnetic field a t a region where B is weakest, it is not surprising that the tubes influence the frequency of each cell. Experiments have been carried out to determine these effects and to investigate what parameters can be adjusted to hold the frequency constant. It has been found t h a t the interrelationships can be represented by an empirical equation of the form

-c, c,,n c,,L, C dd,X . Ln where the C's are constants, D is the diameter of the cavity, and d , is that of the cylindrical drift tubes. This expression shows t h a t to obtain a fixed A with constant values of D and d,, it is necessary for g,/L, to increase as the cell length L, rises along the machine. This procedure may be continued only until g,/L, becomes so large that an intolerable decrease in energy gain is encountered because of the transit-time effect. Alternatively, if A and g,/L, are kept constant, then as L, increases, d , must decrease if D is constant, or D must decrease if d, is constant. Combinations of these possibilities may be used, for each has practical limitations; for example, if the diameter of the drift tubes is reduced too much, the inner holes become so small as not to transmit particles which are somewhat off course and the edges of the tubes become too sharp to hold the voltage. A recent development is to shape the outer contours of the drift tubes so that they resemble footballs rather than cylinders. It is then possible to maintain a constant D and an almost constant d, and g,/L, over a larger increase in L, than is possible with cylindrically shaped tubes. I n order to compensate for irregularities of manufacture, tuning elements are incorporated along the length of the cavity. These may consist of metal rods or balls which are pushed into the cavity, or of sheet copper "blisters" on the walls which may be deformed by pistons or cams to project varying distances, the goal being to keep &, .,, constant along the full length of the machine. @L =

+

+

+

14-3. Economics of Cavity Design I t is shown in texts on RF circuits that the shunt resistance of a cylindrical cavity of radius b and length S, oscillating in the TMolo mode, is given by

and that the quality factor is

\

.

, ,

277

ECONOMICS O F ('AT1 T F DESIGN

Here 6 is the .'skin depth"; the density of currcnt in an AC circuit falls off exponentially with dist:mcc bclow thc surf:icc of the cc~nductor,but losses nlny 1)o computed on the basis of :t uriifo1111C L I I ' I . C ~ c~ l ~ m i l ydown to the (Icpth 8. This quantity is givc~iin I l K S units by the exprcssion 6 = [ p / ( ~ f p ~ , , ) whcrc p is thc resistivity in ohm-nwters, p is the rtll:it,i~.c,pcr~nc:~hihty of the conductor (approximately unity for copper), po = 4~ x henry per meter is the perrncability of space, and f 1s the frequency. \\'it11 c.ol)pcr p = 1.73 x lo--' ohm-meter, so we find 6 = O.O66/f% = 2.2 x 1 0 ~ ~\\'hen ' ~ 6. and b are both cxl~ressedin terins of A (by Eq. (14-10) we have A =: 2.6lb) and when losscs in the disc-shaped ends are neglected (that is, if h << P) , the above relations take the forms ]Ih,

Q

=

T<.LX$4,

(14-35)

where the K's are constants. Thc powcr dissipated in a circuit is related to the peak voltage by the expression P = iT7,,,:!/R,,,, so in the cavity we have

This rather weak dependence of the power on h implies that the choice of wavelength is of little importance. However, linear accelerators are almost always operated in pulses, since the powcr denlands are hxgli (several megawatts for machines in the 50-Mev range), and the cost of the driving circuit is determined less by the powcr than by the magnitude of the encrgy TV delivered per pulse. Hence we now give consideration to this quantity. The voltage in a driven oscillatory circuit is known to rise rapidly a t first and then asyniptotically to approach thc peak r d u e according to the expression (1 4-37) V = Tirn[l - c'xp (act/QX)], so when the tirne is given by t = &A/(Tc), the voltage is 1 - l / e = 63% of the final value, while when t = lO&h/(ac), the figure has risen to 90%. Whatever the length of the pulse t,, it is clear that we may write t, a &A.

(14-38)

Then the encrgy in the pulse is

IV

=

Pt, m PQX.

(14-39)

Substitute for P and Q from Eqs. (14-36'1 and (14-35) to find

The R F energlj per pulse is therefore strongly dependent on the wavelength. A small value of h is desirable from the cost standpornt of electronic equipment,

278

LINEAR ACCELERATORS

but it may imply such small apertures in the drift tubes as to make doubtful the attainment of any useful quantity of projectiles. Further, as will be seen l:tter, focusing magnets often are incorporated within the tubes. In addition, any decrease in general dimensions makes the tolerances of manufacture more rigorous, and if the cavity is not built to specifications it will tend to oscillate in some undesired mode. Note also from Eqs. (14-36) and (14-40) that for a given total voltage, both the RF power and the RF energy per pulse are inversely proportional to the total length S of the cavity. This means that one must compare the price of a long accelerator and relatively inexpensive R F equipment with that of a short machine and costly electronic components. Compromise and judgment are required. 14-4. Phase Stability

I t should be clear that up to this point in the discussion of linear accelerators, interest has been confined to the synchronous particles. These cross the gap a t such a phase of the voltage that the subsequent velocity is just correct to bring them to the next gap a t the same phase as before. But, just as in circular accelerators, only a very few particles reach the gaps a t the synchronous phase; most arrive either sooner or later and these particles will receive the synchronous energy only on the average, through the principle of phase stability, provided that the synchronous phase lies on the appropriate side of the voltage wave. This situation has already been discussed in 8 6-4 and the result for a linac was expressed in Eq. (6-10) :

A particle with momentum in excess of the synchronous value must necessarily take less than the synchronous period to reach the next gap, since the distance between any given pair of gaps is fixed. I n order that the phase of such an ion should return towards the synchronous phase +,, it is necessary that the next acceleration should be smaller, and this will occur if the electric field is increasing a t phase 4,. Such a condition was illustrated in Fig. 6-6. 14-5. Transverse Stability

The situation in a linac, with regard to focusing and defocusing forces arising from the electric field a t the gaps, is similar to that encountered in a cyclotron. This was discussed qualitatively in $7-15, which the reader is urged to peruse once more, with the realization that the axial (vertical) forces produced by the electric field between the dees are now replaced, in the linac, by forces transverse to the line of flight. The shape of the field lines is as shown in Fig. 7-12, where we now consider the electrodes and the electric field to have circular symmetry about the axis. In brief review: for drift tubes open a t each end, the defocusing forces encountered during transit of the second half of the gap will

TRANSVERSE STABILITY

279

be smaller t h a n t h e focusmg forces expcr~encedin the first half (so t h a t a net focusing action occurs) only if the electric field is derre(1sing with time during thc entire transit. This is dinmetricdly opposed to the requirenlc,nts for !)l~asc, st:rhilitv, which necessitates t h r field to bc rising, as seen in the preceding section. P11:ise st:tbility is essential if any significant q u m t i t y of ions is to he :lccclernted, so we must see what can he done to ohtnin transverse stability c w n though the synchronous ions cross when the f i ~ l ( is 1 increasing. Sccond-order (velocity) focusing, mentioned in S 7-15, depends on a chnngc in v c l o c ~ t yduring transit. If thc gap 1s crossed nhilc t h e field is rising and very near t o its peak, the defocusmg forcv of the sc,contl half lasts for a shortcr. time than the slightly mmdxr forming force of thc first half, so t h a t a net convergence c:m result. B u t if t h e synchronous pllase angle is thus positioncd near ~ / 2 it, is clear from t h e discussion of § (5-7 that the acceptance anglc for nonsynchronou. particles is very s ~ u a l land only minute currents can ljc protluccd. Althougll this type of focusing appe:trs to h a l e been responsible for the olwr.ltion of thc. early \T'ideriic 11n:lc and its i l u n ~ e t l ~ a dcsccntl:ints, tc i t i5 cntirely inadcquntc hy modern standards. All present-day proton h e a r accelerator- ohtain trar~svcrhcfocusing either through the use of grids ( a s ~ 1 1 1hc cle+ ~ 1 1 1 ) c ~sl l ~ o ~ - t l yorl hy t h r cwq~lovrlwntof magnets inside the tlrift tuhcs. If m1cnoid:tl windings are used to c ~ e a t e:Llongitudinal magnetic field, particles inoving a t :in :tngle to the axis will perfor111 hc1ic:tl p i t h s and return t o the axis after c:wh revolution. Solenoids within tlrift tuhes Imvc hecn used to product focusing in this manner, hut thcl 1)owcr i.oquirc~cIis I ~ g and c the heat-dissipation prohlem i: forlnitlahle. T h e nlost rcwxit tc~clm~quc is to put quatlnipolc magnets inside t h e tlrift tuhes; hucc~essivemtgnets are so oriented t h a t a net focusing action takes pl:tce, : ~ w s l l be tlescribed in Chapter 15. Since the forces are stronger t h a n with solenoids, the power and heat dissipation difficulties are less severe. \Ye will now consider, in first approxinmtion, thc gcncral prohlv~nof trnnsvcrsc stahihty in linear accclcrntors, since this will give some appreciation of wliat forces ~ n w br t countered by ~ o l ~ n o i ( o1rs q u : ~ d r ~ p o l ( ~ :inti s ; we will thcn discuss how focusing can be obtained by the usc of grids in the. entrance openings of the drift tubes.

Fig. 14-4. Div & = 0 in the region outlined by dots.

Uecauscl of t l ~ cc u r ~ a t u r c 'of t l l ~c l ( ~ t r ~l111e5 c of f o r w i n th(> fringing fitsltl k)etu.ccn drift tubes, :is shown in Fig. 14-4, i t l a apparent t h a t the t r a n s ~ w w (i.c., radial) component of the field varim in nl:-lgnitutle with distancc along thc

280

LINEAR ACCELERATORS

gap. An expression to describe this change can be derived from the divergence of the field, given in cylindrical coordinates R , 6 , z: div E

=

1

a

Ra~ (REB) -

a&. + R-1 a&9 -+5 ae

Now assume that there are no charges present in the region of interest, so that div E

=

0.

(14-43)

This means that we neglect space charge; the number of ions is assumed to be so small that the field is not distorted appreciably by their presence. It also means that the electrodes from which the field lines originate lie outside the region traversed by ions. Thus in Fig. 14-4, div & = 0 in the space outlined by dots. It is clear that &E = 0 and therefore dGe/a0 =O. Therefore the last equation may be written as

2 iR

~ ~ ( R G R ) = R- m~. Here it has been assumed that a&,/& is not a function of R and hence may be taken outside the integral. This assumption is not strictly true, but is more and more valid the smaller the value of R . This analysis therefore holds only for small excursions of the ions from the axis. Integration yields

The radial field component I n is seen to change sign a t the mid-point of a gap with symmetrical field, since 8, increases with z in the first half and decreases in the second, as shown in Fig. 14-4. (The ions move from left to right.) The radial force due to the electric field first focuses and then defocuses:

F,

=

qg,

=

-9--.R a&* 2 az

This force is reckoned as positive if directed outward along increasing R , as occurs in the second half where d&,/az is negative. There is a further radial force which arises from the R F magnetic field encircling the axis of the resonant cavity. (This was omitted in the study of the forces a t the dee gap of a cyclotron, $ 7-15, as it was there negligible.) Like the electric force, it acts only when the particles are in the gaps, since the drift tubes shield their interiors from a time-varying magnetic field. The strength of the field B is related to the time rate of change of the axial electric field F, through the Maxwellian equation curl H = i),which may also be written as curl R I P = ko& the permeability po and the permittivity ko of space being connected by the equation ~ o k - 0= l/c2. Tn the present instance we consider a circular loop centered on and perpendicular to the axis, and the Maxwellian equa-

tion states that the inagnetomotir-e forct. around thc loop is cqu:d to thc tinw rate of change of the electric flux through it:

It follows that

This sliows, as expected, that the :tziiriuthal magnetic field is zcro on tlic axis and strongest a t the walls of tho cylindr~c:dcavity. In order to tlcterinine the direction of tlirs field from elementary considerations, wc iiiay ronsi(icr it as generated by the flow of a current par:tllel to the axis ( a true movcmcnt of

Fig. 14-5. Magnetic focusing force on ' I an off-axis proton between drift tobcs.

0 \

--

-

cliarge along the drift tubes :tnd a displacerncnt current in the gapsi. \\lic~n this current ruoves in the direction of the projectiles, say tow:ird the rc:der, the right-hand rule shows that R is counterclockwise arid hence the foiw on a positively charged ion is toward the axis. This force comes to zero just when 5:defocus FB:focus occel

Fig. 14-6. Relations between electric field, charging currents, and transverse forces.

1

F ,focus 8' FB:defocus

w -

n

decel

forward

I bockword

the axial currents on the tubes have brought thein to the statc of niaxiinu~ii polarization; this is, when the accderatlve clcctric field is :it its peak :tnd tllc axial currrnt is zero. The situation is intlicatcd in Figs. 14-5 and 14-6. By refcrencc to E h l (14-48) it is scen that tlie nitlgnetic force is

As in Eq. (14-16), a positive force acts in the direction of increasing R ant1 is defocusing. This occurs when the axial current is in the bnckwartl direction, a t which timci d ~ , l d tis ncgativc. 'l'hc total radi:tl force 17B = Fr f 1;;7 is givrn hy

282

LINEAR ACCELERATORS

Thus, while the phase of the voltage changes from - ~ / 2 to ~ / 2 thc , clcctric ficld defocuscs and the rnagnctic field focuses. From ~ / to 2 3 ~ / 2the sitwition is just reverscd. We must now determine where the synchronous phase angle must lie in order t h a t the net force may be focusing. A changc in momentum equals the integral of the force over the time in which i t acts. Hence the outward radial momentum h p R gained by a particle in crossing the gap of length g a t velocity v (assumed constant, so dt = dz/v) is given by

F R dz

=

q~ --

2v

v ac ) -+-2 dz. a2 cz at

JYl2 (ag, -n/2

(1 4-51)

Now&, varies not only with time but also with z, so 8, = F, ( z , t ) . Let t , be the instant a t which the sync.hronous ion reaches the center of the gap. Then the time a t which i t has movcd a distance z beyond the center is

t

=

t,

+ z/v,

so that we have

and Then

Carry out the indicated operation and rearrange to find

Therefore Eq. (14-53) may be given as

After integration of the first term, this becomes

Now 8, is a sinusoidal function of time with an amplitude which depends on the axial position, so using Eq. (14-52) we have 8, = G,(z) sin w,t

=

&,(z) sin wrf(t,

+ z/v).

(14-59)

A particle moves from one gap to the next in half a period in a Wideroe linac and in a full period in the Alvarez machine. T o cover both types, we write for

TKXSSV EIISE STAB 1 ~ 1 ~ 7 ;

283

this distance I, =- IVK = 2 a z ! K / ~ ,where ~ I{ = f for the first machine and K = 1 for the second. Hence 2irK/! (14-60)

-i-.

=

WTf

Thcreforc with the help of Eqs. (14-59) and (14-60), Eq. (14-58) becomes

ap,,

=

a)

@ (l--

2~"

rr/2 .-,2

acZ --

a(/, + 2/13)

dx.

T o perform tJhis intcgr:ition, we iliakc. thc, f l ~ r t l ~ assumption ~r t h a t the anil~litutie of thc ficltl innintnins :I constant \-:1!11v along thc gap, t h a t ib. &,,(a) = C,,,. Tlien t l i f f c ~ ~ n t i : i tEq. c (14-59) using this :~ssurnl)tiori:mtl Eqs. (14-GO) ant1 (14-62), t o find

ac2 a t

+

2

2 a Ku

- --

)

L.

E.,, con

(2'p+

&,)j

so t h a t Eq. (14-63) becomes

Carry out the integration m t 1 use the trigonon~etricidentities for t h e sinc of sum and of a difference to obtain

:t

This is t h c final result, for n linac with drift t u l ~ c sopcn at e:wli cncl. A pohitixrc value of Ap,, rcprcwnts a n outn:irtl r:idi:d r n o ~ n m l u i na c y ~ ~ i r cin d trc>nsit of the gap. T h c tlcfocusing t c m ~nit11 cocf5cient unity arises froin tlic c.lcrtric ficl(1, n l ~ i lt(l~~ cfocnsing t c m l with cocfficllmt ,Pcoilici from the :wocintcd 111:ignctic ficltl. I3otli influcncw t l c p n t l cqu:dly or1 t h e length of the gap, the factor K k i n g for the \\7idcr.iie linnc and I for tlie A l w r c z acce1cr:~tor.If tlic net effect is t o bc. focusing, A,,, must be nygative, and this O C C U I . ~o111y if cos 4, is negative. Therefore t o obtain transverse st:ll)ility the syncllronous pliase

284

LINEAR ACCELERATORS

angle must lie in the second quadrant (between ~ / and 2 T ) so the electric field will be falling with time. Unfortunately, as seen in § 14-4, phase stability requires +, to lie in the first quadrant and therefore the two types of stability are incompatible. The incompatibility disappears when P = 1, for then Ap, is zero irrespective of the value of 9,; there is neither transverse focusing or defocusing. Such a situation is closely approximated in electron linear accelerators a t energies over 2 or 3 Mev, but with proton linacs of energy even as high as 50 Mev, where p z a t the final energy, the transverse defocusing forces are severe, particularly in the early gaps where P is very much smaller. The difficulty is fundamental and can be ovcrcome only by some stratagem which supplies extra forces (such as the introduction of magnets within the drift tubes) or which alters the character of the electric fields, as by the use of grids. This latter technique will now be described. 14-6. Transverse Focusing by Grids

Let us leave the exit holes in the drift tubes open, but add grids a t the entrance end of each. In the following analysis it is assumed for simplicity that the grid is so dense that all the lines of force terminate on the grid with no penetration beyond it. (In practice the grid may consist of a few ribbons of tungsten or molybdenum aligned edgewise to the beam, thus affording someFig. 14-7. Schematic representation of the electric field lines when a grid covers the entrance hole of a drift tube.

thing less than perfect shielding.) In the ideal case illustrated in Fig. 14-7, it is clear that the radial component of electric field is always focusing. Wc return to Eq. (14-61), and in the already integrated first term we adjust the value Em(z) to fit the new conditions. At the open exit end of the first tube, where z = -g/2, we set 8, (z) = 0 as before, but a t z = g/2 (just outside the surface of the grid a t the entrance of the second tube) we let 8, ( 2 ) = Em, its assumed constant value along the gap. Then Eq. (14-61) becomes A ~ R =

"R

--

u ~ sin, TKg + 4s - (1 - p)

)

/"' wc+

dz). (14-67) 2v2 ( 7 ZIV) The integral is handled as earlier, in going from Eq. (14-63) to Eq. (14-66), by assuming that the amplitude is constant along the path. The result is

- (1 - ,)[sin

L

+ &)- sin (-F+ +s)]}

(14-68)

We arc concerned here only with proton l i m e s for which P2 is inconsidcrnhlc cwniparctl to u r i ~ t y ITcnw, :11q~rownlatc.1\',

This is the end result. Transverse focusing exists when h p l Cis negative. F o r a vanishingly short gap, this occurs for all valucs of +, from 0 to T . For a finite n K g / L ; so since phase sta= aI
+

stobility

erse stobility

9 Fig. 14-8. The range of stabil~tyin a grid-focusing linac. tomary choice t h a t g / L = 4, particles can be awcleratcd successfully with +, lying bctwwn a/4 :tnd ~ / 2 T. h e cniployment of grids a t the entrance ends of the drift tubes thus makes a proton linac a practical device, although a t thc cxpcnsc of losing perhaps 2.574 to 5 0 9 of tht. p:~~%ic.les by interception on the grid bars.

14-7. Bunchers Because of the limited range of phase in which p:trticles can he accepted in a proton linac I even if magnetic focusing is used, rather t h a n grids), i t is clear tliat there is a waste of p a r t of :t continuous stre:tin of ions entering the linac from its injector. A properly tinlcd chopper whcad of the linac can renlovc these ions, hut there is little point in such a cornplication. I t is much more d r sirable to shift tlie otlicrwise usc~less particles forwlrd or backward, so tliat they will join the small bunches t h a t nil1 be accepted. This can be accomplished by thc use of :t b u n c h u . This consists of n short section of linac (generally containing only onc g a p ) , operating al, the same frequency a s the linac itself and positioneti ;In appropriate tlist:lnce ahead of it. P:trticles t h a t a r e destined to e n t w tlie 1in:tc a t the proper phasc for acceleration pass through the buncher when its ficild is close to zero; earlier ions traverse i t when its field is decelcrntiw, and later ions are speeded up. Consequently when the drift space between huncher and linac has b t m traversed, all three groups arrive almost simultaneously and a t the proper pliusc for acceleration. T h e action is the same as the velocity modu1:ttion employed in klystrons t o form bunches of electrons. Such buncl~ingin s p a w is nccessarily acconlpanied by

286

LINEAR ACCELERATORS

an increase in the spread in energy as the particles enter the linac, but the net vffcct is a gain of intensity a t the output end. 14-8. Debunchers

When a linear accelerator is used as an injector for a synchrotron, the spread in energy of the particles leaving the linac may be too large to permit all of them to be accepted by the synchrotron or, a t the least, may produce an uncomfortably big amplitude of the synchrotron oscillations. This objectionable situation can be alleviated by permitting the ions to drift some meters after leaving the linac, so the fastest particles arrive first a t a debuncher (another onc-gap cavity driven by the linac's oscillator) and the slowest one later. The field in the debuncher retards the fast ions and accelerates the slow ones, thus reducing the energy inhomogeneity ; in a practical case, perhaps to one fifth its fornwr value. A corollary to this effect is a debunching in space of the protons, so that they enter the synchrotron in a more uniform stream. 14-9. Typical Proton Linacs

Like all accelerators, proton linacs have characteristics that make them suitable for some purposes but not for others. An outstanding attribute is the fact that the particles leave the machine of their own accord and in a fairly well collimated beam just as from a Van de Graaff accelerator, with the advantage that the mergy can be vely much greater. The Q of an Alvarez machine is very high (perhaps in the neighborhood of 80,000), so it takes a relatively long timc to build the voltage up to operating value a t each pulse; and therefore with comparatively little furthcr expenditure of energy the syst,em can be kept oscillating for a fairly long time (several hundred microseconds). The necessary input of some megawatts of R F power for an interval of this length is within the capability of modern, though expensive, oscillators and amplifiers. Pulsed operation of this sort is particularly suitable when the linac is used as an injector for a synchrotron. When a pulsed proton linac is employed not as an injector but as an accelerator in its own right, it has the advantage over a cyclotron that the energy is unlin~ited,in principle, but the disadvantage that the time-averaged output currmt is several hundred times smaller. To run an Alvarez linac continually, rathcr than in pulses, necessitates the steady generation of enormous quantities of R F power which ran be obtained only with a very great investment in electronic equipment. As will be noted in some detail further on, this has been done in one instance, but only a t the comparatively low energy of 7 Mev. Roughly speaking, the cost of a synchrocyclotron varies as the cube of the energy, while that of a linac only as the first power, so beyond some critical encrgy the linear machine should be cheaper. But the development of the proton synchrotron has altered the picture a t the superenergy end of the spectrum, while the advent of the high-current and variable-energy FFAG cyclo-

288

LINEAR ACCELERATORS

these increase in length from 1.45 to 6.29 inches; in the second, from 6.29 to 12.35 inches; in the third, from 12.35 to 15.79 inches. Tube diameters decrease along each tank, from 5.52 to 2.42 inches in the first, from 6.57 to 2.54 inches in the second, and from 5.64 to 3.90 inches in the last. The ratio of gap length to cell length is constant a t 0.25. Transverse focusing is supplied by grids. Proton energies a t the ends of the three sections are 10, 40, and 68 Mev, and magnetic deflectors can divert particles of these energies for experimental use. Power requirements for the three tanks are 600, 2200, and 3200 kw a t 202.55 Mc/sec. This is supplied by a master oscillator which drives three amplifier tubes, one for each section. The four tetrodes involved are so-called resnntrons. Each of these consists of two resonant coaxial cavities, one inside the other, the filament structure, control grid, screen grid and anode each forming part of the cylindrical walls. The system operates in 300 microseconds pulses, a t 30 to 60 pulses per second. Time-averaged currents are 0.2, 0.04, and 0.02 microampere a t the three stations. A 50-Mev proton accelerator has been built a t the Rutherford High Energy Laboratory in Harwell, England. This machine also is in three cavities, the energy increments being 10, 20, and 20 Mev. Stability is obtained by grids in the first section, where the drift tubes are very short, and by quadrupole magnets in the others. The 50-Mev proton linacs used as injectors for the Brookhaven 33-Bev alternating-gradient synchrotron and for the 12-Bev zero-gradient synchrotron a t Argonne both employ 124 drift tubes which increase in length from 1.931 to 13.961 inches. The outer diameters range from 8.16 to 5.80 inches, while the inner hole increases from 0.5 to 1.25 inches. All the tubes are shaped somewhat like footballs, for the reasons given earlier. The ratio of gap to cell length changes from 0.231 to 0.2508. The outer cylindrical shells, 31.2 inches in diameter and 110 feet long, are each composed of 11 copper-clad steel cylinders bolted together. The RF power required during pulses is 2.5 megawatts a t a frequency close to 200 Mc/sec. Both machines are driven by oscillator-amplifier units, one superpower triode being employed in the last stage a t Argonne, while several smaller tubes are used a t Brookhaven. Transverse focusing is produced by quadrupole magnets within each drift tube. I n the Brookhaven machine the magnets are pulsed for somewhat over 10 microseconds and need not be cooled, but a t Argonne multiturn injection into the synchrotron is employed, so the beam pulses are 250 microseconds long. Consequently the quadrupoles are activated continually and their windings are formed of water-cooled tubing. A 50-Mev proton linac, rather similar t o t h a t a t Brookhaven, acts as the injector for the 28-Bev alternating-gradient synchrotron a t CERN. I n Russia, a 20-Mev linac is the injector for the 7-Bev synchrotron a t 310scow. There is a 21-Mev machine a t Kharkov and a 10-RIev device a t \Tarsaw. A 9-Mev linac injector is used with the 10-Bev synchrotron a t Dubna.

290

LINEAR ACCELERATORS

A high-intensity linac of the Alvarez type has been built a t Lawrence Radiation Laboratory, Livermore, California. The machine operates a t 48 Mc/sec :tnd produces 100 milliamperes of 7.5-Mev deuterons: This large yiclh is obtained by continuous, rather than pulsed, operation and the input energy is correspondingly high. The beam of ions accounts for 750 kilowatts; R F excitation, 1000 kilowatts; dissipation in the anodes of the drivers, 750 kilowatts; solenoidal focusing magnets in the drift tubes, 4400 kilowatts; auxiliary power, 500 kilowatts: a total of 7400 kilowatts. The injector for the 7-Bev proton synchrotron Nimrod a t Harwell, England, is a 15-Mev linac operating a t 115 Mc/sec. It is 44 feet long and contains 50 drift tubes, of which the first and last are half-length. Focusing is produced by quadrupole magnets. An output current approaching 5 milliamperes is expected during pulses of approximately 2000 microseconds. 14-10. Linear Accelerators for Heavy Ions

Shortly before the FFAG cyclotron was shown t o be feasible, there was considerable interest in linear accelerators designed for particles heavier than protons, and several have been constructed. It is interesting to realize that a single machine can handle a variety of ions, and that the energy per nucleon is fixed, once the machine is designed. Thus if a projectile of rest mass M o contains A nucleons each of rest mass MN, then Mo = A M x and the kinetic energy a t the nth (last) cell, as given by Eq. (14-12), may be written

so that the energy per nucleon is a constant. Whether a particle can be accelerated depends on its charge-to-mass ratio and on the available electric field. Eq. (14-28) shows that there is a constant value of the quantity C given by Eq. (14-26), here repeated:

C

E

9 FA - Ern sin 4.

M,, c2

....

Particles with various values of q / M o can be handled provided Em sin 4 can be raised or lowered so as to hold C constant. Low values of Ern sin + are no problem, but the upper limit is set by the maximum power output of the oscillator or by voltage breakdown and by the fact that it is not profitable to allow sin to approach unity, since the width of the band of phase acceptance then becomes excrcdingly small. I t is evident that massive ions must carry a hrgc charge. Tlic machines built thus far are appropriate for ions with q / M o in cxress of a figure in the neighborhood of 0.2 to 0.3, in units of the electron's c11:lrgc and the proton's mass. It is extremely difficult to obtain adequate quantities of ions with a charge of

+

more than 3 rinlts, in tllc present stat( of dc\elopment of ions sources. Con- c ~ l u c ~ i t l yafter , h i n g genc~r;itctl,thcl ~ ) : ~ r t i c l varc \ brought to several I~unc-lrceL I,(,\ I)y :L (:o(.kcroEt-\Y:~ltol~ : L ( T O ~ ( ~ I : L ~ O: LI I I ( ~: ~ r rthtm 1 1 1 1 ( ~ tinto ~ d tlro l i ~ ~ ~ v y I O I ~liriac ( s o i ~ c t i ~ ~( i~ (d+l il c :I. / L I L ( L C ) T I I I ic, ~ oftc]1 h ~ i l tin two sections. T l i ~ first bring5 ttlc 1):lrticlcs u p to :tl)out 1 A[(>\ I)er nucleon, after which t11c.y I):M tlirougli the s t u p p e r . Tim conbl-ts of :L thin nictallic foil o r a jet of mercury va1)ar whicl~r m y rcniovc t n o niorc. electrons, and t h e projectiles t h c r ~ cntcr the swond section foi f i m l acccler:ltioi~. Tlic hilac a t Berkeley 1)rocluccs Ions such as Iielium, carbon, nitrogen, a n d oxygen a ~ t h:an energy of I 0 Rleo 1)er nuclwn. T h e two sections opcr:ttc a t 70 AIc/sec, the pre-strllqwr hc.lrig 16 fcet i o r ~ gand the post-stripper 90 fcct T h e y cont:m 87 : ~ n d67 drift t~lkws wliicli 1ncre:Lse in length from 1.67 to 19.8 inclrcs, t l ~ cgal) lcnqtlis clr:inging frorn 0 53 to 8 26 inclies Grid focusing 1- u ~ c in l the, firht v ~ t i o and ~ i ~ I I I : I ~ I Y I I I O nxtgnet ~C fucuslng in thc 'second. Tliv t m l < eli;inic~tc~~s alc3 124 and 108 iilrllch Tht. tl(.vicc 01)erutcs in pulse> 3 rnilliwroi~tlslong :inel yictltlh scvc ral nilcro:imperw of c w r e n t , averngetl 01 cr tinic. A r:~t'rrer \ ~ n i i l n rniaeliiric a t Y:ile U n i v c r s ~ t yworks at the s:tnlc frequency and energy, :lltl~oughthcrc arc only 36 drift tubes in tlie first section and 67 in the sccoiitl. I n tlie 1 1 c : i ~ y i o nlinac at the, I_'nivc,islty of Momchestcr, England, t h e first unit corit:llnh 40 drift tuhcs tir~vcnIn the ir1:mncr of :L \Videroe m:~cliinca t 23 ;\Zc/sc~,~1lerc:isthe second sc r~icrn,n l t h 75 tubos, 1s of the Alvarez type and o.icil1:ltc.s :it 7.5 ;\lc/sec Hcrv :lq:lin. grids and cluadrupoles supply r:~dial ~ ~:&o q 10 &rev per nucleon. d a b i l l t y in tltv two 1)arls 7'hc c i ~ c is A Ilea\-y-lon Iinar, 1)uilt in :t smglc scction, is operating a t Kharkov. The. r:ivlty is 18 ~ ~ t c t c rlory, h rc~~orl:ltc~s a t 14 3 l\lc/'we anti coritC~~ri5 101 drift tubes n11ic.h cont:i~rl qriel1 for foru-ing Injclction is a t 2.5 J l e v from a Van tit. (;r:~:ifF :~cceler:~torA slj:~rk-tyjic~\ourcc> 1)rotluccs Ci+, N4+ or 05+ions. The outl)ut of c:t~l)orr~ ) : i r t ~ r l eis\ 40 I ~ I I C I O : L I I ~ ~ (during ~I~F eaclh 400-microsecond ~ ) u l s ca, t :in cncrgy of 1 0 R/lc~pc3r nucleon. 14-11. Electron Linear Accelerators

If :i ~);trtic.lcof c1i:lrge q lrlovcs n d ~ s t a n c eI, in a n electric field I , i t will it is not necessary t h a t t h e field should :ic.cp~i~c :m cncrgy ygL. JIon.c~vc~r, c.xtcnc1 for tlrc full clistnrlcc,; one c:m, in principle, mount t h e two par:tllcl platcs of :i rl~:lrgcdc:tp:iritor :it tlic front nntl rear of a vehicle so t h a t :in ion I)cltnecm tlrcwi d l cslwicnce :i f o ~ m r dforce, h u t i t will never reach t l t ~ front elcctrotlc if the vcl~icleis tlriwn fornard by some outside agency : ~ t such :In inrrc;tsing s p e d tliat t l ~ eparticle always rcniains between t h e plates. T o mtlow thc, projectile with :my given a n m i n t of enerqy, one has the clioicc of using a wc& field and a long distance of tmvel of tlie vehicle, or a strong Geld and :a diort distance.

292

LINEAR ACCELERATORS

This technique goes back beyond recorded history; it was invented by the drivcr of a farm wagon who held a bunch of carrots in front of his horse by means of a long pole. The scheme was applied, stepwise, to particle acceleration in 1924 when Ising suggested that an array of metal tubes be aligned on a common axis and connected by wires of increasing length to a spark gap which discharged a capacitor, the length of the wires being such that each successive tube would receive a negative pulse just as a positively charged ion approached it. The idea was reduced to practice nine years later by Beams, who obtained a voltage multiplication of 6 in this manner. He also made the next advanrc by connecting the spark gap and tubes in series through a delay line, so that again the voltage pulse would arrive a t each tube just before the particles. Measurable quantities of electrons a t 1.3 Mev were obtained. Further attempts a t linear acceleration of electrons did not occur until after the war. It was obviously impractical to use the \Videroe method because of the extreme length of the drift tubes for such light particles. And when the magnetron and klystron became generally available, i t was clear that the Alvarez linac also was unsuitable, since a t h = 10 cm, Eq. (14-10) shows that the outer cylinder would have a radius of only 3.8 c n ~ ,leaving little room for drift tubes with a large enough hole for the passage of an interesting number of electrons. The electron linear accelerator which employs a true traveling wave was a direct outcome of the knowledge of waveguides arising from the development of microwave radar. The subject of waveguide theory is vast and well covered in many texts. We will here be content with a sketchy description of some aspects that are relevant to the task a t hand. It is known that a hollow pipe can act as a transmission line (i.e., a waveguide) for alternating currents, with the added characteristic that the guide behaves as a high-pass filter, since energy is transmitted a t all frequencies from infinity down to a critical cutoff value f,. In other words, wavelengths from zero up to a cutoff value A, are passed. Electromagnetic waves zigzag back and forth across the guide a t the velocity of light, and interference between these waves results in a pattern of electric and magnetic fields of a repetitive nature along the guide. Therefore a guide u~avelengthh, can be assigned to the pattern-for example, the distance between points of net maximum field or of net zero field. The guide wavelength therefore corresponds to the axial distance between points where the phase of the pattern has the same value. A pattern of interest, since it contains an axial component of electric field, is that of the TM,, mode, sketched in longitudinal cross section in Fig. 14-9. For this mode, it can be shown that h, = 2.61b, where b is the inner radius of the cylindrical guide. The associated magnetic field circles the axis, being strongest a t the walls and most concentrated a t the axial positions where the electric lines terminate. The entire array of patterns slides along the tube, say to the right, a t a characteristic phase velocity v,. Consequently if a chnrged

ELECTRON LIhTEAIlACCELERATORS

293

particle of appropriate sign were 1oc:~tc~I a t a pomt such as B in the above figure, i t would be accelerated in the direction of motion of t h e pattern; and if the phasc velocity could continually be adjustcd 1.0 equ:d t h a t of the particle, the projectile would remain indefinitely in this localized accelerating field. Fig. 14-9. Electric field pattern of the TM,,, mode in a cylindrical wnT eguide. Thc cntil.c>pattern s1idt.s along the tube with velocity v,. Froni the closc analogy between this and the situation where a thrill-seeker on a surfboard perchcs on a single ocean wave t h a t carries him along with it, :wcclrrators h s e d on this principle are often called surfboard linacs. Unfortunately, as will now be indicated. t h e phase velocity in such a simple w:~veguideis unsuitcd to the purpose. I t can be shown t h a t the free-space w:~velengtlih (related to t h e frequency of thc oscillat,or by fh = c ) is connected with thc cutoff wavelength A,. anti with t h e guide wavelength A, by thc equ:ltion

from which i t follows t h a t

<

If A A,, then we have h, z A ; b u t if A incre:zscs, then h, exceeds h and becorueh infinite when A =A,. Since the f r e q u m r y is the same no matter what wavelength is considered, i t follows t h a t thv phase velocity v, along the guide is given by

<

Hcncc when A , , then 71, =: c ; and when h = h,, v, becomcs infinite. Thuh thc phase velocity always exceeds t h a t which a n y particle can attain, no matter what its energy, and the pattern of wavcs will rush past the projectile, alternntcly accderating and decelerating it. It m a y be profit:ible t o repeat a rather well-known analogy t h a t shows I l o ~ a phase velocity can rang(. between infinity and tllc wave velocity. Consitlcr "a league-long rollrr thundering on the reef," t h a t is, a n ocean w:~venl)l)roncliing a s t r a i g l ~ tshort line. Let ohservers 1)e spaced along t h e beach. Then if t l ~ c direction of :wIvance of t h e wave is a t a n angle to the shore, the crest of t l ~ e wave will rc~iclithe observers a t succe.;sive instmtx, and evcm though t l ~ c forward velocity of the wave is only a few feet per second, thc crcst c:in arrive a t niclcbly separated observers mlth only a m a l l delay if the direction

294

LINEAR ACCE1,ERATORS

of advance is almost perpendicular to the shore line. For truly perpendicular incidence, the crest reaches all observers sin~ultaneously,so the phase velocity along the beach is infinite. At the other extreme, when the waves travel parallel to the shore, the crest passes the observers with the velocity of the waves themselves. I n the present instance, wavrs with v = c approach the walls of the waveguide a t an angle which depends on the wavelength and on the dimensions of the guide. The waves are reflected a t the walls and hence advance down the guide in a zigzag manner. The interference pattern between two such waves, moving across the guide in opposite directions, gives rise to the maxima and minima of the electric (and magnetic) fields which constitute the phase waves referred to above. When the zigzag paths are so "compressed" that no advance of the original waves occurs along the guide (that is, a t cutoff, when h = A,), then the waves approach the walls perpendicularly and the phase has infinite velocity along the guide. But when h A,, so that the zigzag pattern approaches a straight line, the phase advances with a velocity close to that of light. Nevertheless, there are means by which v, can be lowered. One way is to line the inner surface of the tube with some thickness of a nonconducting material of appropriate dielectric constant, a hole being left for passage of the projectiles. Much serious thought has been given to this. The chief disadvantages are that the dielectric will break down a t the desired high field strengths and it is difficult to remove the heat generated by hysteresis losses. A more practical scheme is to insert into the cylindrical tube a number of closely spaced annular discs of conducting material; these are often called irises. The inner surface of the waveguide is then sharply corrugated, as indicated in Fig. 14-10, and the original tube of diameter 2b now has a

<

Fig. 14-10. Longitudinal cross section of an iris-loaded cylindrical waveguide. central hole of diameter 2a. If the radial dimension b - a is less than a quarter wavelength, each pair of discs adds some inductance to the equivalent network and hence lowers the phase velocity. Values of a and b can be computed to give any desired phase velocity for any specified wavelength. I n using such a loaded guide as a traveling-wave accelerator, A is chosen as small as is consistent with available RF sources, since this reduces the power and the energy per pulse. The thickness of the discs is relatively unimportant, except for mechanical reasons, so it generally lies close to $ inch. The choice of the distance d between corresponding sides of adjacent irises is more in-

-

volved, thc olkinluin h i n g ti! = A / 3 5 , hut for practical reasons in constructing :mil tcsilnq qliort sections, tliv f i ~ u r e(i A/4 is 111orc convenient. This nle:ms that thc p l l : ~ch:lngc~hy 90° fro111 clihc to disc, c,o the structure is s:~iitt o operate on t l ~ c~ / 2~ i ~ o tIlt ~is. only in tlic firqt foot or so of the guide t l ~ t graded contml of the phnqc ~clocaity IS neetlcd, since electrons so rapidly approach the velocity of light. Subsccper~tly,the disc loading can be const:tnt, with v, = c or slightly less. T o prevent Ihe reflection of t h e traveling waves a t the output end of the guide, tlierehy engcndcring st:tntling w : ~ \ e s ,t h e energy can b r tlivertctl into a lossy termination and there t1issil):ttetl. Alternatively, if tlie pulses are widely enough sp:~ceti,the reflected wave will 1)c dissipated in the guide before the next pulse starts. 14-12. Stability in a Traveling-Wave Electron Linac

Since tlie lines of the clccxtric firld : I ~ Cinorc or Icss convex toward t h e axis of the acccler:ttor, i t is clear that the axial coml)oncnt g- is greatest a t the n ~ i d point of t h e wave pattern, a s suggested in Fig. 14-11. T h e arguments of Chapter

E ~ A. 08*C *D Schematic representation of electric field pattern and graph of the ficld strength.

Fig. 14-11.

/ -

~

/--"-.

z

6 on phase st:tbility all apply, though with solnewhat less severity, since a Iiigh-cncrgy eltxctron cannot ch:tnge its velocity ery much. If the energy gain a t points 9 or C is what is rcquirecl to keep tlie pal t i d e advancing a t the specd of the wxvc, tlic pl~asc~-st:~hle l m i t i o n is at, C ratlier t h a n a t A, for electrons bctn-wn A and C gain too much energy and luove forward past C t o Dl ml~ercthey gain too little and Iicncc n ~ back, ~ while c electrons behind A :icquire too little and fall even iurt1lr.r bchintl. T h e nearer the phase-stable angle +, is to the crest, the narrower is the hand of plxtsc stability. (These statements are not contrary to those of $ 6-4, w l i c ~ ci t was said t h a t for a linxc, 4, niust lie in tlie quadrant in n h c l l the ficld increases with time; for if one stands in the 1:ii)oratory and w-atclics t11c nravcts go by, i t will be serm t h a t ph:tsc-slahlc particles are indeed located a t a region in space where t h e field docs increase with time.) T h e 1)hase oscillations behave inuch like those which were considered in Chapter 6 for circular machines; in ])articular, the oscillations undergo adiabatic damping, approaching zero :lmplitude the particle velocity approac.lics c. (lonscqucntly t h e waveguide can be so built t h a t just after injection t h e synchronous phase angle +, lies a t the> front of the wave, a s at

F

296

LINEAR ACCELERATORS

point D. Large phase oscillations occur, and a considerable spread of early and late particles are trapped in the phase-stable region. This damps rapidly, since the energy rises quickly. Later parts of the guide are made of such dimensions that 4, slides back to the crest of the wave, point B, where the now tightly bunched electrons receive more energy per foot of travel. As far as transverse stability is concerned, it is clear that a particle on the leading edge of the wave and somewhat off the axis, a s a t point F in Fig. 14-11, experiences an outward component of electric field and is driven still further away, while an off-course electron on the trailing edge, a t point El is forced back toward the axis. A complete analysis, which will not be developed here, includes the radial force arising from motion of the electron with respect to the R F magnetic field in the waveguide. The over-all results are qualitatively of the same sort as were found in a linac with drift tubes: stability of phase and stability of radial motion are incompatible, and the transverse defocusing force, associated with stability of phase, approaches zero as the velocities of the wave and particle approach c. This occurs so soon in an electron accelerator that only a slight defocusing occurs. (This is often controlled by a longitudinal magnetic field applied externally.) Since the energy gain thereafter appears almost entirely as an increase of mass, a given transverse momentum is accompanied by a reduction in the transverse velocity. As a result, the transverse displacement increases only slowly. An alternative way of looking a t this is based on the relativistic Lorentz contraction of dimensions in the direction of motion. The length of the accelerating tube appears to be enormously reduced to an observer traveling with an electron, so the angular size of the exit hole a t the far end is correspondingly increased. It is worth noting that since the electron's velocity soon reaches an almost constant value, the output energy of the accelerator can be lowered merely by reducing the input power; if the electric field is weaker, the electron simply does not gain so much mass in a given distance, but it still keeps up with the accelerating electric field. As opposed to proton linear accelerators, electron linacs are of variable energy. 14-13. Typical Traveling-Wave Electron Linacs

I n the United States, the development of the traveling-wave electron linear accelerator has taken place largely a t Stanford University under the stimulus of the late W. W. Hansen. The Mark I machine, completed in 1947, produces 6-Mev electrons in a distance of 14 feet, after injection a t 79 kev, a t which energy P has the value 0.5. The waveguide is disc-loaded and graded so that v,,/c ranges from 0.5 to 0.972 in the first 3 feet, while in subsequent sections i t has the values 0.9823, 0.9906, and 0.9967. The system is driven a t one point by a magnetron oscillator delivering a peak' power close to 1 Mw during pulses of 1 microsecond duration, repeated 60 times a second, a t a wavelength of 10 cm. The time-averaged current is 0.05 microampere.

STAN1)ISG-IVAVE ELECTllON LIYEAR ACCELERATORS

297

The Mark 11 accelerator, put into operation in 1949, attains a n energy of 40 Mev, agam in 14 feet. A magnetron oscillator drives a klystron amplifier, 13 Rlw being delivered in 1-m~crosecontl~)ulses,again a t A = 10 cni. Thc. disc-loaded guide is dcsigtwd for v,/c = 1 throughout, even though this ent:uls sonic. loss of acceleration just after inlectlon a t 80 kev because of slipping of the electrons with respect to tlie pliase velocity. 'I'llc internal diameter of thc guide is 20 = 3.261 inches, the central hole 1s of' diameter 2a = 0.872 inch. while the ch:tr:~cteristiclength is t i = 1.033 inches = A/4. Mark 111, also at Stanford, follows its pretlecessors in general construction. I t is 220 feet long and protiuccs 700-hIcv electrons, again with injection a t 80 kev. The RF power is supphrd by 21 klystron amplifiers which are d ~ s tributed a t 10-foot intervals. These operate a t 2856 Mc/sec and produce a peak p o ~ ofr 300 M w dunng pulses of 2-microsecond duration. The repetition rate is 60 per second. The construction of such traveling-wave :~cceleratorsrequires machine-shop work of the very highest caliber, since some thousands of irises must be built and mounted to extremely closc tolerances. The program a t Stanford has also involved the dcs~gnand construction of Llystron amplifiers capable of a power output many t m c s th:~tof the commercially available units developed during the war. The success of the Stanford effort has been so great and the rewards in pure physics so outstanding that, ldans arc now under way for a similar device 10,000 feet long capable of yielding electrons of 45 Bev. This is a figure totally ur1att:~inable in electron synchrotrons because of the prodigious loss of energy associated with thr. centripetal acceleration. No such losscs occur In linear :meleration. Much of the early theory of trawling-wave electron accelerators was carried out in Great Britain, though the emphasis there has been on the construction of machines operating a t the 15- to 25-Mev range. At Orsay, France, a I-Wcv machine is scheduled for coniplet~onin 1960 and there arc reports of an elcctron linac of similar energy a t Kharltov, U.S.S.R. Two electron linear accelerators haye been rtyori,ed in the Ukraine. One is a 30-Mev machine with the phase velocity increasing from 0 . 5 ~to 0 . 9 7 ~ in the first section, while it holds constant values of 0.98c, 0 . 9 9 ~ )and c in the remaining stages, the total length heinq 530 cm. A current of 150 milliamperes is obtained during each I-microsecond pulse, repeated 50 times a second. Thc energy is variable from 10 to 30 Mcv. The second machine is a somewllat similar dcvice which operates hctween 30 and 90 Mev. 14-14. Standing-Wave Electron Linear Accelerators

A slightly different arrangement has hcen employed in a few electron linacs of modest energy. I n early form, these devices employ a succession of resonant cylindrical c a ~ i t i e sin close proximity, all tuned to thr same frequency. I n the machine developed a t Yale, 8 cavities of radius 19.4 em oscillate a t 600 Mc/sec

298

LINEAR ACCELERATORS

in the TMolomode, so the electric fields are entirely axial as in the Alvarez linac, except for curvature a t the holes of 0.9-cm diameter, through which the electrons pass, in the common end discs between pairs of cavities. There is negligible electrolnagnetic coupling between the resonators, and each is driven by a triode amplifier, all of these being excited by one crystal-controlled oscillator a t 4 Mc/sec, which is followed by frequency multipliers. Phase shifters in the feeds to each amplifier are adjusted empirically so that the field in each cavity is accelerative when a pulse of electrons arrives. Injection is a t 5 kev, and the lengths of the first three cavities increase in steps, from 7.5 to 12.5 to 17.5 cm, this latter figure being maintained in the remaining five units. The time-average current of 6-Mev electrons is 0.2 microampere, the electron gun being pulsed for 2 microseconds near the end of each 10-microsecond RF pulse. Radial stability is afforded by velocity focusing in the first two cavities, where the energy changes from 0.005 to 0.86 to 2.28 Mev. Beyond that point the transverse forces are substantially zero, as described earlier. If the holes between such cavities are enlarged until there is appreciable coupling, a succession of cavities then resembles an iris-loaded cylindrical waveguide and may be treated in that manner. When the length d is half a guide wavelength, the reflected waves a t the boundaries set up standing waves in the resonators. Since there is a phase change of 180" between adjacent irises, the system is said to operate in the T mode. Such a standing wave is composed of traveling waves moving in opposite directions. By proper control of the size of the iris, the phase velocity can be adjusted to that of the projectile, which rides with the forward-moving wave just as in the traveling-wave accelerator. The wave running backward dissipates power to no useful end and makes no net change in the energy of the electrons, sincc the accelerative and dccelcrative forces alternate a t a very rapid rate. The original advantage of this type of accelerator was that the accurately tuned cavities, each driven by a separate magnetron oscillator, acted to stabilize the frequency of the magnetrons, which otherwise might have drifted. With the advent of klystron amplifiers of adequate power, this advantage has become less important. The 18-Mev electron accelerator a t the Massachusetts Institutc of Technology is of this type. Injection is from a 2-Mev Van de Graaff machine, and 21 self-excited magnetrons oscillating a t 2800 Mc/sec drive the iris-loaded waveguide in the T mode. To assure synchronism, the magnetrons are powered in succession during each pulse. The machine delivers 1 microampere in pulses 1 microsecond long, the duty cycle being 1 in lo4. References to general papers on linacs will be found on page 367; more specific articles on Widerije, Alvarez, and heavy-ion machines appear on pp. 368-369. For papers on the theory of electron linacs, see pp. 369-370; on particular installations, pp. 370-371; on dielectric-loaded guides, page 371.

15-1. Introduction

I n a magneticsally guided accelerator the particles oscillate in hoth transverse plancb,i, thc c~nvvlopeof their. lnotion h i v g k q ) t wtliin rcasonnhle bounds by the action of thc ficxlti; but when the p:~rticlch:LIT extr:wtsil from the niachint~, the beam will sprcntl, hcroining quite cliffusc 1)y the tiinr, it crosses a room. 1Yh:~tis then nccdctl is some sort of lenS systcrn wlllc.11 will keep t h e lateral d ~ n ~ c n s i o nfrom s growing; if it c:tn bring the p:~rticlcs to a focus, so nluch the kwtter. F o r many years only partial fulfillment of thesc desiderata was availal~le,and this by means of a longitudinal magnetic field ; the vacuum pipe through which the particle,i travel m a y be wound with Lure t o form a long solenoid. Those fcw ions which hal)pen to be traveling p:ar:tllcl to the axis are unaffcctcd by thc field, but tht. majority move a t a n anglc untl 11:~vclongitudinal and radial coinpontmts of velocity. F o r a, trajectory a t 21ngle 6 to thc axis, the r:dial velocity is v sin 6, :inti the resulting r a d ~ u sof curvature is r. = -If71 sln B / ( B q i . T h e combination of this circular motion ahout thc axis and thc longitudinal vclocity brings ahout a helical p:ith. Foeusing a t the end of each t u r n occurs for particles starting :it the snnie angle, by restricting the anglc t o :t single value by the use of diaphragms, this tschniquc~1i:as I ) c m used to measure e l m of electrons. But particles leave a n accelerator with a spread in angle, so no general focusing occurs, although the c~nvclopc~ of thc hc:m is kept within a cylindrical bountiary. Such a method has been used in linear accelerators, thc solcnoitls 1)eing i n o ~ ~ n t cw t l~ t h i nthe drift tubcs of proton instrunlents and outside the w:ivoguidc~of elcctron n~achiner. 15-2. Magnetic Quadrupole Lenses

With the advent of the strong-focusing principle, i t i m m d i a t e l y became evident that t h ~ tcchnique s could t)e applied to t h e containment and focusing of a straight I-warn of ions, in acldition to its use in circular guide fields. As will be shown presently, i t is possible to produce ficlds, cither magnetic or electric, in t h e region :11)out :I beam of particles traveling, say, in the 2 direction, in such a w t y t t ~ focusing t forres arc produced for one tr:tnsvcrse direction .r, 299

300

QUADRUPOLE LENSES

Fig. 15-1. Components of field and force in a magnetic quadrupole. Positive ions approach the reader on paths parallel to the +z axis. and simultaneous defocusing forces in the other direction y. When this is followed by a field-free region and then by a similar field of force which has been rotated 90" about the beam, a reversed action takes place, resulting in an over-all focusing in both planes. A single unit is formed by a quadrupole magnet; two such quadrupoles form a complete lens system. Such a technique is more powerful than solenoidal containment and offers the added advantage of permitting the beam to be focused into a small spot of dimensions dependent on the size of the source. Such systems are of great use in directing particles throughout an experimental area, and quadrupoles have been installed inside the drift tubes of recent linear accelerators. Consider first the case of a magnetic quadrupole. If the source pole is a

RIAGNETIC QUADKUPOLE LENSES

301

rectangular hyperbolic cylindcr (both s l i t ~ t s )and thc sink pole is a congruent one in the other quadrants of the same asymptotic p1:rnes (Fig. 15-l), then it is not difficult to show t h a t the gradients of the field cornponents arc constant and t h a t the forces on a charged p r t i c l e nloving parallel to the surfaces of the cylinders arc proportional to tlie displacelnent,s from the corner formed by the asyn~ptoticplanes. This corner is taken as the 2 axis. If fixed magnetic potcntials T'o and -Vo are applied to thc two hyperbolic cylinders, in such a way that adjaccnt surfaces have opposite polarities and opposed surfaces the same polarity, then tlie potential a t any point (x, 21, zl between the poles is given by V = Gxy, (15-1) where G is a constant soon to be evaluated. Note that V = 0 on the x axis. The components of the field are

and the gradients of the components are

The gradients are identical, constant : m i positive, since B , rises with y and B,, riscs with s. Thc forces on a cl~argedl)articlc moving parallel to the z axis with constant vclocity v, = v are F -- - y l ~ H , ,= - q&x (15-6)

F,

=

yt)13, = qvGy.

(15-7)

These expressions show t h a t the forces :Ire proportional to the gradients ant1 to the displacements. The force F , acts t,o converge the projectile towards the

Fig. 15-2. A quadrupole magnet. The same cross section is maintained along the length.

zy planc, while F , drives i t away from the rx planc. If tlic quatlrupolc i z uf finite length in tlie 2 direction, it acts as :t bic.ylindricd lens wit11 crossed axes,

302

QUADRUPOLE LENSES

Courtesy of Pacific Electric Motor Co.

PLATE 28 Quadrupole lenses with 2-inch aperture and a gradient of 4,500 gauss per inch.

converging in the horizontal x direction and diverging in the vertical y direction. The practical realization of a quadrupole lens may be as indicated in Fig. 15-2, where a circular (sometimes rectangular) iron yoke supports four poles around which are placed the exciting windings. Hyperbolic surfaces are hard to machine, and it has been found experimentally that circular surfaces are almost a s effective, particularly if the particles do not approach the poles too closely. 15-3. Focal Length of a Converging Lens

Let us first calculate the focal length in the horizontal plane. A positively charged ion with constant velocity v, = v moving in the field B, is acted on by the force F, in the direction of decreasing x, as may be seen in Fig. 15-3. The equatiori of motion is

Now with v comt:mt

IW

1i:tve

--

d2x dz"

-

Hence

where the constant K2 is defined as

Here the rigidity Br has been used to rcplacc 1 1 f ? / / q , since i t is a more convenicnt measure of the particle's momentrun. Note t11at we have neglected the

Fig. 15-3. Coordinates and parameters for a converging lens.

extremely small change in v which attends the component of motion in the .r direction. Since the gradient G is positive., so also is R" and the solution of Eq. (15-9) is t h a t of simple harmonic motion given by

x

=

a cos Kz $ h sin Kz,

(15-1 1 )

whence

dx x

f

-

=

dz

-

a K sin Kz

+ bK co,s Kz.

(15-12)

If x = xl and a' =x,' a t the beginning of the rnagnet whcre x = 0, then n = x, and b = xl'/K. Therefore after traversal of the magnet of length L, we find

x x'

= XI cos =

-zlK

+ LLK' sin K L sin K L + xlf cos KL. ,1.

KL

I n matrix form these two expressions are

I sin 0)

(15-13)

(j

- K sin 0 cos 0 where for convenience we set 0

E

RL.

(15-16)

Define the focal length as that distance measured from the downstream surfare of thc lcns at which a n input beam, traveling parallel to the axis but

QUADRUPOLE LENSES

304

displaced from it in the x direction, crosses the axis after traversing the lens (Fig. 15-4a). Hence we must multiply together the matrices which take the particle through the magnet and then through a field-free region of length S

Fig. 15-4. A converging lens can become diverging. (a)

( b)

(which will be set equal to the focal length f, later on). The matrix for the neutral region has been developed in Eq. (3-24).We obtain

1 - sin 0 cos 0 - SK sin 0 K =

(-K

sin 0

+ S cos 0)

(x:)

(15-18)

cos 0

Now write this out as two equations and use the following boundary conditions; the input beam is parallel to the axis, so xl' = 0, and is displaced from the axis by the amount xl,while a t the focal point where S = f, the displacement is x = 0 and the slope has the value x'. Hence we have 0 = xl(cos 0 - fcK sin 0)

x' = -xlK sin 0. The second of these expressions is of no interest to us, but the solution of the first for the focal length of a converging lens yields fc

=

1 K1 cot 0 = cot KL, K

(15-21)

where K has been defined in Eq. (15-10). It is worth while writing this expression out in full, to realize its implications.

As far as the coefficient 1 / K is concerned, the focal length is increased by a particle of large momentum, and it is shortened by a powerful magnetic gradient. When we consider the term cot 8, we see that a long length of magnet, a strong gradient, and a particle of small momentum all act to shorten the . that value, particles focal length, which becomes zero when 8 rises to ~ / 2 At entering parallel to the axis but displaced from it are deflected so strongly as to converge on the axis just as they leave the magnet. If 0 is made to exceed r/2, the focal length becomes negative, which means that the particles cross

tlic axis whon still inside the lens and :we divergent on emerging (see Fig. 15-4M. If 0 ristls further, to lie between T and 3 ~ 1 2 the , lens again bcconies focusing. 15-4. Focal Length of a Diverging Lens I n the vertiral plane, the quadrupole magnet is diverging, as nlay be scvn by reference t o Fig. 15-5; the force acts in the direction of increasing y , so we have

and by an argunient similar to that given earlicr we find

wlicre K 2 -- G,/I Br) as before. The rnotion is divergent and a solution is y = c cosh Kz

so y'

=

2

=

+ (1 sinh Kz,

(15-25)

+ dK cosh Kz.

ch. sinh Kz

(15-26)

With y = y, and y' = y,' a t z = 0, we find c -= y1 and d = yIf/K, so after the particle passcs through the niagnet of length L the new parainctcrs are given by

(

1 - t O s h8 -

sinh O)

K sinh 8 cosh

(;)

8

where, as befort., 8 -- K L . T o find the foc~illength, measured froin the downstream surfucc., :L beam of ions moving parallel to tlie axis and (lisplaced from it is sent first through the lens and then through a field-free region of length S ;

(i :)(cash 9 1

(i) =

.

slnh 8)

K sinh 8 cosh 8 o h0 =

+K

1 . s i n 8 - smh 0

(K sinh 0

Fig. 15-5. Coordinates and parameters for a diverging lens.

1c

t:)

+ S cosh 0)

((::)

115-29)

cosh 8

Fig. 15-6. Diverging lens.

QUADRUPOLE LENSES

306

The input beam is given by yl = yl and yl' = 0. At the focus the displacement is y = 0 and the slope is y', so the matrix equation may be replaced with

+ XK sinh 8)

0

=

yl(cosh 8

y'

=

ylK sinh 0.

Again, only the first of these if of value to us. The particular value X of interest is -8 = -fd, as seen in Fig. 15-6. Hence

If this expression is written out in full, we have f

a -

1 -~ 0 t 8h

- -(-)'

Br dBz/dy

[(Br)

~0th

dB,/dy

$6

L].

(15-33)

Considering the term 1/K, we see t h a t a small momentum and a large gradient make the lens strong, by decreasing its focal length. The value of coth 9 is infinite when 0 = 0 and falls asymptotically to unity as 8 rises. Consequently a diverging quadrupole can never become convergent. 15-5. Focal Length of a Converging-Diverging Pair

Here we are interested in the net focusing properties in the horizontal x direction of a pair of quadrupole lenses of which the second member is rotated 90" about the z axis with respect t o the first, as occurs in practice when two quadrupoles are used to form an altcrnating-gradient lens system. Assume that the first lens focuses in the x direction, so the second one defocuses, as indicated in Fig. 15-7.

-=BEE

Fig. 15-7. Converging-diverging pair of lenses.

The matrix describing traversal of a converging lens and a field-free space has been given in Eq. (15-18), while Eq. (15-29) describes the process of passing through a diverging lens and then through a neutral region. I n Eq. (15-18) we set S = D, the separation of the adjacent faces of the two magnets, while in Eq. (15-29) we change the coordinate from y to x to fit the present situation and let S = F , d be the focal length of the pair, measured from the last surface of the second lens. The subscript cd implies that the converging lens is followed by one that diverges. The indicated product is

FOCAL TJEXGTH OF A DIVERGING-CONVERGIKG PAIR cosh 6

+ F&

sinh 0

drlh 0

K sinh 0

+

F C , i cosh

307

6

cosh 6 1 cos 0 - DK sin 0 - sin 0 K -K sill 0 cos 0

+ 0 cos 0)

5-341

When this is n~ultipliedout, the result will represent, two equations of the forin J:

=

+

(15-33)

Mll~ Mi2xi1

+

( 1 5-36) x' = M21x1 M 2 2 ~ 1 ' . Tlic input parameters of the beanl arc z, = xI and zl'= 0, and a t the focal point we have :c = 0. Therefore Eq. (15-35) becomes

and since MI, depends on Frd, this one equation will evaluate the net focal length of the system; we do not need Eq. (15-36). The value of d l l l is obtaincvl from Eq. (15-34), and Eq. (15-37) becomes 0

=

[(cosh 0 + FCdKsinh 0) (cos 0 DK sin 0) + (&sinh 0 + F.s aosh 0 ( - K -

sin 0)

I

XI.

(1538)

Multiply out, divide numerator and denominator by sin 0 sinh 0, and solve for FCd to obtain -cot 0 coth 0 DK coth-.0 1 FCd= (1 5-39) K cot 6 - I)K2 - K coth 0

+

+

But cot 0 = f,K and coth 0 = -fdK, by Eqs. (15-21) and (15-32), and wlien we divide nunicrator and dcnominator by I<%we obta,in

This gives the net focal length, nicasurcd froni the downstream surface of tlie second magnet, of a pair of quadrupoles of which the first converges and the second diverges in the s direction. 15-6. Focal Length of a Diverging-Converging Pair

As f a r as the y (vertical) niotion is concerned in the same set of quadrupolcs, the particles now pass first through a diverging and then through a converging lens, again separated by the distance D. The matrix formulation may be obtained from Eq, (15-34)) which describes a converging-diverging pair, by changing the coordinates froni x to y, by interclianging the order of tlie matrices, and by substituting D for Fed, and Fac for D. We obtain

308

QUADRUPOLE LENSES cos 0 - Fd,K sin 0 -K sin 0

cosh 0

1

- sin 0

K cos 0

+ D K sinh 0

(K sinh 0

+ Fdccos 0 1

- sinh 0

K

+ D cash 0)

(;:)

(15-41)

cosh 0

On the input side yl = yl and yl' = 0, and a t the output y = 0. By an argument similar to that of the last section, the expression for y becomes

0 = Miiy~. When Mll is coniputed from Eq. (15-41) this becomes 0

=

(cos 0 - Fd,K sin 0) (cosh 0

+ DK sinh 0)

+ ($ sin 0 +

Fdc

> I

cos 0 K sinh 0 y,.

(15-43)

On multiplying out, dividing numerator and denominator by K2 sin 6 sinh 6, and again using Eqs. (15-21) and (15-32), we obtain

Here we have an expression for the net focal length of a pair of quadrupoles of which the first diverges along y and the second converges.

15-7. Image-Object Relations of Converging-Diverging Pair Let the object be a point on the axis a t a distance u from the nearest side of the first lens, assumed to l,e converging in the x direction, while the second lens diverges, as shown in Fig. 15-8.

- 3 K D

Fig. 15-8. Image and object distances.

The matrix representation of the trip through the neutral region of length u and then through the converging lens is analogous to that of Eq. (15-IT), but the order of the two matrices is now reversed: cos 0 -K sin 0

u cos 0 cos 0

-Ksin0

+ K-I sin 0

-uKsin0+cos0

= Mi.

IhIAGE-OBoJECT RELATIOKS OF FOCI'S-1)EFOCI-P PAIR

309

Travcrsnl of the interlens distance D and the t-livcrging magnet iq giwn by :in expression likt Eq. 115-28) , but again wit11 rcwrs:tl of t l ~ corder of the mttriceh: s i r 0) K sinh

e

( ):

cosh 8

0

-

e K sinh e

+ K 6rinh 0 DR sinh 0 + cosh 0

1) ws11 0

)

= M2.

Wc now multiply these togcther in propcr order:

where

M11

=

cosh e c o s e

+

Mzl = K sinh 0 cos 19

+ (DK sinh 0 + cosh 6)(-

K sin 8)

+ ( D R sinh 0 + cosh 8) (-uk'

sin 0

+ cos 0).

The source lies on the axis a t x, = 0 and thc beam leaves it with slope al'. The image is formed on the axis a t x = 0 and the ions arrive with slope x' after traveling a distance v past the second lens. The completc description is MI2

+

uM22)

Msz

(it).

(1 5-52)

This yields the equations

I n the first of these we substitute for M l z and M2:! from Eqs. (15-49) and (15-51), then divide by K%in 8 sinh 6, use Eqs. (15-21) and (15-32) to eliminate cot 8 and coth 0, and ernploy Eqs. (15-40) and (15-44) to introduce Fed and Fdc.Wc define

After all this, Eq. (15-53) becomes

310

QUADRUPOLE LENSES

The subscripts cd have been added to the image distance v to indicate that the converging lens preccdcs the diverging one. T o obtain an alternative expression, we add and subtract Fcdon the right of Eq. (15-56) and obtain

An equivalent to the Newtonian form of the lens equation can be formed from this by multiplying both sides by u - F d c and then rearranging:

+

(15-58) (u - Fde) (ucd - Fcd) = F e d F d c RHere the quantities in parentheses are the image and object distances measured from the focal planes of the lens combination, as may be seen from Fig. 15-8. 15-8. Image-Object Relations of a Diverging-Converging Pair

I n the vertical plane (perpendicular to that just considered), the beam passes first through the diverging lens. An entirely analogous set of steps leads to the following expressions.

(U

- Fcd)(vdc- Fdr)

=

FedFdc

+ R.

(15-61)

15-9. Stigmatic and Astigmatic Images

Since Fcdis not equal to Fdc,as shown by Eqs. (15-40) and (15-44), it is evident that the image distances vcd and vdCare unequal for any arbitrary object distance u. Consequently the system is astigmatic, and a point source forms horizontal and vertical line images a t different distances from the second lcns. In between, the images become ellipses which reduce to a circle a t the mid-point. Sometimes this property is useful. For example, in the extracted beam from a cyclotron, the vertical divergence is from a horizontal line "object" which is separated by some distance from the vertical line "object" from which the horiaontal divergence takes place. A properly positioned pair of quadrupoles outside tne cyclotron can bring the particles to a common focus, for we can imagine the ions to travel backwards from a single object (the actual image) and to form images (the true objects) a t different distances inside the machine. I n other circumstances a stigmatic system is desirable, as when a small beam spot is to be refocused a t some distant region. There is one object distance which yields a stigmatic image; this may be found by equating vPd to vdc, using Eqs. (15-56) and (15-59), for example, and solving for the stigmatic object distance usti,. This gives

ELECTROSTATIC QUADRUPOLES

311

where only the positive sign of tlie squ:ire root is taken sincc the object distance must he positive. This expression can be simplified if we multiply R by 4 (Fbd Fd<)'/4 (Fed FdL)'. It then ~ O ~ ~ Othat W S

+

+

15-10. Electrostatic Quadrupoles

Electrostatic fields also may bc used to form alternating-gradient quadrupole lenses if tlle magnetic poles are replaced by hyperbolic cylindrical elcctrodes maintained a t DC potentials. The forces arc now parallel to the components of the fields, rather than normal to them, and the median planes toward or away from which thc particles are urged are rotated 45" with respect to these planes in the magnetic lens. As before, clioose tlie x and y axes so that the lens converges in tlle za plane and diverges in tllc U i pl:mc, as shown in Fig. 15-9. Let

Fig. 15-9. Components of force in an electlostatic quadrupole.

and 7 replace the previous x and y axes as asyni])totes of the 1iypert)olic surfaces. The electric potential at, :my point (;',17,z) between the electrodes is tlicn

where 0 is tlit. s:xnc for all the equipotrntial surf:ms. The equations of trwnsformation hctn-ccn the two sets of c-oorclinates, Fig. 15-10, are [ = x cos q =

+ $ y sir1 +

-x sin + $ y eos +.

(1 5-65)

(15-66)

312

QUADRUPOLE LENSES

Hence Eq. (15-64) becomes

V

=

+ y sin +) (-x sin + + y cos +) sin + cos + - x y sin2 + + x y cos2 + + y2 sin + cos +).

C(x cos +

I/ = C ( - x 2

Therefore the x component of the electric field is

dV

8, = - = C( - 2x sin

dx

+ cos + - y sin2 + + y cos"),

(15-67)

SO

dE, = -2C sin + cos + dx

=

constant.

Solve this for C and use it in Eq. (15-67) to obtain

But for 4 = 45", tan + = cot 4: so we have

The force on a charged particle is

But the gradient is negative, as shown by Eq. (15-68), so the force has the opposite sign to x and therefore focuses.

Fig. 15-10. Relations between two systems of coordinates.

It is not necessary to derive the equation of motion and to repeat all the arguments on different combinations of focusing and defocusing lenses, provided we can find the conditions under which the electric and magnetic forces are the same. We therefore equate Eq. (15-71) to Eq. (15-6) :

whence

T H E E F F E C T S O F F R I S G I S G FIE1,DS

313

By Eq. (3.5-lo), the restoring force per unit displacement in the a direction is li' = G/ ( B r ) , so we can exprws K 2 in tcrrri$ of t l ~ celectric gradient as

and P i s positive, since d&,/dx is ncg:itive. Thus although the electric force exist:: whetlwr the particle is moving or not, the T,elocity appears in the equation of nlotion hecause of its inclusion in the expression for tlie restoring force. An entirely simi1:tr analysis of the elrctric forces in the y direction gives the expression sn:tlogous to Eq. 115-68) :

dr,

- = 2C sin

dy

4 cos C$

=

con~t,ant,,

( 1 5-74)

ho that this gradient is positive. The electric force P', = qyd&,,ldy thcrcforo has the snmc sign as y and hence tlefocuscs. V'lien it is cquntcti to the, corresponding magnetic force of Eq. (15-7) we find 1hat

and Ii2is again positive. Comparison of Eqs. (15-138) and (15-74) shows that d?,,ldz = - - d ~ , / d ~so, t h a t the K' appwring in Eq. (15-73) is identical w ~ t l i that in Eq. ( 1 5 - 7 5 ) .All the earlier results on focal lengths and lens comhin:~tioris then follow :is hefore, provided that with the electric quadrupole, K iinterpreted as having tlie new meaning As is the case with magnetic quadrul)oles, the hyperbolic surf:ices in electric 1cnst.s are often replaced, in practice, 1 9 7 cylindrical surfaces. 15-11. The Effects of Fringing Fields

It must k)tl pointed out that tht. forcyying discussion has tacitly implied that both the lnagnctic and electric fields terminate in the z direction abruptly at the ends of tho poles. This, of course, is not true; there are fringing fields which extend beyond the quadrupoles. I t is customary to assume that the field remains uniform out to the points where it actually falls to half value. The expressions given earlier then apply if the length of the lens is taken to be this "effective" value. The "effective" soparation of the lenses is then less thiln the physical T-alue,and the calculated focal lengths must be measured froril the effective boundaries. If the focal lengths and the separation are large compared with the apertures, these corrections are negligible, but otherwise they can significantly alter the accuracy of t l ~ ecomputations. Either an experimental measurement of the effective length is required, or adjustments must be ~natlc empirically.

314

QUADRUPOIJ? LENSES

15-12. The Use of Lenses in Matching Two Accelerators

Consider particles traveling in the z direction in a region which focuses both horizontally and vertically. To a first approximation the motions are independent and sinusoidal, and either one is given by an expression of the form y Then the transverse momentum is p,

=

=

y, sin wut

y = Mw,y, cos w,t. M ddt

I n the most general case, a t any particular point along the path there will be particles a t all possible phases of this transverse motion, with amplitudes of oscillation ranging from zero up to a maximum set by the transverse dimension of the evacuated enclosure. A convenient way to vizualize this jumble of orbits is by the use of a phasc diagram, wherein the momentum p, is plottcd against the displacement y. Thus, considcr first a single particle. Its displacement is a sinusoidal function of time, indicated by representative points A, B, C, D in Fig. 15-lla. Its path in the p, vs. y space traces out an ellipse, Fig.

(0)

Fig. 15-11. Transverse motion and corresponding phase plot. 15-llb, with major and minor axes p,, = Mo,y, and y,, since a t A the displacement is zero and the transverse momentum is a maximum, while a t B the displacement reaches its peak and the momentum vanishes, and so on. A particle executing oscillations of the same amplitude but of different phase takes successive positions E, F , G, H a t the same instants a t which the first ion had the positions A, B, C, D. Thus a large number of particles distributed in any arbitrary manner in phase but with the same amplitude are represented by a corresponding number of points in the p, y plane, each point independently describing the same elliptical path. Particles with smaller amplitudes are represented by points which lie within the limiting ellipse; each travels on its own smaller ellipse, as long as its path in real space is sinusoidal. This is indicated in Fig. 15-12. There is a remarkable property about this which is expressed by what is known as Liouville's theorem; the density of points in the phase space does not

USE O F LENSES I N h1ATCIiIKC; TJVO ACCE:LERATORS

315

change. Each point represents a particle, so if none are lost, the region which encompasses them may change in shape. but its area remains fixed. For example, consider what happens if the focusing forces suddenly ceasc, as will occur if the paths of the ions l e d them into a drift space where there are no focusing fields. When this region is entered, the orbits beconlc straiglit

Fig. 15-12. Phase plot of ions with a variety of amplitudes and phases.

lines and each particle continues on wit11 t l ~ esame transverse llioinentuin that i t had a t cntrancc, while its Iatcral displaceincnt increases. This means that the points on the phaw plot craw describing ellipses; each point niaintains a constant value of p, and moves parallcl to the y axis. As a result, the ellipsc bccon~essheared, the more so the longer tlic field-free region is, but its area docs not change. This behavior is suggested in Fig 15-13.

-focusing ----------+

-c--field

free

-/

Fig. 15-13. Orbits and phase plots when particles leave a focusing region and enter a field-free space.

If now the field-free region terminates and the ions enter a second focusing region with ~rropertiesidentical with those of the first (for example), thcn each point in the phasc plane again takes up its independent motion in an ellipse, since the particles in real space recommence their sinusoidal motion. Thc limiting sheared ellipse (and all the ellipses within it) now appears to rotate, as time advances, like a propeller of variable length and width, but keeping its area constant. This is shown in Fig. 15-14. Note in Fig. 15-14b, t h a t tlic inaxill~unlw l u e of p, is nlucli snlaller than it

QUADRUPOLE LENSES

316

was before the drift space was entered, although the extreme value of y is considerably greater. If we wish to send the beam across a room, the configuration of (b) would be a good one a t which to start a second field-free region, for the particles have a minimum of divergence and therefore the beam will not expand laterally very much, although it already has considerable

(a)

(b)

(dl

(c

Fig. 15-14. If the particles re-enter a focusing region when their phase-space pattern is as in ( a ) , the pattern subsequently rotates, as in ( b ) , ( c ) , and ( d ) . breadth. On the other hand, a t the configuration of ( d ) , the beam is narrow, although the particles have considerable divergence in direction. This condition corresponds to a focus and is a suitable one a t which a target might be mounted, if a small bombarded area is desired and if the angle of approach of the projectiles is of no consequence. It is sometimes convenient to consider the phase diagram as the plot of the angle of divergence 0, versus the displacement y. Since w,t = 27rvtlX = 2m/h, where X is the wavelength, we have 27rz y = y m sin w,t = y, sin F(15-78) X

But tan 6, =dy/dz and since 8, is small, it follows that

Or comparing this with the expression for p,, Eq. (15-77), i t is evident that 8, may be substituted for p, by a suitable change of scale in the phase diagram. The fixed limiting area on the 8y, plot is most easily calculated when the major and minor axes of the ellipse lie along the 6, and y axes. The area is then A = u0,,y, and so it has the dimensions of radian X length. This invariant quantity is called the quality of the beam and must be specified in both

USE O F LENSES I N MATCXINC;

rITo ACCE1,ERATORS

317

transverse directions, since i t may not, have the same value in each. (Some writers, in specifying the quality, refer to the radius of the beam and lth tlivcrgmcc from t h axis; others use t l ~ ctlianicter and thc total divcrgence, f r o ~ none sidc to the othcr.) Not spcc~ific.:rllyexpressed but nevertlicless iinplicd is the notion that the quality refers to t: definite number of particles-that is, to a definite number of points in thc phase plane. If the wide hut not very divergent beam indicated in Fig. 15-14b were sent through a narrow slit which reduced the width in the y direction, the qu:tlity would appear to become improved. since the area p,y is reduced; but this is accomplished only a t the expense of a loss of particles, as indicated by the shaded regions in Fig. 15-15.

Fig. 15-15. Shaded areas represents loss of ions in traversing a slit. *

y

Very often i t is necessary to transfer particles from one accclcrator into another, say from a linac to a synchrotron. For si~nplicityof illustration, let us assume t h a t the focusing propcrties and internal clearances are the same in both nlachines. Then if the orbits or~ginatetlin each accelerator, the ellipses in pliase space would be identical for hot11 devices; and consequently if the two ~nacliinescould be intimately joined, particles would be transferred from the linac to the synchrotron without loss. It, would then be said that the emittance of the first michine just matched the acrepfnnre of thc second. B u t in practice t l ~ c r cis always a field-free drift space between the two accrlerators, and this introduces :z mismatch; for, as seen carher, the cllipsc which represents the ions in the linac will become sheared in the drift space and will therefore not coincide with the ellipse appropriate to the synchrotron; and when the ions enter that device, their phase plot will rotatc prc~pcllcrwisc.Since only thosc ions will pass through the synchrotron which lie within that accelerator's el-

f

- No. I

-

t drift

space

-,

a c c e p t a n c e of No. 2,\ ,'\

\

- --

No 2 " - ~ '

Fig. 15-16. Phase plots representing transfer from one accelerator to another, via drift space, with loss of ions.

QUADRUPOLE LENSES

318

lipse, many particles will be lost to the walls, just as the diverging stream of water from the open end of a pipe will not all enter another pipe of the same diameter. The situation is indicated in Fig. 15-16. All could be saved by making the second pipe bigger; that is, by making the aperture of the synchrotron so large that its ellipse can contain the rotating pattern of the particles that enter it. But such a method of avoiding loss is prohibitively expensive, and a much cheaper solution can be found by the introduction of lenses. These act to alter the angle of divergence 6, by an amount proportional to the transverse displacement without changing the displacement itself, so the lens shears the phase plot in a direction parallel to the 8, (or p,) axis, and when this is compounded with the shearing parallel to the y axis which is introduced by the drift space, the result is to reform the ellipse into a shape and orientation identical with what existed initially-i.e., in the present instance, the acceptance shape and orientation of the second accelerator. How this is accomplished is best understood by study of Fig. 15-17. If the identity in fo-

acceleaor

N

drift

+

space

I.

accelerator No. 2

Fig. 15-17. Matching two accelerators by means of one drift space and two lenses. cusing properties and in lateral dimensions holds not only for the y but also for the x dimension, then each lens indicated in the figure must be considered as a pair of appropriately spaced quadrupoles. I n the more realistic case where the cylindrically symmetrical beam from the linac must be matched to the synchrotron in which the vertical and horizontal apertures and restoring forces are different from each other and from those of the linac, the problem is much more complicated. A list of papers on quadrupole lenses will be found on pp. 371-372.

Chapter 16 STOCHASTIC ACCELERATORS -- -

--

--

16-1. Introduction

After reading of ail the effort t h a t has been expended in designing synchrocyclotrons in which the oscillator's frequency is made to follow approximately the frequency of revolution of the projectiles, i t niay come as somewhat of a shock t o learn t h a t successful acceleration ('an he obtained by deliberately making the oscillator frequency jump about in a stochastic manner, t h a t is, by guesswork or a t random. The idea was first suggested by Burshtcin, Veksler, and Kolomenskij in 1948, though not published until 1955, and was put to a successful test in 1959 by Keller, Dick, and Fidecaro. This group obtained a microampere of 4.4-Mev protons in a weak-focusing cyclotron of 5 0 - c n ~diameter by driving the single dee with 5 kilowatts of RF "noise" showing a continuous spectrum of frequencies between 21 and 23 Mc/sec, the root mean square dee potential being 2000 volts. How the scheme works may be introduced by referring to the acceleration process in two conventional m:tchines When a fixed-frequency cyclotron is adjusted to be in resonance with ions leaving the source, they rotate for n ~ o s t of the time on the "wrong1' frequency, the final energy being equivalent to the energy excursion of a "nonsynchronous" particle, in the terminology used when discu~slngphase stability. If the target (and ion source) were removed, ions would s ~ h out l and in, out and in, alternately gaining and losing energy by an amount equal t o the full output value. Lack of synchronization also occurs for nmny of the particles in a syncl~rocyclotron. In 5 6-2 it has bren shown that most of' the successfully accelerated ions undergo synchrotron oscillations of pliase, energy, radius, and revolution frequency. Only twice during each synchrotron oscillation is the revolution frequency the same as that of the oscillator and therefore 'Lcorrect"; most of the time i t is "wrong." But we can equally well say that it is the oscillator frequency which is "wrong," although it always lies within the range of frequencies appropriate for particles of all the energies t h a t the machine can contain. l\'ith these ideas a s background, it brco1nt.s easier to understand how accel319

320

STOCHASTIC ACCELERATORS

eration to full energy of some ions can finally occur if the oscillator runs a t a constant frequency for a short interval, then jumps to another value for a brief time, and then jumps again and again, though with no specified order in the direction or magnitude of the change or in the length of time it operates on any single frequency. It is only necessary that the frequencies used should a t least cover the range that would be required of a conventional synchrocyclotron of the same final energy. Imagine the machine to be full of ions a t different radii and therefore a t different energies. No matter what the oscillator's frequency and phase, some ions will find these parameters to be of just the correct values for synchronous acceleration, others will find them not so far off but that the particles can be trapped in a synchrotron oscillation, while still other ions will be unfavorably disposed and will lose energy. This situation will repeat, but on different groups of particles, whenever the oscillator shifts to a new frequency. To be sure, many ions have bad luck and are driven back to the source and lost, but new ones are always starting out to replace the casualties, so it is not surprising that some survive long enough to reach the target. As a crude analogy, imagine a group of people who start a t a flagpole and walk on paths which radiate away from it across the field; let them reverse the direction of their motion a t random moments, say every time a butterfly or a mosquito comes within three feet. Many people will sooner or later come back to the pole and drop out of the game; but if fresh contestants start every minute, a t the end of the day people will be dotted all over the countryside. A somewhat more precise picture of stochastic acceleration can be obtained by the following argument. Let us examine the position of a particular ion a t moments separated by one period of the oscillator, which runs a t constant frequency for a few cycles. If the energy of the particle is such that its frequency of rotation initially equals that of the oscillator, its positions a t the moments of inspection will lie on a radial line (almost), as is indicated by the dots in

Fig. 16-1. Positions of an accelerated ion at intervals of one oscillator period which is ( a ) synchronous and then becomes ( b ) too long, ( c ) too short.

Fig. 16-la. (We disregard synchrotron and betatron oscillations.) If the period of the oscillator now jumps to a slightly greater value, the pattern of dots will continue as in Fig. 16-lb, and if the period then changes to a value less than the original, the pattern will be that of Fig. 16-lc. An unfavorable phase of

thc vo1t:tgc can hring :ihout deccler:ttion, the loci of the ion's position curving onc way or the other depending on whether the IIF period is too high or too

low. On :dding a siln~pleof ear11 of t l i ~ wsitu:~tionsto what already exists, wc obtain thc 1)attcrn of Fig. 16-2a. Then if straight lines :ire drawn bctwcen the

Fig. 16-2.

( a ) The ion now undergoes two intervals of deceleration. (b) The analogous pattern of "Brownian motion."

end points where the oscillator frequcncy changes, the pattern of Fig. 16-2b is sbtained, wllcre now the arrow points, not in thc direction of motion of tlic particlc, hut in the direction of the pattern since the last change in frequency. I t is clear that the situation bears a striking resemblance to Brownian motion, and in fact the analysis of stochastic acceleration borrows heavily from the theory of diffusion of gases and liquids. An approximate treatment by this nlctllod will 1)e given later on. There are alternative ways of producing the required dee voltage, other than the succet.sion of random frequencies discussed abo\-e. I t is equally cffcctil-c to use "noise"-that is, a "white" spwtrum which simultaneously covers the entire needed 1)andwidth. This condition can be attained by a randoin variation in the amplitude of a single frequency, for :I Fourier analysis shows the existence of :Lband of frequencies. Still another way of obtaining the stochastic condition is to run through the frequency modulation cycle of a convention:il synchrocyclotron oscillator but a t so high a rcpet~tionrate that a synchronous particle cannot keep u p with it. Thus, if the rate of turning of the tuning caparitor of a synchrocyclotron is gradually increased from zero, the output current rises from zero to a m a x i n ~ u nand ~ then falls off ( b u t not quite to nothing) ; thus far, the machine is acting as an ordinary syr~chrocyclotron.If the turning rate is increased further, the current riws gradually; this is the stochastic regime. Since the band widtli of noise is mo,it c.ffcrtlve when it covers the range appropriate for a conl-entional FN cyclotron of the same energy, the generation of frequencies above and below this range is wasteful of power, although some overlap rnay be desirable for practical reasons, such as nonabrupt cutoff of band-pass filters. It is exciting to contemplate the possibility of producing particles a t energies above those attainable in a fixed-frequency cyclotron but without the complication of azimuthally varying fields, for if synchronization with the rotation frequency is not required, thrn a simple weak-focusing magnet is nde-

322

STOCHASTIC ACCELERATORS

quate. But we seldom gain an advantage without cost, and in this case it is the necessity of generating an R F voltage over the frequency band width customary in a synchrocyclotron but a t a considerably higher potential. Since the construction of a large stochastic cyclotron has not yet been attempted (1960), it is too early to assess the practicality of the scheme or to compare its cost with other devices of the same energy. But it would seem that for equal expenditures in R F equipment, the output current of a stochatron probably will be substantially less than that of a fixed-frequency cyclotron, for a machine built within the energy range of the latter. Above this value, it appears that the almost continuous current from a stochatron can exceed the time-averaged output from a synchrocyclotron (because of the latter's low duty cycle), but the generation of the required R F voltage presents formidable problems, whether it is produced in the form of noise or as sequential frequency modulation carried out a t a very high repetition rate. Practical attention is now being given in several laboratories to methods of combining stochastic and synchronous acceleration by modifications of existing machines, as is discussed in the next two Sections. 16-2. Hybrid Accelerators

The low output of particles from a synchrocyclotron is in large part due to the fact that ions can be accepted in phase-stable orbits only during a small fraction of the modulation cycle, since the frequency of the oscillator soon becomes inappropriate for particles starting from rest, as has been described in 5 6-8. A promising type of hybrid accelerator appears possible if a stochastic cyclotron is used as an injector for a synchrocyclotron. Thus, a small auxiliary dee, driven by R F noise over an appropriate band width, may be mounted opposite the single dee of a synchrocyclotron, as indicated in Fig. 16-3.

noise generator

Fig. 16-3. A hybrid accelerator, where a stochastic cyclotron acts as an injector for a synchrocyclotron.

The small dee draws ions from the source continually and accelerates them in a stochastic manner, with the result that particles of all energies from zero u p to the value corresponding to its maximum radius are present a t all times. When the frequency-modulated cycle of the large dee comnlences (its voltage being turned on a t this instant), particles with this spread in energy are available for trapping in phase-stable orbits. The acceptance time of the FM regime is thereby increased, leading to a substantial rise in the output current of each modulation cycle. This scheme is being tried on the 600-Mev synchrocyclotron

THE EQUATIONS O F STOCHASTIC: M O T I O S

323

a t CERN. I t is c q w t e d that the current will rise from 0.1 microampere by a factor of 8 to 10. 16-3. Stochastic Ejection

Bccnuse of the pulsed nature of the current from a synchrocyclotron, the dctccting counters of the users of the machine are sometimes choked by too h r g c a number of events during the output pulse and are starved during the rest. of the FM cycIe. It has been suggested that a more continuous output can olkained hy mounting an additional hollow clcctrode shapcd like a C just inside the final radius of the particllcs and opposite the frequency modulated tlcr (Fig. 16-41. When this (lcee" is driven by noise, it will accelerate and de-

Fig. 16-4. Stochastic ejection from a spnchrocyclotron.

celerate the ions entering it a t the end of each main acceleration interval, with the result that the output from the cec will be sprcad out in time and the current will t)c nlinost rontinuous. 16-4. T h e Equations of Stochastic Motion

For a brief interval after leaving the ion sourw in a cyclotron-like accelerator, thc particles rotate without penetrating into the field-free regions inside the dees, so they are continually acted on both by the magnetic and electric fields, as was discussed in S 7-9. I n actuality, thcre is no abrupt boundary where the electric field terminates, for it gradually drops off until i t is practically zero a t a distance inside a dee equal to the distance D between the "floor" and "roof" of the (lee. It is convenient to idealize the situation by imagining that the electric field is uniform, not only across the dee-to-dee gap but also to a depth of penetr:ition D within each electrode, and that the field then suddenly drops to zero, much as though the dees wercl separated by a distance 2 0 and h:ld dense grids across their open faces. This is indicated in Fig. 16-5. For the initial part of the following discussion, we will be concerned with particle motion for whicli the radial coordinate r is small, to the extent that r < D. The applied electric field Fa across the idealized accelerating gap has, during any hricf interval, a radian frcqucmcy w 511~11that E,(t) = F cos w f

(16-1)

324

STOCHASTIC ACCELERATORS

where the amplitude is written simply as E. This expression can also be written RS

E,(t)

=

+

i~(ejwt e-jot).

(16-2)

I n this form, the field is represented by two vectors, each length %E, which rotate in opposite directions a t w radians per second. If the first vector turns in the same direction as the particle (say counterclockwise) and a t approximately

Fig. 16-5. (a) Electric field distribution between real dees. ( b ) Simplified and idealized field distribution.

Fig. 16-6. A positively charged particle P and the electric vector && rotate counterclockwise. The magnetic field is directed into the paper.

the same rate, it may cause acceleration. A component is tangent to the orbit and a component &, is radial, for the azimuth of the particle P may differ from that of the vector %&bythe phase angle +, a s shown in Fig. 16-6. Therefore =

$5 cos 4

(16-3)

Ee =

3 E sin 4.

(16-4)

E,

The second electric vector, which rotates against the particle's motion, has a tangential component which accelerates and decelerates the particle (twice each revolution if the angular velocities are equal, for example) and hence produces only a small perturbation, which will be neglected. The radial equation of motion is as given in Eq. (2-16) but with the addition of the electric force:

Here vs is the azimuthal velocity. Since d r / d t = v,, the radial velocity, this can also be written as

where the dot indicates the time derivative.

T H E EQUATIONS OF STOCHASTI(> MOTIOX

325

Tlic azimuthal motion is given I)y

where 8 is the azimuthal angle from sonle arbitrary point. B u t we .. .measured . .. h a w V H = rk so t1i:tt 7 i f l = r 8 r 8. Solve this for 6 t ~ n duse = v s / r and = v,, so t h a t Eq. (16-7) becomes

+

It will prove caonvenicnt to introduce a fictitious particle which is imagined to rotate a t radius r under the influence of the magnetic field only, just as though thc dee voltage were turned off. For this particle the basic relation Eq. (2-18) is applicable:

The subscript s is added to indicate that, this would be the synchronous frequcncy to which the oscillator would have to be tuned if the machine were an ordinary syncl~rocyclotron.Conscqucntly we denote this reference particle a s the synchronous ion S. I t s azimuthal veloclty is

by Eq. (2-17), and hence i t depends on the mass-i.e , on the energy. t both the electric and magnetic iSow the actual particle, which is s u b j t ~ to fields, has an azin~uthalvelocity 21e which differs froin the synchronous value uflsby some as yet undetermined amount wl~ic~li we will call u l s . Thus, by Eq. (16-10)

Tlicn for symmetry in notation, we may write tlic radial velocity of the actual particlc as tht: synchronous value (zero) plus an undetermined contribution zc, arising from the elcctrie field:

T o find values for these velocity coinpolx-nts we employ the equations of motion. Thus when we substitute Eqs. (16-11) and (16-12) in Eq. (16-6) and drop the relativc.ly small terms uy, and uyci"r, the result is

(16-8) to find the value of ul,, Before Fk~s.(16-11) and (16-12) arc used in i t is necessary to differentiate 131. (16-1 1 ) in older to determine us. Note that

326

STOCHASTIC ACCELERATORS

even though the magnetic field is static, there is an implicit dependence on the time, since the particles move into a weaker field as they gain energy. (We assume a weak-focusing magnet.) Thus from Eq. (16-11) we find

The second term of this may be written as

where n is the field index defined in Eq. (2-1). Therefore Eq. (16-14) becomes

where, as before, we have written 1: = v,. Now use the values of vg, v,, and ~g given by Eqs. (16-ll) , (16-12), and (16-15) in the azimuthal Eq. (16-81, multiply out, and drop the small terms wo and w,wg/r, to find that

so since n

< 1, we have

Substitue the values of 8, and Ee given by Eqs. (16-3) and (16-4) into Eqs. (16-13) and (16-17) to obtain

W,

t

- sin 4. 2B

(16-19)

The vector sum of ws and w, is the vector w:

(As an alternative expression, we may use the root mean square electric field = & / d 2 , from which it follows that

E,,

w

=

-.&rms (r < D). d 2 ~

This form will be of use later.) By employing Eq. (16-20) in Eqs. (16-18) and (16-19), we find we = -W cos 4 (16-22) w,= w sin 4. (16-23)

THE RASn WTDTH OF HF NOISE

327

We now recall that v f land I P ,are the ax11nut11:d : m i radial velocity con1l)oncnts of the real particle P , ovcr and above thc cwrrcspondlng conlponcnts of the reference particle S which rotates in the magnetic: field only. Then if $ is

Fig. 16-7. Azimuthal relations between the rotating electric field, the real particle P and the fictional particle S.

Fig. 16-8. Further angular relations.

the angle between these particles, a s shown in Fig. 18-7, the equations of motion of the P particle relative t o the S particle are dr d t - wr =

zu sin

(16-24)

C$

rd+

- = ws = - w cos, dt

16-5. T h e Band Width of R F Noise

The foregoing expressions show that if P is to move in a random manner with respect to S, the angle 4 must randomly assume all values between 0 and and the radius vector of 2a. Kow 4 is the angle between the electric vector particle P; let 9, be the angle between ><& and the radius vector of particle S. Then we have (16-26) 4 = - 48.

--

xi;

+

I n order that, 4 should range between 0 and 2a, it is necessary that the phase of the electric vector (from which 4, is irieasurrd) should vary stochastically over the same range. Thus, exprcsscd in terms of the radian frequencies, wc have

4.

=

/w.dt

-- lwdt,

(16-27)

where w is the frequency of the oscillator, as indicated in Fig. 16-8. Although it is adequate to have a random shift covering 2a radians of the phase of the oscillator with respect to the phase of the fictitious synchronous particle rotating a t a particular frequency w,, this must occur for each and every different value of w,. Now w, = q B / M = q B c 2 / ( E o y ) , where y is the total energy in rest-energy units, so that w, varies from the value appropriate for ions just

328

STOCHASTIC ACCELERATORS

leaving the source up to the value appropriate for the output energy. It is therefore apparent that the random noise which drives the dees in a stochastic cyclotron must cover a band of frequencies AW of the same range as the band covered sequentially in an ordinary synchrocyclotron of the same energy. A narrower band of noise is not adequate; a wider band wastes power. 16-6. The Application of Diffusion Theory

If the particle moves in a random manner with respect to the synchronous ion, some quantity equivalent to the mean free path x of Brownian motion should be computable. When a gas atom moves with an average velocity w between collisions, h and the mean free time f a r e related by

and the collisions result in a large change in the direction of motion. Now it is far from clear what changes in direction may be expected for the corresponding velocity u l in the case of a stochastic accelerator when the oscillator frequency alters suddenly. But we should be correct, a t least in order of magnitude, if we suppose that the direction changes by .rr radians when the oscillator frequency changes from its mid-value w to either extreme w f Aw/2. Suppose that a t some instant the vector w is pointing north. The next "collision" may make it turn eastward to an azimuth angle 0 lying anywhere between 0 and 7, or i t may swing westward to some angle between 0 and -T. If t is the time between lLcollisions,"the angular rate of turning of w lies between dO/dt = -T/? and do/& = r / E Borrowing from the analogy of Brownian motion in a gas, where t = h/w, as quoted earlier, these limits become dO/dt = *7w/A. We now assume that these extreme rates of turning of w correspond to the rates a t which the oscillator frequency changes by one half its total band width. That is, given do/dt = f A o / 2 and dO/dt = f nw/X,we set dO/dt = dw/dt and obtain

I n textbooks on kinetic theory it is shown that the diffusion of a gas is described by the expression

4 = div[8 grad PI, dt

(16-30)

where p is the density of atoms and 6 is the coefficient of diffusion. Atoms diffuse along a density gradient from regions where the density is high to where it is low. The coefficient 6 is given by 6 = hw/3, to a first approximation for three-dimensional motion, h being the mean free path and w the average velocity.

THIC APPLICh'I'ION OF DIFFUSION THEORY

329

I n the case of stochastic acceleration, we are concerned with two-tiimcnsion:rl tliffusion, so the coefficient hecomes

Further, our int,erest is not in the gradient of p:trticle density but rather in the gradient of "velocity density," so that in thc steady state the diffusion equation bccomes div

[$grad ((up)1

=

0.

Here p is the number of particlcs per ulrit volun~eon the mid-plane and the quantity in x p t r c brackets is thc flus: the number of particles passing through ' in Eq. (16-20) t h a t A = 2nwlAw; and, unit area per wcond. Xow wc ~ : L v (seen for r < D , Eq. (16-21 I shorn that u1 = g,,,,,/(\/ZB). IIcncc if r < D, neither h nor zc is a furictlon of the radius. They rimy therefore he canceled from Eq. (16-32) which becomes (16-33) div grad p = 0. In cylindrical coordinates this is

Wc arc concc~rncdonly with radial diffusion, so the last two terms drop out. On integrating the first term we obtain

(16-33)

p=alnr+h,

where n and b are const:tnts. I t will prove convenient to define C', and C2 such that n = -C1, and b = Cl In D P2 where D is again the height of a dee, so as to obtain

+

The dctcrmination of C1 and C 2 must be postponed until the diffusion equation 1s applied to thc region where r > D. For the greater part of the :tccelerative process, the ions penetrate into t l ~ c field-free regions within the dees. They arc thcrcforc no longer subject to a continually acting electric field. but receive only two impulses per turn when crossing the dee-to-dce gap. We may neocrthelcss approximate this situation l y imagining that a tangential field I,,, w t s continually, and we may evaluate it hy equating the energy gained per turn under thc. fictitious and real conditions. Let V,,, be the root mean s q w m potenti:~lof cnch dee to ground. Then 4(1Vnnr = 2 ~ r & n . l ,

.o t h a t Em,,

=

2Vm

-----. Ti"

(r

> D)

(16-37)

330

STOCHASTIC ACCELERATORS

Substitute this fictitious field for the continually acting field in Eq. (16-21) to find that

Thus for these large values of r, we have w cc l / r , and so by Eq. (16-29) we find that A a l / r also. Hence Eq. (16-32) becon~cs div

[k

grad

I):(

=

0.

(r

> D)

In cylindrical coordinates we have in general

so since only the radial component is of concern, Eq. (16-39) becomes succcssively div

2

(,)I

r dr r

=

0

and

After integration, this yields

We are now in a position to determine three of the constants in Eqs. (16-36) and (16-42), by employing continuity and boundary conditions. When r = D, both equations must give the same value of p. This results in

Cz = C3D2

+ CgD.

(16-43)

As a second condition, we set p = 0 when r = R, the final radius of the orbits. Equation (16-42) then gives

(This boundary condition still permits a flux a t R. I n general the flux is F = ',A grad (wp);so for r > D, when ho: l / r and wo: l l r , we find F cc ( I l r ) grad (p/r). Thus when Eq. (16-42) is substituted for p and we set r = R, it follows that F = C3/R, which is finite.) The final condition is to equate the fluxes a t r = D.When r < D ,we have seen that w is constant. Hence using Eq. (16-36) for p, the flux is given by

F

:=

3 X grad (wp) = $Xu) grad p

When r > L), by E:qs. ( I 6-38) i d ( 1 0-2!)) n e using Eq. (16-42) for p, the flus is found to he

fild

X = (2.\/21',,,,) 1 (Awbr) ; so,

n7Iicn Eqs. (16-451 and (16-46) are set cqual, with r = D, and we use Eqs. (16-291, (16-211, and (16-37) to eliininatc~A, u3,and I,.,,, the result is

Then Eqe. (16-44) and (16-47) show that,

and when Eqs. (16-48) and (16-47'1 are used in Eq. (16-43), we find t h a t

Consequently Eqs. (16-36) and (16-42) become

(It may be observed t h a t Eq. (16-50) g o ~ to s infinity when r = 0. But the diffusion equation is applicable only after thc particles lmve received some :lcceleration so that r is finite.) I n order to cvaluate the remaining constant Cl, we make use of the fact that the defocusing force causcd by space charge must not exceed the focusing forcc arising froin thc gradient of the magnetic field. T o do this, we must first find the value of I,, the electric field due to space charge, a s a function of the distance 2 above the m i t i - p h e . We have div

r,, = 9pk) -ko

where p(z) is the number of particles per unit volume a t z , and

h-0

is the per-

332

STOCHASTIC ACCELERATORS

mittivity of space. Since the ions are distributed fairly uniformly in the radial direction, but fall off in number quickly as a function of z, thc variation of &, is mostly axial; so to a fair approximation the last expression becomes

The way in which the ions are distributed axially is assumed to be of the form

where a s before, p is the number density of ions on the mid-plane, for this expression has a maximum a t z = 0 and vanishes a t the roof and floor of a dec where z = * D / 2 . When Eq. (16-54) is substituted in Eq. (16-53) and the latter is integrated, the result is

the constant of integration vanishing, since a t the mid-plane &,"; rection and hence is zero. Therefore, for z g D , it follows t h a t

changes di-

As a result, the axial defocusing force on an ion bearing charge q a t elevation x is

T o obtain the magnetic focusing force, we recall the equation of axial motion of a particle under the influence of the magnetic field only:

MZ

+ Mw2nz = 0,

(16-58)

where n is the field index. The magnetic force is F2

=

B wnz -Mw2nz = -M Y-

M

= qwB1rz,

(1 6-59)

where B'= dB/& is the almost constant gradient. The maximum density of space charge which will allow axial stability is found by equating FI to F S . This gives

We now use this expression in Eq. (16-Til), which is valid when r solve for C1, to find

> D,

and

Now Eq. (16-51) may be re-cqresscd a s

:I

function of r / R in the forin

\\'it11 i n c r c a h g v x l u c ~of C 1 ,this plots as tlie f:mily of curved lines shown in Fig. 16-9, assunling for cusc of illustration that Eq. (16-62) is valid all tlic way

Fig. 16-9. The behavior of

p and of

p,,,,, as

:I

function of r / R .

to r = 0. On t l ~ cother. liand, tlie v:tlues of pales, given by Eq. (16-60), plots a s a straight lint. To insure stability at all radii, p should always be less than p,,,,,, so t h a t must be given its ~riiniriluiuvalue. From Eq. (16-61), valid if r > D l i t is seen t h a t the minimuin is C -- k,,~13'IY and since D

< R , this is also

' - q(R - D)'

With this latter value, the curve of p always lies under the line representing p,,,,,,, tlie two hcing tangent at the origin, thus assuring stability against beaiii blow-up. This value of C1 will bc uscd in $4 16-7.

16-7. T h e Calculation of the Current Thc current of ions a t any radius is givcn by the charge per particlc niultiplied by the integral of the flux through the surface of a cylinder of radius r and height I ) , due corisitleration bcing glven to the assumed cosinoidal axial distribution of particle cicmity givcn by Eq. (16-551. \\'e obtain

I

= 2rrp

x- grad 2

(wp)

IDD (z) -

I)/Z

cos

a

d z = SprDh - ( w p ) . ar

(16-65)

ELq)ress A in tlrc for111 given by Eq. (16-291, u' accortling to Eq. (16-38), ant1 as given I)y Eq. (16-51). to find that

334

STOCHASTIC ACCELERATORS

and when C1is substituted from Eq. (16-64), we obtain finally

Here co may be taken as the average-ie., the center value-of the oscillator's radian frequency of noise. Aw is the width of the noise band, B' is the radial gradient of the magnetic field, V,,, is the root mean square potential of each of the two dees with respect to ground, B is the field in webers/m2, R the final radius, and ko is 1/(36r x lo9) farads/m. Note that B' is negative in a weakfocusing cyclotron. This expression is the end result, but care must be used in its interpretation. It has been seen that the band width of RF noise must be as great as the frequency that would have to be covered in a synchrocyclotron of the same energy. If the parameters of a given stochastic cyclotron are adjusted to produce a certain current a t the maximum value of the radius and are kept a t those values, then the above expression shows that if a probe target is pulled out from the center to the periphery of the magnet, the current that strikes i t will be inversely proportional to its radius. This is in agreement with experimental observation, as is also the dependence of the current on the square of the dee voltage. If one wishes to compare the expected current in two machines of different energies, obtained by an increase in the maximum radius, then another step must be made in the analysis, for a greater energy requires a larger value of Aw. Thus, the frequency of a synchronous ion is w, = q B / M = qBc2/E = qBc2/(Eor); so the difference between the initial and final frequencies in a synchrocyclotron is given by

But the ions start from rest, so yi = 1; and in the stochastic machine we must have Aw = w,i - w,, so that

Further,

w

appearing in Eq. (16-67) is given by

I t follox~-sthat

THE CAT,CUT,ATION (IF T H E CITRREXT

335

where we drop T in comparison with 2E0 in order to simplify the ultimate result. TJse of the non-relativistic exprcw-ion for t l ~ :kinetic energy given hy Eq. (1-37) then shows that

and when this is substituted in Eq. (16-67) we find

where the energy is in electron-volts, thc field is in webers per square meter and the lengths are in meters. It will be recallrd that two 180' dees are assumed, eacli a t i7,,,, volts to ground. I n the light of tile approximations involved, this expression is probably valid only to order of magnitude. Furtlicrmore, it has been assumed tacitly t h a t the particles move in a perfect vacuum. I n practice, of course, this is not the case and a substantial loss will occur if the time to reach full energy is comparable with the mean free time between collisions with gas molecules. References to stochastic acceleration appear on page 372-373.

With :I f'cx cxccption>, the niatcrid cluotcd has berri taken only from the rrgular journals, sincc most of the privately print.ed rncmorantla have had a limited distributio~i. BIBLIOGRAPHIES, LISTS OF INSTALLATIONS A N D COLLECTIONS O F DATA ON ACCELERATORS

(OTS = OAicc of Tecimical Srrvireb, U S. D r l ~ t of . Conimrrce, Wa-hmgton 1.5, D C )

E. Thomas, 1'. Mittelniaii, and H. H. Goldsmith. Particle Accclcrators: Bihliogmphg and List of High Encryly Installations. BKL-L-I01 and AECU-31 (July 1, 194S), 55 pages. Bonnir E. Cw1irn:rn. R i b l i o p q d ~ y of Partick Accclcrators. Jul!~1948 t o Decembc~r 1D:iO. UCRL-1238 (llarch, 1951), 54 ptrgrs. Avai1:hlc from OTS; photostat 69.30; microfilm $3.60. Srrgey Sliewchwk. Uiblioyraphg of fJarticle Acrdcrators. .Jnrrr~aryt o D ~ c e m b c rl.%il. 1TCIIL-1951 (Srptrmbrr, 1052), 45 Ibagrr. Avail:lble from OTS, 25 ccntr. .J. E. Thomas, W. L. Krau.di:iar, ant1 I. H:tlpern. "Synchrotrons." At~nrtul R w i e w o f Xuclear Scieucc 1: 175-198 (1952). Lists 23 wrnk-focusing rlectron ,synrhrotron~of 1 Hrv or less, with par:imrtc3rt of the 300-SIw machines at U. of Calif., Cornell, mtl R1.I.T. F. E. Frost and ,T. hl. l'l~tnnni. I'artirlc Accr~leratom:I. Biblio~graphy;TI. List of High Ihrerqy Installutions. 1JCIlL-2C,72 (Sovrrntjer 16, 1954). .4vailal)le from OTS; photostat $1 1.50; nlirrofihn $4.00. E. Amaltli. "CEItN, t h ( ~E t ~ r o l m nClouncil for Nwleur Research." ~ V ~ t o vCoi m m t o , Ser. 10, Vol. 2, Suppl. 339-854 (1955). History of CERN. Lists of Europcan :iccclerators of over 100 ;\lev: 5 h i l t , G planned. E. H. Krause. "I'article Accelerators--Locutions :and Characteristics of 0lwr:itional Rlachincs Tl~rollghout the World." A m r r . Inst. of Physics Handbook. 8.181-8.201. RIcGraw-Hill, 1957. Lists 110 VdG, 38 C-71';1 rvctifietl AC; 43 F F cycl; 15 F h I cycl; 3 microtrons, 25 betatrons, 20 electron sy~iclirotrons;3 proton synchrotrons; 15 electron linac::: 4 proton lin:m, I deuteror~linttc. R. L. Cohen. "Cyclotrons and Synchrocyclotrons." Har(dbuch der f'hysik, XLIV, Springer, I3erli11, 1959. On 111). 162, lfi3, 165 data are given on 47 cyclotroris and 18 synchrocyclotrons, as of 1955. G. A. Br11m:in. I'artick Accelwators. Biblioqrapir~j List of Accelerator Installations. CCIIL-SO50 (,l:~nuary1, 1!)58), 153 p:iges and 9 page Addendum. Available from OTS, $3.00 and $0.50. G. A. Rc1m:tn. "List of I'ulicle Accelerator Installations." Nuclear Itlstr. 3 : 181-217 (1958) ; 5: 129-13'2 (1'350). Lists X ! ) L)C rliacliineri, 56 betatrons, 55 F F and F M cyclotrona, 47 qmchrotrons, 60 1in:ic.-. 337

338

BIBLIOGRAPHIES, LISTS, DATA, CONT'D

F. T. Howard. Cyclotrons and High Energy Accelerators-1958. ORNL-2644 (November 17, 1958), 311 pages. Available from OTS, $5.00. Data sheets on cyclic machines existine; and under construction, design, and study: 76 FF cyclotrons (including 16 sector-focused), 18 FM cyclotrons, 17 proton synchrotrons, 11 electron synchrotrons ( > 0.5 Bev), 10 proton linacs (> 10 Mev), 4 electron linacs (> 0.5 Bev), 4 heavy ion linacs. Anonymous. "Charts of Accelerators and Parameters." Nuclear Engineering 4: No. 37, April, 1959. Data apparently taken from ORNL-2644. F. T Howard, Ed. Sector Focused Cyclotrons. Nuclear Science Series Report NO. 26, Publication 656, National Academy of Sciences-National Research Council, Washington, D.C., 1959. $2.50. On page 291 are given parameters of 15 sector-focused cyclotrons built, building, or under design or study. INTRODUCTORY ARTICLES O N ACCELERATORS I N GENERAL A. K. Solomon. Why Smash Atoms? Harvard University Press, 1940, 174 pages. Penguin Rooks, Inc., 1960. An elementary, accurate and entertaining account of atoms, nuclei, voltage-multipliers, Van de Graaff machines, cyclotrons, and particle-detecting devices, written for the layman. No mathematics. Excellent photographs and drawings. R. V. Langmuir. "Electronuclear Machines." L. N. Ridenour, Ed. In Modern Physics for the Engineer, McGraw-Hill Book Co., 1954. Chapter 7, pp. 173-196. Brief, general discussion of accelerators, with little mathematics. E. H. Krause. "Particle Accelerators." I n Amer. Znst. of Physics Handbook, McGrawHill Book Co., 1957. 8.172-8.201. Very brief description of DC and cyclic circular and linear machines. Many references. Twenty-page list of the world's accelerators. R. R. Wilson. "Particle Accelerators." Scientific American, March 1958, pp. 65-76. A popular account of circular machines. S. Glasstone. Source Book on Atomic Energy, 2d ed. Van Nostrand, 1958. Chapter 9, p p 237-272. Nonmathematical, qualitative description of various types of accelerators. E. XI. RiIcMillan. "Particle Accelerators." I n E. SegrB, Ed. Experimental Nuclear Physics. John Wiley & Sons, 1959. Vol. 111, pp. 639-784. Easily read description of all trchniques of acceleration with considerable quantitative analysis. It. R . Wilson and R. Littauer. Accelerators; Machines of Nuclear Physics. Anchor Books, Doubleday and Co., 1960, 196 pages. This book, one of the Science Study Series, is directed a t the secondary school student. Breezy and informal, it surveys many types of accelerators and the information gained by their use, entirely without mathematics. Good drawings and photographs. DC HIGH VOLTAGE GENERATORS GENERAL REVIEWS

R. ,J. Van de Graaff, J. G. Trump, and W. W. Buechner. "Electrostatic Generators for the Acceleration of Charged Particles." Reports on Progress in Physics 11: 1-18 (1946-47). R. L. Fortescue. "High Voltage Direct Current Generators for Nuclear Research." Progress in Nuclear Physics 1 : 21-36 (1950). E. S. Shire. "Electrostatic Generators." I n The Acceleration of Particles to High Energy, Institute of Physics, London, 1950, pp. 29-35.

DC HIGH VOLTAGE GENERATORS, CONT'D

339

E. Rnltlinger. "K~skntlcngencr:~torcn" H n t d h ~ r r h d c r l'hys11; XLIV, Springer, Bcrl~n, 1959, p p 1-63. 1: C: IIerb. "VIII tlc Crr:~nffCrc~nc*r:~tor~ " Hondbuch d w I'h?/~ikXLIV, Springcr Rcr-

lin, 3959, pp. (it-104. INTERESTIN(: EARLY PAPERS O N

nc

GENERATORS

Velocity l'ositive Ion.." J. D. Corkcroft : ~ n dE. T . S.Wnlton. "Experiment:: with Proc. R o y . Soc.. Lo~rdon.A 12!): 477-489 (1930). 280 kcv protons. J. D. Corkcroft 2nd E. T. S.Walton. "Fl~rtbcrDevelopment in the Method of Ohtaining High Vf,locit>-Positive Ions." Proc. R o c / . Soc.. I,om.lon, A 136: 619-630 (1932). 700 kev proton.<. The rlpvc~lopmmtof this c q ~ i p m c n tpcrmittetl the first nl~clritrtiirintegr:rtioni I)!. man-rn:~tlcprojectiles. This and the prccetling p3pcr are "mnst~." 11. .I. V:m d~ Gr:inff. "A 1,500,000 volt Elertrost:ltir Grncrntor." Phys. R P I ) .38: 19191920 (1931) : ~ l ~ s t r n rThe t . origin:rl Van dr Gr:~aff volt:rgc gcncr:rtor. No tlirrh:trgc tnhe. This sho~lltlhe read hy a11 wers of the sophi4c:ited VdG machines now :iv:ril-

able. 11. A. T3artor1, I:).W. Rfwller, nntl L. C:. Van .9ttn. "A Compact High I'otcntinl Elcctrostatic G~ncratcr."Phys. R e v . 42: 901 (1932) :ihstrnct. First w e of n V t E generator in c o m p r c w d air. No disrhargr t111)e. 11. J . Van d r Crrnnff, K. T. Cornpton, and L. C. Van Atta. "The Electrostatic Prochiction of High Voltages for Nuclear Investigi~tion." I'hg's. Kpv. 43: 149-157 (18:3:1). Plans for the :trcclcrator a t Ro11nd Hill, ;liars., with the use of two 15-ft tliamctrr spheres charged to 2 5 M v with a dischargt. tnhc lwtwecn them. L. C. Tr:m Att:~,I < . .T. Van de Gr:caff, nntl EI. A. T1:trton. "A Kcw Dwign for a High Yoltngr, 1)isch:irgc T11l)e." Ph?ls. R P P .43: 1r18-159 ( 1!)33). 1% of n film cylinder n.itll :I hc1ic:tl ink lirle to divitlc the voltage. Espt~ctrdto h p wet1 :it Round Hill. 1,. C. Van Atta, D. L. Northrnp, C. IT. Van Atta, and R . ,J. V:in de Granff. "The I)(,sign, Operation and P c r f o r m a n c ~of the 1lo11ntlHill Electrost:~ticGenerator." P k y s . Rrv. 49: 761-776 (1936). Att:~i~lnlcnt of 2.4 31v positivf. antl 2.7 1 I r neg:ttive. 5 0 discharge tuhe. L. C. \'an Att:i, D. L. Northrup, I:. ,I. Van tle Gra:lff, and C. AI. Van Atta. "Elcctrohtxtic Generator for Yuclear Iicse:~rch:lt I1.I.T." Reu. Sci. Inst,. 12: 534-545 (1941 ) . Ilonnd Hill equipment m o \ d to 1l.I.T. nntl rchl~iltwith only one termin:d at hidl potential. I'rotluccs 4 ma of ? i - ; \ I t ~ electrons. I n 11se for I I I : I ~years. ~ IZI. A. Tuve, Id. 1:. H:rfstad, antl 0. k h l . "lI1g11 Voltage Teclinicpe for S ~ i c l e a rPh>.sics St~~tlirs." Plr!ls. R P L S4s: . :il5-3:37 (193.5). SIICC(+.FIIIprod~ictionof 1.2-IIev protons with Van tic Graaff generator at atnlospht~ricprw-ure. Scction:rlized clisc11:trgc t ~ ~ b c . R . G. Herl), D. H. Parkinson, and D. \V. Krrst. "-4 Y:rn tlc Gra:iff Electrostatic G(werator Operating Under High Air I'resrurt~." R P I ~Sci. . I t ~ s t r 0: . 261-'X5 (10:15). Attainmcnt of 550-kv :ind of 400-kcv protoo- in conlpxt niachine. R . G. Herb, 1).B. I'arkinson, and D. \V. Kc.r,st. "The Develop~nentm d I'erfor~n:~nrc of an Elcctrortntic (2encr:ltor Operating Ilntler IIigh Air l'rcasurc." Phys. Rev. 51: 75-83 ( 1937 ) . TI(i-l\lcv 1x01ons. D. 13. l'arkinson, R. G. Herh, El. J. Bcrnt,t, and J. L. ATcKibbcn. "Electrostatic Generator Operating Under High Air Pressure-Opcratio~lal Experience and Accessory .4pparatus." I'hys. Rrv. 53: 6.22-630 (1')3X). IJsc of frcon added to compressed air raises proton energy to 2.4 Mev. R . G. Herb, C. RI. Turner, C. ?\I. Hudson, and R . E . Warren. "Electro~tntic Generator with Concentric Electrodes." Phys. Rcz). 58: 579-580 (1940) letter. The concent,ric Faraday cage idea is included. 4.5 AIcv protons.

340

DC HIGH VOLTAGE GENERATORS, CONT'D

T A N D E M ACCELERATORS

A. .J. Dempster. "The Production of High-Velocity Protons by Repeated Accelerations." Phys. Rev. 42: 901-902 (1932) abstract. Energetic protons are neutralized in gas and drift to a negative electrode where gas converts them to positive ions so that they are repelled; voltage doubling by this means is demonstrated. W. S. Bennett. U.S. Patent 2,206,558, July, 1940. Voltage multiplication by change of charge of ion. L. W. Alvarez. "Energy Doubling in DC Accelerators." Rev. Sci. Instr. 22: 705-706 (1951). Improvements in design and technique. K. W. Allen, F. A. Julian, W. D. Allen, A. E. Pyrah, and J. Blears. "The Tandem Generator for the United Kingdom Atomic Energy Authority." Nature 184: 303-305 (1959). 11.5 Mev. J. A. Weinman and J. R. Cameron. "Negative Hydrogen Ion Source." Rev. Sci. Instr. 27: 288-293 (1956). L. E. Collins and A. C. Riviere. "A Negative Hydrogen Ion Injector for a Tandem Electrostatic Generator." Nuclear Instr. and Methods 4: 121-128 (1959). F. H. Kiemann. "Low Energy Particle Accel~ratorsfor Precision Nuclear Physics Research." Nuclear Instr. and Methods 7: 338-349 (1960). Nonmathematical comparison of linacs, variable-energy cyclotrons, and tandem Van de Graaff accelerators. At en~rgiesbelow 20 XIev, the tandem is shown to have important advantages. P. H. Rose, R . P. Bastide, ,4. R. Wittkower, D. 1,. Webb, C. H. Goldic, and J. Shaw. "Injection of Fast Neutral Beams into the Three-Stage Tandem Acrderator." R r v . Sci. Instr. 31: 1052-1053 (1960). 0.15 X amp of protons at 15 hIev & 350 ev. R. J. Van de Graaff. "Tandem Electrostatic Accelerators." Nuclear Instr. and A i ~ t h O ~ 8S : 195-202 ( 1960). P . H. Rose. "The Three-Stage Tandem Accelerator." Nuclear Instr. and Methods 11: 49-62 (1961).

RF HlGH VOLTAGE GENERATORS

G. M. Breit, M. A. Tuve, and 0. Dahl. "A Laboratory Method of Producing High Potentials." Phys. Rev. 35: 51-65 (1930). 5.2 Mv produced by a Tesla coil in pressurized oil. No discharge tube. 1.5 ;\Iv from n Tesla coil. RI. A. Tuve. "Note on the Production of Extremely High Voltages." Phys. Rev. 36: 1576-1578 (1830). The idea of controlled potential distribution is carried further by the use of a set of concentric Faraday cages. D. H. Sloan. "High Voltage Vacuum Tube." Phys. R e v . 43: 213 (1933) abstract. Production of 600 kv by an evacuated, lumped-parameter resonant circuit. D. H. Sloan. "A Radiofrequency High-Voltage Generator." Phys. R e v . 47: 62-71 (1935). An important early paper on evacuated resonant systems, with both lumped and distributed parameters. R. S. Stone, M. S. Livingston, D. H. Sloan, and M. A. Chaffee. "A Radiofrequency High Voltage Apparatus for X-ray Therapy." Radiology 24: 153-159, 298-302 (1935). Application of the above device for generation of x-rays. J. J. Livingood and A. H. Snell. "Search for Radioactivity Induced by 800-Kilovolt Electrons." Phys. Rev. 48: 851-854 (1935). Production of external beam of electrons from an evacuated resonant transformer.

GENERAL REVIEWS O N CYCLIC ACCELERATORS

34 1

T h o v whwh arc devoted to only a s~ngletype are listed later in the correspontling category.

J. D. Corkrroft. "The Cyclotron and Betatron." ,I. Sci. Instr. 21: 189-193 (1944). Ikscription of r q ~ ~ i p m s nand t the I m i r prinviples. L. I. Schiff. "l'rotlnction of Particle Encrgies Rcyond 200 RIcv." R e v . Sci. Instr. 17: 6-14 (1946). Interestinq prrvicw- of things to romp j11i;t after the principle of $I:ISP stn1,ility was : ~ n n o ~ ~ n cAnalysis cd. of losses 1)sradiation. 1'. a. S a l i i h ~ ~ r.'Accelcratorr y. for Heavy Particlrs." Nudr~onics1, No. 3: 34-44 (Nov. 1947). RI. S. Livingston. "I'nrticle Accelerators." Advances i n 1Slectronics 1: 269-316 (1945). Academic Press, Inc., N.Y. Qualitative discussion of weak-focnsing circular and linexr accelcmtors. ,J. Itotblatt and F. K. Goward. The Acceleration of Particles t o High Energy. Institutr of Physics, London, 1950: "The C:yclotron." J. I:., 14-28. Also covers synchroryclotrons and proton synchrot,ron,*.-"Rrtatrons and Synchrotrons." F. K. G., pp. 29-35. -Very good articlcs with practically no mathcmntics. W. de Groot. "Cyclotron and Synchrocyclotron." I-'h,ilips Technical Review 12, 190. 3 : G5-72 (1950). Good elementary exposition of axial, radial, and phase stability. E. ill. IbIcMillan. "High Energy Accelerators." Helvctica Physica Acta 23, Suppl. 3: 11-26 (1950). Verbal description of linac, F F and F M cyclotron, synchrotron. G. F. Chew and B. J. Moyer. "High Energy Accelerators at, the University of California Radiation Laboratory." Amer. J. Phys. 18: 125-136 (1950). Good description of Alvarez linac, 154-inch Fhl cyclotron, elwtron synchrotron. ,J. H. Fremlin and J. S. Gooden. "Cyclic Accelerators." Reports on Progress i n Physics 13: 295-349 (1950). Excellent general discussion of FF and FM cyclotrons, betat'rons, and the synchrotrons under construction a t Birmingham, Brookhaven, Berkeley. Not much mathematics. T. G. Pickavance. "Cyclotrons." Progress in Nuclear Physics 1 : 1-20 (1950). Good brief descriptions of FF and FM cyclotrons. E. L. Chu and L. I. Schiff. "Recent Progress in Accelerators." Annual Review of Nuclear Science 2: 79-92 (1953). Review of DC and Van de Graaff machines, F F and FM cyclotrons, electron and proton sjmchrotrons and linacs. 76 references. M. S. Livingston. High Energy Accelerators. Interscience Publishers, 1954. 154 pages. Semiqunntitative treatment of phase and orhit stability, and general discussion of linear ant3 circular accelerators, including .4G focusing. T. G. Pickavnnce. "Focusing in High Energy Accelrrators." Progress in Nuclear Physics 4 : 142-170 (1955). Nonnlathematical presentation of weak and strong focusing principles in circular machines and linacs. E. 0. Lawrence. "High Current Acce1er:ltors." Sciencr. 12'2: 1127-1132 (1955). Also found in Peaceful Uses of Atomic E n w g y 16: 62-68 (1956). Popular lecture on Bevatron, Thomas cyclotron, spiral-ridge m:ichines, and high-current linacs. Good photographs. J. D. Cockcroft and T. G. Pickavame. "High Energy I'article Accelerators." Endeavor 14: 61-70 (1955). J. B. Adamb. "Acceleration of Nuclear Particles in ( h r v e d Paths. Cyclotron, Synchrocyclotron, and Synchrotron. A Rev~ew."Enginetnng 180: 530-535 (1955). R. Ko1l:tth 7 ' r ~ l c h f ~ n b e s c h l e u ~ 11955, / g e ~ , Vleweg & Sohn, Urnnn-ch~ieg, 2 2 I m p . Excellent treatment of linnrs and circu1:ir acceltmtors.

342

GENERAL REVIEWS O N CYCLIC ACCELERATORS, CONT'D

TV. Walkinshaw. "Acceler~tion of Protons to High Energies." Rmearch 8 : 29.5400 (1955). V. I. Veksler. "Principles of Acceleration of Charged Partirlcs." Soviet J. Atonzic Encrgy 1: 77-83 (1956). English translation. A most lucid account^. T o be fonnd also in Peaceful Uses of Atomic Energg 16: 69-74 (1956). n4. L. Oliphant. "The Acceleration of Protons to Energies Above 10 Bev." Proc. Roy. Soc., London, A234: 441-456 (1956). Qualitative lecture on cvclic accelerators, w n k and strong focusing. and FFA4G.Some detnils on the Canherra machine. P. Kl~nzc. "Zirkl~!nrl)eschlel~niger."Naturwiss. 43: 457-46.5 (1956). Semipopular, scminmthcmatical description of circular accelerators. I?. S. I.i~-inc?.~tou. "Trends in Cyclotron Design." Inrlmtr. Eng. Chem. 48: 1231-1237 (1956). Nontechnical discussion, emphasizing efforts on stronxcr cnrrcnt, mercy homogeneity and variability. Xricf description of proposed 850-Rlev spiral-ridge n~n~hinc. J. P. Hlcmctt. "Acceleration of I'articles to High Energies." Handbook of Physics. 9.153-9.160. hIcGraw-Hill, 1958. Short discnseion of DC and cyclic machines, both circular and linear. Condensed mathematical treatments of orbit and phase stability and ~\-nchronol~s nccclemtion. 1'. I. Vekelcr. "The Present State of the Problcm of Acceleration of Atomic Particles." Souiet Physics 66 (1) No. 1 : 54-61 (1958). English translation. Very readable, nonma,thematical account of present accelerators and fnture possibilities. D. L. Judd. "Conceptual Advances in Accelerators." Annual Review of Nuclear Science 8 : 181-216 (1958). Verbal presentation of present status, present problems, and the new ideas not yet in practice. Well worth reading. G. K. Grccn and F;. D. Courant. "The Proton Synchrotron." Hardbuch der Physik, XLIV. S p r i n g t ~ ,Berlin, 1969. Pp. 218-340. Detailed mathematical theory and many operational and engineering details on both weak- and strong-focusing machines. B. L. Cohen. "Cyclotrons and Synchrocyclotrons." Ha~zdbuch der Physik X I J V . Springer, Berlin, 1959. Pp. 105-169. Comprehensive treatment of orbit and phase stability problems, with discussion of much auxi1i:iry equipment. Tables of data on 47 fixed-frequency cyclotrons and 18 synchrocyclotrons. W. Paul. "Erzeugung von Elementarteilchen in Laboratorium." hTaturwiss. 46, No. 9: 277-283 (1059). Brief reviev of cyclic accelerators, their use in production of mesons, hyperons, and antinucleons. Some data on the Bonn ~ynchrotron. Twenty-three authors write on Gcrinan accelerators and special aspects of the subject and give a list of the world's large machines. Die Atomuirtschaft July-Aug. 1959, pp. 273-349. N. D. Fedorov. "The Cyclotron-A Cyclic Resonance Accelerator of Ions." Atomizdat, Moscow 1960. 88 pages. Includes discussion of AVF machines. I n Russian. M. S. Livingston and J . P. Blewett. Particle Accderators. McGraw-Hill. I n press. SPECIALIZED REPORTS O N NEW ACCELERATORS C E R N Symposium on High Energy Accelcmto~sand Pion Physzcs, lU.iG, Proceedings: Vol. I. E. Regenstreif, Ed. High Energy Accel~rators.CERN, Europcnn Organization for Nuclear Research, Geneva, Switzerland. 576 pages. 40 Swiss francs. Contains many worth-while papers on the new developments. Henceforth referred to as "C'ERN Symposium 1956" The above meeting has been briefly summarized in the follo~vingthree articles: M. H. Blewett. "CEIIS S y i n p o h m on High Energy I'hy&x" Physics Today 11, No. 9: 18-22 (1956).

SPECIALIZED REPORTS O N NEW ACCELERATORS, CONT'D

T. Cr. T'irk:~vnncc. and Cr. H . Stafford. "Iliirh E h r g y Nnrlcnr Ph\..4cs--Rrport CERN S>-niposi~~rn, 19S(i." hntrirt 178 : 1 l(i-1 18 (1 956). .T. B. ad am^. '"Tl~c CETtN Syrnl~osillm,1956.'' ~VuclcarInstr. 1 : 2-9 ( 1 0 5 7 ) .

343

on

A brief rt,port on a conference on acrderstors held in hloscow in ?\1:1y, 1056 is given by

T. G. P~cka\:inceand T . H R. Shyrme. "hlo~cow('onf(1rtmce." Nature 178: 115-118 (1956) Internotco~irrlt'ctt~fcrzricc~or! Hrqh Encrqy rlccc2~ratorsm d Inctrumc~2tatton.('ERLV, 1C)iB. Proc~ctl~nqs. 1,. Komarski, Ed CEIIN, Eliroprm Orp:in~zat~onfor Nuclear Rrsearch, Gmeva, Switzerland, 1!159. i 0 4 p:iqes 50 Sw1.s francs Henceforth rcferred to as

"CERN Symposium 1929" F T Howard, Ed Scctor-Focused C ~ p d o t r o ~ Proceedings ~s. of Sea I 4 m d Conference, Fcbrunry, 1050 SncleCirScicnce Scrlcs rtt.port N u n h e r 26, I1uhlmtlon 656, IVat~ond Ar:itlrm\ of Sc~cncc.b-N,rt~ord Re-earch Counc~l,11a h n g t o n , I) C ( 1 0 5 9 ) . 2!)1 1);1gc> $2 50 Discu-lons of the norltl'. quite hniited experience 151th sector-focused accelerators, present plans and problems.

THEORY OF ORBITS EARLY WORK

11. A. Rethe and Rf. Rose. "Thc RI:nimmn Energy 0btnin:~hlcfrom the Cyclotron." I'hys. Rezi. 52: 1'254-1255 (1937) lettcr. Recognition of thc conscqnences of the rc1:itivistic iucrrastl of 111:1s. 11. E. Itow. "Focusing and llaxirnum Energy of Ions in the Cyclotron." Phys. R e v . 53 : 3!%40S ( 1035). It. R . JVilson. ",2Iagnetic and E1ectro;itatic Focusing in the Cyclotron." Phys. Rev. 53 : 405-420 ( ( 9 3 8 ) . 11. E. I{osc.. "Jlagnetic Ficltl Corrections ill the Cyclotron." l'h ys. 1Zev. 53: 715-719 ( l!MS) . 1i. I t . Wilson. "Theory of thc C1yclotron." J. Appl. I'hys., 1.1: 781-796 (1!)40). HCCO(.NITIOX

L)

OF BE1 APRON ObCILLAIION FREAkI 1:PUClLb

1Y Kc1.1 .md It S c r l m "Electron~c OrL~tsin the Illduction Accelerator." Phys. Rcu. 6 0 . 53-58 ( l O - I l ) .

DISCOVERY O F PHSSE STABILITY

V. I. VekJcr. "A New Method for Accelcation of Relativistic Particles." Comptes IZet~dus(Dolilatly) tle l'dcadtimie des Scir~ncr~s de Z'IIRSS 43, Xo. 8 : 329-331 (1944); 44, Yo. 9: 365-368 (1944). Ihglihh edition-J. of Physics USSR 9, No. 3 : 153-158 (1!)1.5). Thgli4i translation. These tlirce papers tlescribe the basic principles of the r~licrotron,iy~ichrocyclorror~, :ynchrotron, and 111i:w srnl~ility. I<.11. ~lclIill,iri."?'ht> Syriclirot~~on--A1'rol)o:;d High E~irxrgy.4cceler:1tor." lJlr!j,s.It( 1 ) . (is: 11:L131 ( l!)-Ir1) k t t c r ; (i!): 5:i3 ( l W i ) k t t c r . " 1:csoii:lnce Accelcrat ion of C111:1rgc~l I':~rticlca." lJit!p.RcI).7 0 : SO0 (1946) nbstr:ict. C I I C : ~ statement of the l)rol,lwl :111,1 of its solution.

344

THEORY O F ORBITS, CONT'D

FVRTHER THEORY O F ORBITS

11. R . Crme. "The Racetrack: A Proposed Modification of the Synchrotron." Phys. Rev. 69: 542 (1946) letter. D. M. Dennison and T. H. Berlin. "The Stability of Orbits in the Racetrack." Phys. R c v . 69,542-543 (1946) letter. D. M. Dennison and T . H. Berlin. "The Stability of Synchrotron Orbits." Phys. Rev. 70: 58-67 (1946). N. H. Frank. "The Stability of Electron Orbits in the Synchrotron." Phys. Rev. 70: 177-183 (1946). D. Bohm and L. Foldy. "The Theory of the Synchrotron." Phys. Rev. 70: 249-258 (1946). R. Serber. "Orbits of Particles in the Racetrack." Phys. R e v . 70: 434-435 (1946) letter. D. M. Dennison and T. H. Berlin. "Racetrack Stability." Phys. Rev. 70: 764-765 (1946). D. Bohm and L. Foldy. "Theory of the Synchro-Cyclotron." Phys. R e v . 72: 649-661 (1947). R. Q. Twiss and N. H. Frank. "Orbital Stability in a Proton Synchrotron." Rev. Sci. Instr. 20: 1-17 (1949). N. M. Blachman and E. D. Courant. "The Dynamics of a Synchrotron with Straight Sections." Rev. Sci. Instr. 20: 596-601 (1949). T. Teichman. "Beam Oscillations in an F-M Cyclotron." J. Appl. Phys. 21: 1251-1257 (1950). P. C. V. Hough. "Radial Oscillations in the Cyclotron." Rev. Sci. Instr. 24: 42-48 (1953). E . A. Crosbie and M. Hamermesh. "Coupling of Betatron and Phase Oscillations in the Synchrotron." Phys. Rev. 98: 233 (1955) abstract. C. L. Hammer, R. W. Pidd, and K. M. Terwilliger. "Betatron Oscillations in the Synchrotron." R e v . Sci. Instr. 26: 555-556 (1955). P. A. Sturrock. Static and Dynamic Electron Optics-An Account of Pocusing in Lens, Deflector and Accelerator. Cambridge University Press, 1955. 240 pages. A thorough trcatment, bascd largely on the classical methods of Hamilton, Lagrange and Poincar6 as applied to optical systems. L. C. Teng. "Resonant Excitation and Damping of Betatron Oscillations." Bull. Amer. Phys. Soc. 3, No. 2: 102 (1958). GENERAL REVIEWS ON FIXED-FREQUENCY CYCLOTRONS

W. B. Mann. "The Cyclotron and Some of I t s Applications." Reports on Progress in Physics 6: 125-136 (1939).

M. S. Livingston. "Standard Cyclotrons." Annual Review of Nuclear Science 1: 157-162 (1952).

W. B. Mann. The Cyclotron, 4th ed. Methuen, London, 1953. 118 pages. Elemrntary treatment of resonance condition, qualitatwe statement of axial focusing. Rrlef, good discussion of R F circuits. B. L. Cohen. "The Theory of the Fixed-Frequency Cyclotron." R e v . Sci. Instr. 24: 589-601 (1953). M. S. Livingston and E. M. McMillan. "History of the Cyclotron" Physics Today 12, No. 10: 18-34 (1959). Informal reminiscences of the early days in Berkeley. 31.S. Livmgston. "Early Development of Partlcle Accelerators." Amer. J . Phys. 27, S o . 9: 626-629 (1959).

SPECIFIC FIXED-FREQUENCY CYCLOTRONS

345

BERKELEY

E. 0 . Lawrence and N. F Edlsfssn "On the Production of High Speed Protons." Science 72: 376-377 (1930) First announrement of the idea. E. 0. Lawrence and RI. S. L~vlngston."A Nethod for Pro,lucing High Speed Hydrogen Ions Without the Ube of High Voltages." Phys. R e v . 37: 1707 (1931) abstract, 10-inch magnet. 80-Kev H + H .

E. 0. Lawrence and RI. S. Livingston. "The Production of High Speed Protons Without the lJse of High Voltages." Phys. R e v . 38: 834 (193 1) letter. 9-inch, 0.5-Mev H+ :tncl H + H. E. 0. Lawrenct. and RI. S. Livingston. "Prod~~ction of H i d l Spcctl Light Ions Without the Use of High Voltages." Phys. R e v . 40: 19-35 (1932). 11-inch, 1.2-Mev H + . h1. S. Livineston. "High Sneed Ions." Ph14s. Rev. 42: 441-44" (1932) letter. ";-inch, XA-IIS~i ~ 11. + 11. S. Livingston and E. 0. L:~wrrncr."'The A l ~ ~ l t i l dAcceler:ltion r of Ions to Vrry IIigh Spsede." Ph?/s. R e v . 45: 608-612 (1034). 274-inch, 5-RIev H + H . .I. .J. Livingootl. "Rntlioactivity by Bombartlment." Electrot~ics8, No. 11: 421-423, L>

460 (Kov. 1035).

E. 0. Lawrrnce and D. Cooksey. "On the .-21)pnr:1tus for the Rlultiple Acceleration of Light Ions to High Speeds." Phys. R r v . 50: 1131-1140 (1936). 27;-inch, 6.3-JIev ti. First external I ~ e x nFirst . flanged and ga.skc+d p l w dee supports. F. N. D. Kuric. "The Cyclotron: A New Itescarch Tool for Physics and Biology." Gcncral Elrctric Review (,June 1937) pp. 364-272. F. N. D . Kurie. "Prwent-Day Design ant1 Tschnique of the Cyclotron." J. A p p l . Phys. 9: 691-701 (1938). Pole tips incrcawd to 37 inchrs. 8-Mev d . E. 0. Ilawrcncc. "The Rlcdical Cyclotron of 1 he William 11. Crocker Radiation labor:^tory." Science 90: 407-408 (193!)). 60-inch, IfGhle\- d. E. 0 . Lawrcnrr, I,. W.Alvarcz, W. ?\I. ISrobcck, I). Cooksey, D . R . Corson, E. 11. and It. L . Thornton. "1niti:il Performance of the 60-Inch hIcMillan, W. W. Snlish~~ry, Cyclotron at ths Wi1li:lm H. Crocker Ratlia~ionLaboratory, University of California." Phys. R e v . 513: 124 (1039) letter. 16-RIev d.

11. S. Livinpton. "The 1Ingnei1c lie6onance Accelerator." R e v . Sci. Instr. 2 : 55-68 (1936). 16-inch, 2-JIcv I,.

1' G. Krugrr :md G K Green "The C o n d r w t ~ o nand Opcratlon of :I Cyclotron to l'roduce On(. l l i l l ~ o nVolt L)cutcronb " Ph 11s Rerl 51 : 609-705 (1937) 16-1~11 l< 11 L > n i m , W E Ogle, 11. E 1' G Tir~lgcr,(; K (:roctzmger, ,J 11 I<~ch,~rclwn, Sclson, G Srhnarz, J H Grevnc, N C C ' u l i ~ 11 , \V I x e , C E lIrClell,rn, D Sc,lc, 1,loytl S m ~ t h ,:t11t1 F I< T,~ll~rl.~tlge "A O c l o t r o n Which Allo~rsthe Accrlrratcd 1':irtlrlc to I
RI. C. IIcntlerson ant1 hI. G . White. "The Dwign ant1 Opcmtion of a Large Cyclotron." R r l l . Sci. I ~ t s t r .9 : 19-30 (103S). 35-inch, O-l[t~vn . 'This niachine was 1:lrgely destroyed by fire and wa,q rel~uiltafter the war illto :L syl~chrocyclotron.

346

SPECIFIC FIXED-FREQUENCY CYCLOTRONS, CONT'D

TOKYO

Y. Nishina, T . Yasaki, and S. Watnnnbe. "The Instnllation of a Cyclotron" Sci. Papers Inst. for Phys. and Chem. Research, Tokyo, 34: 1658-1668 (1938). 26-inch, 3-Mev d. SWARTHMORE

A. J. Allen, M. B. Sampson, and R. G. Franklin. "The Cyclotron of the Biochemical Research Foundation." J. Franklin Inst. 328: 543-561 (1939). 38-inch, 11-Mev d. Glass dee insulators with parallel-conductor resonant circuit in air. PURDUE

W. J. Henderson, L. P. D. King, J. R. Risser, H. J . Yearian, and J. D. Howe. "The Purdne Cyclotron." J. Franklin Inst. 228: 563-579 (1939). 37.5-inch, 16.5-Mev a. SOVIET UNION

D. G. Alkhazov, M. G. Meshcheriakov, and L. M. Chromchenko, "The Radium Institute Cyclotron-The Arc Type Ion Source." J. Phys. USSR 8 : 56-61 (1944). In English. 4-Mev d. MASSACHUSETTS INSTITUTE O F TECHNOLOGY

?\I.S. Livingston. "The Cyclotron." J. Appl. Phys. 15: 2-19, 128-147 (1944). 42-inch, 12-Mev d. Two coaxial h/4 evacuated dee supports. Good general discussion. STOCKHOLM

13. Atterling. "Design of Acceleration Chamher and Dees for the 225 cm Cyclotron at the Nobel Institute for Physics, Stockholm." Archiv for Fysik 2: 559-569 (1951). H. Atterling and G. Lindstrom. "Notes on the 225 cm Cyclotron a t the Nobel Institute for Physics, Stockholm." Archiv for Fysik 4: 559-563 (1952). H. ,4t terling and G. Lindstrom. "A Fixed-Frequency Cyclotron with 225 cm Pole Diameter." Nature 169: 432-434 (1952). 25-Mev d. H. Atterling and G. Lindstrom. "The 225-cm Cyclotron at the Nobel Institute of Physics, Stockholm." Arkiv for Fysik 15: 483-502 (1959). BIRMINGHAM

Anonymous. "The University of Birmingham Cyclotron." Nature 169: 476-477 (1952). 61;-inch, 20-Mev d. 44-INCH R. J. Jones. "The Oak Ridge 44-inch Cyclotron." Phys. R e v . 91 : 223 (l953), abstract. F. L. Green. "Electromagnetic Shims for Focusing in a Fixed Frequency Cyclotron." Pkys. Rev. 91: 223-224 (1953) abstract. R. S. Livingston and R. J. Jones. "Design Features of a High-Current Cyclotron." Phys. Rev. 94: 1436 (1954) abstract. OAK RIDGE,

63-INCH R. S. Livingston. "Acceleration of Partially Stripped Heavy Ions." Nature 173: 54-57 (1954). Description of 63-inch cyclotron for 25-Mev N3+. OAK RIDGE,

RIDGE, 86-INCH I{. S.Livingston. "The Oak Ridge 86-Inch Cyclotron." Nature, 170: 221-223 (1953). 24-Mev p.

OAK

SPECIFIC FIXED-FREQUENCY CYCLOTRONS, CONT'D

347

r z i w q r { t it(; H

I{. S. I3cmtlcr,.1' 11. Itrilly, A . .T. .kllcn, .T. S...Zrtllur, :tnd .I.H:~~wmnn. "The University of T'ittd)~~rgliScattering Project." R(Y. Sci. Instr. 23: 542-547 (1952). 55-inch, 10-11cv HH-' , 20-RTrv d, 40-\lev a .

F. H. Schmidt, C: W F:mwll, .T E Hmderson, T . ,1. Morgan, and J . F. Streib. "The I'niver,itv of Wn\hlngton 60-Inch (hclotron." Rev. Sci. Instr. 25: 499-510 (1954). 11-RIev 1111 1,22-RIrv d, 44-Rlcv a . 44CLAY

I;:. A. i Sacl:~y" L'Onrle &lcctriqzte 35, NO. 344: 1046-1051 (I!),55). 70-inrli, 11-3lrv p,22-RIev d, 44-Mcv a . A. Rarimid. "Lr. cirrults H F tlu cyclotron." L'Onde dlcctrique 35, No. 344: 1052-1063 (1955).

P.Dc1)ramc. "LICC ~ r l o t r o n(111 C.

SOVIET 1 XION

L. AT. Nrmcnov, S. P. Kdinin, L. F. Kondrachov, E S L h o n o v , A. A. Naumov, V. S. Pnnns\ltk, ?: D Fetlorov, N N Khnlthn, 2nd A A Chnbakov. "1: hIeter Fixrd Frrqiiencv C\clotron" Sovzrt J. i l t o m ~ cE,rcrqq 2, No 2 : 37-41 (1957). Engli-h tr:tnil:itlon. 12 2-Mrv 11, 19.6-Nev d, 39 - ; \ J c v a , I t ' O - 3 l l ~No+. LONDON

J . \I7. Gnllop, D D Yonberg, X ..J. Post, W. B.P o n d , ,J. Sharp, and P. J . Wnterton. "A Cyclotron for llctlirnl Research " Proc. Inst. tor .Elect. E n g . 104, Part n, No. 17: 452-406 ( 1957). 55-lnch, 15-hlcv tl. PRETORIA

,J J . Rurgerjon, J. ,J. I h T o i t , and C . A. .J. Kritzinper. "The Pretoria Cyclotron." Nu-

clcnr Iristr :<: 323-3'25 (1958) 112 cm, 1.5-Vev (3. IIEIDELBERG

R . Rock, A. I)oehnng, J. Jiinecke, 0. Knecht, L Koester, EI. Maier-Leibnitz, Ch. Schinelzer, ant1 17 Schmidt-llohr. "EIII I'c~htfreclwnz Zyklotron m ~ temem Dee." 2. f . angeu3.Ph ys. 10: 49-55 (1958) 101 r m , 13-hIev d, :!G-XIev a.

VARIABLE ENERGY FIXED-FREQUENCY CYCLOTRONS MELBOURNE

D E Caro, L tI Rlartin, and J L Rouse "A Vnr~ablclEnergy Cyclotron" Austral. J P h ~ j s8 306-309 (1955) 2 7-to-1 change in frequrncy m t l ficld In 40-mch mnchine. LOS ALAMOS

R L Thornton, K Thyer, and J M. Petcrson. ' C?clotrons Designed for Prrc~cion Fast-Neutron Cro43ection Lleasurements " Pencrful Usrs of Atomic Energy, 4 : 87-91 (1956 I 42-mch; 3 5-9 LIev protons, 7-17 5 1Iev drutrroni, 10 5-12 RIP> tritons; 5 6-18 k i l ~ g ~ ~8 0-14 ~ i - AIc/>cc. ~

348

VARIABLE ENERGY FIXED-FREQUENCY CYCLOTRONS, CONT'D

K. Royer. "Variable Frequency in the Los Alamos Cyclotron." Sector-Focused Cyclotrons, pp. 171-173,262-265 (1959). ROCHESTER

H. W. Fulhrinht, D. A. Bromley, and J. A. Bruner. "An 8-Mev Variable Energy Cyclotron." Phys. R e v . 99. 854 (1955) abstract. H. W. Fulbright. "The R-F System of the Variable-Energy Cyclotron at the University of Rochester." Sector-Focused Cyclotrons, pp. 174-178 (1959). LIVERMORE

C. J . Taylor, A. Bratenahl, H. P. Hernandez, J. M. Peterson, R. H. Smith, R. Stahl, J. A. Galvin, and I?. K. Mullins. "Livermore Variable-Energy Cyclotron: Magnet, Rndiofrequency, P.rformnnce." Bull. Amer. Phys. Soc. 11, 1 : 178-179 (1956) abstract R . L. Thornton, K. Boyer, and J. M. Peterson. "Cyclotrons Designed for Precision Fast-Neutron Cross Scction Mrasurements." Peaceful Uses of Atomic Energy 4: 87-91 (1956). 90-inch; 2.6-14 Mev protons, 5.2-12.5 Mev deuterons, 7.7-8.3 Mev tritons; 2.6-8.6 kilogauss; 4.0-9.3 Mc/sec. C. J . Taylor and J. M. Peterson. "Livermore Cyclotron Beam Features and Extraction." Sector Focused Cyclotrons, pp. 208-210, 230-233 (1959). TOKYO

H. Kumagai, K. Ono, H. Shono, I. Hayashi, K. Kuroda, and M. Imaizumi. "16-Inch Variable Energy Cyclotron." J . Phys. Soc. Japan 14, No. 1: 1-9 (1959). S. Kikuchi, I . Nonaka, H. Ikeda, H. Kumagai, Y. Saji, J. Sanada, S. Suwa, A. Isoya, I. Hayashi, K. Matsuda, H. Yamaguchi, T. Mikumo, K. Nisimura, T. Karasawa, S. Kobayashi, K. Kikuchi, S. Ito, A. Suzuki, S. Takeuchi, and H. Ogawa. "A 160-cm Synchro- and Variable Energy Ordinary Cyclotron." J. Phys. Soc. Japan 15: 41-59, 527 errata, (1960). At fixed frequency, produces 7.5-15 Mev protons, 15-21 Mev deuterons, 30-42 Mev alphas. With FM, attains 57 Mev protons. H. Kumagai. "On a design of Wide Range Magnet for Cyclotron." Nuclear Instr. and Methods 6: 213-216 (1960). Pole shape which avoids local saturation. Used on both Tokyo machines. ARGONNE

W. J. Ramler, J. L. Yntema, and M. Oselka. "Energy Degrading Focusing of Cyclotron Beam." Nuclear Instr. and Methods 8 : 217-220 (1960). .dse of foils to degrade d and a from 10.8 to 2.5 Mev/nucleon. CYCLOTRON ION SOURCES

J. R. Dunning and H. L. Anderson. "High Frequency Filament Supply for Ion Sources." R e v . Sci. Instr. 8 : 158-159 (1937). Used to avoid distorting forces caused by DC current in presence of magnetic field. The complication of R F heating is often nowadays avoided by using a rugged filament of Q-inch diameter or more. R. R. Wilson. "Formation of Ions in the Cyclotron." Phys. R e v . 56: 459-463 (1939). The effect of various parameters, using a simple filament; prior to use of arc source. E. M. McMillan and W. W. Salisbury. "A Modified Arc Source for the Cyclotron." Phys. R e v . 56: 836 (1939) letter. M. S. Livingston, M. G. Holloway, and C. P. Baker. "A Capillary Ion Source for the Cyclotron." R e v . Sci. I ~ s ~ 10: T . 63-67 (1939).

CYCLOTRON ION SOURCES, CONT'D

349

D. C. Cowic and C . .I. Ka:nntln. "-4rc3 So~lrccwith Dirrct-(l~lrrrnt Fi1:unrnt S111)ply for Winch Cyclolron." R r v . Sci. Instr. 16: 224-225 (1945 1 . Good tlrawing of tlct:rils of construction 11. S. Li14ngston. "Ion So~lrcesfor Cyclotrons." IZev. M o d . Phys. 18: 293-299 ( I W i ) Good review. A. E. Hayes, J r . "R-F ITrating Cycldron Filammts." Phys. R e v . 70: 220 (193Ci) lcttrr. RJT n w of X i 4 trnnrmission line, filament load is purely resistive, t>hm rcducing re:~ctiwcurrents. R . S. Livingston, and R. J. ,Tones. "Biqh Intensity Ion Source for Cyclotrons." R e v . Sci. Instr. 25: 552-557 (19%). R. .I. Jones and A. Znckcr. "Two Ion S o ~ ~ r c for r s the Produrtion of Multiply-Charged Sitrogen Ions." R e u . Sci. I m t r . 25 : 562-566 ( 1954). C. T3. Rlills and C . F. Rnrnrtt. "High-Intcmsity Ton SOII~CP." R ~ u Sci. . Iitstr. 25: 1200-1202 (1934).

11 L. Ile~.nol,lqand A Z ~ ~ r k r"('vclotron r 1011 So~~rcei. for the Production of N 3 + Ionq." R w . Sri. Instr. 26: 894 (1955) lrttcr TIT. F. Stubhins. "An Exprrm~c~ntnl Ion So~lrcefor tlir 184-Inch Cyclotron." Phys. Rw. 0 : 274-275 ( 1955) :&tract. P. Rl. Rlorozov, B N Makov, and h3 S. Ioffe. "Source of hlultiply-Charged Nitrogen Ions for a Cyclotron." Soviet J . Atomic Enerqy 2, No. 3: 327-331 (1957). English translation. N1+ . . . N-i+. CYCLOTRON OSCILLATORS

D. H. Sloan, R . L. Thornton, and F. A. Jmkins. "A I>emountable Power Oscillator Tnbr." R P V .Sci. Instr. 6 : 75-82 (1935). .I. Thrhlls. "Design of Cyclotron Oscillators." R e v . Sci. Instr. 22: 84-92 (1951). F. H. Schmidt, and 11.J. ,Jakol)son. "Cyclotron O~cillatorsand the Shifting Inter-Dcc Crrol~ntlS ~ ~ r f n r cIZev. . " Sci. Instr. 25: 136-139 (1954). K. R. MacKcnzie. "Calculator for Some R F Problems in Accelerator Design." R r v . Sci. Instr. 27 : 580-583 ( 1956). CYCLOTRON ENERGY MEASUREMENT

J. H. Manley and M. J. Jakobson. "Cyclotron Beam Energy Determination by a Timeof-Flight llethod." R e v . Sci. I m t r . 25: 368-389 (1954) C. J . Delbecq, W. J . Ramler, S. R . Rocklin, and 1'. H . Yust,er. "Crystal Techniques for Measuring Cyclotron Ream Energies." R e v . Sci. Instr. 26: 543-546 (1955). GRIDS IN CYCLOTRON DEES

W. B. Powell. "Improving thf, Ch:mcteri~tics of the Cyclotron Beam." Nature 1 5 i : 1045 (1!)56).

V. S. Panasyuk "On the LIotlon of Charged I':~rt~clesin the Central Region of :L Cyclotron." S o v ~ e tJ . Atorn~cEnergy 3, No. 10: 1173-1176 (1957). English transhtion. H. G. Blosser and F. Irwin. "Grid Focusing Studies in Cyclotron Central Region." Bull. A m e r . Phys. Soc. 11, 3 : 180 (1958) abstract. A. H. Mort,on and W. I . 13. Smith. "Improved Cyclotron Performance from ControI of Initial Ion Motion." Nuclear Instr. and Methods 4 : 36-43 (1959). W. I. R. Smith. "Improved Focusing near the Cyclotron Source." Nuclear Instr. and Mrthotls 9 : 49-54 (1960).

350

CYCLOTRONS-MISCELLANY

R. L. Murray and L. T. Ratner. "Electric Fields Within Cyclotron Dem." J. App2. Phys. 24: 67-69 (1953). Use of Schwarz-Chrlstofell tr:msformation to complitc electric field components parallel and perpendicular to magnetic field. A[. J. Jakobson and F. H. Schmidt. "Characteristics of a I'roposed Double Mode Cyclotron." Phys. Rev. 93: 303-305 (1954). Dees <180°, push-pull and push-push modes. W. E. Parkins and E. C. Crittenden, J r . "A Graphical Method for Determining Particle Trajectories." J. Appl. Phys. 17: 447-449 (1946). Useful in deflector calculations. J. Kokame and S. Yamashita. "Electrostatic Deflection of a Cyclotron Ion Beam." J. Phys. Soc. Japan 11: 332-333 (1956). Analytical method of computing o r b ~ tin deflector channel. P. P. Dmitriev, N. N. Krasnov, and E. N. Khaprov. "The Problem of Beam Deflection in a Cyclotron." Soviet J. Atomic Energy 3, No. 7: 778-780 (1957). English translation. Use of axially shaped deflector electrode to reduce radial width of beam on I-m cyclotron. J . J. Went. "Soft Iron for the Electromagnet of a Cyclotron." Philips Tmh Rev. 10: 246-254 ( 1949). F. A. B. Ward. "A Mechanical Model Illustrating the Principle of the Cyclotron." Proc. Phys. Soc. 51: 810-816 (1939). J. S. Miller. "Demonstrating the Principle of the Cyclotron." Amer. J. Phys. 26, No. 7 : 503 ( 1958). M. J . Jakobson and J . H. Manley. "Phase Properties of the Deflected Ion Beam from a Fixed Frequency Cyclotron." Phys. Rev. 95: 600 (1954) abstract. S. D. Bloom. "A New Technique for Observing Cyclotron Phase Grouping." Phys. Rev. 98: 233 (1955) abstract. N. D. Federov. "Phase Relations in a Cyclotron!) Soviet J. Atomic Energy 2, No. 4: 471-473 (1957). English translation. M. Konrad. "Ion Phase Measurements on the Birmingham Cyclotron." R e v . Sci. Instr. 29: 840-845 (1958). Y. Z. Zavenyagin and N. D. Federov. "The Problem of Choosing the Potential Difference Between the Dees of a Cyclotron." Soviet J. Atomic Energy 3, No. 7: 785-790 (1957). English translation. Analytical method of calculating dee voltage in terms of field contour. V. S. Panasyuk. "Possibility of the Technical Application of the Inherent Modulation of the Ion Beam in a Cyclotron." Soviet J. Atomic Energy 3, No. 7: 781-784 (1957). English translation. W. w. Havens, J r . "Pulsed Accelerator Slow Neutron Velocity Spectrometers." Peaceful Uses of Atomic Energy 4 : 74-86 (1956). Describes pulsed arcs in Columbia 36-inch and Brookhaven 60-inch cyclotrons; pulsed axial deflection of Columbia 400-Mev proton synchrocyclotron; pulsing techniques on Yale electron linac and General Electric betatron; Columbia pulsed transformer.

HEAVY I O N S IN FIXED-FREQUENCY CYCLOTRONS

L. W. Alvarez. "High Energy Carbon Nuclei." Phys. Rev. 58: 192-193 (1940) abstract. First detection of 50-Mev lYY+ in 37-inch cyclotron. D. Walker and J. fI. Fremlin. "Acceleration of Heavy Ions to High Energies." Nature 171: 189-191 (1953).

HEAVY I O N S IN FIXED-FREQUENCY CYCLOTRONS, CONT'D

3.5 1

.I. FT. Fremlin, W. T. Link, ant1 K . (.: Stephens. "The .2rcclrr:itinn of 11c:rvy F'iwtl-Frequency Cyclotron. R ~ f tJ. . Appl. I ' h ~ l , ~5 .: 1,57'-l(il ( 1!)51). Gootl (Iisr~issionoS t l i r ~ ~ r o c e s s , I). W:~lkclr. "I-Icnvy Ions of IIigh Energy." l'rogrr'ss in .Nwlcar I'h!/sics 4: 215-'233 (1955).Goocl rwirw ; many referencw. K. E. A. EfTat :md .J. 11. Fremlin. "Second II;rrmonic Acce1cr:rtion in the Cyclotron." J . Sci. Inst. 34: 415-417 (19573. H. A4tterling."The Accclerntion of Heavy Ions in the Stockholm 225-cm Cyclotron." Archiv for Fysil; 7 : 503-506 (1954) ; 15: ,531-558 (1!359). r).Wmlker, Ions in :I

SYNCHROCYCLOTRONS

BERKELEY,

37-INCH

On t h in~tial ~ experiments pcrformrd on the rrhuilt Xi-inch magnet in ortlrr to demon-t rate the fen41111tv of frcrl~~cricv-n~otl~~l:ltc'd operation.

,J R . R ~ c h n r d ~ oI<. n , R . RIarKcnz~e,E ,T Lofarm, anti 13. 'T. Wright. "Frequency Mod111nted Cvclotron " P h ~ j s R. e v . 60: 669-670 (1946) letter F H Schmiclt "5lechan1cal Freqwncy Rlod~il:~tion Syitem as Appl~cdto the Cyclotron " Rev. Sci Irlstr 17: 301-306 (1946) K. R . RIacKenzic, and V 13 JT7mthln:an. "R-IT Sy-tern for Frequency Rlodulntetl Cyclotron " RPV ;s'ci. Instr. 18: 900-907 (1947). .J. R. Richardson, R T Wright, E. J . Lofgrrn, and I3 Peters. "Development of the F r e q ~ ~ e n c1Jothllnted y Cyclotron." Phys. Rcv 7 3 : 424-436 (1948) Exper~mentnl v ~ r ~ f i c a t i oofn 1 heory. BERKEI EY,

184-IN('H, 730 ~ I E V

W. 11. Brobwk, E 0 Lawrrnw, K I1 RlacI<enzit>,E A4 RIclIillan, R . Scrber, D. C Sencll, K A1 S ~ m p w n :lnd , R L Thornton ' ' I n ~ t ~ aI'crformnncc l of the 184-Inch Cyclotron of the U n ~ ~ e r \ of ~ tCj n h f o r n ~" ~Ph ria Rrv 71 : 440-450 (lO47). 200-hIcv deuterons, 400-hlev alpha. K R RIncKenzie. "Cyclotron Frequency Rlotlulat~on." Phys. Rev. 73. 540 (1048) abstract. K. Ii. ilIacKenm. " W ~ d e Range Frequcnc:, LIodulation." Phys. Rcv. 74: 104-105 (1948) letter K 11 I l , l c K e n ~ i r F , H Schmdt, .T 11 lTooti\:lrd, a n d L F Woutcr- " D e i ~ g nof the Rad~ofrequcncySystem for tht. 784-Inch Cyclotron " Rcv. SO. Instr 20: 126-133 ( 1949). L. It. Henrich, D. C. Sewcll, and ,J. Vale. "Operation of the 184-Inch Cyclotron." Rev. Sci. Instr. 120: 887-898 (1949). Comparison of experiment with theory. K. R. RIacICenzic. "The l'rot,on-Deuteron IW Systt>rnfor the Berkeley Synchrocyclotron." Rev. Sci. Instr. 22: 30'2-30'3 (1951). The nlcthocl of obtaining a frequency range of 22.9 to 15.8 Mc/sec for 350-Mev protons and 11.5 to 9.8 Mcjsec for 190-Mev deuterons. R. L. Thornton. "Freqncncy Rlodulrrtion and liadiofrequtlncy Systrm for the Modified Berkeley Cyclotron." C E R h T Symposium 1956, 11p. 413-418. The use of vibrating blades for frequency modulation.

352

SYNCHROCYCLOTRONS, CONT'D

ANGELES, 41-INCH,21-MEV J . W. Burkig, E. L. Hubbard, and K. R. MncKenzie. "Pulsed Oscillator for F-M Cyclotron." Rev. Sci. Instr. 20: 135 (1949) letter. UNIVERSITY O F CALIFORNIA AT LOS

ROCHESTER, 130-INCH,250-MEV S. W. Barnes, A. F . Clark, G. B. Collins, C. L. Oxley, R. L. McCreary, J. B. Platt, and S. N. Van Voorhies. "Note on the Rochester Cyclotron." Phys. Rev. 75: 983 (1949) letter. AMSTERDAM, 71-INCH,2 8 - M ~ v F. A. Heyn. "The Synchrocyclotron at Amsterdam." Philips Tech. Rev. 12, No. 9: 241-256 (19511 ; Philips Tech. Rev. 12, No. 12: 349-364 (1951) ; (with J . J . Burgerjon) Philips Tech. Rev. 14, No. 9: 263-279 (1953). HARWELL, 110-INCH,175-MEV T. G. Pickavance, J. B. Adams, and M. Snowden. "The Harwell Cyclotron." Nature 165: 90-91 (1950). T. G. Pickavance and J. M. Cassells. "High Energy Nuclear Physics a t Harwell." Nature 169: 520-522 (1952).

130-INCH,450-MEV M. Foss, J. G. Fox, R. B. Sutton, and E. C. Creutz. "Design of a Cyclotron Magnet: Pole Design Using a Half-Magnet." Rev. Sci. Instr. 22: 469-472 (1951).

CARNEGIE INSTITUTE O F TECHNOLOGY,

HARVARD, 95-INCH, 168-MEV L. L. Davenport, L. Lavatelli, R. A. Mack, A. J. Potk, and N. F. Ramsey. "A Dee Biasing System for a Frequency Modulated Cyclotron." Rev. Sci. Instr. 22: 601-604 (1951). CHIGAGO, 170-INCH, 450-MEV H. L. Anderson, J. Marshall, L. Kornblith, Jr., L. Schwarcz, and R. H. Miller. "Synchrocyclotron for 450-Mev Protons." Rev. Sci. Instr. 23: 707-728 (1952). PRINCETON, %-INCH, 18-MEV M. G. White, H. W. Fulbright, P . Gugelot, and R. R. Bush. "Redesign of the Princeton Cyclotron to Yield 18-Mev Protons." Phys. Rev. 74: 1242 (1948) abstract. G. Shrank. "Energy Control for External Cyclotron Beam." Rev. Sci. Instr. 26: 677680 (1955). LIVERPOOL, 156-INCH,383-MEV M. J. Moore. "The 156-Inch Cyclotron a t Liverpool." Nature 175: 1012-1015 (1955).

D. W. Fry. "An International Laboratory for Nuclear Research." Nature 172: 646-647 (19.53). T. G. Pickavance. "Synchrocyclotrons and the CERN 600 Mev Machine." Nuovo Cimento Ser. 10, 2, Suppl.: 403-412 (1955). C. J. Bakker. "CERN, European Organization for Nuclear Research." Physics Today, 8, No. 9: 8-13 (Sept. 1955).

SYNCHROCYCLOTRONS, CONT'D

3.53

F T
. tlie V ~ h r a t ~ nC,rpacitor g of a S~nchrocyclotron.' H Bollke "Chlatlni's F ~ g ~ l r eon Phzlips Tech K P E 19: 84-85 (1957-58) F. Krlmen. "The 'Tuning Fork Frrquenrv ~lo(111lntorof the C E R N S~nchrocyclotron." Nuclear lnetr. nnd Aletltods 5: 280-208 (1959). DTTBNA, &M, 680-\/IF,v D. V. Efremov, ?\I. G. RIcshcheriakov, A. L. I l i n t ~ V. , P.Dzhclepov, 1'. P. Ivanov, V. S. Katishev, E. G . Komnr, N . A. XIonoszon, I . FI. Nevinzhskij, B. I . Polyakov, and A. Y. Chestnoi. "Thc Six-lfcter Syrrrhroryrlotron of thc Institute for Nuclear I'roblems at the Acndemy of Sciences of the USSR." So1:ir.t J. Atomic E n ~ r g yNO. 4: 5-12 (1956) Englirh translation. Aha, ('ERN S?lmpoaiirm 1,956, pp. 148-152. .4. I,. Mints, I. H. Xcviazhskij, and 13. I. I'olynkov. "The Ilntlio-Frequrncy System for the 680-Mcv Proton Synchroryclotron." PF:RAV S/lm,posium 1,956, pp. 419-414. V. P.Dzhclepov and H. R I . Ponterorvo. " I ~ ~ ~ s r a r in c l iHigh-Energy Part,icle Physics ~ for Nuclear Physics of thc Performed on the Synchrocyclotron of t h 1,:thor:ltory Joint Institl~tefor Nilclear Ilescarch." Souic't J. A t o w ~ i cEnergy 3, No. 11: 1273-1314 (1057). English translation. Phot,os, tnrgrt :ind heam arrangements, etc. A. E . Ignatenko, V. V. Krivitsky, A. I. hIukhin, E:. I'ontecorvo, A. A. Reut,, and K. I . Tarakanov. "E:xtraction of High-Energy I'art,icle Beams Through the Yoke of thc y 5: 667-671 (1956). Engli.sh Sy~lchroryclotronJlagnet." Soziet J. Atomic E ~ ~ e r gNo. tmnslation.

A. Suzuki, S. Suwa, Y. Sazi, T. Karasawa, T. Yamada, Y. RIiyazawa, and J. Sanndn. "On the External Proton Beam of About 55 Rlev of the INS,J 160-em Synchrocyclotron." Inst. for Nuclear S t u d y , U . of Tokyo, INSJ-23 (Ilecemher 17, 1!)59), 11 pages. BONN, 190-CM,35-MEV DEUTERONS H. Briickmann. "Ein Modulator fur das Synchroayklotron Bonn mit Ferroelektrischer Keramik." Aruclear Instr. and Methods 6: 169-175 (1060). (COLTJMBIA) , 164-INCH,385-RI~v ,J. Rainwater, W. W. Havens, Jr., J . S. Desj:irdins, and J . L. Rosen. "Nevis Synchrocyclotron Slow Neutron Spectrometer." Rrv. Sci. Instr. 31: 481-459 (1960). l'ulsed electric field to deflect protons axially. NEWS

ORSAY, 2.8-M, 157-LIEV C. Bergamaschi, J. C. Brun, A. Cahreepine, R. Gayrand, J . GBnin, H. Langevin-Joliot, N. hlarty, A. Michalowicz, P. Hadvanyi, 31. Rlou, J . Trillac, arid C. Victor. ''Le Synchrocyrlotron de 157-Mw." J. de physique et le radium 21: 305-314 (1960). DEFLECTION FROM FM CYCLOTRONS

W. hl. Powell, L. R. Henr~ch,Q. A. Kerns, I). C. Senell, ,and 13. L. Thornton. "Elcctromagnetlr Deflector for the Beam of the 184-Inch Cyclotron." R e v . Scz. Instr. 19: 506-512 (1948). Q. A. Kerns, R It. Raker, R. F. Edwards, and G . 11. Fsrly. "H~gli-VoltageI'ulhcr for the 184-Inch Cyclotron Electric Deflector." R e v . Sci. Instr. 19: 899-904 (1948).

354

DEFLECTION FROM FM CYCLOTRONS, CONT'D

D. R . IIxmilton and B. J. Lipkin. "On Deflc~t~ion a t n = I in the Synchrocyclotron." R P V Sci. . Inxtr. 22: 783-792 (1951). S. S I I ~-1., Sanada, T. I
J. L. Tuck and L. C. Teng. "Regenerative Deflector for Synchrocyclotron." Phys. Rev. 81: 305 (1951) abst,ract.

K. J. LeCouteur. "The Regenerative Deflector for Synchrocyclotrons." Proc. Phys. Soc., London, B 64: 1073-1084 (1951) ; "Perturbations in the Magnctic Deflector for Synchrocyclotrons." Proc. Phys. Soc., London, B 66: 25-32 (1953). A. V. Crewe and K. J. LeCouteur. "Extracted Proton Beam of the Liverpool 156-Inch Cyclotron." Rev. Sci. Instr. 26: 725 (1955) letter. K. .J. LrCouteur, A. V. Crcmc, and J. W. G. Gregory. "The Extraction of the Beam from the Liverpool Synchrocyclotron." Proc. Roy. Soc., London, A 232: 236-251 (1955). K. J. LeCouteur and S. Lipt,on. "Non-Linear Regenerat,ive Extract'ion of Synchrocyclotron Reams." Phil. Mag. 46: 1265-1280 (1955). A. V. Crewc and U. E . Kruse. "Regenerat,ive Beam Extraction on the Chicago Synchrocyclotron." Rev. Sci. Instr. 27: 5-8 (1956). S. Cohcn and A. V. Crewc. "Regenerative Action in High Energy Accelerators." C E R N Symposium 1956, pp. 140-147. N. F. Verster. "Synchrocyclotron Ejector." C E R N Symposium 1956, pp. 153-154. W. F. Stubbins, E. Kelly, J . T. Vale, and K. Crowe. "184-Inch Cyclotron Beam-Extraction System." Bull. Amer. Phys. Soc. 11, 2: 382 (1957). S. Cohen and A. V. Crewe. "Regenerative Action in High Energy Accelerators." Nuclear Instr. 1: 31-40 (1957). G. Calame, P. F. Cooper, Jr., S. Engelsberg, G. L. Gerstein, A. M. Koehler, A. Kuckes, J . h4. Meadows, K . Strauch, and R. Wilson. "Some Features of Regenerative Deflection and Their Application to the Harvard Synchrocyclotron." Nuclear Instr. 1: 169-182 (1957). K. J . LeCouteur. "The Vertical Motion of Part,icles Through the Regenerative Beam Extractor." Nuclear Instr. 1 : 343-344 (1957). S. Mayo, C. A. Heras, and J. Rosenblatt,. "Regenerative Beam Extraction on the Buenos Aires Synchrocyclotron." Nuclear Instr. 2: 9-12 (1958). W. F. Stubbins. "Design of Regenerative Extractors for Synchrocyclotrons." Rev. Sci. Instr. 29 : 722-725 (1958). J. Rosenblatt and R . J. Slobodrian. "The Magnetic Deflector of the Buenos Aires 180cm Synchrocyclotron Beam." Rev. Sci. Instr. 31: 863-868 (1960).

GENERAL REVIEWS ON CG PROTON SYNCHROTRONS

M. L. Oliphant, J. G. Gooden, and G. S. Hide. "Acceleration of Charged Particles to Very High Energies." Proc. Phys. Soc., London, 59: 666-675 (1947). Qualitative discussion of the projected Birmingham proton synchrotron.

GENERAL REVIEWS ON CG PROTON SYNCHROTRONS, CONT'D

355

J. S. Gooden, 11. H. .Jrnrcn, and J. L. Pymon~ls."Theory of fhc Proton Synchrotron." tliac~lssion. Proc. Ph?ls. Soc.. London, 50: 67T-(iO3 (30-17). Sctnirnnth~n~ntic:rl A[. S. 1,i~inpio11. ~ ~ l ' r o t o Syn(~11roiron." n . 4 ~ 1 / 1 r tRa(~, I ~ ; Pof ~ I ;\'~/r/(,ar , SC~PUC I :C 1601'74 (I952 ) . TTcrybrief tlixursion of mnchincs \wing I~nilt: ~ 13irrningh:~m, t 13rookhnven, Berkeley, C'anl~erra. .J. 1'. I3lcwctt. " l < c v n t Dcvc~loprnont~ on Proton Synchrotrons." Annztnl Review of Nuclrar Scic,ncc I : 1-1" 195-1). 15rit.f tlisc~~,.Conr:rnt. "Tl~cl I'roton Synchrotron." Natitlhuch dcv PIillsilz XLIV. Springclr, Brrlin, 1959. 1'1). 218-340. Detailed lhcory and many engineering details of CG and AG mnchinrs. OPERATING CG PROTON SYNCHROTRONS (1960) BIRMINGHAM,

I REV

L. TT. Hil111:irtl."The Birmingham I'roton Synchrotron." Nlrcleonics '7, No. 4: 30-43 (Ortoljer, l!%O). L. Ilitltlifortl. "1':1cruln1 Systml of the Birnlingham Proton Synchrotron." .I Sci. . Zmtr. 28, Sr~ppl.1 : 47-58 (1951). Anonymol~s."l'rotoli Synrhrotron (IT" the. I!nivf'rsity of Rirminghani." Nature 172: 704-706 ( l!M 1 . L. r'.I1il)l~:rrtl.,.The Ilatlio F r q ~ ~ c n rSystan y of tlic 13irnlingll:im l'roton Synchrotron." J. Sci. 111stt~. 41 : 363-371 ( 1964). 1'. B.LIoon, L. Itidtliford, and J. 1,. Synrontla. "F:xl)~~rimc~nt:~l Clinr;~ctc~rislics of thr I'roton Synchrotron." Proc. Ii'o?l. Soc., Lo~~ilotr. A 2 0 : 201-215 (1!)551. L. I?itltliford, 11. \'. \':in tlrr It:~:ry,:rncl I:. F. Cor. "Porne I k w n S t ~ ~ t l i cwit,h s tht, l n duct ion Elrc~trotlc."l'roc. Ph!l,s. Soc.. Loii~loti,.4 liS: 4'3:)-50'' (I!)%). 13. 1,ctllcy ant1 L. Ititltlifortl. " P c ~ t c ~ r m i ~ x of ~ t iawl'roloii S~mcl~rotron I3cxin Size wit11 A (i!:i SX-S:39 ( 1!)55). a Cryst:\l Scintillator." l'roc. I'ii !IS. Soc.. l,ot~rlo~i. ('. A. Ilnnrrn. "1'rinc~il)lcs ant1 A p ~ ) : ~ r : l tof l ~ sthe I~ljoctionSyhtcm of the Birniingli;rn~ I'roton Pyric~lirotron."J . Sci. Itisti.. :Xi: 52-5s (1!L56). of Injr,ction l'licnonlcn:~ in C. A . 1::11nln, 11. F. C'or, :111(l1'. 1'1.J7:~uglln. i n \ n ; ~ l ~ & i ~ . IN?-106 ( l ! ) X ) . the H i r n ~ i ~ ~ g I'roton h : ~ n ~ Synchrotron." J. Sci. I t ~ ~ s f33: 1'. 11. 1Ioon. " S p c r (7h:lrgr : I ~ 1onizatio11 J I'llcnonlcm in Const:rr~t-Gratlicnt Proton P~.nchrotron~." I'roc. I'll !IS. Soc.. Lot~tloii.-2 ti!): IXi-1Fdi (1936 1. N.E. 1300th ant[ G . JV. Hntchiri-on. "Fine ' J i ~ t l ( ~ - S t r ~ ~ t01'' t l1k:tnls ~ ~ - c from t h t Hinningh:ml I'roton S~.nc~hrotro~l." A l ~ r ~ cIiistr. l ~ ~ c ~I i: ~SO-% (1'357). S. J. C;oltls:rck. "Two-Tnrcyt Ol~crationof :I I'roton S~.~lchrotron." A'uclcjar Itlstr. 1: 90-91 (1957). ." f I . 13. V:tn tlcr I~:I:I!.. "Foewing of t l i ~8y11chrotron Sr:~tierc~tl-On1I ~ o : I I ~A'uclcai~ h s t r . 1 : 351.-353 (1!)57). G . A. I)or:~n,I<. A. I;inl:~y, 11. I?. Slia~.lor~ arid 1 1 . AI. \Vinn. "I<x~r:~riion of I1rotom

356 COSMOTRON,

OPERATING CG PROTON SYNCHROTRONS, CONT'D

3 BEV

RI. S Livingston, J P. Rlewett, G. K. Crcen, and L. J. Haworth. "Design Study for a 3 Rev Proton Synchrotron." Rev. Sci. Instr. 21: 7-22 (1950). The plans for the Cosmotron. Cosmotron etaff. "The Cosmotron." Rev. Sci. Instr. 24: 723-870 (1953). The Cosmotron as built. A very illnminating and detailed account of the design and construction of all components. Required reading for machine builders. F G. Brockman and 11.W. Louwerse. "The Ferroxcube Transformer Core Uqed in the nrookhnven Cosmotron." Philips Tech. Review 15, No. 3: 73-83 (1953). Ferroxcube is a trade name for a particular ferrite. 0. Plccioni, D. Clark, R. Cool, G. Friedlander, and D. Kassner. "External Proton Beam of the Cosmotron." R e v . Sci. Instr. 26: 232-233 (1955) letter. G. B. Collins. "The External Proton Beam of the Cosmotron." C E R N Symposium 1966, DD. 129-132. D:C. Rallm. "A Rapid Beam Ejector for the Cosmotron." Bull. Amer. Phys. Soc. 11, 2: 11 (1957) abstract. E. .T. Roger~."Method of Damping Phase Oscillations in a Synchrotron." Rev. Sci. Instr. 29: Z5-"7 f 19%). 11.Q. Barton. "Measurement of Betatron Oscillation Frequencies in the Cosmotron." Rev. Sci. Instr. 31: 1290-1291 (1960).

W. M. Brobeck. "Design Study for a 10-Bev Magnetic Accelerator." R e v . Sci. Instr. 19: 545-551 (1948). The plans for the Bevatron. The completed machine has not been described in any single article in the usual journals. Vanous aspects are given in the following. E. J. Lofgren. "The Proton Synchrotron." Science 111: 295-300 (1950). Lloyd Smith. "The Bevatron." Scientific American, Feb. 1951, 20-25. Elementary discussion, with photographs taken dur~ngconqtruction. B. T. Wright. "Magnetic Deflector for the Bevatron." Rev. Sci. Instr. 25: 425-431 (1954). B. Cork. "Proton Linac Accelerator for the Bevatron." Rev. Sci. Instr. 26: 210-219 ( 1955). C. 'N. Winningstad. "Generating HF Energy for the 6-Bev Bevatron." Electronics "1, No. 2: 164-169 (1955). E. .T. Lofgren. "Bevatron Operational Experiences." C E R N Symposium 1956, pp. 496503. W. C. Struven. "Bevatron Magnet Pulse Timing System." Electronics 24, No. 6: 160163 (1956). W. M. Brobeck, and W. C. Strwcn. "Bevatron Frequency Measurement System." Electronics 29, Xo. 5: 182-1187 (1956). E. J. Lofgren. "The Bevatron." Proc. Nut. Acad. Sci. U S A 45, No. 4: 451-456 (1959). K. F. Stone and R. J. Force. "Flip-Coil Target Positioner for Use with Accelerator." Rev. Sci. Instr. 30: 787-788 (1959).

H Bruck and R. LCVI-?\landel "Sur le pro~etdu synchrotron & protons de Saclay." N71ovo Czmento, S e r m 10, 2, Suppi 1. 423-441 (1955). Cr Broncs, H Rruck, J Hamelin, G Xe] ret, a r d T Balzmger. "Le Test des blocs tlr 1'6lcctro-alrn.int du synchrotron de S d a ) " 6ltclra1 Instr. 1: 123-132 (1957). G Nej ret and J. Param "Les mcults correcteurs dc synchrotron Saturne." Nuclear Instr. and Methoda 5. 269-265 (1959).

OPERATING CG PROTON SYNCHROTRONS, CONT'D

357

F. Perrin e t :LI. "Le synchrotron h protons 'R:rt~lrne'" L'Otdr ilectrique 30: 421-633 (June 1959). Comprchen+e article on all components. The C'f3R.V Sym,posilrm 1956 contains tiiscussions of various components hy a. n ~ n n l wof Sovirt :mi hors: E . G. Kornar, N. A . RIono~zon,K.S. S t r d t w v , mil G . 31. Feclotov. "Strncturd Fwt i m s of thc II:lgnct," pp. 382-384. -4. A. Zhur:~vlrv,l?. Cr. Koln:~r,I. -4. lloz:~levikij,N.A. Rlonoszon, and A. hI. Stolov. "Mapnctic (Ihnr:rctcristics," 1'11. 330-313. M. A. Gsshc\r, E. G . Komnr, N . A. lIonoszon, F. 11. Spev:~kova, and -4. hI. Stolov. "Magnet Powrr Supply," lip. 378-381. E. G. Komar, I. F. hIalyshev, I s . L. Mikhclis, and A. V. Popkovich. "Vacuum Chamber," pp. 385-386. S. XI. Rubcbinskij, ,4. A. Vasilev, h1. P . Scltlovicli, 1'. F. Kuzmin, a1111S. S. Kurochkin. "1nst:mt:~neous hIeasuremcnts of Magnetic Field a,nd Elcctric Frequency," pp. 40-1-412. A. 12. Mints, S. hr. Rubchinskij, LI. M. Veishcin, F . A. Vodopianov, and A. A. Vasilcv. "Frequency (hntrol System," pp. 429-435.

V. I . Vcksler, I). V. Efremov, A. L. Mints, M. h4. Veisbein, F . A. Vodopianov, M . A. Gashrv, A. I. Zeitllits, 1'. 1'. Ivanov, A. .4. Kolonlenskij, I. F. Llalyshev, N. A. llonoszon, I. Kh. Nrvinzhsltij, V. A. l'rtiibhor, 11. S. I<:~l,inovich,S. 11. T
CANBERRA,

.T W I3l:~mcv "I'hr O r h t a l RIagnet and Pov\i.r Supply of the 10 Gev Proton Synchrotron a t the .lu,trall,in Nat~on:il ITnl\er.ity " PElr'il' S7/mposmrn 1976, I1p 344-3.ih D S Ion: ~ n dD Elliott. "Xlagnrt~c I i o w ~ s111 Corm of Varlow Sh:ipc. " ,1.'uclrar Instr and Methods 5: 133-141 (1959) D r z ~ g nof Eerrlte cores for the accclrr:~tlori cavity.

JYLVANIA, 3 I ~ E V h1. G. Wlxte, 17. C. Shoemaker, and G. K . O'Ne~ll."A 3-Ucv High Intensity Proton Synchrotron." CERLV Symposium 1976. PI). 525-529. F. C. Shoen1:tkcr "Thc Princeton-Pennsylvarila Arcelerator." CERN Symposium IliLi:), pp. 362-365. PRINCETON-PEN N

ARGONNE NATIONAL

LABORATORY, CHI(A(:O, '(z G s," 12 BEV

E. A. Crobljle, 11 I Ferentz, ILL H. Fozs, hI El:imcrrnesh, J . J . Llv~ngoud, d 11 M a r t ~ n and , L C. Tcng "The Argonne 1'7 5 Bev l'roton SJ nclirotron." C'ER'V SIIIIIposium l 9 X , 1117 42-43 I':~rily oboli>te A V Crene "The Argonnc Zero-G~acl~ent STnclirotron (ZGS)." CER,V Srj~upoacrcttr 1tYci9. ~31, 350-361.

358

CG PROTON SYNCHROTRONS UNDER CONSTRUCTION, CONT'D

RUTHERFORD HIGH ENERGY

LABORATORY, HARWELL, NIMROD,"^ BEV

T. G. Pickavance, P. Bowles, and G. W. Dixon. 'LThe 7-Gev Proton Synchrotron." Nuclear Engineering 4, No. 37: 151-168 (1959). A very interesting engineering description. T. G. Pickavance. "7-Gev Proton Synchrotron (Nimrod)." CERN Symposium 1959, pp. 343-346. J. J. Wilkins and A. J. Egginton. "Nimrod-Britain's 7 Gev Proton Synchrotron." Nuclear Energy 14, No. 141: 57-61 (Feb. 1960). CG ELECTRON SYNCHROTRONS GENERAL REVIEWS

J. E. Thomas, Jr., W. L. Kraushaar, and I. Halpern. "Synchrotrons." Annual Revieu of Nuclear Science 1: 175-198 (1952). A particularly informative article, with many engineering details. Contains a list of electron synchrotrons and the parameters of those at Berkeley, Cornell, M.I.T. No theory. R. R. Wilson. "Electron Synchrotrons." Handbuch der Physik XLIV, Springer, Berlin, 1959. Pp. 170-192, 447-449. The most recent general article on the subject. SPECIFIC C G ELECTRON SYNCHROTRONS AND PARTICULAR PROBLEMS

H. R. Crane. "The Racetrack." Phys. Rev. 70: 800-801 (1946) abstract. Brief description of first machine with straight sections. F. K. Goward and D. E. Barnes. "Experimental 8 Mev Synchrotron for Electron Acceleration." Nature 158: 413 (1946). D. W. Fry, J. W. Gallop, F. K. Goward, and J . Dain. "30 Mev Electron Synchrotron." Nature 161: 504-506 (1948). F. R. Elder, A. M. Gurewitsch, R. V. Langmuir, and H. C. Pollock. "A 70-Mev Synchrotron." J. Appl. Phys. 18: 810-818 (1947). J. P. Blewett. "Magnetic Field Configuration Due to Air Core Coils." J. Appl. Phys. 18: 968-976 (1947). J . P. Blewett. "Design of an Air Core Synchrotron." J. Appl. Phys. 18: 976-982 (1947). T . R. Kaiser. "On the Capture of Particles into Synchrotron Orbits." Proc. Phys. Soc., London, A 63: 52-66 (1950). T. 11. Kaiser and J. L. Tuck. "Expcriments on Electron Capture and Phase Stability in a 14-Alev Synchrotron." Proc. Phys. Soc., London, -4 63: 67-74 (1950). 1).W. Fry, J. Dain, 11. H. H. Watson, and 11. E. Payne. "The Design and Operation of a 30-Mev Synchrotron." P ~ o c .I?&. Elt~ct.Eng. 9'7, Part 1, No. 108: 306-319 (1950). F. K. Goward, J . D. Lawson, J . J. Wilkins, and It. Carruthers. "The Design of Electron Synchrotrons." Proc. Inst. Elect. Eng. 97, Part 1, No. 108: 320-339 (1950). W. U. Jonca, K. It. Kratz, .I. L. Lawson, D. H. hliller, 11. D. hliller, G. L. Ragan, J. Itouvina, and 13. G. Voorhies "300-Alev Nonierromagnetic Electron Synchrotron." Rev. Sci. Instr. 26: 809-826 (1955). W. McFarlnne, S. E. Barden, and D. L. Oldroyd. "The Glasgow 340-Mev Synchrotron." Nature 17G: 666-669 (1955). G. Sd\-ini. "The Italian Design of a 1000-Xev Electron Synchrotron. A Comparison Between the Strong and the Weak Focusing." Nuovo Cimento, Ser. 10, 2, Suppl.: 442-458 ( 1955). G. Ghigo and I. F. Quercia. "Field Stabilization in a DC-AC-Excited Magnet of a Synchrotron." Nuclear I m t r . 1 : 57-61 (1057).

CG ELECTRON SYNCHROTRONS CONT'D

359

BETATRONS GENER4L REVIEWS

T .I Wnng. "Thc Betatron." Electro7tzcs 18. 11'8-134 (,Junt> 1935). D W Rcrst "H~stor~c:~I Development of the Betatron " Nature 157: 90-95 (1946) n Well worth rcwi~ng.R~fercncesglrcn on earlv attenlpts at ~ n d u c t ~ onccelernt~on. TI7 I I o ~ l e ~"The - . Eetatron " J Scz Inatr. 2 3 . 277-283 (1946). D \I7. Kerst "The Betatron " Hantlbuch der I1hysaX XLIV Springer, Hcrlm, 1950. Pp. 1O:i-217. Very complete dlscu~sion. SPECIFIC BETATRO F;S

I).W. Icerst. "Acc>eleration of Electrons by Magnetic Induct,ion." Phys. R e v . 58: 841

Acceleration of Electrons by iLIagnetic Induction." I'hys. RCV. 20-3Iillior1 Electron-Volt Bciatror~or Intlwtion Accelerator." Rev. Sci. Irrstr. 13: 387-394 (1942).-"llethocl of Inertwing Rct:ctron Energy." I'hys. R e v . 08: 233-234 (1945 1 letter. Bias. If7. W ~ s i e n ~ l o and r p E. Churlton. "A 100-Mev Induction Electron i\ccelerator." J. Appl. t'hjls. 16: 581-5!)3 (1945). W. F. Weetentlorp. "The Use of Direct Current in Induction Accelerators." J . Appl. f'h U S .G57-Mi0 ( 1045). Bias. L. S. Skaggs, Q. 11. *4lniy, I). \V, Kerst, arid L. H. Lnnzl. "Removnl of the Electron Rcnrn From the Hetatron." Phys. Rcu. 'TO: 03 (1!44(i) letter. E. An~:rltli: ~ n dI-:. Fcrrctti. "On Two Possible Modifications of the Induction Accclerntor." R e v . Sci. Instr. 17: 389-395 (1946). B i : s 1-1. F. Kaiser. "Europe:ln Electron Induction Acct:lerator~." J. Appl. Phys. 18: 1-17 (1947). H. F. Kaiser, D. L. Mock, 0. E . Berg, D. K. Stevens, and \V. E . Harris. "The Kaval 1:caearch Laboratory Bet at ron." h ~ ~ d c a r z i c2:s 4'342 (Fcb. 1'348). Good article with 69 refcrences. 11. W. Koch sntf (2. S. Iiohinson. "Lnitlircctior~al I'ul+: Operation of a Lktntron." Xev. Sci. Instr. 19: 36-39 (1948). A. Bicrmnn and 11. A. Cklr. "lktatron With and Without Iron Yoke." Philips Tech. Reu. 11, No. 3 : 65-'i8 (Sept. 1949). D. W.Kerst,, G. D. Adams, H. W. Koch, and C. S. Robinson. "An 80-hlev llodcl of a 300-Mev Betatron." R e v . Sci. Instr. 21 : 462-480 (1950 1 .-"Operation of a 300 Mcv Betatron." Phys. Rev. 78: 297 (1950) let,ter. The most energetic betatron yet built. E. C. Gregg. "A Flux-Forced Field-Biased Betatron." Rev. Sci. Instr. 22: 176-ls2 (1040) letter.-"The

60: 17-53 (1941).-"A

360

BETATRONS, CONT'D

A. S. Pcnfold and E. L. Garwin. "Rctntron Enerey Calibration by Magnetic Field Measurement." Rev. Sci. Instr. 31: 155-163 (1960). ENERGY LOSS BY RADIATION IN CIRCULAR ACCELERATORS

D. Iwanenko and I. Pomeranchuk. "On the Maximuin Energy Attainable in a Betatron." Phys. Rev. 65: 343 (1944) letter; and also Comptes Rendus (Doklady) URSS 44, No. 8 : 315-316 (1944), English edit,ion.

E. M. McMillan. "Radiation from a Group of Electrons Moving in a Circular Orbit." Phys. Rev. 68: 144-145 (1945) letter.

J. P. Blewett,. "Radiation Losses in the Induction Accelerator." Phys. Rev 69: 87-95 (1946).

L. I . Schiff. "Production of Particle Energies Beyond 200 Rlev." Rev. Sci. Instr. 17: 6-14 (1946).

J. Schwinger. "Electron Radiation in High Energy Accelerators." Plzys. Rev. 70: 798 the Classical Radiation of Accelerated Electrons." Phys. Rev. (1946) abstract.-"On 7 5 : 1812-1925 (1949). F. R. Elder, R. V. Langmuir, and H. C. Pollock. "Radiation from Electrons Accelerated in a Synchrotron." Phys. Rev. 74:52-56 (1948). Aleasurement of spectral distribution of visible light. A. A. Kolomenskij and A. N . Lebedev. "The Effect of Radiation on the hlotion of Relativistic Electrons in a Synchrotron." CERN Symposium 1956, pp. 448-455. K . W. Robinson. "Radiation Effects in Circular Electron Accelerators." Phys. Rev. 11 1 : 373-380 (1958). E. M. Moroz. "Effect of Quantum Fluctuations in the Electron Radiation of the Synchrotron Oscillations." Soviet Phys.-JETP 6 : 1008-1009 (1958), in English. GAS SCATTERING

N. M. Blachrnan and E . D. Courant. "Scattcring of Particles by the Gas in a Synchrotron." Phys. R e v . 7 4 : 140-144 (1948); 7 5 . 315 (1949).

J. M. Greenberg and T. H . Berlin. "Scatter~ngL o s m in the Synchrotron." Rev. Sci. Instr. 22: 293-301 (1951). E . D. Courant. '~ltevisionof Gas Scattering Theory." Rev. SCZ.I m t r . 24: 836-837 (1953). A. N . Matveev. "On the Llagnitude of the Losses of Particles Through Scattering in the Residual Gas in Sjnchrotrons " S o v ~ e Physrcs: t Technzcal l'hysacs 3, No. 8 : 1687-1693 (August 1058), English tramlation. M. J. hloravcaik and J. hI. Sellen, J r . "Gak Scattering in a Strong-Focusing Electron Synchrotron." Rev. Scz. Instr. 26: 1158-1164 (1955). G ~ v e smany early references. M. D. Kostin 'GAS Scatter~ngin a Strong-Focusing Synchrotron." Nuclear Instr. and Methods 4 : 9'3-109 (1958). MICROTRONS

V. I . Veks1er.-See

three papers listed above under Theory of Orbits, Discovery of Phase Stability. P. A. liedhead, H . LeCaine, and W. J. Henderson. "The Electron Cyclotron." Nucleonics 5 : ($0-67 (Oct. 1949).-"The Electron Cyclotron." Can. J. Research 28A: 7391 (1950). C. Schmclzer. "Uber giinstige Betriebszustiinde des Elektronzyklotrone." 2. f . Naturforsch. 12, 5a: 808-817 (1952).

MICROTRONS, CONT'D

361

C. TTrnderson, F. F . H r y ~ n a n n and , 1:. E. Jrnrtings. "I'haw St:rhility in thc \Iicrotron." Proc. 1'hy.s. Soc-.. Lot~don.66R: 41-40 ( 1!)5;3). C. TTenclerwn, F. F. Hcymann, xntl It. E. Jrnnings. "The Dc>sign antl Oprrntion of :I 4.5-1lev 1licrotron." Proc. Phys. Soc.. London. (i6T%:654-664 (1953). J. S. Bcll. "Vrrtiral Focl~singin the P\Iicrotron." Proc. Phys. Soc.. London. GAB: 802804 (1953). H. F . K:li~er."3licrotrons (Electron Cyclotrons) for X and K Band Operation." .I. Franklin Inst. 257: 89-108 (1954). E. Rrannen :md FI. I. S.F e r g ~ ~ s o"nT. h m Extraction in the 1Iicrotron." Rcv. Sci. Iristr. 27 : 833-844 ( 1956). A. Carrelli and F'. Porreca. "1nform:ttion on I: 2.5 M r v hlicrotron." N i t o ~ oCimcnto fi, No. 3: '729-734 (l95'i). D. 11. Zorin, 0 . P. l\Iilov:inov, ant1 A . V. Shalnov. "A Linear-Cyclic .4ccelerator." Sovict J . Atomic Energy 2, Xo. 6: 675-676 (1957 I . English transl:tt,ion. Sugwstion of 180" mamete to direct ions repetitively through linact;. E. 31. RIoroz. "Accelrrntors with Stublr~Joinetl I'articles Trajectories." Soviet J. Atov'c Etiergy 4, No. 3: 323-329 (1958). Engli~htranslation. S~~ggestion similar to the reference al~ove.Considerable analysis. A. Iiohertr. "The Proton Ilicrotron." A n ~ n of l ~Physics 4 : 115-165 (1958). S~~gge;tion of combination of linuc and sectored hcndmg magnets, similar to preceding tn.0 rcferences. F. T'orreca. "On the Maxirnal Energy antl Intensity of the Electrons .4cceleratctl I)\. the AIicrotron." Nuouo Cimento 11 : 284-286 (1!159). A. Paulin. "Eine lIiiglichk(1it tler Stromintensitiits- und Energieerhijhung cles AIikrotrons." Nuclear Inst. and Methods 5 : 107-1 10 ( 1059). D. K. Aitken F. F. Heymann, R . E. Jennings, and P. I. P. Kalmus, "The Design and Construction of :I 29-XIev hl~crotron."Proc. I'hys. Soc. London. 77: 769-785 (l96l). THEORY OF AG SYNCHROTRONS

E. D. Courant, M. S. Livingston, and II. S.Snyder. "The Strong-Focusing Synclirotroll-A S e w High Energy Accr1rr:~tor." I'h ys. Rev. 88: 1190-1196 (193'12). First paper on sul,ject.-l'hys. Reu. 91: 20'2-203 (1953) letter. Acknowledgnlent of X. Christophilos' unpul)lished study of 1950. E. I).Courant. "A 100-Billion Volt Accelerator." Scientific American, May 1953, 40-45. Popular article, written before the design of the Brookhaven AGS had crystallized. Photos of the Cosrnotrori and Birniinghani machine. L. Lundquist. "Stability of Orbits in a Strorlg-Focusing Synchrotron." I'hys. Rev. 91: 981-983 (1952). Solution of Hill':: equation t12x/d%' $- y ( 0 ) x = 0 and study of thc effccts of straight sections. J . B. Adams, hl. G. N . Hine, and .I. D. L a ~ a o n "Effcct . of Magnetic Inhomogcncitics in the Strong-Focusing Synchrotron." Xature 171: 026-927 (1953). Brief rxl)osition on the seriou~effects of irregularities of magnet construction. G. Liiders. "Theory of I'article Orbits in the Alternating-Gradient Synclirotron." Xuovo L'irnetito Ser. 10, 2, Suppl.: 392-40'2 (19%). Out,line of mat,hematical trcatnient of imperfections. E. l'er.iico. "A Theory of the Capture in a High Energy Injected Synchrorroli." Nuovo Cirnerrto Ser. 10, 2, Suppl.: 459-4G!) (1055). L. L. Golden and I). G . Koskarev. "Synclirotron Oscillation in Strong-Focusi~ig Accelerators." Xuovo ('imrtito Srr. 10, 2: 1251-L3IS (l!)55).

362

THEORY OF AG SYNCHROTRONS, CONT'D

V. V. Vladimirskij and E. K. Tarasov. "On the Possibility of Dispensing with a Critical Energy in an Accelerator with Strong-Focusing." I n Some Problems in the Theory of Cyclical Accelerators. USSR Acatl. of Sciences 13-22, 1955. In Russian Suggested use of reversed-field magnets a t intervals close to wavelength of betatron oscillations. I. F. Oslov. "The Non-Linear Theory of Betatron Oscillations in the Strong-Focusing Synchrotron." Nuovo Cimento Ser. 10. 3 : 252-259 ( 1956). E. Bodenstedt. "Analogue Computer for' the ~iffereniialEquation yN f ( x ) y g(x) = 0." Rev. Sci. Instr. 27: 218-221 (1956).

+

+

I n the CERN Symposium 1956: G. K. Green. "Phase Transition in the A.G.S." Pp. 103-106. K. Johnsen. "Effects of Non-Linearities on the Phase Transition." Pp. 106-109. A. Citron. "Ejection Problems in the CERN A.G. Machine." Pp. 137-130. H. G. Hereward, K. Johnsen, and P. Lapostollc. "Problems of Injection." Pp. 179-191. R . Hagedorn, M. G. N. Hine, and A. Schoch. "Non-Linearities in the A.G.S." Pp. 237253. E. D. Courant. "Non-Linearities in the A.G.S." Pp. 254-261. K. Johnsen and C. Schmelzer. "Beam Controlled Acceleration in Synchrotrons." Pp. 395-403. A. A. Kolomenskij. "Elimination of the Critical Energy in Strong Focusing Synchrotrons." Soviet Physzcs, Technical Physics 1, No. 4: 721-730 (1957). English translation. P. Lapostolle. "La focalisation forte dans les acc616rateurs de particules; les synchrotrons & gradients alternks." L'Onde e'lectriqzte 37: 41-47 (1957). E. D. Courant and H. S. Snyder. "Theory of the Alternating Gradient Synchrotron." Annals o j Physics 3: 1-48 (1958). Highly mathematical theory. P. A. Sturrock. Ton-Linear Effects in Alternating Gradient Synchrotrons." Annals of Physics 3: 113-189 (1958). G. K. Green and E. D. Courant. "The Proton Synchrotron." Handbuch der Physik XLIV. Springer, Berlin, 1959. Pp. 300-335. Theory and detailed information on the Brookhaven and CERN machines. M. Barbier and A. Schoch. "Study of Two-Dimensional Non-Linear Oscillation by Means of an Electromechanical Analogue Model, Applied to Particle Motion in Circular Accelerators." Nuclear Instr. and Methods 5: 211-233 (1959). SPECIFIC AG SYNCHROTRONS BROOKHAVEN

R. A. Beth and C. Labky. "The Brookhaven Alternating Gradient Synchrotron." Science 125: 1393-1401 (1958). Good nonmathematictll general description. G. K. Green. "Brookhaven Alternating Gradlent Synchrotron." CERN Symposium 1959, pp. 347-349. Anonymous. "Brookhaven AGS in Operation." Physics Today 13: 74-77 (Oct. 1960). Intcmtlng photographs. BONN

Anonymous. "The Bonn University Electron Synchrot,ron." CERN Symposium 1956, pp. 482-483. CAMBRIDGE (HARVARD-MIT)

hl. S. Livingston. "The Cambridge Electron Acc~lemtor." CERN Symposium 1956, pp. 439-446; 19.i.7, pp. 335-338.

SPECIFIC AG SYNCHROTRONS, CONT'D

363

A. E Barrington, J. Dcklern, and .I. It. Rees "X-1):md Analogue of the Cambridge Electron Accelerator RF Sv*tem " Bull Amer Ph!ys Soc 11, 4: 269 (1959). K W Rohinson "The RF Syatem of the Cambridge &Rev Electron Synchrotron." Bull Amer Phlls. Soc. I I , 4 : 169 (1959). H 11 S y ~ i i t e r".4ccurate hf,~gncticFicld nncl Field Gradlent Measuring Instruments for Djnamlc Low Fields in a Synchrotron Magnet " Nztclear Instr. 4: 44-49 (1959). C E R N

J. B. A t l m s . '.The Alternating Gradient Proton Synchrotron." N u o v o Cimento Scr. 10, 2, Suppl.: 355-374 (1055). E a d y rrad expo~itionof the AG principle with it? :~l)plicationto the C E R N machine. Comp:irison of CG and AG for the 25-Bet- range. A. Citron and 121. G. N. Hine. "Fkperimentnl F:tcilities of the C E R N Proton Synchrotron." Nzrovo Cime,lto Srr. 10, 2 , Sl~ppl.:375-391 (1!135). Estimates of yield:: of d ) n n c l c a r p:~rticles,rjrction tcrhniqllrl;, :mi focusing schcmcss. C. J. Bnkker. "CERN, E n r o p a n Organization for Nuclear Research." Physics Today, 5,S o . 9 : 8-13 (1955). General hi.~toryof the org:inizntion; parameters of the -4GS. Anonymous. "CI':T<S 2 5 Grv Proton Synchrotron." C E R N Symposium 1959. 111). 3s:3-384. Anonymous. ''CERX's 25 Gcv I'roim Sync~llrotron."hTucleclr E~zergy 14, No. 143: 147-151 (,4pril 1960). .I. 13. Adnms. "The CEItN Proton Synchrotron." Xcrturt, 155: 568-572 (1960). First, operation. Brief cliscu~sion of .4G pinciplc. Comp:trison of cost and performance (existing and cxpected) with other m~~lti-I?ev marhines. 1'. Lnpostolle. "I,e synchrotron k proton:: du CEIIN." L'Onde e'lectrique 40: 489-504 (1960). l'hysicd description. No mnthcmntics. CORNELL

A. Silverman, D. Corson, .J. D(aWire, D. I ~ ~ c k Ry ., JIartin, R. SIcDaniel, R . Wilson, and W. Wootlwartl. "Cornrll illternatilig-(~;r:~tlic.ntSynchrotron." Hull. Amer. P h y . Soc. 11, 1 : S!MO (1956). E. Rlalnmucl and A. Silverm~n."3I:~gnctic h l r : ~ s l ~ r f m e non t s the Cornell AltwmtingGradimt Synchrotron." Nucletcr Irrstr. am' illethotls 4: 02-78 (1950). See :dm V:iriow rcm:rrks in I<. It. Wdson. "Electron Synchrotrons." I l a n d b ~ i c h der I'hysil; XLIV. Springer, Berlin, 15159. 1'11. 171-192; 447-4451. 11. E . .Jackson, I<. L. Martin, and J . Waggoner. "Yclf-Excited 150-kilovolt Resonant Cavity for Operation a t 87 Mcgacyc1c.s." R e v . Sci. I w t r . 30: 187-190 (1959). HAMBURG

W. Jentbchke. "L)eutsches Electroncnsynchrotroa." CERIV Symposium 195.9, Pp. 3 3 342. LUND

E. Smirs and 0. Wernholm "Deslgn Stndy of a Strong-Focuhing Electron Synchrotron." Arkiv fijr Pysik 7 : 463-472 (1954). SOVIET U N I O N

E . G. Kom:tr, N.A. ?*Ionoszon, A. &I. Stolov, V. A. Titov, and V. bI. Shekhter. "Experimental Ring-Shaped 200-(350 AIev Strong-Focusing Proton Accelerator." C E R N S y m p o s i u m 13i6, pp. 387-391. V. V. Vladimir&ij, E. ,J. Komar, and A. L. Xlints. "Project of a Proton lting Accelerator for 7 Bcv." C'ERN Sgtnposium, 1956, pp. 118-121.

364

SPECIFIC AG SYNCHROTRONS, CONT'D

V. V. Vladimirskij. "7 Gev Synchrophasotron." C E R N Symposium 1959, pp. 371-372. V. V. Vladimirskij, E. J . Komar, and A. L. Mints. "Main Characteristics of a Projected Strong-Focusing 50-60 Bev Proton Accelerator." C E R N Symposium 1956, pp. 122-125. Anonymous. "50 Gev Synchro-Phasotron." C E R N Symposium 1959, pp. 373-374. THEORY OF SECTOR-FOCUSED ACCELERATORS

L. H. Thomas. "The Paths of Ions in the Cyclotron." Phys. Rev. 54: 580-598 (1938). The long-neglected work. L. I . Schiff. "On the Paths of Ions in the Cyclotron." Phys. Rev. 54: 1114-1115 (1938) letter. T. Ohkawa. Symposium on Nuclear Physics, Univ. of Tokyo, 1953. Suggestion of reversed fields in alternate sectors. K. R. Symon, D. W. Kerst, L. W. Jones, and K. M. Terwilliger. "Fixed-Field illternating Gradient Accelerators." Phys. Rev. 98: 1152-1153 (1955) abstract. Four abstracts which expound the important elaborations of Thomas's idea. A. A. Kolomenskij, V. A. Petukov, and M. S. Rabinovich. "The Ring Phasotron-A New Accelerator for Charged Particles." In Some Problems i n the Theory of Cyclical Accelerators." USSR Acad. of Sciences, 7-12. 1955. In Russian. Suggestion of oppositely directed fields in alternate sectors. L. C. Teng. "Continuous Injection into Circular Ion Accelerators." Rev. Sci. Instr. 27: 106-107 (1956) letter. Suggested use of the regenerative t,echnique to transfer ions from one fixed-field machine to another.-"Constant Frequency hlult'i-Bev FFAG Accelerators." Bull. Amer. Phys. Soc. 1: 178 (1956) abstract. Possible use of high-order terms in keeping betatron frequencies constant in an isochronous accelerator.-"Linear Theory of Betatron Oscillations in Sectoral Cyclotrons." Rev. Sci. Instr. 27: 1051-1058 (1956). An important theoretical paper. D. F. Dempsey. "Third Order Aberration and Focusing with Sector-Shaped Magnetic Fields." Rev. Sci. Instr. 26: 1141-1145 (1955). D. L. Judd. "Theoretical Study of Relativistic Constant Frequency Cyclotron." Phys. Rev. 100: 1804 (1955) abstract. K. R. Symon, D. W. Kerst, L. W. Jones, L. J. Laslett,, and K. 1 4 . Terwilliger. "Fixed Field Aliernating Gradient Particle Accelerators." Phys. Rev. 103: 1837-1859 (1956). Basic theory of many varicties of FFAG machines. A very important paper. I n C E R N Symposium 1956: P. D. Dunn, L. B. Mullett, T. G. Pickavance, W. Walkinshaw, and J . J . Williams. "Acce1er:ttor Studies a t A.E.R.E. Harwell." Pp. 9-31. D. W. Kerjt, K. R. Symon, L. .I. Lnslctt, L. W. Jones, and K. M. Terwilliger. "FixedField Alternating-Gradient Particle Accelerntors." Pp. 32-35. K. It. Symon and A. M. Sesder. "hldhods of Radiofrequency Acceleration in Fixed Field Accderators with Applications." Pp. 44-58. F. Salzman and A. Roberts. "On the Possibility of Designing an Isochronous Accelerator With Constant Betatron Frequencies." P. 59. D. W. Kerct. "Spiral Sector Magnets." Pp. 366-375. E. M. Moroz and M. S. Rabinovich. "Cyclotron with Sectioned Magnet System." Pp. 547-551.

L. J . Laslett. "Fixed-Field Alternating-Gradient Accelerators." Science 124: 781-787 (1956). Semimathematical exposition.

THEORY OF SECTOR-FOCUSED ACCELERATORS, CONT'D

365

A. R o l m t s . "Sornc l'ropertics of Flare and Gradient Forwing in Iccelerator &id? Fi(1ltlr." Bltll A m t r . Ph?ls. Soc. T I , 2, 10: 240 (1957) alH r:wt . A. A . Kolomenskij. "Accelcr:~tor. with S i ~ n i h rOrl~its." Soviet J. of A t o m f c E n c r w :j, No. 1" ]I:)71-1X8 (]9,57). Eng1i.h tran~1:ition.S i ~ = g c ~rercracd ii direction of fidd in : ~ timxltr l sctltors. N.IT. King an11 It'. Walkinsha\v. .'Spirnl I?idgr I'nrtirlc Dj.n:~~nicsAr)l)lic(l to Convcraion of ihc IIar~vellSynchrocyclotron." Xuclcar Instr. 2 : 287-298 (1958). Wcll worth reading. D. L. Jlitld. "Concept~~nl Atlwmces in Accelrrntors." Annual Rwirw of Nuclear Science 8 : 181-216 (1058). Conrise, lucitl ncrount of the present status and future possilities. G. K. C r w n and E . D. Collrnnt. "Thr Proton Synchroton." Halldhuch der Physili XLIV. Springer, ncrlin, 10,50. I'p. 336-337. F. T . H o w m i , 1';rl. Sector-Focrtsrtl Cyclotrons. Proceetling,q of an Informal Conference, Scnl Islantl, (:eorei:i, Frh. 2-4, 1950. S:rtion:d Aratlr~nyof Scienccx-National Resitarch Coimcil, W:tihington, Il.(l. T'~thlic:~tionOM. NurIe:~rScicnccs Series, IZ~port No. 20. 291 p a p s . $2.50. Filled with interesting itlens and descriptions of presctnt and planned machines. SPECIFIC SECTOR-FOCUSED CYCLOTRONS

R H

Smlth and K I{ LIacKenzie "Three 1'hr.e Iladlc~frerluencySystem for Thomas Cyclotrons" Rev Scz Instr 27 485-490 (I95Gl L Ruby, ill Heuslnhveld, &I .Jakol~.on, and 13 H Smith "Studies 151th a Three-Dre Threr-Phn.e Proton Cyclotron " R t v SCI Insti' 27. 490-493 (1956) E L Kellv, Ii V I'vlc, R . L Thornton, ,I R Richardwn, nnd I3 T Wright "Tno Electron llorlels of a Conqtmt F'rcqurnrj R r l a t ~ l s t ~Cyclotron c " R e v . Scz I115tl 27 493-503 (1956) L VC' Jonei, K 11 Tem~lllger,and R 0 Aaulw "Expcrlmcntal Test of thc Fiwl Flelil Alternating Gr:idlent Principle of I',trtlcle Accrlcr~torD e i g n " R e v Sc1 I n \ t ~ 27 651-652 ( 1956) letter Raili:tl v c t o r ring F T Cole, 11 0 Hauhv, L W ,Jonc,i, C 11 I'ructt, and K M Trr\nll~gcr "Electron RIodel Flxfd F ~ e l dAltc.rn:~tlng Gr,rd~rnlAccelcrator " R w Scz l n ~ t r28 403-420 (1957) Radi:rl icctor rlng K 1 2 Tcrwllger, I, W .Jones, and C H Pructt "Be:tn~ Stnchlng Euprrimcnts In an Elcctron Model FFAG Accrler,rtor " R ( z1 hcz Instr 2 8 987-0q7 (1957) Rach,d sector ring. , E. Rlills, T . O h k a ~ t , D. W. K m t , 13. J . Hausman, R . 0. IIashy, L. J. I ~ d c t t F. F. L. Peterson, A. TI. Srssler, J . N.Snytlrr, nntl IV. A. Wallenmepr. "Operation of a Spiral Srctor Flxctl F~eltlAlternat~ngGratlicnt Arcrlerator." R e v . Sci. Instr. 28: 970-971 (l957\ lrtter. D. \IKcrst., T.E. A. Day, H. J . IIausmnn, R . 0. IIaxhy, L. J. Lasktt, F . E. &lillby T. Ohkaw:t, F . L. I'eterson, E. M. Rowe, A. hI. Sessler, .J. N. Snydrr, and W. ,4. Wallcnmcyer. "Electron Modcl of a Spiral Sector Accelerator." Rev. Sci. Instr. 31 : 1076-1106 (1960). F. A. Hcyn and K. T. Khoe. "Operation of a Radial Scctor Fixed Frequency Cyclotron." R e v . Sci. Instr. 29: 662 (1958) letter. First proton machine of this type. R. E. Worshnm, H. G . Blosser, and R. S. Livingston. "Design Description of Oak Ridge Cyclotron Analogue." Bull. Amer. Phys. Soc. 11, 2: 9 (1057) abstract. H. G . Blosser, R . E. Worshnm, C. D. Goodman, 11. S. Livingston, ,T. E. Ma1111, H. AI. R'Ioseley, G. T. Trammel, and T . A. Welton. "Foiir Sector hzirnuihally Varying Field Cyclotron." R e v . Sci. Instr. 29: S19-834 (195d). hlethod of comln~tingorbits

366

SPECIFIC SECTOR-FOCUSED ACCELERATORS, CONT'D

and corresponding coil positions for the Oak Ridge nonferric isochronous electron cyclotron. V. I. Danilov, Yu. N. Denisov, V. P. Dmitrievskij, V. P. Dzelepov, A. A. Glasov, V. V. Kol'ga, A. A. Kropin, Lu Ne-chuan, V. S. Rybalko, L. A. Sarkisyan, A. L. Savenkov, B. I. Zamolodchikov, N. L. Zaplatin, and D. P. Vasilevskaya. "Starting Up of the Cyclotron with Space Variation of the Magnetic Field." Nuclear Instr. and Methods 5: 335-336 (1959) ; also CERN Symposium 1959, pp. 211-225. S. Chatterjee, A. I . Yavin, and J. S. Allen. "University of Illinois Variable Energy Spiral Ridge Cyclotron." Bull. Amer. Phys. Soc. 11, 4: 417 (1959). J. S. Allen, S. Chatterjee, L. E. Ernest, and A. I. Yavin. "A Variable Energy Spiral Ridge Cyclotron." Rev. Sci. Instr. 31 : 813-822 (1960). I n CERN Symposium 1959: J. A. Zavenyagin, R. A. Metschcherov, E . S. Mironov, L. M. Nemenov, and J. A. Kholmovskij. "1.5 Metre Cyclotron with Azimuthally Varying Magnetic Field." Pp. 225-231. J . A. Martin. "A 450 kev Eight-Sector Fixed-Frequency Electron Cyclotron." Pp. 205-2 11. R. 0. Hsxby, L. J. Laslett, F. E. Mills, F. L. Peterson, E. M. Rowe, and W. A. Wallenmeyer. "Experience with a Spiral Sector FFAG Electron Accelerator." Pp. 75-81. 75-MEVOAK RIDGE ISOCHRONOUS CYCLOTRON (0 R I C ) R. S. Livingston and F. T. Howard. "The Oak Ridge Relativistic Isochronous Cvclotron: Part I. ~ e v e l o p n e n tof the Isochronous ~yciotron." Nuclear Instr. and " ~ e t h o d s 6: 1-25 (1959). B. L. Cohkn, H. G. Blosser, E. D. Hudson, R. S. Lord, and R. S. Bender. "Part 11. Magnetic Field Design for the Isochronous Cyclotron." Nuclear Instr. and Methods 6: 105-125 (1960). M. M. Gordon and T. A. Welton. "Part 111. Analysis of Ion Orbits in the Isochronous Cyclotron." Nuclear Instr. and Methods 6: 221-233 (1960). R . H. Bassel and R. S. Bender. "Addendum. Some Recent Results of Orbit Studies." Nuclear Instr. and Methods 6: 234-237 (1960). INTERSECTING BEAMS

D. W. Kerst, F. T. Cole, H. R. Crane, L. W. Jones, L. J . Laslett, T. Ohkawa, A. M. Sewler, K. R. Symon, K. M. Terwilliger, and N. Vogt-Nilsen. "Attainment of Very High Energy by Means of Intersecting Beams of Particles." Phys. Rev. 102: 590-591 (1956) letter. D. W. Kerst. "Properties of Intersecting-Beam Accelerating Systems." CERN Symposium 1956, pp. 36-39. G. K. O'Neill. "The Storage Ring Synchrotron." CERN Symposium 1956, pp. 64-67. -"Storage-Ring Synchrotron: Device for High-Energy Physics Research." Phys. Rev. 102: 1418-1419 (1956) letter. V. A. Petukhov. "Concerning the Possibility of an Experimental Investigation of the Structure of the Electron." Soviet Physics J E T P 5: 317-319 (1957). English translation. Embodies the suggestion of a two-way accelerator for electrons by the use of a sectored magnet with alternately reversed fields. T. Ohkawa. "Two-Beam Fixed-Field Alternating Gradient Accelerator." Rev. Sci. Instr. 29: 108-117 (1958). Independent similar suggestion, with orbit theory given in detail.

INTERSECTING BEAMS, CONT'D

367

A. A. Kolomenskij. "A Symmetric Circular Synchrotron with Oppositely Directed Reams." Soviet Physics J E T P 6 : 231-233 (1958). English translation. G. K . OINeill and E. J. Woods. "Intersecting-Beam Systems with Storage Rings." Phys. Rev. 115: 659-668 (1959).

I n the C E R N Symposium 1959: G. K. O'Neill. "Experimental Utilization of Colliding Beams; St'orage Rings for Electrons and Protons." I'p. 23-28, 125-136. K. R. Symon. "The MURA Two-way Electron A c c e l e r a t ~ r .I'p. ~ ~ 71-74. V. N. Kannunikov, A. A. Kolomenskij, A. N. Lebedev, E . P. Ovchinnikov, A. M . S ~ O ~ O V , V. A. Titov, -4. P. Fateev, and B. N. Yahlokov. "Investigat,ions Connected with the Design of ilccelerators of the Ring-Phasotron Type." Pp. 89-99. )I. Barbier, F . A. Ferger, E . Fischer, P . T. Kirstein, G. L. Munday, M. Morpurgo, RI. .J. Pentz, A. Schoch, A. Sushi, and N. Vogt-Nilsen. "Studies of an Experimental Beam-Stacking Accelerator." Pp. 100-114. A. A. Kolomenskij and A. N. Lebedev. "Certain Beam-Stacking Effects in Fixed-Field hIagnetic Systems." Pp. 115-134. A. Schoch. "A ~~iscusnion of Colliding Beam Technicpcs." Nucleur Instr. and Methods 11: 10-46 (1961). LINEAR ACCELERATORS (TENERAL DISCUSSIONS A N D THEORY

J C Slater "The Des~gnof Linear Accelerniors " Phys Rev 70: 799 (1946) abstract Good bnef statement of the problem J C Slater T h e Design of Lmear Acct4erator5" Rev Mod. Phys 20: 473-518 (1948). Lengthy review, ~ l t h o u much t mathcnlat~ci L Rrillouin "Waves and Electrons Travelling Together-A Compmhon between Travelling Wave Tubes and Lmear Acceler:~tors" Ph ys Rev 74: 90-92 (1948) D W Fry and W. Walk~nshaw"Lmear Accelerators." Reports on Progress zn Phystcs 12. 102-132 ( 1949) Very readable discuwon on standing- and traveling-ware clectron linnrs antl Alx arez t \ pe Man1 references on enrller and abandoned schemrs E RI hlc>I~llan "The Relation Betnee11 l ' h a ~ e Stablllty and First Order Focu-in% in L ~ n p a r .4cceler:1tor\ " Ph ys Rev 80 4!U (1050) General proof of the inroinpat~bilityof 1)h:l.e st:rbil~tyantl tmn\\cr.e focumg a ~ t hopen-cndecl dnft tulx,i 1) W Fr) "The L I ~ P Elcc,tron J~ Accrlernior " P h h p s Tech Rev 14: 1-12 (19.52) Excellent gencxraldcscr~ption J C Slater "Tmear accelerator^ " Annual Revatw oj N ~ ~ c l e uSctence r 1 : 199-206 (1952) I l l u m ~ n a t ~ ndiscuwon, g n ~ t h o u t~nathematics .J P Blenett "Ilad~alFocus~ngIn the Llne,lr Accelerator " Phys Rev 88: 1107-1 199 (1952) ,4nal\ -1s of a very general huac, a ~ t han c,lpositlon of the focubing propertley of qundrupolt, lens+ u ~ t h~~ r eli n K .Johnien Orr the Theory of tho L711cwrA c c d ~ r a t o rChr Ill~chclhensInst. f. V~tlcnsk:rp og ,4ncl~frihet, 1951 135 p'rgr- Vcry thorough ant1 rlgorolli trratment I T P:rnof\h\ '"The I m e n r A c c ~ l e r , ~ t"o rSelcnt~ficA n ~ e r ~ c a Ort n 1954, 40-44 Pop~llnr artlclr on proton and electron I~il,lr*I n t ~ r c ~ - t m~g) l i o t o g r ; l p l ~ T G lt'icknxanc7e "Proton line:^ Accelerators for Nuclear Ilevxrch and thrl A E I< E 600-Xlrv l'rolect " Xuo~loPrmrnto Ser 10, 2 413-422 (1955) General rompnriion of FA1 c ~ c l o t r o nand 11n:rc D i w ~ s ~ ooln focusing and RF problem- of the 1:rttci lieaclabk. Llovd Srn~tll."Linear Accclerator~." Hundhuch der Physik XLIV. Springer, Berlin, 1959, Pp. 341-389. Thorough treatment.

R. Wideriie. "iiber ein neues Prinzip zur Herstellung hoher Spannungen." Arch. Elelctrotrch. 21 : 387-406 (1928). E. 0. Lawrence and D. H. Sloan. "The Production of High Speed Canal Rays Without the Use of High Voltage." Proc. N u t . Acad. Sci. 17: 64-70 (1931). 10 kv applied to 21 drift t~ihesto produce Hg+ at 200 kev. D. H. Sloan and E. 0. Lawrence. "The Production of Heavy High Speed Ions Without the Use of High Voltages." Phys. R e v . 38: 2021-2032 (1031). Hg+ a t 1.26 Mev by use of 30 drift tulxx D . H. Slonn and W. M. Coates. "Ileccnt Advances in the Production of I-Iigh Speed Ions Without the Use of High Voltages." Phys. R e v . 46: 530-542 (1934). IIg+ at 2.85 hlev; 36 drift tubes. B. B. Kinsey. "Attempts at Disintegration Using Lithium Ions." Phys. R e v . 50: 386 (1936) abstract. Wideroe linac to produce I hlev Li+ ions. ALVAREZ LINACS L. W. Alvarez. "The Design of a Proton Linear Accelerator." Ph,ys. R e v . 70: 799-800 (1946) abstract. Brief, clear presentation of planned machine; parameters. G. Gabor. "St:ibilizing Linear Particle Acceler~~tors by Means of Grid Lenses." Nature 159 : 303-304 (1947). L. W. Alvarez, 11. Bradner, J. V. Frank, H. Gordon, J. D. Gow, L. C. Marshall, F. Opprnheimer, W. K. H. Panofsky, C. Richman, and J . R. Woodyard. "Berkeley Proton Linear Accelerator." R e v . Sci. Instr. 26: 111-133 (1955). Details of the 32-Mev proton machine. B. Cork. "Proton Linear-Accelerator Inject,or for the Bevatron." R e v . Sci. Instr. 26: 210-219 (1955). Details of the 10-Mev machine. N. C. Christofilos. "Calculations of Drift Tube Shapcs for Linear Accelerators." Bull. of Computation of Amer. Phys. Soc. 11, 1, No. 6: 290 (1956) abstract.-"Method Drift Tube Shapes." C E R N Symposium 1936, pp. 176-178. H. B. Knowles. "Energy Stabilization of the Bcrkeley Proton Linear Accelerator." R e v . Sci. Instr. 29: 130-136 (1958). E. A. Day, R . P. Featherstone, L. 13. Johnston, E . E. Lampi, E. B. Tucker, and J. H. Williams. "Minnesota lo-, 40-, and 68-blev Proton Linear Accelerator." R e v . Sci. Instr. 20: 457-476 (1958). Rlechanicnl and electrical details. W. W. Salisbury. "The Resnatron." Electronics 19: 92-97 (Feb. 1946). Progenitor of the tetrodes which drive the Minnesota linac. E. Zacchcroni. "The 2.5 M w H. F. Amplifier of the CERN Linear Accelerator." Nuclear Instr. and Methods 5 : 78-89 (1959). W. A. S. Lamb and E. J. Lofgren. "High Current Ion Injector." R e v . Sci. Instr. 27: 907-909 (1956). Continuous currents of protons or deuterons of 0.75 ampere, or pulsed currents of 2 ampere, for linacs. J. P. Blewett. "Linear Accelerator Injectors for Proton Synchrotrons." C E R N Symposium 19.i6,pp. 159-166. HEAVY I O N LINACS C. E. Anderron and K. W. Ehlers. "Ion Source for the Product~onof Multiply-Ch:~rged Heavy Ions." K a v . Sci Instr. 27 : 809-816 (1956). 1t. Beringer, and W. Itall. "Mercury Vapor ,Jet Target and Stripper." R e v . Sci. Instr. 28: 77-79 (1957).

HEAVY I O N LINACS, CONT'D

3ti9

A. A . Plyntto, K . 1;.Kervnlirlxe, and 1. F. Kvnrtskhnv:~.*.Sp:lrk Sourrc for RIultiplyCh:tracrl Ton;." So1'ir~tJ. Atomic h'l~f,rg!/3 , N o

S:

!W-!)2S (I!).57). English transl:~-

ELECTRON LINACS EAR1.Y EFFORTS

G. hing. '(l'rinzip cinrr IIethodc z ~ l rI-Trrstl~ll~~ng vnn Knnnlstrnhlen hoher Voltznhl." Archiv fiir mntematili. n s t r o ~ ~ o moi fysik 18, KO. 30: 1-4 (1924). Suggrstion of sending voltngc pillar to succrs4ivc~tlrift t11l)cs. -1.K.13r:rnls :rut1 11. Trottclr. "Thr Arcrlcr:rtion of Rlcctrons to High Energies." Ph!ls. Rev. 45 : 849-850 ( 10334) letter. I .X-1Icv electrons. L. 13. Snodtly, I i . Trottc.r, IT. H : m , : ~ n d.I. K. Ek:~ms."Impulse Circuits for Ohtsirting :LTime Srpnr:tiion Iletn-(sen t h Appt~:imncr ~ of I'otlmti:~l nt Diffrrent Points in :I Circuit." /. F ~ a ~ t l i l Trrst. i n 2 3 : 55-76 (1!)37 1 . T H E O R Y O F TRATF:I,Ih-(;

\i7A\E

LINACS

W.W. Hnnwn. '.A T y p r of Elcrtric:d Rewl~:ttor." .I. A p p l . Phys. 9: 654-663 (1935). C':rlcul:ttions on cylinders. l)rixnw, sphrrcs. E. I,. Chn :nltl IT. IT. H:nlwn. "Tl~eoryof Disc-Lonclrd JTT:~vrg~~ides." J. A p p l . I'hys. IS: 996-1008 (l!Mi). W. W. Hansen and I:. F. Post. '.On the l\le:ts~~rrment of Cn\.ity Impcdancr." J. Appl. P h y s . 19: 1059-1061 (1948). IT. Hnnsen, and W. I<. Kennedy. '(-Linear Electron Acrrlcrntor." E. L. Ginzton, R O I ISci. . I r ~ s t r 19: . 80-108 (1948). E . L. Chu, aritl W.W. Hanien. "Disc-Loaded W:L\-cGuide-." J. Appl. I'hys. 20: 280-'ST,

R P I )81: . fi55 (1!151) ahtr:lct. R . R. Seal. "D17sign of Linear Elwtron A(w+rtltors with 1lc:lm Landing." J. Appl. Phys. 29: 1019-1024 (1958). E. S.Akeley. "Study of a Certain Type of I
370

ELECTRON LINACS, CONT'D

G. A. Zeitlenok, V. V. Rumiantsev, V. L. Smirnov, L. P. Fumin, V. K. Khokhlov, I. A. Grishacr, and P. hI. Zeidlits "Choice of Rwic Parnmeterc for High Energy Lincnr Electron Accelerators." Soviet J. Atomic Enerq?! 4, No. 5: 583-580 (1058). Engl~sh translation. J . C. Nygard and R . F. Post. "Recent Advances in High Power R/Iicrowave Electron Accelerators for Physics Research." ATnclear Instr. altd Mrthods 11: 126-135 (1961). SPECIFIC TRAVELING WAVE ELECTRON LINACS GREAT BRITAIN

D. W. Fry, R. B. R.-Shersby-Harvie, L. B. Mullet, and W. Walkinshaw. "Travelling Wavc Linear Accelerator for Electrons." Nature 160: 351-352 (1947). 580-krv elrctrons. D. W. Fry, R. B. R.-Shersby-Harvie, and L. B. Mullet. "A Travelling Wave Linear Accelerator for 4 Mev Electrons." Nature 162: 859-861 (1948). R . B. R.-Shersby-Harvie and L. B. Mullett. "A Travelling Wave Lincar Accelerator with R. F. Power Feedback and an Observation on R.F. Absorption hy Gn* in Presence of a Magnetic Field." Proc. Phys. Soc. London, R60: 270-271 (1949). K. Johnsen. "Heavy Beam Loading in Linear Electron Accelerator." Proc. Phys. Soc.. London. B 64: 1062-1067 (1951). C. F. Bareford and M. G. Kelliher. "The 15 Million Electron Volt Linear Accelerator for Harwell." Philips Tech. Rev. 15: 1-26 (1953). C. W. Miller. "Traveling Wave Linear Accelerator for X-Ray Therapy." Nature 171: 297-298 (1053).

J . Rotblat. "The 15 Mev Linear Accelerator at St. Bartholomew's Hospital." Nature 175: 745-747 (1955). Parameters and performance. STA hTFORD

G. E . Becker and D. A. Caswell. "Operation of a Six Mev Linear Electron Accelerator." R e v . Sci. Instr. 22: 402-405 (1951). Mark I. R. F. Poet and N. S. Shirer. "The Stanford Mark 11 Linear Accelerator." Rev. Sci. Instr. 26: 205-209 (1955). 40 blev. M. Chodorow, E. L. Ginzton, W. W. Hansen, R. L. Kyhl, R. B. Neal, and W. K. 11. Panofsky. "Stanford High-Energy Linear Electron Accelerator hlark 111" Rev. Sci. Instr. 26: 134-204 (1955). 700 Mev. Engineering data and considerable theory of beam dynamics. 11. B. Neal and W. K. H. Panofsky. "The Stanford Mark 111 Linear Accelerator and Speculations Concerning t h Multi-Bev ~ Application of Electron Linear Accelerators." C E R N Symposium 1956, pp. 530-546. R. B. Neal. "Stanford Two-Mile Linear Electron Accelerator." C E R N Symposium 19.7.9. pp. 349-359. LIVERMORE

N. A. Austin and S. C. Fultz. "22 Mev Electron Linear Accelerator." Rev. Sci. Instr. 30: 284-289 (1959). SOVIET UNION

0. A. Valdner, 0. S. Milovanov, G. A. Tyagunov, and A. V. Shalnov. "4.5 Mev Linear Electron Accelerator." Soviet J. Atomic Energy 3, No. 7: 771-775 (1957). English translation.

SPECIFIC TRAVELING WAVE ELECTRON LINACS, CONT'D

37 1

K. D. Sinc1nikn~-.I. .I. (:ri,. of Scienrcs of the U h i n i ; r n Soviet Socialist Republic." CiEIZLVS~~tripcrsium l!M?.PI).875-:382. STANDING WAVE ELECTRON LlNACS

12, ,J, I,:~\vtt~n :III(I\T, C . 11:1hn. ~ ~ ~ ~ s ~ ~ c I. DIELECTRIC-LOADED GUIDES

QUADRUPOLES

E. I). Co11r:rnt. 11. 8.Livingston, : I I I ~11. S. Sny11(\r..*1'1teStrong-Foc~lsingSynrlirotroi~ -A Now High Energy hrcclcr:itor." P h y s . 88: 11!)0-I 196 ( 1!)5'1). Inclutlcs first -~~ggcstion and nn:ilysi:: of q11:1tlrl1l)olo Irnsrs. .T. 1'. Rlc\v(>tt."Ratlial Focusing in the Lir1e:rr Arrclrraf or." f'h !is. Rev. 88 : ll9i-ll9O I<('zl.

( 1052).

I,. (:. Trng. ".ilternating Gr:ldient Electrostatic Focn+ing for Li11c:tr Aci&nrtor~." RW. k r . I n s t r . 25: 264-268 (1954). F . R . S h ~ ~ lC. l , E. llacF:~rl:rntl, and RI. N . Rrc~tselicr."Conrcntration of :I Cyclotron Ream by Strong-Focusing Lrnws." Rc7). Sci. Irrstr. 25: 364-3(i7 (1954). I\'. C. Elrnorf,, RI. W. Garrett, I. EL Dayton, F . C. Shoemakrr, and R . F. Mozlej.. .'RIcns~~rcrncnt of Two-Dimensional Fieltls." Reu. Sci. f n s t r . 25: 480-489 (1954). I!. L . 1luhb:rrtl tint1 E. L. Krlly. "Alternating Grrttlient Focusing of ;L Cyclotron Estern:ll Rcan~."Rt~11.Sci. Irrstr. 25: 737-730 (1054). Slichrl-Yvcs T3rrn:rrd. "La fo~alizationforte (lam lee :rcckl6ratcurs linkaires d'ions." drrriulcs tic physique 9: G33-(jS'-! (1054).

3 72

QUADRUPOLES, CONT'D

M. L. Bullock. "Electrostatic Strong-Focusing Lens." Amer. J . Phys. 23: 264-268 (1955). Lloyd Smith and R. L. Gluckstern. "Focusing in Linear Ion Accelerators." R e v . Sci. Instr. 26: 220-228 (1955). Theoretical analysis of focusing by the use of quadrupoles, grids, or solenoids. W. F. Stubbing. "Alternating Gradient Channel Using Permanent Bar Magnets." R e v . Sci. Instr. 26: 666-671 (1955). H. G. Hereward and K. Johnsen. "Alternating Gradient Focusing in Linacs: Computational Results." C E R N Symposium 1956. pp. 167-175. A. D. Vlasov. "Alternating-Gradient Focusing in Linear Accelerators" Soviet J. Atomic Er~ergyYo. 5: 687-694 (1956). English translation. Use of matrices in solution of the problem. P. Levy. "Abaque pour la resolution d'un probl6me de lentilles BICctrique ou magnetique gradient altern6." J. phys. rad. 17: 60-4-61A (1956). P. J . Lynch and D. J. Zaffarano. Tests and Analysis o f Magnetic Quadrupole Lenses. Ames Laboratory, Iowa State College, ISC-927, Aug. 1957 (OTS). This report is the basis of the exposition given in the present volume. K. N. Stepanov and A. A. Sharshanov. "Strong-Focusing in Linear Electron Accelerators." Soviet J . Atomic Energy 2, No. 2: 202-204 (1957). English translation. Calculation of required gradients in auadruvoles. H. Schneider. "kn A.G. Channel with Quadrupole hhgnets." Nuclear Instr. 1: 268-273 ( 1957). H. A. Enge. "Ion Focusing Properties of a Quadrupole Lens Pair." R e v . Sci. Instr. 30: 248-251 (1959). L. N. Hand and W. K. H. Panofsky. "Magnetic Quadrupole with Rectangular Aperture." R e v . Sci. Instr. 30: 927-930 (1959). D. Cohen and A. J. Burger. "Acceleration of Polarized Protons with Strong-Focusing Linear Accelerators." R e v . Sci. Instr. 30: 1134-1135 (1959) letter. J . Roscnblatt. "Design of Alternating Gradient Quadrupole Lenses." Nuclear Instr. and Methods 5: 152-155 (1959). Graphical method of designing lenses when position of object and image are known. P. Grivet and A. Septier. "Les lentilles quadrupolaires magn8tiques." Nuclear Instr. and Methods 6: 126-156.243-275 (1960). A. Septier. "Lentilles quaclrupolaires sans fer B. gradient constant." Nuclear Instr. and Methods 7: 217-218 (1960). A. Septier. "Les lentilles magnetiques quadrupolaires sans fer." J . phys. rad. 21, Suppl. au no. 3: 1A-15A (1960).

our

STOCHASTIC ACCELERATION

E. L. Burshtein, V. I. Veksler, and A. A. Kolomenskij. "A Stochastic Method of Particle Acceleration." Certain Problems o f the Theory of Cyclic Accelerators. USSR Academy of Science>,3-6, 1055. R. Kellcr and K. H. Schmitter. Beam Storage with Stochastic Acceleration and I m provement o f a Synchrocyclotron Beam. CERN Internal Report 58-13. Dec. 1958. R. Keller, L. Dick, and M Fidecaro. "Acc6leration stochastique dans un cyclotron de 5 Mev." Comptes rendus, Acad. des Sciences 248: 3154-3156 (1959). R. Keller. "Le mouvement anharmonlque des ions dans un cyclotron et le diagramme de phase." Nuclear Instr. and Methods 4: 181-188 (1959). A. A. Kolomenskij and A. N. Lebedev. "On the Theory of the Stochastic Method of Particle Acceleration and Beam Stacking." C E R N Symposium 1%9. pp. 184-187.

STOCHASTIC ACCELERATION, CONT'D

373

R . Krllrr "Eup~rlment- on Stocha-t~c .?\rcclerntion" ( ' E R N Symposium 1,959, pp 187-192 31 Rarhwr ''P<~wageOver T m n q ~ t ~ oEncrgy n By 3Iran. of Stochait~cAcccler.1tmn " CERX S ~ / m p o s ~ u r19n 79. j'p 03-639 A. Cabreep~ne."Methodci prrn~ettnntd'ambl~orcrla \ t r w t n r r en temps du faiswiu externe du synchrocyclotron " J . phys. rad. 21: 332-337 (1960).

A U T H O R INDEX FOR I3IB1,IOGRAPHY AND REFERE.NCES

13~~~tin., .J Ir , 369 ( 2 ) Hcrhu, C . E , 370 I3cl11ri,in,G A,, 337 (2) I3el1, J S , Xjl, 371 13erltlcr, It. S., 347, 3GG ('2) Jici~nc~tt, W S , 340 Berg, (2 E , 359 C' , 353 IZerg:~rn:l~cll~, I3erin~yr,I?., 3% lkrlin, T EI , 344 ( 3 ) , 3(i0 Ikrn:irtl, l i l c h ( + - Tca, ~ 372 l k r ~ i u ( l ~ nC'., i , 359 (2) J3ernc,t, E ,J , 339 Ikth, lt A4, 36'2 Beth(,, R.A , 343 H ~ r r n l mA, , 359 l3izzar1, U , 359 (2) 131xl~rr1:m,N 11 , 344, X 0 I3lanlcy, ,J. W , 357 131~~1r*, tJ., 340 Ulcvett, J . I' , 342 ( l l i , :is5 ( 1 ) ,356, 356 ( 2 ) , 3ti0, 367, 30S, 371 Rlcnctt, 11 JI , 3.12 I3loon1, S 11 , 350 Blosor, H G , 349, 365 ( 3 ) , 366 130~41,.l L , 347 I3oclk, 11 ,347 Hothbtedt , E , 312 l3ohm. I ) , 343 ( 2 ) Boll6c~.13 . 1153

376

AUTHOR INDEX

Breit, G. PIT., 340 (2) Bretscher, hI. M., 371 Brillouin, L., 367, 360 Brobeck, W. >I., 345, 351, 356 (2) Brockman, F. G., 356 Bromley, D. A., 348 Bronca, G., 356 Bruck, G. C., 371 Bruck, H., 356 (2) Briickmann, 13, 353 Brun, J. C., 353 Bruner, J. A,, 348 Buechner, W. W., 338 Bullock, M. L., 372 Burger, A. ,J., 372 Burgerjon, J . J., 347, 352 Burkig, J. W., 352 Burshtein, E. L., 372 Bush, R. R., 352 Cabrespine, A., 353, 373 Calame, G., 354 Cameron, J. R., 340 Caplan, D., 369 Caro, D. E., 347 Carrelli, A., 361 Carruthers, R., 358 Cassells, J. AT., 352 Caswell, D. A,, 370 Chaffee, M. A,, 340 Charlton, E., 359 Chatterjee, S., 366 (2) Chernyak, L. L., 369 Chestnoi, A. Y., 353 Chew, G. F., 341 Chodorow, M., 370 Christofilos, N. C., 368 (2) Chromchenko, L. M., 346 Chu, E. L., 341,369 (2) Chubakov, A. A., 347 Citron, A,, 362, 363 Clark, A. F., 352 Clark, D., 356 Coates, W. M., 368 Cockcroft, J. D., 330 ( 2 ) , 341 (2) Coe, R. F., 355 (2) Cohrn, B. L., 337,342,344,36G Cohen, D., 372 Cohen, S., 354 (2) Cohn, G. I., 371 (2) Colby, N. C., 345

Cole, F. T., 365, 366 Collins, G. B., 352, 356 Collins, L. E., 340 Compton, K. T., 339 Cooksey, D., 345 (2) Cool, R., 356 Cooper, P. F. Jr., 354 Corazza, G., 359 (2) Cork, B., 356,368 Corson, D. I?.,345,363 Cortellessa, G., 359 Courant, E. D., 342, 344, 355, 360 (2), 361 ( 3 ) , 362 (3), 365,371 Cowie, D. C., 349 Crane, H. R., 344,358,366 Creutz, E. C., 352 Crewe, A. V., 354 ( 5 ) , 357 Crittenden, E. C. Jr., 350 Crosbie, E. A., 344,357 Cross, W. G., 60 Crowe, K., 354 Cushman, B. E., 337 Dahl, O., 339, 340 Dain, J., 358 (2) Danilov, V. I., 366 Davenport, L. L., 352 Day, E. A., 365,368 Dayton, I. E., 371 Debraine, P., 347 de Groot, W., 341 Dekleva, J., 363 Delbecq, C. J., 349 Demos, P. T., 371 Dempsey, D. F., 364 Dempster, A. J., 340 Denisov, Yu. N., 366 Dcnnison, D. &I.,344 (3) De\i:trdins. ,T. S., 353 De Wire, J., 363 Diambrini, G., 359 (2) Dick, L., 372 Dixon, G. W., 358 Dmitriev, P. P., 350 Dmitrievskij, V. P., 366 Dorhring, A,, 347 Doran, G. A,, 355 Durin, P. D., 364 Dunning, J . R ., 348 Du Toit, J. J., 347 Dzelepov, V. P., 353 (2), 366

AUTHOR I N D E X

377

Edlefqrn, N F , 345 E t h m l s , I?F , 353 Cff'it, K TC 2 , 351 Efrcrliox , 1) Y , 35'3, 357 Ii:gglnton, A $1, 358 Elllerk, K If7, .iOS Eltlcr, F I? , X h , 360 Elllott, D , 357 Elmore, IT C , 3 7 1 Enge, I T .L\ ,372 Engel+herg, S ,354 Ercmenko, E V , 3 7 1 E r n e i t , L $2 , 3Mi Fnrl) , G 1 2 ,353 Fnr\%cll,C RT, 347 F a t r r v , A P , 367 Fr,~thcr-tone,I< P , 368 , 342, 347,350 ( 2 ) Fedorov, S I) Fctlotov, G A1 , 357 Fcrentz, R I I , 3.57 F r r g r r , F A , 367 Fcrguhon, R I S , 361 F e r r e t t ~R , ,359 Fidecnro, ;\I ,372 F i n l a r , E A , 355 Fischer, E , 367 Fle.hcr, G T , 3 7 1 (2) Foldv, L , 344 (2) Force, R ,J , 850 1, ,338 Forteqcur, I? Foss, ,.?I\ 352, 357 Fox, .J. G., 353 Frank, J. V., 368 Frank, N. 11.. 1144 (2) Frankel, S., 371 G., 346 Franklin, I?. Frendin, . l . H., 341, 350,351 (2) Friedlandcr, G., 356 Frost, F. E., 337 F r y , D. W., 352,358 ( 2 ) ,367 ( 2 Fulhright, H. If7.,348 ( 2 ) ,352 Fultz, S. C., 370 F~lmin,L. I'., 370 Gnhor, G., 368 Gallop, J . W., 347, 358 Galvin, J. A,, 348 Garrett, 1 1 . R.,371 Gnrlvin, E. I.., 360 G a r h ~ v31. , A,, 357 (2) (hyrnncl, IZ., 3.53

Grwrr, E'. I,.,346 G r t w l , G . K., 3.22, 345, 355, 356, 362 (31,

Hnf+tnd, L. 11 , 339, 340 H:iqvlorn, 1: , 362 I I a h n , IT7 C., 371 H:111wrn, I , 337, 355 11:11n,W., 369 H : n m ~ l ~ nJ , , X56 H : ~ n w r n w - l ~11 , ., 344, 357 II:lnlllton, D. It., 354 I I : m n w r , C. L , 344 Hand, L. N ,372 fI:in.-cn, ll . IT' , 369 (5), 370 11:1rr1\, W.E., 359 I I : l r \ I[., R B., I?.-Shcr4y-,

Mnvcns, 15'. W. Jr., 350, 353 ITaworth, L. J . , 356 Haxhy, I<.0..365 (4), 366 I~I:l!.arhi, I., 348 (2), 354 fIa!.w, .4. E. Jr., 349

378

AUTHOR INDEX

Henderson, C., 361 (2) Henderson, ,I. E., 347 Henderson, 11.C., 345 Henderson, W. J., 346,360 IIenrich, L. It., 351, 353 Hcras, C. A,, 354 Herb, It. G., 339 (5) IIereward, H. G., 362, 372 Hernandez, H. P., 348 Her~sinkveltl,ill., 365 Heymann, F . F., 361 (31 Hcyn, F. A., 352 (3), 365 Hibbard, L. U., 355 (2) Hide, G. S., 354 Hine, h1. G. N., 361, 362,363 Holloway, hI. G., 348 Hough, P. C. V., 344 Howard, F. T., 338 (2), 343,365,366 Howe, J . D., 346 Hubbard, E. L., 352,371 Hudson, C. M., 339 Hudson, E. D., 347, 366 Hutchinson, G. W., 355 Ignatenko, A. E., 353 Ikeda, H., 348 Imnizumi, M., 348 Ioffe, RI. S., 349 Irwin, F., 349 Ising, G., 369 Isoys, A., 348 Ito, S., 348 Ivanov, P.P., 353,357 Iwanenko, D., 360 Jackson, H. E., 363 Jakobson, M. J., 349 (21,350 (2), 365 Jiinecke, J., 347 Jenkins, F. A,, 349 Jennings, R. E., 361 (3) Jensen, H. H., 355 Jentschke, W., 363 Johneen, K., 362 (3), 367,370, 372 Johnston, L. H., 368 Jones, L. W., 364 (31,365 (3), 366 Jones, R.J., 346 (2), 349 (2) Jones, W. B., 358 Judd, D. L., 342,364,365 Julian, F. A., 340 Kaiser, H. F., 359 (2), 361 Kaiser, T. R., 358 (2)

Knlinin, S. P., 347 IC:dmus, P. I . P., 361 Kannunikov, V. N., 367 Karasawa, T., 348, 353,354 Kxssner, I)., 356 K:rtishev, V. S., 353 Kcllcr, 11., 372 (31, 373 Kelliher, ?\I.G., 370 Kelly, E. L., 354, 365, 371 Kennedy, W. R., 369 Kerns, Q. A., 353 (2) Kerst, D. W., 339 ( 2 ) , 343, 359 ( 8 ) , 364 ( 4 ) , 365 ( 2 ) , 366 (2) 369 Kervalidze, K. N., Khaldin, N. N., 347 Khaprov, E. N., 350 Khoe, K. T., 365 lihokhlov, V. K., 3'70 Kliolmolvskij, J . A,, 366 Kikuchi, K., 348 Kikuchi, S., 348, 354 King, L. P. D., 346 King, N. W., 365 Kinsey, B. B., 368 Kipp, A. F., 371 Kirstein, P. T., 367 Kitaevskij, L. Kh., 371 Knecht, O., 347 Knowles, 1.1. B., 368 Kobayashi, S., 348 Kwh, H. W., 359 (2) Korhlcr, 4 . M., 354 Koestcr, L., 347 Kokame, J., 350 Kol'ga, V. V., 366 Kollath, R.. 341 Komar, E. G., 353, 357 (4), 363 (21, 364 ECondrashov, L. F., 347 Kondratenko, V. V., 371 Konmd, M., 350 Kornblith, L., 352 Koskttrev, I).G., 361 Kostin, RI. D., 360 Kowarski, L., 343 Krasnov, N. N., 350 Kratz, K. R., 358 Krause, E . H., 337,338 Kraushaar, W. I,., 337, 35s Krienen, F., 353 (2)

1.:11111),IT. A. S., Xis I,ani~)i,E. E., X i S I,nngc.\-in-.Toliot . I I., 353 I,nnymuir, I<.V., :33iS, :i5S, Xi0 1,:1nzl, 1,. H., 359

L~q~ostollc, l',, 3x2 (2 ) , 3m L:wky, C., 362 T,nclrtt, L. .J., Xi4 ( 3 ) , Xi6 (2 1 .

(I! 1 I.:lv:lt?lli, L., XI2 I . : ~ ~ v r r n cE. r , O., 341, 3 5 ( S ) , .'k5 1 WS ( 2 1 L:l\vwn, ,I. I)., X A ( I ! 1 , X i 1 :GI I i ~ w t o nE. , Lct)c.tlcv, A. S.,XO, ii(i7 (?), :373 1.r (':1i11c3 H ., 3(jO I,(, Coutcw, Ti. .J., 3.54 ( 5 ) I,ctllcy, li., 3355 IAT, It. \IT., 345 T.t;ri-M:mtlcl, I:., 356 I x v y , I)., 37" I . ( ~ i i sE. , L., 371 I,intistrom, C., 346 (2)

*r.,

I.ink, TT'. T., 351 I ,illkin, H. J . , :<54 Lipton, S., 354 It., 33% 1,itt:111~r, I.ivingootl, .I. J . , 340, 345, 357 Livingston, hI. S., : M I , 341 (2), 343, 344 (31, 3 5 ( ( j ) , M i , :HS, M!), 351, 355, 356, 361 ( 2 ) ,3 2 , 371 Livingston, I:. S., 34", 3% (,'3), 349, ,365 ( 2 ) ,366 I,onrli, B . G., 369, 371 T,ofgrrn, E. -7.: 351 ( 2 i , 356 (31, 368 1 o d , 11. 3i6

s,,

Alan). S.. 354 \Jci'lcll:rn,

C. E., 315

380

AUTHOR INDEX

Mints, A. L., 353 (21,357 (2), 363,364 Mironov, E. S., 347, 366 Mittleman, Y., 337 Miyazawa, Y., 353 Mock, D. L., 359 Moneti, G. C., 359 Monoszon, N. A., 353,357 (4), 363 Moon, P. B., 355 (2) Moore, M. J., 352 Moravcsik, M. J., 360 Morgan, T. J., 347 Moroz, E. M., 360,361,364 Morozov, P. M., 349 Morpurgo, M., 367 Morton, A. H., 349 Moseley, H. M., 365 Moyer, B. J., 341 Mozalevskij, I. A., 357 Mozley, R. F., 371 Mueller, D. W., 339 Mukhin, A. I., 353 Mullett, L. B., 364, 369, 370 (3), 371 Mullins, R. K., 348 Mundsty, G. L., 367 Murray, R . L., 350 Murtas, G. P., 359 (2) Naumov, A. A,, 347 Neal, R . B., 369,370 (3) Nelson, M. E., 345 Nemenov, L. M., 347,366 Neviazhskij, I. H., 353 (2), 357 Neyret, G., 356 (2) Niemann, F. H., 340 Nishina, Y., 346 Nisimura, K., 348, 354 Nonaka, I., 348 Northrup, D. L., 339 (2) Nygard, J. C., 370 Nysiiter, H. M., 363 Oele, H. A,, 359 Ogawa, H., 348,354 Ogle, W. E., 345 Ohkawa, T., 364,365 (2), 366 (2) Oldroyd, D. L., 358 Oliphant, M. L., 342,354 O'Neill, G. K., 357, 366,367 (2) Ono, K., 348 Oppenheimer, F., 368 Oselka, M., 348

Oslov. I. F.. 362 ~ v s y k n i k i vV. , M., 369 Ovchinnikov, E. P., 367 Oxley, C. L., 352 Panasyuk, V. S., 347,349,350 Panofsky, W. K. H., 367, 368, 370 (2), 372 Parain, J., 356 Parkins, W. E., 350 Parkinson, D. B., 339 (3) Paul, W., 342 Paulin, A,, 361 Payne, H. E., 358 Penfold, A. S., 360 Pentz, M. J., 367 Perrin, F., 357 Persico, E., 359, 361 Peters, B., 351 Peterson, F . L., 365 (2), 366 Peterson, J. M., 347, 348 (3) Petukhov, V. A., 357,364,366 Piccioni, O., 356 Pickavance, T . G., 341 (3), 343 (2), 352 (3), 358 (2), 364,367 Pidd, R. W., 344 Platt, ,J. B., 352 Plyutto, A. A., 369 Polyakov, B. I., 353 (2) Pollock, H. C., 358, 360 Pomeranchuk, I., 360 Pontecorvo, B. &I., 353 (2) Popkovich, A. V., 357 Porreca, F., 361 (2) Post, R. F., 369 (2), 370 (2) Post, R . J., 347 Pot&,A. J., 352 Powell, W. B., 347,349 Powell, W. M., 353 Pruett, C. H., 365 (2) Puglisi, RI., 359 (2) Putnam, J. M., 337 I'yle, R. V., 365 Pyrah, A. E., 340 Quercia, I. F., 358, 359 (2) Querzoli, R., 359 (2) Rabinovich, M . S., 357 (2), 364 (2) Radvanvi. P.. 353 Ragan, G.' ~ . , ' 3 5 8 Rahm, D. C., 356

AUTHOR INDEX Srhmplzer, C., 347, 360, 362

Rainwater, J., 353 Rall, W., 368 Rtimler, W. J., 348, 349 I i a n m , C. A., 355 (2) Ramsey, X. I?., 352 Ratner, L. T., 350 Redhead, 1'. -4., 360 Rees, J. R., 363 Regcnstreif, E., 342 317 Reilly, E. M., Iteut, A. A,, 353 Ilevutskij, E. I., 36!r Iieynolcls, H. L., 34!) Richardson, J. It., 345, 351 (2), 365 Richman, C., 368 Riddiford, L., 355 (4) Riou, hl.,353 Riseer, J . R ., 346 I h i e r e , -%. C., 340 Roberts, A., 361, 364, 365 Itobertson, D. S., 357 Robinson, C. S., 35!J (2) I
Sacerdoti, G., 35!1 ( 2 ) Saji, Y., 348, 334 Salisbury, W.W., 341, 345, 348, 368 Sdvini, G., 358, 359 (2) Sulzmnn, F., 3634 S a m p o n , 11. B., 346 Sannda, .I., 318, 3.53, 354 Smnn, G., 359 ( 2 ) S:~rkis\-an,L. A , , :-I66 S a w ~ i k o v-4. , I,., 366 S:lzi, Y., 354 Scag, D., 345 Schiff, L. I., 311 ( 2 ) , 360, 364, 369

Schmidt, F. EI., ,347, 349, 350, 351 (2) Schmidt-Rohr, U., 347 Schmittc:r, K. H., 372 Schneider, H., 372 Schorh, A,, 362 ( 2 ) ,367 (2) Schr~ltz,H. L., 371 Schw:trcz, L.,352 Sclin-arz, G., 345 Schwinger, J., 360 (2) Scri., E., A0 Sc!tlovich, hI. P., 357 Jr., 360 Sellen, J. 1'1. Sel)tier, A,, 372 ( 3 ) Scrher, R., 313, 344,351 SersIer, A. JI., 3G4, 365 (2), 366 Sewll, D. C., 3.31 ( 2 ) ,353 S l d n o v , A. V., 361, 370 J . , 347 Sh;~rl), Sh;\rshanov, A. A , , 372 Slinw, J ., 340 Shnylor, B.R ., 355 S l ~ e k h t cV. ~ , M.,363 Shew2luck, S., 337 Shire, E. S.,338 Sliirrr, P';.S., 370 Shoctmakcr, F. C., 357 (2), 371 Shono, EI., 348 Shrmk, G., 352 Shull, I?. H., 371 S~c:gbd~n, K., 36 S~lverman,.4., 363 (2) Simpson, K. hl., 351 Sinelnikov, K. D., 357, 371 $k:lpge, I,. S., 359 Skyrme, T. B. It., 343 Sl:ttcr, J . C., 367 ( 3 ) , 371 Slonn, I). H., 340 ( 3 ) , 349,368 (3) Slobodrim, R. J., 354 S~ntirs,E., 363 S~nirnov,V. L., 370 Smit,h, B. H., 348, 365 (2) Smith, Lloyd, 345, 356, 367,372 Smith, W. I. 1~1.,349 (2) Snell, A. II., 3.40 S~iocltly,I,. B., 369 Snowtlcn, AI., 352 Snyder, 11. S., 3fil ( 2 ) ,W4 372 Snyder, -1. S.,365 ( 2 ) Solomon, A. K., 338 Sona, 1'. G., 359

381

382

AUTHOIt INDEX

Spevakova, F. M., 357 Stafford,G. H., 343 Stahl, R., 348 Stcpanov, K. N., 372 Stephens, K. G., 351 Stevens, D. K., 359 Stolov, A. M., 357 (3), 363,367 Stone, K. F., 356 Stone, R . S., 340 Strauch, K., 354 Streib, J. F., 347 Streltsov, N. S., 357 Struven, W. C., 356 (2) Stubbins, W. F., 349,354 (2), 372 Sturrock, P. A., 344,362 Suprunenko, V. A., 369 Snsini, A,, 367 Sutton, R. B., 352 Su~va,S., 348, 353, 354 Suzuki, A,, 348,353,354 Svartholm, N., 36 Swihtlrt, J., 369 Symon, K. R., 364 (4), 366,367 Symonds, J. L., 355 (2) Takeuchi, S., 348 Tallmadge, F. K., 345 Tarakanov, K. I., 353 Tarasov, E. K., 362 Taylor, C. .J., 348 (2) Teichman, T., 344 Teillac, J., 353 Teng, L. C., 344, 354, 357, 364 ( 3 ) , 371 Terman, F. E., 271 Trrrall, J . R., 371 Terwilliger, K. M., 344, 364 ( 3 ) , 365 (3), 366 Thomas, E., 337 Thomas, J. E., 337, 358 Thomas, L. H., 364 Thornton, R. L., 345, 347, 348, 349, 351 (2), 353, 365 Titov, V. A., 363,367 Toschi, R., 359 (2) Trammel, G. T., 365 Trotter, H., 369 (2) Trump, J . G., 338 Tuck, .T. L., 354, 355 Tucker, E. B., 365 Turner, C. M., 33!) 'I'urrin, A,, 359 (2)

Tuve, M. A,, 339,340 (3) Twiss, R. Q., 344 Tyagunov, G. A., 370 Valdncr, 0.A., 370 Vale, J. T., 351, 354 Van Atta, C. M., 339 (2) Van Atta, L. C., 339 (5) Van de Graaff, 11. .J., 338,3:<9 (5), 340 Van der Raay, 13.V., 355 (2) Van Voorhies, S. N., 352 Vasilev, A. A., 357 (2) Vasilevskaya, D. I'., 366 Vaughn, T. B., 355 Veisbein, M. hf., 357 (2) Veksler, V. I., 342 (2), 343 (3), 357, 360, 372 Verster, N. F., 354 Victor, C., 353 Vladimirskij, V. V., 362, 363, 364 (2) Vlasov, A. D., 372 Vodopianov, F. A., 357 (2) Vogt-Nilsen, N., 366, 367 Vonberg, D. D., 347 Voorhies, H. G., 358 Wadcy, W. G., 371 Waggoner, J., 363 Waithmnn, V. B., 351 Walker, D., 350, 351 (2) Walkcr, G. R., 371 (2) Walkinshaw, W., 342, 364, 365. 367. 369 Clittllenmcyer, W. A., 365 (2), 366 Walton, E. T. S., 339 (2) Wmg, T. J., 359 Ward, F. A. B., 350 Warren, R. E., 339 Wntanabe, S., 346 ivaterton, P. J., 347 bTatson,13. 13. H., 358 flcijb, D. L., 340 34 Weinman, .J. .4., @dton, T. A,, 365, 66 flent, ,T. J., 330 fernholm, O., 363 Xest, W.D., 371 Wcstendorp, W., 359 (2) White, Rf. G., 345,352,357 Wicher, E. R., 371

S

AIJTHOK INDEX

Tal)lokov, 13. N., 3(;7 Txnadn, T., 353 Yamnguchl, I1 , 318, 354

Zucrhrroni, E., 368 Z:~ffar:~no,D. J., 372 Z:~~noloctcl~ikov, R. I., 366 Zaplatin, S. L., X 6 Za~eny:~gin, J . .4., 366 Z:lveny:tgin, Y. Z., 350 Zeitllits, ,4. I., 357 Zeidlits, 1'. RI., 370 Zclitlcnok, G . A., 370 Z I u ~ r : ~ v lA. ~ vA,, , 357 Zorin, I>.M., 361 Zucker, A .> 319 (2)

383

Adlabat~c damping of betcitron o w I l , ~ tlonb, 39-41 n~, of ~ n c l l r o t r o no ~ c ~ l l s t l o96-100 Alternatmg-gradlent accelerators, 24, 193218 betatron frequenc~es,201-205 comparison w t h CG m~~clilncs, 207 equlllhrlum orhlt, 208-212 field index, 19G, 109, 200 , 212 momentum c o n ~ l m t l o n206, optical analogy, 195 o r h ~ stabi11t> t (nrcktle) diagram, 198 phase btah~llty,205-208 resonancrs, 200 synchrotroni, 11-t of, 214-218 parameter,., 216,217 t r m b ~ t ~ o(crit~c.ll) n encrgy, 206 Alvarez h e a r accelerator (hnac), 7, 270-

29 1 c a v l t ~deslgn, 275278 cell length, 271 examplci, 288-291 for heavy Ions, 290 tmn-it tlme, 273 See also Proton l ~ n a c Amplitude of bctatron oscillat~on., 88 of synchrotron o-clllat~ons,91-93" Srchimetlean hpiral r d g e , 239 Argonne Nat~onalLabor'ctorg , p,ir,tmctery of synchrotron, 157 Average firld Index X ,230-232 In ~hochrononsc\ clotron, 236 In sp~ral-sectorring, 249, 253 relat~ont o momentum compnctlon, 2.31232 Azimuthally-varying-field (AVF) accclcrator, 219 See also Sector-focused acceleratori; Beam splitter, 128 BerkeIey (Lawrence I h i i a t i o n Lalmxtory), heavy ion linac (hilac), 291 3

Berkeley (Laurence Rad1'1t1on L.'I b oratory) (cont ) 1.oc11ronous cyclotron, 219, 244, 246 [roton Imac, 267 + J nchroc) clolron, 149, 150, 153 Ser also Bcvatron Bc t atron, 12, 154-188 hibed, 187 sector-focused, 257-259 Betatron ,icceleratlon In elertron synchrotron, 18'2 In proton synchrotron, 172 13t.t:itron osctllatlons, 12, 24, 32-36 d i a b a t l c damping, 39-41 :mplltutles, 38 irequencley 111 AG machlne, 201-205 In CC: machme, 32-36 ln race track, 52-54 111 w t o r e d ninchlne, 232 In qm-aled machine, 2213-230 In Thomas cyclotron, 224-226 In ZGS, 62-65, 157 ratllal freqwncj and injection process, 175-177 and encrgy hmlt In ~sochronouscyclotron, 23!9 Btbatron, 154 tlr~fttube, 168-171 injector, 287 magnrxt, 154, 155, 158 pnramtstrrh, 156 vacuum chnml~cr,164 Blab voltage on drc, I 15, 128 R m c d betatron, 187 h a - e d pc~ntlulnn~ analog\, 88-90 B~rminghnmIJnlv , ~soclironousc>clotron, 246 proton ~ynchrotron,154 tleflwtlon. 179 drift tube, 169 magnet, 155, 156, 157 magnet power supply, 161

Birmingham Univ., proton synchrotron

(cont.) parameters, 156 space charge limit, 75 vacuum chamhrr, 165 B r vs. kinetic energy, table, 20 chart for isochronous cyclotron>,242 Brookhaven National Laboratory, AG synchrotron, 214 parameters, 217 linac injector, 288 See also Cosmotron Buncher, 285 Calif. Inst. of Tech., electron synchrotron: 182 Canberra, proton synchrotron, 157, 162 Cavity in Alvarez linac, 276-278 in synchrotron, 167 Center of mass energy, 259-263 CERN, AG synchrotron, 214 parameters, 217 linac injector, 288 synchrocyclotron, 149, 150 Charge of electron, 9, 18, 102 Circumference factor, 256 Closed orbit, see Equilibrium orhit Cockcroft-Walton accelerator, 2 Coils of cyclotron magnet, 104 of synchrotron magnet, 155-158 Colorado Univ., isochronous cyclotron, 246,248 Columbia Univ., synchrocyclotron, 149 Const,ant-gradient accelrrators, 24 betatron frequencies, 32-36 comparison with AG machines, 207 equilibrium orbit, 23 field index, 24-28 momentum compaction, 80-84 phase stability, 84-94 resonances, 66-71,74 See also Betatron; Fixed-frequency cyclotron; Nicrotron; Synchrocyclotron; Synchrotron Cornell Univ., AG electron synchrotron, 214 parameters, 216 transition energy, 207 Cosmotron, 154 cavity, 168

Cosnmtron (cont.) ejection, 170 magnet, 154 operatmg point, 71 parameters, 156 synchrotron frequrncy, 1'72 vacuum chamber, l(i4 Coupled resonances, 67 Crit~calenergy, see Trans~tionmergy Current, see Ion current Cyclotron, see Fixed-frcquwcs~ c.1 c!ot ron ; Ihochronous cyclotron; S t o c h a h c cyclotron Cyclotron frequency, 10 Damping, see Adiabatic damping DC accelerators, 2-4 Debuncher, 286 Dee, 8, 109-118 aperture, 109 bias, 117, 128 dummy, 111, 145 feelers, 119, 142 forces, 140-142,280 grids, 119, 128, 142 multiple, 145-147 pullers, 119 transit time, 123-125 voltage in cyclotron, 103, 139 minimum, 134-140 in isochronous cyclotron, 234,237 in stochastic cyclotron, 319, 322, 334 in qmchrocyclotron, 150 Deflector for cyclotron, 125-133 for synchrocyclotron, 152 regenerative, 153 for synchrotron, 179-181 Delft, Technische Hochschule, C. G. synchrotron, 157 vacuum chamber, 165 isochronous cyclotron, 245 Difference resonances, 67 Drift tube in linac, 6, 7, 267, 271 effect on frequency, 276 forces at, 278-284 in synchrotron, 168-171 Dubna, U.S.S.R., CG synclirophasotron, 154, 155, 156, 164 isochronous cyclotron, 245 phasotron, 149 Dummy dee, 111, 145

SUB.JECT I S D E S Eddy currents, 28, 158, I73 E:tlgc focr~sing,55-155 Ilcctron ch:trgc,, 9 , 18, 102 KIILISS, 18 Elcct ron cyclot run, 219, 2'21, 544 Scr a l s o hlicrotron Electron linar, ?!)I-2!)S stal~ility,295 standing W:IW, 297 trarelinq wave, 292-295 e x m p l c ~296 , Electron synchrotron, 12 AC:, 183,214,216 ('G, 181 r:di:ltion loss, 182 S r t also S,~.nrIirotron Elcrtron-\-olt , 18 Electrostatic qn:ictr~qwlw,31 1-31.? Enclrg. gained per tnrn in fixctl-frcyllc-ncy cyclotron, I?:!-125 in irochronow cyclotron, 234,237 in microtron, Ih9, 1!)2 in ~ynrhrocyrlotron,150 in synchrotron, 167 Energy rr:~rhcdin het:~tron,187. IS8 in fisrd-frequency cyclotron, 10. 10'2 inhomogencity, 128 in hilac, 290,291 in isochronous r\.clotron, 239, '24?-'23,5 in lin:rc! elrctron, 296-298 l~roton,287,288 in microtron, IS9 in -ynrhroc>.clotron, 149 in synchrotron, BG, 216, 217 CCT, 156, 157, 181, IS2 E q u i l i h r i ~ ~orbit, m 23 in AGS, 208-21.' in r:tdial-rector ring, 255 in sector-foc~wtlmachine, 220,221 in two-heam :~ccelerator,264 External Ixm~f'rum betatron, 197 from fixed-frt\qr~cncycyclotron, 125-133 from synchrocyclotro~i,152 from synchrotron, 179-181 Ferrite, 168 FFAG accelwator, 519-266 See also Sr(~tor-foc11m1 accelrrator Field index 71 in AG rnnchirie-, 196, 199, 200,204,213,21C,, 217 in CG mnchlr~c~r, 24-28, 31, 31

387

Ficltl intlcx 11 in CG mnrhinrs (co17t.i in bc,t:~trun, 27, 1 S(i in FF ;in(l FA1 cyclotron, 111:ix.v : I ~ I I ( + , "i, 4 1, 68 in mit~rotrori,IS!), in stochastic cyrlotron, 317, 326 in sync-hrotron, 27, 28, 41, 154, l;i(i, 157 in ZGS, 61, 157 in mas. spectron~rtrrs,86 rpncc charge, influence of, 73-75 Fixed-frequency cyclotron, 7-10, 10'2-147 1)e:rrn splitter, 125 I)et:ltron oscillations, 12, 23, 32-36 d:ln~])ing,3 - 4 1 frequencies, 34, Xi initixl a m p l i t ~ ~ t l w 38, c~lrrc~nt, c~xtcrnnl,126 intern:~l,I03 tlw, 8, 100-118; s w ulso I k e deflector, 125-133 energy, masimurn, 10, 10'2 inhomogcnc~ity,128 v:~ri:~tion, 142 w.F k , 20 ficltl intlex, 26, 31 h:~rrnoniracvlcrat ion, 143-145 lityrvy ions, 144 ion sonrcc, 118 ~ii:~gnc+, 8, 103 control, 115 ficltl, typic:~lvnlncs, 20 mode operation, 145-147 ~noincmtnmcompaction, S2 m~~Itip:rrtine, 115 multiple tlecs, 145-147 orbit, mrly, 120-123 cquilil~ri~~ni, 23 selxlration, lX3 o d l ; i t o r freqncncy, 10, 102 projcc,tiles, 102, 14-1 reson:inces, Mi-(;!) Rose rings, 26, 109 ~ccontl-ortlcrfoewing, 141 s c p t ~ u n1213 , shicdtling, 147 shimming gap, 109 shorting bar, 111 synchrotron oscillation, 101 transit time, 123-125 vacullm chamher, 108 varinhle-energ\-, 1.12

388

SUBJECT INDEX

Flxed-frequency cyclotron (cont.) veloc~tyfocuilng, 141 "Flsh" (phase) dlagr:~m,93 Flor~da,U n ~ v, isochronons cyclotron, 248 Flutter, 2'22-224 Focal length of magnet edge, 57-60 of quadrupole, 302-308 Focusing, by foils, in linac, 287 by grids, In linac, 284 by magnet edge, 55-60 by quadrupoles, 299-313 by solenoids, 290, 296, 299 strong (AG) vs. weak (CG), 24 FODO, FOFDOD, 214 Free oscillation, 24; see also Betatron oscillat~on Frrq~tency,cyclotron, 10 Frequency-modulated (FM) cyclotron, ser Synchrocyclotron Frequencl of oac~llntorin AG synchrotron, 183,208,216,217 in electron linac, 296-298 in electron C G synchrotron, 182 in fixed-frequency cyclotron, 10, 102, 111 in isochronol~scyclotron, 221, 235, 237, 24 1-243 in microtron, 189, 192 in proton hnac, 271,277,287-291 in proton CG synchrontron, 156, 157, 165 in spiral-sector ring, 251 in stochastic cyclotron, 319, 327, 334 in synchrocyclotron, 149 Gas scattering, 173 Grids, in dees, 119, 128, 142 in linac, 142, 284 Half magnet in AGS, 213 Hall effect, I66 Hamburg, electron AG synchrotron, 214, 216 Harmonic acceleration, in fixed-frequency cyclotron, 143-145 in isochronous cyclotron, 243 Harmonic order (number), 87 and amplitude of synchrotron o d l a tions, 92, 95 I-larvard-1I.I.T. electron AG synchrotron, 159, 162, 183,214,216

Harwell (Rutherford High Energy Laboratory), CG synchrotron Ntmrod, 157 isochronous ryclotron, 246,248 proton linnc, 288,289,290 Heavy ions, acceleration in cyclotron, 143145 Illinois, Univ., betatron, 187 isochronous cyclotron, 245.246 Injection, into linacs, 269, 275, 255, 291, 296.297 into F M cyclotron by stochastic method, 322 into synchrotron, 7, 12, 173-178,286 Injection energy, 12, 89 error of, 94 influence of, 173 Injector, 7, 12,286 Intersecting beams, 263 accelerator, 264-266 Ion current, in cyclotron, 103 in isochronous cyclotron, 233 in linac, 286,287, 288, 290 in stochastic cyclotron, 322,333-335 in synchrocyclotron, I 1 in synchrotron, 11, 156, 157,216,217 Ion source, 118, 144, 243 Isochronous cyclotron, 233-248 betatron frequencies, 232 conditions for isochronism, 235 dee voltage, 234,237,241 energy limit, 239 momentum compaction, 236 particular machines, 244-248 transition energy, 237 variable energy, 241-243 by ha,rmonic operation, 243 Jump target, 180 Kerst force, 221 Kerst-Serber equations, 33, 36 Kharkov, electron linac, 297 hllac, 291 Cinetic energy, relativistic expression, 14 and available energy, 262 and B r (table), 20 Jaslett force, 221 ~awrence Radiation Laboratory, Berkeley; Livermore

we

SUU.JECT I SDEX

389

Llncar accelcr,ltor ( I ~ n a r ) , set. Alvnrez Illomenturn compaction (cont ) 111 Ilnacy, 86 Ilnac; Electron I m x ; IIllac; Proton and nli~ltlplc-turnmject~on,178 11n:tc ; Witlerbe hnac rcLitlon to average field mdex k , 231, Lmcr, 108 222 Liouv~lle'~ theorcm, 314 m .1)1rd-sector rmg, 250 Lwermore, A1v:rrc.z Imic, 290 31omcntr~m-pc~r1ot1 relat~on In AG maLogar~thrnicqp~rnlr~dge,250 ch~nci,206 Los Alarnoh Sc~cnt~fic Laboratory, Thoinns In CC: m:lchlneq, 84-87 cyclotron, 243 Loi Angelei, Unl\.. of C&f. at, ~ ~ o r h r o - HI ~sochronouicvclotronq, 237 In 11n,lc\,66, 276 nous cyclotron, 245 In spiral-sector rmge, 250-252 proton I~nac,287 Lund Unlv., -1G electron yynchrotron, 103, l l o v o n , proton AGS, 216-218 l I ~ ~ l t ~ p : ~ r t115 ing, 216 X[I'R 4, r,rtl~al-\ertorrmg, 257 n~icrotron.103 -111ral--ecator rlng, 255 t n o - h a m accelerator, 266 Magnet, AGS, l!fii, 197, 215, 215 betatron, 186 cyclotron, 8, 103, 106, 107, 108 isochrono~lscyclotron, 234, 246, '147, '148 microtron, IS!.) Oak Ritlge Xntional Lnl~oratory,isoehror:ttli:zl-scctor ri~ig,255, 257, 258 n o w cyclotrons, 244, 215, 246, 247, spiral-arctor rmg, 248-250, 254 248 storage ring, 263 Operating point, in I G machines, 200 synchrocyclotron, 148 in C:G maclhines, 66, 71, 74 synchrotron, elwt ron, 181 in spir:11-sector rings, 240 proton, 154: EX, 157 ( 'h-hit, carly, in cyclotron, 120-123 T h o ~ ncyclotron, ;~ 220, 221 ([)rl~itst:tl)ility, 22-24, 2 8 4 1 t ~ o - l , ~ : machinr, m 2M, 2ii5 in .4G rn:lcl~ines,105-205 1I:tgnetic qu:tdru])ole~s,2IK)-318 in C X cirrlrs, 32-36 ll;inchestcr, [Jniv., hilnc, 291 in ('(2 race tracks, 52-54 h l a s ~ .Inst. of Technology, electron AGS, in elrrtron linncs, 295-2136 216 in isoc*lirono~~s inachincs, 23!)-241 c,l(>ctronlinac, 298 in proton linac~, 278-285 Thomas cyclot ron, '145 in radial sector machines, 255-259 I l a s s i;l)cctrorrlrtc~r,ctlge-focr~singtylw, 60 in spiral sector mxhincs, 226-230, 248T d? type, 36 250 Matching accolrrators, 314-318 in T11om:ts machines, 219-222, 224-226 JI:itrix, in 201-205 in ZCS, 55, 61-65 in CGS, 44-52 orbits, se1)aration of, in fixed-frequency in CGS r a w track, 52 cyclotron, 133 ml~lti~)licutioll, 43 in isoc+hrononscyclotron, 237 in ql~atlrul)olcs, 303-300 in ~nicrotron,1!)3 in ZGS,62, (i4 in ::yrlc-hroc~yc~lotron, 152 LIicrotron, 18!)-1!)1 S w ulso Llonlentum cornpartion XIinncsotn, I!niv., proton lin:~c-,287 ( ) r s : ~ yclwtrrm , lin:\c, 2!?S S!.nO>c.ill:~tion,w e Uct:ttron osc.ill:~tio~~; llcrtle oprr:~tiunof r y l o t r o n , 14.5-147 chrotron oscillation l l o l n e n t u n ~co1111 );iction, SO-S1 Oscillator frcqucncy, see Frequenrv of in .4G machines, 212 oscillator in CG machines, 81-84

390

SUBJECT INDEX

Parameters, of AG synchrotrons, 216, 217 of CG synchrotrons, 156, 157 Peaking strips, 166 I'ermeability of space, 'i2, 277 Permittivity of spacc, 72, 331 Phase diagram, 90-94 Phase equation, 87 Phase oscillation, 79, 88; see also Synchrotron oscillation Phase shift of betatron oscillation per sector, 51, 52 Phase shift per turn in fixed-frequency cyclotron, 134 Phase stability, 76-101 requirements for, 77 in AG machines, 205-208 in CG machines, 77 in electron linacs, 295 in isochronous cyclotrons, 237 in microtrons, 189 in proton linacs, 86,278 in spiral-sector rings, 250 Phase velocity in electron linac, 292-295 I'haeotron, 149 Piccioni ejection, 179-181 I'ole-face windings, 28,234 Princeton synchrocyclotron, 149 Princeton-Penn. proton CGS, 157, 159, 161 Proton charge and mass, 9, 18 Proton linac, focusing by foils, 287 by grids, 142,284 by quadrupoles, 270,300 by solenoids, 290, 299 1)liase stability, 86, "8 transverse stability, 278-285 velocity (second order) focusing, 279 see also Alvarez linac; Widerije linac Proton synchrotron, 11 AG, 195-197,217 CG, 154, 156, 157 See also Synchrotron Q of cyclotron, 112 of Alvarez linac, 271,276,286 Quadrupoles, electric, 31 1-313 fringing field?, 313 niagnetic, 299-31 1 w e in linncs, 279, 300 in matching accelerator., 314-31s Q11~111ty of bc:1111,316

Race track, 12 betatron frequency, 52-54 of ZGS, 62-65 resonances, 70 rotation frequency, 165 Radial 7-mode stop band, 240,244 Radial-sector (ridge) accelerator, 219, 255-257 Radiation by electrons, 85, 182 Regenerative deflection, 153 Relativity, 10, 13-17 Resonances, in AG machines, 200 in C G machines, 66-75 in isochronous cyclotrons, 240, 241, 245, 248 in spiral-sector rings, 249 Rest mass of electron and proton, 9 Rest mass energy, 14 of electron and proton, 18 Residual field, 28, 159 Rigidity, 19 Rose rings, 26, 109 Saclay, electron CG synchrotron Saturne, 154, 155, 156,165 Second order focusing, in cyclotron, 141 in linac, 279 Sector, in AGS, 201 in CGS, 45 in isochronous cyclotron, 240 Sector-focused accelerators, 219 betatron, 257-259 betatron frequencies, 232 isochronous cyclotron, theory, 233-241 typical, 244-248 radial-sector ring, 255-257 spiral focusing, 226-230 spiral-sector ring, 248-255 Thomas cyclotron, 219, 222,245 Thomas focusing, 224-226 two-beam accelerator, 264-266 Septum, 126, 127, 128, 152,237 Shieldmg, of cyclotrons, 147 of synchrocyclotrons, 153 of synchrotrons, IS1 Shimming gap, 109 Shunt resistance of Alvarez linac, 271, 276 Solenoids, 279, 296, 299 Space charge and irijcction energy, 173 and max. current from synchrotron, 7175

Splral angle, 22G, 235 Spir,il focwng, 22 1, 226-230 S l u r ~ ritlqe, l A r c l i ~ r n t ~ l r .2vY1 ~n, log,irithrnic, 2.70 Sp~ral-sectorring m d ( ~ r , ~ t o24h-252 r, 1)hase sLtt)illt>,230 poss'be design, 152-255 rotahon frequrncv, 251 St,rhllit>, sw O r h t \ t a l ) ~ l ~, t I'll \ 1-0 .t 11111I ~ Y

Stand~ngwave electron llnnc, 297 Stanford Univ , e l t ~ t r o nIin:rc~,2r)A, 207 Stochastic accelemt~on,319-335 Stochast~ceject~on,323 Stochastic lnlcction, 322 Stop band, radial -rr-mode, 240,244 Storage rings, 263 Storrd energ] in n q n e t , 160, 161, 16'' Straight scct~on,12, 23, 144, 145, 1.56. 157 mntr1.c rcprebeni &on, 4,5 Strong-focusing :icceler:tt or., 24, 1%- 107 S)nchrocyclotron, 11, 148-153 hetatron oscillat~ons,24, 32-36 damping, 39-4 1 mitlal amplitiides, 38 current, 11 dee voltage, 150 energ,, 11, 149 inhomogenc~t~ , 98, 99 external beam-, 152 ficltl miex, 26, 3 1, 68 Ion source, 118 magnet, 148 momentum compaction, 82, 151 o~cillatorfrequency, 149 rate of modulation, 150 phase angle, optlmum, 95 phase stability, 76-100 regcnerat~vedeflection, 153 resonances, 200 shlelding, 153 synchrotron osr~llations,151 typ~calmach~nei,149 v:lcluim cham1 Irr, 140 Svnchronou\ p h , ~ angle, ~ c 77 In AG machlne~,207 In CG n~nchint~s, 80 in linaci, 86 Svnchrophniotron, AG, 216, 217 CG, 154, 155, 156, 164 Synchrotron, 11,154

S.,nchrotron (cont .) 1)ct:ltron acc~lrr:ition,172 I ~ct:rtronosc~llat~nri~, .4G. 201-205 C G , 32-30, 51-51, (i2-A5 .lrnplltllt.l~.,35-4 1 c ~ v i t y:rc.cel~~rnt~on, I(i7 currrnt, 11, 156, 157, 216, 217 drift tuhes, 11 :~cceleration,168-171 elrctron nm-hines, AG, 214, 216 CG, 12, 181 energy, 11, 154, 156, 157, 181, 162, 216, 217 gninctl per turn, 167 at injection, 156, 157, 173, 216, 217 inhomogene~ty,OS,99 cc]lnht)r~m orblt, ~ .4Cr, '708-212 CG, 23 eutrrn:il be:ms, 179-181, 215 fic,ld index, AG, l%i, 200, 204, 211, 216, 217 CG, 27, 31, 41 injection method$, 173-178 ~njector,12, 156, 157, 216, 217, 287, 28'3 ni,lgnft, AG, 196, 107 CG, 11, 154-164 nlomentiim ~ ~ o m p a c t ~AoGn , 212 CG, 82-84 n c c k t ~ r( i t a l ~ i l i )t ~tl~agmln,AG, 108 owllator frrqnency, 12, 165, 182, 2P8 period-momcntmn rcl ttlon, 4G, '200 CG, 85 phaqe ~ t a h i l l t >ACr, , 205-208 CG, 76-80 proton mackunri, AG, 214-218 CG, 154, I56 race t r x k , 12 resonances, 66-75,200 shlelding, 181, 215 space charge, 71-75 synchrotron owllatlons, 79, 96-100. 171 tnrgf+, 1'78 t r , ~ n i ~ t i ocrlergv, n 57, 206, 207, 214, 215 216, 217, 218 varulim chnml)er, 164 Synchrotron o ~ r i l l : ~ t i o79, n ~ ,90-03 damping, 96-100 depentlrncc on hmnonic ortlcr, 95 in fixctl frcqi~rnc.~ c>clotrori, 101 In liriac. 86, 278 In microtron, 180

392

SUBJECT

Synchrotron oscillations (cant .) in synchrocyclotron, 151 in synchrotron, 171 Tandem Van de Graaff accelerator, 4, 5 Ta,rget, in betatron, 187 in cyclotron, 103, 125 in synchrotron, 178 Thomas focusing, 219-221, 224-226 Three-phase cyclotron, 146 Tokyo, electron AGS, 216 Total energy , 14 Trace, 49 Transition (critical) energy , in AGS, 87, 206, 207, 214, 215, 216, 217, 218 in isochronous cyclotron, 237 in spiral-sector ring, 251, 252 Transit time, in Alvarez linac, 273, 275 in cyclotron, 123-125 in Wideriie linac, 270 Transverse stability in linac, 278-285 Traveling wave linac, 291-296 typical, 296 Tune, 34, 66 Two-beam accelerator, 264-266 U.S.S.R., electron linacs, 297 hilac, 291 proton linac, 288 phasotron, 149 synchrophasotron, AG, 216, 217, 218 CG, 154, 155, 156, 164 Vacuum chamber of betat’ron, of cyclotron, 108 of linac, Alvarez, 7, 271 electron, 294

186

INDEX Vacuum chamber, of linac (cont.) Wideriie, 6 , 2iO of synchrocyclotron, 149 Van de Graaff accelerator, 2, 3 tandem, 4, 5 Variable energy , in electron linac, 296 in fixrd frequency cyclotron, 142 in isochronous cyclot,ron, 241-244 in microtron, 143 Velocity focusing, in cyclotron, 141 in linac, 279 Walkinshaw resonance, 68 Weak-focusing accelerator, 24 Wentzel-Kramers-Brillouin (WKB) method, 40, 97, 98, 99 Widerr; linear accelerator (linac), 6, 267cell length, 267-269 transit time, 270 See also Proton linac Wronskian, 48 X-rays from betatron, 187, 258 Yale Univ., hilac, 291 electron linac, 297 Zero-gradient synchrotron (ZGS), 61, 162 betatron oscillations, 62-65, 177 injection, 177 magnet, 61, 160, 162-164 momentum compaction, 83, 178 parameters, 157 vacuum chamber, 163, 164 Zurich, Swiss Federal Inst. of Tech., isochronous cyclotron, 248