Linear Accelerators

LINEAR ACCELERATORS edited by PIERRE M. LAPOSTOLLE ISR Division, CERN, Geneva, Switzerland ALBERT L. SEPTIER Instit...

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LINEAR

ACCELERATORS

edited by

PIERRE M. LAPOSTOLLE ISR Division, CERN, Geneva, Switzerland

ALBERT L. SEPTIER Institut

NORTH-HOLLAND

PUBLISHING

d’Electroniyue Fondamentale Fault& des Sciences 91 Orsay, France

COMPANY

-

AMSTERDAM

o NORTH-HOLLAND PUBLISHING COMPANY , AMSTERDAM, 1970 All rights reserved. No part of this book may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise without the prior permission of the Copyright owner. Library of Congress Catalog Card Number: 70-97209 ISBN 7204 0156 9 Printed in The Netherlands

Publishers: NORTH-HOLLAND PUBLISHING COMPANY - AMSTERDAM

Sole Distributors for the Western Hemisphere: WILEY INTERSCIENCE DIVISION JOHN WILEY & SONS, INC. - NEW YORK

Preface

Accelerator science is of relatively recent origin and all the important developments in linear accelerators date from the last decade or two. Although several books exist on circular machines, most of the published material about linear devices has appeared in the form of laboratory reports, very often of an internal nature and not always widely circulated, and in the proceedings of international or national conferences on accelerators. Although the latter constitute an easier source of information, since the communications are grouped together, the material is usually so condensed that it is often only of interest to the specialist. There is a lack of comprehensive and more logical presentation for the student who would like to become acquainted with this field. In the few books which have been written on accelerators in general, there is usually a chapter on linear machines but it is necessarily brief. There exist a few Russian books on the theory of linear accelerators; a book has very recently been published on the big Stanford electron accelerator and in the current issue of Advances in Electronics and Electron Physics (no. 25) a chapter is devoted to ion linear accelerators. There was, however, still a need for a volume which would describe in enough detail theoretical as well as technological aspects of linear accelerators in general and briefly cover the various types constructed and their applications. In order to achieve this aim, two possibilities were available to us: - either to write such a book ourselves after having collected all the available information and had discuss ions with many specialists all over the world; - or to distribute the work between such specialists, asking them to write chapters devoted to their own particular sphere of interest and activity. The second method has been adopted because this seemed to be more reasonable in a field which is in constant evolution, where only those involved are able to judge the relative merits of proposals or achievements described in the literature, pick up the most important concepts and make realistic guesses on possible developments. The drawback in this method was, however, the quasi-impossibility of obtaining a perfect homogeneity in the presentation of the various parts; this book is no proof against such a criticism. Asking the cooperation of almost fifty authors from over three continents also caused difficulties and unavoidable delays. Furthermore, it has to be fully appreciated that, since several phenomena are still

vi

PREFACE

not completely understood, the chapters give a description of the present stage in knowledge only; let us hope that in the years to come more elaborate and exact theories will remove the contradictions which sometimes subsist and give a better description of the experimental observations. It is highly probable that a new edition of this book would include changes in various chapters. We are, nevertheless, convinced that in its present form and despite its imperfections, this book will represent a solid and essential basis for engineers or physicists entering the field of linear accelerators. In this constantly progressing field, a modern and complete documentation is presented either in the developments shown or in the fairly detailed list of references appearing at the end of each chapter. This book should have a chosen place, not only in universities and libraries, but also on the shelves or on the desk of all scientists in laboratories where linear accelerators are either being built or used. We would like to express our thanks to all the authors who agreed to contribute to this volume and to all those who took part in its preparation for their active collaboration. Without their help a succesful book would have been impossible. Geneva - Orsay July 1969

P.M. Lapostolle A.L. Septier

Contents Prefhce

v

Contents vii List of contributors x i x Author index

1181

Subject index

1197

Introduction PART A The history of linear accelerators 1 J.P. BLEWETT 1 Ising's proposal and Wideroe's accelerator 1 2 The accelerators of Beams and of Sloan and Lawrence 4 3 Postwar developments 5 4 The Stanford linear electron accelerators 6 5 SLAC - The Stanford Linear Accelerator Center 9 6 Other linear electron accelerators 10 7 The Berkeley 32-MeV proton linear accelerator 11 8 Other proton linear accelerators 12 9 Heavy ion linear accelerators 13 10 Linear accelerators for high average currents 14 11 Recent developments 15 12 Linear accelerator literature 16 13 Acknowledgments 16

The general theory of linear accelerators

19

H.G. HEREWARD

1 2 3 4 5 6

Phase stability 19 Transverse defocusing 20 Properties of the travelling wave 21 Longitudinal motion 22 Liouville's theorem and adiabatic processes 25 Rf power considerations 28 7 Electron linac structures: periodic structures 30 8 Proton linac structures 33

viii

CONTENTS

Electron Linear Accelerators PART B

THEORY

-- B.l.l

Accelerating structures 39 G.A.LOEW, R.B.NEAL 1 Introduction 39 2 The principles of periodically-loaded structures 40 3 Principal design parameters 47 4 Traveling-wave accelerators 52 5 Standing-wave accelerators 65 6 Superconducting accelerators 76 7 Analytic and computer techniques for structure design 86 8 Empirical design methods 88 9 Summary of characteristics of accelerating structures of existing electron linacs 107

- B.1.2

Particle dynamics 115 R.H. HELM, R. MILLER 1 Introduction 115 2 Injection dynamics 116 3 Long range transport in electron linacs 128

Beam loading and transient behavior in traveling wave electron linear accelerators 147 ~

J.E. LEISS 1 Introduction 147 2 Nature of the beam loading problem 148 3 Beam loading theory based upon diffusion equation 151 4 Beam loading theory including dispersive effects 158

Beam breakup 173 R.H. HELM, G.A. LOEW 1 Introduction 173 2 Physical manifestations and models of beam breakup 175 3 Regenerative beam breakup 183

ELECTRON LINACS 4 Multisection or cumulative beam breakup 198

5 Discussion of starting conditions 210 6 Proposed remedies for various types of accelerators 215 7 Other electromagnetic instabilities in linacs 218

Problems connected with the measurement of accelerated electron beam characteristics 223 R. BERGERE 1 Tentative classification of methods of measurement 223 2 Time-frequency characteristics associated with the perturbation methods 225 3 Beam characteristics associated with the different lobes of the frequency spectrum 227 4 Quantity of information contained in one measurement 229

TECHNOLOGY - B.2.1

Electron injectors 237 P. BRUNET, X . BUFFET 1 Introduction 237 2 Choice of an injector 239 3 Main other sub-assemblies 251 4 Measurements of injector characteristics 256 5 Example of injector assembly 258

Accelerating structures: technology 265 A.L.ELDREDGE, A.V.LISIN, V.G.PRICE 1 Introduction 265 2 Dimensional requirements 266 3 Fabrication methods 271 4 Microwave tuning 290 5 Cooling and temperature control 301 6 Support structures 306

Radiofrequency problem 315 H.HOGG,J.V.LEBACQZ 1 Introduction 315 2 3 4 5 6 7 8

Power requirements 315 Power sources 318 Stability requirements 322 Phasing systems 325 Low energy machines (less than 100 MeV) 329 Medium energy machines (0.1 to 1 .O GeV) 331 High energy machines (above 1.0 GeV) 334

x

B.2.4

CONTENTS

Installations and control 355 L.BURNOD 1 The building 356 2 Principal auxiliary equipment 363 3 Adjustments and controls 374

VARIOUS TYPES

- B.3.1

Use of high energy electron linear accelerators 385 J.BALLAM 1 2 3 4 5 6 7 8 9 10

Introduction 385 Form factors 386 Review of medium energy physics achievements 387 Potential of very high energy linacs 388 Positron acceleration and annihilation beams 388 High energy muon beams 391 Secondary beams 393 Neutral K particles of high energy 393 Radiofrequency separated beams 397 Characteristics of high energy electron linacs as related to experimental problems 397 11 Use of detection apparatus at high energy electron linear accelerators 402 12 Review of high energy physics achievements 406 13 Summary 410

B.3.2

High duty factor electron linear accelerators 415 J. HAIMSON 1 Introduction 415 2 General design considerations 416 3 Principal design parameters 440 4 Rf system 442 5 Accelerator waveguides 445 6 Injection system 458

B.3.3

Low energy, high intensity electron accelerators 471 H. LEBOUTET 1 Introduction 471 2 Radiation chemistry 471 3 Radiography 482 4 Flash radiography. Stroboscopic radiography 486 5 Radiotherapy 489

B.3.4

Linear accelerators as injectors for cyclic machines 495 M.C. CROWLEY-MILLING 1 The electron synchrotron 495

ELECTRON LINACS

2 Production of narrow energy spectrum 497 3 4 5 6 7 8

Effect of variation of parameters 501 Emittance 503 Beam loading effects in the synchrotron 504 Debuncher systems 51 1 Examples of electron synchrotron injectors 514 Injection of positrons 520

Positron accelerators 523 F.AMMAN 1 Introduction 523 2 General remarks o n positron linear accelerators 525 3 Positron yields 528 4 Focusing systems 530 5 Matching systems 536 6 Dephasing of the positrons in the matching and focusing systems 541 7 The converter 544 8 Positron accelerator operation and experimental results 546

The racetrack microtron: an electron accelerator ,for medium energies 553 B.H. WIIK, P.B. WILSON 1 Introduction 553 2 Principles of the racetrack microtron 554 3 Design details for a racetrack microtron 559 4 Conclusion 566

Radiofrequency separators 569 B.W. MONTAGUE 1 Introduction 569 2 Basic principles 569 3 The rf separators of Panofsky and Schnell 570 4 Deflecting waves and structures 572 5 Separator optics 578 6 Engineering aspects 580 7 Present status and future developments 580

di

CONTENTS

Proton Linear Accelerators PART C

THEORY Accelerating Structures C .1. l a

Introduction 585 P. LAPOSTOLLE

C.1.lb

Low and medium energies Alvarez structures 587 A. CARNE 1 Introduction 587 2 The Alvarez structure 588 3 Tolerances 593 4 Effect of individual cell tuning errors 593 5 Optimisation 597

C .1.lc

High energy structures 601 E.A.KNAPP 1 Introduction 601 2 n-mode accelerator structures 606 3 Possible cavity configurations using the &mode 606 4 Specific properties of the fn-mode for a biperiodic system 608 5 The side-coupled accelerator structure 609

C .1.1d

Numerical computation of field distributions 617 A. KATZ 1 lntroduction 617 2 Mathematical formulations 617 3 Discretization of the partial derivative equation 620 4 Expression of boundary conditions 623 5 Diagonal dominance of the A matrix 626 6 Error evaluation 626 7 Convergence rate 628 8 External improvements of convergence rate 628 9 Calculation of electrical parameters 630

-

PROTON LINACS 10 Results 631 11 Non-zero modes 633 12 Effects of the stems and of the aperiodicity of the structure 634

C.1. le

Review and survey o f accelerating structures 637 G.DOME 1 2 3 4 5 6

Shunt impedance and transit time factor 637 Travelling versus standing wave operation 659 Some definitions 661 Structures with resonant coupling 665 Structures with non-resonant coupling 709 Comparison of structures for shunt impedance 728

Particle Dynamics at Low Intensity C. 1.2a

Introduction 739 P. LAPOSTOLLE 1 Differences between proton- and electron-dynamics 739

2 Step-by-step computation 740 3 4 5 6 7

C. 1.2b

Numerical methods. Acceleration by a gap 747 A. CARNE, B. SCHNIZER, P. LAPOSTOLLE, M.PROME 1 2 3 4 5

6 7

8 9 10

C.1.2~

Accelerating structure design 740 Field law 742 Phase law 742 Buncher 744 Debuncher 744

Introduction 747 Field representations 749 Series expansions of the fields 752 Voltage, transit time factor and other beam dynamics coefficients 754 Motion of a proton in a spatially homogeneous time-harmonic field. The perturbation parameter K 757 General considerations on perturbation theory. The thin lens approximation 759 Derivation of the non-relativistic difference equations for energy, phase, radial position and velocity across a gap 761 Non-relativistic difference equations for the change of the dynamical quantities in the first half of a gap 764 Relativistic corrections 765 Application to linac design and computation 767

Focusing 785 M.PRO MI^ 1 The need for focusing 785

xiii

xiv

CONTENTS

2 Elements of the periodic structures theory 786 3 Stability diagrams 788 4 High energy focusing 794

C. 1.2d

Nonlinear efects 797 R.L. GLUCKSTERN 1 General considerations 797 2 Longitudinal phase acceptance 797 3 Coupling between longitudinal and transverse oscillation 799 4 Coupling between the two transverse oscillations 801 5 Misalignments and errors 801 6 Misalignments and couplings 803 7 Effect of space charge 803

Particle Dynamics at High Intensity C. 1.3a

Introduction 805 P. LAPOSTOLLE 1 Beam loading 805 2 Transients 806 3 Space charge effects 806 4 Other effects 806 5 Beam break up 807

C.1.3b

Transients and beam loading efect 809 T. NISHIKAWA 1 Various methods of analysis 809 2 Transients calculation by cylindrical waveguide model 810 3 Normal mode analysis 812 4 Results and comparison with measurements 815 5 Field induced by the beam 818 6 Effects of beam loading 819 7 Methods for improvements 822 8 Conclusive remarks 824

C.1 . 3 ~

Space charge efSects 827 R.L. GLUCKSTERN 1 Introduction 827 2 Longitudinal space charge effect 828 3 Transverse space charge effect 830 4 Self-consistent models 831 5 Numerical models 834 6 Limitations 836

PROTON LINACS

TECHNOLOGY C.2.1

Pre-injectors 837 J.HUGUENIN, B.VOSICKI 1 Introduction 837 2 Ion sources 838 3 Optics 852 4 The accelerating tube 853

C.2.2a

Alvarez structure 879 C.S. TAYLOR 1 Introduction 879 2 Lay-out and brief description 880 3 Requirements 881 4 Actual structures 886 5 Buncher and debuncher 893

C.2.2b

High energy structures. Fabrication and tuning 897 E.A.KNAPP

Radiofrequency problems 905 I Introduction 905 2 Accelerating structure as a resonant load 906 3 The power source 925

4 The coupled system - amplifier and accelerating cavity 927 5 Controls and instrumentation 931

Beam monitoring, accelerator shielding and services 935 A. VAN STEENBERGEN 1 Beam monitoring 935 2 Proton linac shielding 950 3 Linac cooling and vacuum requirements 959 4 Description of a typical installation 966

Use of computers,for controls 971 H . BUTLER 1 Introduction 971 2 Tasks for the computer 971 3 Limitations of the computer 972 4 Advantages of a computer 973 5 Survey of applications 974

xvi

CONTENTS

VARIOUS TYPES Injectors for circular accelerators 977 P. LAPOSTOLLE 1 General layout of injection. Injection devices 977 2 Required characteristics 979 3 Typical performances 980 4 Improvement activities 983 5 Injectors for the generation of synchrotrons with several hundred GeV energy 985

Proton linacs for nuclear physics studies 987 J.M.DICKSON 1 Introduction 987 2 Comparison of injector and nuclear physics linacs 987 3 A nuclear physics machine 988 4 Beam quality 990 5 Variable energy 992 6 Polarized beams 993 7 Time-of-flight deflectors 994 8 Operation 994

Meson production

999

D.E. NAGLE

Intense neutron generator l o l l P. TUNNICLIFFE, G. BARTHOLOMEW 1 Introduction 1011 2 Conceptual arrangement of the Chalk River ING project 3 Accelerator 1013 4 Targets 1015 5 Special features and problems of the accelerator 1017

Possibility o f deuteron acceleration 1025 TH. SLUYTERS, M. REGLER 1 Introduction 1025 2 General considerations 1026 3 Acceleration in the CERN proton synchroton linear accelerator 1027

xvii

HEAVY I O N LINACS

Heavy Ion Linear Accelerators PART D

D. 1

Special problems in heavy ion acceleration 1029 CH. SCHMELZER 1 Introduction 1029 2 Requirements on particle energy and beam properties 1030 3 Ion sources 1031 4 Stripping 1034 5 The pressure transparency 1038 6 Heavy ion accelerators 1042

D.2

Existing and proposed heavy ion linear accelerators 1047 D. BOHNE, CH. SCHMELZER

1 Difference between proton and heavy-particle accelerators 1047 2 Existing machines 1051 3 Proposed heavy-ion linacs 1059

D. 3

Radiofiequency focusing in heavy ion linear accelerators 1073 D. BOUSSARD 1 Introduction 1073 2 Grid focusing 1074 3 Alternating focusing 1079

xviii

Superconducting Linear Accelerators PART E

E. 1

The surface impedance of superconducting metals at high frequencies 1089 A. SEPTIER 1 2 3 4 5

E.2

The historical background 1089 The surface impedance of metals at very low temperatures 1090 Macroscopic theories of high frequency conductivity in superconductors 1092 Recent theories of the rf conductivity 1095 The use of superconductors at uhf 1104

Theory and design of superconducting electron linear accelerators 1107 P.B. WILSON 1 Introduction 1107 2 Theory 1109 3 Design 1120 4 Limits on the gradient and current 1131 5 Application to accelerators 1135

E.3

Superconducting proton linear accelerators and particle separators 1141 A. CITRON, H. SCHOPPER 1 Introduction 1141 2 Accelerating structure 1143 3 Injection 1149 4 Beam dynamics 1150 5 Beam loading 1152 6 Parameters, design features and cost estimates 1163 7 Superconducting particle separators 1167

List of Contributors

F. Amman, Laboratori Nazionali di Frascati del CNEN, Frascati, Italy. J.Ballam, Stanford Linear Accelerator Center, Stanford, California, U.S.A. G. Bartholomew, Atomic Energy of Canada Ltd, Chalk River, Ontario, Canada. R. BergPre, Centre d'Etudes Nucltaires de Saclay, 91 Gif-sur-Yvette, France. J.P. Blewett, Brookhaven National Laboratory, Upton, New York, U.S.A. D.Bohne, lnstitut fur Angewandte Physik der Universitat Heidelberg, Heidelberg, Germany. D. Boussard, Centre EuropCen de Recherches NuclCaires, Geneva, Switzerland. P. Brunet, Laboratoire de 1'Acceltrateur Lineaire, Facultt des Sciences, 91 Orsay* France. X. BufSet, Laboratoire de 1'AcceICrateur Lineaire, FacultC des Sciences, 91 Orsay, France. L. Burnod, Laboratoire de I'AccCltrateur LinCaire, FacultC des Sciences, 91 Orsay, France.

H. Butler, Los Alamos Scientific Laboratory, Los Alamos, New Mexico, U.S.A. A. Carne, Rutherford High Energy Laboratory, Chilton, Didcot, Berkshire, England. A. Citron, Institut fur Experimentelle Kernphysik der Universitat Karlsruhe, Karlsruhe,

Germany.

M.C. Crowley-Milling, Daresbury Nuclear Physics Laboratory, Daresbury, Lancashire, England. J.M. Dickson, Rutherford High Energy Laboratory, Chilton, Didcot, Berkshire, England. G. DGme, Centre EuropCen de Recherches Nucltaires, Geneva, Switzerland.

A.L. Eldredge, Stanford Linear Accelerator Center, Stanford, California, U.S.A.

R.L. Gluckstern, University of Massachusetts, Amherst, Massachusetts, U.S.A. J.Haimson, Massachusetts Institute of Technology, Cambridge, Massachusetts, U.S.A.

xx

LIST OF

R.H. Helm, Stanford Linear Accelerator Center, Stanford, California, U.S.A. H.G. Hereward, Centre EuropCen de Recherches NuclCaires, Geneva, Switzerland. H. Hogg, Stanford Linear Accelerator Center, Stanford, California, U.S.A. J.Huguenin, Centre Europten de Recherches Nucltaires, Geneva, Switzerland. A. Katz, Centre d'Etudes NuclCaires de Saclay, 91 Gif-sur-Yvette, France. E.A.Knapp, Los Alamos Scientific Laboratory, Los Alamos, New Mexico, U.S.A. P.M. Lapostolle, Centre EuropCen de Recherches Nucleaires, Geneva, Switzerland. J.V. Lebacqz, Stanford Linear Accelerator Center, Stanford, California, U.S.A. H. Leboutet, Centre de Recherches Thomson-CSF, Corbeville, 91 Orsay, France. J.E. Leiss, National Bureau of Standards, Washington, D.C., U.S.A. A. V. Lisin, Stanford Linear Accelerator Center, Stanford, California, U.S.A. G.A. Loew, Stanford Linear Accelerator Center, Stanford, California, U.S.A. R. Miller, Stanford Linear Accelerator Center, Stanford, California, U.S.A. B. W. Montague, Centre EuropCen de Recherches NuclCaires, Geneva, Switzerland. D.E. Nagle, Los Alamos Scientific Laboratory, Los Alamos, New Mexico, U.S.A. R.B. Neal, Stanford Linear Accelerator Center, Stanford, California, U.S.A. T. Nishikawa, University of Tokyo, Department of Physics, Faculty of Science, Bunkyo-ku, Tokyo, Japan. V.G.Price, Stanford Linear Accelerator Center, Stanford, California, U.S.A. M.Prome', Centre d'Etudes NuclCaires de Saclay, 91 Gif-sur-Yvette, France. M. Regler, Institut fiir Hochenergiephysik der dsterreichischen Akademie der Wissenschaften, Vienna, Austria. Ch. Schmelzer, Institut fur Angewandte Physik der Universitat Heidelberg, Heidelberg, Germany. B. Schnizer, Centre Europeen de Recherches NuclCaires, Geneva, Switzerland. H. Schopper, Institut fur Experimentelle Kernphysik der Universitat Karlsruhe, Karlsruhe, Germany. A.L. Septier, Institut d'Electronique Fondamentale, FacultC des Sciences, 91 Orsay, France. Th. Sluyters, Brookhaven National Laboratory, Upton, New York, U.S.A. C.S. Taylor, Centre EuropCen de Recherches NuclCaires, Geneva, Switzerland. P. Tunnicliffe, Atomic Energy of Canada Ltd, Chalk River, Ontario, Canada. A. Van Steenbergen, Brookhaven National Laboratory, Upton, New York, U.S.A.

CONTRIBUTORS

B. Voiicki, Centre EuropCen de Recherches NuclCaires, Geneva, Switzerland. B.H. Wiik, High Energy Physics Laboratory, Stanford University, Stanford, California, U.S.A. P.B. Wilson, Stanford Linear Accelerator Center, Stanford, California, U.S.A.

xxi

PART A

Introduction

J. P. BLEWETT H. G . HEREWARD

1

The History of Linear Accelerators

I I

J. P. BLEWETT Brookhaven National Laboratory Upton, New York, U.S.A.

To one uninitiated in the accelerator field it would seem reasonable to expect that a linear accelerator is one in which particles are accelerated on a linear path. This is indeed a necessary condition but not a sufficient one. According to custom a linear accelerator is defined as one in which particles move on a linear path but are accelerated by radiofrequency fields, high energies being achieved by repeated acceleration through steps in energy which, individually, are much smaller than the final energy achieved. The term "linear accelerator" is often abbreviated to "linac", a term which now has general acceptance. The idea of reaching high energies by repeated small pushes is as old as the child's swing. As soon as Maxwell made it clear which electromagnetic fields were theoretically possible it must have occurred to many physicists that particles could be accelerated by radiofrequency fields But technology lagged far behind theoretical understanding. The first formal proposal for a linear accelerator seems to have been in 1924. A number of proposals and experimental devices appeared in the following two decades but a linear accelerator useful for research in nuclear physics was not born until after thevast development of microwave devices during World War 11. Then, within three years practical designs were made for both proton and electron linear accelerators. Progress since then has been rapid and extensive.

I

Ising's proposal and Wideroe's accelerator

The early proposal mentioned above was due to G. Ising [I9241 of Stockholm. It was References p . 17

;c) North-HollanrlPubl. Co., Amsterdam I969

1

2

J.P. BLEWETT

INTRODUCTION

called "The Principle of a Method for the Production of Canal Rays of High Voltage", and was published in the Arkiv fiir Matematik, Astronomi och Fysik. The principle of the device is indicated in fig. 1 reproduced from Ising's paper.* A positive ion beam (then known as "canal rays") emerged from the discharge tube shown at the left of the figure into a sequence of drift tubes. An electrical wavefront from a spark discharge was applied in sequence to the drift tubes. Transmission lines of increasing length delayed the arrival of the wave so that fields appeared across the gaps between drift tubes at the moment that the positive ions arrived. Thus acceleration should take place at every gap.

QII Fig. 1 . Ising's 1924 design for a linear accelerator.

Ising's paper concludes with the following remarks: "It would be premature to discuss now more closely the details and eventual difficulties of the realization of this idea. I hope to carry out some practical experiments soon... ." The proposal was indeed premature. No record appears of the "practical experiments" and it seems probable that Ising was frustrated by the rudimentary state of the electrical art. Maintenance of a steep wavefront through delay lines is not easy and Ising must have lacked any sort of equipment adequate to observe the propagation characteristics and frequency response of his systems. The first successful linear accelerator was described by R. Wideroe [I9281 in a paper that is one of the most significant in accelerator history. In this paper Wideroe described the first working linear accelerator, evolved the basic theory of the betatron and described an unsuccessful attempt to build a betatron. It was Wideroe's discussion of the linear accelerator in this paper that led E. 0. Lawrence to the invention of the cyclotron. It is of interest that in 1967 this versatile inventor is still at work producing ingenious ideas in the laboratory of the Brown-Boveri Company in Switzerland.

* The writer is indebted to M. S. Livingston and to Dover Publications, Inc. for permission to reproduce figures and text from "The Development of High Energy Accelerators", edited by M. S. Livingston (Dover, 1966). This book includes translations of the papers of Ising and Wideroe. Figures 1 and 2 are reproduced from the translated papers.

A. 1

THE HISTORY OF LINEAR ACCELERATORS

3

Wideroe's paper was entitled "A New Principle for the Generation of High Voltages." His accelerating system was similar in appearance to that of Ising but his drift tubes were connected alternately to a source of high frequency waves and to ground. The lengths of the drift tubes and the frequency were chosen so that ions ating phase. When the field was in the opposite (decelerating) direction the ions were shielded by the drift tubes. The configuration of the system is shown in fig. 2. Rather than a gas discharge ion source, Wideroe used a hot filament source of sodium or potassium ions. Three drift tubes were used, the first and third grounded and the central one excited at a frequency of about 1 MHz to voltages of about 25 kV. An electrical deflecting system was used to measure the energy of the accelerated ions. It was shown that, with an rf voltage of 25 kV, ions could gain up to 50 keV of energy. It was clear that this was a process that could be continued to higher energies.

Fig. 2. Wideroe's linear accelerator and its associated circuits.

Wideroe concluded that the process would be even more effective if ions of higher mass were used. In his words "Caesium ions ... should be favorable in this respect. Assuming an accelerating voltage of .. . 170 kV and five accelerating tubes, a voltage of about 1 MV could be attained." Such heavy ions were not those most desired for studies in nuclear physics where the most important experiments were to be performed with protons, deuterons, alpha particles, or electrons. But, at a given energy the lighter particles travel at much higher speeds and higher frequencies are appropriate for their acceleration. Power generators at such high frequencies were not available in 1928 nor for some years thereafter. So, in spite of this promising beginning, useful linear accelerators were still far in the future.

4

J.P. BLEWETT

INTRODUCTION

2 The accelerators of Beams and of Sloan and Lawrence In about 1931 K. H. Kingdon of the General Electric Company suggested an accelerator design to L. B. Snoddy of the University of Virginia. This seems to have been essentially the linear accelerator proposed in 1924 by Tsing although probably neither participant was aware of Ising's paper which had been published in a rather obscure journal. A couple of years later Snoddy together with J. W. Beams reported (Beams and Snoddy [1933]) on a traveling wavefront electron accelerator. In this device a 15 kV impulse was applied in sequence to 15 drift tubes. Electrons at 28 keV injected along the axis of the drift tubes were accelerated to about 90 keV. The work was continued by Beams and H. Trotter 119341 with higher voltage and fewer electrodes. With a 300 kV pulse applied to four electrodes in sequence, electrons were accelerated to 1.3 MeV. The final report from this group (Trotter and Beams [1935]) described the application of a 54 kV pulse from a Marx circuit to a six drift tube system. The peak electron energy reported was 2.5 MeV. As was usual in these early papers the reports reflected increasing optimism on the part of the experimenters. Then, suddenly, there were no more papers. There is no account of any application of the high energy electron beam. Probably the difficulties of maintaining a steep wavefront along what amounted to a loaded transmission line proved to be insuperable. At the University of California, E. 0. Lawrence was inspired by Wideroe's paper not only to invent the cyclotron but, with D. H. Sloan (Sloan and Lawrence [1931]) he constructed a much improved version of Wideroe's linear accelerator. Figure 3 is a sketch of this accelerator. Mercury ions from an arc discharge passed through 30 drift tubes which were connected alternately to the two ends of the inductance in a high frequency (7 MHz) oscillating circuit. Sloan and Lawrence were unable to afford commercially available vacuum tubes in the 20 to 50 kW range; hence they built their own continuously-pumped, water-cooled oscillators. (During the next few years, Sloan with Livingood and others built oscillators for power levels as high as 500 kW cw.) The home-made oscillators sufficed to excite the drift tubes to about 42 kV and the system delivered ions with energies up to 30 times 42 keV or 1.26 MeV. The final energy was measured in an electrostatic deflecting system. Sloan continued this work with W. M. Coates (Sloan and Coates [1934]). In a 36 drift tube accelerator mercury ions were finally accelerated to 2.85 MeV. Coates [I9341 tried bombardment of a number of targets with the mercury ion beam. No nuclear events were reported but X-rays of moderate intensity were observed from some materials. These accelerators, again, were in advance of the electronic art. The lack of sources of high power at very high frequency forced their designers to use heavy ions which were not useful at the energies achieved for nuclear physics. They were soon abandoned in favor of the cyclotron. Other Sloan-Lawrence accelerators were built during the late thirties, for example

A.l


5

at Cornell University and in Japan. The Cornell machine accelerated deuterons, the lightest ions accelerated in a linear accelerator up to that time. After the war, heavyion accelerators for higher energies were built for use in nuclear chemistry. They will receive attention later. Here we note only the fact that the first section of the heavy-ion accelerator a t the University of Manchester (Nassibian et al. [1961]) uses the SloanLawrence structure which, for heavy ions at low energies, presents certain advantages over modern designs. In the first section of the Manchester machine ions are accelerated to energies of about I MeV per nucleon.

w Fig. 3. The linear accelerator of Sloan and Lawrence.

3 Postwar developments At the beginning of World War I1 it was clear that useful linear accelerators for either electrons or protons would not be possible until methods were developed for generation and control of radiofrequency power in the megawatt range at frequencies of hundreds or thousands of megahertz. Some of these developments had begun during the late thirties. The invention of the klystron was announced in 1937 and conventional vacuum tubes had been pushed t o higher and higher frequencies by reducing their sizes and hence their power outputs. The major wartime advances were made under the pressure for radar protection for Great Britain and under the expanding pressures for radar for tactical uses and for countermeasures against enemy radar. Most important of the radar power sources

6

J.P.BLEWETT

INTRODUCTION

was the multianode magnetron which speedily reached multikilowatt power levels first at 3000 MHz, then at 10000 MHz, and finally at 30000 MHz. Such levels were not maintained continuously but were pulsed at low duty cycles as was appropriate for radar systems. For use at frequencies in the 100 to 1000 MHz range, triodes and tetrodes were developed but could not reach the power levels attainable with magnetrons. Paralleling these advances were major improvements for controlling and monitoring microwave systems. By the war's end it was clear that linear accelerators could be designed for useful operation and several laboratories started immediately on developments that led to the linear accelerators of today. Two of these were particularly important. At Stanford a group under W. W. Hansen proceeded toward construction of an electron machine, a project of which they had dreamed since prewar day. At the University of California design studies began under L. W. Alvarez; these studies were to result in the first successful proton linear accelerator. These two important projects will be presented in some detail in sections 4 and 7.

4 The Stanford linear electron accelerators The interest at Stanford in electron acceleration dates back to the mid-1930's when W. W. Hansen became intrigued with the properties of cavity resonators which could be excited to very high electric fields without the expenditure of unreasonable quantities of radiofrequency power. At the time Hansen's hope was that electrons could easily be accelerated to a few MeV for use in generating X-rays. He studied the then relatively unknown properties of certain cavity resonators which he christened "rumbatrons". In 1938 he became distracted from acceleration of electrons when he participated with the Varian brothers in furthering the development of the klystron. The klystron is an amplifier in which the mechanism of amplification involves "velocity modulation", a process that had been discussed in the literature since 1935. In the klystron, for the first time, cavity resonators were used as the input and output circuits. The first successful operation of klystrons was described in the literature in 1939 (Varian and Varian [1939]). Also collaborating with the Varian brothers were two graduate students, J. R. Woodyard and E. L. Ginzton, both of whom were to take leading roles in Stanford's postwar programs. Several klystrons were built for power applications but, in its wartime applications, the klystron was of primary importance as a source of milliwatts at radar frequencies for use as a signal generator or local oscillator. A second discouraging development was Kerst's successful betatron which appeared in 1939. This was obviously a cheap source of electrons of many tens of MeV. To match its performance a radiofrequency accelerator even using cavity resonators would require enormous quantities of high frequency power at frequencies then unattainable at high power levels. By the end of the war the situation was quite different. Several new power sources

A.l

THE HISTORY O F LINEAR ACCELERATORS

7

were available. Sloan and Marshall had developed a tetrode (the "resnatron", see section 8) specifically for use in electron linacs. The development of the magnetron made linear accelerators feasible for energies higher than the betatron could hope to attain. Billion volt accelerators appeared somewhat remote since 300 or so megawatts of pulsed power appeared to be required and single magnetron outputs had reached only a couple of megawatts. It was not clear then that many magnetrons could be synchronized with the necessary precision. The Stanford group noted the fact that development of power sources was necessary and, in 1946, began to build small linear accelerators using the available magnetrons. Simultaneously E. L. Ginzton, M. Chodorow, S. Sonkin and several others began a program aimed at development of sources for higher power and better stability than seemed possible with magnetrons. Hansen had surrounded himself with an extremely capable group. The work proceeded rapidly and by 1947 a design study was complete and a first accelerator had been built and tested (Ginzton, Hansen and Kennedy [1948]). It was shown that, since the electrons would be traveling over most of their path at extremely relativistic velocities, virtually no focusing would be necessary. A simple accelerating structure was chosen; this was simply a waveguide excited in a transverse magnetic mode (which includes an axial electric field). The phase velocity in the guide was reduced to the velocity of light by loading periodically with iris diaphragms. Frequency was chosen to be about 3000 MHz since power requirements decrease as frequency is increased but this was the highest frequency for which power in the megawatts was available. The accelerator described in this paper was about 9 ft long. In it electrons had been accelerated to 4.5 MeV. This accelerator, later to become known as the "Stanford Mark I", was later extended to 14 ft and reached a peak energy of 6 MeV (Becker and Caswell [I 95 I]). It was powered by a single magnetron which yielded 0.9 MW. Encouraged by this quick success Hansen and his associates submitted to the Ofice of Naval Research in 1948 a proposal for a billion volt accelerator to be 220 ft long. This rather daring proposal was approved with a rapidity that is amazing to present-day proposal writers and construction of the machine (to be called the Stanford Mark 111) was begun in 1949. As a prototype for the 1 GeV machine, a second short accelerator was built. This was the Stanford Mark I1 (Post and Shiren [1955]). It was about 12 ft long and, when eventually supplied with about 20 MW of rf power, it accelerated electrons to almost 40 MeV. A very powerful team attacked the design and construction of the billion volt accelerator. Important technical leadership was provided by R. L. Kyhl. A multiplicity of theoretical problems, many fundamental to the success of the machine, had been under attack for some years by E. L. Chu in collaboration with Hansen. In 1951, Chu produced a 237-page report which contained virtually the whole detailed theory of the electron linear accelerator (Chu [1951]). Two main topics were treated exhaustively, first, the theory of the field patterns in an iris-loaded waveguide and second, the dynamics of electrons in such fields. One of the most important of Chu's

8

J.P. BLEWETT

INTRODUCTION

contributions was an analysis of permissible dimensional tolerances in long electron linacs. As a result of his analysis, tolerances of 0.0002 of an inch were set on several important dimensions (diameter of the waveguide, diameter of the hole in the iris, and thickness of the iris). These are unusually severe tolerances and require very refined methods of fabrication. In the search for improved rf sources the Stanford group adopted earlier recommendations made by Hansen who had been confident that the klystron was capable of design for operation at multimegawatt levels. The development of the high-power klystron started in 1946 by Chodorow, Ginzton, Neilsen, and Sonkin [I9531 finally was successful in 1949. Chodorow had analyzed the operation of the klystron when its electrons have relativistic velocities and had shown that it should still be operable. Very long extrapolations were made from the low energy designs of klystrons that had been built previously and many difficulties were discovered and circumvented. It seemed essential to separate the vacuum system of the klystron and the accelerator so that it was necessary to pass the high output power through a "window" which gave endless trouble with rf breakdown. Klystron life initially was short but improved steadily. During the construction of the Mark I11 important changes took place in the staff. In 1949 the group was saddened by the death of Hansen. He was only forty years old. A minor consolation was the fact that he lived long enough to see successful operation of the first high-power klystron. Direction of the Mark I1 and the Mark 111 accelerator projects was undertaken by E. L. Ginzton who brought both machines to successful completion. In 1951 W. K. H. Panofsky moved to Stanford from the University of California. He was to take a leading role in Stanford's future programs. By 1951 30 ft of the 220 ft were completed. The accelerator worked from the moment of the first test. With three klystrons each supplying 8 MW, the 30-ft accelerator produced 75-MeV electrons. Ten-foot sections were added at a rate slowed down by the fact that an active experimental program was started almost immediately. By 1953 the length had reached 210 ft and the electron energy was over 600 MeV. This energy limit was set by the capability of the klystrons to yield output power. The Mark 111 accelerator was described in a 70-page paper in the February, 1955 "Linear Accelerator Issue" of the Review of Scientific Instruments (Chodorow et al. [1955]). This paper includes, in a long footnote, a valuable history of linear accelerator projects at Stanford. The experimental program included R. Hofstadter's historic measurements on the form factors of the proton and the neutron for which he was awarded the Nobel Prize in 1961. Five years later a 90-ft extension was added and the peak energy reached 900 MeV. In 1963 a major improvement program was instituted including dismantling of the whole machine and replacement of the accelerating sections by new units of improved design. This resulted in much improved performance and in 1964 the 300-ft Mark I11 machine reached its present operating energy of 1.2 GeV.

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5 SLAC


-

9

The Stanford Linear Accelerator Center

The success of the Mark I11 accelerator soon inspired speculation about a step to even higher energies. Hofstadter was the first to suggest consideration of a machine 10 to 20 times larger than the Mark 111. In 1955 small, informal discussions began of a new "Project M" (whether M stood for "Monster" or "Multi-BeV" is no longer certain). On April 10, 1956 the first official Project M meeting was held at the home of W. K. H. Panofsky. The group included Hofstadter, Ginzton, R. B. Neal who was later to be directly responsible for SLAC's construction, and a number of other Stanford physicists. It was agreed that a possible accelerator might be two miles long for initial operation at 15 GeV with eventual expansion (by addition of more klystrons) to 45 GeV. One year later a "Proposal for a Two-Mile Linear Electron Accelerator" was submitted to three agencies of the U.S. Government. The cost of the 15-GeV accelerator was to be of the order of $100 million. This was sufficiently higher than the cost of any previous accelerator that it caused a considerable amount of excitement in Washington. During the next couple of years panels advisory to the National Science Foundation and to the Atomic Energy Commission recommended construction, as did President Eisenhower. Hearings were held before the Joint Congressional Committee on Atomic Energy. For the first time the nation's program in high energy physics became a political issue and as such it was debated for several years. Finally, the project was approved by the AEC in 1962 and construction commenced. The design study had been directed by Ginzton who resigned in 1961 to become Chairman of the Board of Varian Associates. He was s~~cceeded by Panofsky, under whose capable direction the project moved swiftly toward a successful conclusion. The new accelerator was to be the central facility in a new national laboratory, christened the Stanford Linear Accelerator Center (SLAC). Its initial energy had been raised to 20 GeV. The machine was to be completed in 1966. In 1962 when the project was initiated the SLAC staff already included 170 people and the design had been worked out in considerable detail. Hence it was possible to proceed directly into construction. The staff was expanded rapidly to about 1000 people and the work moved ahead speedily and on schedule. In May of 1966 the two-mile machine was complete and a beam of electrons traveled through its whole length to emerge with an energy of 10 GeV. Within two weeks this was raised to over 18 GeV, and by January of 1967 the energy passed 20 GeV. During the construction period extensive preparation was made for the experimental program which began as soon as the machine was turned on. A major program is now in progress under the direction of high energy physicists from Stanford and about ten universities throughout the United States. A number of descriptions of the two-mile machine and its operation will be found in the proceedings of accelerator conferences collected at the end of this chapter.

10

J.P. BLEWETT

INTRODUCTION

It is also described in the monograph The Stanford Two-Mile Accelerator (Benjamin, New York).

6 Other linear electron accelerators Although, in the writer's opinion, the major contributions to the linear electron accelerator art have been made at Stanford, many other laboratories have played an important part in the development. In the paper of Ginzton, Hansen and Kennedy [1948], the authors note the existence of programs at the General Electric Company in Schenectady, M.I.T., Purdue, the Radiophysics Laboratory in Australia, the Telecommunications Research Establishment (T.R.E.) in England, the University of Virginia, and Yale. Of these the most important were the programs at T.R.E., M.I.T. and Yale. At T.R.E. a group headed by D. W. Fry and including W. Walkinshaw, L. B. Mullett, R. B. R-Shersby-Harvie and others made design studies that paralleled those proceeding at Stanford. Their results confirmed, or, in some cases, preceded the Stanford conclusions. Several electron linear accelerators were built by this group which soon moved to the Atomic Energy Research Establishment (A.E.R.E.) at Harwell. Several lines of investigation not pursued at Stanford were undertaken at A.E.R.E., for example the use of dielectric loading rather than iris loading, and the possibility of saving power in a traveling wave system by feeding back the power emerging at the high energy end of the waveguide to combine with the input power. References to the work at T.R.E. and at A.E.R.E. will be found in the paper of Chodorow et al. 119551. At M.I.T. the attack was led by J. C. Slater, who made an independent and exhaustive analysis of beam dynamics and accelerator structures (Slater [1948]). Under Slater's supervision, A. Kip, W. Bostick, P. Demos and others built a linear accelerator powered by a number of magnetrons and solved successfully the problems of keeping several magnetrons in phase. At Yale, under the direction of H. L. Schultz, a 10-MeV accelerator of a different design was constructed (Schultz and Wadey [1951]). Eight independent cavities operated at 600 MHz were separately driven by eight triodes. The same type of excitation was used more recently in the Los Alamos PHERMEX (Pulsed High Energy Radiographic Machine Emitting X-rays) accelerator (Boyd et al. [1965]). This machine, designed and now in use for flash radiography, includes three cavities excited by separate amplifiers at 50 MHz. Beam currents of 30 amperes are accelerated during pulses of 0.1 or 0.2 microseconds. During the period between 1949 and 1953 studies of electron and proton linear accelerators were carried on in the laboratory of P. Grivet in Paris by a group that included one of the editors of the present volume. Several electron accelerators were built and operated using 3000 MHz magnetrons as power sources; both traveling

A.l


11

wave machines (Vastel [1952]) and standing wave machines (Sarazin [1953]) were investigated. This work led to construction of a 5-MeV traveling wave accelerator that found application in French hospitals for use in X-ray therapy. In more recent years electron linac development has taken two directions. Electron linacs of a few tens of MeV have proved very useful in radiography and in medical applications. They have become commercially available from a number of sources. The second direction has been the push to higher energies, for the most part in government-supported institutions. At Orsay (near Paris), for example, a linear accelerator for between 1 and 2 GeV has recently been operated at energies above I GeV. Although conventional in concept this machine has several unconventional and ingenious mechanical design features. At Kharkov, in the U.S.S.R., a series of electron linacs has culminated in a machine operating at 2 GeV.

7 The Berkeley 32-MeV proton linear accelerator The second major postwar development took place at the Radiation Laboratory of the University of California where Alvarez assembled a very able group of physicists and engineers to undertake the design and construction of a proton linear accelerator. A most important achievement and one which later had profound effects on accelerator work both at Berkeley and at Stanford was Alvarez' success in persuading Panofsky to leave the Bell Telephone Laboratories and return to academic life at Berkeley where he remained until his move to Stanford in 1951. The problems in designing a proton linac differ materially from those to be met in an electron linac; in most cases they are more severe. At I MeV the velocity of a proton is less than 5 % of that of light. It is impractical to slow down phase velocities in waveguides to so low a value so some new accelerating system must be invented. At low phase velocities and high frequencies a radial variation in field appears which is quite intolerable at frequencies much above 200 MHz so cavities for acceleration of protons become much bulkier than those for electron acceleration. Finally, although the recently discovered principle of phase stability indicated that protons could automatically stay in phase with accelerating fields up to indefinitely high energies, it was speedily noted that protons stable in phase were automatically unstable in radial motion (in charge-free space) so that very soon after injection it could be expected that the beam would disappear due to radial defocusing. The Berkeley group set about solving these problems. Alvarez already had decided that the accelerating field should be set up in a long cylindrical cavity using the same transverse magnetic mode that is used in the smaller waveguides of the electron linac. The protons would pass along the axis of the cavity through a series of drift tubes of increasing length which would shield them from the rf field when it was in the decelerating phase. This is not the same system as was used by Sloan and Lawrence; in that system the particles passed through a drift tube in one half of an rf period and

12

J.P. BLEWETT

INTRODUCTION

fields at half of the gaps were opposite in direction to those at the others. In the Berkeley system protons require a full rf period to pass through a drift tube and all gap fields have the same direction. Operation at reduced frequency was not a bad problem. In fact, it happened that a considerable quantity of war surplus radar components usable at 200 MHz were available and could be adapted to power the new linac. The focusing problem was troublesome. It seemed that it could be solved by inserting dense grids across the upstream end of the bore hole of each drift tube-thin foils would be even better. So an attempt was made to mount beryllium foils less than a micron thick in the appropriate positions. Injection energy was chosen to be 4 MeV (from an electrostatic accelerator of energy higher than had been reached in any such machine then in existence), high enough that scattering in the foils would not be serious. But the foils were quickly destroyed by radiofrequency discharges and had to be replaced by heavy grids. Since these grids intercepted a good deal of charge, they were made rather sketchy and so had poor, though adequate, optical properties. Thus the major problems were solved and a machine 40 ft long was constructed. The rf cavity was a light copper structure about 3 ft in diameter; it was enclosed in a steel vacuum tank. Construction began in 1945 and the first 32-MeV proton beam was observed in 1947. A description of the Berkeley machine appeared in the "Linear Accelerator Issue" of the Review of Scientific Instruments (Alvarez et al. [1955]). This important accelerator has been the prototype for most subsequent proton and heavy ion linear accelerators. Only one basic change has been incorporated in more modern linear accelerators. It was shown at Brookhaven (Blewett [1952]) that alternating-gradient focusing could be used in linear accelerators to provide radial focusing with good beam optics and without interception of beam. The first test of this principle was on the Berkeley linac where electrostatic alternating-gradient focusing was included inside the drift tubes only a few months after announcement of the principle. Since rather large electric fields are necessary, subsequent linac designers have preferred to include magnetic quadrupoles in the drift tubes and to use magnetic alternating-gradient focusing. The historic 32-MeV accelerator at Berkeley was dismantled in 1958 and moved to the University of Southern California.

8 Other proton linear accelerators Until 1967, the highest energy reached in a proton linear accelerator was at the University of Minnesota where J. H. Williams and his associates constructed a machine for a maximum energy of 68 MeV (Day et al. [1958]). This machine differed from the first Berkeley linac in several ways. It was divided into three sections, the first accelerating to 10 MeV, the second to 40 MeV, and the third to 68 MeV. The injection energy was only 500 keV, thus simplifying markedly the problems of the

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13

injector. Finally. it was powered by tetrodes nicknamed "resnatrons" and patterned on similar tubes designed before the war for use on electron linear accelerators and improved during the war for use in radar countermeasures. The inventor of the resnatron is D. H. Sloan, who, some years before, had collaborated with Lawrence on a n early linear accelerator. Construction a t Minnesota began in 1949. By 1953 the machine was operating at 40 MeV and experimental programs at both 10 a n d 40 MeV were under way. By 1958 the full energy had been achieved. Since then this machine has been a very useful tool in many studies of nuclear structure. Linear accelerators are more expensive than cyclotrons or synchrotrons; hence very few have been built for use in nuclear physics. The proton linac has, however, found an important application as a n injector for proton sychrotrons. A 20-MeV linac serves as the injector for the 6-GeV Bevatron at Berkeley, a 9-MeV linac is the injector for the 10-GeV "synchrophasotron" at Dubna in the U.S.S.R., a 15-MeV linac injects into the 7-GeV "Nimrod" synchrotron in the U.K., and 50-MeV linacs provide injected protons for the 12.5-GeV ZGS at Argonne, the 28-GeV PS at C E R N , and the 33-GeV A G S at Brookhaven. During the summer of 1967 the Minnesota record was surpassed at Serpukhov in the U.S.S.R. where a 100-MeV linac came into operation. It is now in use as the injector for the Soviet Union's 76-GeV proton synchrotron. All of these machines use cavities and drift tubes similar to those found in the first postwar machine at Berkeley. Modern linacs combine the rf cavity and vacuum tank, using copper-clad steel tanks. Almost all use magnetic alternating-gradient focusing. The problems of rf power sources have largely been solved by industrial developments both in America and in Europe. Relatively reliable sealed vacuum tubes are now available for use around 200 MHz at levels of several megawatts. The largest linac injector under construction in 1967 is the 200-MeV machine which is to be the new injector for the Brookhaven AGS. It is scheduled for completion in 1971. It seems probable that a similar machine will be used as part of the injection system for the 200-GeV synchrotron to be built at Weston, Illinois.

9 Heavy ion linear accelerators For use in nuclear chemistry (for example for production of transuranic elements) several heavy ion linear accelerators have been built. Two virtually identical machines resulted from a design study held in 1953-54 a t Berkeley. The participants were a Yale group headed by R. Beringer and a University of California group headed by C. M. Van Atta. The accelerators were to be capable of accelerating ions in the mass range from helium to argon to energies of about 10 MeV per nucleon. In these machines a first stage of rf acceleration carries ions from a 500-keV injector to energies of I MeV per nucleon. The beamis then passed through a "stripper"

14

J.P. BLEWETT

INTRODUCTION

which may be a thin foil or a gas jet. During passage through the stripper the degree of ionization is raised by stripping of electrons from the ions. Subsequent acceleration is more effective because of the higher ionization and the beam is carried through a poststripper cavity to energies of about 10 MeV per nucleon. For heavy ions even lower frequencies are appropriate. The Yale and Berkeley machines are operated at 70 MHz; commercial vacuum tubes at the necessary power levels are available for this range. Because of the lower frequency the accelerating cavities are large in diameter (about 9 ft). The poststripper tank is about 90 ft long. The very large size of these tanks presented some problems in transportation, particularly around the hairpin turns in the road leading up the steep hill to the Berkeley Radiation Laboratory. Fortunately these were overcome and both machines have been in successful operation since 1958 (Hubbard et al. [1961]). Another heavy ion accelerator for essentially the same final energy but built in three tanks has already been mentioned in section 2. This is the accelerator at the University of Manchester whose prestripper accelerator uses the configuration of Sloan and Lawrence. A fourth heavy ion linear accelerator, again for a final energy of 10 MeV per nucleon, was built in the linear accelerator center at Kharkov in the U.S.S.R. during the period between 1958 and 1963.

10 Linear accelerators for high average currents At the Livermore branch of the Lawrence Radiation Laboratory several accelerators have been built for acceleration of continuous proton or deuteron currents of an appreciable fraction of an ampere. The first of these had its design based in 1948 on the maximum frequency for which high-power continuously-operating vacuum tubes were available. This was judged to be about 12 MHz and a tank 60 ft in diameter and 60 ft long was built. The drift tubes were so large that a short section of railroad track was constructed to run directly into the tank to facilitate their installation. The difficulties associated with this operation are easily imagined and it was only as a result of a major effort that continuous currents of 50 mA were accelerated to 10 MeV. Pulsed currents as high as 225 mA were attained. Industrial vacuum tube developments soon made it possible to raise operating frequency to about 50 MHz and the 60 ft tank was abandoned in favor of a series of smaller cavities. In this, the so-called A-48 accelerator, continuous currents of 75 mA of protons at 3.75 MeV or 30 mA of deuterons at 7.5 MeV were achieved. This effort was abandoned in 1958. During the period since about 1960 considerable interest has been manifested both in the United States and in Europe in accelerators to produce intense beams of mesons. Design requirements were for average proton currents of about 1 mA at energies between 500 and 1000 MeV. A lengthy and somewhat heated debate ensued between proponents of linear accelerators (Yale and Los Alamos) and of various

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15

sorts of cyclotrons [Oak Ridge, UCLA, and ETH (Zurich)]. In the United States a decision was made in 1967 in favor of Los Alamos and the linear accelerator; construction has been started on a "Los Alamos Meson Physics Facility" (LAMPF). The LAMPF accelerator is to be about 2500 ft long and will have a final energy of about 800 MeV. This machine will incorporate a new feature previously proposed at Brookhaven and at Yale. At an energy between 100 and 200 MeV the drift tube accelerator system loses efficiency until it is equalled by an iris-loaded waveguide system operated at a higher frequency. At Los Alamos the first 100 MeV will be of conventional drift tube design operating at 200 MHz. Above 100 MeV a change will be made to iris-loaded waveguide operating at 800 MHz. The principle of operation in this range is somewhat different from that used in electron linacs; it will be discussed elsewhere in this book (chapter C.l.1~). The most daring proposal for a high current linac comes from the Chalk River Laboratory of Atomic Energy of Canada, Limited. This is for a 1000-MeV linear accelerator to yield a continuous proton current of 65 mA. The purpose of this device is the production of enormous quantities of thermal neutrons, higher by a factor of ten than available from existing nuclear reactors. It has been named the ''Intense Neutron Generator" (ING). In 1966 a proposal was submitted to the Canadian Government. Since ING is estimated to cost about 3 150 million its approval is being considered somewhat deliberately. The I N G would be similar in principle to the LAMPF but would be subdivided into more accelerating sections because of the enormous power requirements. Its target is to be a rapidly flowing eutectic mixture of liquid lead and bismuth; it is in this target that the neutrons are produced. They will be thermalized by a heavy water moderator as in a reactor.

I 1 Recent developments In 1967 three important lines of development were evident. The first was the progress toward higher intensity in both proton and electron linacs. Peak currents of the order of 100 mA were reached in machines of both types. The old grid-focused proton linacs accelerated currents of a few hundred microamperes. With the first alternatinggradient-focused machine at Brookhaven it was hoped that a peak of 5 mA could be reached. That machine now runs at 30 or 40 mA. Peak currents approaching 150 mA have been accelerated in the 50-MeV linac at CERN. Under these conditions beamloading effects begin to appear and space-charge limits are approached. Much theoretical work during the early 1960's at CERN, Brookhaven, Los Alamos, and in the U.S.S.R. has led to a rather complete understanding of beam dynamics under conditions of high charge density. When electron linacs are used for acceleration of beams in the range of hundreds of milliamperes an unpleasant phenomenon appears; modes with transverse fields are

16

J.P. BLEWETT

INTRODUCTION

excited and the beam is lost. This phenomenon is now well understood and countermeasures are available. The second important development relates to accelerating structures. The doubly periodic structure (see chs. C . l . 1 ~and C.l.le) has resulted in considerable improvement in stability both in waveguide and in drift tube structures. This development, primarily at Brookhaven, Los Alamos, and CERN, is still in progress at the time of writing. Finally, work now in progress on superconducting resonators is aimed at linear accelerators for continuous operation. From experiments at Stanford, Karlsruhe, Brookhaven and elsewhere it seems that cavity losses can be reduced to negligible levels. Then the operation of linacs in short bursts of high intensity can revert to continuous operation at moderate intensity with consequent major improvement in the linac as a research tool.

12 Linear accelerator literature Until about 1955 progress on linear accelerators was recorded in the open literature. Since that time, however, there has been an increasing trend toward exchanges of reports only among the fraternity of linear accelerator builders. These reports become slightly more available when they are presented at accelerator conferences whose proceedings are published. Much information about linacs is to be found in the proceedings of the six International Accelerator Conferences held in 1956 and 1959 at CERN, in 1961 at Brookhaven, in 1963 at Dubna, U.S.S.R., in 1965 at Frascati, Italy, and in 1967 at Cambridge, Mass. The last of these conferences was held at Yerevan, U.S.S.R., in 1969. Further information will be found in the proceedings of three Accelerator Conferences held in Washington, D.C. in 1965, 1967, and 1969. Beginning in 1961, six Linear Accelerator Conferences have been held in the United States. They were held in 1961 and 1962 at Brookhaven, in 1963 at Yale, in 1964 at MURA, Madison, Wisconsin, in 1966 at Los Alamos, and at Brookhaven in May of 1968. Finally, a good example has been set by the SLAC staff who have prepared for publication a monograph on the two-mile accelerator (see section 5). It is to be hoped that similar publications will eventually become available describing other major accelerator installations.

13 Acknowledgments The author of this chapter is deeply indebted for many helpful comments and suggestions to L. Alvarez, E. L. Ginzton, W. H. K. Panofsky and G. W. Wheeler.

THE HISTORY O F LINEAR ACCELERATORS

References Articles in periodicals and Laboratory Reports

ALVAREZ, L. W., H. BRADNER, J. V. FRANCK,H. GORDON,J. D. COW, L. C. MARSHALL, F. OPPENC. RICHMAN, and J. R. WOODYARD, 1955, Rev. Sci. Instr. 26, 111. HEIMER, W. K. H. PANOFSKY, 1933, Phys. Rev. 44, 784. BEAMS, J. W. and L. B. SNODDY, J. W. and H. TROTTER, 1934, Phys. Rev. 45, 849. BEAMS, BECKER, G . E. and D. A. CASWELL, 1951, Rev. Sci. Instr. 22, 402. BLEWETT, J. P., 1952, Phys. Rev. 88, 1197. and D. VENABLE, 1965, Rev Sci Instr. 36, 1401. BOYD,T. J., Jr., B. T. ROGERS,F. R. TESCHE, M., E. L. GINZTON,I. R. NEILSEN,and S. SONKIN,1953, Proc. I.R.E. 41, 1584. CHODOROW, CHODOROW, M., E. L. GINZTON,W. W. HANSEN, R. L. KYHL,R. B. NEAL,W. K. H. PANOFSKY, and the Staff of the W. W. Hansen Laboratories of Physics, 1955, Rev. Sci. Instr. 26, 134. CHU,E. L., 1951, Report No. 140, Microwave Laboratory, Stanford University. COATES, W. M., 1934, Phys. Rev. 46, 542. DAY,E. A,, R. P. FEATHERSTONE, L. H. JOHNSON,E. E. LAMPI,E. B. TUCKER,and J. H. WILLIAMS, 1958, Rev. Sci. Instr. 29, 457. GINZTON, E. L., W. W. HANSEN, and W. R. KENNEDY, 1948, Rev. Sci. Instr. 19, 89. E. L., W. R. BAKER,K. W. EHLERS,H. S. GORDON,R. M. MAIN,N. J. NORRIS,R. PETERS, HUBBARD, L, SMITH, C. M. VAN ATTA, F. VOELKER,C. E. ANDERSON,R. BERINGER, R. L. GLUCKSTERN, W. J. KNOX,M. S. MALKIN,A. R. QUINTON,L. SCHWARCZ, and G . W. WHEELER,1961, Rev. Sci. Instr. 32, 621. I s r ~ cG., , 1924, Arkiv for Matematik, Astronomi och Fysik 18, 1. NASSIBIAN, G., J. R. J. BENNETT, D. BROADBENT, S. DEVONS, R. W. R. HOISINGTON, and V. E. MILLER, 1961, Rev. Sci. Instr. 32, 1316. POST,R. F. and N. S. SHIREN,1955, Rev. Sci. Instr. 26,205. SARAZIN, A., 1953, Ann. de Radioelectricitt 8, 134 and 228. SCHULTZ, H. L. and W. G. WADEY,1951, Rev. Sci. Instr. 22, 383. J . C., 1948, Rev. Mod. Phys. 20, 473. SLATER, 1931, Phys. Rev. 38, 2021. SLOAN, D . H. and E. 0. LAWRENCE, SLOAN, D . H. and W. M. COATES,1934, Phys. Rev. 46, 539. TROTTER, H. and J. W. BEAMS,1935, Phys. Rev. 47, 641. VARIAN,R. H. and S. VARIAN,1939, J. Appl. Phys. 10, 321. J., 1952, Ann. de Radioelectricite 6, 20. VASTEL, WIDEROE, R., 1928, Archiv fiir Elektrotechnik 21, 387.

Conference Proceedings International Accrlerator Conferences:

CERN Symposiun~on High Energy Accelerators and Pion Physics, 1956, edited by E. Regenstreif (published by CERN, Geneva, Switzerland) 567 pp. lnternational Conference on High-Energy Accelerators and Instrumentation, 1959, edited by L. Kowarski (published by CERN, Geneva, Switzerland) 705 pp. Proceedings of the lnternational Conference o n High Energy Accelerators, 1961, edited by M. H. Blewett (published under the auspices of the Division of Technical Information, U.S. Atomic Energy Commission) 490 pp. Proceedings of the International Conference o n High Energy Accelerators, 1963, edited by A. A. Kolomensky (Atomizdat, Moscow, U.S.S.R ) 1092 pp. These proceedings are partly in Russian and partly in English (see next entry). A two-volume translation into English of all of the papers in the preceding item has been prepared by the U.S. Atomic Energy Commission and is available from the Clearinghouse for Federal Scientific and Technical Information, National Bureau of Standards, U.S. Department of Commerce, Springfield, Virginia.

18

J.P.BLEWETT

INTRODUCTION

Fifth International Conference on High Energy Accelerators, 1965, edited by M. Grilli (published by the Comitato Nazionale per 1'Energia Nucleare, Rome, Italy) 752 pp. Sixth International Conference on High Energy Accelerators, 1967, edited by R. A. Mack (published as Cambridge Electron Accelerator Report CEAL-2000) 702 pp. (This report is also available from the NBS Clearinghouse.)

. ... ,-

' , ?, . ,

.

.'

.

. ,.,

.'

'

National Accelerator Conferences: The First National Particle Accelerator Conference, 1965, IEEE Trans. Nucl. Sci. NS-12, No. 3, 1-1073. U.S. National Particle Accelerator Conference, 1967, IEEE Trans. Nucl. Sci. NS-14, No. 3, 1-1233. Particle Accelerator Conference, 1969, IEEE Trans. Nucl. Sci. (in press). I

Lirieay ~.heleratorConferences: Minute$ gf the Linear Accelerator Conference held at Brookhaven National Laboratory during April, f.961, edited by A. van Steenbergen (published as Brookhaven Accelerator Department Report x.4 AvS-1) 109 pp. (out of print). Minutes of tHe Conference on Linear Accelerators for High Energies, 1962, edited by A. van Steenbergen, H. B. Knowles, S. Ohnuma, and C. K. Sinclair (published as Brookhaven Report BNL 6511) '385 pp. (out of print). Minutes of .the Conference on Proton Linear Accelerators at Yale University, 1963, edited by H. B. Knowles (published by the Physics Department of Yale University, New Haven, Conn. 1964) 479 pp. Minutes of the 1964 Conference on Proton Linear Accelerators held at Midwestern Universities Research Association, 1964, edited by F. E. Mills, C. D. Curtis, D. A. Swenson, and D. E. Young (published as MURA Report No. 714) 629 pp. (This report is available from the Clearinghouse for Federal Scientific and Technical Information, National Bureau of Standards, U.S. Department of Commerce, Springfield, Virginia.) Proceedings of the 1966 Linear Accelerator Conference held at the Los Alamos Scientific Laboratory, 1966, edited by J. W. McDonald (published as Los Alamos Report LA-3609) 516 pp. (This report is also available from the NBS Clearinghouse.) Proceedings of the 1968 Proton Linear Accelerator Conference held at the Brookhaven National Laboratory, 1968, edited by J. W. Bittner [published as Brookhaven Report BNL 50120 (C-54)] 807 pp. (This report is also available from the NBS Clearinghouse.)

2

The General Theory of Linear Accelerators H. G . HEREWARD CERN, Geneva, Switzerlund

I

Phase stability

Linear accelerators range from ones where the particles are accelerated by a succession of distinct gaps to those where a travelling wave produces a smooth, continuous, force. We look first at the elementary principles of the travelling-wave accelerator, for most of its concepts and results can readily be extended to the general case. The first requirement is to have a wave with an electric field component in the direction of the beam, and with a wave velocity equal to the intended speed of the particles. Particles on the appropriate half of the wave will be accelerated, and, if the peak electric field exceeds .a .certain threshold depending on the rate of increase of wave velocity, there will be two phases (fig. 1) at which a particle can stay, moving along with the wave and gaining velocity at the same rate as the wave. In the neighbourhood of cp, there is "phase stability", since a pargcle ahead of it will gain energy more slowly, become slower than the wave, and drift back towards cp,; while a particle which is a little behind receives some extra acceleration and will tend to catch up. The result is that particles which start not too far from cp, - and not too far from the correct velocity - will oscillate about the synchronous phase cp,, gaining energy on the average at the correct rate. The same kind of argument shows that -9, is an unstable point; particles a little way from it will be driven still further away. The principle of phase stability was invented, in connection with the synchrotron, by Veksler [1945], and by McMillan [1945]. The first mentions of it for the linac case seem to be Slater [1946], and Alvarez [1946]. As the velocity of light is approached the particle velocity becomes almost in0 ~Vorth-HollandPubl. Co., Amsterdam

References p. 37

1969

19

20

H.G.HEREWARD

INTRODUCTION

dependent of its energy, and the frequency of these phase oscillations becomes so low that they practically do not occur. Then there is no threshold, and one aims to put all the particles close to the crest of the wave, where they will accumulate the maximum of energy and the minimum of energy spread. In practice proton and heavy-ion linacs operate in the regime of phase oscillations, while electron linacs have vrv c at least over most of their length. This makes an important difference to their operational characteristics: a section of proton linac will only accelerate over the range of energies for which it is designed, and its rf level must be above threshold; while a section of v = c electron linac will accelerate any sufficiently relativistic particle at any rf level.

Fig. 1. The accelerating wave is considered to be moving from left to right. The broken line indicates the field necessary for keeping up with the wave.

2 Tvansverse defocusing This phase stability, or longitudinal focusing, that can be obtained by accelerating on the front of a travelling wave, must unfortunately be paid for in the transverse motion. We shall obtain more detailed formulae in the next sections; for the moment consider the situation in a frame in which the wave and the synchronous particle are at rest, so that the force can be derived from an electrostatic potential. Neglecting the space-charge of the beam itself, and supposing that there is no other charge density within the beam, we have divE = 0 ,

(1)

and multiplied by -elm this can be expressed o:+o;+o:=o,

where o,is the frequency of small-amplitude phase oscillations in the neighbourhood of the synchronous particle. By choosing to work at the phase-stable cp, we make o, real, so the transverse motion must have at least one negative 02, and be unstable. The Lorentz transformation to the laboratory frame changes all frequencies by a factor y-l=J(l -P2), so leaves the relation (2) valid.

THE GENERAL THEORY OF LINEAR ACCELERATORS

A.2

21

In practice some means of transverse focusing is provided, but the defocusing contribution of the rf wave is still, in most cases, determined by this equation. A general proof of this incompatibility between longitudinal and transverse stability has been given by McMillan [1950]. I t is mentioned by Alvarez [1946], who proposed to provide focusing by use of thin metal foils in the beam - so that one no longer has divE = 0.Later the foils were replaced by grids. Alvarez et a/. 119551 discuss rf defocusing and the effect of grids or foils in terms of the fields in a single accelerator gap.

At the present day quadrupole magnets are the most important means of focusing in linacs. They are dealt with in chs. B.1.2 and C.1.2~.

3 Properties of the travelling wave The electromagnetic field can be expressed as the sum of a TE field with E, =0, and a T M field with B, =O. In the case of cylindrical symmetry these fields are independent and only the TM is of interest for acceleration. The helix is a structure in which the TE and T M fields are inevitably coupled, but the Bzhas anegligible effect o n the motion (Johnsen and Dahl [1951]). Cylindrical symmetry is also lacking inseveral other types of structure, such as the cross-bar and the multistem (ch. C.l.le), and the input couplers for corrugated waveguide (ch. B.l.l and B.1.2), for reasons connected with power flow; but the field asymmetry in the central region occupied by the beam can be kept small. There are the special structures of ch. D.3, where the asymmetry has a n important effect on the transverse motion; and the unsymmetrical modes which are used in separators (ch. B.3.7), and may cause beam break-up in linacs (ch. B. 1.4), must also be mentioned.

For the TM field with cylindrical symmetry and no sources, we have in polar coordinates ( r , 9, z )

and Maxwell's equations become

-

dB,

az

1 a (rB,) r

dr

z

c-'~,

= cC2 E=

H.G. HEREWARD

22

INTRODUCTION

One can eliminate B, between (5) and (6), and, with (4), obtain the wave equation for Er or E,. The variables can be separated: in particular, if the rf has frequency 0 / 2 n and we consider a wave with phase velocity pc=v we can put E , = f ( r) cos ( a t - o z / o ) and (4), (5), (6) require that f ( r ) shall satisfy

and therefore is

where I, is the modified Bessel function I, ( x ) = 1

+ x2/4 + x4/64 + ... .

(12) The other field components are then given too. In particular, the radial force on a charge e having velocity pc in the z-direction is 1 e(Er - PCB,) = - EI, -

Y

(g)

sin ( w f - o z / u )

with I, ( x ) = x/2

+ x3/16 +

. . a .

(14)

Note that we derived these expressions from the assumptions (3) and Maxwell's equations. Consequently they are valid for a TM wave of velocity PC, with cylindrical symmetry and in empty space, regardless of the ndture of the surrounding rf structure that carries it. f If the wave velocity varies with z one must write

in place of wt - wz/v; then the expressions remain good approximations provided the variation is slow (Chu [1951]). A slow variation of E is also admissible.

4 Longitudinal motion Following Lloyd Smith [I9591 we use z, rather than t, as the independent variable. The equations of motion for a particle on the axis are

A.2


dW dz -

= eE, = e E

23

cos (at -

where W is the total particle energy, mo its rest mass, y, = w/moc2,and we have introduced k = w / c = 2n/lo.We now suppose that there exists a synchronous particle which has velocity equal to that of the wave*, and that d W, = e E cos i p s . dz

(18)

--

The value of E has to be suitably related to the variation of /3, along the accelerator to make this possible. For any other particle (16) gives

where we have put W = W,+ A W and work to first order in A W; and (17) becomest dA W dz

-

-

= eE (cos q - cos cp,)

.

The two first-order differential equations (19) and (20), or their second-order equivalent

govern the longitudinal oscillations. Three aspects repay study, the frequency for small amplitudes, the range of stability, and the effect of variation of coefficients. For small amplitudes the equations may be linearised by putting cp - q, = d i p and working to first order in Aq. Then (20) becomes dW = - e E sin i p s . Aq. dz

--

If the coefficients p,3Y;3 and E sin ips were constant, (19) and (22) would determine simple-harmonic oscillations of Acp and A W with wavelength A, given by

(

1

-

BhZmoc2 - -)I ~ , / 2 n= - keE sin q,

In fact the quantities vary with z ; provided the variation is slow (23) is still a good approximation, and these phase oscillations have a slowly varying wavelength. The

*

With some care it is possible to generalise to a slowly changing ps. t It is mathematically inconsistent t o work to first order in A W and then keep the full nonlinear expression on the right of eq. (20), but in practical cases A W is small compared with the kinetic energy even for particles oscillating to the extreme limits of stability.

H.G. HEREWARD

24

INTRODUCTION

form is

with, in general, A and a slowly varying too. We see that eE sincp, must be negative, corresponding to acceleration on the front of the wave in fig. 1, for otherwise the solutions will be of hyperbolic type. The decreasing frequency of the oscillations with increasing y is also evident from (23). It is convenient to consider such particles as tracing out small ellipses in a Aq, A W plane, with the ratio of axes amplitude of A W amplitude of Acp

=

[eE(- sin g,) j?:y:mOc2/klr.

To investigate the stability limits we again treat the parameters as constants, but instead of linearising we look for a constant of motion with validity extending to large amplitudes. The method is classical: we multiply (21) by dAq/dz, and using primes for d/dz, P s3 Y 3 Acp' Acp" = - keE ( C O S-~coscp,) Acp', moc

,

S

and integrate with respect to z

fP:y: ( A ~ ' ) '= -

4 1

moc keE --

m 0c which can also be written k 2p:y:moc

(Aw

(cos g

(sin cp

-

-

cos p,) dip

g cos cp,

+ C)

) +~eE (sin cp - cp cos cp, + C) = 0.

For each value of the constant C this formula gives a possible trajectory in the cp, A W plane, as illustrated in fig. 2. One sees that the stationary but unstable point at cp = - g, is confirmed, and furthermore that this point determines the limiting amplitude of phase oscillations. The special trajectory obtained by putting C = (sin cp,

-

g, cos 9,)

is a separatrix, and bounds the region of stable oscillations. The maximum stable energy excursion is readily found by putting q=qs on the separatrix: one gets A W = i2 [k-1eEmo~2/ls3ys3(qs cosvs- sinqs)]$. (29) The limiting stable excursion in phase towards the negative side can also be obtained from the separatrix, but only by numerical solution. It is equal to qs for q, small, and only 3 % less at qs = - 30'. The numerical properties of this stable region, particularly its area, have been extensively tabulated, but usually in the notation appropriate to the phase oscillations in synchrotrons (Gumowski [1967]).

A.2


25

It is common in the literature to refer to this stable region as a "bucket": the idea is that one can put particles in it, in order to lift them to a higher energy. One is not obliged to fill the bucket with particles. A reasonably compact area that is filled with particles being accelerated is called a "bunch".

The effect of the slow variation with z of the coefficients in the phase-oscillation equation is best discussed in the light of the notions of the next section.

Fig. 2. At the top, the accelerating field as function of phase, the synchronous energy gain WIs shown as a broken line. Next, the effective potential well, obtained by integrating cEz - W's. Bottom, phase-plane trajectories for such a potential, including the separatrix.

5 Liouville's theorem and adiabatic processes The factor j:y: certainly varies along the linear accelerator, and there is often a n interest in considering variable E and cp, too. For some special cases there are still

H.G.HEREWARD

26

INTRODUCTION

explicit solutions of the longitudinal equations, but it is usually more informative to appeal to Liouville's theorem and the notions of adiabaticity. Liouville's theorem states that in a Hamiltonian system the area of phase-space occupied by any set of particles, and the particle density in phase-space in the neighbourhood of any given particle, are constants of motion. Otherwise expressed: the particles move in phase-space like an incompressible fluid. It is proved in textbooks of statistical mechanics (Tolman [1938]). The theorem is more familiar in a phase-space of momenta and coordinates, with time as the independent variable, but nothing prevents us using it in the plane of energy and time (or phase, since o is constant) with z as independent variable. We see that (19) and (20) have precisely the form of canonical equations

if we take as Hamiltonian

H=-

k

,

( A w)' - eE (sin q - cp cos 9,). 2P,3?+0c Therefore Liouville's theorem applies to the motion in the Aq, A W plane. Taken alone, Liouville's theorem is useful mainly for making negative statements. For example: one cannots contrive that the product of energy-spread and phasespread of a bunch decreases steadily along the length of a linac. The concept of phase-space conservation for the transverse motion in accelerators became current during the development of the strong-focusing idea - see, for example Goward [1953]. The formulation in terms of Liouville's theorem for the longitudinal motion in linacs was by the present writer (Hereward [1954]). The idea of an adiabatic process can be illustrated by considering a bunch of particles bounded by one of the oval "trajectories" of fig. 2. If the parameters were constant, they would stay inside this curve, going round with their phase-oscillation frequencies. When the parameters change, the shapes of the curves change too, and the details of the particle motion become quite complicated; but, for sufficiently slow changes, the boundary of the bunch will at each moment be one of the new "trajectories", calculated from the current values of parameters. To illustrate the use of this, consider those particles which enter a linac within one of the small ellipses of eq. (25). With subscript 1 to indicate the beginning of the linac, this ellipse has axis ratio:

And, if the changes are slow enough to be adiabatic, at the end of the linac the same particles will be in an ellipse with axis ratio: AW, A rp2

--

= [eE,(-

sincp,, ,).&

y:,2

moc2/kjt.

(33)

A.2


27

With adiabaticity giving us the shape, we appeal to Liouville to tell us the size:

and the amount phase-spread damping follows

The importance of this method lies in its reliability and its generality. For Liouville's theorem a Hamiltonian must exist, but it is not necessary to know it accurately: in contrast, other methods may give quite wrong results for the damping if certain small terms are neglected (Carne et al. [I965]). To use the adiabatic law we need only the equations of motion at the beginning and the end of the process; how they change in between is irrelevant so long as it is slow. If we have sufficient data about trajectory shapes and areas, the method remains usable for non-linear amplitudes.

Fig. 3. The "golf-club" acceptance diagram. Particles coming from outside the bucket pass the unstable fixed point at a ; by the time they come back to this neighbourhood at b the bucket has expanded and they find they are inside.

There is, however, a limit. Particles close to the separatrix pass close to the stagnation point at - cp,, and have a very low phase oscillation frequency: however slowly the parameters change, some of these particles will find it too fast to be adiabatic. The area within the separatrix of fig. 2 is, at constant E and q,, proportional to &*, as one may see from (29); and this typically increases relatively fast at the beginning of a linac. One result of this is the "golf-club" shape of acceptance region computed by R. Taylor [1959]. When the area of a stable region expands, Liouville's theorem indicates that it must acquire the extra area from the surroundings, and the way this takes place is shown in fig. 3.

H.G.HEREWARD

28

INTRODUCTION

6 Rf power considerations The requirements in rf power of linear accelerators are usually enough to make considerations of power economy play a big part in the accelerator design. Since the power dissipated per unit length of a structure is proportional to the square of the rf field in it, a useful characteristic of the structure is the shunt impedance per unit length, defined by

where E is the amplitude of the useful travelling wave. Thus the energy gain of an accelerator, or section of an accelerator, say e v , is given by eV

= eEL

cos cp,

= ~ ( z , P L ) cos + cp,

(37)

where L is its length and P is the power dissipated. Outside the field of linear accelerators the term shunt impedance is conventionally defined somewhat differently; the r.m.s. rather than the peak field is used, making Zs a factor two less. Also our use of E, the useful wave amplitude, in (36) means that any adverse transit-time factor for accelerating gaps is effectively included in our Zs, so does not appear in (37). On the other hand cosp, appears in (37), rather than being assimilated into Zs, because it is not a property of the structure (see also ch. C.l.le).

The cost theorem of Slater [I9481 is based on (37) and the simple assumption that the cost of a linear accelerator can be expressed as the sum of certain fixed costs, a cost proportional to length, and a cost proportional to the rf power required: Cost

=a

+ bL + cP.

(38)

Minimising this with respect to P and L, subject to the energy gain held constant, gives

that is, at the minimum the length-cost and the power-cost are equal. Rather commonly such a calculation yields a short accelerator operating above the electric breakdown limit, showing that in fact minimum cost will be obtained by choosing a field strength as close to the breakdown limit as reliability arguments will permit, and that the length-cost will be greater than the power-cost. Another quantity of great importance is stored energy energy d~ssipatedper radian of rf

Q=-----

A.2


29

and the closely related time-constant

Q w

-

stored energy power dissipated

Apart from some special cases, such as superconducting linacs, or others with high beam loading, the installed rf power is not much more than the dissipation at operating level, and the rf supply is usually pulsed, so it will take a time like Q / o at the beginning of each pulse to provide the stored energy of the structure. The value of Q also determines the frequency tolerance; the details are different for different arrangements, but it is always true that a frequency error of one part in Q corresponds to a phase error of one radian after a time equal to one time-constant Q/w; and a phase error of one radian must be regarded as large. It is clear that any change in a structure that just reduces the losses will raise both the Q and the shunt impedance; the reduced power requirement is an advantage, offset a little by the extra filling time. One should also note the advantages of keeping the stored energy low; if one can choose a structure with the same shunt impedance but less stored energy, the power requirement will be the same but the filling time and tolerances will be easier. So, a t constant shunt impedance one prefers a low Q. The quantity ZJQ is independent of the losses, and is frequently used as a factor of merit, to measure how low is the stored energy at given accelerating field level. The perturbation technique, which compares IE l 2 at each measured point with the total cavity stored energy, is essentially a method of obtaining Z,/Q (see ch. B.l.l). The choice of operating frequency is influenced by many practical matters, including the availability of rf power equipment, and has to be considered for particular cases; but it is worth mentioning the scaling laws that apply if one changes the rf wavelength of a linac, keeping its length and energy range constant. Rather obviously: transverse dimensions (including aperture) K A , volume KL', electromagnetic fields KAO, total stored energy c c A 2 . In the skin-effect regime the surface resistance of copper is proportional to Apt, then: surface area rcA, total dissipated power K L ~ , shunt impedance K X * , energy time-constant ~ / w c c A * Q , x A ? The group velocity of a travelling-wave structure scales unchanged, so strict geometric scaling cannot keep Q/w and the propagation time in proportion; one solution is to divide the linac into sections of length proportional to A*. The wavelength for phase oscillations and the accepted energy spread scale like At[eqs. (24) and (29)l. One would like to scale the transverse focusing system in such a way as to keep q (ch. C . 1 . 2 ~ )constant; this would hold constant the ratio between applied focusing and rf defocusing [eq. (2)]. Then the wavelength of the transverse oscillations scales hke I.* and ideally the period of the focusing magnet system should d o the same, but this probably conflicts with the period of the rf structure. By disregarding this limieach tation we can at least have an idea of the trends: the aperture scales with i, transverse acceptance like A+, and the magnet pole-tip field scales unchanged. If the beam emittances are scaled to follow the acceptances, one can keep constant the

30

H.G. HEREWARD

INTRODUCTION

relative strength of space-charge defocusing (both transverse and longitudinal) by scaling the beam current acI1, but the brightness then decreases with increasing I. There is an optimum wavelength if current and brightness are both desired.

7 Electron linac structures: periodic structures An accelerating wave of the form (9) cannot be propagated in a uniform cylindrical metal waveguide, because the boundary condition E,=O on the cylinder wall is incompatible with (1 1) - at least, unless the velocity of the wave is greater than c. The wave may be slowed down by loading the tube with a dielectric (Fry and Walkinshaw [1948]), but practically all existing electron linacs employ disc-loaded waveguide, consisting of metal discs with central holes (irises) equally spaced along the tube (fig. 4), the whole being made of copper. There are several different ways of describing the behaviour of such a structure, and it will be discussed more fully in later chapters, but many of its most interesting features are due to the property - which it shares with most other types of linac structure - of being periodic. We follow Slater [I9481 in giving a general outline of wave propagation in periodic structures, based on Floquet's theorem, and taking the structure to be lossless. This theorem may be stated: in a given mode of oscillation of the structure, at a

Fig. 4. Typical disc-loaded accelerator waveguide. The dimensions (in mm) are those of the Stanford Mark I11 machine.


A.2

31

given frequency, the values of the fields are multiplied by a given complex constant when we move down the structure by one period. This results from the fact that if the whole structure is displaced along the z-axis, by one period d, it coincides with the original structure. There are certain intervals of o , called stopbands, where this complex constant is in fact real, and not 1. Then one has "evanescent modes" and there is no propagation. But the pass-bands are the part of the spectrum that interest us: here the complex constant has unity modulus, so can be written as exp(-j$), and $ is interpreted as the phase-shift of the travelling wave per period of the structure. The curve of w as a function of $ is called the dispersion characteristic; in fig. 5 we sketch it for the lowest pass-band of a disc-loaded structure. It is usual to draw such a diagram only for Ocity as the particle, i.e., for all practical purposes, at the velocity of light. There e .-\lo ways in which this effect should, in principle, be obtainable. One is illustrs .d in fig. l a where a sinusoidal electric field wave is propagating in a cylindric . d~avzguidein the TM,, mode. It is representative of a traveling-wave type of structure. ii synchronism between the particles* and the wave crests could be

* In this 4 apter, it ,ill be aswmed that the process of particle bunching, described in ch. B.1.2, has already Liken place and t' ~t these bunches of small phase extent are spaced a free wavelength 1.0 apart.

B.l.l

ACCELERATING STRUCTURES

41

obtained, the energy gained by the particles over a length I, neglecting attenuation, would be V = &I. The other is illustrated in fig. 1b which shows an array of "pillbox" cavities. It is representative of a standing-wave type of structure. Each cavity is individually supplied with rf power and resonates in the TM,,, mode. No power is actually flowing in the beam direction. However, if the phases of the individual power sources are arranged as shown, the particles will be submitted to a cumulative force. This results from the fact that the square wave is made to appear to travel in synchronism with the particles. The magnitude of the net accelerating field is reduced slightly

Fig. la. Unloaded TMol cylindrical waveguide showing mode pattern and traveling-wave axial sinusoidal electric field amplitude.

Fig. lb. Chain of independently powered TMolo cavity resonators and example of in;iantaneous axial electric field amplitude along z-axis for 7r/2 phase shitt pcr cavity.

G.A.LOEW, R.B.NEAL

42

ELECTRON LINACS: THEORY

in relation to the square wave amplitude Eo because of two factors. The first is a transit time factor which comes about because the field in the cavity is changing sinusoidally with time while the particles are traversing it; its magnitude can easily be shown to be equal to sin [(rill,) ( d - t ) ] / [ ( n / I , )( d - t ) ] , where d is the distance between cavity centers, t is the drift distance between cavities, and 1, is the guide wavelength. The second factor is due to the fact that the particles are shielded from the field when they pass through the drift tube from one cavity to the next; this factor is simply [1 - ( t l d ) ] . Thus, the energy gained by the particles over a length I would be

V

= Eol [I - (tld)] sin [ n ( d -

t)/A,]/[z(d - t)/A,].

In practice, while both of the above models give good insight into how linear accelerators work, neither is ever built in this simple form. As will be shown below, the uniform waveguide does not work as an accelerator because the velocity of the wave crests, generally called the phase velocity, up, is always greater than the velocity of

L BEAM

Fig. lc. Periodic disk-loaded TMol waveguide and example of traveling-wave axial electric field amplitude along z-axis for 7c/2 phase shift per period.

1 \

I.1

light. The simple array of short cavity resonators is not very desirable because of the excessive complexity and cost due to the numerous individual power feeds. The actual solution is a compromise between the two concepts. It is illustrated by the periodically loaded structure shown in fig. lc. As will be shown later, periodic metal irises of this type do not provide the only method of reducing the phase velocity to the velocity of light but they we the basis of probably the simplest and certainly the almost universally used config 1. .tion fo electron linac structures. 1 he pro .aracteristics of a uniform waveguide are given by the solution of the wavc equatl 11

B.1.1


43

where k = w / c and o is the angular frequency. For a waveguide of cylindrical symmetry with coordinates 8, r and z and the TM,, mode which has a component E, in the direction of particle acceleration, the electric and magnetic fields are given by the well-known solutions (see for example, Ramo and Whinnery [1953]): E, = E,Jo ( k , r ) exp [j ( o t - Bz)] E, = jqEo [ l - (wcjw)2]t J , (k,r) exp [j (wt - S z ) ] No = jEoJ, ( k , r ) exp [j ( a t - P z ) ] where J o and J, are Bessel functions, to, is the cut-off frequency, kc = w,/c, fl= o/vp, P2 = k 2- k:, k = w / c and q is the intrinsic impedance of the medium. For these solutions to exist, the propagation constant p must be real; this condition is fulfilled only if up > c.

I

Fig. 2. Brillouin (or

(,I-/$

/

diagram showing propagation characteristics for uniform and periodically loaded structures.

These properties are illustrated by the hyperbola shown in fig. 2. The slope of any line joining the or~ginto a point on the hyperbola gives the value of up for that particular value of w . The slope doldp of the hyperbola at that point is the group velocity v , which, in most cases, is also equal to the energy velocity ( ' -r a deeper insight into this relationship, see Brillouin [1960]). As seen, the group velocity is lower than the velocity of light but as up approaches c from above, u, approaches c from below. As a result, the value of E for a given power input P is relatively low since E 2 is proportional to Plv,. Both these disadvantages lead n t~ rally to the concept of the periodic structure of fig. Ic. It can be seen intu e!y thp the p: iodic obstacles are small, their effect on the fields will be negligible ,ept n -n the guide half-wavelength or a multiple thereof becomes equal to the c .k spac rlg and reflections from successive obstacles add to create a series of sto land* ~ ~ h i are rh

44

G.A.LOEW, R.B.NEAL


separated by pass-bands. This causes the hyperbola to have breaks and to fold over as a function of frequency so that v,+O at the boundaries of the pass-bands. As the apertures of the disks become smaller, the stop-bands become wider and the passbands narrower. Intersection of the v p = c line with the lowest pass-band of the w-p diagram can be made to occur where desired. As an example, for a frequency corresponding to a disk spacing d of /1/4 (n/2 phase shift per period) as shown in fig. Ic, the axial field configuration is between a sine wave and a square wave. This field configuration matches the periodic boundary conditions and thus can be Fourier analyzed in terms of space harmonics (sometimes called Hartree harmonics) of relative amplitudes an where n is an integer between - co and co. It can be shown (e.g., see Chu [1951]) that the total E, field in the lowest pass-band can be expressed as

+

where p,=p, + 2nn/d, p, is the propagation constant of the fundamental space harmonic and kfn= k 2 - p:. That these relations hold is a direct consequence of field analysis and of Floquet's Theorem which states that "for a given mode of propagation at a given steady-state frequency, the fields at one cross-section differ from those one period away by only a complex constant." (For a comprehensive discussion, see for example, Watkins [1958].) There are several interesting results which may be obtained from eq. (4). One is that the phase shift per period for the fundamental or n=O space harmonic is pod and that the phase shift per period of any other space harmonic differs from Pod by an integral number of 2n. Thus a _ , , the term which usually has the second largest amplitude, has a phase shift P-,d=p,d-2n and a,,, the term with the third largest amplitude has a value fl+,d=Pod+2n, etc. The successive harmonics are represented by the continuous (solid line) branches shown in fig. 2. At a given frequency w,, the slopes of all these branches are equal. This means that all space harmonics propagate with the same group velocity, dwldp, for a given w,. Each harmonic, however, has a different phase velocity and only one, generally a,, is made synchronous with the electrons and acts cumulatively during the acceleration process. Design optimization of a given structure then consists, among other considerations, in choosing the value of Bod (i.e., 2n/n, where n is the number of disks per wavelength) giving the highest axial field strength for a given input power, and in enhancing the value of a, with respect to all other harmonics, i.e., maximizing the expression a i / x a:. This point will be elaborated further in § 8. Another important result obtainable from eq. (4) is that for a given direction of power propagation, say from left to right, the negative branches ( r = ~ . .., - 3, - 2, - 1) have positive group velocity like the positive branches, but their phase velocities up, are of opposite sign. The n=O branch for the standard disk-loaded waveguide has a propagation from right to left.)

B.l.l


45

Structures of this kind where the group and phase velocities for n=O are of the same sign are called forward-wave structures. When the group and phase velocities have opposite signs, they are called backward-wave structures. Forward-wave structures result when the coupling from cell to cell is predominantly electric whereas the coupling for backward-wave structures is predominantly magnetic. As already mentioned, practically all existing electron linacs are of the metal disk-loaded type. However, many other types of structures, both of the forward- and backward-wave type, have been proposed and studied. Some of them are shown in fig. 3. I n summary, even though these structures may present some advantages, none of them have so far proven to be as practicable as the metal disk-loaded waveguide. Another important feature which results from eq. (4) is that when the fundamental harmonic (n=O) is traveling at v , = c , P o = k and kro=O. As a result, J , ( k r o r ) is independent of r and E, has no radial variation. Hence, even for a beam of finite radial dimensions, all electrons are subjected to the same accelerating field and the final energy spectrum is not deteriorated. Referring to fig. 2, it should also be emphasized that there are a n infinity of passbands and stop-bands (only two pass-bands are shown). Some of the higher bands can be traced to branches of the TM,, modes, others to the TE,, modes, and many are actually of the hybrid "HEM" type. These modes are not used for acceleration but they can have very detrimental effects on beam transmission for medium and high current machines. For example, the backward-wave H E M , , mode shown here can be coupled to the beam and amplified, resulting in transverse beam breakup of either the regenerative or cumulative types. These effects are discussed in detail in ch. B. 1.4. The field configurations discussed so far have assumed a steady-state travelingwave with negligible attenuation. More generally, it can be shown that if a structure of length I with input at z = 0 is made resonant by a n appropriate shorting plane at z = / , there will be a n infinity of successive internal reflections. Assuming a reflection coefficient Q for all waves bouncing back inside the structure* at z=O and summing all forward traveling terms of the form exp ( j (cot --P,z) - r z ) and backward traveling terms of the form exp ( j [ o t - P, (I- z) - P,I] - a / - cc (1- z)) where a is the attenuation per unit length, one obtains a steady-state standing-wave field profile of the form n-

Esw

+ oo

C

=

[(2a, ejWt-"')/(I - ~ e - ~ " ' )cosh ] [a ( 1

-

z) + jpn(l - z)] .

(5)

n+-m

It can be recognized that when al=O and n+

ESW =

Q

=0, this expression reduces to

+m

1

2an cos cot cos /In(1 - Z)

n--ao

which is the simple expression for a standing wave in a matched traveling-wave structure which has been shorted a t the end. O n the other hand, it can be shown that

*

Conversely, an incoming wave incident onto the structure at z = 0 will also see a reflection coefficient Q and the transmitted fraction will be proportional to (1 - e2)t

46

G.A.LOEW, R.B.NEAL


for a given value of al, the standing-wave structure is critically coupled if the value of Q is adjusted to be equal to e-2a', in which case expression (5) becomes n+

ESW= (1 - e-4a1)-

+ a,

C n+

2a, ejm'-"' cosh [a (1 - z)

+ jpn(1 - z)] .

(7)

- a,

As will be seen later, this expression can be useful in gaining an understanding of standing-wave and superconducting structures. FORWARD-WAVE STRUCTURES

I. DISK-LOADED STRUCTURE

2. VENTILATED STRUCTURE

3. CENTIPEDE STRUCTURE

4. RECTANGULAR SLAB

BACKWARD -WAVE

S.'>UNGLE GYM"

7. RING 8 BAR STRUCTURE

STRUCTURES

6. SLOTTED DISK STRUCTURE

8. LOADED EASITRON

Fig. 3. Periodic structures proposed for electron linacs.

B.l.l


47

Before concluding this discussion, it should be recalled that, in principle, there are other methods by which the phase velocity in a waveguide can be reduced to the velocity of light. An obvious technique is to incorporate a continuous cylindrical sleeve of dielectric inside the waveguide until v,=c (see, for example, Flesher and Cohn [1951]). The main disadvantage of this solution is that the field concentration then takes place in the dielectric and not along the beam axis. Another disadvantage is that the internal surface of the ceramic must be covered with a thin metallic coat in order to prevent electron collection on the walls. This disadvantage is also inherent in the periodic ceramic structure first proposed by R.-Shersby-Harvie [I9481 and more recently elaborated by Walker and West [1957].

3 Principal design parameters In this paragraph, the principal parameters which are useful in the design of electron linear accelerators will be defined and briefly discussed. These parameters will be needed in the following sections of this chapter which treat various types of electron linear accelerators. Shunt impedance. The shunt impedance per unit length is the characteristic which measures the excellence of a structure as an accelerator. It will be designated here by the symbol r, and is defined as follows:

where E,, is the amplitude of the space harmonic component of the axial electric field of the wave having a phase velocity equal to and in the same direction as the electron being accelerated and dP/dz is the rf power dissipated per unit length in the structure. Thus, a high value of shunt impedance per unit length is desirable since it means that a high accelerating field can be obtained with a given expenditure of rf power per unit length. As discussed later in 5 8, the measurement of r, takes into account the fact that there are an infinite number of space harmonics in the accelerator structure, only one of which (or two, exceptionally, in the case of the n-mode) travels at the electron velocity and produces useful acceleration over a n extended length. The other space harmonics merely perturb the motion of the electrons as they pass through the accelerator and produce no net effect. However, these harmonics are necessary to satisfy the boundary conditions in the accelerator structure. The factor of merit Q. The "unloaded"* Q is a measure of the merit of an rf cavity as a resonator. It is defined as the ratio of the energy stored in the cavity to the energy lost due to rf dissipation in the cavity per radian of the rf cycle. Since both of these quantities are increased by the same factor if several identical cavities are

* On the other hand, the "loaded" Q, QL, is the effective Q of the cavity considering the added losses due to coupling to external systems and due to beam loading.

48

G.A.LOEW, R.B.NEAL

ELECTRON LINACS : THEORY

considered together, the definition of Q which is convenient in linear accelerator design may be written: where w is the rf energy stored per unit length, dP/dz is the rf power loss per unit length and o is the angular frequency of the rf power. The ratio ro/Q. This is a very basic parameter in microwave cavities and structures. It may be obtained from eqs. (8) and (9): ro/Q = ~ & / w w .

(10)

Thus, ro/Q is seen to be a measure of the square of the electric field in the structure in terms of the energy stored per unit length of the structure. Since w, the energy stored per unit length is proportional to E& times the crosssectional area of the structure (which is proportional to &'), it is evident from eq. (10) that Thus, for an accelerator structure having a particular geometrical configuration, ro/Q depends only on the operating frequency. The dimensions of the structure are, of course, scaled as o-'.At a given operating frequency, the magnitude of r,/Q is independent of the choice of material from which the cavity is fabricated and of the surface finish of the cavity walls. Group velocity. As discussed in 5 2, the group velocity, v,, is the velocity at which rf energy flows through the accelerator. The group velocity, u,, depends strongly upon the ratio of the diameter, 2a, of the disk aperture to the diameter, 26, of the accelerator cavity. An approximate formula for the normalized group velocity is

where c is the velocity of light and K is a constant depending upon the number of disks per wavelength and the disk thickness and configuration. Group velocity is an important accelerator parameter for three reasons: 1. The "filling time", i.e., the time required to fill the accelerator with rf energy, depends upon the group velocity. For a simple traveling-wave accelerator of length I, the filling time is just l/v,. A high value of group velocity is preferred from the viewpoints of minimizing the time the rf pulse must be on, and of minimizing the transient time due to beam loading (see ch. B.1.3). 2. The power flowing in the structure and the energy stored per unit length of the structure are inter-related through the group velocity as follows:

Since, as previously discussed, w is proportional to E;,, it is clear that a low value of v, is preferred from the viewpoint of obtaining maximum accelerating fields for a given power flow.

B.l.l


49

3. As will be shown in 9 4 and Q: 8, the values of r,, Q , and r,/Q depend upon the choice of u, for the accelerator structure. In general, decreasing u, results in an increase in r,, a decrease in Q and thus a n increase in r,/Q. Therefore, a low value of u, is preferred from the viewpoint of obtaining high values of r, and r,/Q. The above considerations indicate that the choice of the group velocity, u,, must be a compromise depending upon the particular requirements of the accelerator being designed. This subject will be discussed further in Q: 4. Frequency. Almost all of the basic accelerator parameters have frequency dependence. The choice of frequency is therefore one of the first and most important tasks when undertaking the design of a new accelerator. Some of the significant considerations in the selection of the operating frequency are as follows: 1. It is desirable that the shunt impedance per unit length, r,, be high so that the amount of rf power required for a given energy gain will not be excessive. The dependence of r , on frequency may be determined using the defining relation, eq. (8). The rf power loss per unit length is proportional to the product of the square of the wall current and the wall resistance, R, per unit length, i.e.,

The axial electric field is proportional to the wall current divided by the radius b of the accelerator structure, i.e.,

and the wall resistance R per unit length is equal to the resistivity Q of the wall material divided by the area of the surface through which the current is flowing. Thus,

where 6 is the skin depth which is given by

and p is the permeability of the walls. Combining the last 4 equations with eq. (8) and noting that h ~ o - 'yield the result

Thus, from the viewpoint of conserving rf power, the operating frequency should be chosen to be as high as practicable. However, since the rf power from available single sources drops off more rapidly with frequency than r, increases with frequency, it is not likely that extremely high frequencies will be advantageous. 2. The diameter of the aperture in the accelerator structure through which the electron ; thus, lower frequencies are preferred from the beam is transmitted varies as 6' viewpoint of accelerating beams through a long accelerator with minimum interception.

50

G.A.LOEW, R.B. NEAL


3. In some applications, it is desired to obtain very intense (high power) electron beams at a given energy. The use of lower frequencies makes more rf power and stored energy available for conversion to beam power. 4. From scaling arguments, it can be shown that the filling time in an optimized accelerator varies as This is an inducement to use a design frequency as high as practicable to increase the fraction of the rf pulse length available for accelerating electrons. 5. The relative dimensional and frequency tolerances, 6qlq and 6 o / o which must be held in the fabrication and operation of the accelerator are proportional to mi, while the absolute values of 6 q and 6 2 are proportional to o*.Since the degree of difficulty in meeting frequency and fabricational requirements is usually a function of the relative rather than the absolute tolerances, the higher operating frequency ranges are preferred from this point of view. (See Chodorow et al. [1955].) From the above considerations it is obvious that the choice of frequency is not simple or clear-cut. The choice is usually based upon obtaining the best balance among technical and cost factors. As noted in 9 1, virtually all electron linear accelerators built to date have operated at design frequencies between 700 and 10000 MHz. The largest number have been designed to operate at frequencies near 3 000 MHz. Operating mode. As discussed in 9 8, the shunt impedance ro depends not only upon the operating frequency but also upon the operating mode which is defined as the phase shift per cavity of the accelerating structure. Thus, the operating mode is given by

where n is the number of disks per guide wavelength. For ease of measurement, n is usually chosen to be an integral value. Most electron linear accelerators have been designed for n = 2 , 3, or 4 , i.e., for @,=IT, 27113, or 7112. As will be noted in 9 8, the optimumvalue of n decreases as the disk thickness increases. A value of n = 3 ( @ , = 27113) is about optimum for accelerator structures having practical geometries. Attenuation constant. The variation of electric-field strength and rf power flow in an accelerator structure is given by

and

where a ( z ) is the attenuation in nepers per unit length. From eqs. (20) and (21), it follows that the electric field strength and the power flowing in the accelerator at a point distant z from the input are given by


B.l.l

51

and z

where E, and Po are the values of electric field and power at the input (z=O). The attenuation constant, z, is defined as the one-way attenuation, expressed in nepers, in an accelerator structure of length I. By this definition,

In a structure of uniform modular dimensions (constant impedance), a is constant thro~~ghout and z = cxl.

(25)

The magnitude of cx is a strong function of the diameter of the disk aperture. Increasing the aperture diameter increases the group velocity and decreases a. Thus, the desired value of z can be obtained by proper selection of the aperture diameter and the distance, I, between feeds. The shunt impedance per unit length, r,, varies slowly with the aperture diameter; thus, it is clear from eqs. (8) and (21) that the electric field strength Eo, varies approximately as (2aP)i. For a given power flow P, IdPldzl can be made large by designing for a high value of a. However, too high a value of a will result in excessive attenuation of the power and thus low fields in the output end of the structure. Too low a value of a will result in low fields throughout and excessive loss of power at the output end of the accelerator (assuming a resistive termination is used). A compromise value of z is clearly required to optimize the electron energy gain in the traveling-wave accelerator structure. The determination of the optimum value of z will be discussed further in Q: 4. Rf pulse length and repetition rate. In many physics applications, it is desirable to have as large a value of beam duty cycle (defined as the product of beam pulse length and pulse repetition rate) as possible in order to improve experimental statistics. The maximum practical value of the beam duty cycle is determined by economic considerations. The rf duty cycle D r f must be larger than the beam duty cycle D B in order to allow time for filling the accelerator with rf energy during each pulse before the electron beam is injected. The ratio of beam to rf duty cycles is given by

where tF and t,, are the filling time and rf pulse length, respectively. For a given beam duty cycle, the required rf duty cycle is obviously reduced by making the filling time as small as practicable.

52

G.A.LOEW, R.B.NEAL


From eq. (26) it follows that

Thus, a fractional increase in the rf duty cycle permits an even larger fractional increase in the beam duty cycle. Most electron linear accelerators constructed to date have beam duty cycles between 5 x lo-' and 5 x However, conventional electron linear accelerators with duty cycles of 2% (at 350 MeV) and 5.8 % (at 220 MeV) are now under construction at the Centre d7Etudes NuclCaires de Saclay (France) (Sokolovsky [1966], Leboutet et a]. [1966]) and at the Massachusetts Institute of Technology (USA) (Bertozzi et al. [1967]), respectively. A similar accelerator with a duty cycle of 10% (at 250 MeV) is being planned at the Instituut Voor Kernphysisch Onderzoek (Holland) (De Vries [1967]). All three of these machines will be capable of 50 to 100% higher energies when operating at reduced duty cycles. The superconducting electron accelerator discussed in 5 6 and more completely in ch. E.2 has, in principle, sufficiently low rf losses that a duty cycle approaching unity should be feasible. In fact, the promise of high duty cycle at high energies is the main reason for interest in this type of accelerator.

4 Traveling-wave accelerators

,

In the standing-wave accelerator, there is two-way propagation of waves while in the traveling-wave accelerator the wave propagates only one way. In most of the machines which have been constructed to date, the rf structure has served the dual purpose of waveguide and accelerator. This arrangement is mainly a matter of simplicity and economy. It should be remarked that a traveling-wave accelerator does not necessarily dmply power flow along the accelerator. One can construct a traveling-wave acceler,{' ator with isolated, separately fed cavities. In a strict sense, the distinction between the traveling-wave and standing-wave accelerators is whether the fields have only a ej(wt-Bz)or ej(wt+pz) dependence as in the traveling-wave case or a superposition of both of these forms as in the standing-wave case. Nevertheless, because of its obvious advantages, the combined accelerator-waveguide structure has been and will likely remain the most popular form of linear accelerating device for electrons. It is to this type of accelerator that consideration will be given here. Constant impedance accelerator. The traveling-wave accelerator case where the power transferred to the electron beam is negligibly small compared to the rf losses in the walls of the accelerating structure will now be considered. Beam loading in linacs is discussed in ch. B. 1.3. Assume that an amount of rf power Po is introduced at the input of a travelingwave accelerator. If the structure is properly designed, the power will propagate through the structure with a phase velocity equal to the electron velocity which, over


B.l.l

53

most of the accelerator length, is essentially the velocity of light. In the discussion which follows, a uniform (constant impedance) rf structure in which the fundamental space harmonic travels a t the velocity of light will, for simplicity, be assumed. From the definition of Q [eq. (9)] and from the relationship between P, kt3, and v, [eq. (13)], the power loss per unit length, the power flow, and the electric field equations may be written:

The above formulations are convenient since w, v,, and Q are parameters which can be readily measured in test cavities as discussed in 5 8. Comparing eqs. (21) and (28), it is noted that

The energy gained by a n electron situated a t a n angle 6 with respect to the crest of the traveling wave can be obtained by integrating eq. (30) over the accelerator length, 1. The result is

V

=

Eol [(I

-

eCr)/z] cos 0

(32)

where T = x l is the total rf attenuation in nepers in the accelerator structure. Using eqs. (8) and (28), eq. (32) may be written completely in terms of measurable quantities:

v = (~T)"(I

- e-')/TI (~,r,l)+ cosO

(33)

where Po is the input rf power. Equation (33) shows that the energy gain depends upon the attenuation parameter, T, which for a fixed accelerator length is a strong function of the aperture diameter of the loading disks in the accelerator structure. Since ro varies very slowly with cavity dimensions, it may be regarded as constant when the energy gain V in eq. (33) is maximized with respect to T. The condition for maximum energy gain is found to be:

which is satisfied for

With this optimum value of T, the energy gain from eq. (33) becomes:

From eq. (36), one would expect that as long as x is decreased to maintain T = 1.26, the energy would increase as l*. However, since r , decreases with increasing v, (i.e.,

54

G.A.LOEW, R.B.NEAL


with decreasing a), the energy of the optimized accelerator increases less rapidly than l3 for fixed power input. The dependence of r, upon v,/c is shown in fig. 4. The quantity v/[(P,~,I)~ cos01 from eq. (33) is shown vs. z in fig. 5. The maximum is observed to be quite broad so that it is not necessary to design exactly for z = 1.26 to obtain good results. A lower value of z is preferable in most cases for beam loading considerations (see ch. B.1.3), to reduce filling time, and to relax frequency and dimensional tolerances. It is instructive to calculate what would be gained in energy by extending an optimized accelerator to an infinite length. As 1 becomes very large and M is main-

I

I

I

0.1

0.2

0.3

0 0

0.4

vg/c

Fig. 4. Dependence of shunt impedance per unit length, ro, upon the normalized group velocity v g / c measured for 4 2 mode; 6 is the skin depth.

1

0

~

0.2

1

0.4

~

0.6

0.8

1

1.0

~

1.2

1

1.4

~

1.6

1

1.8

~

2.0

r

Fig. 5. Normalized energy gain in a constant impedance traveling-wave accelerator versus attenuation parameter z.

1

~

1

B.l.l


55

tained constant, assuming cosd = 1, eq. (33) becomes:

so that

Thus, the electron energy would be increased only 39% by increasing the length of the optimized accelerator to infinity. Filling time. Using the definition of the attenuation parameter, z = wl/2vgQ,the time required to fill the constant impedance accelerator with rf energy is given by

Thus, the filling time of an accelerator optimized in accordance with eq. (35) is

Energy stored in accelerator. The total amount of rf energy contained in the accelerator, W, at the end of the filling time may be obtained by integrating the energy stored per unit length w [eq. (13)] over the length 1 of the accelerator using eq. (29),

The ratio W/Pot,is the fraction of the input rf energy which remains available in the accelerator after the filling time to accelerate electrons. This ratio shown in fig. 6 vs. z

T Fig. 6. Fraction of the input rf energy remaining in the traveling-wave accelerator structure at the end of the filling time.

56

G.A.LOEW, R.B.NEAL


is the upper limit of the conversion efficiency from rf energy to electron beam energy. At T = 1.26, W/P,t, is 0.36. It is clear that, where energy conversion efficiency is the primary criterion, lower values of T are indicated. This will become even more evident when the subject of beam loading is discussed in ch. B. 1.3. Feedback of rf power. During the early British work on linear electron accelerators, a method was proposed by R.-Shersby-Harvie and Mullett [I9491 which provided for feedback of the residual rf power from the end of the accelerator and combination of this power in suitable phase relationship with the input power from the rf source by means of a waveguide bridge circuit. In this manner, the rf power entering the accelerator can be increased t o a value above that available from the source. The principle of rf power feedback is shown in fig. 7. An rf bridge is used to combine the power P, from the external power source with the power fed back from the end of the accelerator. The combined power is then fed into the accelerator. The

4 @

POWER

'4

RESISTIVE LOAD

RF BRIDGE

SOURCE

2's

&PHASE SHIFTER \

Po= P s ( I + g )

II

I

ACCELERATOR

Fig. 7. Schematic diagram illustrating feedback principle. Relations shown represent steady state conditions with correct bridge ratio.

bridge ratio, which will be designated by the symbol g, is defined as the ratio of the powers which the bridge is designed to combine. When the ratio of the feedback power to the source power is equal to the bridge ratio, g, and when the feedback phase is properly adjusted, the power input to the accelerator will be (1 +g) times the source power and the power to the external load will be zero. For example, if the bridge ratio is unity and if the power entering the accelerator is attenuated to one-half by the time it recirculates to the bridge, the steady state power entering the accelerator will be twice the source power. Steady state conditions are not attained instantaneously. Initially, the source power, P,, entering the bridge is divided, a fraction, 1/(1 +g), going to the accelerator and a fraction, g/(l +g), going to the external load. The residual power at the end of the accelerator is combined in the bridge with the source power. With proper phase relations, the result will be an increase in the amount of power entering the accelerator and a decrease in the amount entering the external load. As the recirculation of power proceeds, the power to the accelerator will continue to increase and the power to the


B.l.l

57

load will continue to decrease until steady state conditions are reached. When this occurs, the power to the accelerator will be greater than the source power by a factor which depends upon the attenuation in the feedback loop and upon the bridge ratio. If the attenuation in the accelerator structure (including beam loading) and feedback loop is such that the feedback power is equal to the input power, Po divided by x2,(where x23 I ) , then it is clear from fig. 7 that the bridge ratio, g , which will cause the steady state power to the resistive load to be zero is given by

P,(1 + g ) l x 2 = g p , or y = 1/(s2- I ) .

When the condition specified by eq. (42) is met, the steady state power build-up ratio in the accelerator will be

From eq. (43), it is noted that a large build-up ratio results when x2 is small (close to I), i.e., when a large fraction of the input rf power is fed back to the bridge. A graph of POIPSvs. 1 / x 2is shown in fig. 8. I n the same figure, the fractions of the source power which are fed to the accelerator and to the resistive load are shown when the bridge ratio, g , is unity. Comparison of these curves shows that a simple bridge of

Fig. 8. Power build-up ratio vs. fraction of accelerator input power fed back to bridge. Cases shown

are for ideal bridge ratio [y = I/(x2- I ) ] and for unity bridge ratio ( g = 1).

58

G.A.LOEW, R.B. NEAL


ratio g = 1 can be used over a considerable range of loop attenuation without serious decrease in power build-up. For this reason, the majority of accelerators using the feedback principle have employed bridges of unity ratio. Examples of bridges which have unity ratios are the standard rectangular waveguide "magic tee", the ringcircuit "magic tee", and the short slot hybrid coupler. A bridge having a ratio which is adjustable by a simple mechanical motion (known as the "circular magic tee") was used (Bareford and Kelliher 119531) with the 15 MeV accelerator constructed by Mullard Research Laboratories and installed in 1952 at the British Atomic Energy Research Establishment at Harwell. An adjustable bridge of this type is useful in maintaining maximum accelerator performance over a wide range of beam loading. The use of such a device, however, increases the complexities of accelerator construction and operation. The power build-up in the accelerator with feedback is a stepwise process with intervals between steps equal to the loop transit time. If the ideal bridge ratio given by eq. (42) is used, the build-up process requires a larger number of transits through the loop as the bridge ratio g is increased (and loop attenuation decreased). The number of transits required to build up to a given fraction of the steady state value is roughly proportional to g. For example, when g = 1 (x=2*), the power builds up to 75% of the steady state value after two transits through the loop, whereas when g =4 [ x = (1.25)"], eight transits are required to build up to the 75% level. The actual time to build up the circulating power is roughly independent of the bridge ratio since the transit time decreases as g increases (and x decreases) because of the increased value of group velocity in the accelerator structure. If a bridge of fixed ratio is used, the number of transits required to reach a given fraction of the steady state value is roughly proportional to l / x Z .The transient behavior of POIPSfor a system employing a unity ratio bridge is shown in fig. 9 for three values of x.

NUMBER OF TRANSITS

Fig. 9. Percent of maximum power build-up vs. number of transits around feedback loop. Bridge ratio g = 1 in all cases. Different values of loop attenuation for each curve.

B.1.1


59

If the attenuation in the accelerator with feedback is due mainly to rf wall losses, i.e., there is negligible beam loading, the build-up ratio given in eq. (43) can be expressed as

The energy gain can then be obtained by inserting eq. (44) into eq. (33), i.e., (omitting the cos 8 term) :

When z approaches zero, the first term in eq. (45) approaches unity, which is its maximum value. This should be compared with the coefficient of 0.903 given in eq. (36) for the optimum single feed case. Actually, the single feed case without feedback compares more favorably than indicated in the example just given. In the process of making the attenuation parameter r very small, it is necessary to open up the aperture, thereby increasing the group velocity, v,; this causes r,, the shunt impedance per unit length, to decrease as shown in fig. 4 so that the net effect of using small z and feedback may actually be a loss in beam energy compared to the optimized single feed case. On the other hand, at the optimum single feed value of z = 1.26, the use of feedback would change the value of the coefficient in the energy equation from 0.903 to 0.941, an increase of 4%. This relatively small improvement would be obtained by incurring the additional expense and operational complexity resulting from the provision of extra waveguide components in the feedback loop, including the power combining bridge, phase shifters, etc. The above discussion shows that there is little advantage from the viewpoint of energy gain in employing feedback. A more involved analysis leads to the same conclusion in the beam loaded case. The main incentive which led the early British workers to the use of feedback was the less restrictive frequency, temperature, and dimensional tolerances resulting from the lower value of z which is permitted when feedback is employed. This can be understood by the following simple argument. The number of rf cycles introduced into the accelerator structure during the filling time is given by the frequency times the filling time, i.e., n=ft,. A frequency shift 6f (or equivalent temperature or dimensional error) will result in a change 6n in the number of entering cycles, i.e., 6n = t , 6f. But according to eq. (39), t , = 2Qz/cc,; thus, the resulting phase shift in the wave at z = l is

Since the phase shifter in the feedback loop can be adjusted to give optimum input phase conditions, the phase error in the feedback loop does not build up during the many transits around the loop beyond the value given by eq. (46). As an example, compare an accelerator with feedback with z=0.126 with the optimized single feed

60

G.A.LOEW, R.B.NEAL


accelerator with r = 1.26. Assume further that v,/c = 0.1 and 0.01, respectively. According to eq. (46), the accelerator without feedback must control the frequency (and associated temperature and dimensional tolerances) 10 times more accurately than the feedback accelerator to maintain beam energy to the same accuracy. For T =0.126 in the feedback case, the coefficient in eq. (45) is very close to unity compared to 0.903 in the case without feedback. However, as may be noted from fig. 4, the value of r, is reduced in the feedback case by about 40% compared to the value in the case without feedback. The net result, in this example, is that the accelerator with feedback has about 90% of the energy of the accelerator without feedback. Feedback found its main use in comparatively short single feed accelerators used for medical and radiographic purposes and employing relatively unstable magnetron oscillators as the power source. Because of the additional mechanical and operational complexities, it has rarely been used in machines having more than a single feed. In recent years, the use of klystron amplifiers fed by frequency-stabilized rf driver systems, the development of more precise fabricational and tuning techniques for accelerator structures, and the employment of accurate temperature stabilized cooling systems have made the use of feedback techniques unnecessary and undesirable. Possibly, the advent of cryogenic accelerators with their inherent low rf losses will again cause accelerator designers to turn to feedback principles. However, at this time, the use of standing-wave structures seems to be favored in the prototype cryogenic machines. (See chs. E.2 and E.3.) The constant gradient accelerator. Because of its uniform modular cross section, the rf power and electric field in a constant impedance accelerator decay exponentially with distance z as shown in eqs. (29) and (30). The electric field is plotted versus zll in the sloping curve of fig. 10. The maximum electric field, which occurs at z=0, is higher than the average field along the length of the structure by a factor which depends upon the value of the attenuation parameter z. This situation can be demon-

Fig. 10. Electric field strength versus normalized distance z/l in constant impedance and constant gradient accelerator structures. r = 0.57.

B.l.l


16

strated by a simple rearrangement of eq. (32) as follows (dropping the cosO term):

E o / ( V i l )= T/(I - eCT).

(47)

The ratio of peak to average electric field is shown in fig. 1 1 . By suitable variation of the transverse modular dimensions, it is feasible to design a n accelerator in which the axial electric fields are essentially constant along the length of the structure. This type of structure is called a constant gradient accelerator. By definition, the ratio of peak to average electric field in this structure is unity as shown in fig. 1 1 . The basis for the design of a constant gradient accelerator is provided by

-r =

WtF/20

Fig. 1 I . Ratio of peak to average electric fields in constant gradient accelerator structures.

eq. (8). Since, as noted previously, r , changes but slowly with the modular dimensions of the structure, it can be regarded as a constant.* Then the requirement for constant gradient along the structure becomes dP/dz

= constant.

(48)

Thus, if the rf power input to the accelerator is P, and the power escaping from the accelerator at z = I is P,, then the requirement of eq. (48) is satisfied if the power flow

*

The final results can be corrected for the minor variation of electric field due to the variation in

r,,, if desired. However, it should be noted that the variation of modular dimensions calculated on the assumption of constant ro actually is such as to cause ro to increase slowly with increasing z.

This causes a slight increase in the axial field strength with z at zero beam loading. With a small amount of beam loading, the fields then become nearly constant throughout the accelerator length.

G.A. LOEW, R.B. NEAL

62


in the accelerator is given by P = Po - (Po - P,) 211.

(49)

With the substitution, P,/Po =e-2', eq. (49) becomes

Two equivalent expressions for dP/dz may be obtained from eqs. (49) and (50):

and

While the power dissipated per unit length, dP/dz, in a constant gradient structure is constant over the length of the structure, dP/dz is far from being constant in the constant impedance structure, as shown by eqs. (28) and (29). In fact, the ratio of power loss at the input end to that at the output end of an optimized (z= 1.26) constant impedance structure is about 12 to 1. Such a large variation of power dissipation complicates the problem of maintaining the temperature constant over the entire accelerator structure. Variation of group velocity. Eliminating dP/dz between eqs. (28) and (52), substituting eq. (50) for P, and solving for v, yields:

Eq. (53) shows the required variation in v, to produce a constant gradient characteristic. It may be noted from eqs. (50) and (53) that P and v, decrease in the same linear manner and by the same ratio between z = 0 and z = I. Filling time. The filling time of the constant gradient accelerator may be found by integrating dt=dz/u, between the limits 0 and 1 using eq. (53) for v,:

Comparing eqs. (39) and (54), one notes that the filling times of the constant impedance and the constant gradient accelerators are equal if the two machines are designed to have the same values of the attenuation parameter, z. Energy stored. The energy stored in the accelerator structure at the end of the filling time may be obtained by integrating w =Plugfrom z = 0 to z = 1 using eqs. (50) and (53) for P and v,, and eq. (54). The result is:

A comparison of eqs. (41) and (55) shows that the stored energies in the constant


B.l.l

63

impedance and constant gradient accelerators are also equal when both have the same attenuation parameter, 7. While the simple expression 7 = ccl is suitable to define s in the constant impedance accelerator since cc is constant throughout, it is not suitable in the constant gradient case where cc varies with z. A more general definition suitable for both types of accelerators is given by eqs. (39) and (54): 7 =

wtF/2Q

(56)

where it is assumed that Q is the "average" value in each structure. Like r,, Q in the constant gradient structure varies slowly with the cavity dimensions and it is usually satisfactory to regard it as constant. Electric Jield strength and energy gain. The axial field strength in the constant gradient accelerator may be obtained using eqs. (8) and (52). It is given by E

=

[(I

-

eC2') P,r,/l]*

(57)

and is constant throughout the accelerator length (for negligible beam loading). The energy gain is obtained by integrating the electric field over the accelerator length, I.e., V = [(I - e-") ~ , r , l ] + .

(58)

A comparison of the unloaded energies of theconstant impedance accelerator [eq. (33)] and the constant gradient accelerator [eq. (58)] is shown in fig. 12 vs. s. It may be noted that the energy gain in the constant gradient accelerator is slightly higher than that of the constant impedance accelerator for all values of 7 but the advantage is greater for the higher values of s. Other characteristics of the constant gradient accelerator with negligible beam loading are given by Neal [1953]. Beam loading in the constant gradient accelerator is studied by Neal [I9581 and a comparison of the constant gradient and constant impedance accelerators under beam loading conditions is given by Neal [1961]. Design details of the constant gradient sections of the Stanford two-mile electron linear accelerator are given by Borghi et a/. [1966]. Several other advantages of the constant gradient structure are discussed in the above references. Among its advantages, not yet mentioned, relative to the constant impedance structure, are its lower sensitivity to frequency deviations, its higher maximum conversion efficiency (ratio of maximum electron beam power to input rf power), its lower beam-loading derivative, and its decreased susceptibility to beam breakup phenomena (see ch. B. 1.4). Multiplefeed accelerator. Because of the limited amount of power available from a single rf source and the impossibility of extending the accelerator length indefinitely, all electron linacs with output energies above about 50 MeV have used multiple feeds. Indeed, many lower energy accelerators designed for heavy beam loading are provided with multiple feeds. The multiple feed accelerator is usually comprised of N identical


G.A.LOEW, R.B.NEAL

61

single feed accelerators in tandem, each fed by an identical rf power source. From either eq. (33) or eq. (58) the energy gain from one of these single feed sections may be written V

=

K (~,r,2)+

(59)

where K is a function of z. When N of these sections are joined in proper phase to form a single accelerator, the total energy gain V;. is given by V,

=NK

where P, length.

( ~ , r , l ) += K [(NP,) r, ( N l ) l t

= NP,

=K

(P,J-,L)+

(60)

is the total input rf power and L = NI is the total effective accelerator

CONSTANT

GRADIENT

CONSTANT IMPEDANCE STRUCTURE

Fig. 12. Comparison of the unloaded energies of the constant impedance and the constant gradient structures versus T = WIF/ZQ.

From eq. (60), it is clear that a given total energy V, can be obtained from an accelerator having a shunt impedance per unit length r , for a specific product of total power P, and total length L. For example, a high total power and a short length, or conversely, a low total power and a long length may be used. The values of P, and L are usually chosen on the basis of economic considerations. In general, minimum total cost results when the cost associated with supplying rf power is equal to the cost associated with providing the accelerator length. In some machines such as the SLAC two-mile accelerator, a low power to length ratio has purposely been chosen (Borghi ~t al. 119661) in the initial design in order to provide for later increase in energy by adding more power sources along the length. Fortunately, the curve of cost vs.

B.l.l


65

power/length is flat enough that little penalty is paid for a reasonable deviation from the optimum P,/L ratio. For a given total power P,, the selection of the number of individual power sources and the number of rf feeds depends upon several considerations: 1. As discussed previously, the shunt impedance per unit length, r , , is increased moderately as the group velocity is decreased. Therefore, closely spaced feeds which require low group velocity (high a ) structures are favored from the viewpoint of obtaining high shunt impedance. 2. It is generally more economical to provide a given total rf power from a small number of high power rf sources than from a large number of low power rf sources. The reason for this fact is that the cost of a microwave tube depends more strongly upon the number of steps required in its fabrication and processing than upon its physical size and power output. 3. If the feeds are too closely spaced, it may be necessary to split the power from each rf source (if the number of sources is chosen on the basis of consideration No. 2 above) into two or more parts. Such splitting leads to increased cost and complexity of the waveguide system. 4. A large number of feeds increases the costs of the instrumentation and control system and increases operation difficulties. 5. The accelerator aperture is decreased as the feed interval decreases for a fixed value of T. This may lead to problems with beam transmission through a long accelerator. Consideration (I) above favors a large number of feeds while considerations (2) to ( 5 ) favor a small number of feeds. The final choice of feed interval must be based upon a detailed study of economic and technical factors. Various accelerators have been designed with feed intervals ranging from about 1 to 10 meters.

5 Standing-~~ave accelerators Elementary considerations. The traveling-wave accelerator utilizes an axially nonresonant waveguide in the sense that the wave travels in only one direction. O n the other hand, the standing-wave accelerator makes use of a waveguide which is axially resonant so that there are waves traveling in two directions. In the simple travelingwave case, the accelerator waveguide can either be terminated with a matched resistive load or if the rf pulse length is not greater than twice the filling time (i.e., twice the time required for the wave to propagate the length of the section at the group velocity), the wave can simply be allowed to reflect from the end of the accelerator guide without affecting the performance of the accelerator or the power source. The fields in the traveling-wave accelerator may be considered as building up "in space" while those in the standing-wave accelerator build up "in time". In the traveling-wave case, the rf pulse must be on for a period of time to "fill" the accelerator with energy flowing

G.A. LOEW, R.B.NEAL

66


at the group velocity in the structure before the electron beam is turned on. Analogously, in the standing-wave accelerator, a period of time must be allowed for the fields to build up in the resonant structure to essentially equilibrium values before the electrons are injected. A simple example is helpful in gaining insight into the behavior of the standingwave accelerator. Consider an accelerator containing a forward wave with a fundamental space harmonic of amplitude A which has a phase velocity equal to the electron velocity and a backward wave with a fundamental space harmonic of amplitude B which has the same (but negative) phase velocity. For generality, it will be assumed that A f B , due, for example, to lossy conditions at the end of the structure. For simplicity, other space harmonics which have no net effect on the electron energy gain will be neglected. The two waves under consideration are then given by expressions of the form

E,

= A sin ( o t - Poz)

(61)

and When both of these waves are present, a standing wave which is a combination of E, and EB will be set up, i.e.,

E = E,

+ EB = ( A - B) (sin o t c o ~ / ? ~-z (A ) + B) (cos o t sin Poz).

(63) By differentiation of eq. (63), it can easily be shown that successive maxima and minima of the standing-wave pattern occur at one-quarter wavelength intervals along the length of the structure. They are given by

and are located at E,,,:

P,z

= (2n

+ 1) 7c/2

(66)

wheren=O, l , 2 , 3,.... Figures 13a and 13b show the forward and backward waves at successive points in time to, t,, t,, and t,. Figure 13c shows the combination of these waves forming a standing-wave pattern with maxima and minima given by eqs. (64) and (65) located at fixed values of pOzas given by eqs. (66) and (67). Now suppose that an electron traveling at essentially the velocity of light is injected into the accelerator in which the standing-wave field of eq. (63) exists. This electron will reach a point distant z from the input at a time Substituting eq. (68) into eq. (63) and integrating the resulting expression over the


B.l.l

67

accelerator length, I, yields the energy gain of the electron V

= A1

sin of, - Bl (sin Pol/Pol)[sin (Pol

+ wt,)] .

In order for the accelerator structure to be axially resonant, it is necessary that its length be an integral number of half-wavelengths, i.e., POI=Nn. With this substitution, eq. (69) becomes : Vs,

= A1

sin of,.

(70)

Maximum energy is obtained if the electron is injected into the accelerator when s h o t , = 1, i.e., at, = (4m l)n/2 where m = 0, 1, 2, 3,. . . . This situation is illustrated in fig. 13c. The electron is injected at a time to and a t

+

-

FORWARD WAVE

s--- BACKWARD WAVE

ELECTRON POSITIONS AT J , " . ' .

tn

tl

(2

FIELD ENVELOPE

t,

STANDING WAVE

I

I

I

I

I

I

I

I

0

TI2

7T

3T/2

2lT

5lT/2

37T

7T/2

I 87T

Poz

Fig. 13. (a) Combination of forward wave of amplitude A and, (b), backward wave of amplitude B in a n accelerator structure. The resultant standing-wave pattern is shown in (c). Successive points in time are indicated as t o , t ~t t,, ts. The standing wave has maxima of amplitude I A f BI and minima of amplitude \ A - BI at fixed positions as indicated. The average accelerating field amplitude is equal to A.

68



Poz=O and sees the field J A - B ( . At time t,, the electron has reached the point P,z=~1/2 and sees the field ( A + BJ. In general, the electron is subjected to the fields denoted by the top curve of fig. 13c which has an average value of A as shown in the figure and as given by eq. (70). From the above discussion, it is clear that only the wave having a phase velocity equal to and in the same direction as the accelerated electron contributes to the energy gain. The ratio of peak field in the structure to the average accelerating field is ( A + B)/A. In general, when A z B, this ratio is ~ 2 This . is a disadvantage when high accelerating fields are required because of the increased possibility of internal arcing. The use of the TC mode improves this situation somewhat because in this mode, all the cavities are fully excited and, as discussed later in this §, both the forward and backward traveling powers carry a space harmonic (n = 0 and n = - 1, respectively) which is synchronous with the electrons. In this case, the ratio is about 1.6 to 1. Steady-state operation. Consider further the simple traveling-wave structure with attenuation parameter z and assume that the wave is reflected at the output end. The resulting backward wave will return to the input end undergoing further attenuation in field strength by a factor e-" except in the TC mode, this backward wave is not useful in accelerating electrons in the forward direction. At the input end, the backward wave will be partially reflected and partially transmitted out through the input coupler toward the rf source. If the length of the accelerator section is an integral number of half-wavelengths, the wave reflected at the input end will have the correct phase to contribute to the acceleration of electrons. The above process of reflection at each end of the section will continue until equilibrium is reached. If the input coupler is correctly matched for this resonant condition, none of the backward power will, after steady state is reached, pass out through the input coupler toward the source; all of the input power will be absorbed in the structure. The general behavior is quite similar to that of the feedback accelerator discussed in $ 4 except that in this case the accelerator structure itself rather than an external feedback loop serves to return the residual rf power to the input end. Following the same reasoning used in the feedback case, the power build-up becomes

which is the expression analogous to eq. (44). The steady state energy gain is therefore given by a slightly modified form of eq. (49," i.e.,

+

where the symbols have the same meaning as before and where the (1 e-2T)-) term accounts for the attenuation of the wave going in the backward direction through the accelerator section.

* Equation (72)can also be obtained by integrating the expression for the field ESW given by eq. (7), using only the fundamental space harmonic. In this case a0 = (2aroPo)t and PO= (1 - Q ~ Ps ) where for critical coupling Q is adjusted to be equal to e-2az.

B.l.l


69

From eq. (72) it is obvious that the standing-wave accelerator gives a slightly higher energy gain than the corresponding structure used as a traveling-wave accelerator. However, this is accomplished only at the expense of a n increased filling time due to the required multiple reflections in the structure in the standing-wave case. The relative energy gain of these accelerator types will be further discussed later in this section. The limiting values of the energy gain in the standing-wave accelerator for very large and very small T are found from eq. (72) to be: V,,,

= (2~~r,l/.c)'

(73)

which, as expected, is the same as the limiting value of energy gain in the single feed case given by eq. (33), and Vr+,

= (Psrol/2)'.

(74)

Equation (74) states that, when the attenuation in the structure is very low, only one half the power from the source is effective in accelerating electrons (except in the n mode case); the other half is expended in maintaining the backward traveling wave which does not contribute to the energy gain. Shunt impedance and related design considerations. The value of the shunt impedance r, depends upon the mode of operation. As shown in 5 8, the maximum value of r, in a simple periodic disk-loaded structure operating as a traveling-wave accelerator occurs for a phase shift per cavity between rr/2 and 2n/3, the larger phase shift being favored as the disk thickness increases. When the structure is operated as a standing-wave accelerator, the number of space harmonics is doubled due to the presence of the backward wave. As the operating frequency approaches the upper cut-off frequency (n mode), the forward and backward space harmonics coalesce and the fields add in phase to give resultant fields with twice the amplitude of the individual harmonics. For this reason, the n mode* has a higher effective shunt impedance than any of the other operating modes as shown in fig. 27. Although the n mode is preferred from the viewpoint of obtaining maximum shunt impedance in a resonant structure, there are a number of problems which are to some extent all related to the fact that group velocity is 0 at n mode (see also ch. C. I . l b). For a given number of cavities in a n accelerator structure, the separation of modes is greatest in the center of the pass-band (n/2 mode) and least at the edges of the passband (0 and rr modes). The mode separation can be calculated using the results of coupled circuit theory obtained by Nagle [1964]. The dispersion relation for a linear array of N coupled circuits is &cot

=

1 + y cos(nq/N)

(75)

* Another way of looking at the rr mode is that both the ao space harmonic of the incident wave and the a 1 space harmonic of the reflected wave travel forward, in synchronism with the electrons. Thus, in this case, the shunt impedance r x to be substituted into eq. (74) is the sum of ro and r-1 or 2ro since when they coalesce, ro = r - I . The rr-points plotted in fig. 27 which can be substituted into eq. (74) are by definition only valid for standing-wave operation (see also ch. C.l.le, 5 1.5).

70

G.A.LOEW, R.B.NEAL


where y is the coefficient of coupling between circuits, q is the number of the mode with resonant frequency w,, and w , is the resonant frequency of the individual circuits. In general, nq/N is the phase shift per cell. Thus, the zero, n/2, and n modes occur for q=O, N/2, and N, respectively. Altogether, there are N+ 1 resonances." The separation of the lower and upper cut-off frequencies (0 and n modes) defines the bandwidth (BW) of the structure. From eq. (75) the bandwidth for small y is found to be

Thus, y is the normalized bandwidth of the coupled resonators. For the present discussion concerning the n mode, the separation of the N mode ( n ) and the ( N - I) mode is pertinent. From eq. (75) this separation is found to be

whereas the separation of the n/2 mode from its nearest neighbors is

Thus, for a given structure, the separation of modes near the 4 2 mode is 2N/n times as great as the separation near the n mode. Operation at a particular mode without excitation of its neighboring modes requires that the Q of the structure be sufficiently high to prevent overlapping of the mode resonances. From eqs. (77) and (78) the requirements for the n and 7112 modes are clearly

and

For example, if Q = lo4 and y = lo-', the number of cavities in the n mode structure cannot exceed z 16 without giving concern with respect to mode interaction. For the same Q and y, the 7112 mode structure can have z 157 cavities without overlapping of modes. Thus, the number of cavities in the n mode is much more severely limited than for the n/2 mode (and, in general, for other modes not too distant from the center of the pass-band). In a long accelerator utilizing the n mode, the number of feed points must be correspondingly increased. In the n mode structure, the phase shift from cavity to cavity is n radians in the lossless case. However, in the practical case with loss, energy must be supplied through excitation of adjacent modes to compensate for the losses and there are resultant phase changes in the cavities. Thus, strictly speaking, operation in the 71 mode is not possible. The phase deviation from n radians per cavity is given by (Nagle [1964],

* Equation (75) is valid for coupled cavities with half-cavity terminations in which case there are N full cavities and two half cavities giving N 1 resonances. When full-cavity terminations are used, there are N cavities and N resonances. In the latter case, the cosine term in eq. (75) becomes cos [zql(N 1)l.

+

+


B.l.l

71

Knapp [1964], and Smith [1966]):

Thus, the phase shift over the entire length of the structure is A@,,,

.,

= (1 - Y)+

WQY.

(82)

For the same values of Q and y as in the previous exampIe and for N = 16, the net phase shift is ~ 2 . radians 5 which is obviously excessive. To reduce the phase shift to a tolerable level, the value of the coupling coefficient y (i.e., the bandwidth) of the structure would have to be increased. This would, in turn, lead to a decrease in the shunt impedance of the structure. The n-mode structure is similarly quite sensitive to beam loading variations and to dimensional errors. On the other hand, the n/2 mode is much less sensitive to these effects and to cavity losses. For this mode, the effect of losses in the structure and the effect of beam loading in a correctly phased section is merely a reduction of field strength without change of phase. It is for these reasons that most resonant accelerators now under construction or under consideration utilize the n/2 structure. Means have been devised to improve the shunt impedance of this mode so that it approaches that of the n mode. One technique consists of using a linear bi-periodic disk-loaded structure with alternate long and short spacings between disks (Giordano [1965]). The short cavity is adjusted to have the same resonant frequency as the long cavity. The end terminations are placed so that the null in the standing-wave pattern occurs in the short cavities and the amplitude of the synchronous space harmonic is thereby increased. Another technique (Knapp [1964]) completely removes the resonant coupling cavity from the beam line and places it at the side of the structure where it now serves to couple two adjacent in-line cavities through slots in the mutual outer walls (see ch. C. 1. lc). Thus, the structure as a whole operates in the n/2 mode, but the in-line cavities are effectively operating in the n mode. While this structure is mechanically complicated, it has an even higher shunt impedance than the linear biperiodic structure (see also ch. C. 1. le). Pulsed operation of standing-wave accelerators. The values of accelerating field given in eqs. (72)-(74) for a standing-wave accelerator are for steady state conditions. Actually, most accelerators are run with short rf pulses and the fields in the cavities must build up from zero to the steady state values. The fields in a low loss structure build up .according to the expression given by Slater [I 9501:

EIE, = 1 - e p m ( / 2 Q ~

(83)

where E/E, is the ratio of instantaneous field strength to the steady state value, and QLis the loaded Q of the cavity. When impedance relations are adjusted for maximum steady state power transfer from the source to the cavity, Q,=+Q, The build-up of fields under these conditions is shown in fig. 14 where E/E, is plotted against ot/Q,. At ot/Q, = 2 the field has risen to 86.5 percent of the steady state value. For example,

72

G.A.LOEW, R.B.NEAL


with w= 1.79 x 10" radianslsec (corresponding to A,= 10.5 cm) and Qo= 17000, this occurs at a time t z 1.9 microseconds. At ot/Q, = 1 the fields have reached 63.2 percent of the final values. Therefore, if the beam is injected at tz0.95 microseconds and is on for one microsecond, the output electron energy at the end of the pulse will be about 1.37 times as high as at the beginning. Beam loading considerations have been neglected in this discussion and will tend to moderate the energy rise as rf power is transferred to the beam during the pulse. Another way in which the output electrons could be made more nearly monoenergetic might be to vary the relative phase angle 0 between the electrons and the wave during the pulse such that E cosB remains substantially constant.

Fig. 14. Buildup in resonant cavity under matched conditions

(QL = +Qo).

The feeding of rf power into a resonant structure such as a standing-wave accelerator requires considerable care. The steady state impedance of the cavity as viewed from the position of the detuned short is given approximately by Slater [1950]:

where Z is the impedance, Z, the characteristic impedance of the waveguide, Q,,, the external Q of the cavity which represents the losses from the cavity to the external waveguide, Qo the unloaded Q, o the angular frequency at which the measurement is being made, and coo the resonant frequency of the structure. At resonance, eq. (84) states that Z/Zo = Q0/Qe,,. The standing-wave ratio in the input waveguide will then be Zo/Zor Z/Zo, whichever is greater than unity. These two conditions correspond to "undercoupling" and "overcoupling", respectively, of the waveguide and cavity. When Z = Z o (i.e., Q, = Q,,,), the cavity is said to be "critically coupled". In this case, the cavity is perfectly "matched" to the waveguide at resonance; the standing-wave ratio is unity and none of the input power is reflected.

B.I.l


73

Equation (84) gives the cavity impedance under steady state conditions. However, during build-up of power in the cavity, the impedance varies, starting at zero at t=O and approaching the steady state value as t approaches infinity. During this transient period the impedance is given by (Slater [1950]):

z/zo

-

[I

Q,:

When Qo = Q,,,

Z / Zo

-

(1

- e-"~t"2Q~]/[~,'

= 2Q,,

- e-wo'/Qo

+

Q,:

ep"ot/2Q~].

(85)

this becomes

)/(I

+ eCmotiQ0).

(86)

A plot of Z / Z o ~ ~ n d these e r conditions is shown in fig. 14. It may be noted that the impedance builds up even more slowly than the fields. For example, at wot/Qo=2, it has risen to 76.2% of the steady state value. As just noted, the accelerator sections appear as short circuits at the beginning of the rf pulse. A self-excited oscillator cannot easily be used to supply power to such a load as it cannot start well when confronted with a short-circuited load (A. Sarazin [1953]; J.Vaste1 [1952]). A power amplifier can supply power, but at the beginning of the pulse all of the incident power will be reflected, and some of the power will continue to be reflected during the entire transient period. There are several disadvantages to this behavior: 1) the varying load causes the amplifier power output to vary during the pulse; 2) the changing power output may contribute to a greater energy spread in the accelerated electrons; 3) the reflected power may cause excessive electric gradients in the transmission line resulting in damage to such components as rf windows, etc. (see also ch. C.2.3). There are several ways in which the effective load as seen by the power source may be altered. All of these ways result in some loss in useful power. For example, resistive material (pads) can be added in the transmission line which at least partially isolate the cavities from the power source. T o be effective, this method will result in the loss of a fraction of the incident power. A better method, in principle, consists of the use of a unidirectional power absorber such as a ferrite isolator in the line. Such a device appears as an attenuator of large magnitude for waves propagating in the opposite direction, i.e., for reflected power traveling toward the power source. Thus, it can effectively decouple the load from the source without excessive loss of forward power. Rf isolators which are capable of handling several megawatts of incident peak power are available commercially. Such devices, however, are not yet capable of handling the highest powers ( > 2 0 megawatts) which have been used with accelerators. At these power levels, the isolators would probably have to be designed to operate in high vacuum. Comparison of energy gains ,from standing-wave and traveling-w,ave accelerators. Equation (33) gives the steady state energy gain obtained from a constant impedance traveling-wave accelerator having a n attenuation parameter T ; eq. (72) gives the steady state gain obtained when the same structure is operated as a standing-wave accelerator. Operation in the standing-wave mode gives higher steady state energy gains than operation in the traveling-wave mode for all values of z. The factor of

74



improvement is (1 -e-4')-t, i.e., the square root of the build-up factor given in eq. (71). The relative improvement is greater at the lower values of z. The higher energies obtainable by operating the accelerator in a standing-wave rather than in a travelingwave mode are achievable only at the expense of significantly longer filling times. The normalized steady state energy gains for traveling-wave and standing-wave operation in the same accelerator structure are shown in fig. 15.

Fig. 15. Normalized steady state energy gains for traveling-wave and standing-wave operation in the same constant impedance accelerator structure versus attenuation parameter .r.

As noted earlier, the n mode has the highest shunt impedance of all the operating modes because of the coalescence of the forward and backward space harmonics. It is therefore of interest to compare the energy gain of a standing-wave accelerator operating in this mode with the energy gain of a traveling-wave accelerator operating in the 2x13 mode which is near optimum for traveling-wave operation. In making this comparison, it will be assumed that the same length of each type of accelerator is supplied with the same amount of rf power, that each is operating at P= 1.0 with the same disk aperture, and that the difficulty in supplying power to the standing-wave structure has been overcome by the use of idealized lossless isolators or other suitable devices. The ratio of the energy of the standing-wave accelerator given by eq. (74) [modified by the transient build-up factor given in eq. (83)] to the energy of the traveling-wave accelerator given by eq. (33) (with the 1 -e-" term replaced by 1 - e - ~ ~ g t l l = 1- e - < ~ f / 2 Qto~ ~show build-up during the filling time) is: = 3 2 ) [(I - e-wf'Qsw)/(l- e-Wt12QTW)] (rsw / rTW )+ VSW/VTW

(87)

where the subscripts SW and TW have been added to quantities pertaining to the standing-wave and traveling-wave cases, respectively. It has been assumed that matched conditions exist at steady state for the standing-wave accelerator (i.e., Q=2Q,)

B.l.l


75

and that difficulties with adjacent mode interactions can be overcome. The value of tz 1.26 should be used to optimize the steady state energy of the traveling-wave accelerator. I n using eq. (87). the value of t in the numerator is allowed to run from 0 to cc but the value of t in the denominator runs only from 0 to I/v, a t which time the traveling-wave structure is completely filled with energy. T o go further, experimental values of the shunt impedance r and Q are needed. The following values have been measured at Stanford for the TC mode (standing-wave) and 2 ~ 1 3(traveling-wave) cases at a frequency of 2856 MHz. -- -

Mode n

-

--

--

--

-

--

8,

t

d

2u

2b

Q

r (MQlrn)

1.0

0.230

2.068"

0.8225"

3.232"

17200

100

oK/c

0

t~ (psec) -

In the above table, P, is the phase velocity divided by the velocity of light, t is the disk thickness, d t h e spacing between disk centers, 2a the diameter of the disk aperture, 26 the inside diameter of the cavity, v,/c the normalized group velocity, and t , the filling time. The value of r used for the 2 ~ 1 3mode is that measured for the fundamental (n=O) space harmonic. For the n mode, the value of r used is the sum of the values measured for the n=O and n = - 1 space harmonics (see footnote on p. 69). Using the above tabulated values of r and Q for the two modes of operation, the energy build-ups (V/Vq,,,,, ,,,) vs. time are plotted in fig. 16 for the standing-wave (T mode) and traveling-wave (2x13 mode) cases. In addition, the energy ratio of the two accelerators (Vsw/VTw)vs. time is plotted in the same figure. It is noted that, except a t very low values of t when the standing-wave structure has a slight margin, the traveling-wave structure builds up more quickly than the standing-wave structure. At the end of the filling time of the traveling-wave structure, its energy is about 12% h ~ g h e rthan the energy of the standing-wave structure at the same time. The travelingwave structure continues to have higher energy up to a pulse length of ~ 3 . psec 5 when the two accelerators have equal energies. A t steady state in the standing-wave structure, its energy is about 3 % higher than the energy of the traveling-wave structure. F o r the assumed values of z, w , v, and Q(,,,,,. the length of each of the two 3 of the standing-wave structures is 4.9 meters which is equivalent to ~ 9 cavities structure. However, according to eq. (79), (using Q,= 17200 and assuming y ~ 0 . 0 1 ) the TC mode standing-wave structure should not have more than ~ 2 cavities 0 in order to avoid interactions with adjacent modes. Thus, to achieve the performance calculated in the above example, the n mode accelerator would have to be divided into 5 o r more independent lengths (each about 1 meter long) and each length must have a n independent rf feed. The resulting complications and cost of these independent feeds greatly favor the choice of a traveling-wave structure.

76

G.A.LOEW, R.B.NEAL


Fig. 16. Comparison of energies of standing-wave and traveling-wave accelerators versus time during build-up of fields. n mode and 2n/3 mode operation are assumed for the standing-wave and travelingwave accelerators, respectively. Other assumptions: equal power to each accelerator, B = 1.0,f = 2856 MHz, and same disk aperture. Shunt impedances, ro/Q and v g / c included in calculation are experimentally obtained.

6 Superconducting accelerators As noted earlier in this chapter, the acceleration of electrons to useful energies by means of accelerators operating in the usual temperature ranges requires the supplying of peak rf power measured in megawatts. The cost of power sources having the needed capacity and the cost of the power being consumed limits the beam duty cycle that can feasibly be achieved to values which are usually quite low as pointed out in 9 3. The application of accelerators to various industrial and research tasks is complicated or greatly limited by low duty cycle. In recent years, the possibility of operating accelerators at extremely low temperatures near absolute zero where the electrical conductivity of certain materials such as niobium and lead is very high, has been investigated; e.g., see Banford and Stafford [1961], Wilson [1963], Wilson et al. [1963], Wilson and Schwettman [1965], Schwettman et al. [1965], Smith et al. [1966], Schwettman et al. 119671, and Weaver et al. 119671. The low rf losses at these depressed temperatures and the consequent low peak power requirements indicate that duty cycles approaching unity may be practicable. Only a few remarks on this new and promising


B.l.l

77

application of cryogenics will be given for completeness in this chapter. A detailed account of this work and a complete bibliography is given in Part E of this book. The rf power required to accelerate electrons to a given energy is inversely proportional to the Q of the accelerating structure. At temperatures near absolute zero, the theoretical Q dependence is given by (Wilson and Schwettman [1965]):

where 28 is the gap in the energy spectrum of the superconductor, T is the absolute temperature, u the operating frequency, and k is Boltzmann's constant (1.38 x 10-l6 erg/deg C). Values of Q near lo9 have been measured at 2856 MHz at temperatures between 4.2"K and 2°K (Wilson and Schwettman [1965]) for electroplated lead cavities in the TE,, mode. Because of the exponential dependence shown in eq. (88), the theoretical Q increases rapidly with decreasing T. In principle, an order of magnitude improvement in Q can be obtained by reducing the temperature 1 to 2 degrees at these low levels. The actual improvement is limited by the residual resistance of the electroplated surface. According to eq. (88) the Q of a superconducting cavity varies as w p 2 at a given temperature. This behavior is different from that experienced at room temperatures where, as discussed in 5 3, Q varies as w - ' . Since the ratio r , , / Q , which is a purely geometrical factor independent of surfaces losses, is proportional to w , the shunt impedance r , in a superconducting cavity varies as u p ' .Thus, the use of low frequencies is favored with superconducting accelerators whereas higher frequencies are advantageous for accelerators operating at room temperatures. The decrease in frequency is limited by the increasing cross section of the accelerator structure. It appears that practicable structure sizes will set the lower frequency limit to a few hundred megahertz. The energy gains from a traveling-wave and a standing-wave accelerator in the case of negligible beam loading are given by eqs. (33) and (72), respectively. For superconducting accelerators, where z is very small, these equations become:

,

where P, is the source power and P, is the power dissipated in the accelerator structure of length I. Thus, for the same energy gain, it appears that the power dissipated in the standing-wave structure must be twice that dissipated in the traveling-wave structure, assuming both structures have the same lengths and shunt impedances. In actual fact, as already noted, it is possible to make up for this disadvantage by using the TC mode or the n/2 mode in a bi-periodic structure for which the shunt impedances are considerably increased. From another viewpoint, for a given net energy gain, the ratio of peak to average fields in the standing-wave structure is up to two times as high as

78



this ratio in the traveling-wave structure. This consideration gives an advantage to the traveling-wave structure for superconducting accelerator applications. However, because of the inherently very low rf losses, traveling-wave superconducting accelerators must employ the feedback principle in order to utilize fully the available rf power and achieve high accelerating efficiency. Traveling-wave superconducting structure with feedback. The unloaded energy of the traveling-wave accelerator with optimum feedback is given by eq. (45). Equation (45) is valid in the room temperature case where the attenuation in the feedback loop is usually negligible compared to the attenuation in the accelerator structure itself. This is not true in the case of the superconducting accelerators unless the feedback loop is refrigerated along with the accelerator structure. If the attenuation in the feedback loop is y nepers, the no-load energy with optimum feedback is given by * g optimum; VTWFB = (27)* [(I - e-')/TI [l - e-2(T+y)]-t(P,r,l)* (i=o

)

The bracketed term which gives the energy build-up ratio due to feedback is reduced by the loop attenuation y. When y is negligible, eq. (91) becomes identical to eq. (45). When the beam current is turned on, the beam energy decreases from the no-load value in accordance with the equation:

where Vn= ~ / ( ~ , r , l ) and * in= (i2r,l/~,)) are the normalized beam energy and beam current, respectively. When z and y are small, eq. (92) may be written:

( r and y small)

The conversion efficiency defined as v ] = Vi/P,, i.e., the fraction of the input rf power which is converted to beam power, can be obtained by multiplying Vn by in as follows:

Curves of Vn and q are shown in fig. 17 vs. in. Two sets of curves are shown in fig. 17. One set is based on y/z= lo4 [case (a)] which is typical of feedback through a loop which is at room temperature while the accelerator structure is supercooled.

*

A more complete t r z a t m n t of the traveling-wave superconducting accelerator with feedback with derivations of the basic equations presented here is given by Neal [I9681 (see also ch. E.2).

B.l.1


79

The other set is based on y / z = 0 . 1 [case (b)] which is a rough approximation* for the case where the feedback loop as well as the accelerator structure are supercooled. From a comparison of these curves, several observations may be made. For fixed rf power input, the theoretical no-load energy is about 95 times higher for case (b) where y / =~ 0.1 than for case (a) where y / z = lo4. As the beam current increases, u] increases much more rapidly for case (b) than for case (a) and approaches 100% for relatively small values of current. Similarly, the beam energy for case (b) drops off much more rapidly with increasing current. At higher values of beam current, the

q [CASE (b)]

Fig. 17. Normalized beam energy (Vn) and beam conversion efficiency (u) for traveling-wave superconducting accelerator with optimum feedback versus normalized beam current (in). Accelerator nepers. Feedback loop attenuation parameter, y =- 5 x 10-3 attenuation parameter, .r = 5 x nepers [case (a)] and y = 5 x nepers [case (b)).

values of beam energy and u] for both cases approach the same values. Case (b) is obviously superior where the attainment of high energy is paramount. However, the very large value of the beam loading derivative in this case requires that the current be maintained constant with high accuracy in order to achieve energy stability.

* At room temperatures, the attenuation in nepers per unit length is typically 50 to 100 times as high in the accelerator structure as in a similar length of ordinary waveguide. Because of the presence of bends and the re-combining bridge in the feedback loop, the more conservative estimate of 10 is being used in this example.

80

G.A.LOEW, R.B.NEAL


The above discussion is based upon optimum feedback which implies a bridge ratio which can be varied to suit any degree of beam loading. When a fixed bridge ratio g is utilized, the general expression for beam energy is: 1-

eCY[(I - e-')/(2z)*] in

(TWFB) (g fixed)

(95)

where Vn and in are the normalized beam energy and beam current as previously defined. The conversion efficiency q is given, as previously stated, by the product of V, and in. Unlike the case with optimum feedback where q continues to approach unity as in increases, for fixed bridge ratio g there is a value of beam current which results in maximum q. The condition for q,,, (for small z) is:

When the condition of eq. (96) is met, the conversion efficiency becomes:

At maximum conversion efficiency the beam energy is reduced to one-half of the no-load value. In general, the beam energy is given by:

is the beam current resulting in maximum where Vo is the no-load energy and iqmax conversion efficiency. The normalized beam energy V, and beam conversion efficiency q are plotted in fig. 18 vs. in for 3 values of the bridge ratio g (g= lo4, lo5, and 9.08 x lo5). The latter value of g gives maximum no-load energy. For ease of comparison, the same values 17. It o f z a n d y ( z = 5 ~ 1 0 - ~y=5x10~8)havebeenassumedasforcase(b)offig. ; is noted that a large bridge ratio results in a high value of the no-load energy and a high value of beam loading derivative. Also, for large g, the conversion efficiency q peaks at a lower value of beam current and the maximum efficiency is less than for smaller values of g. The dashed lines in fig. 18 represent the values of Vn and q for the optimum feedback case. They are the envelopes of all the possible cases of fixed g. The total power dissipated in the accelerator and in the feedback loop including the accelerator is just P, (1 - q). The fraction of the rf power which is dissipated in the accelerator structure alone during operation at maximum efficiency can be found by setting eq. (89) equal to one-half of the no-load energy obtained from eq. (95). The result is

Filling time of traveling-wave structure. An analysis of the transient build-up of the superconducting accelerator with feedback (see Neal [1968]) shows that the number


B.l.l

81

of transits around the loop required to build up to the fraction (1 - l/e), i.e., to 63.2% of the magnitude of the steady state electric field in the accelerator structure is

Since the time for a single transit is very close to I/v, and since from eqs. (25) and (3 l), 1jvg=(2Q/w) T, the filling time of the structure is given by

When the structure is designed for negligible beam loading, the relationship giving maximum energy and conversion efficiency is (1 129)= z y. Thus, the filling time for this case is (2Q/w) ~g = (Q/w) [z/(z + y)]. For heavier design loading, (I/&) 9 ( 2 y) and thus, the filling time approaches (2Q/w) (2zg).

+

0

2

4

6

8 n

10

+

12

14

16

Fig. 18. Normahzed beam energy (V,,) and beam conversion efficiency (11) for traveling-wave superconducting accelerator with feedback versus normalized beam current (;,,). Curves are shown for 3 fixed bridge ratios. Accelerator attenuation parameter, r = 5 x nepers. Feedback loop attenuation parameter, y 5 x 10 nepers.

-

Using eqs. (95) and (96), the filling time may alternately be written:

Thus, the filling time is reduced as the beam current giving maximum conversion efficiency is increased (and the corresponding beam energy decreased). As an example, nepers, Q = lo9, and w=1.79x 10'' assume g = lo5, r = 5 x l o p 7 nepers, y = 5 x rad/sec ( f = 2856 MHz). Then, from eq. (96) one obtains in(,",.

ment between the trends of these curves, however, is quite good and emphasizes two important facts. One is that the maximum shunt impedance for a traveling-wave accelerator is in the neighborhood of n=3, or 2n/3 phase shift per cavity, the other is that the highest shunt impedance is obtained for the n mode or standing-wave case where the synchronous shunt impedance is the summation of both the n=O and n = - 1 space harmonics. The disadvantages of this mode have already been discussed in detail in 9 5. The accuracy of the n mode point is somewhat in question because of the difficulties involved in the space harmonic and Q measurements for this case. Figure 29 summarizes the design parameters of the SLAC constant-gradient continuous taper structure. It shows the range of variation of 2a, 2b, r , and v,/c. Similar plots

Fig. 29. Variation of 26, 2a, v,/c and the shunt impedance ro (corrected for the fundamental space harmonic) as a function of length (zll) along SLAC 3.05 meter constant-gradient section for

t l l o = 5.57 x

have been obtained by other designers at Varian Associates (Palo Alto, California) and C.S.F. (Corbeville, France) for other constant-gradient accelerators using discontinuous tapers. (For parameter ranges, see table 1.) The coupler problem. One of the cavity configurations most commonly used to couple power into and out of an accelerator structure is shown in figs. 30a and 31a. In this design, the transition from the rectangular waveguide is made by means of a matching iris, the exact size and plane of which are adjusted to obtain the correct

Y


B.l.l

97

match. The coupler cavity is a regular accelerator cavity. the fields of which are in the proper phase relationship with those in the rest of the structure to produce acceleration. In other designs, particularly those originating at C.S.F. (France) (see fig. 30b), the coupler cavity is different from the other cavities in that it is much narrower and does not contribute substantially to electron acceleration. Its advantage is that the standing-wave pattern set up in the cavity can perhaps be controlled more closely and the distance over which it acts on the incoming electrons can be shortened. This feature is often important in buncher design. Another variation of this type with similar advantages is the "door-knob" coupler.

I

Input RF

Accelerating

I

w,

Aperture

Matching and Tuning Plunger

a ) SLAC DESIGN

b ) C.S.F. DESIGN

Fig. 30. Coupler designs.

One feature that most of these couplers seem to have in common is that the lateral aperture introduces both a n amplitude asymmetry and a phase shift in the axial electric field Ez as a function of the transverse coordinate x. (See fig. 3 1 a,) The asymmetry can be of the order of 10"/,nd the phase shift of the order of 1.5" over the beam aperture. It can induce beam rotation in low energy sections when magnetic focusing is used (Bergkre et al. [1965]). T o understand the effect of these asymmetries on the beam, let the spatial variation of E, be of the form EZ = [ E Z , ,

+ ( A E / 2 a ) x] exp [ j ( A @ / 2 n )x]

( 1 25)

where AE and A@ are the amplitude and phase variations in the x-direction over the

98



aperture diameter 2a averaged from 0 to d, the cavity length. Then, to firs^. order, dE,/ax

=j

(A@/2a)E,,

exp [j(A@/2a)x ]

+ (AEI2a) exp [j(A@/2a)x ] .

(126)

From Maxwell's equations, By and hence the Lorentz force F, are bo:h in time quadrature with dE,/ax. Referring to fig. 31b, the deflecting force F:, caused by the

RECTANGULAR WAVEGUIDE

(1

CROSS-SECTION OF COUPLER CAVITY r

'i /-

ACCELERATING FORCE

DEFLECTING FORCE DUE TO AMPLITUDE ASYMMETRY

DEFLECTING FORCE DUE TO PHASE ASYMMETRY

b )

FORCE COMPONENTS DUE TO AMPLITUDE A N D PHASE ASYMMETRY I N T H E COUPLER CAVITIES

MATCHING IRIS

X

c )

COUPLER OFFSET TO CORRECT A M P L I T U D E ASYMMETRY

Fig. 31. Effects of coupler design and example of technique used to correct for amplitude asymmetry effect.

B.1.1


99

phase asymmetry and proportional to (A@/2a)E,, , is in phase with the accelerating force F,; it produces a net deflection on a bunch traveling on the crest of the wave. On the other hand, the force F:, caused by the amplitude asymmetry and proportional to 1/2a(AE/Ez,,) E,,,, is in quadrature with the accelerating force F,; its effect, as shown in an exaggerated form in fig. 31b is to spread transversely a bunch traveling on the accelerating wavecrest. Using simple electrodynamics, it can be shown that the transverse momentum Gp, imparted to an electron traveling at an angle 8 with respect to the accelerating wavecrest through the coupler cavity of length d is given by 6p,

=

[eE,, ,id/4rrac] [ A @ cos 0 + (AEIE,, ,) sin 81 .

(127)

In a short accelerator, these effects are of no serious consequence because the angle 8 can be monitored and kept close to zero and the steering effect can be compensated for. In a long accelerator where several beams of different energies may be accelerated simultaneously by different numbers of klystrons, the difficulties may be more serious. To remedy these difficulties, several measures can be taken. One of them consists of offsetting the coupler cavity as shown in fig. 31c. By a proper adjustment of the offset (of the order of 2 to 4 mm at S-band), the value of AE/E can be reduced to 0.1% or less. The phase shift across the axis could in principle be eliminated by feeding the coupler symmetrically from both sides or be compensated for by tilting the coupler cavity by an appropriate angle with respect to the axial direction of the rest of the section. Since these solutions pose difficult fabrication problems, a simpler technique was used at SLAC. It consists of feeding successive accelerator sections from alternate sides and furthermore, of letting the effect of the input and output couplers partially cancel each other by having them on the same side. (For further details and results see Borghi et al. [I9661 and SLAC-50 [1965].) With these remedies, steering effects can be kept to a minimum. Matching and tuning. If the process of machining parts and fabricating an accelerator structure could be controlled with perfect accuracy, a section such as the one shown in fig. 32 would be ready for use immediately after manufacturing. In practice, such a control is difficult and uneconomical. As discussed in ch. B.2.2, most manufacturers prefer to build their structures to relaxed tolerances and end up correcting the imperfections through final tuning and matching. The process of t~lning is intended to adjust the phase shift of every cavity to the design value. It generally consists of changing the cavity dimensions by adjusting the degree of insertion of a small tuning plunger or by applying a controlled external mechanical pressure to impart a permanent deformation to the cavity wall. Matching is done by adjusting the dimensions of the coupler cavities described in the previous section of this chapter so as to insure a low standing-wave ratio at the input and output of the structure. As will be shown briefly below, one of the main difficulties stems from the fact that the two processes are not independent. The problem of tuning and matching an accelerator section can best be ~lnderstood in the broader context of the general problem of measuring impedances at microwave

100

G.A.LOEW, R.B.NEAL


Fig. 32. SLAC ten-foot constant gradient accelerator section.

frequencies through one or several junctions. The junctions in this case are the couplers. The main tool for these measurements is the Smith chart. Comprehensive descriptions of the pertinent techniques can be found, for example, in Deschamps [1953], Chodorow et al. [1955], Mallory [1955], Wentworth and Barthel [1956], Ginzton [1957], Gallagher [1960], BelbCoch [1961], Westbrook [I9631 and Borghi et al. [1966]. The most fundamental idea, common to all these references is the following: the


B.l.l

101

D METER

0GENERATOR

{kt, SLOTTED

MOVABLE SHORT

LOAD

\ OUTPUT COUPLER

DETUNING PLUNGER

l NPUT COUPLER

a ) ILLUSTRATION OF VARIOUS ASPECTS OF NODAL

SHIFT TECHNIQUE

MATCHED LOAD PHASE

BRIDGE

I

I I

I I

L------

GENERATOR

\-h&

MICROMETER

//

INPUT COUPLER

b ) PHASE AND AMPLITUDE

/

BEAD ON THREAD

/

U

WEIGHT

OUTPUT COUPLER

M E A S U R E M E N T OF F I E L D

Fig. 33. Experimental set-ups used for matching, tuning and checking field uniformity of accelerating structure.

102

G A L O E W , R.B.NEAL


impedance of a network measured directly is related to the impedance of the same network as seen through a junction by a bilinear and conformal transformation which characterizes the junction. Consequently, the corresponding plots on a Smith chart obey the rules of conformal mapping, namely, that circles transform into circles and angles are preserved. Referring to fig. 33a, the goal of the matching and tuning operation is to obtain a V.S.W.R. equal to unity when the detuning plunger is removed and the output coupler is connected to the load. This condition can fail to be realized for several reasons: mismatch of the output coupler; mismatch of the input coupler; or internal reflections due to imperfections of the periodic structure. All three conditions must be cured individually since compensating one with the other would not result in a uniformly traveling wave of maximum possible amplitude. The measurement is greatly simplified when the frequency is such that the phase shift per cavity is a submultiple of 2n, typically n/2 or 2x13. The input impedance can be measured for two possible conditions. The first consists of connecting the output coupler to a movable short and of measuring the input impedance for known displacements of the movable short, typically 418 or 4/12. The points a, b, c, d, e, f which, seen beyond the output coupler, lie equally spaced on the rim of the chart (see fig. 34a), transform to a circle of reduced size (dependent on the loss in the structure) when referred to a plane corresponding to the matching iris of the input coupler (generally obtained by detuning the coupler cavity with a plunger). The transformation through the output coupler only has the effect of reshuffling the points on the rim of the chart. For a mismatched but perfect periodic structure, two useful points, (I) and (S), can be obtained through this construction. (M) being the geometric center of the circle, the iconocenter (I), constructed as shown, is the input impedance obtained if the output coupler is connected to the matched load. (S), the so-called Smith center, represents the input impedance that would be obtained if a matched load were placed in the periodic structure, downstream of the input coupler. It is obtained by drawing the line MC which intersects the circle at o, and o,,, (where o designates the V.S.W.R.) and finding the point (S) which verifies the relation CS=(o,,,/o,,,)~ [or, conversely, ( o ~ ~ ~ cif Tthe~ circle , ~ ) does ~ not encompass the center of the chart]. With a little experience, the experimenter can deduce from the location of (S) whether the coupler is under- or overcoupled. The proper action, namely increasing or decreasing the size of the matching iris, will then produce a match, i.e., bring points (M), (I) and (S) to the center of the chart. If there were no other unknown, the entire procedure would be straightforward. The difficulty arises from the fact that very often the periodic structure has irregularities which create unknown reflections which add to the remain unknowns is the coupler cavity itself, the size and phase shift of which depends on the aperture of the matching iris. This interaction significantly complicates the entire procedure, particularly for designs with phase shifts per cavity other than n/2.


B.1.1

RADIALLY

-.-- TolmO

S C A L E D PARAMETERS

q= Lo.0

103

5

8

a

a

' i ' ' h Y " i

s '

, , , , , a , , ,b,

8

rb ' i ' k b ' m

Fig. 34a. Nodal shift plot obtained with movable short at output coupler. I n example shown, neither input nor output coupler is matched.

104

G.A.LOEW, R.B.NEAL


IMPEDANCE OR ADMITTANCE COORDINATES

Fig. 34b. Nodal shift plot obtained with detuning plunger inside structure. Points C, 1 , 2, 3, 4 . . . correspond to perfectly matched and tuned structure. Points C, l', 2', 3', 4' ... represent imperfect tune with perfect match. Points C, I", 2", 3" ... represent perfect tune with imperfect match.

B.I.1


105

At this point, one can consider the second type of measurement which consists of moving the d e t ~ ~ n i nplunger g inside the structure as shown in fig. 33a. If the coupler is perfectly matched and all cavities have 120" of phase shift (see the 2x13-mode example of fig. 34b), pulling the plunger away from the coupler causes the plotted null points to rotate clockwise by 240 electrical degrees on the Smith chart for each cavity length through which the plunger is moved. Starting with the reference point at C obtained by detuning the coupler cavity, points 1, 2, 3, 4 ... move clockwise from 0" to 120" (&I3 or 240" on the chart), from 120" to 240°, from 240" to 0°, and so on. For the first cavities, the V.S.W.R. is nearly infinite and the points are almost on the rim of the chart. Farther down the section, the V.S.W.R. decreases because of the net attenuation. Thus, the plotted points move toward the center of the chart. When a systematic phase shift error exists because of an inaccurate choice of frequency, or a systematic error in fabrication, points 1, 2, 3, 4 . . . appear for example as l', 2', 3', 4' ... . If on the other hand, the coupler is mismatched, the points appear as 1 ",2", 3", 4". .. . At this stage, it is essential to proceed carefully, by successive iterations. Three techniques can be used. The first is to match and tune the coupler by itself using the method suggested by Kyhl [1960], and described by Westbrook [1963]. The second is to assume that at least cavity No. 1 beyond the coupler has the correct phase shift and obtain the iconocenter for points C, I and 2 using the technique described by Wentworth and Barthel [1956]. The point obtained for the plunger in cavity ( 1 ) indicates whether the size of the coupler cavity is approximately correct. Point (2) indicates if the matching iris has approximately the correct width. Having made the proper iterative corrections, it is then possible to move the plunger down the entire structure and tune all cavities for 120" phase shift. The third method is to obtain the impedance circle and Smith center (S), as described earlier, for points such as 8". 9", 10". Moving farther along in the structure is necessary because the first points are still on the rim of the chart and point (S) is hard to obtain from a , and a,,,. The operation of bringing point (S) to the center of the chart is then achieved by iteratively matching the coupler and tuning the successive intermediate cavities. This process is sometimes simplified by replacing the detuning plunger by a partially matched load which reduces the size of the circle and facilitates finding the Smith center (S) (see Gallagher [1960]). The exact choice between these various techniques varies from one laboratory to another and depends, among other factors, on the number of accelerator sections to be fabricated and the degree of control exercised over the manufacturing process. Thus, for example, if the resonant frequency of every cavity is verified and adjusted before final assembly, some of the above steps can be eliminated. Special care must also be taken that the above measurements are performed in a carefully controlled environment. Further details on these techniques can be found in ch. B.2.2. After matching and tuning, it is often of interest, as a final check, to obtain a complete traveling-wave field plot for the entire accelerator section. This plot can be

G.A.LOEW, R.B.NEAL


0.24

90

0.22

80

0.20

70

0.18

60

0.16

50

0.14

40

0 12

30

0 10

20

0.08

10

0 0.2 CENTER OF CAVITY

0.4

0.6

0.8

1.0

1.2

1.4

1.6 ,1.8 2.0 CE~TER OF Dl SK

2.2

2.4

8"

0

Fig. 35. Amplitude and phase of traveling axial electric field given by measurement of amplitude (SLAC 2x13-mode cavity, where d = cavity and phase of square root of bead reflection coefficient (T). length, t = disk thickness.)

obtained by drawing a small metallic or dielectric needle through the structure as shown in fig. 33b. The relative field amplitude is obtained by measuring either the reflection coefficient caused by the wave reflected from the bead, or the phase shift across the section caused by the forward-scattered wave from the bead. The phase of the field can be measured by noting the phase of the reflection coefficient of the reflected wave. Figure 35 shows the amplitude and phase of the traveling-wave field in a SLAC 2x13-mode cavity. It is seen that the phase is fairly "stationary" in the middle of the cavity but varies rapidly through the disk. Similarly, the amplitude undergoes maxima and minima. These are of course due to the space harmonics. It is interesting to note that one can also recheck the relative amplitude of the space harmonics by using both these amplitude and phase plots. Thus, applying eq. (4) on axis ( r = O ) and integrating over the length of one cavity, one obtains

ai/x

a: can be obtained to a few percent accuracy by calculating only Generally, three or four terms. An example which gives further insight into how three of these space harmonics add and subtract is shown in fig. 36. Notice that if P0d=2rc/3, p,z


B.l.l

107

can be rewritten as (1 + 3n)P,z. At z =+d, a,, and a _ , are normal to a,. At z= +d they are again collinear but a,, and a _ ,are of opposite sign to a,. This final field amplitude and phase plot can be of particular help and interest in a constant-gradient structure where doubt may remain, until final assembly, as to whether the desired field uniformity has been achieved.

@

DWELL POINT;z=O;

Po z

a QUADRATURE POINT ; z @

DISK EDGE POINT; z =

=O

$ ; Po z = 3 0 "

9 ; Poz = 6 0 "

Fig. 36. Illustration of how space harmonics add and subtract in a typical SLAC section. It is assumed here that all an vectors are positive.

9 Summary of characteristics of accelerating structures of existing electron linacs To conclude, it seems instructive to show, through some concrete examples, how the theory and design of electron linac structures described in this chapter have in practice

108

G.A.LOEW, R.B.NEAL


Summary of accelerating structure characteristics for selected electron linacs(') --- - --

Linac identification -

-

--

Type of construction Operating frequency (MHz)

-

-

-

-

-

17 MeV 150 MeV M.I.T. N.B.S. (Cambridge, U.S.A.) (Maryland, U.S.A.) ----Constant Standing-wave impedance --

2802

1299

7

9

Number of sections Length of sections, I (m) Phase shift per cavity Field attenuation, nl (nepers) Shunt impedance for fundamental space harmonic, ro (megohms/m)

0.36 44 (31

Number of constant group velocity segments within one section

1

1

Group velocity (v,/c)

0

0.0048

Q Filling time (4) (psec)

Waveguide diameter, 2b (cm)

Iris aperture diameter, 2a (cm)

Disk thickness (cm) Klystron peak output power rating (MW) Average number of accelerator sections per klystron Rf duty cycle Average energy gradient at nominal operating level (keV/cm)

0.3175

1.341

0.25 (Magnetron)

5

113 (3 Magnetrons/section)

1

-

2.4

x

2.18 27

x

w 70

I . Quantities in brackets indicate that more than one value or range of values are applicable. 2. Some of the numbers given in this column were actually published for the 360-MeV Kharkov accelerator. It is believed that these numbers are the same for the 2-GeV machine.

--

--

B.l.l


-

--

-

-

P A

450 MeV Adone mjector (Frascat~,Italy) -

-

Approximately constant gradlent -

--

-

-

57.3 +56.8 50.7 -+ 56.1 49.8+56.8

(0.54,0.57, 1.19)

2.3 GeV (Orsay, France)

2 GeV ( 4 (Kharkov, U.S.S.R.)

20 GeV SLAC (Stanford, U S.A.)

Approximately constant gradient

Constant impedance

Constant gradlent

-

--

-

-

45 old (42, 52) - new

43 1: 3

(52 ; . 60)

3 - old 6-new

I

Continuous taper

0.375

0.83

3.09

2.622 + 1.9235

unknown

0.5842

-

= 0.93-

old

8.1968+8.210 8.415 + 8.2276 8.4546+8.2100

unknown

2.0338 +2.0892 2.8385 +2.1747 2.9565 -+2.0892

(2.7 + 1.9)- old (2.96 -t 2.11) ) (2.8 -t 1.82) \ - new

0.5842

109

0.3000

3. In this column, the shunt impedance includes the contribution of both the n = O and n = - l space harmonics. 4. For the standing-wave case, the filling time is defined as 2 Q ~ / w Q/co where QI. is the loaded Q . For the traveling-wave cases, it is defined as /lug.

-

110

G.A.LOEW, R.B.NEAL


been applied. Ten years ago, a compilation of all electron linac structures would have been a fairly easy task. Today it is not feasible because of the multitude and variety of accelerators that have been built. Furthermore, certain special types of electron linacs will be described in later chapters of this book and will not be covered here. For this reason table 1 presents the characteristics of only a few selected accelerating structures. This selection has been made to obtain a fairly representative coverage of designs in terms of time of construction, countries, types, etc. Low energy and/or high intensity, high duty cycle machines such as the A.L.S. (Saclay, France), and M.I.T. (Cambridge, U.S.A.) accelerators presently under construction and superconducting accelerators in the planning stage are not included here (see ch. B.3.2). The 17 MeV M.I.T. accelerator in the first column is included because it was one of the early operational machines (see Demos et al. [1952]) and it offers an interesting contrast in that it is the only accelerator operating in the standing-wave 7c mode, all others in this table being of the traveling-wave type. The 150 MeV N.B.S. (Maryland, U.S.A.) accelerator was built by the High Voltage Engineering Corp., Burlington, Mass. A comprehensive description was given by Leiss [1966]. It provides a typical example of an L-band accelerator. The 450 MeV Adone Injector, built by Varian Associates, Palo Alto, California, was designed for electron as well as for positron acceleration. The injector and the first three sections are built for high current acceleration, the last eight sections for high energy gain. This difference in purpose explains the difference between the two types of structures. (See, for example, Nunan 119651 and Haimson [1965].) The 2.3 GeV Orsay linac was built in two stages. The first part consisting of 22 sections and giving an energy of 1.25 GeV has been operational since approximately 1959. (See, for example, Z'Onde Electrique [1963].) The second part, an extension installed upstream of the first part, became operational in 1968. Similarly to the Adone Injector, it contains seven sections for high current acceleration designed for positron generation followed by eight sections for high energy gain. (For a description of this extension, see MClard [1967].) The 2 GeV Kharkov accelerator was perhaps the first linac of the new generation of extra long machines. Some of its characteristics were very similar to the earlier 360 MeV Kharkov accelerator. Both these machines are described in some detail by Val'ter et a/. [1963]. Finally, the last column contains some of the characteristics of the 20 GeV SLAC accelerator, which became operational in 1966. It has been described in numerous documents (for example, see Borghi et al. [I9661 or Neal [1967]). It is believed to be the first accelerator in which the constant gradient field was obtained by a smooth continuous tapered structure. Its energy can be gradually expanded to 40 GeV by increasing the number and power output of the klystrons. The SLAC accelerator was in many ways the logical outcome of experience gained at Stanford over a period of twenty years with shorter accelerators and particularly with the 1.2 GeV Mark 111 (see Chodorow et al. [1955]). This machine originally operated in the 7112 constant-

B.l.l


111

i m p e d a n c e traveling-wave mode a n d w a s later c o n v e r t e d to o p e r a t e w i t h S L A C - t y p e (2x13, c o n s t a n t gradient) sections. P l a n s a r e presently being m a d e to rebuild i t i n t o a

I

G e V superconducting accelerator. W h i l e this t a b l e illustrates a wide variety o f designs, t h e f e a t u r e m o s t striking to

be t h e a m a z i n g similarity between the structures used i n a l l these accelerators. Aside from the L-band N.B.S. m a c h i n e , t h e range of variation of t h e v a r i o u s p a r a m e t e r s is extremely n a r r o w . It seems to illustrate t h e f a c t that d u r i n g any given period o f time, all designers, regardless of geographical location, use the best k n o w n ideas until a new d r a s t i c technological jump is made. W h e t h e r the advent of the reader

may

well

t h e s u p e r c o n d u c t i n g linac is i n f a c t this drastic j u m p will b e f o r t h e historians to decide

at a l a t e r date. References

BANFORD, A. P. and H. G. STAFFORD, 1961, J. Nucl. Energy, Part C 3, 287. BAREFORD, C. F. and M. G. KELLIHER, 1953, Philips Tech. Rev. 15, No. 1 , pp. 1-26. BELBEOCH, R., 1961, Problemes PosCs par I'Adaptation du Coupleur d'EntrCe d'une Structure en Guide Chargk par des Iris, Rapport LAL-13, Laboratoire de I'AcceICrateur Lineaire, Orsay, France. B~.RGERE, R., H. BEIL,and A. VEYSSIERE, 1965, Rapport CEA, R 2843. Centre d'Etudes Nucltaires, Saclay, France. BERTOZZI, W., J. HAIMSON, C. P. SARGENT, and W. T U T C H I N E T 1967, ~ , The MIT High Duty, High Intensity 400 MeV Linear Electron Accelerator, IEEETrans. Nucl. Sci. NS-14, No. 3, pp. 191-196. BEVENSEE, R. M., 1964, Electromagnetic Slow Wave Systems (John Wiley and Sons, Inc., New York, London and Sydney). BORGHI, R. P., A. L. ELDREDGE, G. A. LOEW,and R. B. NEAL,1966, In: Young, L., ed., Advances in Microwaves 1 (Academic Press, New York and London), pp. 1-73. BRILLOUIN, L., 1946, Wave Propagation in Periodic Structures, 2nd ed. (Dover Publ. Inc., McGrawHill Book Co., Inc., New York). BRILLOUIN, L., 1960, Wave Propagation and Group Velocity (Academic Press, New York and London). BUECHNER, W. W., R. J. VANDE GRAAFF, A. SPERDUTO, L. R. MCINTOSH, and E. A. BURRILL, 1947, Rev. Sci. Instr. 18, No. 10, pp. 754-766. CHODOROW, M.. E. L. GINZTON, W. W. HANSEN, R. L. KYHL,R. B. NEAL,and W. K. H. PANOFSKY, 1955, Rev. Sci. Instr. 26, No. 2, pp. 134-204. 1947, J. Appl. Phys. 8, 996. CHU,E. L. and W. W. HANSEN, 1949, J. Appl. Phys. 20, 280. CHU,E. L. and W. W. HANSEN, CHU,E. L., 1951, The Theory of Linear Electron Accelerators, Report No. 140, Microwave Laboratory, Stanford University, Stanford, California. DAHL,P. F., K. JELLETT, G. PARZEN,S. GIORDANO, and J. P. HANNWACKER, 1966, Linac Cavity Field Calculations, Proceedings of the 1966 Linear Accelerator Conference, LA-3609, Los Alamos Scientific Laboratory, Los Alamos, New Mexico, pp. 115-1 18. P. T., A. F. KIP, and J . C. SLATER, 1952, J. Appl. Phys. 23, No. 1, 53-65. DEMOS, DESCHAMPS, G. A,, 1953, J. Appl. Phys. 24, 1046. D E VRIES,C., 1967, Existing and Proposed Linear Electron Accelerators at the Institute for Nuclear Research, Amsterdam, Proceedings of the Meeting on the Use of Electron Linear Accelerators for Nuclear Structure Study (to be published), Laboratori Nazionali di Frascati (Italy). G. T. and G. 1. COHN,1951, Dielectric Loading for Waveguide Linear Accelerators, TransFLESHER, actions of the American Institute of Electrical Engineers, 70, pp. 887-893. FRY,D. W. and W. WALKINSHAW, 1948-9, Rep. Progr. Physics 12, 102-132. W. J., 1960, Measurement Techniques for Periodic Structures, M Report NO. 205, GALLAGHER, Stanford Linear Accelerator Center, Stanford University, Stanford, California.

112

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GINZTON, E. L., 1957, Microwave Measurements (McGraw-Hill Book Company, Inc., New York). S., 1965, IEEE Trans. Nucl. Sci. NS-12, No. 3, pp. 213-216. GIORDANO, J., 1965, IEEE Trans. Nucl. Sci. NS-12, No. 3, pp. 499-507. HAIMSON, HELM,R., 1966, Computer Study of Wave Propagation, Beam Loading and Beam Blowup in the SLAC Accelerator, Proceedings of the 1966 Linear Accelerator Conference, LA-3609, Los Alamos Scientific Laboratory, Los Alamos, New Mexico, pp. 254-266. HOYT,C. H., 1966, Designing Resonant Cavities with the LALA Computer Program, Proceedmgs of the 1966 Linear Accelerator Conference, LA-3609, Los Alamos Scientific Laboratory, Los Alamos, New Mexico, pp. 119-124. KNAPP,E., 1964, 800 MeV R F Structures, Proceedings of the 1964 Linear Accelerator Conference, MURA 714, pp. 31-59. (Available from the Clearinghouse for Federal Scientific and Technical Information, National Bureau of Standards, U.S. Department of Commerce, Springfield, Virginia). KYHL,R. L., 1960, Department of Electrical Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts (private communication). LAWTON, E. J. and W. C. HAHN,1948, J. Appl. Phys. 19, 648. LEBOUTET, H., 1963, Problemes Physiques de la Construction de 1'Accelerateur 1 GeV, I'Onde Electrique, No. 435, Vol. 43, 599. H., G. AZAM,R. BENSUSSAN, G. BLAISE, M. DELMON, J. P. MANGIN, J. SOFFER, 1966, Some LEBOUTET, Characteristic Problems of the New High Duty Cycle Saclay Electron Linac, Proceedings of the 1966 Linear Accelerator Conference, LA-3609, Los Alamos Scientific Laboratory, Los Alamos, New Mexico, pp. 426-434. LEISS,J. E., 1966, The NBS Linac, Proceedings of the 1966 Linear Accelerator Conference, LA-3609, Los Alamos Scientific Laboratory, Los Alamos, New Mexico, pp. 20-27. L'ONDEELECTRIQUE, 1963, L'Acctltrateur Lineaire D'Orsay, Vol. 43, No. 435. MALLORY, K. B., 1955, A Comparison of the Predicted and Observed Performances of a Billion-Volt Electron Accelerator, HEPL Report No. 46, W. W. Hansen Laboratories of Physics, Stanford University, Stanford, California. MELARD, L., 1967, Note Technique No. 83.A.8, Sections Accelbatrices D'Orsay, LM/MLN No. 016167, Laboratoire de 1'Acci.Ierateur LinCaire Service "Faisceau", Orsay, France. NAGLE,D. E., 1964, Coupled Resonator Model of Linear Accelerator Tanks, Proceedings of the 1964 Linear Accelerator Conference, MURA 714, pp. 21-30. (Available from the Clearinghouse for Federal Scientific and Technical Information, National Bureau of Standards, U S . Department of Commerce, Springfield, Virginia.) NAGLE,D. E., E. A. KNAPP,and B. C. KNAPP,1967, A Coupled Resonantor Model for Standing Wave Accelerator Tanks, University of California Los Alamos ScientificLaboratory, Los Alamos, New Mexico (to be published). NEAL,R. B., 1953, A High Energy Linear Electron Accelerator, M. L. Report No. 185, Microwave Laboratory, W. W. Hansen Laboratories of Physics, Stanford University, Stanford, California, pp. 255-266. NEAL,R. B., 1958, Theory of the Constant Gradient Linear Electron Accelerator, M. L. Report No. 513, Microwave Laboratory, W. W. Hansen Laboratories of Physics, Stanford University, Stanford, California. NEAL,R. B., 1961, Comparison of the Constant Gradient and Uniform Accelerator Structures, M Report No. 259, W. W. Hansen Laboratories of Physics, Stanford University, Stanford, California. NEAL,R. B., 1967, IEEE Trans. Nucl. Sci. NS-14, No. 3, pp. 705-720. NEAL,R. B., 1968, Consideration of the Use of Feedback in a Traveling Wave Superconductmg Accelerator, TN-68-1, Stanford Linear Accelerator Center, Stanford, California. NISHIKAWA, T., S. GIORDANO, and D. CARTER, 1966, Rev. Sci. Instr. 37, No. 5, pp. 652-661. NUNAN, C. S., 1965, IEEE Trans. Nucl. Sci. NS-12, No. 3, pp. 465-469. 1953, Fields and Waves in Modern Radio, 2nd ed. (John Wiley and RAMO,S. and J. R. WHINNERY, Sons, Inc., New York; Chapman and Hall, Ltd., London). ROBSON, P. N., 1956, Fourier Series Representations of Dispersion Curves for Circular Corrugated Waveguide Used in Traveling-Wave Linear Accelerators, Report No. 5 105, Metropolitan-Vickers Electrical Company, Ltd., Manchester, England.

B.l.l


113

SARAZIN, A,, 1953, Annales de Radioelectricit6 8, 134. R.-SHERSBY-HARVIE, R. B. and L. B. MULLETT, 1949, Proc. Phys. Soc. (London) B62, 270. R.-SHERSBY-HARVIE, R. B., 1948, Nature 162, p. 890. H. A,, P. B. WILSON,and G. Y. CHURILOV,1965, Measurements at High Electric SCHWETTMAN, Field Strengths on Superconducting Accelerator Cavities, Proceedings of the Fifth International Conference on High Energy Accelerators (Frascati), Comitato Nazionale per I'Energia Nucleare, Rome, Italy, pp. 690-692. H. A,, J. P. TURNEAURE, W. M. FAIRBANK, T. I. SMITH,M. S. MCASHAN, P. B. WILSON, SCHWETTMAN. 1967, IEEE Trans. Nucl. Sci. NS-14, No. 3, pp. 336-344. and E. E. CHAMBERS, SLAC-50, 1965, Consolidation of Results of Preliminary Beam Tests with Sectors 1 and 2, Stanford Linear Accelerator Center, Stanford University, Stanford, California. SLATER, J . C., 1950, Microwave Electronics (D. Van Nostrand Company, Inc., New York), Chapters 4 and 5. SMITH,T. I., 1966, Standing Wave Modes in a Superconducting Linear Accelerator, HEPL 437, High Energy Physics Laboratory, Stanford University, Stanford, California. SMITH,T. I., H. A. SCHWETTMAN, W. M. FAIRBANK, and P. B. WILSON,1966, Stanford's Superconducting Accelerator Program, Proceedings of the 1966 Linear Accelerator Conference, LA-3609, Los Alamos Scientific Laboratory, Los Alamos, New Mexico, pp. 491 -493. SOKOLOVSKY, A,, 1966, The New High Duty Cycle High Current 600 MeV Electron-Positron Linear Accelerator of Saclay, Proceedings of the 1966 Linear Accelerator Conference, LA-3609, Los Alamos Scientific Laboratory, Los Alamos, New Mexico, pp. 417-425. VAL'TER, A. K., I. A. GRISHAEV, E. V. EREMENKO, V. V. KONDRATENKO, G. A. ZEITLENOK, G. F. KUZNETSOV, L. M. LEVIN,1. F. MALYSHEV, V. V. RUMYANTSEV, A. N. SEMENOV, F. F. TURKIN,and V. K. KHOKHLOV, 1963, The 2 GeV Traveling-Wave Linear Accelerator, International Conference on High Energy Accelerators, Dubna, U.S.S.R., Conf-114, Book 1, pp. 520-525. VASTEL, J., 1952, Annales de Radioelectricite 6, 20. WALKER, G. B. and N. D. WEST, 1957, IEEE Proceedings 104C. p. 381. D. A,, 1958, Topics in Electromagnetic Theory (John Wiley and Sons, Inc., New York; WATKINS, Chapman and Hall, Ltd., London), Chapter 1. WEAVER, J. N., T. I. SMITH,and P. B. WILSON,1967, IEEE Trans. Nucl. Sci. NS-14, No. 3, pp. 345-349. 1956, IRE Transactions on Microwave Theory and TechWENTWORTH, F. L. and D. R. BARTHEL, niques, M T T 4 , p. 173. WESTBROOK, E., 1963, Microwave Impedance Matching of Feed Waveguides to the Disk-Loaded Accelerator Structure Operating in the 2n/3 mode, TN-63-103, Stanford Linear Accelerator Center, Stanford, California. WILSON,P. B., 1963, Nucl. Instr. Methods 20, 336. and W. M. FAIRBANK,1963, Status of Research at Stanford WILSON,P. B., H. A. SCHWETTMAN University on Superconducting Electron Linacs, Proceedings of the International Conference on High Energy Accelerators, Dubna (Atomizdat, Moscow, 1964), p. 535. 1965, IEEE Trans. Nucl. Sci. NS-12, No. 3, pp. 1045-1052. WILSON,P. B. and H. A. SCHWETTMAN,

ELA THEORY

1.2

B.l

Particle Dynamics R. H. HELM R. MILLER Stanford Linear Accelerator Center Stanford, California, U.S.A.

1 Introduction 1.1 Summary and scope

Although the particle dynamics of electron linacs and proton linacs are in principle completely analogous, great quantitative and practical contrasts arise from the vastly greater charge-to-mass ratio of the electron. Thus to gain an energy increment equal to its rest energy in an accelerating gradient of 50 kV/cm, an electron would travel only about 10 cm, while a proton would require nearly 200 m. Consequently in an electron machine even of modest energy, the particles are highly relativistic over most of the length; transverse rf forces virtually disappear because of the cancellation of electric and magnetic forces at relativistic velocities, and longitudinal bunching motions become negligible because of the increasing "longitudinal mass" proportional to Y 3 ; the problems of acceleration and beam transport are essentially decoupled. The high elm of the electron also has consequences in the injection region. The rapid initial acceleration of the electrons is accompanied by the possibility of rather efficient capture and bunching in constant phase-velocity sections, and this property is often exploited to greatly simplify buncher design. Although bunchers employing tapered phase-velocity sections have higher theoretical capture efficiencies, they are rather inflexible in operation and are seldom used. Thus the concepts of synchronous phase velocity and synchronous phase angle often are irrelevant in electron machines. The present chapter, which is concerned with the particle dynamics of electron linacs specifically, is divided into the following topics: (1) Injection dynamics, including general philosophy of buncher design, gun 0 Norfh-HollunilPubl. Co., Amsferdam 1969

References p. 145

115

R.H. HELM, R. MILLER

116


optics, prebunching, longitudinal and transverse motion in the capture region, and focusing. (2) Asymptotic high energy dynamics. (3) External focusing requirements for high energy transport, especially in very long machines. (4) Transverse perturbing effects such as misalignments and stray magnetic fields.

1.2 Units and conventions Dimensional equations will be expressed nominally in MKS units. However it will be convenient to introduce several special definitions which will simplify the equations of motion; in particular the symbol p (and its rectangular components p,, p,, p,) will designate normalized momenta, i.e.,

P

--YB

-

(1)

and normalized field quantities d and B? will be defined by

d

e~/rnc~

(2)

The scale factors mc=O.5ll MeVlc, mc2/e=0.51 1 MV, and mc/e= 1.705 x lo-' weber/m= 1.705 kG-cm will be useful occasionally. Note that 8 and 99 have units of (length)-'. The symbols y and P have their usual significance as normalized energy and velocity. The coordinate system will always be chosen such that the z-axis is the accelerator symmetry axis.

2.1 Longitudinal motion As stated in ch. B.l.l, the longitudinal electric fields in a periodic traveling wave accelerator structure having cylindrical symmetry can be written* Q)

E,

= exp j ( a t -

k,z) n=-m

- jEn& (k,,r) exp (- 2jnnzld)

(4)

where d is the periodic length, k, is the propagation constant of the fundamental space harmonic, and

k& = k 2 - ( k , + 2nr1/d)~,

*

k = 2n/A

For a discussion of the fields in periodic structures see Slater [I9501or Chu [1951].

PARTICLE DYNAMICS

B.1.2

117

In addition, when d = n / k , and IE(_n-,,I = IE,,], eq. (4) is the expansion for a standing wave in a periodic structure with a period which is an integer submultiple of d (i.e., n,n / 2 , ... mode). If the origin of z is chosen to be a plane of symmetry of the structure, the E,, are $1 real. The gain in energy per unit length in the paraxial approximation is just e times the real part of EZ,so dy/d:

b,J, ( k ~sin) (0 + 2 x n z l d )

=n

where t',, = eE,/mc2 and

Bp is the

normalized phase velocity and has been included in the integrand to cover structures in which the phase velocity is a slow function of position. While eqs. (5) and (6) are suitable for numerical integration of the longitudinal motion of low current beams in either traveling wave or standing wave accelerators, considerable physical insight can be gained by making simplifying assumptions or approximations which permit analytic solutions. The first of these is that B=Bp so that 8 does not vary rapidly over wide ranges. In this case, the force from the n=O space harmonic dominates the motion of the particle, since the forces from all other space harmonics oscillate rapidly with a sin ( 0 + 2nnzld) dependence. This approximation which is excellent in the relativistic limit is only fair in the injection region. The equations resulting from neglecting the other space harmonics and assuming that r, r', and 4' are first order quantities are*

These can be combined into a form which is integrable for ,!Ip and t', constant:

Integration by inspection and substitution of y2 = p 2 f 1 yields the equation for the particle orbits in phase space

where A is the constant of integration.

*

The reference of phase 0 is such that the electric field is 0 for 0 =O.

R.H.HELM, R. MILLER

118


Plots of eq. (9) due to Slater [I9481 are given in figs. 1 and 2 for Bp < 1 and Pp= 1 respectively.* It follows from the symmetry of the cosine that orbits are symmetrical about Q = O (the phase stable field null); a particle which passes through a point (p,, 0) will also pass through the point (p,, - B), if it stays within the structure. It is evident from eq. (8) that phase extrema, Q = Om, occur when PP= ( y 2 - I)) y = P. Thus, for bound orbits

+

and

k cos 8 - cos 8, = -Pp&o

+ I)* - (1 - pi)+ - bpP] .

Fig. 1. Longitudinal phase space for Bp< 1.

"

*

- Fig. 2.

Longitudinal phase space for 8, = 1.

A very complete discussion of this analytic solution is given by DGme [1960].

-'

PARTICLE

B.1.2

DYNAMICS

119

2.1.1 Phase velocity less than c Consider the case when p, < c shown in fig. 1. Each value of cos 8, gives a different orbit. If cosO,< - 1 then the orbit is not closed and if the initial momentum PO corresponds to ljp, the particle will continually advance with respect to the wave as in orbit u2. If cosB,= - 1, the particle velocity p will approach the phase velocity p, as the phase approaches _+n. The point @=/I,, 8= +z) is a point of unstable equilibrium. In this case, however, since the force from the fundamental space harmonic vanishes, it is not valid to neglect the fields from the other space harmonics. When 1 >cosO,> - I the particles execute an oscillatory motion in P and 0. It can be shown that for small excursions in p and 0 the orbits approximate ellipses centered around (/I=flp, tI=O). If an ensemble of particles with a relatively large phase extent d(Ir and small momentum extent dp, enters such a structure, then after traversing $ of a phase oscillation it will have a small phase extent AO, and large momentum extent Ap,. If the ratio of phase excursion to momentum excursion in the structure is QZ/p2 the bunching action can be described (Miller et al. [1965]) in the small orbit limit by

A03

AP, =

2 do,

AP, .

Thus if (HJP~) t,, the sum of the two solutions given by these examples, eq. (3.18) plus (3.20), reduces to the steady state solution found previously, eq. (3.6). The two examples given above allow a large class of solutions to the transient beam loading problem. Since the energy gains from the fields produced by the rf power source and from the fields induced by beam loading are independent, we can generate different solutions corresponding to different magnitudes of beam current and different beam turn-on times by suitable sums of these solutions and time shifts. An example of one such set of beam loading solutions is given in fig. 3.

B.1.3

BEAM LOADING AND TRANSIENT BEHAVIOR

157

Figure 3 illustrates the degree to which energy spread due to transient beam loading can be overcome by turning on the beam current during the rf filling time. This example is calculated for the beam current corresponding to maximum power transfer t o the beam, i,, and for different beam turn-on times after rf turn-on. It illustrates that by suitable timing of beam turn-on the energy spread of the beam caused by transients can be greatly reduced.

Fig. 3. Plot of relative energy gain in the transient region versus time in units of the rf filling time tr for a specific accelerator (IL := 0.6, i = it,,). The parameter 6 is the fraction of a filling time after rf turn-on at which beam is started. These solutions are the sum of eqs. (3.18) and (3.20) with a time shift of 6te in the beam loading term (3.20).

3.3 Other waveguide structures The expressions developed above have been calculated for the case of waveguides whose structure is constant throughout the length of the guide. Many linear electron accelerators are now built with "constant gradient" waveguides whose structure are

158

J.E. LEISS


not constant with length, but rather change t o compensate for attenuation or a certain amount of beam loading. These waveguides are usually of two forms. In the first and most common case the parameters of the guide are not changed continuously, but rather the waveguide is composed of short sections of constant structure waveguide with suitable matching elements. For this case beam loading theory can be calculated using the expressions given above, taking the fields from one element of the guide, with appropriate amplitude change for shunt impedance change, as the input field for the next element of the waveguide. In the second case of constant gradient waveguide the pxameters of the guide are changed continuously. For this case the differential equation for the field, eq. (3.3) or eq. (3.1 1) must now include the variation in shunt impedance r, group velocity, u,, and attenuation constant, I, with Z. Transient beam loading calculations for this case, similar to those developed above for the constant structure waveguide, have been developed by Burnod [ I 9611. The largest influence of the constant gradient structure on the effects thus far considered is an enhanced influence of beam loading upon the rf field toward the end of the accelerating guide (Haimson [1965]). For large beam currents, these fields can be so large as to exceed the voltage breakdown limits of the structure and thus lead to serious sparking.

4 Beam loading theory including dispersive eflects The previous analysis rests upon assumptions which are strictly valid only for a continuous wave accelerator. These assumptions are that the rf field in the accelerator is at one precise frequency, and that sharply rising waveforms are transmitted in the accelerator without distortion. Due to the pulsed nature of these accelerators, the first of these assumptions is necessarily incorrect since frequency spreads are inherent in switching transients. The second assumption can not be strictly true since the accelerator waveguide is essentially a band pass filter. It is still true, however, to the extent that the frequency spread in the rf pulse is all contained within the band pass of the accelerator, and to the extent that the phase shift varies linearly with the frequency over the frequency range of interest. It is further assumed in the previous analysis, although reasonably well hidden, that the particle beam contains only one frequency component. Due to the very tight phase bunching in linear accelerators, this assumption is substantially incorrect. The beam is in fact very rich in frequency structure and is tightly bunched in space. In this seciion we present an approach to the calculation of linear accelerator performance based upon very general properties of wave propagation in coupled periodic systems and upon the linearity conditions discussed previously (Leiss [I 9621, Leiss et al. [I 9621). Two general classes of solution to the linear accelerator problem will be presented ;

B.1.3


159

however the method of calculation of both will be the same. In the first solution the specific dispersive properties of the narrow band-pass accelerator structure are included as well as the bunched nature of the particle beam. In the second treatment. the dispersive properties of the waveguide are ignored, but the bunched nature of the particle beam is retained. For this second type solution, the linear accelerator theory resulting corresponds essentially to the solutions found from the power diffusion equation as regards the fundamental accelerating mode. For higher spatial harmonics and for excitation of other waveguide pass bands the beam loading predictions are different. A signal of constant frequency in any linear system can be thought of as passing from one point (a) to another point (b) in the system by means of a complex phase shift o r propagation constant T,.,(w) such that (signal b) = efra."'"'(signal a ) . The real part of f ( w ) indicates any attenuation of the signal, while the imaginary part indicates the signal phase shift. The general nature of the dispersion diagram for a disk-loaded waveguide is indicated in fig. 4. The imaginary part P, of the complex phase shift, f, in the waveguide is plotted against frequency w for unit length of the waveguide. (In the convention indicated in fig. 4 and in the treatment which follows the unit length of waveguide is taken as twice the distance between loading disks.) A t a given frequency within a pass-band, the propagation constant f ( o ) can have several values:

and ,

=

( )+4

,

for p < 0

(4.2)

where T,,,(w) is the complex phase shift for 0yr;

where we have separated f, into its real and imaginary parts,

From eq. (4.21) it is d e a r that there is no phase or amplitude modulation in this case, just a pulse traveling in the waveguide with time delay T:, attenuation constant sc,, and phase shift fl, per unit distance in the waveguide. Figures 5 and 6 show rf pulse transmission calculated for an S band waveguide designed to operate in the 2x13 mode (o,- wo)/wc=+.This waveguide has 86 cavities, w,/2n= 19.66 M H z and an attenuation length of 0.739 nepers in the 2n/3 mode. Similar calculations for a constant gradient waveguide have been made by Helm. Plots similar to fig. 5 for this case are shown in Helm [1966]. In fig. 5 the amplitude Zq(t), eq. (4.21), is plotted against w,t at different points in the waveguide. The other curves shown correspond to solutions given by Leks et a/. [I9621 in which the rise time of the input pulse is varied. Fig. 6 indicates the variation of Z q ( t ) and E,(t) with q for this same waveguide for the specific time o c t = 8 2 . The dotted curve is the amplitude Z , ( t ) and the solid curve is E,(t). The difference between the two is caused by transients in phase.

J,E.LEISS

164


,'

'0

10

20

30

40

50

60

70

80

90

100

TIM in micro seconds from r t turn on.

I0

120

IN

140

150

160

170

180

190

200

Fig. 5. Pulse transmission (Leiss et al. [1962])in a specific 2n/3 mode S-band accelerator of 86 disks for different values of input risetime, plotted against time in units of act.The amplitude Z(q, t ) is shown for the signal at different points in the waveguide. The solid curve corresponds to the solutions found in eqs. (4.16) to (4.20). Z q O ( t is ) calculated using eq. (4.21). Dispersive calculations similar to these but for a constant gradient waveguide are given in Helm [1966].

4.2 Energy gain at vanishing beam current In this section we calculate the energy gain of an electron in the linac waveguide for the case where the beam current is vanishingly small, i.e. where beam loading can be neglected. We assume the electrons to be completely relativistic. To determine the transit time of signals thus far it has not been necessary to know the longitudinal dimensions of the waveguide, only the dispersive properties. We establish the connection between unit distance in the waveguide and particle transit

(transit time of particle to point q) =- q

(4.23)

One can use any frequency o in eq. (4.23) as long as the appropriate value of

B.1.3


165

Fig. 6. Similar t o fig. 5 but indicating the variation in rf amplitude with distance down the waveguide for a specific time o,,t 82. The dotted curve is the amplitude Z,(t). The solid curve is the field E,(t) which synchronous particles would see. The difference between the two is caused by phase oscillations.

-

$ , is used. It is generally convenient to use the operating frequency

w, of the accelerator. Thus referring t o eq. (4.22), if n + $, - =0, the particle motion is synchronous with the wave motion. For accelerators operating in the n/2 mode (wo=w,), t j O = O ; while for operation in the 2x13 mode tjo = n/3. If an electron enters the beginning of the waveguide, q = O , a t time t , it will arrive at point q at time t +q(n+tjo)/w,. The energy gain, V ( t ) , by an electron in passing through the waveguide is given by the integral over the waveguide of the field which the electron sees.

where q, is the length of the waveguide.

166


J.E.LEISS

As an example we calculate the electron energy gain using ~ : ( t of ) eq. (4.21). vO(t) =

f

e-@.. cos [coat

+ q (n+ (1.,

- &,)I dq,

1

in terms of A ( z ) as in eq. (10). The stability of the system may be investigated by looking for eigensolutions of the form X = t ( z ) eiA(')+"; the eigenfunctions then are solutions of

subject to the boundary conditions X(0, z) = 0, X' ( 0 , z)=O, and E ( L , z)=0 which

190

R.H.HELM, G.A.LOEW


are equivalent to t ( 0 ) = tl(0)= tl'(L)=O. In the special case of a coasting beam [p(z)=constant], 5 has solutions of the form epzsuch that

The general solution for t is a linear combination of the three solutions corresponding to the three roots of eq. (21). Application of the three boundary conditions 5(0)= 5' (0) = 5'' (L) = 0 then yields a determinantal equation which relates the growth rate A to the beam current I,. Hirakawa [I9641 obtained an equivalent result by a much more rigorous method which also includes the case of axial magnetic focusing. He gives numerical results, based on the analogous problem in conventional backward-wave oscillators, which seem to agree at least qualitatively with experimental observations on the regenerative beam breakup. Bjorkholm and Hyneman [I9651 obtained essentially the real part of eqs. (1 8) and (19) and gave computer results which show very nicely the asymptotic eigenfunction behavior of the fields and deflections above the breakup threshold. They also reported that they obtained good quantitative agreement with experimental breakup data on a particular machine (see also fig. 13). A very similar formulation is given by Voskresensky and Koroza [1965], Voskresensky et al. [1966a], and Voskresensky and Serebriakov [1967]. The latter paper gives numerical solutions for the eigenfunction problem including the case of axial magnetic focusing, and indicates that the breakup growth at a particular beam current can be completely suppressed if a sufficiently large focusing field is used.

3.3 Standing-wave breakup In the foregoing discussion it has been assumed explicitly that the fields propagate as pure traveling waves without reflections from the ends of the structures. In actual linacs, however, this is seldom the case; the rf couplers are generally not well matched for the transverse mode and in fact are usually completely decoupled at some particular polarization. Furthermore, if one considers machines which were built before the early 1960's, one finds that most of these operated at the n/2 accelerating mode in which the synchronous point in the deflecting mode was about 3x14; the group velocity was relatively high, the sections were rather short (particularly in machines designed for high-beam currents) and the round-trip attenuation (wL/Qv,) was small (typically a few tenths of a neper). Thus, the buildup of regenerative oscillations was probably affected by standing waves in the guide. On the other hand, in more recent machines in which the accelerating mode is usually 2x13, the synchronous point in the deflecting passband is near the n mode. Thus, the breakup phenomenon in this case can only build up through a standingwave effect.

B.1.4

BEAM BREAKUP

191

Starting current. The case of regenerative breakup in standing-wave proton linacs has been analyzed by Gluckstern [1964], Gluckstern et al. [1965], and Gluckstern and Butler [1965]. Wilson [I9671 has considered electron linacs with particular reference t o superconducting machines (see chapter E.2). A simple calculation of the starting current, equivalent to Wilson's treatment, is given here. In analogy with eq. (9), let the field be defined such that

where the forward and backward waves are assumed to have equal amplitudes, i.e., (Ef( = I Eb( ; the phase relation between E, and E, and the normal mode constraints on /I depend on the boundary conditions at the ends of the structure, which in general are not well defined. The mechanism of energy dissipation is now just the wall loss given by

in which the cross terms between the forward and backward waves vanish through integration because they contain factors of exp (f2ipz). The power generated by the beam is given formally by eqs. (7) and (8), with the field defined by eq. (22). In this case the contribution of the backward wave does not vanish exactly, but may be shown to be quite small compared to the forward-wave contribution.* Thus if one again considers the constant energy case, eq. (1 1) applies and one has

By equating Pb to Pw one obtains the starting current for regenerative standing wave oscillations :

Here also the effect of acceleration may roughly be taken into account by multiplying expression (24) by the factor F, ( p l p , ) given by eq. (1 6). (Gluckstern [I9641 calculates a slightly different factor which, however, gives very similar numerical results.)

* The case of the n-mode resonance is excluded for the moment.

** Equation (24) differs formally from the result of Wilson [I9671 by a factor of 2. This is because here we use the "local" shunt impedance defined by r , = lE12/(toU/Q)at any point in the structure, while Wilson uses an effective "standing-wave" shunt impedance equivalent to (rL)sw = I EfI2LIPa.

R.H. HELM, G.A.LOEW

192


In the special case of the z-mode, it may be shown that, because the forward wave is coherent with a space harmonic of the backward wave, the starting current given by eq. (24) should be reduced by a factor of two. In general, the normal mode constraints on the propagation constant P are such that the phase slip function g , ( L 6P) is not exactly maximized. However, because the mode spacing is typically IP j + - Pjl = z / L , it may be shown that there is always at least one mode for which g, is within ~ 7 0 % of its maximum value. The effect of a "resonance" such that the breakup frequency is a harmonic of the beam bunching frequency has been considered. Estimates by Gluckstern [I9641 and numerical computations by Gluckstern and Butler [I9651 show that the breakup threshold would be reduced by no more than 30% to 50% by a bunching resonance. It should be realized, however, that the starting noise might be enhanced very greatly, since a misalignment of the beam would be equivalent to a transverse modulation as a result of the resonance. Estimate of growth rate. To estimate the rate of buildup of the standing-wave oscillations, one should note that energy conservation in a differential length of the structure, in the absence of power propagation, may be expressed as aU/at +(o/Q) U = aP,/az. In terms of the preceding theory (eqs. (22) through (24)), this expression results in

Thus

E I o e"

where

A=

2Q

(10

-

I).

Is

This estimate is valid only if the growth rate, A, is much smaller than the mode spacing, i.e., if

Otherwise the traveling-wave type of analysis, which has been discussed earlier, is more appropriate.

3.4 Transient coupled resonator formulation The transient theory which was outlined in a previous section of 8 3, may be inadequate under certain conditions. For example, the group velocity may vary appreciably within the frequency band associated with the transient phenomenon. In particular, the assumption of a well-defined group velocity breaks down completely if the synchronous frequency in the deflecting band is near the z-mode, as is the case for the 2x13 accelerating mode.

BEAM BREAKUP

8.1.4

193

The method of expansion of the fields into coupled-resonator normal modes* is one means of solving the n-mode difficulty, but it is not readily adaptable to transient traveling-wave problems. We give here a slightly different and less rigorous but very adaptable version of the coupled resonator formulation. Let the accelerator structure be represented as a linear lattice of loosely coupled resonant cavities (cells) (see fig. 12). The fields in a given cell, say the nth cell, may be represented as a n expansion in the normal cavity modes (Slater [1950]). Assume (a) that only the solenoidal fields are relevant, and (b) that for frequencies near a particular passband the fields may be characterized by a single (perturbed) mode. Thus, in terms of Fourier-transformed** quantities, the vector potential in the nth cell may be defined as

with

For convenience, the characteristic mode function is assumed to be normalized:

) proceeding in analogy Starting with the wave equation (V2 + w ~ / c A~ = ) - ( 4 ~ 1J~and

rEXTEbJJ; nth 7

n- I , n

COUPLING

Fig. 12. Typical unit cells and geometric nomenclature in coupled resonator formulation. 1

* See, e.g., Gluckstern [1964], Gluckstern and Butler [1965], Gluckstern et a / . [1965], and Gluckstern et al. [1966].

**

The Fourier transform of a function A ( t ) will be denoted A ( w ) .

R.H.HELM, G.A.LOEW

194


with Slater [I9501 we find the dispersion equation

( - w2 + iww,,/~.+ o:) A. - ~'(K."-'A.-,

+ K."'~A,,+,)= 47rc I - Rd cell

~ .

(27)

Here the right-hand side and the first three terms on the left follow directly from Slater's theory, with

J

cell

J

wall

where 6=~(21zow,)-~is the skin depth; the terms in K,"" arise from boundary matching at the holes, and may be treated by small-hole coupling theory * (see, e.g., Bevensee [1964]). If the properties of adjacent cells are nearly the same, we may assume that K,"' ' = K,", ,. The dispersion equation may also be interpreted phenomenologically. Thus, if we consider a structure which is uniform so that K,"" =K,"-' K, it is evident that in the case of J=O, eq. (27) has solutions of the form ei("'*"') where 0 =Dl, the complex phase shift per cell, is given by

-

+ iwwn/Qn- w2)/w2. In the usual case where 1 Kl < 1, Q91, and Iw - wnl/on'? IKI, we may write K cos f3 = +(mi

(284

Thus, to the extent that the attenuation term may be neglected and that K and onare independent of frequency, the relation of o to 0 is a simple cosine curve with K as the relative half-bandwidth and wn as the midband frequency. Nishikawa et al. [I9661 give analogous relations between the coupling parameters and the dispersion curve in the case of biperiodic structures. It will be assumed that the current J consists of a beam of electrons traveling in the +z direction with velocity v, so that

and

where t e ( z ) = ~ ~ z ' / v ( z land ) ; that the wave function $, appropriate to the HEM,, mode, may be represented near the axis by

* The characteristic frequency wn will also be perturbed by a fraction on the order of Knn*l, but this need not be expressed explicitly here.

BEAM BREAKUP

B.1.4

195

Substituting eqs. (29) and (30) in (27), keeping only terms of first order in (w - on)/wn, w/Qn and K,"", imposing causality and reality conditions, and transforming back to the time domain, we find

where A, is evaluated along the electron trajectory and the explicit time dependence has been expressed in terms of the starting time, T = t - J ' , dz'/u(z'); the other definitions are:

+

6tn,+z(ln +In_+ ,)/(un v,, ,) is the electron transit time per cell ; q is the impedance of free space (4nlc in gaussian units or 377 ohms); I(T) is the beam current; xn(z) is the centroid of the beam in the nth cell;

The transverse momentum impulse per cell is (Panofsky and Wenzel [1956])

so that the beam displacement is given by

where the half-integer subscripts refer to values at cell boundaries, approximated by averages of the adjacent mid-cell values. The longitudinal momentum p (z) is assumed to be determined by the fields in the accelerating (TM,,) mode. The form factor F, is related to the transverse shunt impedance, given by eq. (2), as follows: ~ (ril/Q>, = 2 r l ( w , / ~ )F?

.

(35)

Equations (31) and (34) describe the transient coupled-resonator formulation of beam breakup, subject to arbitrary initial and boundary conditions. For example, the following typical boundary conditions could be specified at the output boundary, n=N:

A

(T - S t + )

=A

)

(shorting plane at z N ++);

196

R.H.HELM, G.A.LOEW

AN+ (Z - dtN++)= AN+ (Z or AN+,(z)

-

0


+ StN++)

(longitudinal short in N

(shorting plane at zN);

+ 1 cell).

Equivalent specifications for the input boundary are analogous. Beam boundary conditions might include static misalignment, arbitrary transverse rf modulation, or the effect of the breakup interaction in preceding accelerator sections. Typical initial conditions might be A,(O) = 0 for all n. The coupled resonator formulation in general cannot be solved analytically. By introducing suitable restrictions or limiting approximations, one can go either to the normal mode theory (see, e.g., Gluckstern [1964]), or to the continuous transmission

I

II 21 C E L L NUMBER, n

31

Fig. 13. Computed regenerative breakup in a short uniform n/2-mode linac with characteristic impedance terminations.Beam current: 350 mA; all other conditions as described by Bjorkholm and Hyneman [1965]. The small ripples arise from excitation of a field component with backward phase velocity.

BEAM BREAKUP

B.1.4

197

line formulation discussed previously (eqs. (18) and (19)). Numerical computations are required, except in rather trivial cases, in order t o describe the breakup under highly transient conditions. A computer solution of an approximate form of the coupled resonator breakup equations is described by Helm [1966]. Figure 13 shows the result of a typical computation for the structure described by Bjorkholm and Hyneman [1965]. This was a uniform n/2 mode, 2856 MHz accelerator, 0.8 m in length, with injection energy of 80 keV; the measured parameters in the HEM band were w0/2n = 4250 MHz, t),(wo)z -0.019c, Q = 9800; derived parameters used in the present calculation were toJ2nz4285 MHz, w , ~ , " " / 2 n z -50.6 MHz, and r,/Q=30.8 ohms/cm (by eq. (14)). The computed results at a beam current of 350 mA closely match Bjorkholm and Hyneman's computation at 175 mA, the discrepancy apparently arising from an error of a factor of two in Bjorkholm and Hyneman's calculation of the stored energy. Figure 14 shows a computation for the same structure, with boundary conditions of

1

I

0.6 psec

II 21 CELL NUMBER, n

31

Fig. 14. Computed regenerative breakup in a short n/2-mode linac with shorted ends. All other conditions the same as in fig. 13. Note characteristic normal mode pattern.

198

R.H. HELM, G.A. LOEW

ELECTRON LTNACS : THEORY

mid-iris shorting planes rather than characteristic impedance terminations. Here the breakup shows a characteristic normal mode standing-wave pattern. Figure 15 shows a similar computation for the SLAC structure described in chapter B.l.l, fig. 29 and in this chapter, 5 2, figs. 5, 6 and 8. In this case a shorting plane is assumed at the input boundary, and downstream termination results from the tapered structure becoming cut-off, for the dominant breakup frequency, at about the 10th cell. The field pattern and asymptotic frequency (w4148.4 MHz) are quite close to one of the normal modes of the structure as shown in fig. 8.

I

0

I

I

I

20 CELL NUMBER

'10

I

I

I

30

Fig. 15. Computed regenerative breakup in first 30 cells of the SLAC tapered structure for 3 A beam.

4 Multisection or cumulative beam breakup 4.1 Analysis of multisection beam breakup As has been discussed in 5 2, the rnultisection type of beam breakup results from the cumulative interaction of the beam with an entire array of separate accelerator sections which are electrically isolated, the signal being carried from section to section by the beam. The effect manifests itself at beam currents well below the threshold for regenerative breakup. This effect may be divided roughly into two parts: 1. In the first few sections of the machine, where the energy is low, regeneration may be important. In this case the interaction should be treated in detail by use of one of the transient theories outlined previously. This effect appears to be especially

BEAM BREAKUP

B.1.4

199

relevant in machines such as the 2-GeV Kharkov linac which have rather long uniform sections in which the regenerative starting current at injection energy may be of the order of a few tens of milliamperes. 2. In the remainder (whichmight actually include the entire machine), regeneration is unimportant and we may employ a much simpler model. Such a model assumes that the effect of an entire accelerator section, or even a group of several sections, is equivalent t o an impulse at a single point. This description applies particularly to machines such as SLAC, which use tapered sections in which the synchronous length at any frequency is very short. Only the latter case will be considered in detail here, although one must remember that the impulse model may not be strictly valid in the first few accelerator sections. Sitigle cavity model. We assume that the field in the H E M , , passband in a given accelerator section may be expanded in normal modes*, each characterized by a n , a characteristic mode pattern frequency oJ,a phase shift per cell O ~ = f l ~ Iand the field associated with the jth mode is now given in the nth cell of the section by

where *,, u' and r are as defined earlier (9 3 and fig. 12). For example, in a periodic structure of N identical cells terminated by mid-iris shorting planes, we would have f l j l = d j = , j n / ~where O < j < N; X: K sin(/?-4) O j ; the frequency spectrum o'=o(/?j) would be determined by the o-fl curve. In aperiodic structures, the mode pattern, frequency spectrum and phase shift may be determined by rf measurements (see fgs. 5, 6 and 8 ; also see Altenmueller et al. [1966], Loew [1967]) or by computer techniques (Helm [ I 9661). The mode functions will be assumed to be normalized:

Xi

I n analogy with the Slater theory used in the previous section, the following differential equation is now found:

Here an impulse approximation has been used; the beam deflection xn(r) has been assumed to vary only slightly over the length of one accelerator section and has been replaced by a mean value x ( z , 7 ) . It may be shown that FJ, the form factor for the entire section, is given by

where Fn, the form factor for one cell, has been defined by eq. (32), 46 is an arbitrary

*

See, e.g., Nagle [I9641 and Nishikawa [1965].

R.H. HELM, G.A. LOEW

200


phase angle which should be chosen to maximize the interaction, and t,=J:; dzlu. The value of Qj, the unloaded Q of the structure, is given by

l / Q j = C (x3"Qn. n

(40)

For a uniform structure, Qj= Q, by virtue of eq. (37). The transverse momentum impulse, found by integrating (e/c)dA,/dx along an electron trajectory, may be shown to be

It will be convenient to define a new field variable V j ( z ) such that

oi

is the "ringing" frequency defined by wd = w j [1 - ( 1 / 2 ~ j ) ~ ] $It. follows where from eqs. (38) and (41) that

and

where the total shunt impedance R{ is defined by

R { / Q ~= 2~ ( ~ / (d~ )j .~ ) ~

(45)

Equations (43) and (44) describe the equivalent thin-cavity initeraction of an accelerator section. In general R, is related to the shunt impedance per unit length by an effective length L',,:

where r , may be estimated, e.g., by eq. (14). In a uniform structure, the effective length is typically L,, =+T2L (or T 2 L at n. mode) where T=(sin +cr)/(+cr), L= total length, and cr is the total phase slip of the beam relative to the wave. In non-uniform structures the effective length may be found by rf measurements or by computer methods (Helm [1966]). Computer solution of multisection breakup. An actual multisection accelerator consists of many successive beam transport elements such as drift and accelerator sections, lenses, and the deflecting "cavities" described above. The most straightforward method of numerical analysis of the cumulative breakup is to ray-trace each beam bunch through the system. For this analysis, it is convenient to modify the

BEAM BREAKUP

B.1.4

201

preceding results slightly. Let us drop the normal mode index j and replace w j and w { (which are practically equal) by w , , noting that different modes may be treated by superposition of distinct "cavities" rewritten as

in the computation. Equation (44) may be

I

V(T)

=

C1

S

I ( t ' ) X ( Z ,t') exp [iG,

0

(T

- t')] dt'

(46)

where the initial condition V(O)=O

has been assumed; 6 ,= w , (1 +i/2Q) and C , = ~ ? R , / ~ C QNow, . if we express the beam current explicitly as a series of deltafunction bunches, i.e.,

where to is the bunching period, then by using (46) the following recursion relationship may be found:

in which for convenience the functional time dependences have been redefined as bunch number m rather than mt,. The beam transformation for the mth bunch from a point z- just ahead of the "cavity" t o z,, just beyond the "cavity", is

Thus eqs. (48) and (49) represent the effect of a single "cavity" located a t point z, in a form ideal for iterative solution by computer. The effects of other beam transport elements are described by well-known transformations (see, e.g., in chapter B.1.2). A computer program which simulates arbitrary configurations of individual "cavities", accelerator and drift sections, lenses and other components has been used extensively in analyzing the multisection breakup a t SLAC. Examples of computer results and comparison with experiment are given in 4 4.2 (see also Helm et al. [I9681 and Farinholt et al. [1967]). Asymptotic transient behavior. Considerable insight into the behavior of the multisection breakup may be gained by use of a very simple model employed by Panofsky [1966] (see also Voskresensky et al. [1966b]). A number of assumptions are made in this model : (1) All the interacting "cavities" are assumed identical and only a single resonant mode w, is considered. (2) The beam current I is taken to be constant for T > O , and zero for 760. (3) The beam bunching is considered incoherent with the breakup frequency, i.e., o,/o, is not a ratio of small integers. (4) Variations in the amplitudes of fields and deflections are "slow" compared to the frequency w,. ( 5 ) The motion is relativistic, so that u z c and p z m c y . (6) The relative growth in deflection amplitude per accelerator section is small.

202

R.H.HELM. G.A.LOEW


We consider only the component of the deflection which has frequencies near o,, and write

x ( z , z)

= Re ( X

( z , 7) exp [ia ,z])

(50)

where X is a slowly varying complex amplitude. Then, going to a continuous approximation by defining Ap, ( z , z ) = L (dldz) p, ( z , T ) , where L is the spacing between cavities, using eqs. (41) and (46), and neglecting focusing, we find

2 az ( y

z)

= - iCI

j

X ( z , r') drf where

C=

ew:~, 4 r n c ' ~ c' ~

0

A term containing Jb X* ( z , t ') exp [- 2iw, t'ldt' has been neglected because of the rapidly varying phase factor in the integrand. Equation (51) is exactly equivalent to the result of Panofsky [1966]. To obtain an asymptotic solution of eq. (51), we first take the Laplace transform,

where s is the transform variable and X ( z ; s) is the transform of X ( z , t). In the special case when y is a linear function of z, eq. (52) has solutions in Bessel functions of complex arguments. However, in order to retain generality, it is convenient to consider y as slowly varying (but otherwise arbitrary) and use the WKB approximation for the harmonic oscillator with slowly varying parameters. The solution is of the form

X ( Z ; S ) cc (sly)* exp [+ i ( i ~ l l sg)(~z ) ]

(53)

where 2

g (z)=

J Y - + ( ~ 'dzf ) . 20

The asymptotic solution may be found by performing the inverse transformation by the method of "steepest descent". The result corresponding to the boundary condition X ( z o , z ) =6(z), x l ( z o , z ) = 0 is (see Panofsky [1966])

x ( z , z ) = Re ( X exp [ i ~T ,] ) z X,

exp {(I

-

i/43)Fe + (io, - o , / 2 ~z ))

+ C.C.

(55)

where

g is defined by eq. (54) and X ~ C C I ~ ~ - ~ Equation ~ * ~ - *(55) . is asymptotic in the sense that Fe& 1 is assumed. Because expression (55) is asymptotic, extrapolation to T = O is not vslid and the coefficient X o is not equal to X ( z o , T). Also, because the

BEAM BREAKUP

8.1.4

203

result represents the impulse response of the system, it may not apply at late times for which w , ~ / 2 Q ?F,. Within these limitations, eq. (55) describes the essential features of the multisection breakup and is a valuable guide to the scaling laws in the parameters I , T, and y(z). Numerical solutions of eq. (51) are given by Bander [1966]. Comparison of the present results with experimental data is given in a later part of this section. Effect of.focusitig. Panofsky [I9661 extended the smooth-force treatment to include the effect of external focusing. By an obvious modification, eq. (52) becomes

where k is the betatron wave number of the focusing system. The WKB solution is analogous to eq. (53); however, the inverse Laplace transformation becomes impossible in closed form. By expanding and keeping only terms of first order in k 2 , Panofsky obtained an expression formally the same as eq. (55) but with a modified e-folding factor given by

where K (z)

-J

,.

?*(z') k 2 ( 2 ' ) dz'

and the other quantities are as defined previously. Comparison with computer calculations indicates that the above expression is correct in the limit of very weak focusing. Steady-state solution. In long duty-cycle machines such as the proposed superconducting linacs * and several normal-conducting machines ** now under construction, the pulse lengths will be long compared to the decay time (2Q/o,) of the breakup mode. Hence the analytic results in the last two sections, which represent essentially the impulse response, are no longer valid. In special cases one may find solutions which are valid from short times up to z $2Q/w, (see Leboutet [I 966]), but computer solutions seem to be more useful in treating this intermediate region. However, we may readily find steady-state solutions (Helm [1967]; Wilson [1967]; Panofsky and Bander [1968]). Let us redefine the complex amplitude X ( z , co) such that

where w l = o , +6w is a constant driving frequency. We assume that I ( c o ) = I =

* **

See chapters E.2 and E.3. See chapter B.3.2.

201

R.H.HELM, G.A.LOEW


constant. Equation (51) then yields

which is formally the same as (52). The solution is given by (53) with s replaced by i 6w - w1/2Q. It may be shown that the maximum value of the e-folding factor occurs for 6 0 = - ( o , / 2 ~ ) J + ; we then find

X (z)K y-* exp [(I - i

44)Fss]

where

and g ( z ) is defined by eq. (54). Focusing may be taken into account in the same approximation as was used previously in the transient case. The result is that the e-folding factor F,', in the presence of focusing is

4.2

Experimental evidence and comparison with theory

As this book is being written, most of the available quantitative experimental evidence against which the above analytic and computational studies can be verified stems from SLAC. Hence the comparison between theory and experiments which is presented below is entirely restricted to the transient type of breakup. As will be seen, however, agreement is close enough to warrant confidence in the general theories that have been developed. Figure 16 shows three examples of beam profiles at the end of the SLAC accelerator for a 1.6 psec pulse. As seen, at 35 mA, below the breakup threshold, the maximum beam spot diameter is of the order of 0.5 cm. For this current and with the degree of focusing that is used along the machine, the electrons do not strike the accelerator walls. When the current is increased to 45 mA, breakup appears at first in the vertical plane. As has been discussed earlier, this is due to the fact that in the SLAC accelerator sections, the couplers are in the horizontal plane and the value of Q, is roughly 1.5 times higher than that of Q,. At higher currents yet (75 mA), the direction of breakup becomes random. A large fraction of the electrons in the late part of the pulse strike the walls before they reach the end of the machine. Breakup occurs before multiple rf reflections have time to build up in each accelerator section because the one-way travel time in the resonant part of the SLAC sections is of the

B.1.4

BEAM BREAKCP

205

order of 50 nsec. Under these conditions, a number of frequencies are probably excited and carried by the beam. As has been discussed above, the asymptotic solution proposed by Panofsky and others gives easily understandable scaling laws in the absence of focusing. These have been verified under a number of conditions. For example, for constant acceleration (g= y'z and z$z,), the dominant term in the exponential of eq. ( 5 5 ) can be integrated, giving the relation

where the various parameters have been defined earlier and where it is assumed that the beam begins to scrape the accelerator walls (I.D. = 1.7 c m ) when the defection J * has grown with respect to the initial value yo by an e-folding of the order of 15. Notice that for a typical SLAC pulse (1.5 psec). the loss term wri2Q which must be subtracted from the dominant term is of the order of 2-5 depending on the exact Q

Fig. 16. Beam profiles at the end of the SLAC accelerator for three different currents (Energy: 16 GeV, beam pulse width: 1.6 psec). The large grid width is 1 cm. (a) Current below breakup, 35 mA, (b) current for predominantly vertical breakup, 45 mA, (c) current where breakup begins to occur in random direction, 75 mA.

* Earlier in this chapter, the breakup coordinate is referred to as x, with the understanding that x could be any transverse direction. From here on, note is taken of the fact that the predominant polarization at SLAC is y, or vertical.

206

R.H.HELM, G.A.LOEW


Fig. 17. Beam breakup current vs inverse length in the case of weak focusing. The deviation from a straight line of the computed curve is due to the inclusion of some focusing in the calculation. Pulse length, 1.5 psec. Energy gradient, 2.2 MeV/m. Betatron wavelength, = 900 m. Sector length, w 100 m.

I

2

3

4

5

6

ENERGY G R A D I E N T ( M e V / m e t e r )

Fig. 18. Beam breakup current vs energy gradient in the case of weak focusing. Uniform acceleration. Pulse length, 1.5 psec. Accelerator length, 3050 m. Betatron wavelength, w 1500 m.

BEAM BREAKUP

B.1.4

207

that is assumed. For a constant pulse length r, it is seen that the plots of I vs. l/z and dyldz would be straight lines. The results were verified under weak focusing conditions, shortly after turn-on of the accelerator. They are shown in figs. 17 and 18. In the presence of medium or strong focusing, the simple scaling laws break down and one must resort to the computer studies. Experimental data can be taken and presented in a variety of ways. For example, fig. 19 gives a plot of transmitted charge

-COMPUTED - -- COMPUTED - - - COMPUTED

I NORMAL MODE 2 NORMAL MODES 3 NORMAL MODES

.+..*.. EXPERIMENTAL

.+..+. EXPERIMENTAL

0

0 2

0 4

0 6 PULSE

0 8 WIDTH,

10

12

CHARGE CURRENT

14

16

psec

Fig. 19. SLAC beam breakup charge and current vs time to breakup. The experimental points were obtained in August 1967 under the strongest available focusing conditions and at an energy of 16 GeV.

per pulse or current vs. time to breakup. The experimental points were obtained under the strongest focusing conditions presently usable on the machine. These conditions include a system of singlets every 40 feet in the first six sectors ( ~ 6 0 meters) 0 of the accelerator and of strong doublets at the end of each sector from there on. With this system, which came into final operation roughly one year after SLAC turn-on, the corresponding betatron wavelengths are approximately 150 and 400 meters, respectively. Notice that for a pulse length of 1.6 psec, the maximum current transmitted t o date is approximately 42 mA, still somewhat short of the 50 mA current originally

208

R.H.HELM, G.A.LOEW


specified but more than twice as large as the current obtained when the machine was first turned on (see for example Neal and Panofsky [I9661 or Altenmueller et al. [1966]). This improvement is entirely due to the new focusing system. The computed curves were obtained by successively including 1, 2 or 3 of the normal modes in the

/ I

M E A S U R E D AT SECTOR 21 M E A S U R E D A T SECTOR 29

A X

S E C T O R LENGTH I

I

- I00

meters

I

Fig. 20. Transverse beam envelope amplitude ( y ( as a function of ( 1 ~ ) )Energy . gradient: 5.25 MeV/m, beam pulse width: 1.6 psec, medium focusing.

calculation. It is seen that the effect of additional modes is increasingly marked as the pulses become shorter. In the limit of very short pulses or in the case of bunching at low subharmonics, a large number of the normal modes act cooperatively and it becomes difficult to obtain computer agreement with experiment. Figure 20 shows a set of curves of beam breakup "gain" giving the transverse beam

BEAM BREAKUP

B.1.4

209

envelope amplitude 14'1 as a function of ( I Z ) ~ .The experimental points were derived indirectly from the amplitudes of 4140 MHz signals induced by the beam in cavities of the type illustrated in fig. 9 and installed at the ends of sectors 5, 13, 21 and 29. In all cases, a constant external excitation of microwave power at 4140 MHz was injected onto the beam a t the beginning of the accelerator. While the curves are almost straight lines as predicted by the scaling law given in eq. (65), there are four distinct plots depending on the value of z that was used. The computer results show very good agreement. Two different values of R,/Q were assumed and the computed points were normalized to the experimental points in the lo-' cm range.

30

20

0 MEASURED

10

A COMPUTED

PULSE LENGTH = I 5p see

I

0

I I I I I I Ol-rr 0 2 7 ~ 0 3 ~0 4 7 0 5 7 0 6 7 ~ BETATRON PHASE SHIFT PER SECTOR ( k B S )

Fig. 21. Beam breakup current vs betatron phase shift per sector. These data were taken at Sector 19 in February 1967, before installation of the 40-foot singlets.

The effect of focusing can be clearly recognized if, for example, one measures the beam breakup current at a given value of z and .t while varying the energy gradient and the quadrupole current. Figure 21 shows a set of curves of current a t breakup vs. betatron phase shift per sector, k,S, where S is the length of a sector. The value of k,S was measured experimentally. I t could also be computed. A reasonable approximation is given by c o s k s S z 1 -S/2fwhere f is the focal length of the quadrupole

210

R.H.HELM. G.A.LOEW


doublets. Results for four different values of energy gradient are shown in fig. 21. The agreement between experiment and computation is excellent.

5 Discussion of starting conditions Both the regenerative and the cumulative multisection types of beam breakup are triggered by an initial excitation which manifests itself in a small transverse modulation of the electron bunches at the breakup frequency. The amplitude yo of this modulation is then caused to grow according to the mechanisms and laws which have been described earlier in this chapter. From the speculative as well as the remedial poinl: of

SHOCK EXCITATION BY OFF AXIS BUNCH

- -.

ON AXlS TRANSVERSE SHOT NOISE FROM GUN \.hAVhVn HEM,, COMPONENT OFF AXlS LONGITUDINAL SHOT NOISE FROM GUN

?VnAL

'#-

kT THERMAL CAVITY EXCITATION

EXTERNAL NOISE SOURCE (KLYSTRON)

.$.+

Fig. 22. S c h e m a t i c i l l u s t r a t i o n s of p o s s i b l e initial b e a m b r e a k u p e x c i t a t i o n s

view, it is desirable to understand the origins of this initial excitation. Intuitively, three possible sources come to mind. They are illustrated in fig. 22. The first is shock excitation due to the beam current Z for which the equivalent source power would be proportional to 1'. The second is gun shot noise (transverse or longitudinal) for which the noise power would be proportional to I. The third is rf excitation of the accelerator cavities, either thermal or generated by the klystrons, for which the noise power would be independent of I. Before entering into a discussion of these effects, two points should be made. The first is that, as this book is being written, a definitive answer to these questions has not yet been found. The second is that independently of the nature of the answer, it can be asserted that it would take a very significant

B.1.4

BEAM BREAKUP

211

reduction in noise power before the effect on the breakup threshold would become noticeable. Hence, in the cumulative case, assuming an exponential growth of the deflection of the form =y,eF, it is easily seen that a reduction in noise power R in dB corresponds to a relative increase, T, in beam breakup threshold given by

Thus, for example, letting R%20 dB and F= 20, it is seen that T= 1.39, giving less than 40% improvement in current. The discussion which appears below is based exclusively on experiments carried out at SLAC. The only other published measurements on the subject can be found in Jarvis et 01. [1965] and Vishnyakov et 01. [1967a]; their results however, giving rf

COMPUTED POWER INDUCED BY l 8 m A BEAM, ARBITRARI L Y NORMALIZED

4 1 4 0 MHz POWER AT KLYSTRON OUTPUT R E W I R E D TO AFFECT PULSE SHORTENING A T THE END OF SECTOR 19 BEAM ENERGY 2 2 GeV BEAM CURRENT

* )

I

,

, I , [

I8mA

l , l , 1 , 1 , 1 , {

I

1-4 1-8 2 - 4 2-8 3-4 5 8 4 4 4-8 5-4 5 - 8 6-4 6 - 8 SECTOR-KLYSTRON NUMBER I

40

1000

2 0 0 0 DISTANCE,FEET

Fig. 23. Power at 4140 M H z injected into SLAC accelerator as a function of klystron position. For each point, the corresponding klystron was removed and the injected power was increased until pulse shortening observed at Sector 19 was affected. (Data taken in February 1967, before completion of installation of 40-foot singlets; beam accelerated up to 2.2 GeV; beam current: 19 mA).

212

R.H.HELM, G.A.LOEW


starting levels below watts, are difficult to compare with the SLAC results because of differences in experimental setups. Two basic types of experiments have been carried out at SLAC, the first to obtain an order of magnitude of the noise power, the second to understand its origin. Figure 23 shows measured and computed levels of power at 4140 MHz injected into the SLACcaccelerator sections through the output waveguides of the high power klystrons. The level of recorded power injected a t a given location is that which was required to affect the breakup threshold at sector 19. Near the beginning of the

9-

.8-

SHOT NOISE

w

5-

KLYSTRONS OR OTHER R F EXCITATION

2

o

z

W

a

3-

0

0

10

20

30

40

50

60

BEAM CURRENT, mA

Fig. 24. Relative noise source power as a function of beam current at Sector 29 (Energy: 16 GeV, beam pulse length: 1.6 psec). Graph shows experimental points and curves corresponding to three main hypotheses.

accelerator, i.e., at about the 120 MeV point, this level is of the order of 10 pwatts. This value sets an upper limit to the allowable noise power from the klystrons, above which the klystrons certainly become the dominant driving source. On the other hand, it is not clear what exact fraction of this 10 pwatt injected power actually acts on the beam because the transmission loss of the rectangular waveguide and the coupling coefficient of the accelerator coupler are not exactly known and vary somewhat along

B.1.4

BEAM BREAKUP

213

the machine. Given a n arbitrary normalization, the agreement between computation and measurement is reasonable. The results of the second type of experiment are given in fig. 24. The object of this experiment was t o obtain a plot of relative noise power as a function of beam current in order t o determine which of the mechanisms illustrated in fig. 22 is dominant. The experiment was p2rformed as follows. Forty feet downstream of the injector, power Po a t 4140 MHz was injected into a C-band cavity (fig. 9). A t the end of sector 5 , roughly 500 meters downstream, beam-induced power P, was extracted from a similar cavity. In the absence of any external excitation, this induced power P, was measured for a given beam current. Then, the injected power Po was increased until P, was roughly doubled. This measurement was repeated over a range of current. Because that fraction of injected power Po which actually acts on the electrons must be of the same order of magnitude as the natural noise power carried by the beam, a relative, normalized measure of noise can be inferred. The results shown in fig. 24 d o not allow one to distinguish whether the noise power is quadratic with current (shock excitation) or linear (shot noise). However, it can be concluded with fair certainty that this power is not independent of current, and hence, that neither klystron noise nor cavity thermal excitation are dominant. This conclusion is further confirmed by two facts. One is an experiment in which the first ten high power klystron outputs were fitted with 40 dB filters at 4140 and 4428 MHz: no visible effect on the breakup threshold was found. The second is a simple estimate of thermal noise excitation. Assuming that initial noise of energy kT is responsible for setting up fields in one accelerator section with a value of R J Q of 400 ohms, the resultant transverse momentum is approximately

Using kT=0.4 x lop2' joule at 300°K one obtains p,, = 2 x 10-lo MeV/c. Comparing this value with a normal transverse momentum acceptance of approximately 0.2 MeVlc at the end of the accelerator, it is seen that a gain of lo9 would be required. This value is two orders of magnitude greater than the present estimate. Additional information about the nature of the noise source a t 4140 MHz can be gained by counting the number of pulses of induced power as a function of pulse height. Two cases are shown in fig. 25. If the normal beam modulation is due to the statistical noise in gun emission, the current amplitude should exhibit a half gaussian distribution. Since the pulse height distribution is proportional to the square of the current amplitudes, it should have an exponential distribution. This seems to be borne out by fig. 25a. The sharp peak in fig. 25b was obtained in the presence of some external stimulation injected onto the beam downstream of the injector. This result strengthens the shot noise hypothesis. This hypothesis is further strengthened by the fact that an improvement in beam breakup threshold of about 12% was found by increasing the cathode filament power from the temperature-limited to the space-

214

R.H.HELM, G.A.LOEW


Fig. 25. Number of counts vs pulse height of rectified 4140 MHz power induced in a cavity, 500 meters downstream of the injectcr. (a) Spontaneous beam breakup, 23 mA, (b) Stimulated beam breakup, 18 mA.

charge-limited region. Unfortunately, under normal conditions, the SLAC gun is already operating under space-charge-limited conditions and straightforward improvements do not seem easily obtainable. Using the usual formulas for temperature-limited shot noise power in a band-width f/Q, the transverse and longitudinal effects shown in fig. 22 can be lumped to express the initial "equivalent" mean square deflection:

8.1.4

BEAM BREAKUP

215

where f is the breakup frequency and r is the radius of the beam whose center of charge is offset by 6. Taking an equivalent lumped value for the bracket of (0.1 ~ r n ) ~ , one finds a value of yo of the order of cm, in good agreement with the lo6 to 10' beam breakup gain which would cause the beam radius to grow to 1 cm over the length of the accelerator. This initial level of excitation, however, is of the same order of magnitude as that which can be computed for shock excitation due to initial misalignment. Assuming a time to breakup As= 1.6 psec, an initial misalignment 6=0.01 cm and a breakup frequencyf'=4140 MHz, the equivalent noise modulation can be estimated roughly as

Experimentally, however, it has been found that the sensitivity of the breakup threshold to steering at the beginning of the accelerator is not as large as expected. Admittedly, these experiments and others using chopped beams at a subharmonic frequency are not yet conclusive. In conclusion, it must be said that an overall understanding of the beam breakup excitation problem has not yet been obtained.

6 Proposed remedies for various types of' accelerators As discussed in the introduction of this chapter, the problem of beam breakup has now been recognized as a limiting factor in the design of many of the new linacs. Unless an unforeseeable breakthrough is made, electron and possibly also proton linac instabilities will have t o be coped with whenever a given combination of current, pulse length and accelerator length is exceeded. The remedies that suggest themselves depend to a considerable extent on the type of linac under consideration, and on its economics. The types of remedies which will be discussed below can be divided into two categories: rf cures and focusing cures. lmprovements to be expected from an initial noise reduction have been described earlier. As discussed, knowledge in this area is still incomplete but as this book is written, the expectations are limited.

6.1 Rf cures Rf cures, in turn, can be divided into two sub-categories: structure modifications and feedback or rf cancellation. Because beam breakup is caused by interaction with the H E M , , mode, one of the most obvious remedies is t o attempt to change its propagation characteristics. One such remedy is to resort t o "Q-spoiling" techniques, i.e., selectively coupling power out of the H E M , , fields, for example by using large cut-off holes a t both ends of the

216

R.H.HELM, G.A.LOEW


accelerator structure. Another remedy is to displace the o-B diagram of the HEM,, mode by modifying the modular dimensions of the cavities or perturbing their symmetries, e.g., by means of radial slots (see Levin et al. [1967]). In short accelerators, one can shorten the effective length of each section by using several shorter sub-sections (see for example Vishnyakov et al. [1967b]) or by brea!yo. Using SLAC parameters with a rough guess of 6=2 cm, and assuming an amplification of e4.6= 100 corresponding to growth from an initial misalignment of 0.01 cm, one finds a current on the order of I .3 A to produce beam loss at z= 3 km. Thus, the limiting current for the “wake field” breakup is probably at least an order of magnitude greater than for the HEM,, effect.

References ALTENMUELLER, 0. H., R. R. LARSEN, and G. A. LOEW, 1964, Rev. Sci. Instr. 35,438. ALTENMUELLER, 0. H., E. V. FARII\‘HCLT, Z. D. F ARKAS, W. B. HERRMANNSFELDT, H. A. HOGG, R. F. KOONTZ, C. J. KRUSE, G. A. LOEW, and R. H. MILLER, 1966, Beam Breakup Experiments at SLAC, Proceedings of the 1966 Linear Accelerator Conference, LA-3609, Los Alamos Scientific Laboratory, Los Alamos, New Mexico, pp. 267-272. ARTEMOV, V. J., Y. M. B AZAJEV, V. I. B ELOGLAZOV, V. A. VISHNYAKOV, I. A. GRISHAEV, P. S. GOROHOVATSKY, A. S. DOVBUSH , Y. I. D~BROLYBOV, G. K. DEMYANENKO, A. I. ZYKOV, G. I. IVANOV, V. M. KOBEZSKY, B. B. KONDRATENKO, L. A. MAKHNENKO, V. B. MYFEL,~. 1. MYAKOTA, V. V. PETRENKO, B. A. PONOMARENKO, V. 1. P OPENKO, K. S. RUBSOV, and V. A. SKUBKO, 1967, The 360 MeV and 2 GeV Electron Accelerators of the Ukrainian Physical-Technical Institute, Proceedings of the 1967 Sixth International Conference on High Energy Accelerators, CEAL2000, Cambridge Electron Accelerator, Cambridge, Massachusetts, pp. 408-412. B ANDER, M., 1966, Solution of the Beam Break-Up Equation, TN-66-28, Stanford Linear Accelerator Center, Stanford University, Stanford, California. (See also Panofsky and Bander [1968]). B ERTOZZI, W., J. HAIMSON, C . P . SARGENT, and W. TURCHINETZ, 1967, IEEE Trans. NucI. Sci. NS-14, No. 3, pp. 191-196. B EVENSEE, R. M., 1964, Electromagnetic Slow Wave Systems (John Wiley and Sons, Inc., New York, London and Sydney). B JORKHOLM, J. E., and R. F. HYNEMAN, 1965, IEEE Trans. Electron Devices, ED-12, 281. BOAG, J. W.,andC. W. MILLER, 1959,TechnicalAspectsofa High-Current Electron-PulseGenerator, Proceedings of the Second International Conference, Geneva, Switzerland (Pergamon Press, London, New York, Paris and Los Angeles). BROWN, K. L., A. L. ELDREDGE, R. H. HEIM, J. H. JASBERG, J. V. LEBACQZ, G. A. LOEW, R. F. MOZLEY, R. B. NEAL, W. K. H. PANOFSKY, and T. F. TURNER , 1961, Linear Electron Accelerator Progress at Stanford University, 1961 International Conference on High Energy Accelerators, Brookhaven National Laboratory, New York. CHENG, G. H. H., 1964, Pulse Shortening in Electron Linear Accelerators, M. S C. Thesis, University of California, Berkeley, California. C HU, E. L., 1966, A Crude Estimate of the Starting Current for Linear Accelerator Beam Blowup in the Presence ofan Axial Magnetic Field, TN-66-l 7, Stanford Linear Accelerator Center,stanford California. C OURANT, E. D. and A. M. SESSLER, 1966, Rev. Sci. Instr. 36, No. 11, 1579. C ROWLEY-MILLING, M. C., T. R. JARVIS, C. W. MILLER, and G. SAXON, 1961, Nature 191, 48

220

R.H.HELM, G.A.LOEW


CROWLEY-MILLING, M. C., 1962, A 40-MeV Electron Accelerator for Germany, AEI Engineering 2, No. 2. FARINHOLT, E. V., R. H. HELM,H. A. HOGG,R. F. KOONTZ,G . A. LOEW,and R. H. MILLER,1967, Progress Report on Beam Break-Up at SLAC, Proceedings of the Sixth International Conference on High Energy Accelerators, Cambridge Electron Accelerator, Cambridge, Massachusetts, CEAL-2000, pp. 90-98. Y., Hybrid EH Guided Waves and Their Application to R F Separators of Very High GARAULT, Energy Particles (to be published in Advances in Microwaves). GLUCKSTERN, R. L., 1964, Transverse Beam Blow-up in Standing Wave Linacs, Proceedings of the MURA Conference on Linear Accelerators, Stoughton, Wisconsin, p. 186. Also published as Internal Report AADD-38, Accelerator Dept., Brookhaven National Laboratory, July 1964. R. L., H. S. BUTLER,and K. JACOBS,1965, Interaction of a Bunched Beam with GLUCKSTERN, Transverse Modes in R F Cavities, Proceedings of the Fifth International Conference on High Energy Accelerators, Frascati, Italy. R. L., and H. S. BUTLER, 1965, lEEE Trans. Nucl. Sci. NS-12, No. 3; p. 607. GLUCKSTERN, GLUCKSTERN, R. L., D. E. NAGLE,and W. M. VISSCHER, 1966, A Note on Transverse Beam Instabilities in Multisection Linacs, Proceedings of the 1966 Linear Accelerator Conference, LA-3609, Los Alamos Scientific Labcratory, Los Alamos, New Mexico; pp. 281-283. 1965, Rev. Sci. Instr. 36, 1788. HAHN,H. and H. J. HALAMA, 1966, Nucl. Instr. Methods 45, 141. HAHN,H. and H. J. HALAMA, HAIMSON, J., 1965, IEEE Trans. Nucl. Sci. NS-12, No. 3; pp. 996-1011. HELM,R. H., 1966, Computer Study of Wave Propagation, Beam Loading and Beam Blowup at the SLAC Accelerator, Proceedings of the 1966 Linear Accelerator Conference, LA-3609, Los Alamos Scientific Laboratory, Los Alamos, New Mexico; pp. 254-266. HELM,R. H., 1967, Preliminary Estimate of Beam Blowup for a Superconducting Electron Linac, TN-67-6, Stanford Linear Accelerator Center, Stanford, California. HELM,R. H., G . A. LOEW,and W. K. H. PANOFSKY, 1968, SLAC Handbook (W. A. Benjamin, Inc., New York) Chapter 7. HIRAKAWA, H., 1964, Jap. J. Appl. Phys. 3, No. 1, 27. 1965, Proc. IEEE (London) 112, No. 9; JARVIS, T. R., G. SAXON,and M. C. CROWLEY-MILLING, p. 1795. KELLIHER, M. G., and R. BEADLE, 1960, Nature 187, 1099. LEBOUTET, H ., 1966, Auto-deflexion du Faisceau d'un Acctltrateur LinBaire, Rapport Interne, INT. 2217-HL/ml/md, CSF, France. LEVIN,V. M., V. L. SMIRNOV, and L. P. FOMIN, 1967, On the Possibility of Production of Large Pulse Currents in Linear Electron Accelerators Using Homogeneous Structure, Proceedings of the 'Sixth International Conference on High Energy Accelerators, Cambridge Electron Accelerator, Cambridge, Massachusetts, CEAL-2000, pp. A-39-41. LOEW,G. A,, 1967, IEEE Trans. Nucl. Sci. NS-14, No. 3; pp. 529-540. M.L. Report No. 581, 1959, W. W. Hansen Laboratories of Physics, Stanford University, Stanford, California. MELARD, L., 1967, Note Technique No. 83.A.8, Sections Accel6ratrices d'Orsay, LM/MLN No. 016167, Laboratoire de I' AccClCrateur Lintaire Service "Faisceau", Orsay, France. MORTON, P. L., V. K. NIELand A. M. SESSLER, 1966, J. Appl. Phys. 37, No. 10, 3875. NAGLE,D. E., 1964, Coupled Resonator Model of Linear Accelerator Tanks, Minutes of 1964 Conference on Proton Linear Accelerators, MURA-714; p. 21. NISHIKAWA, T., 1965, Normal Mode Analysis of Standing Wave linacs - Field Excitation and Beam Loading in Linac Cavities, Internal Report AADD-87, Accelerator Department, Brookhaven National Laboratory, November 1965. NISHIKAWA, T., S. GIORDANO, and D. CARTER, 1966, Rev. Sci. Instr. 37, No. 5, 652. NEAL,R. B., and W. K. H . PANOFSKY, 1966, Science 152, 1353. NYGARD, J. C., and R. F. POST,1961, Nucl. Instr. Methods 2, 126. PANOFSKY, W. K. H., and W. A. WENZEL,1956, Rev. Sci. Instr. 27, 967. PANOFSKY, W. K. H., 1966, Transient Behavior of Beam Break-Up, TN-66-27, Stanford Linear Accelerator Center, Stanford, California. (See also Panofsky and Bander [1968]). PANOFSKY, W. K. H., and M. BANDER, 1968, Rev. Sci. Instr. 39, No. 2, 206.

B.1.4

BEAM BREAKUP

221

SAXON, G., T . R. JARVIS, and I. WHITE,1963, Proc. lEEE (London) 110 (8); p. 1365. SESSI-ER, A,, 1966, Lecture given at the Stanford Linear Accelerator Center, Stanford, California (unpublished). SLATER, J. C., 1950, Microwave Electronics (D. Van Nostrand Co., New York). VALTER, A. K., I. A. GRISHAEV,E. V. EREMENKO, J . I. DOBROLYBOV, A. I. ZYKOV,V. V. KONDRATENKO, G . F. KUZNETZOV,J. M. BAZAJEV,F. A. GOROSHOVATSKY, V. B. MUFEL, N. I. MOTCHESHNIKOV, V. V. PETRENKO, V. I. KOLOSOV, N . A. KOVALENKO, G . A. ZEITLENOK, V. M. E. G . KOMAR,V. V. RUMJANTZEV, F. F. T U R K I and N V. K. CHOCHLOV, LEVIN,I. F. MALISHEV, 1965, 2-GeV Traveling Wave Electron Accelerator, Fifth International Conference on High Energy Accelerators, Frascati, Italy; pp. 233-238. VISHNYAKOV, V.A.,A. I. ZYKOV,1. A.GRISHAYEV, and V. M. GRIZ:-IKO, 1967a, Effect of an External R F Source on the Stability of the Current Pulse in a Linear Accelerator, No. 306/VE-069, Physics and Engineering lnstitute of the Academy of Sciences, Kharkov, U.S.S.R. and V. M. GRIZHKO,1967b, Increasing the VISHNYAKOV, V. A,, A. I. ZYKOV,I. A. GRISHAYEV, Current Pulse Limit in a Linear Electron Accelerator, No. 308/VE-071, Physics and Engineering lnstitute of the Academy of Sciences, Kharkov, U.S.S.R. and L. A. M A K H N ~ N K1967c, O, Questions VISHNYAKOV, V. A., A. I . ZYKOV,I. A. GRISHAYEV, Concerning the Increase of the Limiting Current in Multisection Linear Accelerators, No. 3091 VE-072, Physics and Engineering lnstitute of the Academy of Sciences, Kharkov, U.S.S.R. G. V., and V. I. KOROZA,1965, The Transverse Motion of the Particles in Linear VOSKKESENSKY, Electron Accelerator Caused by the Action of Nonsymmetric Wave, Proceedings of Fifth lnternational Conference on High Energy Accelerators, Frascati, Italy. G . V., V. I. KOROZA,and Y. N. SEREBRYAKOV, VOSKRESENSKY, 1966a, Transverse Instabilities of a Beam in a Linear Accelerator Prior to Increa5ing Injection Current, Uskoriteli (Accelerators), Vol. 8, Moscow lnstitute of Engineering Physics, Atomizdat, Moscow, U.S.S.R. G . V., V. I. KOROZA, and Y. N. SEREBRYAKOV, 1966b, Toward the Investigation ofthe VGSKRESENSKY, Radial Instability of a Beam in a Linear Electron Accelerator, Uskoriteli (Accelerators), Vol. 9, Moscow Institute of Engineering Physics, Atomizdat, Moscow, U.S.S.R. 1967, Dynamics of Transverse Instability of a Beam VOSKRESENSKY, G . V., and Y. N. SEREBRYAKOV, in Linear Electron Accelerator, Proceedings of the Sixth International Conference on High Energy Accelerators, Cambridge Electron Accelerator, Cambridge, Massachusetts, CEAL-2000; p. A-46. WILSON, P. B., 1963, A Study of Beam Blowup in Electron Linacs, HEPL-297 (Rev. A), High Energy Physics Laboratory, Stanford University, Stanford, California. WILSON,P. B., 1967, A Simple Analysis of Cumulative Beam Break-Up For the Steady-State Case, HEPL-TN-67-8, High Energy Physics Laboratory, Stanford University, Stanford, California.

ELA THEORY

1.5

B.l

Problems connected with the Measurement of Accelerated Electron Beam Characteristics R. BERGERE Centre d2Etudes Nucleaires de Saclay 91 Gif-sur-Y v e t t France

A detailed description of the possible methods for measuring accelerated electron beam parameters can, of course, not be given in such a short chapter. In particular, we shall not discuss the technological problems raised by different types of measuring instruments. More details will be given in chapter B.2.4. Moreover, the references are mainly cited as typical examples and do not constitute an exhaustive bibliography on the subject. This chapter merely proposes a classification of certain general problems encountered whenever accelerated electron beam characteristics are to be measured: dectrical characteristics (mean current, peak current I,(t) during length of pulse, total charge of elementary electron bunch, density distribution along accelerating sinusoidal electric field, etc.); geometrical characteristics (position of center of gravity for a cylindrical slice of accelerated charges with reference to linac axis, charge density distribution within such a slice, etc.); mechanical characteristics (beam emittance, energy spectrum of accelerated electrons).

I

Tentative classlJication of methods of measurement

Two types of measuring instruments can at once be discerned; those converting all the available beam energy into signal energy and those doing so only partially. (0North-Holland

Publ. Co., Amsterdam 1969

Refer~ncesp. 234

223

224

R. BERGERE


Hence we can classify the measuring methods according to the same criteria and discuss in the following subsections four groups of methods.

1.1 Total interception of electron beam

For instance, we can measure a beam current by means of total interception in a Faraday cup (Brown et al. [1956], Kretschko et al. [1962], Isabelle [1963]); also the magnetic analysis of an electron beam energy spectrum, where all the electrons not falling within the energy interval (E-AE, E+AE) are absorbed in the analysing instrument itself. Clearly, in most such measurements the total electron beam energy is not exclusively converted into signal energy but is partly lost due to radiation and heating effects in the measuring device itself.

1.2 Partial interception of electron beam Partial interception is here defined as the passage of an electron beam through a thin screen or detector of thickness Ax, atomic number 2, atomic mass A and M g/cm2 in which an energy loss of AE occurs. Upon traversing such a screen, the mean energy loss due to radiation and ionization is respectively (dE/dx), Ax and ( d E / d ~ Ax. )~ The (dE/dx), Ax energy has been used to operate Cerenkov and beam position monitors whereas the (dE/dx), Ax part has often been used in current monitors (Tautfest et al. [1955], Taimuty et al. [1961], Vanhuyse et al. [1962]), position monitors (Okabe et al. [1965]) and even energy monitors (by means of secondary electron emission). Unfortunately the angular dispersion given by

(where 0 is the angle between electrons emerging from such a thin monitor and the linac axis, and 8, =2.5 x ZM*/EA*) prohibits the installation of such monitors at the end of each section in a linac. For Ax=O.l mm A1 screen, we have for 15 MeV radians. electrons 6, =2.2 x

1.3 Real-photon emission Using standard notation we can write: fi,=flc (c, is the electron velocity) and Y =(I - P 2 ) -+. Let cb,, and cfl, be the parallel and perpendicular components of the acceleration

B.1.5

MEASUREMENT PROBLEMS

225

with respect to the velocity flc. For a n electron having velocity Be. the radiated power due to real photon emission can then be written as:

For linear accelerations (i.e. p,=O) this radiated power is practically zero. Such real photon emission caused by periodic transversal deflections (magnetic ondulator) o r similar, non-periodic phenomena (magnetic deflections) can provide information concerning the beam current, the energy or the phase bunching of the electrons (T.I.D. [1962], Motz [1964]). We can add to this method the backscattering of visible photons after antiCompton interaction with accelerated electrons, the energy of the photons being shifted towards higher energies, by a n amount depending on the electron energy (Kulikov et a/. [1964]).

1.4 Perturbation of associated Coulomb field These methods d o not intercept the electron beam at all. They merely detect the perturbations in the propagation of the Coulomb fields associated with accelerated electrons by means of appropriately chosen electrodes. Such methods can in fact be used for observing the evolution of beam characteristics, section by section, perturbing the electron beam itself as little as possible. The remainder of this paper will essentially consider the latter methods only.

2 Time-fvequency characteristics associated with the perturbation methods T o a n electrode, at rest in the laboratory system, the more a n electron is relativistic, the more its associated Coulomb field, moving with the electron along the Oz axis, will appear to be concentrated in the transverse plane (xOy). Thus the observed electric field EL associated with an electron having a velocity r = 6,=PC at the moment t in the laboratory system can be written as:

where (1) refers to an observation at a distance Ay from the electron trajectory and (2)

226

R. BERGERE


to an observation at a distance Az along the accelerator axis from the actual electron position. It follows that the fields interacting with an electrode placed at a transverse distance Ay from the electron beam ought to exhibit the same time and frequency structure as the accelerated beam current i,(t) itself, at least for frequencies smaller than y/lc/Ay (for 50 MeV electrons at Ay=3 cm, ypc/Ay= loi2 Hz). Moreover, it can be shown (Bergkre et al. 119621) that the Lorentz-Fitzgerald contraction just compensates for the increase in field strength given by equation (1). Thus, in practice, the potential difference created across the electrode, or the magnetic flux created inside it, will reflect the electrical or (and) geometrical characteristics of the electron beam without providing us with any information as to the energy spectrum of the latter. In fact, an observer looking at the beam passing from one section to the other, would observe a succession of elementary electron bunches of duration Az at the accelerating frequency v = z-' (AT z). If these bunches had been accelerated from t = - oo (fig. 1) onwards, the current could then have been written as: +m ib(t) = C j ( t - nz) = n=-m

or putting J ( f ) the Fourier transform of the current j(t) representing a single bunch:

Figure 2 shows that we obtain a repetition of pulses having a width At (At S z and a few nanosec< At I : and > 2 v . Figure 8 shows the gain in information when detecting the envelope of a very weak signal (a photon pulse created by electrons hitting a target) first in lobe [I] @a), then [2] (8b) and finally [3] (8~).

Fig. 7

Thus we see that the quantity of information available for evaluation depends rather on the signal-to-noise ratio of the complete system (detecting electrode+ transmission l i n e ~ m e a s u r i n gdevices) than on the sensitivity a of the detecting electrode alone. Of course, we could increase a by improving the gain of the coupling between electrode and beam. This is what happens when we detect a signal S, ( t ) by means of a U H F cavity instead of with a non-periodic loop. Unfortunately, a simultaneous decrease in the bandwidth B leaves us with practically no increase in information at all (except when looking for mean values). When trying to improve the measurements made on the evolution of a beam characteristic during a pulse length A t , the best approach seems to be an increase of the quantity of information available by appropriate shielding of transmission and

234

R. BERGERE


Fig. 8

measuring devices, rather t h m an increase in sensitivity a t the expense of bandwidth of the detecting electrode as such. References B E R G ~ RR., E , 1966, Etude des proprietks du faisceau d'electrons d'un accelerateur lineaire au moyerl des champs BlectromagnCtiques associks a ce faisceau, Doctoral Thesis, Paris Orsay. R., E. DELEZENNE and A. VEYSSI~RE, 1962, Etude de moniteurs de courant pour un accelkBERG~RE, rateur lineaire d'electrons, Nucl. Instr. Meth. 15, 327.

B.1.5

MEASUREMENT PROBLEMS

235

BERGERE, R., A. VEYSSI~RE and S. ROLAND,1964, Ondes millimetriques rayonnees par le faisceau d'electrons d'un accelerateur lintaire a electrons, Proc. 5eme Congres Tubes Hyperfrequences, Paris. BESS,L., 1. OVADIA and J. VALASSIS, 1959, External Beam Current Monitor for Linear Accelerators, Rev. Sci. Instr. 30, 985. 1956, Faraday Cup Monitors for High Energy Electron Beams, BROWN,K. L., and G. W. TAUTFEST, Rev. Sci. Instr. 27, 696. BRUNET, P., 1963, Etude et realisation d'un systeme de reglage automatique des phases hyperfrequences sur un accelerateur lineaire. Rapport LAL-31, Ecole Normale Superieure. ISABELLE, D . B., 1962, La mesure de I'intensite de courant produit par un accelerateur linkaire, Onde ~ l e c t r i q u eno. 421 354. ISABELLE,D. B., 1963, Beam Current Monitors for a High Energy Linac, SLAC report no. 25, part I. JOHNSON, K., T. KLIPPERT and W. J. RAMLER, 1961, A Non-intercepting Accelerator Beam Position Sensor, Nucl. Instr. Meth. 14, 125. KRETSCHKO, J., D. HARDER and W. POHLIT,1962, Absolutme~sungder Teilchenflussdichte schneller Elektronen mit einem Faraday-Kafig, Nucl. Instr. Meth. 16, 29. KULIKOV, 0. F., Y. Y. TELNOV, E. I. FILIPPOVand M. N. YAKIMENKO, 1964, Compton Effect on Moving Electrons, Phys. Letters 13, 4. LARSEN, R. S., and H. A. WOODS,1965, Position Monitoring Electronics for the Stanford Linear Accelerator, IEEE Trans. Nucl. Sci. NS-12, 862. MOTZ,H., 1964, Application de la radiation d'un ondulateur a la mesure de la focalisation d'un linac, Conferences INSTN, Saclay. MOTZ,H., 1956, Cerenkov and Ondulator Radiation, Trans. I.R.E., PGAP 4, 374. 1965, Beam Position Monitors for Accelerators, Rev. Sci. Instr. 36, 97. OKABE,S.. and T. TABATA, TAIMUTY, S. I., and B. S. DEAVER, 1961, Transmission Current Monitor for High Energy Electron Beams, Rev. Sci. Instr. 32, 1098. G. W., and H. R. FESTER,1955, A Non-saturable High-energy Beam Monitor. Rev. Sci. TAUTFEST, Instr. 26, 229. T.I.D. REPORT,1962, Linear Electron Accelerator Studies. Statute report, Stanford University no. 14 180. VANHUYSE, V. I., E. D. WATTECAMPS, R. E. VAN DE VIJVER,and G . I. VANPRAET,1962, Secondary Emission Beam Monitor for 0.5 to 3.5 MeV Electrons, Nucl. Instr. Meth. 15, 59.

ELA

2.1

TECHNOLOGY

B.2

Electron Injectors P. BRUNET X. BUFFET Laboratoire de I'Acctle'rateur Lintaire Faculte' des Sciences, 91-Orsay, France

I 1.1

Introduction Definition

The injector is the linac section where electrons are so produced and injected into the linac proper as to enable their capture and acceleration. It includes the following main elements : ( I) an electron gun (the source); (2) a more or less sophisticated structure for bunching (one bunch per T or 1) and accelerating the electrons to a velocity Be = ve/cz 1 ; (3) a focusing system acting in opposition to the radial deflection forces on the low energy electron beam. The end of the injector is not well defined in terms of physics, but it may be defined in terms of the structure as the place where 1 - P e < l o p 3 (P,% l), which corresponds to about 12 MeV. The structure at this point is considered the end of the injector.

1.2 Main characteristics and quality (a) The average energy V of the electrons. At the injector end, V is about 12 MeV, so that the electrons are synchronous with the wave in the following sections. (b) Intensity of the beam injected into the accelerator. In this chapter we consider only injectors for a linac of low or medium current, that is, currents lying in the range O North-HollandPubl. Co., Amsterdam 1969

Refermces p. 263

237

238

P.BRUNET, X.BUFFET

ELECTRON LINACS: TECHNOLOGY

O< IOU,< 500 mA (at the injector end) with a range of pulse widths 0.5 ,us< z< lops. (c) Beam intensity acceptance of the injector. The ratio A = I,,,/I,,, defines the quality of the injector in terms of captured electrons: A 3 40% means a good injector; A 2 60jl/,, a very good injector. , be as small as possible. It is (d) Phase spread of the bunches. The spread A C ) ~ ~should in fact constant through the entire linac where u, =c=constant, and the minimal "current-energy" spectrum width at the end of the linac is ( ~ 4 ) ~ / 8 .

A4,,, < 10": good injector + A V / V < 0.4% Aq5,,, < 5": very good injector + A V/V < 0.1%. (e) Radial emittance R (see fig. 1). The radial emittance can be defined as the surface surrounding all the abscissa points r (distance from the axis) and the ordinate p, (radial momentum). If R is small at the injector end, a narrow beam can be obtained through the entire linac (see chapter B.1.2). This provides the following main ad-

Fig. 1. Typical radial emittance surface.

Fig. 2. Emittance limits for a good injector ( V rn 12 MeV).

ELECTRON INJECTORS

B.2.1

239

vantages: (1) all electrons in a given transverse section are subject to the same field (narrow spectrum), (2) all electrons are captured by the linac (good transmission and no structural damage). For a good injector, the output emittance may be inscribed in a rectangle (fig. 2): r= f2.5 mm, p,= k2.5 x 10-3p, at 12 MeV. (f) Energy dispersion at the end of the injector. In fact, A V is conserved through the entire linac and can provide a substantial relative spread of the spectrum in short accelerators. On the other hand, for good focusing, A V/ V should be small before the end of the injector. A V / V < 10%: good injector

A V /V < 5% : very good injector.

2

Choice of an injector

2.1

Buncher design

The buncher is designed so as to produce a beam having the most favorable representative phase volume. The action of the structure on the particles is essentially characterized by the time variation of the accelerating field along the axis of propagation:

The functions E(z) or ~ ( z and ) B,(z) can be defined as generating functions for the electron phase and energy in the buncher. cx=eEl/rnoc2,cj?, is the phase velocity of the wave, y = rn/rno. The volume that best characterizes the beam during bunching can be defined in the orthogonal coordinate system 0 4 , Oy, Or, Op,, 0 8 , Op,; z is the independent variable along the propagation axis. The trajectories in this space are determined by a system of coupled differential equations whose solution is almost impossible in practice. The following simplifying assumptions are therefore made in order to uncouple the longitudinal and transverse momenta: the generating functions depend only on the variable z in the neighborhood of the axis, and the diameter of the beam is constant throughout the structure. The latter assumption requires focusing fields computed a posteriori, since the electron energy is known at all points in the buncher. 2.1.1

Choice of theoretical cc (z) and 3/, ( z ) functions

The practical determination of the accelerating field and the phase variation in the structure is generally made by successive approximations. The first step consists of

240

P. BRUNET, X. BUFFET

ELECTRON LINACS : TECHNOLOGY

selecting simple generating functions consistent with the initial energy of the electrons : bw(0) will be of the same order of magnitude but greater than P,(O) (see fig. 3). For be(0)=0.6, b,(z) can thus be taken equal to 1 after the beginning of the buncher (curves 1). For lower Pe(0) velocities, it is suitable to consider a function bw(z) that starts with a value less than 1 and increases to 1 (curves 2). In such cases, the order of magnitude of a(z) will be known a priori; it will be such that the velocity of the electrons at z, is sufficient : be(z,) 0.9 : y (z,) = y (0)

+ kzle~,/m0c2.

Fig. 3. Theoretical Bw(z),a(z), q5 (z) generating functions.

The coefficientk is less than 1 and expresses the electron lag behind the wave, characteristic of electrons that cannot acquire the maximal energy during the bunching. With a somewhat arbitrary choice of the a and b, functions, the equations of motion are solved by computer; this involves the numerical determination of the functions $, [z, $,(0), y (O)] and y [z, $,(0), y (O)] on the basis of the following equations :

1d V -

-

e dz

= E, (z) sin 4 (z);

The functions $,(z) and y (z) are computed step by step, with the step A z chosen so as to provide the accuracy desired.

ELECTRON INJECTORS

B.2.1

241

For a pair of values [q5,(0), y(O)] corresponding to the initial conditions for an electron, the functions 4,(z) and y(z) are then known at all points in the buncher. For other initial 4,(0) and y(0) values the evolution of the beam in the buncher can be found by repeating the computation. The initial condition q5,(0) is assigned a certain number of values between 0 and 2n; the initial condition y (0) is assigned discrete values corresponding to the highvoltage range of the gun.

2.1.2 Representations of' the movement (a) The first possible representation follows directly from the computed results: q5,= 4,(z) and De=De(z), see fig. 4. This representation has the advantage of per-

1

\d(o) = constant

z )

Fig. 4. Electron velocity and phase first

representation.

Fig. 5. Electron velocity and phase second representation.

mitting a comparison of the phase velocity and the electron velocity. It also reveals the phase limit towards which the bunches tend. It is then possible to modify the a and p, functions in order to improve the efficiency of the buncher. (b) The second representation consists of plotting the output energies and phases as a function of the corresponding input values (fig. 5). It is then easy to compute the bunching rate, the phase spread of the bunches, their positions with respect to the wave, and the energy dispersion of the bunch at the injector end. It is extremely important to make sure that the solution found by this method is

242

P.BRL.NET, X.BUFFET


highly efficient for a substantial range of Eo and y (0) values. For example, the bunching should remain correct when Eo is varied between 110 and 150 kV/cm and the high gun voltage between 80 and 120 kV. 2.1.3

Effects due to the intensity

It is interesting to estimate the effects of certain phenomena that can become very important at high beam intensities. (a) Beam loading modifies E(z) in the structure. In a cross section traversed by a beam at Be = 1, the power due to the rf source and the beam are independent: the two effects (fields and energy) add to each other linearly. In the general case where the wave and the beam are synchronous, the accelerating field Eo and the field induced by the beam Ei are opposite in phase at all times. Conversely, in a buncher, where the bunches phase varies as a function of z and the accelerating field, the amplitude and phase of Z(z) and Ei (z) depend on E, (z). The modification of the field in the structure, small in the initial cavities, has almost no effect on the efficiency of rapid bunching, but the phase limit and the energy of the bunches are changed. In a long buncher where the bunching is slow, beam loading also affects the efficiency of the bunching. It is relatively simple to compute the field Ei(z) by assuming that at all points z the total charge of the bunch is concentrated in the phase of the central electron. Under the additional assumption that the movement is synchronous, the term corresponding to beam loading can be introduced into the energy equation in the following form : 1 dV(z) --

= Eo (z)

sin 4 (z) - Ei (z, I) cos 8(z)

e dz where O(z) is the phase shift between the given electron and the center of the bunch. (b) The space charge couples the longitudinal and transverse motions. The forces increase rapidly during the bunching; with increasing energy they then decrease when the bunching is achieved. Introduction of the space charge forces into the computation of the trajectories involves serious problems and requires a certain number of assumptions. Indeed the field acting on an electron during formation of the bunch depends on the beam intensity, the instantaneous length A4(z) and the energy of the bunch, and the distribution of the charges in the bunch volume. This leads us to assume firstly that the diameter of the bunch and its charge distribution do not vary during the motion. A distribution function and a constant beam diameter are then chosen. The (accelerating or decelerating) field acting on a given electron in the bunch can then be computed as a function of I, A$(z) and the phase difference between the electron and the center of the bunch (- + A 4 < O < ++A4).

ELECTRON INJECTORS

B.2.1

243

For example, the space charge field acting o n the electrons a t the edge of the bunch (8= k t A 4 ) in the course of the bunching can be as high as about 106 V/m, with I = l A, V = 1.5 MeV, A4=4" for a beam diameter of 3 mm. In order to take the intensity effects into account in the bunching computation, it becomes necessary to use a more elaborate program capable not only of computing the individual electron trajectories but also the global development of the bunch length A4(z), the position O(z) of a given electron in the bunch, and the space charge field E,,. This field for example is found by the computer as a function of A4(z), 0, y and 1. Thus the computer now processes the more complete equations: I dV(z) e dz -

2.1.4

=

E , (z) sin 4 (z)

-

E, (z, I ) cos O(z)

+ E,, (A4,O, y, I )

sin 0 (z) (sin O(z)l -

Effect of' the space harmonics

Up to now we have considered the buncher as a structure to be designed in conformity with the theoretical generating functions (amplitude and phase). It is known that because of the geometric discontinuities in the iris diaphragms, these theoretical functions are modified by space harmonics. It is therefore appropriate to measure the real functions a(z) and B,(z), either for a mock-up or for a structure already realized. This measurement can be made by a perturbation method. By moving a n obstacle along or parallel to the buncher axis, the values of the reflection coefficient Q and of the phase shift Y between the incident and reflected waves are measured as a function of z. This gives the functions: E(z)

=

kgk(z) and

@,(z)

= +Y(z)

(Steele [1966]) which leads to the actual functions: a ( z ) = s(,,,,,~

Q* (z) -

Qfa,

and

P,(Z)

=

2n

----

-

A, d@, (z)/dz

'

These functions (cf. fig. 6) are introduced into the program either by tables of numerical values, by approximations in terms of analytic functions of z, or by the coefficients of the first harmonics after expansion in a Fourier series. It can easily be seen that the effect of these harmonics is small in a normal linac section where the electrons already have a velocity very close to c. In particular, the energy gained is almost unaffected. In a buncher the problem is found on the level of the input coupler. The waves there are generally semi-stationary (the phase varies very little over a length > 1 cm in zones where the amplitude of the field is already high), that is, j, is very much greater than its rated value.

244



The electrons, rather than being captured in this zone by the field, rapidly slip behind the wave and are sometimes slightly dispersed in phase. The intended perfect synchronism between the wave and the electrons is destroyed, and the velocity P,(z) is at all times less than p, = (z) (Haimson [1965b]). It should be noted that the induced field Ei(z) is modified in the same way as E(z), with their relative amplitude remaining the same.

Fig. 6. Actual electric field and phase variations according to Haimson [1965b].

2.1.5

Transverse motion

Upon leaving the gun the beam has a certain emittance surface in the (r,p,) plane. This emittance, possibly deformed by an optical system before the buncher, could extend beyond the diameter of the allowed beam zone of the structure. In addition to this divergence, due essentially to the gun emission, we have the defocusing effect of the space charge and the sf field in the structure. These effects decrease fairly

B.2.1

ELECTRON INJECTORS

245

rapidly with the energy: The radial force of the space charge acting on an electron located at the edge of the bunchers of diameter 2r can be written as follows:

Similarly, the force due to the rf field can be maximized by the following expression:

This makes it necessary to provide focusing forces for assuring a beam of small size. The computation for the magnetic fields needed to compensate exactly the defocusing forces is not easy. It would be necessary to solve an equation of the following type:

where the last term represents the external magnetic focusing force. We therefore generally limit ourselves to determine these forces by excess as a function of the development of the bunches (fig. 7) and provide a system of coils and solenoids giving the most reliable and flexible focusing possible.

Fig. 7. Typical axial magnetic field pattern.

When interested in high beam intensities, we could be tempted to allow the beam to spread radially during the bunching, so as to decrease the axial space charge forces. But it would then be necessary to take into account the variations of E ( z ) as a function of the distance r to the beam axis during the computation for the bunching. It is then possible to set up a program (SLAC) operating in terms of the quantities 4, y, r andp,.

2.2 Various types of rf buncher structures 2.2.1

General properties

A buncher is characterized by the parameters V, A V / V , A 4 , defined in section 1.2, to

246

P.BRUNET, X. BUFFET


which the bunching efficiency b can be added: b = A ~ i n / A 4 0 u where ,, is the phase domain that at the input includes all the electrons found in A$,,, at the end; b is a function of 4 and Ac$,,. The values that will be given are the maximal values of b as a function of 4 . For comparison, b will be given for Acji, = 60". All the parameters are of course coupled; however it may be possible to separate them for individual optimization, by complicating the technology according to the following general remarks : The theoretical (Dame [1960], Hirel [1961], Lapostolle [1966]) and practical (section 2.1) computations show that the electrons can be coupled to the rf wave with p w = l or P w < l . In the first case, b is small, but there are no phase oscillations; in the second case, b can be very much greater, but 8, is greater than 8, at certain points; this leads to phase oscillations that are smaller in amplitude and frequency the closer PWis to 1 . The phase oscillations are disturbing to the extent that they increase A V / V and reduce the operating range. (a) In a long buncher (1 to 2 m) of low field, "time" is available to finely control the bunching process by shaping the a(z) and Bw(z) functions. Moreover, low gun voltages can be used along with low rf powers; this would simplify the technology and therefore lower the cost. On the other hand, a non-relativistic beam must be controlled over a long distance, that is, the rf defocusing forces and possible longitudinal and radial space charge effects must be compensated (compare this situation with the conditions found in the case of proton linacs). Moreover, the beam loading increases rapidly and considerably perturbs the functions a(z) and P,(z). Thus, this solution is limited to currents of the order of hundreds of mA. (b) In a short buncher (about 10 cm) having high field, the electrons are accelerated very rapidly; after several centimeters, it is no longer possible to control the bunching. It is therefore necessary to study the configuration of the fields in the input coupler (Haimson [1965b]) in great detail; the technology becomes complicated when the gun voltage and the rf fields (which are moreover limited by the breakdown voltage) increase, and the cost increases rapidly. For reasons opposite to the preceding case, this type of buncher is more suitable for medium and high currents; in particular, the beam loading can be neglected during the bunching. It is important that A V/ V be small throughout the bunching. This in fact provides the following advantages : (1) Good beam geometry, since the magnetic focusing systems are chromatic. (2) A cumulative process, since all the electrons in a well-focused beam remain on the axis and are subject to the same electric field. (3) Reduction in the interaction with the deflecting modes, when the electrons remain on the axis. The acceptance A can be artificially increased by providing a prebunching cavity and A+,,, artificially reduced by limiting A$in using an rf chopper (section 3). The optimal phase limit is near 7112 ( V maximum and A V / V minimum) but some-

B.2.1

ELECTRON INJECTORS

247

what less, so that the slower electrons precede the bunch; this narrows the A 4 due to slipping. The phase limit can be controlled by a buncher consisting of several structures whose phases can be shifted with respect t o each other. 2.2.2

Special properties. Performances

Let us now review four types of structures used in existing linacs (see also ch. B.3.4). (a) pw= constant = 1 : This solution, which is the easiest to handle theoretically, is the least used. It consists of directly injecting the electrons from the gun into a standard linac section. The electrons are captured only if cr is large, and A is better the higher U , (Lapostolle [1966]). However, bunching occurs only if the initial velocity of the electrons cp, is very much less than c ; in fact, at the limit, for fi,= 1, a 360' phase would be acceptable, and A4,,, would be 180". The values used in practice are: 80 < U, < 150 kV,

0.5 < /I, < 0.64

and

3 < cr < 4 (U,: gun voltage).

This system is used a t Stanford (M.111) (S-band) with a prebunching cavity. It operates a t U,= 80 kV using a 3 m section identical to those of the SLAC but mounted backwards in order to obtain high input fields (a,= 3). The a(z) then rapidly decreases with increase in z. The theoretical curves then give A=40% and b=7.5. The NBS injector also operates according t o this principle, with a prebunching cavity and a 2.5 m constant-impedance section: U, = 150 kV,

cr = 4 (L band).

The theoretical curves for the section alone, considered as buncher, give (Leiss [1959, 19661):

A > 50% and

b

=6

(A$,,,

=

10").

The value of b is small, but thanks to the high values of cc and U , this injector conserves its properties u p to currents of several amperes. The experimental performances of these two injectors have to the best of our knowledge not been published. (b) pw< 1 varying slowly, U, ION, (C.S.F.) : The electrons can be captured with U, and x very much lower, if p, < I . Azam, Mangin and Tronc (C.S.F.) systematically studied the low-U, bunchers and built the Saclay linac (ALS) injector in this form (Mangin and Tronc [1964, 19651). Capture conditions: If U, decreases, p,, must be decreased, as the maximum current lies in the neighborhood of /3,,=/3,,; the optical properties of the gun a t 25 kV and the difficulties involved in realizing the structures a t pw50% and decreases asymptotically to 50% (see fig. 8). Bunching: In the light of the general remarks above, it would be advantageous to use a large a to realize short bunchers; however this presents two difficulties.

Fig. 8. General buncher intensity acceptance versus gun high voltage and electric field magnitude.

Technologically, high fields cannot be maintained in a structure having low P,; indeed, E,(r) is then minimal on the axis, and the ratio E,(max)/E,(r=O) increases when b, decreases (Tronc [1966]): it is equal to 1 for P,= 1 and about 4 for bW=0.3 (see fig. 9). The allowable maximal a before breakdown is therefore three to four times lower than for a standard section. Moreover, a high a would lead to rapid variation of p, ( z ) and therefore to structural discontinuities that are difficult to make consistent with good rf adaptation.

Fig. 9. Longitudinal electric field radial distribution.

Moreover, if low AV/V is desired in order to limit the phase oscillations, the acceleration must be very progressive; this requires a long buncher and a low field. A compromise must be made between the longitudinal compression, which is maximal in the neighborhood of 4=O, and the energy gain, which is maximal for 4 =n/2. For the Saclay linac, the value +=60° (which is not critical) was chosen for

B.2.1

ELECTRON INJECTORS

249

the synchronous electrons; this gives the B,(z) and a(z) functions taking into account the limitations mentioned above. The following values were adopted for the ALS buncher ( n / 2 mode, S-band): 35 k V < U o < 4 0 kV a(z)=0.7 and almost constant 0.45 < /lw (z) < 0.99 total length 1.5 m V = 5 MeV A V/V: 25% of the injected electrons in A V/V= 3%, 50% in A V/V= 10% b= 15+A~,,,=4" A z 4 0 % to 50%. The main characteristics were verified by tests (with a prebunching cavity): I = 2 0 mA, i n p ~ power: ~t P = 0 . 8 MW V = 6 MeV, A V / V= 1.2% at half height A4,,,=4.5" for 50% of the electrons 90% of the electrons pass through a 3 mm iris, (pr/pz),,,ax< The adjustments are not critical, and V is independent of P to the first order, since the variation of 4 compensates that of cx. The bunching continues in the first standard section, which according to our definition is still part of the injector. The phase limit can then be controlled by the phase shifter of the section input.

(c) P,,, variable by levels, U , average =80 k V (SLAC). Beyond P,=0.5, it is easier t o capture the electrons. Starting with the solution (a), the team of Miller (SLAC) has improved h and solved the phase limit problems by providing a very short buncher (I wavelength) before the Pw= 1 structure [P, = constant = 0.75, a t low field (a = 0.6)] (Miller [I 9651). At the output of the buncher the electrons are grouped with a A4 already relatively small and only slightly accelerated (2 to 300 keV). All the electrons, subject to a high field (a = 2.5) in the following section at p, = 1, are captured and rapidly reach their phase limit, which can be optimized by the section phase shifter. This solution is a good improvement upon (a). The value of U , is moderate, which in particular permits the use of a triode gun. The rf power to be controlled in the buncher itself is low (0.6 kW), and the next section is a standard one. However, the low-field problems are still present, though over a very much shorter length. In particular, the output performance may no longer be adequate for I>300 mA. The buncher section has the following theoretical performances: 40% < A < 50% h=15 V= 35 MeV. A V/V< 1%, without beam loading.


250


Systematic tests were carried out with a short, 0.5-m section and a prebunching cavity. The following performance has been published (Miller [1965]): I = 5 mA480% of the current in A4,,,= 1.2" 1=117mA+llOmA in 5.1" V= 6 MeV, A V/ V at half height = 1.8% beam diameter: 90% of the electrons within 5 mm (~rlpz)max= 3 x

a,

(d) varying rapidly, U , high (Varian). In order to take advantage of high fields while preserving the condition P, =/I,, it is necessary to start with a high value of U,; Haimson and Numan (Varian) studied solutions of this type systematically for medium and high current accelerators and applied them in particular to the accelerator of the University of Saskatchewan (Canada) and that of the University of Frascati (Italy) (Haimson [I 962, l965aI). Starting with 0.55 . DC

\

BEAM LEVEL

-do+-+-60 o I

-180

I

60

1

ao

62 - 0 0,

' 3 2 - '30 > Fig. 10. Departure-arrival phase relationship for prebunched electron beam .Current-arrival phase relationship for prebunched electron beam.

benefit from ideal bunching and limit-phase conditions. The curves of section 2 show that the maximum allowable shift is 10". Since a small variation Avo in vo at the structure input leaves the shape of the prebunch unaltered to the first order, the 10" tolerance can be reduced to the phase of the central electron (@,). A@,/@, = - A/3,/P0

with A@, = 10"

= 2rcj36

rad.

Withthe previous orders of magnitude, @, x Ion, A@, < 10°leadstoAPo/Po< 5 x Example of a compromise: R= 2, D = 30 cm, Lo = 10 cm, Po =0.6+U,/Uo=60,/,, input P= 3 kW.

ELECTRON INJECTORS

8.2.1

253

3.2 Rf chopper In order to design a phase gate with adjustable width and position it is possible to introduce a radial displacement depending on the phase, limiting it by an iris that blocks the undesirable part of the beam. Several systems have been developed: 3.2.1

Dejecting cavity

A rectangular rf cavity (TM,,, mode) displaces the electrons radially to the distance L, according to the function d = ( k ~ * ~ / psin@ , ) where P is the input power of the cavity; p,, the longitudinal momentum of the electron at the gun output; @, the phase of the electron with respect to the rf wave. The electrons pass without deflection two times per period (@ = O and @ = n). If fchoPper = $facce,eratOr a slot located at L perpendicular to the radial motion can select a bunch of width A 4 once per period in L P ~ Q is the geometric width of the slot. A$ is such a way that A ~ = Q ~ , / ~ where adjusted by the parameter P (attenuator) and 4 (phase shifter) at the cavity input. In addition to the use of a frequency 4f of fixed phase with respect to f this system has the disadvantage of causing maximum dispersion of the electrons. Indeed, the p, vanishes for the selected electrons, but Ap,/A@ is maximum. Haimson [I9621 uses a variant that provides the electrons with radial "polarization" at the gun output, by a fixed magnetic deflection d m ; the cavity, which then operates at the frequency f, returns the electrons to the axis once per period. Thewidth A 4 is controlled by P or dm;the phase @, by a phase shifter at the cavity input. The electrons are accepted in the neighborhood of p,=O and A p , / A 4 = O . The input power of such a cavity is of the order of several kilowatts.

Fig. 1 1 . Chopping mechanism for a T M m cavity with "polarization".

254

P.BRUNET, X.BUFFET


3.2.2 'Keyhole' (Mangin [Sept. 19651)

C.S.F. has also perfected a chopping system : It consists of two coupled parallelepiped rf cavities operating at the frequency of the accelerator in the TE,,, mode; they are mounted on the same axis 0,. Only the fields Ex and H, exist in the neighborhood of the beam (x=y=O). If a continuous magnetic field B, is applied, the entire beam moves uniformly in the z-direction but describes a circle in the plane perpendicular to 0,; in this way, a phase condition between the two cavities is realized such that the beam out of the second cavity is centered on the 0, axis and parallel to it. The electron trajectory projection on the plane separating the two cavities is a circle centered on 0,. The angular position of an electron on this circle is its rf phase, to within a constant. In order to select the electrons included in an interval 4,-6, =A$, it suffices to make a V-shaped orifice in the wall (angular opening A$). The corner of the V is on the 0, axis; where one makes a cylindrical hole for the passage of the beam when chopping is not used ("keyhole"). This system is used in the ALS with A$ = 90". Disadvantages: A$ cannot be adjusted, and the powers involved are relatively k-igh (10 kW for U , = 40 kV, B, = 600 to 1000 gauss). Advantages: The beam leaves withp,=O, and the apparatus is short (10 to 15 cm). The radial repulsive forces of the electrons are low since r is increased, and they are compensated by B,. The rf chopping suffices for providing bunches that are as short as wanted. This solution, which in itself is very simple, is not realistic for the normal intensities required in classical accelerators; the situation is quite the opposite for a cw (superconductivity) accelerator with a current equal to 1 mA.

3.3 Electron gun

Electron guns used in standard accelerators are characterized by their high-quality optics and their flexibility rather than by their emission power (perveance of about 0.1 x is generally sufficient). We shall not discuss the design of a gun in detail here, but rather review the requirements for a good gun.

3.3.1 Electron dynamics The most important parameter is the voltage U,; e U , is in fact the initial energy of the electrons entering the rf structure. It should be sufficiently high for the electron velocity to be of the order of the lowest phase velocity that can practically be realized in rf structures (=0.3c-+30 kV), but it must also be sufficiently low so as t o allow a

ELECTRON INJECTORS

B.2.1

255

subsequent velocity modulation sufficient for the bunching (0.8c+ 150 kV): 30 kV < U , < 150 kV. But U, must also be absolutely constant throughout the pulse (modulation) and from pulse to pulse (jitter), so that the initial velocities are the same for all the emitted electrons. The computations of section 3.1 show that the following condition must be satisfied: dfio/fio 70 dB; this completely avoids perturbing the beam during normal operation. Injector No. 1 can be controlled as an independent accelerator (energy, current, bunch). After finding a good adjustment, one turns off the deflecting structure and magnet supply currents, rigorously cancelling out the possible magnetic field. The beam is injected into the accelerator. Two adaptation parameters are then available: (1) A phase shifter at the klystron input adjusts the phase of the current to the rest of the accelerator. (2) The lens Lo adapts the injector emittance to the accelerator acceptance.

References BERGERE, R., 1966, Etudes des proprietes du faisceau d'electrons d'un accelerateur lineaire au moyen des champs electromagnttiques associes a ce faisceau (Study of the properties of a linear accelerator electron beam by the electromagnetic fields associated with this beam). Rapport C.E.A., R 3126. R., 1967, Mesure de I'extension en phase des paquets d'electrons (Measurement of the CHEHAB, phase spread of the electron bunches). Laboratoire de I'Acctltrateur Lineaire, Orsay (France). Rapports LAL 1180 et 1193. G., D~ME , 1960, Electron bunching by uniform sections of disk loaded wave guide. Project M. Report N' 242-H, Stanford University (December 1960). I~AIMSO J.,N ,1962, Some aspects of electron beam optics and X-ray production with the linear accelerator. IEEE Trans. Nucl. Sci. NS-9 121 p. 32 (April 1962). MAIMSON, J., 1965a, A comparison of theoretical and actual beam performance of a 140 MeV linear accelerator at the University of Saskatchewan (Saskatoon, Canada). IEEE Trans. Nucl. Sci. NS-12 [3] p. 547 (June 1965). HAIMSON, J., 1965b, Space harmonics, space charge and transient field effects in high current electron linear accelerators. Varian (August 1965).

264

P.BRUNET, X. BUFFET


HIREL,R., 1961, A study of the electron bunching problem. Project M, Report No. 269, Stanford University (June 1961). LAPOSTOLLE, P., 1966, Introduction a 1'6tude des accCl~rateurslineaires (Introduction to the design of linear accelerators). Course given at the INSTN, Saclay (France) CERN report 1966-20. LEISS,E., 1959, Injection systems for linear electron accelerators. National Bureau of Standards, Washington, D.C. (January 1959). LEKS,E., 1966, The NBS linac. Linear Accelerator Conference, Los Alamos Scientific Laboratory (Oct. 1966), LA 3609, p. 20. MANGIN, J. P. and D. TRONC,1964, Phases et vitesses d'6lectrons semi relativistes dam un champ Clectrique alternatif. Application aux bunchers (Semi-relativistic electron phases and speeds in an alternating electric field. Application to bunchers). Rapport interne 2035, C.S.F., Domaine de Corbeville. 91 - Orsay (France) (Sept. 1964). J. P., 1965, Principe d'un hacheur de faisceau (Principle of a beam chopper). Rapport MANGIN, interne 2074 (Sept. 1965). C.S.F., Domaine de Corbeville. 91 - Orsay (France). MANGIN, J. P. and D. TRONC,1965, Traitement par machine arithmetique de calculs sur structures acceleratrices (Digital computer processing for accelerating structures). C.S.F., CEPEC, Orsay (France). Rapport interne 2077 (Oct. 1965). MILLER,R. H. et al., 1965, The Slac injector. IEEE Trans. Nucl. Sci. NS-12 131, p. 804 (June 1965). STEELE, C., 1966, A non-resonant perturbation theory. IEEE Trans. Microwave Theory and Techniques, MTT-14, pp. 70-74 (Feb. 1966). TRONC,D., 1966, Groupeurs d'accelkrateurs lineaires sous forte puissance H F (High rf power linear accelerator bunchers). Rapport interne 2738 (April 1966) C.S.F., Domaine de Corbeville. 91 Orsay (France). VARIAN Associates, 1965, Technical Description of Orsay injection system (Oct. 1965). WARNECKE, R. and P. GUBNARD, 1953, Tubes i modulation de vitesse (Speed modulation tubes) (Edition Gauthier et Villars).

ELA

2.2

1

TECHNOLOGY

B.2

Accelerating Structures

A. L. ELDREDGE* A. V. LTSIN

V. G . PRICE Stanford Linear Accelerator Center Stanford, California, U.S.A.

I

Introduction

The performance that can be achieved in an accelerator is determined by a number of factors. Included among these are the shunt impedance, the operating vacuum level, the conformance of cavity dimensions to design values and the material, surface finish and Q of the cavities comprising the accelerator. Each of these factors depends upon the methods and materials used in the construction. An improvement in accelerator performance has been observed over the past two decades as a result of the many innovations in fabrication techniques and the refinements in materials that have taken place during this period. Early accelerator parts, for example, were made of brass and were soft-soldered together. The performance obtained was relatively poor. In later machines, copper of increasingly higher purity has been used resulting in lower surface resistivity and hence lower losses. With better machining and assembly procedures, the surfaces have become cleaner and smoother permitting a higher permissible gradient for a given cavity geometry. Some of the accelerators built in recent years(e.g. SLAC) are large and sophisticated. It has been essential in the construction of these machines to select and develop economic methods and tooling to reduce costs. This will be particularly true with new

*

Presently at Varian Associates

$ North-Holland Publ. C o . , Amsterdam 1969

References p . 312

266

A.L. ELDREDGE ET AL.

ELECTRON LINACS : TECHNOLOGY

cryogenic accelerators now in the design stage. These machines require the development of new techniques to offset the very high cost of materials. The fabrication methods used successfully in the construction of accelerators at several centers are discussed in this chapter. Although the techniques used at Stanford are emphasized because of the first-hand experience of the authors, the methods used at some other locations are included where documented information has been available. No attempt was made to review the techniques used, for example, in the Russian accelerators, because of insufficient information. The reader is referred to ch. B.l.l and ch. B.3.2 for additional discussion of accelerator fabrication methods. To construct an accelerator capable of optimum performance from the standpoint of freedom from inherent phasing errors, it is necessary to make tuning adjustments to correct for the effects of machining tolerances or minor errors in design or manufacture. The relationship between these errors and the performance that would be obtained without tuning is given. It may be seen that although tuning adjustments must be made, it is necessary to maintain very small dimensional tolerances to limit the amount of the adjustment. The boundary walls of the SLAC cavities are deformed mechanically in the tuning operation. The apparatus used for this deformation is described. The dimensions of accelerator cavities vary with temperature. Consideration is given in this chapter to the significance of precise control of the metal temperature in reducing energy loss to a minimum. The problems of steering and focusing the beam in a linear accelerator are simplified if careful attention is given to the system of support and alignment. A description of the methods of support and alignment used at accelerator centers such as SLAC and Orsay is provided. Pre-operational tests on the SLAC accelerator sections were made at high rf power levels to outgas the section and to measure performance prior to installation. The apparatus used in these tests is described.

2 Dimensional requirements Optimum performance of an accelerator is possible only when a precise phase relationship is maintained between the driving rf field and bunched electrons throughout the acceleration length. In a disk-loaded waveguide, this relationship is strongly dependent upon the dimensions of the cavities which form the accelerator, and any errors in these dimensions that are uncompensated by tuning can cause a fractional loss in energy. The cavity dimensions will be incorrect for the design frequency if errors have been made in machining, fabrication, or in the choice of operating temperature. The subject of cavity dimensional tolerances has received detailed treatment by Post [1950], Chu [1951], Neal [1953], Chodorow et al. [1955], Chu and Ginzton [1955], Slater [I 9481, and Shersby-Harvie [1948].

B.2.2


267

It is not possible to machine or fabricate parts with zero tolerance, hence, a basic problem in accelerator design is to specify the permissible tolerances. This specification is a compromise between high cost of fabrication and the allowable fractional loss of energy. The best performance is achieved usually at the lowest cost by making tuning adjustments to compensate for dimensional errors rather than by machining to ultra-close tolerances. However, to reduce the amount of time-consuming tuning work and the degree to which the cavities must be deformed to make them meet electrical specifications, the untuned cavity dimensions must be very nearly correct. Fabrication errors can be systematic or random. Systematic errors result from poor design, improperly calibrated tools and measuring equipment, and temperature deviation. Small systematic errors which remain undetected during the fabrication of single section accelerators can be compensated for by adjusting either the operating temperature or the frequency. Random errors result from human error and the accuracy, precision, and reliability of measuring instruments. Chu [I9511 shows that deviations from design dimensions in accelerator cavities may be analyzed by treating the deviations as shunt susceptances on an otherwise uniform transmission line. These susceptances decrease the amplitude of the rf field and produce phase shift errors. Both of these effects cause a reduction from the design value in the energy gained by the accelerated electrons. However, the fractional loss of energy due to the phase shift error can often be offset, in part, by adjusting the driving frequency to correspond to the average resonant frequency of the cavities. Neal [I9531 derives eq. (1) which assumes that the frequency has been chosen to produce a zero net phase shift error in an accelerator section.

where @,=probable phase error in radians for one cavity due to dimensional errors N =number of cavities in an accelerator section fed from one source V =energy of the electrons r = o t f / 2 Q= the attenuation parameter in nepers o = 2nf, =the radian frequency Q =the cavity loss factor t , =the accelerator section filling time f, =the operating frequency in the fundamental space harmonic mode. The first bracketed expression in eq. (1) gives the fractional electron energy loss as a function of attenuation in the accelerator section. The second bracketed expression gives the fractional energy loss due to random phase shift deviations when the net phase shift error over the accelerator section is zero. As z, the attenuation parameter, approaches zero the fractional energy loss approaches +N@:. The value of 0: is equal to the sum of its constituent parts, one for each cavity

268

A.L.ELDREDGE ET AL.


dimension, namely: 0:

=

@ti=

+ @tZb) + @td)+ 0;).

PI

The dimensions, q,, are identified in fig. 1. The phase shift Oqiresulting from an error in dimension q iis just the variation of electrical length of the cavity, i.e.: O ,(radians)

=S

61 (2nd) ( 300 MeV

--

g ~ m - ~ cm

g cm+

cm

g cm-2

cm

Lead Copper

Heavy concrete (d 4.5)

-

Barium loaded concrete (d 3.5) -.

-

-

Ordinary concrete (d = 2.2) Earth (d = 1.7) Water -

--

Air

Correspondence between biological doses (in millirems) and particle flux density (particles/cm2) for different types of particles. Equivalent dose

Nature of the particles

-

m~ll~rems .- -p.~rn-~ -

---

-

-

7x

Medium energy neutrons 0.5< En< 15 MeV (dose due to multiple collisions in tissues)

4 x 10-5

High-energy neutrons En> 80 MeV accompanied thick shielding ( I , , large: large thickness necessary for a dose due to multiple collisions)

millirems/hours . p . ~ m -s-I ~

-

-

-

High energy neutrons En> 100 MeV alone (dose from simple collisions)

by secondaries in equilibrium with them outside

Equivalent dose-rate

--

-

1.5 x at E = 200 MeV 4x at E = 700 MeV

-

0.54 1.44

y's of several MeV -

The charged particles (protons, electron-positron pairs, pions) very quickly lose their energy, either by ionization, by bremsstrahlung, or by other electromagnetic interactions. The lifetime of neutral pions is very short (10-l6 sec).

362

ELECTRON LINACS: TECHNOLO GY

L.BURNOD

The photons will be rapidly absorbed by the shielding, since the length of electromagnetic absorption is 5 times shorter than that of nuclear absorption. Therefore, the shielding will be calculated chiefly in terms of the high-energy neutrons, which are created by photoproduction or by secondary reactions between a pion and a nucleon. It turns out that the neutron production is weighted approximately by a factor of l/k2 up to k = E , , and since the absorption of particles of energy less than 200 MeV becomes very great as a function of the thickness of the shielding, the most important range of energy for the neutrons produced will be between 200 and 500 MeV (Dardenne [1967], De Staebler [1962]). 1.2.3 Calculation of protective shielding We will not attempt here to describe the complex calculations and experiments which have been done in order to determine the shielding (Dedrick and Clark [1962], De Staebler [1961], [1962]). We shall only indicate the initial steps of a simplified method of calculation (De Staebler [1963]).

ELECT

Fig. 3. Shielding for an igloo geometry.

This method supposes that neutrons of energy T are produced in a thick target from a certain number of interactions with cross-section oi and these neutrons are attenuated in the shielding according to their mean free path i ( T ) (fig. 3). Knowing the experimental cross-sections for each reaction one can evaluate the sum for all the reactions. If d 2 n / d dl2 ~ represents the yield of high energy neutrons, differential in energy T and in solid angle SZ at an angle 8, arising by all these reactions from the absorption of one electron in the target, the flux of neutrons emerging from a shielding of thickness H i s , per electron

If the shielding is thick enough to stop neutrons with an initial energy less than

INSTALLATIONS AND CONTROL

B.2.4

363

150 MeV, one may take for i ( T ) its asymptotic value 1, given for several materials in table 2. Then the calculation of the flux of neutrons is simplified:

Then, outside the shielding the flux of neutrons thus obtained at a distance R, in meters, from the target is converted to the level of radiation by the factor (1.4 x R ' rem/cm2 hr. This factor takes also account of the secondary radiation emerging from the shielding in equilibrium with the primary high-energy neutrons. This simplified method gives sufficiently accurate results to estimate the shielding.

2 Principal auxiliary equipment 2.1

Electrical power sources and concept of efficiency

For accelerators of low intensity all the rf power will be dissipated in the structure. One obtains the maximum energy, which is in this case the interesting parameter, with constant gradient accelerators, for which the total peak rf power varies as the square of the unloaded energy. In fact, the power given to the beam becomes important for accelerators of medium energy and high intensity, and it can be seen that the total peak rf power furnished to different accelerators (Amman [1967], Averill [1967], Azam [1967], Blanc-Lapierre and Burnod [1967], Kruse and Loew [1967], Leiss [1967], Valter [1965], Wilson [I 9671) increases almost as their unloaded energy. The mean relationship, taken from fig. 4 is

PI, = 0.57 ~ 0 0 . ~ '

(PI, in MW, V, in MeV).

On the other hand, the peak current is often limited, for various reasons (beam blowup, for example), in the different accelerators to values lower than those giving the maximum power transfer to the beam. Thus, we find a much greater dispersion in the values of peak power transferred to the beam, and the efficiency: Peak power of the beam rf peak power -

-

-

- -

varies a great deal. It reaches the interesting value of 71% for NBS, is between 20 and 35% for the other machines (Amman [1967], Averill [1967], Azam [1967], Wilson [1967]), and is about 15% for the machines presently limited by the beam blow-up (BBU) (Blanc-Lapierre and Burnod [1967], Kruse and Loew [1967]).


As for the average rf efficiency Average power of the beam Average rf power the NBS machine once more leads with a percentage of 59%, while the others have between 6 and 25%.

5 000

2 OOC

1 OOC

-3

50C

E 20, 0

a + .. L

%

1OC

a 50

2C

10

C

I

I

I

I

I

I

I

02

0.5

1

2

5

10

20

V,(GeV)

Fig. 4

The "efficiency of the modulator-klystron combination" Average rf power in kW Apparent power in kVA at modulator input is situated between 14 and 26% for all the machines.

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365

Defining the average power of the accelerator alone as being the sum of the alternating apparent power in kVA necessary to the functioning of the accelerator, as opposed t o the apparent power required for experimentation (deflector systems and experimental areas), one may define: The "electrical efficiency of the accelerator alone" Average power of the beam --

--

--

-

Average power of the accelerator alone which stay between 0.5% and 2% for all the machines except NBS, where it reaches

4.6%. At last, the power necessary for experimentation is, according to the various laboratories, of the order of the installed power for the accelerator, and often even several times greater.

2.2 Temperature and cooling The energy V gained by an electron in one section is a function of the copper temperature T , by the field E, the length L of the section and the phase 4 between the electron and the peak of the wave, as follows

2.2.1

Influence of the phase

For the operating frequency of the accelerator fo,one temperature To corresponds to the synchronous operation v4= c for which the electron is in phase at 4 (z, To)=O. For T = To+AT, the structure is expanded and the dispersion curve is modified. In first order one finds the same characteristic v4=c for the same mode at a frequency J'=fo A.f such that A f /f= - VAT, where v is the coefficient of linear thermal expansion ( v = 1.6 x l o p 5for copper). But this frequency-temperature compensation is not always easy to obtain in the following cases: - Injected rf power different between two sections, which necessitates a different temperature for each section. - Rf power dissipated in the copper per unit length varying according to z, which necessitates different thermal exchange areas or different water flows according to z. This is one of the advantages of the constant field sections for which the losses in the copper per unit length (dP/dz) are constant. In high-current accelerators, if it is desired that the losses in the copper be kept constant, structures must be fashioned for which the field E, is constant for the nominal current I.

+

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- The cooling water, which generally arrives at one end of the section, is heated all

along the structure. One is led to calculate the flow for which the temperature rise remains small and to combine the input and output circuits to maintain as nearly as possible a constant temperature of the copper. For these reasons, the frequency-temperature compensation is never perfectly realized and the functioning point on the dispersion diagram moves from A(wo,To)

Fig. 5

to B(wo, T) (fig. 5). There results a modification of the phase velocity and thus of the phase, which, in the first order, is expressed (Brunet [1963a, b, 19651) A$ (z, T)

= wozzvAT,

where z, is the filling time of the structure at the abscissa z,

Through z,, A$(z) is a function of the chosen type of structure. It is obvious that this variation is important. For example, for a section such that: sec, v(copper)= 1.6 x A$=0.3AT, namely 17"1°C. w0=271x 3 x lo9, z,= This characteristic is used to measure precisely, by the difference in the phase of the wave between the input and the output, the rise in temperature of a structure. 2.2.2

InJluence of thejield and of the length

The field E, depends also on the chosen type of structure. For example, in the case of


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367

the constant gradient accelerator and in the x / 2 mode

Po does not depend on T, and neither does v, in first order for such a mode. E, only , is depends on T through ( r / ~ ) *that

It can be seen that in eq. (3) the dependence of the field on AT is negligible compared t o that produced by the phase. The same goes for the effect of T on the length, since AL/L= PAT.With these approximations, the relation expressed in (3) can be written:

V ( L , T ) = eE

1:

cos (o0vATz,) dz .

For the same section as that preceding, this relation shows that V ( T ) / V ( T o decreases ) by I, 3.8 and 15.5% for 1 , 2 and 4 "C resp. In conclusion, for a given structure, one calculates the relation (4). From the tolerance permitted in the energy, one deduces that for the heating of the walls of the structure. By the flow of cooling water and its distribution along the section, one strives to keep this heating independent ofthe abscissa z. For that, it is necessary to take account of the temperature gradients between the water at input and a t output, between the water and the copper, and across the copper. It is evidently advantageous to reduce t o the minimum the total temperature gradient between the water at input and the walls of the structure, in order to reduce the effects due to dispersion of power to be dissipated. For existing accelerators with a power t o be dissipated of the order of 2 kW/m, this temperature gradient can be maintained at less than 2 ° C . In such a case, the temperature control of the section becomes that of the water a t the input. With the apparatus schematically shown in fig. 6, which is regulated by the valve admitting cold water, a stability of k0.1 "C on the water a t input can be attained (Liot [1967]).

2.3

Vacuum

The vacuum in a linear electron accelerator must be very good for two reasons: t o reduce the beam scattering by molecules of residual gas, and to permit the propagation of intense electromagnetic fields.


2.3.1

Beamscattering

In order to maintain the angle of the beam scattering from the molecules of residual gas to a minimum at the end of an accelerator of several hundred meters' length, the pressure must be kept lower than torr. 2.3.2

Strong electromagneticjelds

The electrical field which can scarcely be exceeded is of the order of 15 MV/m. This strong field may extract electrons, which under certain conditions are accelerated and strike other surface areas, giving rise to liberation of adsorbed gas, which in turn may produce uncontrollable breakdowns (Bergkre and Veyssikre [1964]).

Cooling tower

Section

Section

Sect ion

Fig. 6. A typical cooling installation.

For lower values of the field, there may already exist a multipactor effect between two surfaces, and hence here again, outgassing, ionization and arcing. The instantaneous heat dissipation in the superficial layer due to the circulation of rf energy can also liberate gases and produce the same effects. Thus, it is necessary to maintain not only a good vacuum limit, but also a high pumpingspeed. However, the latter is necessarily low (< 3 l/sec.m) in an iris structure functioning in S-band. This low rate may lead to a limitation of the length of the sections. Certain expedients, however, may increase this limit. They consist in drilling small holes in the end of each cavity and introducing the stack of the cavities into a vacuum enclosure, from which is effected a differential pumping. The rate of pumping may thus be increased by a factor of 20.

H.2.4


369

The conduction of the rf power wave-guide between the klystron and the section is similarly poor in S-band (< 15 I/sec.m). For reasons of maintenance, the klystron is often installed in an accessible zone with beam on; the distance from klystron to section is then long and may necessitate a costly pumping system. For high peak or average rf power, the wave-guide will be pressurized. for example with SF,, providing the danger of a window failure toward the section be low. It is equally important to reduce the extent of outgassing due to rf, by baking under vacuum, sections and guides, before applying the rf power and by eliminating materials which are apt to outgas. The technique generally adopted is to assemble the various pieces with metallic seals or welded bellows. For radiation damage and high vacuum even the vacuum valves will be equipped with metallic seals. In choosing a pumping system, the use of impurities liable to outgas is avoided. For this reason, pumping by oil diffusion with cold-trapping is abandoned in favor of ionic pumping by titanium. Since the pumping speeds of titanium pumps reach their nominal values for pressures of the order of torr, the forepumping is generally effected with cryosorption pumps, or with a combination of rotary pumps and cold-trapped oil-diffusion pumps, or with a combination of primary pumps and turbomolecular pumps. The accelerator pipe is thereafter isolated from the forepumping system. With these precautions, the vacuum in the absence of rf is of the order of a few lo-' torr, and operating with rf and beam, it reaches about twice this pressure (Conviser [I 9651, Liot [I 9671).

2.4

Beam optics

After the injector, the optics of which has been examined in chs. B. 1.2 and B.2.1, the increase in transverse momentum (due to space-charge forces and to transverse components of the accelerator field) becomes negligible, while the electrons are ultrarelativistic. It has been seen that with an adequate vacuum, the beam scattering on molecules of residual gas is also negligible. In the chapter concerning beam blow-up by induced deflection modes, it was shown that this effect is reduced by proper focusing. In the case where this blow-up is not produced, the transverse momentum may be further modified by any defect of alignment or transverse stray field. To obtain a beam transport with a minimum of loss between the injector exit and the entry of the energy-analyzing system, it is therefore necessary to: - achieve and control a precise alignment of the accelerator; -- shield the accelerator and drift tubes against exterior stray fields; - compensate the residual defects of alignment or of shielding by a localized steering; - maintain the beam emittance inside the accelerator admittance by lenses of confinement.

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2.4.1 Alignment

Defects of alignment may be reduced to the two situations depicted in fig. 7a and b. The case shown in fig. 7a produces a reduction of the area available to the beam and the optical systems of revolution (lenses) centered on the geometrical axis are no longer centered on the axis of the beam. The case shown in fig. 7b will similarly reduce the area available to the beam in the absence of correction by a coil situated at the defect. However, one wants to limit the number of coils as the current necessary for the correction is a function of the beam energy at this point and the correction introduces a chromatic dispersion of the beam trajectories of which the angle is cc= e(AP/P), where APIP is the momentum dispersion of the beam at A. For these reasons one attempts to maintain everywhere the separation of geometrical and optical axes less than 1 mm.

Fig. 7

This necessitates a considerable mechanical rigidity of the accelerating structures. If by construction one succeeds in centering the cavities relative to each other with great precision (0.01 mm), the structure in copper so constituted would not be sufficiently rigid as soon as its length exceeded 1 m. Thus, one is obliged to support longer sections at several points upon a very rigid girder. Alignment control by the axis of the accelerator is not possible during its operation. Even at the time of its installation, an optical sighting across the irises along lengths greater than several dozen meters is equally imprecise because of diffraction, and does not permit the localization of defects. Therefore, the accelerator must be aligned with respect to a visual axis situated outside the structure. This axis will be interdependent with the rigid girder. In a preliminary stage, with classical optical techniques (theodolite), the axis of the irises of each section will be rendered parallel to an axis fixed relative to its rigid support. Afterward it remains only to align all these exterior axes. For accelerators of several hundred meters one may reach the required accuracy with optical techniques. It is then preferable to separate the operations of levelling (bubble level) and of alignment (theodolite). These sightings can even be effected in air, reducing the effects of distortion and of image translation by refraction in the atmosphere or by temperature gradients, placing enough fiducial marks to enable

B.2.4


371

sightings over short distances (several dozen meters) and by checking the consistency among all these sightings. For longer accelerators, the SLAC team has perfected a very precise method using a laser (Herrmannsfeldt [1965, 19661). Three-meter sections are mounted four at a time and aligned on a rigid support consisting of a 60-cm metallic hollow pipe, the axis of which serves as a reference for alignment. These girders are placed end to end and pumped to a vacuum of a few torr. From the high energy end a laser beam (cw, He-Ne, 6328 A) is directed along the axis of the tube. Inside of each support a removable Fresnel lens images the source on a detector located at the other end of the accelerator. These lenses are thus different from support to support. The detector used is a photomultiplier tube receiving the image of the source through a movable slit. The signal derived from the output of the phototube indicates the center of the image with a precision of +0.02 mm. The transverse position of the center of the image indicates the error in the position of the lens-target relative to its theoretical position constituted by the axis between the laser and the detector center. (The support jacks are adjusted to correct the alignment.) Experience has shown that finally the axis of the accelerator at SLAC can be aligned with a precision of +0.5 mm. 2.4.2

Shielding

Stray magnetic fields have different causes; they can be constant: earth magnetic field, effect of nearby magnetic materials (reinforced concrete, support, framework), dc power distribution lines; or they can be alternating or pulsed: ac power distribution lines, ground currents. Protection against stray magnetic fields is obtained by: - Magnetic shielding: the accelerator structure is enveloped in a tube of magnetic material. Fig. 8 shows the efficiency of different materials commonly used and the influence on its efficiency of the treatment undergone by the material (Borie et al. [I 9671). - Compensation: by means of coils situated in two mutually perpendicular planes symmetrically placed relative to the axis (fig. 9). These coils are fed by a fixed current able to create on the axis a field opposed to the average stray field. One uses them if necessary to increase the efficiency of magnetic shieldings, and in the place of magnetic shieldings when these are difficult to install, for example, at the exterior of solenoids used for focusing positrons. 2.4.3

Steering

Some steering dipoles in two mutually perpendicular planes should be provided locally to compensate faults occurring in the alignment of accelerator sections and of

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I

12

0.05

I

I

01

0.2

I

I

0.5 1 Field (gauss)

I

I

2

5

Fig. 8. Plot of magnetic shielding efficiency. Thickness (mm) -

1 2 3 4 5

--

Diameter of cylinder (mm)

-

Mumetal sheets Annealed mumetal sheets Brazed and annealed mumetal sheets Hypernom sheets Brazed soft iron

0.9 0.9 0.9 0.4 2

325 325 325 318 200

Fig. 9. Degaussing wires.

optical elements of revolution situated on the beam axis (lenses, doublets, triplets), as well as faults in the shielding and in the symmetry of the rf couplers at the input of the sections. Their adjustment is a function of the energy of the beam and is determined from beam position monitors.

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373

2.4.4 Confinement. Beam transport

In the hypervolume of the 6-dimensional phase space of a linear accelerator, one can separate the variables and consider the three two-dimensional spaces (p,, x), (p,, y), (p,, z ) with the following approximations: the particles are ultra-relativistic, their trajectories are paraxial and contained in a median plane, the optical elements may be reduced to thin quadrupole lenses. The emittance of the beam at the output of the injector for energies of 40 MeV has been measured at SLAC and has typical values in the planes (p,, x) and (p,, y) of 2 to 4 x lo-' (m,c/cm) depending on the intensity between 15 and 300 mA (Miller [1966]). The confinement of the beam consists in choosing the optics in such a way as to maintain this emittance within the admittance of the different elements of the accelerator (Ah-Hot and Buffet [1967]). The transfer matrices in the space Ix, dxldzl or 1 y, dyldzl relative to a drift space d, to a lens of convergence c, and to an accelerator section L are respectively

with

where m, and m, are the relativistic masses at the input and the output of the section (in an accelerating section, a < 1 and I< L). The determinant of the transfer matrix of an accelerator section is x , which is different from 1. Liouville's theorem does not apply therefore to the two-dimensional phase spaces (x, dxldz) or (y, dyldz), the area being reduced by cc as an accelerator section is traversed. If the accelerator has not to operate at full energy, one can benefit from reduction of a and of L by applying power to the first sections in preference to the last sections. Thus it is at the beginning of the accelerator near the output of the injector that the matching of the admittance of the beam to the admittance of the accelerator is the most delicate. Suppose that the emittance of the accelerator at the accelerating structure input is contained in the ellipse I of fig. 10. In the structure this ellipse is transformed progressively into the ellipses 2 and then 3. Without a lens, the ellipse tends toward the ellipse 4, leading to a beam loss outside of the limits imposed by &a, with a 6 R the radius of the iris. The lens ought to have a convergence sufficient to swing the ellipse

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3 such that its major axis has again a negative slope (ellipse 5). These lenses generally consist of quadrupoles grouped in doublets or triplets.

3 Adjustments and controls The last category of equipment remaining to be described is that used for control of the beam. The adjustments made on a linear accelerator consist in optimizing a large number of input parameters to obtain the two most important output parameters which are the energy V and the current I in an energy band A V centered at V. These two output parameters are not independent; they are related by a curve (dlld V) (V): the spectrum, characteristic of the accelerator adjustment.

I Fig. 10. Phase space along the linac in the xx' plane.

The qualities of a spectrum are 1) its sharpness which permits reduction of background outside of the useful energy band and increases the width and amplitude of the current pulse and accordingly the efficiency: Power transferred to the beam in the useful energy band Rf power and 2) its stability from pulse to pulse which determines the beam time available for experiments.

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375

For the latter it is interesting t o know the influence on the energy spectrum of each of the parameters to deduce their required stability, in order to provide the corresponding short-term automatic control and t o determine a method of adjustment based on relations between the different parameters.

3.1 Possible method of adjustment. Adjustment parameters

3.1.1 Fixed parameters Obviously it will be advantageous to fix prior to beam turn-on the maximum number of parameters common to all foreseen operating conditions. This is the case, for example, for the frequency and the temperature of the structure which are related to each other and which influence the energy spectrum. These will be fixed and stabilised at the nominal values of the synchronous operating regime, i.e., the values which give u+=c for the desired mode, in sections other than the injector. These values only need to be checked. If the temperature rise of the copper is not negligible, one may be forced t o modify the temperature of the cooling water according to the rf power and the injected current in order to maintain the structure a t constant temperature. The phase and amplitude stabilities ( A @ / @ and AP,,/P,,, respectively) at the input to each section including the injector are parameters which influence the spectrum; during the pulse they affect the width, from pulse to pulse they affect the stability. But these parameters are not easily modified during a pulse or between two pulses. Moreover, in certain cases as with hard-tube modulators, a feedback of the output P,, is possible. Generally it is sufficient to reduce this instability margin to a fixed value by such techniques as "deQing system" for modulator delay lines voltage and as stabilizing the heater and focusing power supplies for the klystrons. Within this margin one considers these parameters as random functions. The relative or absolute stability of each of the optical elements can similarly be determined theoretically or experimentally, independently of the operating conditions.

3.1.2 Parameters preadjustable depending on-desired beam characteristics

..The diverse possibilities of large accelerators have already been mentioned. T o this considerable flexibility of usage correspond different operating conditions, depending on the particular beam characteristics required. For example, a run can be characterized by: - a pulse repetition rate; - a beam-analyzing system for a given target area;

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- a number of sections filled with rf;

- a gun if several injection points are possible; - the type of accelerated particle, e- or e+ ; for e+ one may be required to select one

of several converters located at different places depending on the energy and current desired ; - an energy and analyzed current. Evidently for each run there exist: - A number of "binary" conditions: such a condition or function either exists or does not exist. Example: Power supplies on or off, valves open or closed, vacuum good or bad, cooling water flow sufficient or not, rooms insecure with beam on which are closed or not, etc. - A number of "analog" conditions which, once the foregoing conditions are met, will determine the values of the two output parameters. Examples: (1) The sum of the rf power to be delivered by the N klystrons. This sum is related to the current and to the energy at the maximum of the spectrum by the relation of "beam loading", ( a ~ , $ -RI); a and R are known characteristics of the structure. (2) The ensemble of the optics of confinement, steering, and energy analysis. Between two optical elements it is necessary to know the transfer matrix in which enters the input energy and the energy gain. These elements are also sensitive to the width of the spectrum due to chromatic aberrations. The numerical values to be introduced into the transfer matrices of steering are not well known theoretically, since they take into account defects of alignment or revolution and of field irregularities. They will be determined experimentally. If the input couplers on the sections create deflecting fields, the steering becomes also a function of the phase 4 between the electron and the wave; it is then difficult to preadjust without beam.

V=C~

3.1.3 Parameters adjustable with beam - The current I,, at the output of the injector. - The extension in phase 6 of the bunches at the output of the injector. - The optimum phase 4 for each section between the rf wave and the beam

(4 = 0 in synchronous operating regime). For the large accelerators, the difficulty of regulating 4 according to the energy V at the output has led to automatic phasing such as that described in ch. B.2.3. - The relative value of the energy drop due to transient beam loading depending on the accelerated intensity. It is known that this effect can be reduced by certain tricks, such as injection before the end of the rf filling time; the timing of the current relative to the rf is thus a function of the accelerated current. - According to the precision of the preadjustments, an improvement with beam of the steering and of the rf power of the last element used may be required.

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377

3.2 The various detectors The measurement of the many above-mentioned parameters necessitates the use of appropriate detectors. The detectors measure "classical" physical quantities: - Currents and voltages, - Magnetic fields, - Frequency, - Temperature and heating, - Water flow and pressure, - Vacuum, - Rf peak power, peak amplitude, phase comparison, as well as quantities more peculiar to the beam. These detectors are abundantly described in the literature, and the laws covering their operation are indicated in ch. B. 1.5. We recall briefly here several beam-allied detectors among those most frequently used. 3.2.1 Intensity-destructive measurements Faraday cup. The Faraday cup is designed to measure absolutely the intensity by a

total interception of the beam. It must be thick enough to collect all the charge. For example, to obtain a precision of 0.1% on the collected current, at least 42 radiation lengths are necessary at 500 MeV, and 46X, at 20 GeV (Isabelle [1963]). Absorbing all the power of the beam, it must nevertheless remain sufficiently cool and totally insulated from the exterior. Secondary emission monitor (S.E.M.). This is constituted of a set of thin screens, generally of aluminum foil, interposed into the beam line. These foils are connected alternately to two different potentials, in order to collect the secondary electrons created by beam-produced ionization in the screens. The order of magnitude of the secondary current collected between two aluminum foils is 2.5% of the primary current traversing them. Its calibration depends on the incident energy of the beam, on the state of the foil surfaces, and on the vacuum. It gives a relative meas~~rement and must be standardized by comparison with a Faraday cup. 3.2.2

Non-destructive methods associated with the beam's electromagnetic j?eld,for intensity measurement

Ferrite toroids. These toroids are made of magnetic materials with a maximum frequency response of the order of 10 Mc/s. Thanks to a secondary winding, they function as current transformer by detection of the magnetic energy of the beam pulse. A second winding connected t o a known power source permits verification of calibration. These are used as detectors of peak or average current. Rf'cavities. These cavities, of the same type as klystron cavities, tuned to t h e

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fundamental frequency of the accelerator, are the rf detectors most frequently used among all those which detect one of the fields radiated by electron bunches. The rf signal extracted by a loop from the cavity is sent to a detector fashioned of a crystal or a diode. A signal, function of the peak current, is thus obtained by a law dependent on the law of detection.

3.2.3 Measurement of the beam phase The above rf cavities furnish also a reference of phase related to the central phase of the electron bunches. They may be used by comparison with the rf wave phase feeding the accelerator structure to realize an automatic phasing (Brunet [1963]).

3.2.4 Measurement of the extension in phase Accepting for this measurement a modification of the adjustments, this measurement can be made by varying the central phase of the bunches relative to the accelerator wave so as to compare the corresponding variations of the energy spectrum. One may also at the level of injection introduce a deflecting structure at the frequency of the accelerator, and study, on a screen or after a collimator, the deflection of the beam which is a function of the central bunch phase and of its phase extension (Chehab [1967]). A non-destructive apparatus based on the study of the frequency spectrum of the beam permits the measurement of the distribution of the current according to the phase in the electron bunch (Berghe [1967]).

3.2.5

Measurement of the position of the beam

Ionization chamber. To detect whether the beam is striking the walls of the accelerator, one may use a procedure which consists of a long coaxial ionization chamber parallel to the accelerator and situated near it. The propagation time on this line of a pulse produced by the ionization permits the localization of the steering defect or of the beam blow-up (Panofsky [1963]). Rf cavity in a TM,,, mode. In such a rectangular cavity tuned to the accelerator frequency, the electrical field induced by the beam is zero if the barycenter of the charges is at the center. It varies in amplitude and in phase according to the position of this barycenter compared with the transversal axis of the cavity, and according to the intensity of the beam. Comparing this rf signal with that of a rf intensitymeasuring cavity, it is possible to find the sign and the magnitude of the deviation of the beam axis with respect to the center of the cavity. The two cavities may be placed in two perpendicular planes and thus the deviation in x and in y may be observed (Brunet and Williams 119641, Neal [1965, 19661).

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379

3.2.6 Measurement of /he rransverse dimensions and of the emirtance

The transverse dimension of the beam may be found by destructive procedures, such as the interposition of fluorescent screens examined by television cameras or by collimators of variable dimensions followed by intensity detectors. The study, by microdensitometer, of the darkening of glass plates, exposed for a limited time to the beam, gives the density of distribution in the transverse plane. In the same manner a collimator or a slit, of small dimensions with respect to the transverse dimensions of the beam, and situated a t a distance r from the center, will permit, by examination of glass plates situated after a drift space, the measurement of phase space in the plane (r, r ' ) by varying r .

3.2.7

Energy analysis of the beam

This analysis is generally obtained by magnetic systems. A first magnet disperses the beam in momentum in a plane P and thus in energy for ultrarelativistic particles. An analyzing slit, situated at the focal distance and perpendicular to the plane P, blocks all but the desired energy band fA V/Vocentered at energy Vo.Other magnets situated after the slit "clean" the beam selected by the slit and with appropriate optics focus this beam in the target area. A large number of extraction system types have been achieved (Panofsky [1961]). By introducing in the focal plane a set of n S.E.M.'s, each one yielding a current proportional to the number of electrons situated in an energy range AV/Vo, it is possible to obtain a reproduction of the energy spectrum of the beam.

3.3 Data processing From acquaintance with measurement and detectors, it is perceived that information can be classified under the following 4 shapes: 3.3.1

Binary information

This information is the easiest to process either by an operator (indicator light on or off), or by a specialized logic or a computer. Consequently, it is advantageous to extract as much information as possible in this form. Thus, it is frequently important t o know only if a parameter has or has not crossed a given threshold (e.g. vacuum, flow), or if it is contained between two thresholds, or if it is within a tolerated range of stabilization (e.g. rates of regulations of current, frequency, temperature). By transmitting t o the control room the only binary information resulting from the comparison with respect to the threshold or to the range of stabilization, one may

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L.BURNOD

.,


achieve an appropriate servo-control locally. This is often less onerous, because of the large distances, than the transmission of all the analog information necessary t a establish (by sampling) a central servo-control for several identical systems. Likewise, when a failure is produced on a sub-system and necessitates a repair a t the site, it is useful to have locally an indication of the principal faults, and it suffices to collect in the control room the binary information corresponding to all the conditions necessary to the functioning of this sub-system (e.g., modulators). These data may correspond to conditions or functions indispensable for a given run (for example: open valve in the beam trajectory). In this case the data must be regrouped in adequate interlock systems. They function as safety and readout devices. They may, in other cases, operate simply to facilitate a run (for example: a confinement lens may be turned off in case of defect, giving a slight loss of beam). Here these data serve only as readout information. They may also serve as remote control for all functions which are binary: opening of a valve, positioning of a screen or radiator, power supply turn-on. These data proceed from different electrical circuits each having its own protection. Also the inputs for these data in the logic processing circuits are themselves generally of very high impedance: Electromechanical relays, or some solid-state circuits such as the electro-optical circuits (photo-diodes associated with photoemitting diodes).

3.3.2

Data varying slowly relative to the pulse length

These data may correspond : - to the slow drifts of parameters: frequency, temperature, drifts of current. In this case it has been clearly seen to be advantageous to transform them as much as possible into binary information. - to analog values of adjustment appropriate to a run which have already been defined. These permit one either to verify that a parameter has been adjusted to the desired value, or to make the adjustment by remote control from a given element, say a potentiometer. These data emerge generally in the form of a voltage from different detectors. It is profitable to digitalize these data in order to 1) increase the stability of the readout and facilitate its presentation: digital voltmeter, 2) increase the stability of remote control by a binary coded command (preferable to an analog voltage because of the level of background and distortions), 3) process these data by a computer.

3.3.3 Pulsed data at video frequency These data correspond to the parameters whose evolution it is interesting to follow during the pulse or from pulse to pulse. They are relative:

13.2.4

INSTALLATIONS rAND CONTROL

381

to the pulsed amplifiers: trigger pulses, charge pulses from the modulator delay lines, current and voltage pulses applied to the klystron cathodes, pulses of rf peak power at the outputs of the klystrons and of the sections, pulses of rf peak power reflected toward the klystron by a possible mismatch of the section; - and to the beam: peak intensity, along the accelerator and after energy analysis, position, distribution in energy (spectrum). Knowing the difficulty to establish a feedback during the pulse duration of the order of a microsecond, the pulsed video data are particularly utilized during the pulse qualitatively or statistically: detection of breakdown, of spurious oscillations, localization of the causes of reflection or rf power in a section, etc. On the other hand, one may consider the entire pulse or even a fraction of the pulse determined by a gate circuit and examine either its integral (for example, the quantity of electricity contained in the beam) or its peak value (for example, the level of power reflected toward the klystron). Thus, all the time between two pulses (a few ms) is used to process this information: for example, to compare it with a critical threshold and operate a safety device. One may equally well, by increasing the integration time-constant over several pulses, transform the pulsed information into slow data, in order to know the slow drifts of beam intensity, spectrum, of rf power. For this purpose it is more economical to effect the integration locally in order to avoid the transfer of the information by coaxial cable.

-

3.3.4

Rj'data

These are used when one wishes to examine a bunch of electrons: phase extension, position, comparison of phase between the bunch and the wave or between two waves. These data are processed directly by the shortest possible hyperfrequency circuits: cavities, comparators, then transformed by detection into pulsed signals, themselves in turn processed as described previously.

3.3.5 Possibilities,for use of a computer From the above method for adjusting a linac and classifying information coming from it, one can see a number of opportunities for the use of a computer in controlling a linac (Johnston [1965], Mallory [1966]; see also ch. (2.2.5). Adjusting the lir~ac Readout and recording of binary data, and analog data after digitizing. Updating and storage of a library of operating data as a function of output parameters : energy, current. Calculation and recording of the parameters resulting from the relationship between several input parameters.

382

L.BURNOD


Establishment of laws of variation or of correlation between little-known parameters (for example, study of statistical variations of amplitude jitter). Observation of the parameters of a run with respect to a model run, or to threshold and levels of adjustment, with recording of the deviations and the tendencies of the parameters. Preadjustment of a run by comparison with a model run: Address binary or analog commands to the various functions, with control of the execution and interlock between the different conditions. Assistance in overall operation Assistance in repair by diagnostic research with the help of a test logic sequence stored in the memory. Assistance in preventive maintenance by the recording of the very slow evolution of some parameters (for example, klystron impedance). Knowledge of the rate of time during which the beam is running, determination of the minimum time for maintenance, development, or adjustment; determination of the minimum costs of exploitation and maintenance according to the lifetimes of the elements in different operating standards. Once these various programs have been tested successfully, it is possible to foresee the control in real time of a linac, which appears to be the ultimate phase of control.

References AH HOT, L. M. and X. BUFFET,1967, Etude de la Focalisation d'Electrons Ultra-relativistes dans un Accc5lCrateur LinCaire, L.A.L. 1178, Laboratoire de 1'Accelerateur Linkaire, Orsay (France). AMMAN, F., 1967, Private Communication, Laboratori Nazionali di Frascati, Frascati (Rome), Italy. AVERILL, R. J., 1967, Private Communication, Cambridge Electron Accelerator Laboratory, Harvard University, Cambridge, Massachusetts (U.S.A.). AZAM,G., 1967, Private Communication about "1'Accelerateur Lintaire de Saclay" - CSF - Corbeville - Orsay (France) and "I'Accelerateur LinCaire de Mayence" (Germany). BERGERE, R. and A. VEYSSIERE, 1964, Electrons Emis par Effet de Champs, Nucl. Instr. and Methods 30, 309. BERGERE, R., 1967, Etude des Proprietts du Faisceau d'Electrons d'un Accel6rateur Lineaire au Moyen des Champs Electromagnetiques Associts a ce Faisceau, C.E.A.R. 3126, Saclay (France). BLANC-LAPIERRE, A. and L. BURNOD,1967, Private Communication, Laboratoire de 1'Acctlerateur Lineaire, Orsay (France). BORIE,H., R. CHEHAB and M. PLAGES,1967, Private Communication, Laboratoire de l'Acc6lerateur Lineaire, Orsay (France). BRUNET, P., 1963a, Etude et Realisation d'un Systeme de Reglage Automatique des Phases Hyperfrequences sur un Acctlerateur Lintaire, L.A.L. 31, Laboratoire de l'Acc618rateur LinCaire, Orsay (France). BRUNET, P., 1963b, ProprittCs de la Phase de I'Onde Acctl6ratrice par Rapport aux Electrons, 1'Onde Electrique, No. 435, p. 678. BRUNET, P., 1965, First Order Non-Synchronous Operation of an Electron Linac, Proceedings of the Fifth International Conference on High Energy Accelerators, CNEN, Rome, p. 595. BRUNET, P., and C. B. WILLIAMS,1964, Microwave Beam Position Monitors, SLAC-TN 64-45, Stanford, California (U.S.A.).

B.2.4

INSTALLATIONS ARD CONTROL

383

CHEHAB, R.. 1967, Mesure de ['Extension en Phase des Paquets d'Electrons, L.A.L. 1180, Laboratoire de I'Accelerateur Lineaire, Orsay (France). C ~ N V I S ES. RR , . , 1965, The SLAC Vacuum System, SLAC-PUB 90, SLAC, Stanford, California (U.S.A.). G ., 1967, Private Communication, Laboratoirede 1'Acci.lerateur Lineaire, Orsay (France). DARDENNE, I ~ D R I C K ., G . , and H . H . CLARK,1962, Elementary Calculations of the Transverse Shielding, M . Report 296, SLAC, Stanford, California (U.S.A.). I X STAEBLER, H., 1961, A Review of Transverse Shielding Requirements for the SLAC, M. Report No. 262, SLAC, Stanford, California (U.S.A.). IIE STAEBLER, H., 1962, Transverse Radiation Shielding for the SLAC, Report No. 9, SLAC, Stanford, California (U.S.A.). I X STAEBLER, H., 1963, Simple Shielding Calculations, Internal Memorandum, T N 63-68, SLAC, Stanford, California (U.S.A.). W. B., 1965, Linac Alignment Technique, IEEE Particle Accelerator Conference, HERRMANNSFELDT, Washington, D.C. (U.S.A.) p. 9. HERRMANNSFELDT, W. B., 1966, SLAC Alignment System, Proceedings of the 1966 Linear Accelerator Conference, USAEC, Los Alamos (U.S.A.) p. 163. D. B., 1963, Beam Current Monitors for a High Energy Linac. Report No. 25, SLAC, ISABELLE, Stanford, California (U.S.A.). JOHNSTON, L. H., 1965, Instrumentation and Controls for the SLAC, IEEE Particle Acceleratcr Conference, Washington, D.C. (U.S.A.) p. 44. KKUSE, C. and G. LOEW,1967, Private Communication, Stanford Linear Accelerator Center, Stanford, California (U.S.A.). LEISS,J., 1967, Private Communication, National Bureau of Standards, Washington. D.C. (U.S.A.). LIOT,R., 1967, Private Conlmunication, Laboratoire de I'Accelerateur LinCaire, Orsay (France). MALLORY, K. B., 1966, Some Effects of (not Having) Computer Control for the SLAC, Proceedings of the 1966 Linear Accelerator Conference USAEC, Los Alamos (U.S.A.) p. 479. MILLER,R . H., 1966, Measurements of the SLAC Injector, Proceedings of the 1966 Linear Accelerator Conference, USAEC, Los Alamos (U.S.A.) p. 65. NEAL,R. B., 1965, Status of Design at SLAC, Fifth International Conference on High Energy Accelerators, Frascati (Rome) Italy. NEAL,R. B., 1966, Initial Operating Experience and Performance on the SLAC Accelerator, Proceedings of the 1966 Linear Accelerator Conference, USAEC, Los Alamos (U.S.A.) p. 210. PANOFSKY, W. H., 1961, Linear Electron Accelerator Progress at Stanford University, Proceedings of the International Conference on High Energy Accelerators, USAEC, Brookhaven (U.S.A.) p. 79. W. H., 1963, Progress Report on the Stanford Two-Mile Linear Accelerator, Proceedings PANOFSKY, of the International Conference on High Energy Accelerators, Dubna (U.S.S.R.) p. 407. VALTER, A. K., 1965, 2 GeV Traveling Wave Electron Accelerator, Fifth International Conference on High Energy Accelerators, Frascati (Rome) Italy, p. 233. WILSON,P. B., 1967, Private Communication, High Energy Physics Laboratory, Stanford California (U.S.A.).

ELA VARIOUS TYPES

3.1

1

B.3

Use of High Energy Electron Linear Accelerators * J. BALLAM Stanford Linear Accelerator Center Stanford, California, U.S.A.

I

Introduction

There are four operating high energy electron linear accelerators in existence. Two of these are at Stanford University, Stanford, California, one operating at 1.1 GeV at the High Energy Physics Laboratory (HEPL) (for a design description see Chodorow et al. [1955]), and the other at 20 GeV at the Stanford Linear Accelerator Center (for a detailed description, see Neal [1968]). The remaining two are at the Laboratoire de l'Acck1Crate~1rLinkaire at Orsay (see L'Onde Electrique [1963]) operating at 1.7 GeV, and at the Ukrainian Physical-Technical Institute, Kharkov, at 2 GeV (see Val'ter et al. [1963]). There are no high energy machines in design or construction at the present time.** Since the 1 to 2 GeV machines have been in operation for a long time (the HEPL accelerator has been used for physics experiments since 1952) and both theaccelerators and their application in high energy physics have been well documented, only a brief qualitative description of their program shall be given here and most of the chapter will deal with the SLAC program. The era of high energy physics with electron linear accelerators began with the operation of the halfway station of the Stanford machine in 1952. The very first series of experiments published in 1953 were a precursor of a long list of those to come as predicted by Schiff [I9491 and begun by Hofstadter et al. [1953]. These accelerators have in the past and still are being used for three main purposes:

* Work supported by the U.S. Atomic Energy Commission. ** See note added in proof, p. 413. 0 North-HollanrlPubl. Co., Amsterdam 1969

Referenrcs p. 412

385

386

J.BALLAM

ELECTRON LINACS: VARIOUS TYPES

a. Electron-nucleon scattering. b. Photon-nucleon interactions. c. Electron-nucleus scattering. More recently the Orsay machine has been used as an injector to an electronpositron storage ring. Initially it was thought that checks on the validity of quantum electrodynamics would be one of the main contributions of these accelerators, but the subsequent discovery of the complexity of the nucleon structure has favored the use of other techniques for this purpose such as the measurement of the g-factor of the muon, electron-electron scattering in colliding beams and wide angle electron-positron pair production. Thus it has turned out that their main contribution has been the study of the so-called form factors for the charge andcurrent distributions of the proton, neutron and of the various atomic nuclei (for an excellent summary see Hofstadter [1957]).

2 Form factors Since the concept of form factor as used in high energy electron physics is common to many experiments and the results are often interpreted in terms of this parameter, a brief description and definition will be given here. For an advanced treatment the reader is referred to Bjorken and Drell [1965]. If the particle to be studied is conceived to have a charge located at a point in space then the scattering of an electron from it is commonly described by the formula

where E,, is the incident electron energy, 13 is the scattering angle in the laboratory system, Ze is the charge of the scattering center, and M is its rest energy (the spin is assumed to be zero). This formula is commonly referred to as the "Mott-scattering formula". It treats relativistic incident particles and is therefore applicable to electron-nucleon scattering. For low energy massive incoming particles such as helium nuclei the familiar Rutherford formula is the direct analog of eq. (1). Since the scattering centers do indeed have structure, as determined by experiment (Hofstadter 119571) eq. (1) has to be multiplied by a factor which reduces the scattering probability as some function of the momentum transferred to the proton by the incoming electron. This multiplier is called the form factor since its value does depend on the charge and current distribution in the scattering center. The equation which best describes the present experimental situation is the Rosenbluth [I9501 formula

B.3.1

USE OF ELECTRON LINACS

387

where q 2 / 4 ~ is 2 the square of the normalized invariant 4-momentum transfer to the proton in (GeVlc)', G, is the electron form factor of the nucleon, GM is the magnetic form factor of the nucleon, 0 is the laboratory scattering angle, M is the mass of the nucleon in GeV; and with the implied assumption that only one photon is exchanged during the scattering process. The central problem in electron-nucleon scattering has been to measure the dependence of G, and G, on q 2 for both proton and neutron. A check which is always made as t o the validity of eq. (2) is to plot

for fixed q2 and see if it is in fact a straight line.

3 Review of medium energy physics achievements The main result of the work on elastic scattering was to show that the measured cross sections fall off very rapidly as a function of q2 (Hofstadter [1957]). The data is consistent with the form

where A is a normalization parameter. AZ is < 1 (GeVIc)'. For practical purposes this limits the measurement of elastic scattering for ~ . Janssens electrons from the 1 GeV class of accelerator to q2 < 1.2 ( G ~ V / C )(See et al. [I9651 and Lehmann et al. [1962].) Since the investigation of elastic scattering is now essentially complete, much of the current work at the medium energy machines is in the field of nuclear physics because many high lying nucleon states of low transition probability can only be reached by the high intensity afforded by medium energy electron linear accelerators. The medium energy machines have also achieved some significant results in the fields of photoproduction of single n mesons, inelastic electron scattering, as well as threshold behavior in the photoproduction of the meson (Prepost et al. [1965]), a narrow meson resonance whose mass is 550 MeV. In summary the work of the medium energy accelerator has accomplished the following : a. Elastic scattering from protons. Cross sections measured to accuracy of about 3% up to qZ of 1.2 (GeV/c)'. b. Quasi-elastic scattering from neutrons. Cross sections measured up to q2 of 1.36 (GeV/c)' with about the same accuracy. c. Study of single photoproduction of 71' and no from threshold through the first resonance including measurements of the polarization of the recoil proton.

388

J.BALLAM


d. Study of the charge distribution of most of the atomic nuclei. e. Study of individual nucleonic states and the properties of the collective model of the nucleus (review articles were written by Isabelle [I9671 and Walecka [I 9671 and [l966]). f. Photoproduction of the low-lying meson and nucleon resonances.

4 Potential of very high energy linacs In order to extend the work on elastic and inelastic scattering to higher momentum transfers and to provide the means for doing extensive photoproduction experiments involving strange particles and multibody final states the SLAC machine at 20 GeV was conceived. The principles of its design have been described elsewhere in this book and in Neal [1968]. Even though the justification given above was sufficient, it has turned out that such a machine is much richer in possibilities than was originally envisaged. As a matter of fact further study revealed the following potentialities: a. High intensity positron beams. b. Muon beams derived from a localized source. c. Beams of high energy neutral kayons accompanied by a relatively low neutron background. d. Secondary particle beams of strongly interacting particles (pions and kayons). These have opened up new areas of particle physics research, which have heretofore been the exclusive province of the proton machines, while at the same time providing the electron and photon beams needed to extend the fundamental work begun at lower energies. One can now say that the high energy electron linear accelerator can be used for extensive studies of electromagnetic, strong and weak interactions - a very rare combination indeed.

5 Positron acceleration and annihilation beams Positron acceleration was first achieved at the Stanford 1 GeV machine (see Pine and Yount 119621) by placing a converter followed by a solenoid about half way down the accelerator. Electrons at 350 MeV were put on the converter and low energy positrons (3 to 15 MeV) were focussed into the waveguide which was then reverse phased to accelerate positrons. About 0.1% of the electron current was thus converted into positrons of energy 300 MeV. This experience led to an elaborate development program at SLAC. The general approach is the same as used previously. The converter and source assembly was placed one-third of the way down the machine. This choice was somewhat arbitrary and was partially determined by the amount of power the converter could handle without physical disintegration. The second consideration was that pulse-to-pulse operation of electrons down the full length of the machine


389

/71

SPECIAL QUADRUFQLE MULTIPLETS AT TAPERED SPACING

SOLENOID

TO BEAM

0SOLENOID ("25111

\

STANDARD SECTOR DOUBl PUAORUWLES I SECTORS

POSITRON RADIATOR I" 8 5 f l POINT IN SECTOR I I I

Fig. 1. Schematic of magnetic optics and cross section of wand target for the positron source. The low energy positrons are adiabatically captured by the combination of tapered and uniform solenoid magnets surrounding the converter. Then quadrupoles and triplets are spaced along the length of the accelerator to provide focussing for the positrons which have a larger initial transverse momentum than d o electrons. The wand target is shown as a n alternate for the wheel (as described in the text) and allows alternate pulses of positrons and electrons to be accelerated depending o n whether or not the electron beam is allowed to hit the wand.

390

ELECTRON LINACS : VARIOUS TYPES

J.BALLAM

followed by positrons demanded that the electron beam be of high momentum in order to minimize the effect of the positron focussing solenoids on the electrons. Furthermore, enough out-of-phase electrons were present during the positron pulse so that an rf deflector had to be built in order to eliminate them (see ch. B.3.5). The resulting positron source system is shown in fig. 1. The converter is a 7.6 cm diameter wheel of copper rotating at 200 rpm and capable of absorbing an average beam power of 100 kW. The solenoids calculated by Helm [I9621 and designed by H. Brechna provide an adiabatically damped axial field which captures positrons of energies between 5 and 15 MeV over an angular range of 0.03-0.2 radians. The capture efficiency, expressed as the amount of captured positrons divided by the number of incident electrons, is 5% at the converter, and this reduces to 2% at the end of the accelerator. The wand shown in fig. 1 is an oscillatory converter which drops through the electron beam at frequencies like once or twice per second and is used to provide positrons for bubble chamber operation. A photograph of the wheel and its operating mechanism is shown in fig. 2. The calculation and main design parameters for the source were mainly done by H. DeStaebler and J. Pine (see Pine [1963]).

aE

DEFLECTOR LIN!FORM FIELD SOLENOID

\

ACTUATOR FOR INSERTING ZnS SCREEN INTO BEAM LINE

SPECIAL QUADRUPOLE TRIPLET S-1

FAST VALVE

No 4

Fig. 2. Positron source installed in the accelerator. This is a view somewhat downstream from the converter, showing the uniform solenoid, rf deflector and the first quadrupole triplet. The fast vacuum valve will close in several milliseconds to protect the main accelerator sections from vacuum failures in the positron source.

B.3.1


391

The highest currents so far achieved have been 0.6 microamperes (360 pps x 10" positrons per pulse) average current at 12 GeV measured at the end of the accelerator. The energy spread is about 2% and the current at the experiment depends on the energy precision desired. These positron currents are as large as electron currents achieved in the I to 2 GeV accelerators. Once the positron beam was established the next step was to establish a monochromatic photon beam. This is generated by observing the photons from the annihilation-in-flight of the positrons by electrons in a liquid hydrogen target into two photons. If the angle of emission of one of the photons is 0, its energy k , and the energy of the positrons E,, then

where m is the rest energy of the electron. Backgrounds to this process come from

However, the intensity of these backgrounds decreases with angle much faster than does the monochromatic beam, and it turns out that as long as the angle of observation corresponds to k / E ,

where 6, and 6, are the input and asymptotic phase positions, respectively, for a single particle of velocity Doc injected into a up= c waveguide having a longitudinal electric field fundamental space harmonic amplitude of E,. With an injection phase angle corresponding to the zero field position (6, =0) and an asymptotic phase spread of either one degree or two degrees, this equation indicates that the permissible spread of energy (AVO)and velocity (A&) at injection must be limited as shown in table 8. For example, to obtain one degree asymptotic phase spreads when using injection energies of 100, 200 or 400 keV and optimum values of E,, the energy and velocity spreads at injection must be limited to 6.0, 8.8, and 13.4 keV and 1.22, 0.9 and 0.55 percent of c, respectively. It should be noted that these values are obtained from a simple binding field theory which does not include the effects of cavity space harmonics, entry fringe field or the presence of charge.

J. HAIMSON

460


TABLE 8 Permissible energy and velocity spread versus asymptotic phase spread * for injection at 60 = 0 into up = c waveguide -

-

-

-

Vo

Eo

(keV)

(kV/cm)

-

-

-

-

-

-

-

Energy spread A V O (keV)

---for A s a = 1" for Ada = 2

Velocity spread Abo

-

*

for A6a

=IO

for A s a

-

2"

This does not include phase spread effects due to space harmonics, fringe field or presence of charge.

It has been shown (Haimson [1966a]), for 2n/3 waveguide, that space harmonics cause broadening of both the bunch and the energy spectrum and, for a phase velocity of light circuit, this broadening increases as the injection energy is reduced. It has also been shown that if bunches are to be located ultimately at the crest of the travelling wave, then injection into a 1 coupler cavity has to occur during a decelerating phase which causes the injected bunch to be first retarded and then accelerated prior to arrival at the mid-plane of the cavity. This interaction takes place in a field environment similar to that of a standing wave, i.e., in the presence of two equal sets of opposing space harmonics; and the full complement of space harmonics constituting the main travelling wave is not encountered until further penetration into the circuit has occurred.

a,=

4'

-a 2

0

2 ,,

SINGLE PARTICLES INJECTION AT 8, = OD INTO vp = c WAVEGUIDE

A

- NOT MODIFIED

0

-

MOOlFlED FOR INPUT COUPLER SPACE HARMONICS

0

5 TOLERABLE

10 15 INJECTION ENERGY

20

SPREAD I N keV TO GIVE As,= l o

Fig. 14. Effect of space harmonics and fringing field on permissible injection velocity spread, for 2n/3, up = c input couplers, to achieve an asymptotic narrow phase spread.

B.3.2

HIGH DUTY FACTOR ELECTRON LINACS

461

Injection analyses using up = c accelerator waveguides and taking into a c c o ~ ~the nt spatial harmonic content of the input coupler, as well as the entry fringe field effect, indicate that for a given emergent phase spread, the tolerable values of energy spread at injection are considerably lower than those given by the above binding field formula. Some of these lower values, as modified by space harmonic analyses, are shown in brackets in table 8. The modified values of permissible injection energy spread decrease with reduced gun potential, as before, but in accordance with a much narrower range of permissible velocity modulation. A comparison of the tolerable AVOand A j , values for a fixed injection phase angle and a range of gun potentials, to give a one degree emergent phase spread, is shown in fig. 14. Curve A refers to values obtained from the binding field equation and curve B to the res~lltsof analyses which take into account the above mentioned input coupler effects. This comparison presents a rather pessimistic outlook for narrow phase width performance at moderate currents and low gun potentials because the velocity spread at injection must be very low, and it provides a possible explanation of the common observance that a small change of prebuncher drive affects beam energy spectra far more than predicted by the simple binding field theory. The table 8 and fig. 14 data refer to phase spreads between single particles injected at the same phase but with different energies while the fig. 15 curves show typical emergent phase angles (6) for a 7 MeV buncher waveguide, for a range of entry phase angles (4,) at the input coupler fringe field, and for three closely related injection energies. These data also relate to single particle injection and include the effects of input coupler space harmonics. (With the phase convention used, the maximum accelerating field is established at the center of the input coupler cavity at a time corresponding to 4 = -90°, and entry phase decreases with time, i.e., d 4 / d t = - 2n/j,, where p,c is the electron velocity and t=z//Z,.) Single particle phase plots of this nature are not directly applicable, however, to the phase orbital behaviour of individual particles in a cohesive bunch. Electron bunching analyses which take into account the effects of beam loading, such as reactive phase shift (see 52.5.3) and space charge, indicate that the phase orbital characteristics of intra-bunch particles are unlike those of the fig. 15 ballistic derived curves. As an example, the broken curve superimposed on the fig. 15 data shows the emergent phase width for the hypothetical case of a 25 mA, 10 degree bunch injected with uniform charge density at an energy of 400 key. The tilting effect and broadening of the emergent phase spread are mainly due to the influence of space charge in the initial portion of the circuit, and the slight advance in bunch position is due to reactive phase shift effects. (The last two thirds of the buncher waveguide is designed as a velocity of light structure with an equilibrium orbit at approximately 6 = - 65 degrees at Po = 2.5 MW.) For short bunches, these phase characteristics appear to be affected most by phase location of the centroid, phase width at injection, total charge and beam diameter. (Under some circumstances, a highly compressed bunch with a narrow injection phase spread can result in a broadening of the emergent phase angle.) Although each beam loading condition has

462

J. HAIMSON

ELECTRON LINACS: VARIOUS TYPES CORRECTED FOR INPUT COUPLER SPACE HARMONICS

L

'

A+.= loo

'

p;

0.95+ 1

2 m m B E A M DIAM.

j7

PHASE ANGLE AT ENTRY TO INPUT COUPLER FRINGE FIELD (+:

)

Fig. 15. Phase orbit characteristics, corrected for input coupler space harmonics, for an unloaded buncher waveguide with an initial phase velocity of u,/c = 0.95.

to be analyzed separately, in general, the achievement of a narrow emergent phase spread depends on not only satisfying the stringent A& injection requirements discussed above, but also on maintaining the bunch in a sufficiently high field gradient to counteract the effects of space charge (see 32.5.4). The above conclusions, pessimistic as they appeared for the case of an emergent 2 degree bunch, formed the basis for specifying an injection bunch width between 5 and 10 degrees, for the buncher waveguide, with a velocity modulation content of not greater than 5 0 . 4 percent of c ; and this led to the selection of a high energy chopperprebuncher injection system.

6.2 Selection of gun potential and drift space parameters

Since the biased chopper-prebuncher principle has been presented in earlier reports and discussed in connection with a variety of previously constructed machines (Haimson [1962, 1965c, 1966b]),it will be assumed that the reader is familiar with this


B.3.2

463

means of obtaining a well-defined rf bunch of adjustable phase width. When 120 degrees of beam length per rf cycle (P0A0/3)is correctly phased to traverse the prebuncher gap during field reversal, the electrons will be subjected to velocity modulating fields which closely approximate the ideal saw-tooth environment required for linear bunch compression in the subsequently located drift space in the absence of space charge. (For biased-chopper operation, this results in a transmission efficiency of approximately 20 percent.) If, for the present, we disregard the effects of radial and longitudinal space charge fields, then the total dBo and AVOthat must be introduced by the prebuncher-gap, for bunch compression from 120 to 10 degrees over a given drift space length S, can be determined from the simple time-of-flight expression (42 - 40) = 41 + 2~S[(PolBm)- ll/(Po&) where 4, and ( 4 2 -40) are the departure and normalized arrival phases, respectively, and p0c and P,c are the velocities of the unmodulated and modulated electrons, respectively. [The electron energy (V, in keV), as obtained for a given prebuncher gap condition and departure phase, can be Furthermore, by imposing a related to its velocity by (V,/5l 1) + 1= (1 -P:)-?] restriction based on practical requirements, namely, that for variations of gun potential of k0.1 percent the phase modulation at the end of the drift space shall be no greater than 1.5 degrees, the maximum permissible drift length will be defined by the gun potential. This relationship is developed in 42.5.1 and, for the above voltage variation, (y - 1) ( y 2 - l)-*/AO, where A 4 , is the permay be written as A@, = 2nS x missible phase modulation at the end of the drift space, and y is defined by the energy of the source electrons. The fig. 16 curves show the relationship of S, A & and AVO,as

+

0

200 300 400 500 BEAM ENERGY V, IN keV

100

600

Fig. 16. Prebuncher drift space parameters for 12 to 1 bunch compression, in the absence of space charge.

464

J.HAIMSON


obtained from the above equations, for a range of beam energies up to 500 keV. For systems operating at low beam energies, these graphs indicate that quite small values of energy spread (AVO 1.5 percent). As the gun potential increases, however, the longer drift lengths and larger energy spreads are accompanied by reduced values of velocity spread. If now space charge effects are taken into consideration, it is evident that bunch terminal field strengths as low as 1 kV/cm, acting over a few centimeters, will, for the lower beam energies, seriously perturb the bunching process. To determine the importance of space charge effects in counteracting the velocity modulation produced by the prebuncher, we can consider the commonly adopted drift space configuration in which the beam waist is located at the prebuncher, and after velocity modulation, the bunch is allowed to expand freely before passing through a thin lens assembly and refocusing through a small aperture into the buncher waveguide at the end of the drift space. Thus, toward the end of the drift space, the bunch dimensions are reduced both in the longitudinal and radial direction. For low beam energies, and moderate currents, to overcome the high space charge forces generated by this volume compression, it is essential that the drift space lengths shown in fig. 16 be drastically reduced and the prebuncher drive increased. At higher beam energies, not only are the space charge forces reduced, but the higher values of AVO required for bunching can more readily dominate the space charge potentials. It should be noted that, for high duty factor machines, lower values of perveance allow the beam to be expanded over a larger proportion of the drift length prior to refocusing (see fig. 18), i.e., the radial compression occurs over a proportionally shorter length of drift space. For a given operational condition, to assess space charge de-bunching, and prior to considering the results of detailed computations, a reasonable estimate of gross effects can be made by selecting a suitable bunch model and determining the maximum value of longitudinal field strength. For example, fig. 17b shows the relationship between maximum longitudinal field and bunch length for a constant diameter bunch and several values of beam energy. A spheroidal bunch model of non-uniform charge density, shown in fig. 17a, was chosen as a good approximation of practical conditions as determined by profilometer scans and rf bunch chopping and transverse sweeping measurements. The fig. 17b curves indicate that, with a 2 mm diameter, 100 keV beam, any attempt to drift a 25 mA, 5-10 degree bunch over a distance of several centimeters would result in larger de-bunching potentials than the prebuncher A Vovalues of fig. 16. Of course, the prebuncher drive could be increased and the drift space foreshortened as is common with higher current injection systems (see for example, Haimson [1962]); but under these conditions, the correct compensation of space charge forces becomes a very sensitive function of beam current and lens settings, and prevention of longitudinal phase orbit crossovers, an important requirement for optimum bunching in subsequently located waveguide, cannot be assured. On the other hand, with a 400 keV beam of the same diameter, the fig. 17b data


B.3.2

465

indicate that bunching down to 5 or 10 degrees over relatively long distances would result in space charge potentials much lower than the AVO prebunching value shown for this higher injection energy in fig. 16. Under these conditions, because of the dominance of the prebuncher drive, accurate compensation of space charge fields no longer plays an important role in obtaining a small bunch width, and it can be ex-

BUNCH D I A M = 2 m m I =25rnA

h

0

2

4 6 8 10 12 14 16 18 TOTAL BUNCH LENGTH IN DEGREES

20

Fig. 17. (a) Electron bunch model of non-uniform charge density and spheroidal geometry. (b)iDependence of space charge maximum longitudinal field on bunch length and beam energy.

pected that high voltage systems of this type will be relatively insensitive to different settings of beam current. Furthermore, because of the low perveance and linear compression characteristics, reliance can be placed on the reproducibility of a small but useful reduction, by the space charge fields, of the A p , initially introduced by the prebuncher.

466

J.HAIMSON


6.3 Overall configuration and results of detailed analysis A schematic layout of the injection system is shown in fig. 18. Detailed calculations of radial and longitudinal phase space in the drift length region, including space charge effects, were conducted for a range of parameters; and some bunching results corresponding to the fig. 18 positions marked A, B and C are reproduced in fig. 19. These data refer to a 25 mA, 400 keV bunched beam with an initial geometry and nonuniform charge density distribution as shown in fig. 17a, an initial and final diameter of 2 mm, an initial biased-chopped bunch length of 2.9 cm (120" of poA0 at 2856 MHz), prebuncher peak field strength of 18.5 kV/cm, and a 850 gauss lens 91 cm

a

4 0 0 kV ELECTRON GUN

RF CHOPPER AND DC BIAS MAGNET

PREBUNCHER AND COLLlMArlNG

I

.R..F INPUT

ASSEMBLY

INJECTOR SCHEMATIC LAYOUT

Fig. 18. Schematic layout of high beam energy, chopper-prebuncher injection system.

from the prebuncher. The bunch was divided into a multiplicity of cylindrical and annular elements, the boundaries and centroids of which can be seen in the fig. 19 illustrations. This technique (see for example, Crandall [I9661 and Haimson and Mecklenburg [1967]) assumes that rotational symmetry is conserved about the beam centerline and that the charge density distribution in the azimuthal direction remains constant. The fig. 19 results show that, after 132 cm of drift, a bunch length of 118 degrees at the prebuncher departure plane is reduced to 9 degrees, and the total AVO is reduced correspondingly from 29.1 to 18.5 keV, i.e., an initial total Ap0 of 1.2 percent is reduced to 0.8 percent of c. At this location, while the regular pattern of annuli has


467

468

J.HAIMSON


been disturbed due to non-linearities, the maximum space charge longitudinal field is less than 0.5 kV/cm. Bunch compression continues with increasing drift distance, and computer results indicate that at 140 cm of drift the total phase width is reduced to 5 degrees, and because of increasing space charge fields, the total AVO and A/?, are decreased to 12 keV and 0.5 percent of c, respectively. These results suggested that, at injection into the buncher waveguide, the narrow phase width and velocity spread

CORRECTED FOR INPUT COUPLER SPACE HARMONICS

INJECTION

Fig. 20. Buncher waveguide phase orbit plot for injection conditions as obtained from the drift space computations and for the rf conditions listed in fig. 15.

requirements specified in $6.1 could be satisfied by this concept and, in fact, that care should be taken to avoid over compression of the bunch phase width, especially at high duty cycle, because of an incompatibly low field gradient in the buncher waveguide. In summary, the fig. 20 phase orbit data shows the computed phase spread emergent from the buncher waveguide (see position D, fig. 18) for an injected electron bunch of non-uniform charge density and velocity distribution, as obtained from the


469

References ALLEN, W. D. and J. L. SYMONDS, 1946, Proc. Phys. Soc. (London) 59, 622. 0. H., E. V. FARINHOLT, Z. D. FARKAS, W. B. HERRMANNSFELDT, H. A. HQGG, ALTENMUELLER, R. F. KOONTZ,C. J. KRUSE, G. A. LOEWand R. H. MILLER,1966, Beam breakup experiments at SLAC, Proc. 1966 Linear Accelerator Conference, Los Alamos, LA Report 3609, p. 267. AYERS,W. R., E. L. C H Uand W. J. GALLAGHER, 1957, Measurements of interaction impedance in periodic circuits, Internal Memorandum, W. W. Hansen Laboratories of Physics, Stanford University, Stanford, M.L. Report 403. BEAN,B. R., 1962, Proc. I R E 50, 260. R. and A. VEYSSIERE, 1964, Nucl. Instr. and Meth. 30, 309. BERGERE, HERTOZZI, W., J. HAIMSON, C. P. SARGENTand W. TURCKINETZ, 1967, IEEE Trans. Nucl. Sci. NS-14, no. 3, 191. and B. M. BAILEY, 1967, A study of removal of RF-generated heat from BROOKS,F. P., J. M. DAVIN rectangular waveguides in the MIT-400-MeV linac using one- and two-pass cooling arrangements, Internal Report, LNS-Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts. BURNOD, L., 1963, L'Onde Electrique, L'Accelerateur Lineaire d'Orsay, XLIII no. 435, p. 688. CHODOROW, M., E. L. GINZTON, W. W. HANSEN,R. L. KYHL,R. B. NEALand W. K. H. PANOFSKY, 1955, Rev. Sci. Instr. 26, no. 2, 134. K. R., 1966, A numerical experiment o n space-charge effects, Proc. 1966 Linear AcCRANDALL, celerator Conference, Los Alamos, LA Report 3609, p. 233. I ~ E M OP. S ,T., A. F. KIP and J. C. SLATER,1952, J. Appl. Phys. 23, no. 1, 53. FARKAS, Z. D., C. J. KRUSE,G . A. LOEWand R. A. MCCONNELL, 1967, IEEE Trans. Nucl. Sci. NS-14, no. 3, 223. GINZTON, E. L., 1957, Microwave Measurements (McGraw-Hill, New York) p. 435. J., 1962, IRE Trans. Nucl. Sci. NS-9, no. 2, 32. HAIMSON, J., 1965a, IEEE Trans. Nucl. Sci. NS-12, no. 3, 996. HAIMSON, J., 1965b, Nucl. Instr. and Meth. 33, 93. HAIMSON, HAIMSON, J., 1965c, IEEE Trans. Nucl. Sci. NS-12, no. 3, 499. J., 1966a, Nucl. Instr. and Meth. 39, 13. HAIMSON, J., 1966b, Optimization criteria for standing wave transverse magnetic deflection cavities, HAIMSON, Proc. 1966 Linear Accelerator Conference, Los Alamos, LA Report 3609, p. 303. J. and I. BRODIE,1963, Nature 199, no. 4895, 795. HAIMSON, HAIMSON, J. and L. KATZ,1965, IEEE Trans. Nucl. Sci. NS-12, no. 3, 457. J . and B. MECKLENBURG, 1967, IEEE Trans. Nucl. Sci. NS-14, no. 3, 586. HAIMSON, HELM,R., 1966, Computer study of wave propagation, beam loading and beam blowup in the SLAC accelerator, Proc. 1966 Linear Accelerator Conference, Los Alamos, LA Report 3609, p. 254. JEAN,R., 1963, L'Onde Electrique, L'Accelerateur LinCaire d'Orsay, XLIlI no. 435, p. 623. C. R. EMIGH,L. N. ENGEL,J. M. POTTERand W. J. SHLAER,1966, KNAPP,E. A., P. W. ALLISON, Accelerating structure research at Los Alamos, Proc. 1966 Linear Accelerator Conference, Los Alamos, LA Report 3609, p. 83. H., G . AZAM,R. BENSUSSAN, G . BLAISE,M. DELMON, J . P. MANGINand J . SOFFER,1966, LEBOUTET, Somecharacteristic problems of the New High Duty Cycle Saclay Electron Linac, Proc. 1966 Linear Accelerator Conference, Los Alamos, LA Report 3609, p. 426. LEISS,J. E., 1965, IEEE Trans. Nucl. Sci. NS-12, nJ. 3, 566. LOEW,G. A,, 1967, IEEE Trans. Nucl. Sci. NS-14, no. 3, 529. R. H., J. BERKand T. 0. MCKINNEY,1967, IEEE Trans. Nucl. Sci. NS-14, no. 3, 98. MILLER, MIT RADIATION LABORATORY SERIESNO. 11, 1947, Technique of Microwave Measurements (ed. McGraw-Hill, New York) p. 391. C. G. MONTGOMERY; NAKAMURA, M., 1968, Japan. J. Appl. Phys. 7, no. 2, 146. NEAL,R. B., 1958, Theory of the constant gradient linear electron accelerator, W. W. Hansen Laboratories of Physics, Stanford University, Stanford, M.L. Report 513. NEAL,R. B. and W. K. H. PANOFSKY, 1966, Science 152, 1353.

470

J.HAIMSON

ELECTRON LINACS: VARIOUS TYPE5

ROBSON, P. N., 1956, Fourier series representations of dispersion curves for circular corrugated waveguide used in travelling-wave linear accelerators, Report no. 5105, Metropolitan-Vickers Electrical Company, Ltd., Manchester, England. SLAC-32, 1964, Two-mile accelerator project, Quarterly Status Report, 1 April to 30 June 1964, Stanford Linear Accelerator Center, Stanford University, Stanford. J. C., 1950, Microwave Electronics (D. Van Nostrand Company, Inc., New York) p. 80. SLATER, SOKOLOVSKY, A., 1966, The new high duty cycle high current 600 MeV electron-positron linear accelerator of Saclay, Proc. 1966 Linear Accelerator Conference, Los Alamos, LA Report 3609, p. 417. C. W., 1966, IEEE Trans. Microwave Theory and Techniques, MTT-14, no. 2, 70. STEELE, J., 1966, Microwave phase dependence of the accelerator sections and the waveguide WEAVER, network, TN-66-6, Stanford Linear Accelerator Center, Stanford University, Stanford. WEAVER, J. and R. ALVAREZ,1966, IEEE Trans. Microwave Theory and Techniques, MTT-14, no. 12, 623.

ELA VARIOUS TYPES

3.3

1

B.3

Low Energy, High Intensity Electron Accelerators H. LEBOUTET Centre de Recherches Thonz~on-CSF Corbeville, 91 - Orsay, France

1

Introduction

This chapter deals with "small" electron linacs, covering the energy range from approximately 3 to 40 MeV, which are, or can be used, for industrial applications. In the low energy region, they compete with dc generators. The maximum energy depends on the special use of each machine: below 10 MeV if neutron production has to be avoided, or in the 20-40 MeV range if penetration capability of X-rays is a major requirement. These accelerators are not different in principle or in technology from high energy accelerators. They ordinarily use only one high power rf generator, magnetron or klystron. They are "compact", and in some cases, modulator and accelerator are housed in the same enclosure, which can be moved, or displaced, and one of the problems is finding the best mechanical arrangement for a given use. The accelerating waveguide is made of one single buncher section, for machines up to 10 MeV, or two sections for higher energy or easier energy control. Peak currents in the 1 to 5 amperes range do not create special new problems compared to higher energy "physics" accelerators. Specific problems however, are met in designing windows and targets, because of the high average current transported by those low energy, high average power beams.

2 Radiation chemistry As sources of high energy particles, electron linacs are competitive with other conReferences p . 493

North-Holland Puhl. C o . , Amsterdam 1969

47 1

472

H. LEBOUTET


ventional radiation sources, i.e. radioelements or nuclear reactors, or lower energy dc accelerators. They can provide a much more powerful beam than radioelements and are easier to use than a reactor. They have attained now a degree of industrial development which makes them versatile and reliable. Their main uses are in radiography, radiotherapy and radiochemistry.

2.1 Production of radioisotopes Accelerators can play an important role in the techniques of preparation of radioisotopes, essentially because they can deliver the same elements as those produced by a nuclear reactor, and in addition other radioelements which cannot be created by other means, because the threshold energy is above the neutron energy in a reactor. The reactions generally involved are (yp) and (yn) from bremsstrahlung, or neutron capture, if the accelerator is made as a neutron generator. According to published data (McGregor [1958]), the maximum of the cross sections lies between 12 and 25 MeV depending on atomic mass (fig. 1). Hence an accelerator of 25 to 30 MeV is well suited for such a use. However, no accelerator has

Fig. 1. Choice of parameters for different uses.

B.3.3

LOW ENERGY, HIGH INTENSITY ELECTRON LINACS

473

been built especially for this purpose, but this is envisaged as an important by-product of the new generation of medical accelerators in this energy range. Other types of accelerators, like small cyclotrons, accelerating protons, deuterons, and helium-3 ions in the 15-30 MeV range, look actually a very promising tool for this production, and may in the future be a cheaper source of radioisotopes than electron linacs.

2.2 Chemistry Compared to conventional chemistry, radiation chemistry can be characterized by the fact that radiation gives a new way to produce reacting species, the reactions themselves obeying classical rules. Consequently, chemistry under radiation has a certain versatility due to its capacity to produce reacting species and use them in different environments, and in a very wide domain of pressures and temperatures. The interest arises because one can obtain, by this mass reaction, products whose formation energy is too great to be obtained by conventional methods in usable quantities. Radiation chemistry uses essentially electron ionization and excitation effects due to the passage of charged particles through matter. One gets: single ionization: AB+AB+ + e dissociation and ionization: AB-t A + +BO+e molecular excitation: AB+AB* The products (positive ions, free radicals, excited molecules) interact and recombine and give excited molecules. The chemical reactions induced by the radiation are in fact those which take place in these post-reactions. Order of magnitude of the time involved : (1) Mechanical effect: The energy of the radiation is dissipated by displacement, excitation, ionization of molecules: lo-' sec. (2) Evolution of primaries: 10-12 sec. (3) Chemical species and reactions: IO-'sec. The main chemical effects of radiation are: (1) Franck-Rabinovitch effect (cracking). The rupture of a chemical link at the center of a big molecule gives two free radicals, which are not mobile enough to be much displaced, and recombine later. But if one of the radicals is small, it can be displaced and not recombined, and the long chain of carbon hybrids can give directly light components, methane, propane or butane. One can, therefore, obtain the cracking at normal pressure and temperature without any catalytic agent. For example, the polysterene molecule is particularly stable and requires 2 keV to be cut, but the polyethylene molecule requires only 50 eV. (2) Degradation. At higher doses all polymers are degraded, by rupture of chain. Teflon and polyvinyl are the most sensitive.

474

H. LEBOUTET


(3) Reticulation and polymerization occur as post-reactions. (4) Biological effects. Biological effects of the radiations, adequately dosed, can be used to sterilize certain products.

2.3 Choice of the source and nature of the radiations The foregoing considerations concerning radiation chemistry are, generally speaking, available for sterilization; however, one may consider some other factors: (1) Concerning the cost of irradiation, conditions are less critical, on account of the interest involved in the process. (2) The risk of contamination by neutrons of the products and parts of the equipment (permanently exposed) prohibits the use of nuclear reactors or some radioactive sources, and limits the energy level of the primary particles. (3) Safety of operation of the machine is an important factor (a breakdown in the installation may have serious consequences especially when perishable products are involved). Sources meeting these requirements are essentially some radioactive sources and electrons accelerators. The electron accelerator has the advantage that it can be switched off when necessary and presents no danger when it is shut down. Furthermore, it gives a very good control of the beam, both in position and intensity, and, consequently, easier utilization. Efficiency of the use of all the available power of an accelerator beam into the product can be as high as 80 or 90%, whereas a radioactive source cannot reach more than 5 to 15 percent. The dose needed is also a decisive factor to favor the choice of the electron accelerator. Radioactive sources reaching megacuries are difficult to manipulate, particularly in sterilization devices, where, to be competitive, they should be simple and flexible. The electron beam from an accelerator may be used directly, or can be transformed into X-rays in case of sterilization of the surface, to avoid undesirable reactions in the core of the material. In that case, electrons or X-rays of low energy will be utilized.

2.4 Doses required Table 1 (Swallow [1960]) gives typical doses necessary for various treatments. In certain cases, doses required to destroy undesirable organisms may be reduced by combining radiation effects with other effects (heat or antibiotics for example). Applications are : (1) Sterilization under radiation applied to drugs and medical goods. Sterilization under radiation presents some advantages compared to classical sterilization by heat.


B.3.3

475

TABLE1 Doses requ~redfor treatment of food and drugs (Swallow [1960]) -

Treatment -

-

-

-

Dose (rads) -

- --

--

--

--

-

-

-

-

Complete elimination of micro-organisms; requirement for safe commercial use Combination of irradiation with other methods of processing Extension of food storage life under refrigeration Elimination of insects, Trichinella spiralis, etc. Prevention of sprouting of potatoes --

--

-

--

-

I Megarad corresponds to a n absorbed energy of 10 joules per gram of product.

Lethal effects occur at low temperature; this allows to treat substances such as antibiotics that would not stand heating. An appreciable thickness of material may be treated inside containers made of plastic, glass or metal. The process can be continuous, inexpensive, and made with a minimum of operations. (2) Food preservation. Food sterilization obtained by heating leads to a change in the taste of the food to be preserved; also, vitamins are partially destroyed. The same effects occur with sterilization under radiation but are different from those observed with conventional methods. In addition to a change of taste, probably due

Fig. 2a. Scheme of a n industrial irradiation plant.

476

H.LEBOUTET


to oxidation of the surface, a change in color can be observed. For example, sugar becomes pinkish. By modifying irradiation conditions, or by adding small quantities of chemicals, these inconvenients can be minimized. In certain cases these defects even disappear when irradiation is made below 0°C. Irradiation of cereals and vegetables improves notably their preservation. Gamma-rays slow down multiplication of the insects when doses between 15000 to 20000 rads are applied. A dose

Fig. 2b. Irradiation plant at Orsay (France).

of 64000 rads will kill the insects, avoiding grains to be attacked by weevils. Preservation of potatoes is notably improved after irradiation by 5000 to 20000 rads. Growing of the sprouts is stopped, external aspect is improved while the taste is not modified. After 18 months, the loss of weight is only 20% for irradiated instead of 55% for non-irradiated potatoes. Although sterilization of food by irradiation is still in the experimental stage, experimental plants exist in several countries (fig. 2a).

B.3.3


477

Since it is essential that this operation be made a t energies lower than the neutron threshold, "low energy" accelerators (Van de Graaff type, or insulated transformers, or dc generators) which can deliver large amounts of beam power, seem to be cheaper than travelling-wave type accelerators. But their energy is practically limited to 3 or 4 MeV, and the penetration depth of the electrons is in some instances slightly too small to penetrate thick targets. The 6-10 MeV level is presently covered only by linacs, with powers in the range of 5 to 20 kW. Figure 2b represents an irradiation plant installed at Orsay (France). Principal features: Pulse duration: 3 ps Energy: 5 to 10 MeV Repetition rate: 700 Current: up to 1 mA Width of beam sweep: 60 cm Length of accelerator: 1.8 m Frequency of sweep magnet: 5 Hz Rf power: 8 M W

2.5 Energy cost: factor G Let us assume that an average of 25 eV is required to break a chemical bond. In order t o modify a grammolecule, 25 x 6 x eV are needed (2.4 x lo6 joules). 1 Mrad is 10 joules per gram, so for a chemical species of molecular mass equal to 100, 1 Mrad will modify a fraction 0.042% of the molecules. The efficiency o f a reaction is characterized by its factor G, which is the number of transformations (molecules of free radicals) for 100 eV absorbed by the system. For absolute measurement of radiation dose, decomposition of S0,Fe with a G value of 15.9 is sometimes used. Table 2 gives typical values of G for some reactions. When the factor G is known for a given process, the cost of the operation can be evaluated from the overall cost of the high energy radiation and from the efficiency of the use of the radiation into the reacting elements. Up to the present time radiation chemistry has not been competitive with conventional means, except in very few instances, for example reticulation of polyethylene. Reticulated polyethylene is heat resistant (it transforms into a plastic material similar to neoprene above 120°C) and has a high breakdown voltage. Irradiation of standard electric cables with polyethylene insulation increases the voltage and temperature breakdown limits. Possible interest are irradiation of plastic-impregnated wood, which can open a whole set of new products of presently unknown characteristics, modification of semiconductors, production of non-bioresistant detergents, new resins, paints, etc.

2.6 Technical arrangement Problem of sweeping the beam: The actual machines have a rather wide energy spectrum compared to more sophisticated accelerators used for physics experiments,

478

H. LEBOUTET


Typical values for some reactions -.

Reactions

Value of G

Decomposition of hydrocarbons: formation of peroxide formation of carbonyl compounds formation of acids Decomposition of benzene : into phenol into aldehyde Polymerization of benzene: liquid (electrons) gaseous (rays) Decomposition of acetic acid Decomposition of acetic acid in presence of OZ Decomposition of ethylic alcohol of ethylic alcohol in presence of Oz Formation of CsH5C.Cl3 from CCh $ CsHs Formation of aniline from NH3 CsHs Chlorination of aromatic hydrocarbons Dehydrogenase: alcoholic phosphoglyceraldehyde carboxypeptidase aminoacid oxidase

+

because general considerations on the cost of the machine do not allow very elaborate systems of prebunching and also the waveguide is used in the region of maximum power transfer from rf to beam. Output electrons can lie within 5 to 10% total energy bandwidth. If any means of deviation is used, some beam spread is obtained in the outer regions and the beam density is lower; this effect has to be corrected by special magnetic systems, or complex sweep generators. Problem of belt displacement: Since the energy stored in the sweeping magnet is generally too great to allow a fast return and one way sweeping (like in television receivers) a compromise between spot dimension, sweeping frequency f, trigger frequency of the accelerator F, and belt velocity v must be made to insure a uniform dose over the useful width (see fig. 3). Number of spots per line: E/2f, Condition of lateral overlapping: FD/2f % L, Condition of longitudinal overlapping: fD % u. Typical values are: Beam repetition rate F: 500 to 1000 Hz, Sweep frequency f: 5 to 10 Hz, Belt velocity v: 3 to 10 meterlminute.

8.3.3

LOW ENERGY, HIGH INTENSITY ELECTRON LZNACS

479

Problem of the window: The large output window through which the beam leaves the vacuum vessel is a delicate part of the system. It must stand a high overall dose, and resist to increased chemical action from ozone, oxygen and nitrogen compounds formed in the atmosphere by the beam itself. It must also be capable of absorbing and evacuating the heat generated by the beam. The output power density is then limited essentially by the possibility of heat dissipation in the target or through the window.

Fig. 3. Sweeping of the beam.

Let us assume (fig. 4) that a beam of radius r , falls on a plate of thickness x, (ordinarily x, is short compared to the range of the electrons). The power dissipation in the target increases from the entering face and then decreases, but in this low energy region the build-up effect is not very important, and for this computation made for

Fig. 4. X-ray production.

comparison purposes, we will assume that the energy is released uniformly within the volume of the cylinder of radius r, and length x,. The beam comes as short bursts, a few microseconds long, with a repetition rate of several hundreds per second. One

480

H. LEBOUTET


gets an order of magnitude of the heat transfer by the solution of the heat equation:

(C is the volume heat capacity; K is the heat conductivity). A one-dimensional solution for a step function applied to the region is AT(x, t ) = 1 - erf [x (c/K)* (~cz)-*] which shows that the heat front appears half height at a distance 0.03 mm after 10 microseconds in copper. It can be assumed that all the power dissipated by the beam during one pulse is absorbed locally by the target material, and the instantaneous increase in temperature depends only on the specific heat of the target or window. The energy loss is about 2 MeV/gram cm2 for all materials. The instantaneous i ncrease in temperature is then: AT=

-

Q 2 x 1 0 ~ ~ x , r n 2 x I O ~ joules . rnCrrrix, n r i jouleslg ~ N

The capacity of absorbing big bursts of charges is then proportional to the product of fusion temperature and specific heat. Table 3 gives the value of this parameter for a certain number of metals. Obviously the best metal is beryllium, and far behind, tungsten and molybdenum. However, the low Z of beryllium makes it a very poor gamma converter and difficulty of machining prevents its use as window material. This does not take into account heat dissipation by radiation which is significant only at very high temperatures, and applicable to tungsten, molybdenum and tantalum; this increases the instantaneous heat capacity of those materials, but in fact the great mechanical stress which would accompany the local high temperatures will cause a mechanical fatigue, which will limit the life of the window or target, so that an important safety factor has to be taken for all applications to insure long operation time. Steady state: The most common geometries correspond to a sheet of metal cooled either around the beam impact by radial heat flow, or across it, with one side cooled by a fluid: water, mercury or sodium (fig. 5). In this case, the heat production is still proportional to the density of the material, and the factor of merit will be: 9=

heat conductivity x fusion temperature density

It can be seen from table 3 that beryllium is still by far the best material. Then come aluminum and copper, then tungsten and silver. Platinum is not better than lead. Stainless steel is very poor; since most parts of vacuum envelopes and pipes are made of stainless steel, great care must be taken against any possibility of a powerful beam striking the walls; water cooled copper linings are ordinarily used. In practice, currents up to 200 to 400 pA can be handled on a target or window,

Metal

Specific heat

Density (Jig)

(J/cm7)

Heat conductivity (J/cm "C)

Fusion temperature Tf ("C)

Factor of merit for separate pulses TrC

Factor of merit for steady state -

Copper Tungsten Molybdenum Beryllium Titanium Aluminum Silver Platinum Lead Tantalum Gold Mercury (boiling) Stainless steel

500 350 380 1030 168 530 370 59 58 100 I60 16 30

"Half value heat front" for I0 ps (10-"m) --

3.5 2.8 2.4 2 1.34 3 4 1.6 2.7

S < m

Z

0 **

2

3

?!2 H

482

H. LEBOUTET


WATER COOLING

=T

Fig. 5. Target.

with beam dimension of the order of 1 mm2. Above that value, it is necessary to use more sophisticated systems, like rotating targets, or fast moving liquid metal targets.

3 Radiography Non-destructive testing of metallic objects of great thickness is made in general by ultrasonic testing or X-rays produced by radioisotopes. However these techniques are not sufficiently accurate for very thick samples, and use of linacs has been widely spread these last ten years. In this field, they are in competition with betatrons, over which they have the superiority of much higher output at a lower energy. Two conditions have to be met simultaneously: a very small source and a good contrast. It is also interesting to have a wide field to reduce the number of exposures. The dimension of the photon source depends on the dimension of the electron beam, of the penetration distance, and the electron diffusion in the target. The exposure time depends on the intensity of the source, the thickness of the sample, the distance to the source, the energy spectrum of the radiations, the nature of reinforcing

B.3.3


0

'

5

10

15

20

25

30

MeV

Fig. 6a. Intensity of radiation on the axis per watt of beam.

4

I o

l

RELATIVE DOSE

I 2 00

I

I

1

-

PO0

600

6'

Fig. 6b. X-ray intensity vs. angle.

483

484

H. LEBOUTET


screens, and the quality of the photographic plate. Depending on radiation energy and photographic plate, the impression of the photographic image requires between to joules/cm2. Brightness of the source: The photon yield depends on the energy of the primary beam. Fig. 6a shows the intensity of radiation on the axis, per watt of beam, for a gold target of optimum thickness, compiled from published data (McGregor [1958]). Table 3 shows that tungsten is preferable to gold as target material for high power beam, but X-ray yield is lower because of the lower 2. Fig. 6b shows the angular dependence of X-ray intensity. If the flux intensity has to be made uniform on a wide area by a filter, it can be seen that a thicker target might improve the radiation, at wide angle, at the same time as the central flux is attenuated. However no published data on thick target wide angle emission confirms these theoretical results. Furthermore these curves are only approximate, the shape of the radiation lobe can be modified substantially, by the geometry of the electron beam; for example, a strong convergence adds to the width of the lobe, but decreases the dose on the axis. Exposure time: The sensitivity of the emulsions to photons of a few MeV is low, and reinforcing screens are necessary to reduce exposure time. These screens are high Z thin sheets, placed very close to the emulsion. The Compton or photoelectric electrons or pairs of lower energy have a much higher ionizing coefficient than the primary photons, and increase the action of the emulsion. Radiographic unsharpness: Main contributions to unsharpness are: (I) inherent unsharpness of the film associated with a reinforcing screen, (2) source dimension, (3) scattering of radiation in the material under test. No complete investigation on the magnitude of points 1 and 3 has been made. Fig. 7a gives published values of increase of film unsharpness with the energy of the primary electrons. The effect of source dimension is purely geometrical (fig. 7b). The way the three imperfections combine to give the total unsharpness is difficult to define accurately. Intuitively one source can be neglected ifits effect is much smaller than the others. Halmshaw (quoted by Cusick and Haimson [1961]) indicates that they can be neglected if they are smaller than one half the largest. Therefore, in ordinary applications of high energy sources to radiography, where film and diffusion unsharpness may be 1 mm or more, no improvements are to be expected from very small focal spots, and the practical value lies between I and 3 mm, lower limit depending on target cooling capability. Examples: Several constructors have built this type of accelerator. They all work at S band and 10 MeV, except a model at X band (3 cm) 6 MeV, made by Vickers (England). The increase in shunt impedance obtained at 3 cm wavelength permits a very short structure, of small dimensions, which makes a light and easy to handle apparatus, well adapted to the use in a factory. The rf source is a 3 to 5 MW magnetron, or a klystron amplifier. Since the magnetron requires a lower voltage and has a higher efficiency than the klystron amplifier, and needs no driver, the modulator can be more compact and lighter. But its life is substantially shorter, and there exist


B.3.3

rnrn

1.5

A

FILM SCREEN UNSHARPNESS

I

I

! i - 1

Fig. 7a. Film-screen unsharpness vs. energy (Cusick and Haimson [1961]).

1

I

1

PROJECTED IMAGE

Fig. 7b. Film unsharpness vs. source size (UF corresponds to S=O).

485

486

H. LEBOUTET


the problems of replacement, and fine frequency tuning. All these models can be moved on wheels, or attached to a crane to irradiate in various directions. Fig. 8 shows one possible arrangement.

4 Flash radiography. Stroboscopic radiography For the study of very fast moving parts, inaccessible during experiment, such as detonating explosives, explosive driven shock waves, extreme states of matter, evolution of jets, interaction of jets with jets, cratering, spalling, some special types of machines have been built, which can deliver a succession of very short pulses, 30 to 100 ns long, with a distance between pulses of a fraction of a microsecond. Each

I

A

/iIll;ii ~1011

X-ray beam

Fig. 8. Rotational movements of V 7706 Linac.

pulse carries several joules of energy, and delivers an intense photon beam. The sequence of pulses can give a stroboscopic photograph of the fast moving phenomena under study. The characteristics of the required gamma ray impose the values of E and q (energy and charge) to be accelerated in each pulse. Typical values range from one to several hundred joules, at energies between 10 and 50 MeV. Assuming an optical efficiency ratio y for the accelerator and beam transport system, each pulse of the beam removes from the rf an energy Eqr] joule. Without entering here in the controversy on the maximum fraction of stored energy a pulsed beam can remove from a line or resonator, if an appreciable uniformity of E and q from pulse to pulse is required, only two possibilities remain:


B.3.3

487

either the accelerator can be refilled between two pulses, and each pulse can take all the available energy, or each pulse removes only a very small fraction of the total stored power (fig. 9). The stored energy per unit length in an iris-loaded structure of shunt impedance r is:

r/Q increases as 1 /Iw for homothetic cavities, W , is then proportional to A2 for the same value of the gradient, but the arcing limit has also to be introduced. (If the approximate law VE=constant was supposed to remain valid in the frequency range, then the product E ~ / Z would be independent of I every time the arcing limit is obtained, but the energy density will still increase as 3,. In fact, for rf voltages and large transit

t

BEAM

A

STORED

CURRENT

LOW FREOUENCY

r f ENERGY

5 BAND TIME ps 8

,

10

Fig. 9. 50 MHz

f5

20

25

accelerator rf pulses time pattern.

angles, this law is-no longer valid. Kilpatrick's criterion is a better estimate, which comes approximately to E= constant for practical situations.) A typical value for W, at S band would be 2 joules/meter, at 140 kV/cm. The maximum charge which can be practically accelerated is thus a fraction of lo-' coulomb per pulse in that frequency band. Filling time: If a "steady state" is achieved after the first few pulses, the power removed by the beam at each pulse is refurnished by the amplifier, with an efficiency of the filling process R, A=O and only the first term in parentheses in eq. (5.3) must be retained so that:

for

The case of point sources has been integrated by Ferlenghi and Mango [1963], who obtain: R ~ E A, A +n A +n Q A E = 2@)' [arctg7 tg - - arctg tg A 1-k ; I l + k

-1

If a positron energy band from $ E , to m is considered, (5.6) becomes: QAE = 2 n

R~ -

Y :An

E,.

Eqs. (5.6), (5.7) have to be used, instead of (5.1), (5.3), (5.5) whenever e/R A / I ) ; b) in a low energy positron accelerator, when a small final energy spread is required, it is convenient to operate with A/;I=@/R4 1: about half of the intensity is within an energy band of the order of 4eEC/nR(compare (5.5) and (5.1)); c) for a constant value of I and R (which means also constant value of final emittance), the positron intensity increases linearly with R/Q,if one can make A small enough to keep I/A= R/e; this is true only as far as the source density can be considered uniform with angles. When the largest emission angles, equal to R/y,A for converters outside

B.3.5

POSITRON ACCELERATORS

539

of the lens field, and to (R+,o)/y,A for converters immersed in the field, become 5 (see section 3.2) or bigger, the gain in intensity tapers comparable with 0 ~ 0 . 3 rad off, with decreasing values of A , and eventually vanishes when RlyJ is about 0.7. Taking into account also the dephasing in the matching lens, which is proportional to R2/Ay,, see eq. (6.5), it turns out that there is not much to be gained in making R/y,A larger than j to f. d ) for high energy positron linacs, where the energy spread at the converter does not contribute appreciably to the final relative energy spread, a n d a beam with larger emittance can be used, a further increase in intensity can be obtained decreasing the value of A, while leaving A at the limit set in point c) above; the maximum is obtained for A= A , in which case the focusing system (if it is a solenoid) is achromatic, a n d the source radius can be made equal to the iris radius. Such a n arrangement would be expensive (as it would require a very long solenoid at high field), but it could allow a n increase in acceptance of about one order of magnitude, as compared to the best results so far obtained, and a much higher thermal limit for the primary beam power, owing to the larger source dimensions.

5.2 Adiabatic matching Helm [3962b] has proposed to use, as a matching section, an adiabatically tapered solenoid, where the characteristic length changes slowly from A at the input end, to 1. a t the output, with A %= eE, cos 9 exp (ikz - iwt) = eE,

from which one sees that, for a wave travelling a t the velocity of light, the deflection force is uniform and aberration-free over the aperture. Although the deflection force arises from a combination of electric and magnetic forces, it can conveniently be expressed in terms of an equivalent electric field. The pioperties of hybrid modes have been investigated extensively by Garault [1962, 19641, Hahn [I9631 and Henoch [1958], in addition to the other references previously cited in this section. Hahn and Halama [1965b] have described the waveguide system used in the Brookhaven rf separator, and discussed in detail the choice of

576

B.W. MONTAGUE

ELECTRON LMACS: VARIOUS TYPES

parameters. Falnes [I9621 has studied hybrid modes in connection with plasma experiments. The analysis of the Ell/Hll mode including higher Fourier space harmonics has been carried out by Hereward and Bell [I9631 for digital computation, and computational results are given by Bell and Hereward [1965]. The first experimental demonstration of the deflecting properties of the Ell/Hll hybrid mode was made in 1963 at Orsay by Hire1 [1964], using 2 MeV electrons. Shortly afterwards, electrons in the 30-50 MeV range were deflected at Stanford by Larsen et al. [1963]. More recent work at Stanford on hybrid mode structures is described by Loew and Altenmueller [1965]. The nomenclature of hybrid modes is far from uniform, other authors using variously HEM, HM, HE, EH, etc. Fields with similar properties have been discussed by Hereward [I9581 and Blewett [1959], for structures of more complicated geometry; Gabillard [I9611 has investigated the properties of such structures. The iris-loaded circular waveguide, however, appears to be superior in performance and ease of fabrication to either of these and is used in all microwave particle separators that have so far been built or projected. Hybrid modes can exhibit some curious properties compared with the simpler accelerating modes. Depending on the choice of waveguide parameters, the group velocity can be either positive or negative, and the dispersion curve can even be nonmonotonic, as shown in fig. 3. This results from the fact that the Poynting vector may have the opposite direction in the outer region of the aperture from that near the axis. It can be shown that the behaviour of a waveguide mode with a non-monotonic dispersion curve cannot be represented by a 4-terminal network; an analogue network must have at least 6 terminals to portray such features. The choice of optimum parameters for an rf separator deflecting waveguide is considerably more complex than in the case of an electron linac. This is partly because an rf separator design is subject to many external parameters such as beam emittance, operating energy, beam intensity, acceptable contamination, beam layout etc.; most of these parameters can change from one experiment to the next. Another difference is that the more complicated field configuration of a hybrid mode gives rise to much interdependence between parameters. Recently, an extensive systematic study of these factors has been carried out by Bernard et al. [1968], taking account not only of rf power, deflecting field, length and group velocity of the waveguide, phase advance per cell etc., but also of the rf breakdown properties of the structure. It is not possible in a short review to summarise all the results of this work, but it is worth quoting one or two examples. It is found, for instance, that the ratio Ep/Eo,where Ep is the peak electric field on the iris surface and E, the equivalent deflecting field, is very strongly dependent on group velocity. In fact, the choice of parameters for the original CERN deflecting waveguide, though made in the light of the best knowledge available at the time (1961), appears to have

RADIOFREQUENCY SEPARATORS

B.3.7

577

been an unlucky one, since the power is limited to about 16 MW by rf breakdown. The Brookhaven waveguide, with a slightly smaller aperture and a negative group velocity of about the same magnitude as that of the CERN structure, has a substantially higher breakdown limit. Another interesting feature is that a waveguide operated in standing-wave mode Frequency

MHz

c - mode cutoff

2r

-

t I

------t

phase shift per c e l l

Fig. 3. Brillouin diagram of the original C E R N waveguide.

does not necessarily have twice the peak field as when operated in travelling-wave a t the same E,. The ratio can be appreciably less than 2, a n d under some conditions it might even be preferable t o choose a standing-wave structure with a fairly large negative group velocity t o minimise E,/E,. Normally, however, travelling-wave operation is used. I n contrast to most linacs, rf separator waveguide is optimised on a basis of

578

B.W.MONTAGUE

ELECTRON LINACS : VARIOUS TYPES

breakdown limitation rather than rf power economy, since the performance is so strongly conditioned by external parameters as mentioned above. As a consequence, the concept of structure shunt impedance is of doubtful value and it is more useful to compare structures on the basis of the peak transverse momentum per unit length imparted to the beam by unit power flux. This figure of merit, which is effectively the square root of the axial impedance, lies typically in the range 1.4 to 2.0 (MeVlc) m-' (MW)-) for S-band waveguide and for reasonable values of other parameters.

5 Separator optics The secondary beam transport channel for an rf separator can be divided into a number of sections: a first momentum analyser, a beam matching section, the rf mass-separation stage proper, a second beam matching section and a final momentum analyser. Only the mass-separation stage will be considered here; the other sections are basically similar to those used in dc separated beams, though differing somewhat in detail. More detailed discussions of rf separator optics are given for example by Montague [I9681 and by Keil (De Raad, Minten and Keil [1966]). In fig. 1 the optical arrangement between the cavities was simplified to a single lens imaging R F l into RF2 with unity magnification. In practice the matching of the two transverse phase planes imposes further restrictions such that the transfer matrix between RF1 and RF2 should be of the form

in both vertical and horizontal planes. Since the vertical plane, in which mass separation takes place, is the more critical, we arrange for a,, to be zero in this plane for the central momentum. Then the fundamental properties of magnetic lens systems prevent a,, being simultaneously zero in the horizontal plane. Also, the momentum dependence of the matrix elements limits the momentum bite which can be transmitted without serious loss of acceptance. The required form of the transfer matrix can be obtained with a triplet or with more elaborate quadrupole combinations. The phase-plane diagram taken at the rf deflector centres associated with such a system is shown in fig. 4. The aperture limiting lines at the entry and exit of an rf deflector form a diamond-shaped acceptance region, and for the simplest optical system the useful acceptance is given by the inscribed rectangle. The unseparated beam entering RF1 is collimated so as to occupy the doubly-hatched region and deflection in R F l extends the particle distribution to occupy the singly-hatched area. The transformation between R F l and RF2 simply rotates the phase-plane diagram through n. Finally, due to the time-of-flight difference, wanted and unwanted particles are influenced differently by the deflection in RF2, the wanted particles having their


B.3.7

579

distribution extended into the unshaded part of the rectangle, whilst unwanted particles are brought back to occupy only the original area in the phase-plane. This doubly-hatched region is intercepted by the central beam stopper, whose transformed "shadow" is shown in fig. 4. In the foregoing it has been assumed that the rf cavity acts as a point deflector. It

//

I \n/,

MIXED DEFLECTED BEAM DEFLECTED BEAM AND UNWANTED PARTICLES

BEAM

'

X

STOPPER

Fig. 4. Phase-plane diagram at cavity centre.

can be shown (Montague [1968]) that the action of a long deflecting cavity can be represented by the transformation :

sin u U

where the (X, X') are expressed in normalised units, cpO is the phase angle of the particle at the equivalent centre of the deflector, and u is half the phase slip of the particle relative to the deflecting wave along the length of the deflector. For one particle type and momentum, u can be made exactly zero, and in practice can be approximately zero for all particles of interest, in which case the matrix reduces to: 0

1 sin p0

which represents a point deflection sinusoidal in time.

580

B.W.MONTAGUE


For the detailed design of rf separated beams, a number of other factors have to be taken into account. These include the performance away from design momentum, chromatic aberrations in the beam transport elements, anisochronism due both to finite momentum bite and to beam divergence angles, tolerances on various elements etc.

6 Engineering aspects In many respects the practical design of a microwave rf separator resembles closely that of an electron linac in the 10 to 20 MeV range. There are, however, certain differences, which are of some importance from the engineering viewpoint. Although an rf sepalator has no beam-loading problems, some requirements are stricter than a normal electron linac. Since secondary particle beams typically have rather high ratios of unwanted to wanted particle fluxes, the primary object of any separator design is to determine, as precisely as possible, the fate of the unwanted particles. As in most accelerator designs, some of the basic parameters of the system are determined by the availability of high-power rf sources. For separation in the energy range of about 7 to 50 CeV, S-band klystrons meet the necessary requirements, but at higher energies it will probably be necessary to go to shorter wavelengths. Despite the somewhat strong dependence of phase shift upon h.t. voltage that is characteristic of klystrons, in practice no serious difficulties are experienced in obtaining the necessary stability of phase. In fact, the whole problem of maintaining the required phasing tolerances of about f2 degrees over distances of several hundred wavelengths has turned out to be much easier than anticipated. Other engineering problems are substantially the same as in electron linac practice and are described elsewhere in this book (see ch. B.2.2). The fabrication of deflecting waveguide is very similar to that of accelerating waveguide, with the additional feature that means may be provided for imposing the polarisation plane of the mode. Vacuum requirements are determined, as in small electron linacs, mainly by the high rf power level and the need to minimise risk of voltage breakdown.

7 Present status and future developments The first high-energy rf separator came into operation at CERN in January 1965 (Bramham et al. [1965]). The equipment has operated successfully in the separation of K- and K + at around 10 CeVlc, anti-protons at 12 GeV/c and n f up to 16 GeV/c. The CERN separator operating at 2855 MHz is of the Panofsky/Schnell type, and has been described by Bell et al. [1963b]. The two-cavity version was used initially, but in June 1967 the three-cavity version was successfully brought into operation (Bernard et al. [1967]).

B.3.7


581

The rf separator at the Brookhaven National Laboratory (Hahn and Halama [1965a], Foelsche et al. [1967]), also of the Panofsky/Schnell type, started operation early in 1966. Using two cavities, it is similar to the CERN separator in most important respects. During the first few months of operation it successfully separated K- and K + at 7.4 and 12.8 GeVlc and has since separated z + up to 18.5 GeV/c. At the Joint Institute for Nuclear Research, Dubna, U.S.S.R., a Veksler-type separator has been completed for use with the 10 GeV Synchrophasotron. The system, described by Vaghin et al. [I9631 uses internal rebunching at about 150 MHz, the separator proper consisting of a single deflecting section of 16 separately fed cavities (Davydov et al. [1963]). An rf separated beam is planned at Stanford Linear Accelerator Center for the 20 GeV electron linac. The linac primary beam bunches are of very short duration, about k 5 degrees of phase at 2856 MHz, permitting the design of an extremely efficient and flexible rf separator (Larsen et al. [1963]). The AG proton synchrotron at Serpukhov, U.S.S.R., has now operated up to 76 GeV. Studies have already been made on an rf separated beam for this accelerator (Bernard et al. [1966]) and are continuing. The increasing probability that ultra-high energy proton synchrotrons will be built in the U.S.A. and in Europe has stimulated interest in rf separation at energies of 100 GeV or more. In the same context, there will be a need for long-pulse rf separators for use with counter experiments at these high energies, in view of the greatly increased secondary particle fluxes anticipated. The only foreseeable economic way of building long-pulse separators for such energies is to make them superconducting; recent progress in rf superconductivity suggests that superconducting separators should be feasible in a few years from now (see ch. E.2 and E.3). The problems of superconducting and ultra-high energy rf separators have recently been examined in some detail in a number of reports collected together in ECFA [1967]. The main extrapolations from present techniques that will be necessary for the 100 GeV energy range are the use of higher frequencies, probably in X-band ( A z 3 cm), and the control of phase to a precision of one to two orders of magnitude better than that necessary for existing rf separators. Although this latter problem looks rather formidable, several possibilities exist for dealing with it, and it is reasonable to say that the phasing difficulties at 100 GeV appear no worse today than those at 10 GeV appeared in 1960. In rf superconductivity, the main problem is to obtain in loaded waveguide structures the same magnitude of improvement factor for losses that has already been obtained in simple cavities. Although many problems remain to be solved, progress in this field is very encouraging and there seems to be good hope that superconducting rf separators will become part of the normal equipment for research in high-energy physics.

582

B.W. MONTAGUE


References BELL,J. S. and J. M. BRUCE,1952, "A Semi-circular Waveguide Loaded by a Semi-circular Slotted Sheet", AERE-Rpt. T/M 57. BELL,M., P. BRAMHAM, and B. W. MONTAGUE, 1963a, "Pulse Shortening in Electron Linear Accelerators and El1 type Modes", Nature 198, 277. BELL,M., P. BRAMHAM, R. D. FORTUNE,E. KEIL and B. W. MONTAGUE, 1963b, "RF Particle Separators", Proc. Int. Conf. on High-Energy Accelerators, Dubna, 798. BELL,M. and H. G. HEREWARD, 1965, "Disc-Loaded Deflecting Waveguide Computations", CERN Rpt. 65-37. BERNARD, P., P. LAZEYRAS, H. LENGELER and A . SAMOILOV, 1966, "Preliminary Study of a Radiofrequency Separated Beam SU for the Serpukhov Accelerator", Rpt. CERN/TC/Beam 66-2. BERNARD, P., P. LAZEYRAS, H. LENGELER and V. VAGHIN,1967, "The New Three-Cavity R F Separator of CERN", Proc. Int. Conf. on High-Energy Accelerators, Cambridge, Mass., 190. and V. VAGHIN,1968, "On the Design of Disc-Loaded Waveguides for BERNARD, P., H. LENGELER R F Separators", CERN Rpt. 68-30. J. P., 1959, "A Radio-Frequency Mass Separator for Complete Separation of HighBLEWETT, Energy Particle Beams", Proc. Int. Conf. on High-Energy Accelerators, CERN, 422. BRAMHAM, P., R. D. FORTUNE, E. KEIL, H. LENGELER, B. W. MONTAGUE and W. W. NEALE,1965, "The CERN Radio Frequency Particle Separator - First Operation", Physics Letters 15, 290. BRILLOUIN, L., 1948, "Wave Guides for Slow Waves", J. Appl. Phys. 19, 1023. BROWN,K. L., A. L. ELDREDGE, R. H. HELM,J. H. JASBERG,J. V. LEBACQZ,G . A. LOEW, R. F. MOZELEY, R. B. NEAL, W. K. H. PANOFSKY and T. F. TURNER,1961, "Linear Electron Accelerator Progress at Stanford University", Proc. Int. Conf. on High-Energy Accelerators, Brookhaven, 79. M. C., T. R. JARVIS,C. W. MILLERand G. SAXON,1961, "Pulse Shortening in CROWLEY-MILLING, Electron Linear Accelerators", Nature 191, 483. DAVYDOV, M. S., L. G. DORFMAN, V. V. EKIMOV, V. B. ZALMANZON, G . A. ZEITLENOK, V. M. LEVIN, I. F. MALYSHEV, I. G. PETELIN,V. I. PETRUNIN, V. A. POPOV,N. F. TRUSHIN, I. G. UMANSKY, 1963, "Deflecting System of a 5 GeV/c Anti-proton Channel", Proc. Int. and I. I. FINKELSTEIN, Conf. on High-Energy Accelerators, Dubna, 791. DE RAAD,B., A. MINTENand E. KEIL, 1966, "Lectures on Beam Optics; 111, Radio-Frequency Separated Beams", CERN Rpt. 66-21. ECFA, 1967, European Committee for Future Accelerators, "Utilization Studies for a 300 GeV Proton Synchrotron", CERNIECFA 67/16 Vol. 11. J., 1962, "Hybrid Modes of a Circular Cylindrical Cavity containing a Unidirectionally FALNES, Conducting Coaxial Sheet", Symposium on Electromagnetic Theory and Antennas, Copenhagen, (Pergamon Press 1963). J. LACH,T. LUDLAM and J. SANDWEISS, 1967, "RadioFOELSCHE, H. W. J., H. HAHN,H. J. HALAMA, frequency Separated Beam at the AGS", Rev. Sci. Instr. 38, 879. GABILLARD, R., 1961, "Probleme general de la deflection d'une particule chargee de haute energie par un champ electron~agnetique", Inst. Radiotechnique Lille, Rpts. IREL 61-1 and 2. GARAULT, Y., 1962, Comptes Rendus 254, 843, "Propriete generale d'un type d'onde se propageant dans le vide a la vitesse de la lumiere et utilisable pour la deflexion de particules ultrarelativistes"; 254, 1391, "Solutions coherentes de l'equation des ondes en coordonnes cylindriques, valables quelle que soit la vitesse de phase"; 255, 2920, "Ondes electromagnetiques du type E H dans un guide cylindrique circulaire charge par des iris metalliques". Y., 1964, "Etude d'une classe d'ondes electromagnetiques guidees: les ondes EH. ApGARAULT, plication aux structures deflectrices pour les separateurs A onde progressive de particules relativistes", CERN Rpt. 64-43. HAHN,H., 1963, "Deflecting Mode in Circular Iris-loaded Waveguides", Rev. Sci. Instr. 34, 1094. 1965a, "The Brookhaven Radiofrequency Beam Separator", Proc. HAHN,H. and H. J. HALAMA, Int. Conf. on High-Energy Accelerators, Frascati, 548. HAHN,H. and H. J. HALAMA, 1965b, "Design of the Deflector for the R F Beam Separator at the Brookhaven AGS", Rev. Sci. Instr. 36, 1788.

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HENOCH, B. T., 1958, "Investigations of the Disc-loaded and Helical Waveguides", Trans. Roy. Inst. Tech., Stockholm, No. 129. H. G . , 1958, "Particle Separation Using Internal Bunching (Veksler Separator)", CERN HEREWARD, Rpt. PS/lnt. TH/58--8. HEREWARD, H. G . and M. BELL,1963, "Disc-Loaded Deflecting Waveguide", CERN Rpt. 63-33. HIREL,R., 1964, "Separateur de particules rapides au moyen de champs electromagnitiques H.F.", Nucl. Instr. and Methods 26, 90. JARVIS, T. R., G . SAXON and M . C. CROWLEY-MILLING, 1965, "Experin~entalObservations of Pulseshortening in a Linear Accelerator Waveguide", Proc. IEE 112, 1795. KELLIHER, M. G . and R. BEADLE, 1960, "Pulse Shortening in Electron Linear Accelerators", Nature 187, 1099. LAPOSTOLL~, P., 1960, "Sur la deflection d'une particule rapide par un champ tlectromagnt-tique", Rpt. CERN PS/lnt. AR/60-2. LARSEN, R. R., 0.A. ALTENMUELLER and G . A. LOEW, 1963, "Investigations of Travelling-Wave Separators for the Stanford Two-Mile Linear Accelerator", Proc. Int. Conf. o n High-Energy Accelerators, Dubna, 804. Lorw, G. A. and 0. A. ALTENMUELLER, 1965, "Design and Applications of R F Deflecting Structures at SLAC", Proc. Int. Conf. on High-Energy Accelerators, Frascati, 551. MONTAGUE, B. W., 1960, "RF Travelling Wave Particle Separators; a Matrix Formalism and General Phase Space Properties", CERN Rpt. PS/Int. AR/PSep/60-I. MONTAGUE, B. W., 1968, "Particle Separation at High Energies - 11. Radio-Frequency Separation", "Progress in Nuclear Techniques and Instrunlentation, Vol. 111" (North-Holland). NYGARD, J. C. and R. F. POST, 1961, "Recent Advances in High Power Microwave Electron Accelerators for Physics Research", Nucl. Instr. and Methods 11, 126. PANOFSKY, W. K. H., 1956, "A Mass-sensitive Deflector for High-energy Particles", Stanford Rpt. HEPL-82. PANOFSKY, W. K. H., 1959, "Experience with a Microwave Separator", Proc. Int. Conf. on HighEnergy Accelerators and Instrun~entation,CERN, 428. PANOFSKY, W. K. H. and W. A. WENZEL,1956, "Some Considerations concerning the Transverse Deflection of Charged Particles in Radio-Frequency Fields", Rev. Sci. Instr. 27, 967. SAXON, G., T. R. JARVIS and I. WHITE,1963, "Angular-dependent Modes in Circular Corrugated Waveguide", Proc. IEE 110, 1365. SCHNELL, W., 1961, "Discussion of a Radio-Frequency Particle Sepsrator for the CERN Proton Synchrotron", CERN Rpt. 61-5. S. V. MUKHIN, V. A. PETUKHOV, VAGHIN, V. A,, V. I. V E K S L ~V. R , N. ZUBAREV, A. B. KUZNETSOV, V. L. STEPANYUK and K. V. CHEKHLOV, 1963, V. A. P o ~ o v ,N. B. R U B I N1., N. SEMENYUSHKIN, "The 5 GeVlc Electrodynamic Separator of Anti-Protons", Proc. Int. Conf. on High-Energy Accelerators, Dubna, 788. VEKSLER, V. I., 1958, "Starting up of a 10 BeV Synchrophasotron and First Results of Physical Research", Trans. Sec. Int. Conf. o n Peaceful Uses of Atomic Energy, Geneva, A/CONF/I SIP/2229, 7. WALKINSHAW, W. and J. S. BELL,1951, "Review of Theory of Metal Loaded Linear Accelerator", AERE Rpt. G R 675.

PART C

Proton

Linear Accelerators G. BARTHOLOMEW H. BUTLER A. CARNE J. M. DICKSON

G. DOME R. L. GLUCKSTERN

J. HUGUENIN A. KATZ

E. A. KNAPP P.M. LAPOSTOLLE

D. E. NAGLE T. NISHIKAWA

M. PROME M. REGLER

B. SCHNIZER TH. SLUYTERS

C. S. TAYLOR P. TUNNICLIFFE

A. VAN STEENBERGEN B. VOSICKI

PLA THEORY

I

C.l


1.la

Introduction P. LAPOSTOLLE CERN, Gcweva,Switzrrland

Electrons in accelerators travel most of the time at a velocity very close to the velocity of light, i.e. B= VICE1. Even at injection into the linear accelerator the value of B is large, about 0.5. On the contrary, protons are very much slower, and a p of 0.5 corresponds to 150 MeV, an energy which has not yet been reached in proton linear accelerators. At injection in most of the present machines, Bz0.03-0.04 (i.e. 500-750 keV). As a consequence the structures to use must be different. Only for the very high energies of several hundred MeV which are now being considered a certain analogy might exist for the two types of machines. Nevertheless, even there the rate of change of p will remain very slow compared with electrons, and it will never be possible to use a constant velocity structure of fl= I. The best that might be achieved would be the use of short tanks of constant p, with /3 being progressively increased from tank to tank. The acceleration of slow particles leads to structures which differ completely from those for electrons. The first difference relates to the frequency. As will be seen in chapter C. 1.2bthe transit time factor, T, which reduces the energy gain that any particle can receive through an accelerating gap, is worse on-axis than off-axis. The dependence is given by the function I,(k,r), where

The difference between electrons and protons appears clearly. For electrons k,=0: and I, (k,r) is independent of r. For protons, it is usual practice to keep the difference in T between the axis and beam edge less than 25%. This leads to (for /3 small)

8 North-HollandPubl.

Co., Amsterdam 1969

586

P. LAPOSTOLLE

PROTON LINACS: THEORY

For 1 small, the angular frequency o must also be small. Frequencies of 100 MHz to 200 MHz are normally used. The second difference between protons and electrons is the use of resonant cavities or tanks instead of travelling wave structures. This is to some extent due to the lower frequency used, but also to the type of structure, as will be described in the next chapter. A good structure must have various properties. It must have a high shunt impedance in order not to require too much power for acceleration, the shunt impedance (usually expressed in MQ/m) being defined as the ratio of the square of the accelerating electric field to the Joule losses per unit length of structure. But it should also have a large bandwidth. This characteristic is useful not so much for offering a larger mode spacing, so avoiding a risk of interference between the operating mode and its neighbours: it is useful mainly for the small sensitivity to errors of construction and the transient properties it gives. These are of particular importance as will be seen in chapter C. 1.3. A good structure must also be good from the technological point of view. Easy fabrication, high breakdown level, enough room to include focusing elements are important aspects which have to be taken into account in the choice of a structure. A general synopsis of various aspects of structure properties is given in chapter C.l.le.

PLA THEORY

I

C.l


Low and Medium Energies

l.lb

Alvarez Structures A. CARNE Rutherford High Energy Laboratory Chilton, Didcot, Berks., England

I

I

5 !

Introduction

In this chapter are discussed low and medium energy structures. For historical reasons, the Alvarez structure will be discussed first. All accelerating structures, be they travelling or standing wave type, are (at least) quasiperiodic, that is, they have planes of symmetry with respect to the axis of particle travel, and as such have ranges of operating frequency (passbands) alternating with stopbands. The Alvarez structure operates at the lowest frequency of its passband, in the zero-mode of the E,, passband, i.e. the E,,, mode. The reason for this was purely one of efficiency, i.e. maximal shunt impedance. At the time of its design, indeed until quite recently, this was an all important reason, and dispersion characteristics which define the behaviour of the structure with respect to tolerance, transients, and particularly beam loading, were secondary. With modern equipment for linacs, where currents of perhaps 200 mA are required to be accelerated, the dispersion (dwldk, where w is the angular frequency, k is the phase change per unit length) has assumed greater importance. Because of this it will be seen (as already indicated in the Introduction C.l.la)) that structures with low dispersion (i.e. doldk large, or large bandwidth) are nowadays sought, and found, not in end-of-passband operation, but effectively in centreband operation achieved by joining adjacent passbands together. In this way a high shunt impedance common to zero (or n) mode standing wave operation is retained together with low dispersion. Structures having all these properties are the multi-stem, post-loaded, and cross-bar structures for low and medium energies; and for higher energies, the side-coupled cavity, ring loaded, and possibly the cloverleaf structures (all to be discussed in ch. C.l.le). 0 Xorth-Holland Publ. Co., Amsterdam

References p. 599

1969

587

588

A.CARNE


2 The Alvarez structure The structure of the original Berkeley proton linear accelerator*, usually called the Alvarez Structure after its proposer in 1946, (Alvarez et al. [1955]), is basically a. long cylindrical cavity excited in the fundamental E,,, mode and loaded along its axis with cylindrical drift tubes, as in fig. 1. The Alvarez structure can be regarded as a quasi-periodic structure where the drift tubes provide the loading to produce a synchronous space harmonic of sufficient amplitude, or as a structure with infinite phase velocity (since it is at standing wave E,,, cut-off) where the drift tubes shield the particles from the rf field when it is in the wrong phase for acceleration. These alternatives are almost identical, and become closer at higher energies where the fractional velocity change becomes smaller.

Fig. 1. The Alvarez structure. Division into unit cells and half unit cells.

It can be seen from fig. I that the structure has approximate planes of symmetry at mid-gap (planes AA') and mid-drift tube (planes BB'). Thus, as with all 'periodic' structures, the 'tank' may be regarded as a system of collinear unit cells (doubly re-entrant cavities BB'-BB') or even half unit cells (singly re-entrant cavities AA'BB'), each oscillating at the same phase and frequency in the fundamental E,,, (perturbed) mode, the zero mode. The distance between gap centres is equal to the distance travelled by a reference particle in n rf cycles (the n2n mode of acceleration), i.e. A-A=Ir=nPr/l (n is usually unity, but can be greater than unity for special + Br)l =n (P,-, ++A/?)l, reasons); and the length of a cell is then B-B = L, = in where p, is the output p of cell r, and 1is the free space wavelength. Lengths defined in this way ensure that a reference particle arrives at the gap centres at constant phase, since the gaps themselves are at the same phase. (More precise and flexible definitions for L, and I,, taking into account particle dynamics are discussed in Carne, Lapostollc [1966]). Since successive cells increase in length, other dimensions of the unit cell must vary to maintain resonance. Thus currents on either side of the partitions do not exactly cancel, and the field distribution of the whole tank is not exactly the same as that if the partitions BB' (or AA') were actually present. However, removal of the

*

The Alvarez structure has also been used for the acceleration of heavy ions in Berkeley and in Yale University (Hubbard et al. [1961]; see also chapter D.2).

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589

partitions in effect produces large coupling between the cells, and providing the cells are initially resonant at the same frequency, the difference with and without partitions is practically negligible (except, of course, for the large saving of rf power due to the removal of the partitions. See also chs. C.1.Id and C.1.2a). For the purposes of design, then, it is sufficient to study the characteristics of the unit cell of fig. 2, except of course for discussion of length and tolerances of the whole tank. The parameters a, d, D, R,, Ri, g/L, A, and eventually L, the total tank length, must be chosen as for any linac structure as a compromise between maximal shunt impedance, freedom from voltage breakdown, good beam dynamics, ability of the drift tubes to contain focussing devices when required, ease of manufacture, and

Fig. 2. The Alvarez Structure. The unit cell.

tolerarlces. These criteria are all related; though, as will be discussed later, what is considered the correct order of importance depends to a large extent on the final application of the linac. The unit cell parameters are thus also related, but above all, they must additionally satisfy the condition of resonance. Relations between parameters have been studied both experimentally and theoretically, but a few general remarks here are worthwhile. Historically A has always been of the order 1.5 metre, and this appears to be close to the optimum (only three operating proton linacs have different wavelengths, the 15 MeV injector for Nimrod with A=2.6 metre, and the 20 and 100 MeV injectors of the ITEP and Serpukhov synchrotrons, with A=2 metre). The aperture, 2a, is chosen initially from the requirements of the pre-injector (at 500-800 keV), for quadrupoles and good dynamics (i.e.

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A.CARNE

good transit time factor T, and small phase-to-radial motion coupling), but its effect on frequency is only second order. Gap g usually lies in the range 0.2 < g/L < 0.4, i.e. from the limiting gap field at one end to poor transit time factor at the other. Drift tube diameter, d, must not be too small to house quadrupoles (say 12-14 cm) but not too large to avoid poor shunt impedance. A large Ro is good for freedom from voltage breakdown, and T, but not for housing quadrupoles; except possibly at low energy (Q), and 4 is the angle of rotation between adjacent slots (see fig. 19b). When 4 is not zero the geometrical period of the structure includes several cavities, but the electrical period (with phase shift 8) is still a single cavity. When the slots in adjacent discs are aligned (i.e. when 4 = 0), y 2 x 1 3- P- Q at 8 =0, and y is only slightly larger than unity in short cells: at their 0 mode resonance, the aligned slots are thus quite close to half a wavelength long. Although the dispersion relation (60) with y represents the experimental data well, up to and including the second slot resonance (which happens at a frequency 3 times as high as the first slot resonance), for frequencies below 2 times the first slot resonance this relation is sufficiently well approximated by the simpler one (Allen [l959]) :

The condition for confluence between the cavity passband and the first slot passband at 0 mode is

which corresponds to a slot angle of 74.850" y. Since the correction factor y is always larger than unity, the slot angle must be larger than 75" for confluence to occur. This confluence has been discussed, as an example, by Mc Diarmid et al. [1968]. Equivalent circuit. The equation (61) may be identified * with the dispersion relation

*

Although the two wl's represent slightly different frequencies.

REVIEW AND SURVEY OF ACCELERATING STRUCTURES

C.l.le

685

of the circuit of fig. 20, where k , = O and k , = Q cos$/(l +P):therefore, this circuit" may be taken as equivalent to a structure coupled by long slots. It is seen that LOCO represents the cavity resonance, L,C, represents the slot resonance, and the mutual inductance M I represents the coupling between adjacent slots. This coupling depends on the angle 4 between adjacent slots. By varying $, it is thus possible t o modify strongly the bandwidth of the slot passband: the bandwidth is very small when $ =Oo (adjacent slots aligned) and becomes maximum when 4 = 180" (one slot per disc, rotated 180" f r o m one disc to the next). If they are two slots per disc (see fig. 19b), the maximum bandwidth occurs when 4 = 90" because the case 4 = 180" is then undistinguishable from +=Oo. This is confirmed in fig. 19c, which shows measured dispersion curves for 4 =0° and 4 = 90".

Fig. 20. A different version of the equivalent circuit of fig. 8.2a for resonantly coupled cavities. The arrows indicate the positive directions for the electric currents. This circuit is obtained from that of fig. 8.2a with the substitution

The dispersion relation reads

where

The two frequencies which coalesce at confluence are given by

With ko= kl -0, this is the equivalent circuit proposed by Allen [I9591 and Chodorow [1960]. When LI == co, the dispersion relation becomes

+

where Co/(Co 2C1) is the electric coupling coefficient, and ko is the magnetic coupling coefficient. The influence of these coupling coefficients on the passband is evident.

*

Or the equivalent version of fig. 8.2a.

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G.D~ME


For any frequency, eq. (60) may be converted into:

where the summation extends over all the slot resonances. This is the dispersion relation for a circuit similar to that of fig. 21a, but having an infinite number of parallel resonant circuits in the T-arm. For frequencies below the third slot resonance, it is sufficient to retain only the first two terms of the series, which corresponds to the equivalent circuit of fig. 21. This figure gives also the dispersion equation as modified

Fig. 21. (a) Equivalent circuit of slot coupled cavities, including the next higher slot resonance. The arrows indicate the positive directions for the electric currents. Dispersion relation:

(b) Dispersion curves of a slot coupled structure, including the second slot passband (wz m 3 4 .

by the presence of coupling between adjacent slots, and the dispersion curves of the structure. There is now a third passband, corresponding to the second slot resonance. When all the coupling coefficients k equal zero, there is a frequency in between o,and o2which makes cosfl infinite: at this frequency (which obviously is in a stopband) cose changes sign, making the two adjacent passbands being of the same type, forward

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or backward wave. Small couplings do not alter this basic behaviour of the dispersion curves. which is shown in fig. 21 b, and which has been verified experimentally (Allen [1959]). Such a behaviour is characteristic of circuits where there is a frequency which makes cos0 (and therefore the attenuation) infinite if all mutual inductances are zero. For example, in the circuit of fig. 22a which generalizes that of fig. 20, the resonance frequency tuz of LzCz produces an infinite attenuation: therefore this "stop" frequency separates two adjacent passbands which are of the same type, forward or backward wave. Such a "stop" frequency with infinite attenuation is called "partial resonance" frequency by Curnow [I964 and 19651. In the presence of a small coupling coefficient kz, a narrow passband arises around tuz, as shown in fig. 22b. In the equivalent circuit of fig. 22a, LlCl represents the coupling element while the network LOCO.LzCz represents the cavity (the resonant frequency of the uncoupled cavity is obtained when short-circuiting LICI). With CObeing infinite and all the mutual inductances being zero, this circuit is equivalent to the "in-line partially coupled" circuit proposed by Curnow for representing the modified long-slot coupled structure of Allen. This structure (Allen [1959]; Allen, Kino and Williams [1960]) is a long slot coupled structure with all slots aligned, where fins are added to the outer cylinder halfway between two adjacent slots, covering the centre part of the slot length. The fins introduce a partial resonance frequency which modifies the shape of both passbands and lowers the 0 mode of the slot passband; for certain fin sizes the partial resonance frequency falls in between the cavity and the slot passbands, producing a slot passband of the forward wave type (see fig. 22b).

Fig. 22. (a) Addition of a resonant element LzCz producing a partial resonance, into the equivalent circuit of fig. 20. The arrows indicate the positive directions for the electric currents. Dispersion relation:

(b) Dispersion curves, when kz is negative. For the modified long-slot coupled structure, tun =O.

The slotted iris structure with sectoral slots. In order to increase the coupling coefficient B, the slots may be widened, yielding slotted iris structures with sectoral slots (see fig. 23). Measurements on such structures with drift tubes have been carried on at Brookhaven (Giordano [1964]), Berkeley and Karlsruhe (Bauer et al. [1965]; Eschelbacher et al. [1966]). The models used had four identical slots with a slot angle smaller than 75": the cavity passband is then the lower passband. The shunt impedance was

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measured at n mode of this passband (i.e. the lowest cut-off frequency of the structure), yielding values at 800 MHz which increased from 20 to 30 MQ/m for proton energies ranging from 150 to 600 MeV. For each particular structure, there is an optimum slotwidth beyond which the electric coupling more than cancels the increase in magnetic coupling, resulting in an overall decrease of the bandwidth. The TC mode operation of the slotted iris structure suffers from the drawback of small mode spacing around the operating mode. From the point of view of compensation, this structure should be operated at 0 mode with confluence: it would then become very similar to an Alvarez structure compensated with multistems (which will be described later on). When the slot width becomes comparable with the cell length, the fringing electric field of the slot produces an additional coupling between adjacent slots, and the dispersion relation (61), which was derived for narrow slots, no longer applies.

1

Sbt Width

Slot Angle

Fig. 23. Cross-sections of the slotted iris structure with sectoral slots.

Nevertheless, the results concerning the bandwidth of the slot passband remain qualitatively correct, even when the partition disc reduces to a bar or a single stem supporting a drift tube: therefore, the bar passband is wider for the cross-bar structure of fig. 18c than for the parallel bar structure of fig. 18b. Similarly, the stem passband of fig. 18d has a much larger bandwidth than when the stems are all parallel. As a first approximation, such a rotation of the slots or stems leaves the +n mode unchanged in both passbands: indeed, there is a +n standing wave mode where every other slot is unexcited and does not affect its (excited) adjacent slots. This shows up in fig. 19c for a slot-coupled structure, and has been verified for all the multistem structures.

The cloverleaf structure. In the structures coupled by peripheral slots, the lower

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passband is of the backward wave type. This would produce backward wave oscillations if no precautions were taken when such a structure is used in a travellingwave tube amplifier (Harris [1964]). Therefore, efforts have been made in order to find a structure which, although coupled by slots, would have a lower passband of the forward wave type (Chodorow et al. [1957]). This means that confluence between the first two passbands should occur in zr mode, or that the cavity mode which remains unperturbed by the slots should be a zr mode. In other words, the component of magnetic field which is parallel to the slot should have the same direction on both sides of the slot in n mode. When the successive cavities, in which the dominant E,,, mode is excited, are deduced from each other by a translation along the axis, this happens to be the case in 0 mode but it is clearly impossible in n mode. A possible way of having the parallel components of magnetic field in the same direction on both sides of the slot in n mode, is to cut the slots radially and to make the basic cavity asymmetrical with respect to the plane defined by a slot and the structure axis; finally the periodicity operation

Fig. 24. The cloverleaf structure (Chodorow et al. [1957]). (a) Exploded view. (b) Magnetic field Magnetic field lines in odd cells. pattern. +- Magnetic field lines in even cells. - - + - - -

must involve not only a translation along the axis, but also a mirror symmetry about the plane of the slot (Chodorow et al. [1957]). In such a structure, when the parallel components of magnetic field are the same on both sides of the slot, the axial electric field is opposite in two adjacent cells, making the phase shift per cell equal to n. However, the geometrical period of the structure contains two cells, and this is also the electrical period, except if the mirror symmetry about the plane of a slot is equivalent to a rotation of the structure about its axis; in this particular case, the electrical period is still one cell. The cloverleaf structure is an example of the latter case, sketched in fig. 24. The diaphragm between two adjacent cavities contains eight identical radial slots at 45" to each other. The basic cavity consists of a cylindrical pipe with the walls indented between every other pair of adjacent slots; these indentations are commonly called "noses". The shape of the wall, similar to a four-leaf clover, suggested the name "cloverleaf". For this structure the mirror symmetry about a slot plane is equivalent to a 45" rotation about the axis: hence the electrical period is just one cavity. The

690

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magnetic field, being parallel to the cavity wall, has large components parallel to the eight slots: it thus provides a large coupling between cells. The basic field configuration used in the cavities is still the E,,, mode, as perturbed by the noses; this configuration is unperturbed by the slots when there is a n. phase shift from cell to cell, as can be seen in fig. 24b. Therefore, confluence may be achieved at n. mode by making the slots about one free space half-wavelength long at the cavity n mode frequency (Chodorow et al. [1957]; Carne [1963]; Knapp [1964]). The lower passband, which is of the forward wave type, may have a bandwidth around 25%. As a slot coupled structure, the cloverleaf is represented by the equivalent circuit of fig. 8.2b, where the extra n. phase shift per period due to the 45" rotation from cell to cell is merely obtained by reversing the wires. With drift tubes for increasing the transit time factor and a cavity diameter of 35 cm, the cloverleaf structure at 805 MHz has an experimental effective shunt impedance of 17, 21, 29, 32 MQ/m respectively for proton energies of 45, 150,250,700 MeV. It is worth remembering that at n. mode in a terminated structure, the slots (as coupling elements) are not excited, and so they do

Fig. 25. Cloverleaf and osculating tetrastyle (Bert [1966]).

not present a risk of electrical breakdown. In fact, a cloverleaf accelerator tank performed very well at high power in Los Alamos (Copenhagen et a]. [1966]). The main disadvantage of the cloverleaf structure is its mechanical complexity. However, since the region of high magnetic field is the central region including the nose tips, it is possible to modify the noses near the outer cylinder, where the field is weak, without affecting appreciably the electrical characteristics of the structure. In fact, the noses may be replaced by circular posts which are osculatory to the nose tips (see fig. 25): the osculating tetrastyle structure is much easier to make than the original cloverleaf and still presents the same electrical properties (Bert [1966]). 4.3.5

The coupling element is one or several stems

These structures are a particular case of the slot-coupled structures, where the slots have become so wide that the diaphragm is reduced to one or several stems supporting a drift tube (see fig. 18). As for the slot-coupled structures, confluence is possible at

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0 mode between a lower "stem" passband which is of the backward wave type, and an upper "cavity" passband which is of the forward wave type.

The rnultisten?structure. This structure was first proposed by Giordano at Brookhaven (Giordano et al. [1966]). The cavity field has again the E,,, mode configuration, while the stems produce a lower passband which may become confluent at 0 mode with the cavity passband; this structure is thus an Alvarez structure which can be compensated by a suitable arrangement of the stems, bringing the stem 0 mode resonance in coincidence with the Alvarez 0 mode frequency. We first observe that this latter frequency is little affected by the presence of the stems, which are at right angle to the electric field of the Alvarez 0 mode; moreover the frequency shift of the

Fig. 26. (a, b, c) Successive transformations of a single-stem structure with short cells at 0 mode. (d) Cross-section of the uniform structure 26c (one-stem configuration). +, conduction current.

Alvarez 0 mode due to the stems is readily evaluated by using Slater's perturbation formula (Parzen [1967]). The problem of finding the best stem configuration thus reduces to that of finding the frequency of the stem 0 mode resonance. The case of infinitely short cells. The case of infinitely short cells (very low proton energy) is particularly simple. Consider first (fig. 26a) that each drift tube is supported by one stem, all stems being parallel. At the stem 0 mode, all stems carry in-phase identical currents and all drift tubes are charged with identical electric charges. In

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this case there is little field in the gaps between the drift tubes: the frequency is not altered very much if the gaps are filled in with metal, thus yielding the arrangement of fig. 26b where a coaxial inner conductor having the same diameter as the drift tubes is periodically supported by the stems. Now, if the distance L between adjacent stems is very small compared with the cavity diameter, the stems may be replaced by a thin partition wall extending between the inner and outer cylinders, as in fig. 26c. The electromagnetic fields (and the frequency) of the stem 0 mode in the original drift tube structure are thus approximately the same as those of a 0 mode in a uniform (cylindrical) coaxial waveguide with a partition wall representing the stems. The latter uniform waveguide mode must produce radial currents flowing in the same direction on both sides of the partition

Fig. 27. Two stems configuration: a < 4 < 2n. v+/n = 1, 3, 5, ... or slot length //(+A) w 1, 3, 5, I is taken at the average radius between +d and 30. (a) Structure with period L . (b) Structure with period 2L. -stems on even drift tubes. ----- stems on odd drift tubes.

...

wall: hence it is a H,, mode at cut-off, with Hz varying as cosvcp where v is a halfinteger (see fig. 26d). Since the frequency of a H,,, mode goes up with v and p, the lowest possible mode corresponds to v = + and p = 1. Therefore, in the case of one stem per drift tube and very short cells, a good approximation to the stem 0 mode frequency is given by the frequency of the H+,, mode in the corresponding uniform coaxial waveguide. The current distribution along the stems is approximated by the radial variation of Hz in the uniform coaxial waveguide, i.e. by Bessel functions of order v and argument wrlc. For p = 1, Hz has no zero between th; irner and outer cylinders: therefore the current keeps a constant sign along the stems, with its amplitude decreasing steadily from the outer cylinder to the drift tube. When there are two stems per drift tube, the corresponding uniform coaxial waveguide has two partition walls at an angle 4 (see fig. 27a). In this case v+ must be a multiple of z*, which corresponds to an azimuthal resonance between the two partition walls. Such a resonance is similar to a slot resonance: the length of the (wide) slot, taken at the average radius between inner and outer cylinders, is then close to a multiple of fl.The lowest H,,,, mode frequency is thus obtained when the

* For the stem resonance being coupled to the cylindrically symmetrical EOImode, v# must be an odd multiple of a, so that the currents in the stems have the same radial sign.

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slot is half a wavelength long (i.e. for v = n/4) and when p = 1, which corresponds to a pure azimuthal resonance without radial resonance; the currents in the two stems have then the same radial sign, as shown in fig. 27a. The frequency of the H,,,, mode in the coaxial partitioned waveguide or what is nearly the same, the frequency of the stem 0 mode in a short cell structure, is thus easily varied by varying the angle 4 between the stems, a smaller angle producing a higher frequency. But two angles are defined by the two stems: one is smaller, the other larger than n. Therefore, in a 2-stem structure, the coupling element has two resonant frequencies, and it might be expected that two passbands are associated with stem resonances. Measurements have indeed shown the existence of the lower passband, corresponding to the larger angle ( 4 > n ) between stems (D6me et al.

Fig. 28. Three stems configuration: 2x13 < 4 < n. (a) Structure with period L. (b) Structure with period 2L. s t e m s on even drift tubes. - - - - - stems on odd drift tubes.

[1967]). Although no experimental evidence has ever been found for a higher stem passband corresponding to the smaller angle between stems, the cavity passband behaves as if such a second stem passband were present (D8me and White [I 9681 ; see also later on, compensated multistem structures). Nevertheless, in the following we shall consider only the possibility of confluence with the lower stem passband: therefore, only the largest angle formed by the stems needs consideration. As a consequence, in a 2-stem structure the maximum frequency of the stem 0 mode is obtained with the two stems equally spaced at 180" (i.e. v= I). This maximum frequency of the stem 0 mode in a 2-stem structure is still lower than the frequency of the E,,, mode (D6me et al. [1967]). Therefore, confluence can be achieved only when the largest angle between stems is less than n,which requires a N-stem structure with N > 2 . Figs. 28a and 29a show symmetrical stem configurations for 3 and 4 stems per drift tube: such symmetrical configurations provide two coupling elements with the same resonant frequency, which produces an increased coupling between stem and cavity modes. The largest angle between stems is a minimum when the stems are equally spaced at 2nlN: the maximum frequency of the stem 0 mode is thus obtained with equally spaced stems (yielding v = + N ) . Hence, by increasing the number of stems per drift tube, it is possible t o raise indefinitely the frequency of the stem 0 mode in a structure with very short cells. In fact, the frequency of the Hf mode in the coaxial partitioned waveguide is about 1.2

,,

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times the frequency of the cavity E,,, mode, whereas the frequency of the H,,, mode is about 1.5 times the latter (Dame et al. [1967]). Therefore, a value of N = 3 or 4 should be sufficient in practice. The case of aJinite cell length. Strictly speaking, the above results apply only to infinitely short cells and, because the stem diameter d, must be less than L, also to infinitely thin stems. When L is finite, it may be expected that, if thick enough, the stems will still be equivalent to a continuous partition wall at 0 mode. In fact, the effect of stem diameter may be investigated by considering a stem as the inner conductor of a transmission line, the outer "conductor" being constituted by two magnetic walls at a distance +L,and by the cavity wall which also short-circuits this line at the foot of the stem; the stem line is terminated at its other end by the drift tube capacitance. This point of view is consistent with the current distribution along the stems deduced from the H,,, mode in the coaxial partitioned waveguide.

Fig. 29. Four stems configuration: 2n/4 < 4 6 rr. (a) Structure with period L. (b, c, d) Structures with period 2L. -stems on even drift tubes. -----stems on odd drift tubes.

When LID+ 1, such an analysis (Dame et al. [1967]) yields a relation of the type

(a,,frequency of the stem 0 mode) where f is an ever increasing function of its argument, different for each stem configuration. This general relation yields two important results : 1) For a given LID, the stem 0 mode frequency steadily increases with the stem diameter d,; it rises continuously from 0 for infinitely thin stems, to the value corre-

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spondingtoinfinitely short cells(L/D=O)when thestemsare so thick that 7id,/L z 1 .The frequency of the H , , , mode thus appears as an upper limit for the stem 0 mode. which can be reached only with thick stems. Consequently, structures with 3 or 4 equally spaced stems can be compensated with an appropriate choice of stem diameter. This was discovered experimentally by Giordano on a 4-stem structure when he put flares on the stems, which is equivalent t o varying their diameter (Giordano et al. [1966]). 2) For a given d J D , the stem 0 mode frequency steadily decreases when the cell length L increases. This means that, in order t o maintain local confluence along a variable energy structure with a given stem configuration, the stem diameter must increase (rapidly) with proton energy. The use of a superperiodpL. The derivation of the stem 0 mode frequency limit in a structure with infinitely short cells holds as well when the structure has a period p L ( p integer) involving less than N stems per drift tube, as long as the cross-sectional projections of all stems contained in a period pL yield a N-stern pattern. Such structures are mechanically simpler because they need less stems on each drift tube: on the other hand, in order to produce the same 0 mode frequency with a longer period, the stems must be much thicker than in a N-stem structure with period L. The simplest multiple period is 2L; examples of such structures are also shown in figs. 27, 28, 29, next to the corresponding single period structures. The structures of figs. 28b, 29b, c, d, having the same angle 4 and the same period 2L, have practically the same stem 0 mode frequency *. When 4 =90°, the bars in fig. 29d alternate at 90" from one drift tube t o the next: the corresponding structure is called a cross-bar structure. Another way of creating a superperiod p L is t o put N stems on every pth drift tube, while supporting the other drift tubes with a single stem. The larger N, the larger p may be for a given stem diameter at a given energy. This method has been proposed by Giordano [I9681 for compensating an Alvarez structure around 20 MeV with ,Y=4 and p = 3 o r 4. As a first approximation, the stem diameter appropriate for confluence in a multiple period structure, may be obtained from eq. (62) when L is replaced by pL. Since 7idJL is generally smaller than unity, the following comparison is obtained between structures with single and multiple period. -

-

Structure with period L -

Stem diameter Number of stems per drift tube Number of stems x stem diameter

-

-

-

Structure with period pL ( p > 1 )

-

ds N Nds

* It should be noticed that the structures of figs. 29b and 29c have an electrical period L.

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Since the rf losses induced on the stems by the magnetic field of the E,,, mode are proportional to the stem diameter, the last line shows that the total rf losses on the stems are always larger in a compensated structure with multiple period. Nevertheless, such structures must be used when the corresponding single period structure requires a stem diameter so thin that it would be impractical. Compensated multistem structures. From the above, it is clear that an Alvarez structure can be compensated by using a 3- or 4-stem pattern with an appropriate stem diameter in a structure with period L or 2L. For minimum rf losses on the stems, thin stems would be desirable; but for mechanical reasons, a stem diameter of 2 cm at 200 MHz seems a minimum. Even with this small diameter, each stem increases the rf losses by 24.5% at 0.76 MeV and by 3.75% at 200 MeV (University of Wisconsin [1967]). At the beginning of the linac, double period structures should be used because single period structures would require a stem diameter smaller than 2 cm for compensation. As the energy increases, the double period structures should be replaced by single period ones as soon as the stem diameter for compensation reaches 2 cm in the latter structures.

Fig. 30. Multistem structure with two stems per Energy range Stem diameter for compensation at 200 MHz Angle n between stems Stem losses/Drift tube and outer wall losses Effective shunt impedance at 200 MHz

drift tube. 0.75-8 MeV 2.0-2.0 cm 45"-90" 0.49-0.19 17.9-40.5 MQ/m

Very low energies (0.75-8 MeV). From eq. (62), the effect of dJL and of LID on the stem 0 mode resonance appears only through the product(pL/D) log [nd,/(pL)]. The influence of the stem diameter and of the cell length is thus very small at very low energies (LID6 I), and the stem 0 mode frequency is very close to the frequency of the H,,, mode in the corresponding coaxial partitioned waveguide, whichever stem diameter is used. Therefore, at the beginning of the linac one may choose d,=2 cm with the double period 3-stem configuration of fig. 28b or preferably with one amongst the 4-stem configurations of figs. 29b, 29c or 29d. For mechanical simplicity, the structure of fig. 29c has been selected in fig. 30. For a ratio d/D of 0.178, the maximum angle 4 = n / v derived from the frequency of the H,,, mode in the corresponding coaxial partitioned waveguide is 143" for confluence, whereas the optimum experimental value is about 135" throughout the energy range 0.75-5 MeV (D8me et al.

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[1967]). In order to maintain local compensation a t energies above 5 MeV, the angle should be decreased progressively until it reaches the minimum value flr for a 4-stem configuration (around 8 MeV with d, = 2 cm). In the energy range 0.75-5 MeV, measurements have also been made on a single period 2-stem structure (fig. 27a), in order to investigate the influence of the smaller angle between stems (Dame and White [1968]). This structure has been compared with the structure of fig. 29b, where the two stems on each drift tube form the same angle, as in the single period structure, but alternate a t 180' from one drift tube to the next. In the double period structure the stem passband corresponds to an angle 4 smaller than n; in the single period structure it corresponds to an angle (2n-4) larger than n. Nevertheless, the cavity passbands around the E,,, mode are the same in both structures for 4 = 1 SO0, l65", 1 50°, 135" (the value for confluence), whereas they are entirely different for 4=12O0, 105", 90'. Although q5=13S0 produces the largest mode spacing around the E , , , mode in both structures, only the double period structure appears to be stabilized against perturbations; the single period structure is then very sensitive to perturbations but becomes stabilized when 4 = 150". No explanation has yet been found for this behaviour of the 2-stem structure. On the other hand, the stem configurations which have been used so far in Alvarez linacs are 1 stem per drift tube or 2 stems a t 90°, all stems being parallel. For suc'i stem angles ( 4 = 27t or +lr) the stem passband is so much lower than the E,, cavity passband, that its influence on the latter is very small: with respect t o the no-stem case, the effect of the stems on mode-spacing around the E,,, mode is almost negligible.

4

Low energies (8-20 M e V ) . Above 8 MeV the stem pattern should remain, with 4 = f n , that of fig. 29b, c or d (the last one being a cross-bar structure), but the stem diameter appropriate for compensation grows rapidly with energy (see fig. 31). Another possibility is the double period structure of fig. 32, where the four stems are put on the same drift tube, but only on every other drift tube. This gives the structure an electrical period 2 L , instead of L for all the other multistem structures of figs. 30 to 34. Medium energies (20-45 M e V ) and high energies (45-200 M e V ) . A single period structure may then be used, with 3 stems a t 120" below 45 MeV (see fig. 33) and 4 stems at 90" above 45 MeV (see fig. 34). For all multistem structures, the group velocity at conjuence is close to 0.35c, which corresponds to a very large coupling between stem and cavity modes (Dame et al. [1967]). Stem diameter,for local conjuence. The figs. 30 t o 34 also give the values of stem diameter necessary for local compensation, as deduced from measurements (Dame et al. [1967]; D6me and White [1968]; Giordano, private communication), and the stem losses as a fraction of the rf losses in a n Alvarez structure without any stem (University of Wisconsin [1967]); the stem losses are computed on the assumption that in a given type of structure all stems have the same diameter. Although the stem

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Fig. 31. Cross-bar structure in 0 mode. Energy range 8-17 MeV Stem diameter for compensation at 200 MHz 2.0-5.6 cm Stem losses/Drift tube and outer wall losses 0.19-0.38 Effective shunt impedance at 200 MHz 40.5-38.3 MQ/m

Fig. 32. Multistem structure with four stems on every other drift tube. Energy range 10-17 MeV Stem diameter for compensation a t 200 MHz 2.1-3.7 cm Stem losses/Drift tube and outer wall losses 0.22-0.32 Effective shunt impedance at 200 MHz 42.6-40.0 MQ/m

Fig. 33. Multistem structure with three stems per Energy range Stem diameter for compensation at 200 MHz Stem losses/Drift tube and outer wall losses Effective shunt impedance at 200 MHz

drift tube. 17-45 MeV 2.0-5.4 cm 0.21-0.40 43.7-29.9 MQ/m

Fig. 34. Multistem structure with four stems per drift tube. Energy range 45-100-200 MeV Stem diameter for compensation at 200 MHz 2.2-5.8-1 1.5 cm Stem losses/Drift tube and outer wall losses 0.21-0.47-0.86 Effective shunt impedance at 200 MHz 34.6-19.0-8.1 MQ/m

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losses may be relatively large, especially a t the very low energy end of the linac, it should be kept in mind that one or two stems are needed anyway t o support each drift tube. In the early measurements on a scale model of the CERN linac tank 2 (energy range 10-30 MeV), a constant stem diameter in a cross-bar structure was used throughout the tank, producing confluence only on the average. In such a case the local frequency of the stem 0 mode steadily decreases from the low energy end t o the high energy end of the tank. Local compensation occurs only for a middle cell, the short cells being overcompensated and the long cells being undercompensated. From the local dispersion curves shown in fig. 35c, it is seen that near the E,,, mode w,, the tank is partially cut off a t the long cell end when w <ma, and at the short cell end when w>o,. This appears clearly in the measured variation of E: obtained by a bead perturbation technique* along the CERN tank 2 model as compensated on the

Fig. 35. Local dispersion curves for a variable L structure with constant stem diameter ( L I < Lz). (a) Structure undercompensated for L I , Lz. (b) Structure overcompensated for L I , L2. (c) Structure overcompensated for L I , undercompensated for Lz (this is the case of a structure which is compensated on the average).

average with a cross-bar structure (see fig. 36): the zeros of the field become more widely spaced towards the cut-off region, where the field decays t o the end plate without passing through another zero. On the other hand, when a structure is compensated locally, i.e. when there is no stopband throughout the tank, there is no cut-off region in the modes around the E,,, mode: in fact, this property provides the best criterium for determining experimentally the stem diameters which are appropriate for local compensation (D6me and White [1968]). Since the effect of perturbations in a tank is to excite the neighbouring modes, a good field stability can be obtained ,-

*

The perturbation techniques are described in ch. B.I.1, section 8, Cold tests, eq. (120).


Fig. 36. Variation of Ez2along a scale model of the CERN linac tank 2 (41 cells), showing partial cut-off near the Alvarez 0 mode. The positive values of B refer to the upper passband, while the negative values of B refer to the lower passband (DBme and White [1967]).

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only if the modes around 0 = 0 in the upper and in the lower passband are able to cancel each other, which requires that they be cut off nowhere along the tank or, what is equivalent, that the structure be compensated locally throughout its length. Instead of adjusting the stem diameter exactly for local confluence, the fine tuning of the stem 0 mode resonance may be achieved with plungers put on the cavity wall, in the plane of the stems (Giordano [I 9681). Equivalent circuit. A multistem structure is expected to be represented by the equivalent circuit of fig. 20, where L,, C , represent the inductance of a drift tube and its capacitance to the outer cylinder, C , represents the capacitance of a gap and L, represents the inductance of the stems. This equivalent circuit has been used by Nishikawa (Batchelor et al. [1967]) for the multistem drift tube structures measured by Giordano in Brookhaven at high energies, but it fails t o represent correctly the experimental dispersion curves a t very low energies *. Shunt impedance. From the data of figs. 30 t o 34, it is seen that the stem losses in a compensated multistem structure are by no means negligible. In the worst case, i.e. the 4-stem structure between 100 and 200 MeV, they could be reduced by using 6 stems a t 60°, but this still increases the mechanical complexity of the structure. Such considerations led the Los Alamos group t o propose another kind of coupling element between the drift tubes: the post couplers. the post-coupled structure. This structure was first proposed at Los Alamos (Swenson et al. [1967]) in order t o reduce the stem losses in a multistem structure. These large losses come essentially from the necessity of increasing the number of stems and their diameter at higher energies, in order t o maintain the stem 0 mode frequency as high as the E,,, mode. This situation can be improved in two ways: 1) The stems are replaced by flat flares which are normal to the structure axis. This leads back to a slotted iris structure with sectoral slots, confluent a t 0 mode; the losses on the slotted diaphragm are substantially smaller than those on thick stems. So far this possibility has not been investigated. 2) The stems are replaced by posts, which extend from the outer cylinder towards the drift tube without touching the latter. The frequency of the post 0 mode is thus higher than that of the corresponding stem 0 mode; moreover it can be raised to infinity by shortening the posts t o zero length. Therefore, if in a compensated multistem structure, the stems are replaced by posts with the same diameter, the structure becomes overcompensated; compensation may be reached again either by decreasing the post diameter or by using fewer posts per drift tube, eventually going t o a double period structure, i.e. going backwards from the stem (or post) configuration of fig. 34 to those of the preceding figures. By appropriately shortening the posts, it is thus possible t o use the stem configuration of fig. 30 for any proton energy; even more, it

* This could be due to the fact that the equivalent circuit of fig. 20 does not take into account a second next nearest neighbour coupling between stems, which is certainly not negligible when the stems are alternating at 180' from one drift tube to the next.

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becomes possible to use the otherwise undercompensated stem configurations of fig. 27 or of fig. 26d. The last configurations with 4 = n are the simplest ones. They correspond respectively to: a) one post in the middle of each drift tube, all posts being parallel (fig. 26d); b) one post per drift tube, each post alternating at 180" from one drift tube to the next (fig. 27b with $ = 71); c) two posts at 180' per drift tube (fig. 27a with 4 = n). In standing wave, the +n mode of the post passband is the same for cases a) and b), since every other post is then unexcited. As for the slot coupled and the multistem structures, alternating the coupling element at 180" from one drift tube to the next widens the coupling element passband, leaving its +n mode practically unchanged. This results in a larger mode spacing around 0 mode at confluence, thus in a better stabilized structure. Therefore, when there is only one post per drift tube, the posts should alternate at 180' from one drift tube to the next. Each drift tube must be supported by a stem, which is put at 90" with respect to the posts in order to minimize the coupling between posts and stems. This leads t o the post-coupled structure of

Fig. 37. Post-coupled structure. Energy range Assumed stem and post diamcter for compensation at 200 MHz Stem and post losses/Drift tube and outer wall losses Effective shunt impedance at 200 MHz

0.75-45-100-200 MeV 2.0-5.0-5.0-5.0 cm 0.49-0.24-0.20-0.19 17.9-33.6-23.3-12.6 MQ/m

fig. 37, which was found to be the most favourable one by the Los Alamos group (Swenson et al. [1967]). By analogy with the multistem structure, it is expected that if the post diameter and length are kept constant in a variable energy structure, the post 0 mode frequency will go down when the cell length increases. Therefore, in order to maintain local confluence along the structure, the posts (whose diameter is assumed to be kept constant) must be progressively shortened towards the higher proton energies. Nevertheless, this variation with energy is slow, and the gap between post and drift tube remains always a small fraction of the post length, at least for practical post diameters*. Assuming that the post and stem diameters are chosen equal and of a convenient size **, fig. 37 gives an estimate of the stem and post losses; this estimate should be a

*

If the posts become thinner, the gap between post and drift tube should be made smaller to maintain confluence (Los Alamos measurements, June 1967). * * In practice, the post diameter may be smaller than the stem diameter.

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little pessimistic, because the rf losses on a post have been taken as equal to the losses on a stem with the same diameter. In a terminated structure, the posts are excited at confluence only in case of power flow, tuning errors and transients; these excitations are normally much t o o small in order t o produce voltage breakdown between post and drift tube. Similarly. the transverse electric field set u p on the axis of the accelerating gaps by post excitation (see fig. 38) has been measured in Los Alamos using a needle perturbation technique (Schneider and Swenson [1968]): it is also much too small to be of any concern for the radial motion of the particles. Moreover, this transverse electric field could be suppressed entirely by using two posts at 180' per drift tube.

Fig. 38. Field pattern of the post 0 mode at confluence.

The coupling between post and cavity modes is not as strong as the coupling between stem and cavity modes in a multistem structure as a result, the group velocity czt confluence in post-coupled structures does not exceed 0.10c, in contrast with 0 . 3 5 ~ in multistem structures. This difference appears clearly in fig. 39, which shows typical dispersion curves of a multistem and of a post-coupled structure. Nevertheless. the group velocity obtained with post-coupled structures is large enough to provide good field strbility against perturbations in practice. In order to achieve local confluence along a variable energy tank, the two cavities at the ends of the tank are detuned in opposite ways (so that the overall resonant frequency of the tank is unchanged) and the posts are tuned until the gap field is no longer perturbed by a large detuning of the end-cells (Swenson et al. [1967]). Typically, for a structure around 50 MeV, with & 1 % detuning of the end-cells and 0.5% accuracy on the measured axial field, the stopband width can be reduced locally to about 0.1 % of the operating frequency; this corresponds to an accuracy of about 0.3 mm on the post length at 200 MHz*. The post length can easily be adjusted by a screw assembly; alternatively the post length may be kept constant throughout the accelerator tank, and the frequency adjusted by movable collars located near the ends of the posts (Schneider and Murdoch [1968]). Adjustment of the acceleratingjield. When a structure is compensated locally, the

* See Tuning errors a n d f i l d stabiliry in section 4.2.

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axial electric field is so insensitive to mechanical errors that it approaches very nearly the theoretical field for a perfectly machined structure; on the other hand this theoretical electric field, when averaged over each cell, is practically constant along the structure (Dame and Lapostolle 119681). If for the beam dynamics, or because of perturbations such as drive loops, vacuum ports, etc., it is desired to act on the field distribution in a compensated structure, detuning of the accelerating cells is no longer

Fig. 39. Comparison of the dispersion curves for compensated multistem and post-coupled structures. (a) Dispersion curve of a uniform 4-stem structure at confluence (Los Alamos Scientific Laboratory, 1967). Proton energy: 92.7 MeV. At 0 mode, v,/c = 0.35. (b) Dispersion curve of a uniform post-coupled structure at confluence (Schneider and Murdoch [1968]). Proton energy: 55.5 MeV. At 0 mode, v,/c = 0.09.

effective (as it would be in an uncompensated structure), while detuning of the coupling cells would destroy the local compensation. The only way to affect the field in the main cells is then to act on the symmetry of the coupling element so that its coupling coefficients are different for the two adjacent accelerating cells (Knapp, Knapp and Potter [1968, eq. 201). This might be done by shifting the post axially away from the center of the drift tube, but is much more easily achieved by rotating an eccentric tab at the end of the post (Swenson et al. [I 9671) : it is sufficient in practice that the extension of the tab be equal to the post diameter (see fig. 40). The new field distribution obtained with the tabs is also stable against tuning errors in the main

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cells. When the tab is rotated from its symmetrical position, the post becomes excited; in order to keep this post excitation small, the tabs should not be used to induce more than a few percent variation in the main accelerating field. Although the post excitation could be reduced t o zero by suitably detuning the main cells, it is not advisable to use this possibility because it would require local tuners, which otherwise can be completely omitted in a compensated structure.

eccentric tab

Fig. 40. Post-coupled structure with eccentric tabs.

Equivalent circuit. A post-coupled structure is expected t o be represented by the equivalent circuit of fig. 41, where as in fig. 20 Lo, C, represent the inductance of a drift tube and its capacitance to the outer cylinder, Co represents the capacitance of a gap between drift tubes, L, represents the inductance of the supporting stem and L,, C, represent respectively the inductance of a post and its capacitance t o the drift tube. This equivalent circuit has been proposed by Nishikawa [1967, unpublished]. From this equivalent circuit, a post coupled structure should have four passbands instead of the two passbands in a multistem structure: one passband corresponds t o the Lo, C, resonance (it is the E,, mode passband), another (narrow) passband corresponds t o the L,, C , series resonance, where the post behaves as a short circuit between the drift tube and the outer cylinder (it is the post passband); finally two passbands arise from the L,, C , stem resonance as split by the resonant post. Indeed, at frequencies below the L,, C, post resonance, the post behaves as a capacitance which lowers the resonant frequency of L,, C , ; at frequencies above the post resonance, the post behaves as an inductance which raises the resonant frequency of L,, C,. There result two post and stem passbands (of the backward wave type), which are separated in frequency by both the L,C, and the L,C, resonances. The frequency of the L,C, single stem resonance is always much lower than that of the E,,,, mode; therefore, if the L2C2 post resonance is also made lower in frequency than the E,,, mode (which is always possible by reducing the gap between post and drift tube), the upper post and stem passband can become confluent a t 0 mode with the E,, passband (see fig. 41b). Until now the measurements of dispersion curves of post-coupled structures have been limited to these two passbands. Shunt impedance. Estimates of the shunt impedance for post-coupled structures are given in fig. 37, taking the post diameter as equal to that of the supporting stem. These figures could be increased slightly by reducing the post diameter.

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Fig. 41. (a) Equivalent circuit of a post-coupled structure. The arrows indicate the positive directions for the electric currents. Dispersion relation:

where

In a post-coupled Alvarez structure, ko> 0 and kz > 0 for posts all parallel, kl > 0 for stems all parallel, kz < 0 for posts alternating at 180° kl < 0 for stems alternating at 180°, (b) Dispersion curves, when kz is negative.

The cross-bar structure in .rr mode. In all the above structures, the field pattern which is used for particle acceleration is essentially the E,,, resonance of a chain of drift tube loaded cavities, where two adjacent cavities are excited either in phase or out of phase. This mode, therefore, was called a cavity mode. The coupling element resonances were useful for compensating the structure, but did not play a role for the acceleration of particles. On the other hand, the stem resonances, by bringing electric charges on the drift tubes, produce a useful accelerating field in the gaps between drift tubes at all modes but the 0 mode. In standing waves, the only modes where the amplitude of the accelerating field is constant from gap to gap are the 0, $n and rt modes*. In all multistem structures, the stem resonances in +n or n mode could thus be used for particle acceleration. Two structures based on this principle are worth mentioning because of their extremely large bandwidth: the cross-bar, and the interdigital line. The uniform or symmetrical cross-bar structure. Consider first the uniform cross-bar structure of fig. 31. The symmetry planes of this structure are the cross-sectional planes passing through the axes of the bars. In order to leave the structure unpertur-

+

* In standing waves, the amplitude of the gap field varies as Icos(m0 yl)l along the structure, and this is independent of m only in three cases: 0 = 0 , 0 = 3n with u, = + i n , or 0 = n.

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bed, any metallic plate used as a termination must be located in a symmetry plane: hence it must cut a bar in two halves. Because of the antisymmetry about a conducting plane, the electric currents which flow along the two half-bars must have opposite directions, making the total longitudinal current along the bar equal to zero: this bar is thus unexcited. If there is a rr phase shift per electrical period of the structure, i.e. a rc phase shift between two adjacent drift tubes, then the net current is zero in all bars as soon as it is zero in one: this means that the rr mode of the bar passband cannot be excited in a terminated cross-bar structure. On the other hand, in the +rr mode, all bars which are parallel t o a terminating bar d o not carry any net current, whereas the bars which are perpendicular to a terminating bar may be excited. Therefore, if a cross-bar structure is terminated with two metallic plates which short-circuit two parallel bars, the whole set of bars parallel t o the end bars will remain unexcited in the frr mode, while the set of bars orthogonal to the end bars will be excited. This was apparent in the early measurements on the cross-bar structure (Carne [1963]). The field pattern in the i n mode is unaffected by metallic planes passing through the unexcited bars. With this picture, a current-carrying stem appears as the inner conductor of a transmission line, the outer conductor being constituted by the two adjacent cross-sectional planes and by the cavity wall which also short-circuits this line at the foot of the stem; each stem line is terminated at its other end by the drift tube capacitance. This point of view, combined with a semi-empirical formula for the drift tube capacitance, allows the shunt-impedance and the Q of the structure to be evaluated (Carne et al. [1965]). The unexcited bars have a very small influence on the fields a t the jn mode, but they become excited in the neighbouring modes and therefore affect the dispersion curve strongly. The bar passband is of the backward wave type and may be very wide. As in the slot-coupled structures, the bandwidth is a minimum when all bars are parallel, and reaches its maximum value for a 90" rotation of alternate bars. Moreover, as in all multistem structures, the bar 0 mode frequency steadily increases with the stem diameter. whereas the frequency of the +n mode increases much more slowly. Therefore, increasing the bar diameter steadily increases the bandwidth of the bar passband. For practical bar diameters and energies above 100 MeV, the relative bandwidth may be larger than 5 w , resulting in a group velocity around 0.13-0.20~at in mode. This large bandwidth is due t o the 90" rotation of the bars from cell to cell, as in the slot-coupled structure (see fig. 18c). The asymmetrical cross-bar structure. S o far we have considered a uniform structure in +rr mode, which is insensitive to perturbations but is not optimized for shunt impedance. For travelling waves in +n mode, the shunt impedance is twice the value for standing waves, which is a t once obvious in the uniform cross-bar structure because both sets of bars are then equally excited and contribute equally to particle acceleration. In fact, the very high values of shunt impedance given in early reports on the cross-bar structure (Carne [1963]) refer to travelling wave operation. If there were no drift tubes, the TEM waves propagating along one set of parallel

bars with a n phase shift between two adjacent parallel bars would produce on the z-axis an electric field of the form (see fig. 42b) Ez

-

1

cos (+nz/L)

=2

sin t n n cos (+nnz/L).

The first space-harmonic, cos (+nz/L), is synchronous with the particles. It is obvious that the transit time factor

will be improved if E, is concentrated around z=0, i.e. around the unexcited bars. This is obtained by putting long drift tubes on the current-carrying bars, and by shortening as much as possible the drift tubes on the unexcited bars: the short drift tube length is then reduced to the unexcited bar diameter, both being chosen as small as is practical (say, between 2 and 4 cm at 200 MHz). This leads to the asymmetrical structure of fig. 42a. The electrical period of the structure is now 2L, and the mode is n. The x mode is indeed favourable for a good shunt impedance, but at the same time it introduces a stopband in the dispersion curve.

Conductm current

-L

0

L

(b)

Fig. 42. (a) Cross-bar structure in a mode. 100-300 MeV Energy range Main-bar diameter at 200 MHz 8-1 0 cm Effective shunt impedance at 200 MHz 18.3-15.4 MQ/m (b) Field pattern of the bar n mode.

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The solution to this difficulty is to make the n modes of both sets of bars resonant at the same frequency, thereby producing confluence at n mode. Such a cross-bar structure appears as a resonantly coupled structure, where the coupling element between the current-carrying bars is the other set of bars. Since the thin (unexcited) bars have a larger characteristic impedance than the current-carrying bars, their n mode resonant frequency is in general lower than that of the thick bars. Therefore, confluence can easily be achieved by the use of tuners (deformable wall) at the base of the thin bars: when pushed inwards, the tuners will effectively shorten these bars, and thus increase their n mode frequency. As in the multistem and post-coupled structures, the base tuners can be adjusted in an accelerator tank until the axial field is no longer affected by perturbations introduced at the ends of the tank. Shunt impedance. Considering the main bars as coaxial resonators allows the shunt impedance and the Q of the cross-bar structure to be computed in n mode. Up to 92% of the theoretical Q has been achieved in practical models (Carne et al. [1965]; D6me and White [1968]), whereas the measured Z/Q values were always 10-20% higher than theoretical. Despite this discrepancy in absolute value of Z/Q, the computed values can be used to optimize the geometry of the structure for maximum shunt impedance. The drift tube diameter was chosen as small as is compatible with the housing of focussing quadrupoles, i.e. about 12 cm at 200 MHz. At energies higher than 80 MeV, and up to P= 1, a very near optimum geometry may be taken as 66 cm for the cavity diameter, and 8 to 13.5 cm for the main bar diameter. The computed maximum shunt impedance drops slowly from 18.5 MQ/m at 80 MeV to 15.4 MQ/m at 300 MeV, and tends asymptotically to 10.6 MQ/m at relativistic energies. Finally it should be mentioned that a full scale model of a cross-bar cell for 140 MeV protons has successfully produced an average axial electric field of 4.6 MV/m (D8me and White [1968]). Compared with a post-coupled Alvarez structure, a cross-bar structure in n mode has a smaller diameter (66 cm against 84 cm for the Alvarez structure at 200 MHz) and features a definitely better shunt impedance with a lower Q at energies above 140 MeV.

5 Structures with non-resonant coupling 5.1 Structures of the Sloan-Lawrence type

The first linear accelerator which accelerated ions to a kinetic energy larger than 1 MeV per ion was built by Sloan and Lawrence [1931], following a principle suggested by Wideroe [1928]. As sketched in fig. 43, it is a row of drift tubes which are alternatively connected to the two terminals of a high voltage rf generator. For synchronism. the distance between two successive gaps is L=+/U. This structure may be viewed as being operated with a TL phase shift per electrical period L or a 2n phase shift per

710


G.D~ME

w Fig. 43. The Sloan-Lawrence lumped parameter structure.

geometrical period 2L: in principle, thers is no rf phase shift along the twin line connected to the high voltage terminals. However, when the line length becomes an appreciable fraction of an rf wavelength, the effects of a finite propagation time (which are enhanced by the capacitive loading of the twin line due to the drift tubes) become sensible, resulting in voltage standing waves along the line. The corresponding variations of the accelerating voltage along the structure may be reduced by bending the low voltage parts out of the main accelerating line (see fig. 44), but they cannot be suppressed (Nassibian et al. [1961]; Bieth et al. [1966]). Instead of a twin line, a coaxial system may also be used (Bohne [1968]). When the secondary winding inductance in fig. 43 takes the form of one turn of a very wide tape which is bent as a cylinder, the Sloan-Lawrence structure goes over into the distributed parameter structure of fig. 45b, with currents flowing azimuthally around the cylinder. These currents produce a magnetic field parallel to the axis of the structure: hence the name "H-type accelerating system" (Zeidlits et al. [1962]; Bomko et al. [1965]). If two such structures are put in parallel and then combined into a single cylinder (figs. 45c and 45d) there results the so-called "interdigital line".

Fig. 44. The Sloan-Lawrence twin line, fed by auxiliary lines at right angle.

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Fig. 45. Evolution from a resonant circuit with lumped parameters to a resonant cavity structure of the interdigital type (Zeidlits et al. [1962]).

5.1.1

The interdigital line in n mode

The interdigital line is a drift tube structure where each drift tube is supported by a single (round or flat) stem and the stems alternate at 180" from one drift tube to the next (see fig. 46a). The electrical period of the structure is L, whereas the geometrical period is 2L. This structure may also be considered as a slot-coupled structure where the peripheral slot is rotated 180" between two adjacent cavities (see fig. 18d). As such, it should have a wide slot (or stem) passband: in fact, amongst all multistem structures, it probably has the largest normalized bandwidth of the stem passband (around 96%), resulting in a very low frequency for the n mode stem resonance. In i n mode of the stem passband, the interdigital line could be used in much the same way as previously described for the uniform cross-bar structure. Nevertheless, since the currents which bring electric charges on a drift tube must flow through one stem compared with two stems in the cross-bar, it is expected that the interdigital line will have a lower shunt impedance than the cross-bar*. Therefore, the main interest of the interdigital line comes from its very low frequency in n mode. Since the cell length in n mode operation must be equal to 4/32 where I. is the free space wavelength, this structure is especially suitable for accelerating particles with a small /3, i.e. low energy protons or heavy ions. In order to estimate the resonant frequency and the shunt impedance in n mode, the field pattern shown in fig. 46b suggests t o represent the parallel stems by a continuous conducting wall, and the drift tubes by a distributed capacitance C , per unit length of the structure (see

* A similar argument applies t o the "organ-pipe" structure. This structure is essentially the single stem structure of fig. 26a, operated in n mode of the stem passband (Blewett [1956]). The shunt impedance of such a structure in n mode is smaller than that of a structure where the drift tubes are supported by parallel full bars, and the latter in turn is comparable t o the shunt impedance of the asymmetrical cross-bar structure having the same geometrical period. The asymmetrical cross-bar structure is not only superior to the organ-pipe structure for shunt impedance in TC mode; it is also by far superior for the bandwidth of the stem passband.

712

G.D~ME


fig. 46c). The n mode stem resonance is thus approximated by the lower (H mode) cut-off frequency of a uniform ridged circular waveguide *. As in all structures of the Sloan-Lawrence type, the shunt impedance varies roughly as o)CTfp-' (Pottier [1957]; Zeidlits et al. [1962]). In order to decrease C , , the drift tube diameter should be as small as compatible with the housing of focussing quadrupoles. With a tank diameter of 1.5 m (0.151) and a drift tube diameter of 10 cm, the shunt impedance may be as high as 80 MQ/m at 30 MHz for a

(a)

(b)

Lconduction

current

Fig. 46. Interdigital line. Energy range 0.5-10 MeV 50 MQ/m Average effective shunt impedance at 60 MHz (a) Cut-away view. (b) Periodic field pattern of the stem n mode. (c) Idealized cross-section in n mode.

particle energy around 1 MeV/nucleon, making the interdigital line very attractive for a heavy ion linac (Bomko et al. [1965]; Chabert et al. [1968])** (see ch. D.2). Since C , is a decreasing function of L = 381, the drop of shunt impedance with energy is not as pronounced as fi-'. Nevertheless, it is rapid enough to make this structure uninteresting for a proton linac above 10 MeV (Blewett [1956]; Bomko et al. [1965]). Since the symmetry planes of the interdigital line pass through the stem axes, it is not possible to excite the n mode stem resonance in a terminated structure. Therefore,

* This justifies the name "H-type structure" which is sometimes also given to the interdigital line (Zeidlits et al. [1962]; Bomko et al. [1965]). This nomenclature is avoided here because it should as well be applied to all multistem structures in the stem passband (the asymmetrical cross-bar structure in n mode, for example). ** In these references, the Russian definition of shunt impedance (the circuit-theory shunt impedance) is used, and the transit time factor is not included. The values quoted here are thus obtained when multiplying the former ones by 2 T i .


C.l.le

713

in a tank of total length 1= N L made up of N full cells, the mode n - ( n / N )is used because it is closest to the n mode. As a consequence, the amplitude of the average axial electric field per cell varies as (- 1)"-'sin (nzll)where m numbers the accelerating gaps ( M I = 1, 2,. . ., N ) *. In order to avoid the reduction of standing wave shunt impedance by a factor 2 with respect to n mode operation, the particles are not synchronized with the n - ( n / N )mode direct travelling wave, but with a fictitious n mode wave (i.e. L=@A rigorously). This enables the reflected wave, which contains a space-harmonic with a phase shift per cell n+ ( n / N ) ,to contribute almost equally to the acceleration of the particles: the standing wave shunt impedance in rr-(TIIN) mode is then reduced only by a factor 8/n2 with respect to n mode operation.** This factor is already included i n the figures for shunt impedance just quoted. Finally, it should be noticed that the interdigital line in n - ( n / N ) mode is not a resonantly coupled structure. Nevertheless, it is not too sensitive to perturbations because of the wide mode spacing which is associated with its extremely large bandwidth. The symmetrical cross-bar structure could also be used in the n - ( n / N ) mode of the bar resonance. For the same dimensions, its shunt impedance would be higher than that of the interdigital line, but its resonant frequency would be roughly twice the frequency of the latter: this would make the cross-bar structure in n - ( T I N ) mode unpractically big at the low frequencies which are needed in a heavy ion linac. 5.1.2

The H-type structure in 2n mode

This structure was encountered in fig. 45b in an unshielded version; the shielded version appears in fig. 47a. It is essentially a Sloan-Lawrence structure where the inductance is distributed along the axis, in the form of a cylinder which is slotted axially. The drift tubes are connected alternatively to the two edges of the slot, and are charged with opposite electric charges by azimuthal currents flowing around the cylinder (see fig. 47b). This current distribution corresponds to a 2n phase shift for the electric field in a geometrical period 2L of the structure, and to a 0 phase shift for H z . In order to satisfy the boundary conditions for a uniform axial magnetic field Hz along the structure, the magnetic field lines turn around the inner cylinder at both ends of the structure and close themselves in loops by running along the outer shield:

* In a (x- E ) standing wave mode, the average axial field per cell varies as cos(n- E)z/L. When the structure is terminated in full cells, the voltage per cell is sampled in the gaps at distances z ( m - +) L ; therefore the amplitude of the average field per cell varies as

-

COS(Z- E)z/L = cos(n

-

-

E) (m - +)

-

sinn(m

-

4)sine(m - 4)

-

(-

sin(~z/L).

With E n/N, we have sin(ez/L) = sin(nz/l). In a varying B structure, L is to be considered as the average cell length along the structure. ** This is an important example of a n accelerator where the particles are not synchronized with a space-harmonic whose phase propagates in the same direction and with the same velocity as the beam (see section 1.6).

714

GDOME


this allows the amplitude of the average axial electric field per cell to be nearly constant down the structure (Teplyakov et al. [1967]), in contrast with the interdigital line where the end plates impose the condition I I , = 0 at both ends and therefore a sin(nz/l) variation of Hz along the structure. Apart from this difference, the H-type structure is one half of an interdigital line which has been split through the plane containing the stems. Viewed from the top, the field pattern appears the same as in fig. 46b. With a tank diameter of 0.4 m (0.2il), the shunt impedance is about 0.7P-2 MQ/m at 150 MHz (Teplyakov et al. [1967])*. Therefore, when /3 is smaller than 0.13 (which corresponds to 8 MeV per nucleon), the H-type structure has a higher shunt impedance than the Alvarez structure whereas its outer diameter is 3-4 times smaller at the same frequency: this structure is is thus suitable for a heavy ion linac. The drift tubes may be made hollow and small if a self-focussing electrode is inserted in the middle of the gaps (Teplyakov et al. [l967]) **. Nevertheless, it should be kept in mind that the H-type structure in 27r mode is not resonantly coupled and thus it cannot be compensated.

Conduction current (b)

Fig. 47. The H-type structure (Teplyakov et al. [1967]). Energy range 0.5-50 MeV 600-7 MQ/m Effective shunt impedance at 150 MHz (a) Cut-away view (not to scale). (b) Idealized cross-section in 2n mode.

5.2 The washer-loaded structure This structure is mentioned here for the sake of completeness, because it has been proposed for the 200 to 800 MeV part of the linac injector for the Russian 1000 GeV cybernetic accelerator project (Andreev et al. [1965, 19691; Murin et al. [1967]). The structure consists essentially of a row of equidistant circular washers which are aligned along the axis of a circular pipe (see fig. 48). In contrast with a disc-loaded waveguide, the basic field configuration between two washers is a EO2, mode. Therefore, there is

* The definition of shunt impedance used in this reference is 4 that used here, and the transit time factor is not included. ** For this focussing system, see ch. D.3.

C.l.le


715

a distance r , from the cavity axis where the longitudinal electric field E= vanishes*, and where it is thus possible to insert two or more metallic (or dielectric) rods parallel t o the axis, which will support the washers without disturbing the fields appreciably. The full-circumference slot between the washers and the outer cylinder produces a strong coupling between cells, giving the structure a large bandwidth of about 40% in the lower passband, which is of the forward wave type. For maximum shunt impedance, the washer-loaded structure is operated in n mode, where the washer planes are symmetry planes for the electric field. Consequently some voltage is developed between the washer edges and the outer wall of the cavity; but the peak electric field on a washer edge remains of the same magnitude as the peak electric field on the axis (Andreev et al. [I 9651). Because of the larger spacing between

Supportmg Rod

/

Fig. 48. The washer-loaded structure (Andreev et al. [1965,1969]).

washers (L=+flA), the shunt impedance increases rapidly with energy. For a cavity diameter of 1.3271"" and a hole diameter of 0.1672, the effective shunt impedance of the washer-loaded structure with drift tubes a t 1000 MHz is 5, 20, 26, 36 MR/m respectively for proton energies of 45, 150, 250, 700 MeV. Above 150 MeV, these figures are very close t o the shunt impedance of the cloverleaf structure with a cavity diameter of 0.9401. and a hole diameter of 0.0852. Hence, for the same shunt impedance a t high energies and a slightly larger outside diameter, the washer-loaded structure presents a wider beam aperture and a larger bandwidth than the cloverleaf structure, together with manufacturing simplicity. Nevertheless, as a resonantly

* When there are no drift tubes, this distance is such that 2nrlli. >, 2.405 (the equality holds only in 0 mode). ** This value applies t o a structure without drift tubes.

716

G.DOME


coupled structure, the cloverleaf retains the advantage of getting compensated when the slot resonant frequency is made equal to the cavity resonant frequency in n mode.

5.3 The disc-loaded structure This structure is widely used in travelling wave electron linacs (see ch. B.1.1), and could thus also be used for high energy proton linacs. It consists of a circular pipe which is loaded at regular intervals by metallic discs having a central iris as beam aperture. With the coupling entirely due to the irises, the passband is of the forward wave type. For a good shunt impedance, the bandwidth is narrow (a fraction of 1%) because the coupling cannot be increased by increasing the iris diameter, without at

Fig. 49. Travelling wave shunt impedance of a disc-loaded structure versus the operating mode ( B = 1 , g / L = 1).

the same time decreasing the electric field on the axis (Carter et al. 119661). The field configurationin eachcell is essentially that of an E,,, mode, with its phase shifting rather abruptly from cell to cell upon traversal of the iris. With this picture, which simply neglects the effect of the aperture in the disc, an estimate of the shunt impedance is easily obtained. Fig. 49 shows the variation of shunt impedance for travelling waves, as a function of the phase shift per cell 0 for a given /i'and a given frequency. Although the shunt impedance for travelling waves drops towards the n mode because of a deteriorating transit-time factor, for standing waves, TC mode operation still gives the largest shunt impedance because it avoids the reduction by a factor 2* which occurs in all modes but the 0 and TC modes with respect to the travelling wave case. Because of its very small mode spacing, the disc-loaded structure operated in n mode is very sensitive to tuning errors and transients. On the other hand, it can easily be compensated by introducing a resonant coupling between cells: this may be

*

When the total attenuation al in an accelerator section of length I is negligible (see section 1.5).


C.l.le

717

a reversed loop (pill-box resonator), another cavity (side-coupled or alternating periodic structure), o r a peripheral slot (slotted iris). Since the coupling element is not excited in a terminated perfect structure, it does not affect the shunt impedance too much, so that all the above compensated structures have nearly the same shunt impedance as the disc-loaded waveguide in n mode. At 880 MHz. with an iris diameter 2a=0.151, this measured shunt impedance increases from 12 MQ/m for P=0.55 to 30 MQ/m for p= 1 (Giordano 119631; Batchelor [1962]). The closer spacing of the discs towards the low energies produces a substantial drop of shunt impedance, not only because of the increased losses per unit length of the structure, but also because the fields are then concentrated more in between thediscs, away from the accelerator axis. The disc-loaded structure with drift tubes a t 800 MHz. When drift tubes are added in a disc-loaded waveguide, they not only improve the transit time factor, but at the same time they keep a good part of the electric energy stored around the accelerator axis at any p. As a result, the calculated z mode shunt impedance of such a structure at 805 MHz varies from 40 to 45 MQ/m for P increasing from 0.65 to 1.0 (Knapp [1965]). A further 10% increase in shunt impedance is still obtained by rounding off the sharp corners between the discs and the outer cylinder (Nagle [1965]; Hoyt el al. [I 9661; Knapp et al. [1966]). Because of non-theoretical losses introduced by surface roughness of the walls, losses at joints, and losses in the coupling region, the measured Q is on the average 15 % lower than theoretical *. Accordingly, the experimental values of shunt impedance for a disc-loaded structure with drift tubes operated in n mode at 805 MHz vary from 34 to 38 MR/m for flat discs, and from 38 to 42 MQ/m for shaped discs. These values correspond to a drift tube diameter d = 6-7 cm and a hole diameter 2 a = 3-4 cm, i.e. to ratios c l / i z 0.18 and 2a/Az0.09. The optimum ratio g / L is about 0.5, which yields a transit time factor T around 0.8 (Hoyt [1966]). The disc-loaded structure with drift tubes a t 200 MHz. The relevant figures for a disc-loaded structure with drift tubes operated in n mode at 201.25 MHz are easily obtained from the Alvarez cell calculations in 277 mode (University of Wisconsin [1967]), by noticing that a cell which is designed for 2n mode operation ** at some /I, may as well be operated in n mode at D,=2D2,: the only difference is that the drift tube plates are essential to permit a n phase shift from cell to cell, but the field pattern in each cell remains that of the E,,, mode. By virtue of the relation fL

S

E , cos ( o z / / I c )dz

-+L

* See ch. C.1 .lc: The side-coupled accelerator structure. ** For the sake of precision, the mode is designated here as the phase shift per cell of the space. harmonic which is synchronous with the particles.

718

G.DOME


the transit time factor T, for n mode operation is substantially better than for 271 mode operation. In the energy range above 100 MeV in n mode (i.e. 0.428 l may be small in comparison with those due to n= 1. Under this assumption SL(k,r) may be approximated by the remaining three terms.

5

Motion of a proton in a spacially homogeneous time-harmonic Jield. The perturbation parameter x

+

The homogeneous field E, = El cos (cp cp,), (ot= cp), often serves as a simple model of an accelerating gap. It is only an approximate solution of Maxwell's equations. The solution of the one-dimensional equation of motion, md=eE,, is: kz=cp-x[(~sincp~+cos(cp+cp,)-coscpO], with

and

It is assumed that at wt=cp=O the particle is at the centre (z=O) and has velocity 2,. In a proton linac gap, typically x < 0.1 [ w / 2 ~ =200 MHz, E, z 14 MV/m, in 3 0 . 0 3 5 ~ ( ~ 0 . MeV)], 5 in a proton synchrotron accelerating gap x < 0.01 [o/2?c= 3 MHz, E, ~ 0 . MV/m, 1 i o3 0 . 3 ~(z50 MeV)]. In an electron linac x 5 1; therefore the following considerations apply only to protons or heavy particles. In order to get cp = cp ( z ) , eq. (I) is inverted by successive approximations:

758

A.CARNE ET AL.


rp(,)(z) corresponds to the motion of a free particle, the influence of the field is taken into account iteratively. The resulting expressions are best displayed by expanding them in a Taylor's series with regard to x : ( ~ ( 2 (z) )

+

= (P(O)(z) x ( P y z ) = kz

+ x (P 2

+ x [cos (kz + qO)

+

(z)

+ x3 ... + kz sin yo]

cos (P, q0) - sin (P,]

-

- xZq(')(z) [sin (kz

(2)

.

In this and the following sections subscripts indicate the order of iteration ; superscripts denote the order of derivation with regard to x (or equally the power of x accompanying the expression), e.g. ~ ( Z ~ ( z ) = ( d 2 c p / d x 2 ) x ~ o = ( d 2 c p ~ 2 ) / dWith ~ 2 ) x ~the o. Taylor's expansion of rp = rp(,, (z) into powers of x, the equation of motion integrates to give the energy gain across the gap:

= eVIToh(k)cosq,

+ eVlx.

cos(3kg) -

I

sin (kg) --

kg

sin(2qo)

+ ev1x'...

where To, (k)= sin(+kg)/(+kg) 6 1 is the transit time factor [eq. (4.6), ko =0] for a field homogeneous in the z- and the r-direction [therefore no I,(ka)]. It accounts for the fact that the field strength is changing during the time the particle crosses the gap, so that it cannot acquire a kinetic energy equivalent to the "instantaneous potential difference" eV, cos (P,. It would gain this amount only if its velocity were so great (or the width of the gap so small) that the transit time were small compared to the period of the field. In practice: 471 < k g < +r, 0.96 2 To, 3 0.83. The real To(k) is divided by Io(k,a) 3 1. Moreover, in cavities designed for higher energies the electrical field along the axis exhibits a strong depression in the centre, and this further diminishes To [e.g. To = 0.56 (195 MeV)], Swenson [1964]). Equation (6) represents a power series in x. By use of eq. (4) all other dynamical variables may assume a similar form, and, since it is fair to assume that the order of magnitude of each term is essentially determined by the power in x (the accompanying coefficient not increasing in magnitude with increasing order) it becomes possible to estimate the accuracy achieved. The condition that the perturbation parameter should be small, x 4 I, (eEl/o) < mi,, can be interpreted as follows : The average force eE, times the time I/o, which equals the impact of the field on the free particle, should be small in comparison with the momentum of the free particle. In a numerical investigation (Schnizer [I 966]), repeating the iterations of eq. (4) sufficiently often for a given z gave the corresponding value of the time-angle q ( z ) to any desired accuracy, and finally the exact energy gain Awe,. Comparing the approximate values gave: 1 < A Wl/A We, 6 1.01, and 1 r 1 1 Start

I L~enlin

Con s t r u c t ~ o n

I

w Print

Fig. 8. Flow chart of a beam dynamics program.

The results of CLASS are of great practical interest, so it is worthwhile to consider some of the possibilities in some detail. There are mainly two kinds of questions to be asked: 1. What must be the characteristics of an incoming beam to be accelerated by the linac? (this leads to the notion of linac admittance); 2. A linac and input beam given, what are the characteristics of the beam at output? (this leads to the notion of beam emittance). It is here that "particle statistics" become useful. Generally sets of particles are dealt with having different x, x', y, y', W , cp coordinates: but since varying all six coordinates in the same set of particles would result in too large numbers of particles,

776


A.CARNE ET AL.

it is better to deal with subsets having two or three free coordinates only, the others being held constant. For example, the question of the linac admittance on the axis may arise: x, x', y, y' will be set equal to zero, and a set of particles located at the nodes of a mesh in the W, cp plane are to be tested. A figure like fig. 9 may be obtained, where the letters denote particles which are accepted, whilst the points denote particles which have been tested, but which have not been accepted by the accelerating field. Fig. 9 is

-OI

t

. . . . .

.

.

.

G

.

G

.

G

s

G

s

G

s

G

G

.

.

.

.

.

.

.

.

Fig. 9. Longitudinal admittance on the axis at 0.75 MeV; rejected, G, S, H,

.

.

* ,0 , X, +, accepted.

derived from computations made for a linac accelerating protons from 0.75 MeV up to 20 MeV (the new Saturne injector). Fig. 10 shows the particle positions at 20 MeV. Another question which may arise is that of determining radial admittance. For example, the admittance in the x, x' plane may be found by tracing y =y'= 0, W= Wo, cp=cpo, as the energy and phase of a particle chosen from within the longitudinal admittance area of fig. 9. The result is given in fig. 11, while fig. 12 shows the particle positions at 20 MeV. Had other values for Wo and cp, been taken, other admittance

C.1.2b

NUMERICAL METHODS. ACCELERATION BY A GAP

777

shapes would have been found, obtained from fig. I I by rotating slightly around the point x = x' = 0. Practically, W and cp are not independent: between them exists a relation determined by the preinjector and buncher (if any). The W; cp plane at the input of the linac may only be filled along a curve, being a straight line A W=O if a preinjector alone is used, and the classical S shaped curve if a buncher is used. In addition to the x, x' scan just mentioned, (W,, cp,) variation can be taken into account by multiplying the number of particles by, say 50, and taking 50 pairs of (W,, cp,) values along the W , cp "filling" curve. There results a curve similar to fig. 13, which is the superposition of 50 figures like fig. I I , which was valid for one (W,, cp,) only. Rep~esentationwith

Fig. 10. Position at 20 MeV of particles accepted as shown in fig. 9.

letters is clearly no longer adequate, and must be replaced by some labelling numbers. For example, a number n may define particles, whose coordinates are x, x', as suitable for acceleration up to final energy, for n pairs of W, cp values. Also this number n multiplied by two gives the x,x' efficiency in percent for y=yl=O particles. Alternatively, an input beam may be given by (say) 200 sets of x, x', y, y' values and 50 pairs of W, cp values and the emittance is required. At the output, the xx',yy', Micp planes will be divided into a large number of cells (like squares on a chess board).

778

A.CARNE ET AL.

I

-6.6


I -4.4

I

-2.2

I

I

I

I

0

2.2

4.4

6.6

x (mm)

Fig. 11. xx' admittance at 0.75 MeV for y =y' = 0, W = WO,9 = P O particles (WO,90 are defined in fig. 9).

Fig. 12. Position at 20 MeV of particles accepted as shown in fig. 11.

779


C.1.2b

Each computed particle "hits" the three planes on three cells. An occupation number is associated with each cell. All these numbers are zero at the beginning. Each time a cell is "hit" by a particle the occupation number is increased by unity. An improvement consisted in attaching a weight P to each particle, P being related to beam density a t the input of the linac, and increasing the occupation number by P instead of unity.

0

0

Cl

(I

0

0

5 5 8 4 ~ 2 10

0

0

0

0

0 0

0 0

0 0

0 O 3

0

0

0 0

0

0

0 0

0 0

0

0

0 0

0 0 0 0

10

0

0

0

0

0 0

0 O

1:0 3 40 1 4 37 1 ? 0x

o 0

o 0

0

0

o 0

0

0

0

0

0 X 18 4 1 1 0 Z h I R 0 04 1 x 2 2 I 19 38

:~

7

54:

0I

-2.2

-4.4

0 0

O 0 0

0

0

0 0

0

o

o0 [I

0 u

0 0 0 0 (1

0

o 0 O

0

0

0

0

0

0

0

0

0

u

0

0

0

0

0

0

22

0

0

0

(1

-66

0 0 0

66

44

x (mm)

- -

Fig. 13. xx' admittance at 0.75 MeV for y y' 0 particles. 50 ( WO,YO) pairs are taken. Plotted numbers multiplied by two give the xx' efficiency (%,)for y =,v' 0 particles.

1

I0

-U o E

-

-E

-X

-

~

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 I1 0 0 0 0 0 0 O 0

0

0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ~ 0

~

-

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 5 0 0 0 0 0 0 0 0 0 3 1 2 0 0 0 0 0 0 0 0 1 2 3 1 1 0 0 0 0 0 0 0 0 5 17 I ? 27 0 0 0 0 0 0 0 0 2 O X 9 4 9 3 0 0 0 0 0 0 4 ? 1 4 1 9 6 3 0 2 0 11 3 2 2 0 5 6 3 0 3 7 0 2 1 4 0 0 0 0 89 20 0 0 0 0 3 25 70 370 4'13 5 0 0 0 1 3 1 1 3 3 2 0 3 0 0 27 12 17 i 0 0 R 6 X I ? h I ? R 1 9 0 1 1 2 3 1 1 3 3 6 3 3 5 1 0 1 2 3 0 0 0 0 0 0 0 0 5 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 O O ~ O O O O O O O 0 0 0 0 0 0 0 0 0 0 0

0

0

0

0

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

I

I

I

1

I

-8.4 -6.8 -52

0 0 0 0

0 0 0 0

0 I

0 0 0 0

0 I

0 0 0 0

0 I

0 0 0 0

0 l

0 0 0 0

0 I

0 0 0 0

0 I

-3.6 -2.0 -0.4 0.4 x (mm)

0 0 0 0

0 I

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 5 3 3 3 6 1 3 1 1 9 1 2 8 1 2 6 6 8 8 0 300 320 113 13 0 0 3 7 0 7 0 2 5 3 0 0 5 3 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 U 0 0 0 O O O O O O 0 0 0 0 0 0

0 0 0 0

0 I

2.0

0 0 0 0

0 I

0 0 0 0

0 I

3.6

l

0

0

0 0 0 0

0 0 0 0

I

I

52

0 0 0 0 1 0 0 0 0 0 0 0

0

0

0 0 0 0 0 0 0 0 0 0 O 0

0 0 0 0 0 0 0 0 0 0 U 0

0

0

0 0 0 0

0 0 0 0

I

I

6.8

0 0 0 0 0 0 0 0 0 0 0 0

O

0 0 0 0 0 /

8.4

Fig. 14. xx' emittance at 20 MeV for a given input beam. Nurnbers are proportional to beam density

0

780

A.CARNE ET AL.


All the particles having been treated, the resulting numbers are written inside corresponding cells, giving emittance density diagrams such as figs. 14 and 15, describing x, x' and W, cp at final energy (20 MeV). The above procedure gives a fair idea of output beam properties. It will be noticed that, while the input W, cp plane was populated only along a curve, the output W, cp

Fig. 15. Longitudinal emittance at 20 MeV for a given input beam. Numbers are proportional to beam density.

C.1.2b


781

plane is populated over a certain area. This is due to coupling effects between the six coordinates. Fig. 15 is very convenient for beam debunching analysis. Kn principle, it is necessary to compute particle trajectories along the debunching drift space and through the debuncher cavity itself, to get particle coordinates after the debuncher. This tedious process probably cannot be avoided if it is desired to take into account space charge effects; but as far as low intensity beams are concerned, or for rough estimation the following suffices. The debuncher is required to reduce the beam energy spread. What is of interest then is which energy range will be found in the beam after the debuncher.

Fig. 16. At the linac output a particle has coordinates Wl, ( o l whose scale is given by the rectangular axes. Coordinates of the same point with respect to the curvilinear net give Wz, (oz which the same particle has after the debuncher. When superimposed on fig. 15 this figure permits a quick analysis of beam debunching.

It is then possible to perform the computation backward, once for all, for each desirable debuncher setting: i.e. to find which W , cp plane area particles must be located at the output of the linac in order to be located after the debuncher in the f 2 0 keV, f 40 keV ... areas of the W , 9 plane. In this way, a set of curves like those of fig. 16 is found. The curves of figs. 15 and 16 are plotted in the same scale and may be superposed to see the effect of beam debunching. The several statistics presented here are of course not exhaustive; many other statistics may be devised, and programmed if needed.

782

A.CARNE ET AL.


TABLE 1 Nonrelativistic change of longitudinal kinetic energy, phase, transverse velocity and reduced radial position across a linac gap

A W = W + - W- = eVoToIocos cp + eV, ( d l d k ) (T,krll) r' sin c p . A@ = cp, - cp- = a k ( d / d k ) (Tolo)sincp - a k ( d 2 / d k 2 )( T o k r I l )r' cos c p . Ar' = r', - r'_ = - aT,kI,/k, sin cp + a [(dldk) (T,kl,) - Tol,]r1 cos c p . A r' = J+ - J - = - a ( d l d k ) ( T o k I l / k r )cos cp - a [ ( d 2 / d k 2 )(TOkI;)- ( d l d k ) (Tolo)] r' sin cp. -

--

TABLE 2 The change of the same quantities in the relativistic case

(W+ - W - ) ,

=

(@+ - cp-), ( 1 - p:)-* (r: - r l ), ( 1 - pi)-*

= (r'+ -

pi)-*

= (i+ -

( Y + - F-)r ( 1 -

= (@+ -

---

*

W+ - W-

+ [ - e V o T o k ( I l / k r )r' sincp] .*

@-) ( 1 - ( k i / k 2 ) ]- ctlt(ki/k2)To( I ,jk,) r'cos c p . r:) { I - ( k t / k 2 ) }+ a ( k i i k 2 ) To ( I ; - I,) r' cos cp. Y - ) ( 1 - ( k i / k 2 ) )- a ( k i / k 2 )To(l,/k,) cos cp - a ( k i / k 2 )( d l d k )(2T01i - Tolo)r' sin cp .

- -

--

--

-

Adding the term in the square brackets gives the gain in total energy. TABLE 3 Change of dynamical quantities in the first half of a gap

Aw, = +AW + $eVo [ S , sin cp - ( d / d k ) (as,/ar) r' cos cp] . A@, = $A@ - +ak [(dS,/dk) cos cp ( d 2 / d k 2 )(dS,/ar) r' sin cp] . Ar; = 4Ar' - +a [ S , cos cp + { ( d l d k )(dS,/dr) S,) r' sin cp] . A i l = +A? + +a [(dS,/dk) sin cp - { ( d 2 / d k 2 )(aS,/ar) + d S , / d k ) r' cos cp]

+

+

Common to all three tables: a

= +eVo/W;

W

m = rest mass; 1 - k i / k 2 = k:/k2 z 1 -

= +rn(dz/dt)i;

k i / k 2 = ( d z / d t ) i / c 2z 2:;;

pi.

T h e arguments k = olioof To ( k ) , k r ( k ) r, of I, (k,r,), k and r0 o f S, ( k , r,), Sr( k , r,) and the subscript o f cp,, r6 and a/&, have been dropped. In all the expressions o f tables 1-3 all these parameters refer t o mid-gap values. For the above set o f difference equations it was supposed that 8' = 0, the change t o Cartesian transverse variables can b e easily accomplished b y the transformations:

,

+

2 2 r2 = x + y , rr' = ( X X ' y y ' ) , x = r ( 1 ) , y' = r ( y ) 2 = P (xlr), 7 = r ( y l r ) , Ax' = Ar' ( x l r ) , Ay' = Art ( ~ / r ) , A 2 = A J ( x l r ) , Ay

=AJ

(ylr).


783

References ALVAREZ, L. W., H. BRADNER, J. V. FRANCK, H. GORDON,J. D. COW, L. C. MARSHALL, F. OPPENHEIMER, W.H.K. PANOFSKY, CH. RICHMAN and J.R. WOODYARD, 1955, Rev. Sci. Instr. 26, 111. BELL,J. S., 1954, Linear accelerator phase oscillations, AEREJ Rept. TIM 116. CARNE, A. and P. M. LAPOSTOLLE, 1966, Design equations in an Alvarez-type proton linear accelerator, Linear Accelerator Conference 1966, LA Rept. 3609, p. 201. and M. PROME,1966, Accurate beam dynamics equations in proton CARNE, A., P. M. LAPOSTOLLE linear accelerators, ed. M. Grilli, 5th Conf. on high energy accelerators, Frascati, 1965, Comit. Naz. Energia Nucl. (Roma, 1966). p. 656. H. S., 1930, Introduction to the theory of Fourier's series and integrals, 3rd ed. (Dover CARSLAW, Publ.), sect. 104, 106. P. M., 1965, Equations de la dynamique des particules dans un accelerateur lineaire LAPOSTOLLE, protons, CERN Rept. AR/Int. SG/5-11. P. M., 1966, lntroduction a l'etude des acct-16rateurslintaires, CERN Rept. 66-20, p. 38. LAPOSTOLLE, 1968, Numerical calculations of linear accelerator cavities, CERN MARTINI, M. and D. J. WARNER, Rept. 68-1 1. PANOFSKY, W. H., 1951, Linear accelerator beam dynamics, Lawrence Radiation Lab. Rept. UCRL 1216. PROME,M., 1965, Trajectoires de particules dans un accelCrateur lineaire a protons, remarques sur les equations des intervalles d'accelerations, CEN Saclay Rept. SEFS T D 65/12. PROME,M., 1966, Design of a 20 MeV proton linear accelerator, new injector for Saturne, Linear Accelerator Conference 1966, LA Rept. 3609, p. 403. PROM^, M., 1967, Determination de 1'acceli.rateur lineaire de 20 MeV, nouvel injecteur du synchrotron Saturne, choix de parametres electriques, dynamique des particules, C E N Saclay Rept. 3261. RICH,M., 1964, Beam dynamics calculations for Alvarez-type linear accelerators, Linear Accelerator Conference 1964, MURA Rept. 714, p. 341. SCHNIZER, B., 1966, Motion of a particle in a spatially homogeneous time-harmonic field, C E R N Rept. ISR 300/LIN/66-36. SCHNIZER, B., 1969, General properties of fields and beam dynamics in an accelerating gap, CERN Rept. 69-3. SCHNIZER, B., 1970, Hamiltonian mechanics with a space coordinate as independent variable. Canonical thin lens approximation for a n accelerating gap (to be published as a CERN yellow report). J. A,, 1951, Electromagnetic theory (McGraw-Hill, New York) pp. 28, 349. STRATTON, SWENSON, D. A., 1964, Application of calculated fields to the study of particle dynamics, Linear Accelerator Conference 1964, MURA Rept. 714, p. 328. SWENSON, D. A,, 1967, Generation of geometrical dimensions for drift tube linacs, private communication, LA Rept. MP-3/DAS-I. D. A., D. E. YOUNGand B. AUSTIN,1966, Comparison of the particle motions ascalculated SWENSON, by two different dynamics programs, Linear Accelerator Conf. 1966, LA Rept. 3609, p. 229. TAYLOR, R., 1959, Calculation of drift-tube dimensions in linac injector for the 7 GeV Harwell proton synchrotron, AERE Rept. R 3012.

PLA THEORY

C.1

2 Particle Dynamics at Low Intensity

M. PROME CENS, France

1 The need for focusing The accurate calculation of focusing and output beam properties requires rather sophisticated means of computation such as the computer codes for trajectories in terms of the six phase space coordinates as described previously. However, such codes can only be useful tools if a guide providing a qualitative understanding of the problem is available to their users. Thus it is intended here to describe such guides. These guides are based upon physical considerations, and assume some more or less valid approximations; they nevertheless led to the design of proton linacs before the computer era. Focusing devices are needed because the gaps have normally an overall defocusing effect (chapter A.2). Chapter D.3 will describe various methods devised to overcome this difficulty: grids, alternating phase focusing, non-axially symmetrical drift ~ L bes. I For proton linacs, however, it is now usually accepted that an additional focusing system must be installed to provide radial stability. As an Alvarez structure leads to a cavity diameter of about one meter, this additional system obviously must be located inside the cavity, that is inside the drift tubes. It may consist of solenoids, electrostatic quadrupoles, or magnetic quadrupoles. Magnetic quadrupoles are now exclusively employed: several schemes may be conceived, differing one from the other by the number of quadrupoles, and their sign sequence. More precisely, F being a region where the quadrupolar field is focusing in one transverse direction, D a defocusing region, and 0 the gaps with their associated field free regions, the following periods are used: Q ~Vorth-HollandPubl. C o . , Amsterdam 1969

References p. 795

785

786

M.PROME

FODO FOFODODO


also referred to as also referred to as

+-+++--

FOFDOD FOD the last two having been studied or used by Kapchinsky [1966]. The following sections will be devoted to the analysis and comparison of these various periods. But before, we shall recall some properties of Hill's equation and periodic structures as developed in the theory of strong focusing synchrotrons (Courant and Snyder [1958]).

2 Elements of the periodic structures theory The radial motion can be described by its projections on xz and yz, which are two symmetry planes of the quadrupole system; in ideal quadrupoles systems where a quadrupole is twisted around the z-axis with respect to its neighbours by either zero or ninety degrees, the xz and yz motions are uncoupled. So functions describing the motion have properties which are the same in either planes; in this section r will denote either x or y. The radial motion is described by a Hill's equation:

where k is a periodic function of period L. Let us consider the transfer matrix which transfers r and drldz from point A (z= z,) to point B ( z = z , L), one period away.

+

This matrix has a determinant equal to unity, because Hill's equation does not contain any first derivative terms. If such a matrix has a trace of absolute value larger than two, it can be written: chp

+ cc s h p PB,s h p shp

c h p - ccshp

I

The nth power of this matrix can be obtained immediately by replacing p by np, therefore the radial motion is obviously unstable. On the contrary, matrices having a trace of absolute value smaller than two

FOCUSING

787

can be written : cos p

+ cx sin p Pr

sinp

B , , sin p

cosp-rsinp

I

and, as the rule to get the ~ t power h remains the same, they typify stable radial motion. The remainder of this section will be restricted to this kind of matrix. It can be shown that ,Br is a periodic solution of:

with the same period L as k(z), and that a is related to

Pr by:

We call p the radial oscillation phase change per period, and it is related to

Pr by:

It should be noticed that, while cx and P, are functions of z, ,LL is not, since 8, is periodrc with periodicity L. So far, for a given arrangement of quadrupoles and given strength of these quadrupoles, we have derived a convenient way to see whether or not the radial motion is stable: compute the trace of a transfer matrix over a whole period, beginning anywhere, and see if its absolute value is smaller than two. Much more can be known about sets of trajectories by using the K and ,B, functions. It can be proved that any trajectory may be represented by:

where A and 6 are two parameters fixed by initial conditions at z=z,, and

Notice that if z=z,+ L, * = p , which justifies the name given to p. All particles having the same value A , but any 6, will have an envelope of trajectories given by: r = A ( P , ( z ) ) f The p, function may be found easily by computing several transfer matrices beginning at several points in the range z,, z, L. If a is the radius of the aperture of the drift tube, r must never become greater than a (except in

+

788

M.PROM~?


the gap, but this is impossible because P, is never maximum inside a gap); the maximum permissible value for A is then: A,,, = a//li,,, where P,,, is the maximum value of pr over one period. Such limit trajectories are described by the equation: r = ( a / ~ i , JP f cos ($

+ 6).

By eliminating (1,!+6) between this equation and the expression for r', the first derivative of r with respect to z, there appears the invariant: (1IPr)

[r2 +

+ 8rr')21 = a21Pmax.

Limit particles, so called because their trajectories graze the drift tubes, may be represented for a given z in an r', r phase diagram by an ellipse (fig. 1) the equation of which is the quadratic form above. Its area has a value A=za2/Pm,,, which is called admittance of the accelerator; it should be noted that the shape of the ellipse changes depending on the z-value for which it has been plotted, but its area is a constant.

Fig. 1. rr' positions of limit particles.

3 Stability diagrams In this section the guide for which we mentioned the need in the first section will be presented. The first approximation, and the most important one, is that the radial motion will be computed only for the reference particle, that is for a particle reaching the middle of the successive gaps with the phase angle for which the accelerator has been computed. Other particles will oscillate around the reference particle within a maximum range 2q, - q ; one way to take this oscillation into account is to chose a radial focusing operating point for the reference particle so that the operating point of any other possible particle never crosses the stability boundaries, but this is probably not necessary; the overall radial stability is not lost if a particle during only a short fraction of its phase oscillation goes out of the static stability boundary. Taking, as another approximation, that there is no coupling between the xz and yz planes results in particular in neglecting quadrupole fringing fields. Due to fringing

FOCUSING

C.1.2~

789

fields, angular variations Ax' and Ay' are in fact depending upon coupling terms among which the most important are the products xyy' and xyx' respectively (chapter C. 1.2d). The approximation is not too far from reality, at least if we are interested in beam emittance only and for linac apertures and beam emittances as existing in present linacs. In fact, the axially symmetrical gap does couple the two motions, since A x ' and Ay' depend upon the distance to the axis, ( x ' + ~ ' ) ~Nevertheless . the treatment is valid if we limit the description of the xx' motion to y = y l = O and vice versa. We can then deal with second order matrices. (Notice that in the so-called helicoidal focusing, Tran [1964], where a quadrupole is twisted forty five degrees with respect to its neighbours, fourth order matrices have to be considered.) For the sake of simplicity in the calculations, it will be assumed that the quadrupole gradients are constant over the lengths L, and vanish elsewhere. According to what has been said previously, the gaps may be treated in various ways. Some authors approximate them by a distributed force applied all along the axis; others approximate them by a single thin lens located at the middle of the gap; this is the Smith and Gluckstern [I9551 method; we will describe it in more detail. Other methods lead to similar results. The following references give a comprehensive review of the problem : Teng [I 9541, Montague [1958], Regenstreif [I 9601, Lapostolle [1966], Livingood [1961], Vlassov [1965], Kapchinsky [I9661 and Sura [1966]. Referring to chapters A.2 and C. 1.2b, since P,

=

{m,/(l

-

fi2)*} v, drldz, with v,

= fic,

then, from chapter C. 1.2b, eq. (10.20)

,sin cp sin ('wg/u) (1 - p2)', where the effect of the magnetic field has been added by a multiplication factor (1 - p Z ) (table 2 of chapter C.1.2b) and input and exit velocities v i and v, have been both replaced by the single value u; the suffix of the phase has also been dropped. Let LC be the cell length, L,=PA; the magnetic. period increases with increasing energy, since the cell length increases. As we wish to deal with a periodic structure, we will take the longitudinal coordinate Z = z / L , instead of z ; then the period becomes a constant, -

A(:;)=

r

LE,L, sin q sin {ng/(jA)}(1 mope

-

j2)i .

Let us introduce now a dimensionless coefficient A , characteristic of the defocusing effect of the gap: A

=-

1

r

A

();:

=

eE1k

mop c

sin cp sin ( n g / ( P i ) )(1 - p2)'

.

790

M.PROME


thus, the transfer matrix of the gap, with the Z-coordinate is:

while the quadrupole transfer matrices are : cos e

sin 6

- 8, sin 8

cosO

sh 0

where :

B' and L, being respectively the quadrupole gradient and the quadrupole length, 8, is characteristic of the quadrupole strength; 8, and 9 are dimensionless quantities. The last elementary matrix to introduce is the transfer matrix of a field free region:

The application of the method to the FODO scheme will show how it works; the transfer matrix from A to B (fig. 2) is the product: M = M,M,M~M,M,M,M+M,,

I . F A

1

I

A'

D

I

I B

F I 8'

Fig. 2. FODO scheme showing origins for transfer matrix.

the a and /3 will be plotted by computing matrices such as M', the transfer matrix from A' to B', and so on. Numerical results given later will be valid only for: Lq/Lc=+,which is approximately the minimum filling factor one obtains in proton linacs. We will compare the FODO (+ - -) and the FOFODODO (+ - -) schemes. FOD and FOFDOD schemes suppose two quadrupoles with opposite signs in the same drift tube; information on these schemes will be found in Kapchinsky [1966]. As some resonances between longitudinal and radial motions are to be avoided (Regenstreif [1960]) it will be convenient to introduce a parameter q defined by:

+

+

FOCUSING

C.1.2~

791

Q,.: radial oscillation angular frequency;

RZ:longitudinal oscillation angular frequency. q may be simply related t o p and A ; T being the rf period:

in the Q, = i p / T , in the

Q, = +p/T,

+ + -

++

-

-

scheme,

- scheme.

It can be shown that (chapter A.2):

then Q~/A =-2 / ~ ' ,

and p = 2q (- 2A)*, 1-1 = 49 ( - 2A)*,

in the in the

+ + - scheme, + + - scheme. -

-

(Note that A is negative since cp is negative.)

S t a b l l l t y boundary -

S t a b l l t y boundary

/

-%A

',%h\Ai!i

S t a b ~ l l t yd ~ a g r a m Mode

Stab~lity dragram + - + - Mode

++--

*

L

25

Fig.3.

+

-

i - scheme: stability diagram, q and curves. -

-020

Fig. 4.

-015

+ $- -

-

-010

-005

0 A

scheme: stability diagram, /Llrax curves.

q and

792

M.PROME


Figures 3 and 4 show the stability diagram in the two schemes here considered.* Figures 5 and 7 show the a and jl, functions in the + + - - schemes and the - + - scheme respectively, while fig. 6 shows several shapes of the linac admittances curves at 2 MeV with a radius of aperture a=0.75 cm for the + + - scheme.

+

1

I

Fig. 5.

++

--

I

I

I

I

I

scheme: a ( Z ) and B(Z) functions for A = -0.07, Oo2 = 1.15.

w POINT B

A

r'(mrad)

50,-

POINT C

rhm)

POINT A

Fig. 6.

*

++

--

scheme: various admittances at 2 MeV; points A, B, C, D are defined in fig. 5.

The y of Smith and Gluckstern [I9551 is approximately our Bmax, with the terminology of Courant and Snyder [1958]; their ty is approximately (Dmax/Bmin)t.

C.1.2~

FOCUSING

793

From figures 3 and 4, it is obvious that the + - + - scheme allows smaller values of + + - - one; the admittance will be larger; but it requires larger values of o;, which means larger quadrupole gradients. Practically, the points representing successive quadrupoles on diagrams 3 and 4 will be located on a line such as the dotted line shown in fig. 8 ; for example let us

P,,, than the

p,.

U N I T LENGTH= ONE C E L L I

Fig. 7.

+ + -

-

scheme: a(Z) and b(Z) functions for A

Amin

nref

1

I

I

0

= -0.07,

Amax

Fig. 8. Operating line for quadrupole system

0oZ= 1.85.

794

M.PROME


state that P represents the first quadrupole; this point is valid for the reference particle only; other particles will oscillate around P ; it is desirable that this oscillation, which ranges from A,,, to A,,, should not cross the stability boundary though short incursions in the unstable region do not seem hazardous. As far as resonances are concerned, computations carried out at CERN (Hereward and Johnsen [1956]) showed that q=0.5 is to be avoided, but that a loss of particles at q = 1 is scarcely perceptible; although a quick crossing of the curve q=0.5 may not be very dangerous, it is more advisable to start above q=0.5 as does the dotted line PQ shown in fig. 8; this line is oriented in the direction along which P,,, decreases, which means that q increases slightly up to 1 or 1.5; the advantage of decreasing P,,, while the energy increases would be of no importance for a perfectly aligned machine, since the admittance is determined only by the low energy end of the linac (the radius of aperture a is the smallest at low energy because of transit time factor considerations); but decreasing P,,, (as well as increasing a) allows a certain amount of misalignment without reducing the admittance. In principle all the quadrupole gradients are different: the gradients fall off smoothly with quadrupole number; below 5-10 MeV, the rf defocusing is still large and the gradients normally decrease rapidly; for higher energies the variation slackens down. For energies above 10 MeV it even becomes possible and may be advantageous in practice to make the quadrupole power supply simpler in design by arranging the successive quadrupoles in groups having the same absolute gradient. Another interesting property of the + - + - scheme is illustrated by space charge considerations. The Pma,/j3,in ratio from figs. 5 and 7 can be seen to be greater in the - - scheme than in the + - + - one, which means that the beam periodically exhibits a radial (for instance x) extension much smaller in the + + - - scheme (x is replaced by y half a z period away). In this case, the space charge forces may become very important. - - or - - will be determined by technological Practically, the choice of possibilities (magnetic core saturation, cooling); if high gradient quadrupoles can be manufactured, + - + - is preferable. Notice that the problem of determining the radial focusing is closely related to the choice of the synchronous operating phase angle; this appears obviously as a compromise to be made between radial and longitudinal acceptance.

++

++

+ +

4 High energy focusing What has been presented in the last section describes the situation for Alvarez structures up to 50 MeV; at higher energies, the defocusing forces due to the gaps become small enough to allow savings in quadrupoles to be made. In the case of short enough cavities, locating quadrupoles between cavities could be considered but without going that far, it could be rather easy to locate a quadrupole every third drift

C.1.2~

FOCUSING

795

tube only up to I00 MeV, then every sixth drift tube up to 200 MeV (Yale [I9641 and Brookhaven [I 9641). The Alvarez structure is no longer the most efficient at higher energies; structures which are then usually considered yield sections about 3 m long; focusing devices (quadrupole doublets or triplets) can then be advantageously located between sections. Other possible configurations can be derived from the general schemes presented in fig. 9. Both doublet and triplet configuration may ensure focusing in these high energy structures; for perfectly well aligned machines, the admittances one can achieve with either doublets o r triplets are quite comparable; in any case, they are practically always larger than the beam emittance. The choice between doublets and triplets is based on the following consideration: triplets are more expensive, require more space and power consumption than doublets; doublets lead to more difficult transverse

~

F

n

D

n

F

n

I U U U I

r

1

D n

F

n

D

n

1

Fig. 9. Focusing schemes at high energy.

alignment, especially as far as tilt is concerned; in particular, bench alignment offers no improvement for them as it does for triplets (Ohnuma [1964]). It is generally believed, however, that the doublet tolerances required can be met practically, and that doublets must then be preferred (Ohnuma [1964], Jakobson and Visscher [1964], Visscher [1965]). For energies higher than 1 GeV, placing doublets only every 2 or 3 rf sections has been considered in order to save quadrupoles; for instance in a 3 GeV project (Parain [1963]) the conclusion reached was that one would need to install either 179 constant gradient doublets (one per rf section), or 67 doublets whose gradients vary in the ratio 2 to 1.

References BROOKHAVEN,1964, A proposal for increasing the intensity of the alternating gradient synchrotron at the Brookhaven National Laboratory, BNL Rept. 7956. COURANT, E. D. and H. S. SNYDER, 1958, Theory of the alternating gradient synchrotron. Ann. Phys. 3, 1 . HEREWARD, H. G . and K. JOHNSEN, 1956, Alternating gradient focusing in linacs: con~putational results, CERN Synip. on high energy accelerators and pion physics, 1956, p. 167. JAKOBSON,M., and W. M. VISSCHER,1964, Numerical study of particle dynamics in a high energy proton linear accelerator, Linear Accelerator Conf. 1964, MURA Rept. 714, p. 283. KAPCHINSKY, I. M., 1966, Dynamics of particles in linear accelerators (Russian ed., Atomizdat, Moscow).

796

M.PROME


LAPOSTOLLE, P., 1966, Introduction k l'Ctude des accelCrateurs lidaires, CERN Rept. 66-20. LIVINGOOD, J. J., 1961, Principles of cyclic particle accelerators (Van Nostrand, New York). MONTAGUE, B. W., 1958, Linac exit beam parameters, CERN Rept. PS/Int/Lin 5 8 4 . OHNUMA, S., 1964, Particle motions and the focusing system in proton linacs, Linear Accelerator Conf. 1964, MURA Rept. 714, p. 266. PARAIN, J., 1963, Focalisation radiale dans un accelkrateur lineaire a protons entre 200 MeV et 3 GeV, CERN Rept. AR/Int. SG/63-12. REGENSTREIF, E., 1960, The CERN Proton Synchrotron, CERN Rept. 60-26. SMITH, L. and R. GLUCKSTERN, 1955, Focusing in linear ion accelerators, Rev. Sci. Instr. 26,220. SURA,J., 1966, Choice of the parameters of the focusing channel of the linear proton accelerator in Swierk (Russian ed., Warsaw, 1966). TENG,L. C., 1954, Alternating gradient electrostatic focusing for linear accelerators, Rev. Sci. Instr. 25, 264. TRAN,D. T., 1964, Focusing of an Alvarez linac by use of magnetic quadrupoles oriented at 45 degrees, Linear Accelerator Conf. 1964, MURA Rept. 714, p. 309. VISSCHER, W. M., 1965, Optimization of magnetic lenses for waveguide portion of a proton linac, IEEE Trans. Nucl. Sci. NS-14 (3) p. 617. VLASSOV, A. D., 1965, Theory of linear accelerators (Russian ed., Atomizdat, Moscow). YALE,1964, Study on high intensity proton accelerators, Yale Rept. Y 12.

PLA THEORY 2

C.l

Particle Dynamics at Low Intensity

1.2d 1 Nonlinear Effects

1

R. L. GLUCKSTERN

I

University of Massachuseiis, U.S.A.

1 General considerations The usual analytic treatment of orbital motion in a proton linac divides itself into two parts: In the first, one considers the longitudinal motion of an axial particle and obtains (chapter A.2) a linearized oscillation frequency R or wavelength A,, eq. (23), the limits of stability for no acceleration, eq. (29) or fig. 2, and an estimate of the damping of the linearized oscillations using the adiabatic invariant for the longitudinal motion, eq. (35). In the second, one considers the above solution for the longitudinal motion and arrives at a linearized treatment of the transverse motion, whose oscillation frequency is assumed to be much greaterthan that ofthelongitudinal motion. Numerical orbit computations are then used to include the many neglected effects, including space charge forces between particles. Although a general analytic treatment is hopelessly complicated, it is nevertheless possible to include a semi-analytic treatment of some non-linear effects dismissed in arriving at the results of chapter A.2. Several of these are discussed in what follows. The primary emphasis, particularly in discussing transverse oscillations, is to include those effects which cause a deterioration of beam quality.

2 Longitudinal phase acceptance The analysis of the longitudinal motion in a proton linac leads to an equation identical to that for the motion of a loaded pendulum, eq. (21) of chapter A.2. Although the parameter us or fl, (and occasionally E,Tand 4,) varies along the linac, it is customarily treated as constant in order to obtain an analytic solution of the equation of motion. Adiabatic variation is then invoked to determine the damping of the oscillations, but C North-Holland

Publ. Co., Amsterdam 1969

References p. 804

797

R.L. GLUCKSTERN

798


the "fish-shaped" longitudinal phase acceptance is determined without considering this variation. Numerical solutions, including the variation in v, due to acceleration, lead to a prediction of an altered region of phase acceptance with increased area (for example Taylor [1959]; West [1962]; Blewett [1963]). An approximate analysis of the change (Gluckstern [1967a]) is outlined below. Putting x=4-4, the Hamiltonian for the longitudinal motion can be written as

H (x, P, P,) = 3p2IPl + V (x) where the potential V(X) is given by

(1)

9

V (x) =

2neEoT [sin (4, MC'A -

+ X)

-

sin 4, - x cos 4,] r

The last form for V(X) is an approximate one valid for small x and 4,. During the course of the first oscillation, H will change by an amount

As a result of eq. (3), the total "energy" H decreases during acceleration and a particle with x and dx/ds initially outside the "static" phase stable region may lose sufficient energy to be trapped by the barrier in V(x) at x ~ 2 I 4 , Iand be captured. The result for the limits of phase stability may be put in dimensionless form for small 1$,1 by using the injection variables x and y defined by Y' x = xiIl4sl, One then obtains

/+,I3

= ( 6 ~ i ) '( 6 x M ~ ~ l P :

eEoT).

2

y2(x)=(2-x)'(l+x)+~/y(u)du, where E =

(4)

(5)

X

3{ ~ ~ E o T A / ( ~ ~ 14,1))' M ~ ~ P2.1 , {AP/(& 14,l))',

(6)

with APIP, being the fractional change in velocity per cell at injection. The integral equation (5) may be solved approximately for small E by using y (x) = 12 - xl(l+ x)* on the right side of eq. (5). The analytic results are shown in graphical form in fig. 1 for several values of E. The circled points correspond to numerical integration of the original equation of motion for ~ = 0 . 1 2 . It is clear that the analytic results for small E, which are in reasonable agreement with these computations, can serve as a useful guide for predicting the altered limits of phase stability. An interesting interpretation of the effects of acceleration has been offered by Nishikawa [1967]. He suggests that the primary effect of a parameter change takes

C.1.2d

NONLINEAR EFFECTS

799

place during the first phase oscillation, and that one can make a realistic estimate of the increased acceptance by using the static results for a velocity corresponding to approximately one phase oscillation after injection. This prescription can be shown to be consistent with the result obtained above for small E . Moreover it sheds some light on the large increase in acceptance when space charge is included, since the first oscillation now carries the particles to much higher velocity.

Fig. 1 . Longitudinal phase acceptance of a proton linac. I @ :Calculation from Blewett [I9631 for e=0.12.

3

Coupling between longitudinal and transverse oscillation

The primary mechanism for the coupling of the longitudinal and transverse oscillations consists of I) the dependence of the transverse rf defocussing impulse on the longitudinal phase and 2) the dependence of the energy gain on transverse displacement [radial dependence of the transit time factor, as indicated in ch. C. 1.2b, eq. (4.3) or (10.6)].* The effect of this coupling on the transverse motion has been investigated analytically (for example Gluckstern [1966a, b, c]; Ohnuma [1966b]) and

* There is also a coupling due to the momentum dependence of the quadrupole focussing force and to a longitudinal force in the quadrupoles coming from the transverse velocity. These effects are the dominant ones in circular accelerators, but are small compared to the above couplings in proton linacs.

800

R.L. GLUCKSTERN


computationally (Swenson, unpublished results, and Chasman [1966]). Both treatments agree and indicate that the effect is most important at injection and can lead to a substantial deterioration of the transverse beam quality. The main effect can be readily derived from the non-relativistic coupled equations with smoothed transverse focussing and linearized longitudinal motion:

where kt and k, are the wave numbers of the transverse and longitudinal oscillations respectively. The coupling terms (actually a single term in the Hamiltonian) are seen to decrease rapidly as the energy increases. A first order treatment consists of putting the uncoupled solutions for y and x into the right side of eq. (7), averaging out high frequency terms, and integrating under the assumption of slowly varying parameters. The transverse amplitude then changes by an amount

where x0 is the amplitude of the longitudinal oscillation and 4,, 4, are the starting "phases" of the transverse and longitudinal oscillations. For a given value of $,, eq. (9) represents an elliptical distortion of the transverse phase space. For a collection of starting values of 4,, there is therefore an apparent increase of the transverse amplitude by the amount

This is its asymptotic value as s-+co; for intermediate values of s, before the oscillations die down, it will reach a value almost twice as large. The result for both transverse directions is identical, and 6R/R, the apparent growth of the transverse amplitude, is just that given in eq. (10). Equation (10) has been obtained by neglecting (2k,+k,)-' with respect to (2kt-k,)-'. Moreover, there are higher coupling terms in eq. (7) and (8). However these turn out to be important (Gluckstern [1966b]) only if one crosses the resonance kt =k, quite close to injection. The estimate in eq. (10) is otherwise quite reliable. An alternate treatment of this coupling (Ohnuma [1966a]) starts with the assumption k, 4 k, and obtains the standard stability region for the transverse oscillations in the variables 8' (proportional to quadrupole gradient) and A , the rf defocussing term which is proportional to sin4 (see chapter C.1.2~).One then constructs strong focussing amplitude parameters like the cc, j3, y of Courant and Snyder 119581 and uses

C.1.2d

NONLINEAR EFFECTS

801

the theory of adiabatic invariants to obtain the change in transverse amplitude due to the oscillations of 4. This method works best for k , + k , , but it does include some account of the strong focussing character of the transverse oscillation. The results of both methods are however in general agreement.

4

Coupling between the two transverse oscillations

Both the fringing fields of the quadrupoles (Chasman [1966]; Regenstreif [1966]; Gluckstern [1966c] and Gluckstern et al. [1967]) and higher order multipole components of the focussing field (Gluckstern [1967b]) can lead t o systematic coupling between the oscillations in the two transverse directions. Were it not for the fact that the two transverse oscillation frequencies are usually chosen to be identical, this coupling would be unimportant. The equality of the two frequencies however permits a large transfer of energy between the two modes of oscillation. Nevertheless the quantity (X2 y2)), representing the maximum radial excursion of the beam particles, turns out t o be an approximate invariant under these couplings. This result is apparently an example of the general invariant

+

q

~

-2 p

~ =2constant,

(1 1)

when the resonance is pv,-qv,=r (for example Green and Courant [1959]) with p, q, r being integers (Gluckstern [1967b]). I t is possible to obtain solutions for the equation of motion for x and y in the presence of x-y coupling (Chasman [I9661 ; Regenstreif [I 9661 ; Gluckstern [1966c] and Gluckstern et al. [1967]) which give the extent and rate of the changes in the amplitudes X and Y. These may be useful in those applications where the individual oscillation amplitudes are important (as contrasted with the radius) such as in beam transport systems.

5

Misalignments and errors

Another important contributor to changes in the transverse beam quality is the class of effects which go under the usual heading of random "misalignments and errors" (Smith and Gluckstern [1955]). These include errors in transverse placement and orientation of quadrupoles, rotation errors of individual quadrupoles about the beam axis and errors in field strengths of the quadrupoles. Such effects are important in alternating gradient systems since transverse stability arises from a delicate balance of focussing and defocussing forces, which are upset by random errors. The considerations are however different from those in circular machines (Courant and Snyder [ I 9581) where the same errors are encountered on each revolution. Here each error is encountered only once.

802

R.L. GLUCKSTERN


The equation for the transverse motion in the presence of magnetic displacement errors, A (s), is

where K(s) is proportional to the quadrupole gradient which is a function of axial position. If A (s) represents a random transverse displacement and tilt of each magnet, it is clear that the beam will be displaced coherently (without a change in shape) by an amount which will average to zero (over a large enough sample of different linacs). In a particular linac, this beam displacement will have an rms value (and a width related to the number of random components which go into this displacement) which can be shown to be

where we have assumed that the individual An are uncorrelated and each has an rms value A . The angle On= K ~ I , is the oscillation "angle" for the nthmagnet of length In, and p, is the strong focussing parameter of Courant and Snyder 119581. The effect of magnet misalignments can be greatly reduced by assembling doublets or triplets with high precision and then aligning these assemblies in the linac. In this case the individual An are correlated and the alternation of the sign of Kn causes a large reduction in the final oscillation amplitude (by one or two powers of PA/&). If there are random errors in rotation of each magnet with an rms angle E , there will be amplitude growth rather than displacement of the beam as a whole. The change in radial oscillation amplitude can be shown to be

For random errors in field strength of rms fractional value o, the result is

Once again the effect can be reduced by correlating the errors of adjacent magnets. Similar results can readily be obtained for other errors, such as magnet tilt about either transverse axis, random harmonic components to the magnetic field errors, errors in positioning the magnets along the beam axis. In general those errors which give impulses independent of the transverse position of the particles lead primarily to beam displacement such as that in eq. (1 3). Those which do depend on the transverse position lead to distortions in the transverse phase space which give an apparent growth in the transverse phase space area, such as that corresponding to eqs. (14) and (15).

NONLINEAR EFFECTS

803

6 Misalignments and couplings As discussed in the previous section, errors in the transverse alignment of the magnets cause transverse displacement of the beam as a whole, which can in principle be removed by steering magnets. And coupling of the two transverse directions, discussed in an earlier section, leads t o no growth in the amplitude of the radial oscillation. On the other hand, simultaneous presence of both of these effects n,illlead to a deterioration in beam quality, since a transverse displacement of the beam appears to be an increase of the amplitude of radial oscillation. A quantitative estimate of this effect may be made by adding the x-y coupling term to eq. (1 5) (with a coefficient which decreases as the energy increases) and finding that term in the solution which is proportional to both A and the coupling term. The result is a growth in beam radius substantially below that in eq. (13), since the coupling term decreases considerably before a sizable beam displacement has taken place. A similar calculation can be made (Gluckstern [1966a]) for the simultaneous effect of misalignment and longitudinal-transverse coupling. That part of the amplitude proportional to both A and the coupling leads to an additional increase in the amplitude of the transverse oscillation which may become important.

7 EfSect of space charge Analysis of the contribution of space charge forces to the motion and growth in size of linac beams is made extremely difficult by the non-linear, coupled nature of the forces and by the non-uniform charge distribution of the beam (Nishikawa and Okumura [1967]). This will be discussed in chapter C. 1 . 3 ~ A . t this point however, we wish to use only the obvious fact that all particle oscillation frequencies are reduced by space charge, and to explore the effect of this change on the results of the previous sections. a. Longitudinal phase acceptance: The reduction in the longitudinal oscillation frequency causes the fish-shaped region to be narrower vertically, which implies a reduction in the allowed energy spread. The effect of acceleration on the phase acceptance is estimated, as previously indicated, by considering a static configuration at a velocity corresponding to that after one phase oscillation. This will occur at a velocity much higher than in the "low-current" case, and we can therefore expect acceleration to be of greater importance when large space charge forces exist. Detailed orbit computations confirm this conclusion. b. Longitudinal-transverse coupling: The factor k: in eqs. (7), (8), (9), (10) represents the accelerating gradient and remains unchanged in the presence of space charge, while the kt and k, in the denominator of eqs. (9) and (10) will be decreased. One therefore expects an increased growth of transverse phase space due to coupling, in the presence of space charge. This result must however be used with caution, since there are also contributions to the longitudinal-transverse coupling from the non-

804

R.L.GLUCKSTERN


linear space charge force itself and from the non-uniform charge distribution within the bunches. c. The conclusions about the constancy of X2 + Y 2 in the presence of x-y coupling appear to be unaffected by the existence of space charge forces, unless the altered oscillation wave lengths correspond to troublesome "sum" resonances. d. The beam displacement due to misalignments, corresponding to eq. (13), is to be calculated using the "low-current" values of the oscillation wave length, since the oscillation is coherent and unaffected by all except image type space charge forces. e. The growths due to simultaneous misalignments and couplings are generally incoherent and should be calculated using values of oscillation frequencies which are reduced by the space charge, leading to larger amplitude growths. In general those phenomena which lead to coherent beam displacement will be unchanged, while those which lead to incoherent transverse beam growth will be enhanced, and will require smaller errors to remain within reasonable bounds.

Bibliography BLEWETT, J. P., 1963, Phase acceptance and bunching in the AGS linac, BNL Rept. AD-JPB-18. CHASMAN, R., 1966, Numerical calculations of coupling effects in a low energy proton linac, Linear accelerator Conf. 1966, LA Rept. 3609, p. 224. 1958, Theory of the alternating gradient synchrotron, Ann. Phys. 3, 1. COURANT, E. and H. SNYDER, GLUCKSTERN, R., 1966a, Note on coupled particle motion with errors, LASL Rept. MP 4/RLG5. GLUCKSTERN, R., 1966b, Transverse beam growth due to longitudinal coupling in linear accelerators, Linear accelerator Conf. 1966, LA Rept. 3609, p. 207. GLUCKSTERN, R., 1966c, The effect of quadrupole fringing fields on coupling in linacs, Linear accelerator Conf. 1966, LA Rept. 3609, p. 250. GLUCKSTERN, R., 1967a, Phase acceptance in linacs, 6th. Conf. on high energy accelerators, Cambridge, 1967, p. 153. GLUCKSTERN, R., 1967b,Transverse beam quality in linacs as affected by couplings and non linearities, misalignments and errors, and space charge, 6th Conf. on high energy accelerators, Cambridge, 1967, p. 157. R., R. STEVENS and P. ALLISON,1967, Coupling between transverse oscillations in GLUCKSTERN, linacs, LASL Rept. MP-DO/2. GREEN, G. and E. COURANT, 1959, The proton synchrotron, Handbuch der Physik, Vol. 44 (Springer, Berlin) p. 312. NISHIKAWA, T., 1967, Analytic approach to the space charge effect on the longitudinal phase motions in linear accelerators, Tokyo Rept. SJC A-67-1. NISHIKAWA, T. and S. OKUMURA, 1967, Longitudinal space charge effects in proton linacs, 6th Conf. on high energy accelerators, Cambridge, 1967, pp. 162, 166. OHNUMA, S., 1966a, Coupled motion in low energy proton linacs, Yale internal report Y-14. OHNUMA, S., 196613,Coupled motion in low energy proton linear accelerator, Linear accelerator Conf. 1966, LA Rept. 3609, p. 214. E. and P. TANGUY, 1966, Phase space representation of aberrations due to longitudinal REGENSTREIF, fields in quadrupoles, Linear accelerator Conf. 1966, LA Rept. 3609, p. 245. SMITH, L. and R. GLUCKSTERN, 1955, Focusing in linear ion accelerators, Rev. Sci. Instr. 26,220. TAYLOR, R., 1959, Calculation of drift-tube dimensions in linac injector for the 7 GeV Harwell proton synchrotron, AERE Rept. R-3012. WEST,N., 1962, The Nimrod Injector. Linear Accelerator for High Energies Conference, Brookhaven, Aug. 1962, p. 38.

PLA

THEORY

C.l

3 Particle Dynamics at High Intensity

I

CERN, Geneva, Switzerland P. LAPoSToLLE

When the intensity is increased in a linear accelerator, various phenomena can appear. They are all however a consequence of the presence of large quantities of charge or large charge densities in the accelerating structure. The presence of charges modifies the electromagnetic field distribution as a consequence of Maxwell's equations. The exact computation of such an effect is however tedious and approximations have to be made. The field changes occur in fact not only in the close neighbourhood of the charges but in the whole structure.

1 Beam loading This effect in the whole structure, or "distant effect" in particular does not depend much from the detailed charge distribution in the bunches. It is legitimate to assume that to a good approximation it is the same as if all the charges of the bunches were concentrated in their centre or evenly distributed inside small volumes. This effect is usually called beam loading. The passage of charges absorbs a certain power from the electromagnetic field, power which is used for acceleration. This concept of power consumption is not, however, the most appropriate to describe beam loading since it is a complicated function of the electromagnetic power fed into the cavities. It seems better to consider the beam as a current source which can induce fields. A bunched beam can be replaced with the help of Fourier analysis by a dc current and fundamental and harmonic components. The fundamental current produces the usual beam loading. As a consequence of the linearity of Maxwell's equations it induces fields which add linearly to the accelerating fields fed externally into the 0 North-Holland Publ. Co., Amsterdam

1969

805

806

P. LAPOSTOLLE


cavities. This beam loading field is phase related to the bunches and has a component in opposition with the accelerating field as soon as there is acceleration. In order to compensate for it, additional power, both active and reactive, has to be fed into the cavities if, as it is normally the case, one tries to keep the accelerating field to its nominal value. This is now commonly done on high intensity accelerators.

2 Transients The main difficulty with beam loading is related to transient effects. Linear accelerator operation is usually pulsed. The front edge of a beam pulse is normally sharp and in the high Q cavities which are used, the beam induces transients which extend over periods of appreciable length. In order to compensate beam loading fields, one tries to feed some extra power into the cavities; this also takes time to distribute over a whole structure through transient phenomena which do not exhibit the same behaviour as for beam loading since compensation is usually fed from a single point whereas beam loading is distributed along the whole of each cavity. The Alvarez structure usually operates on the zero-mode where group velocity is zero and energy velocity rather small, as discussed in detail in chapter C.1.3b. That explains all the efforts which are made to try to find an alternative or at least a derivative to the Alvarez cavities where the group velocity should be large.

3 Space charge eflects Besides the distant effect of the charges upon the rf fields in the cavities there is also a "local effect" upon the fields in the neighbourhood of the bunches. This one is to some extent not much dependent on the accelerating structure except from image effects on the walls of the drift tubes. Charged particles create around them a space charge jield which is repulsive for other particles of the same sign. Bunches tend to become larger and the efficiency of focusing and longitudinal stability are reduced. To overcome these effects, stronger focusing and higher accelerating fields are required. This space charge effect is studied in chapter C.1.3~. Experimental results of high intensity operation show however in practice, various additional effects as rounding off of the beam and noticeable emittance blow up, as well in the transverse as in the longitudinal direction; these are certainly a consequence of space charge forces probably through couplings and nonlinearities; they are not yet however fully analyzed nor understood.

4 Other eflects In a bunched beam the fundamental component is responsible for the usual beam

C.1.3a

INTRODUCTION

807

loading. There are, in addition to it, a number of harmonics which might excite higher order modes. In present linacs, however, these harmonic frequencies have probably always fallen outside the bandwidth of these modes since nothing of such an excitation has been ever noticed. This danger nevertheless potentially exists. Eventually there is the dc component of the beam. If the beam was not tightly bunched and if particles could slip from bunch t o bunch, this dc component would be of some effect. It might shift the resonant frequency of the cavities since it would behave as a medium of dielectric constant different from 1. Its permittivity should even be complex, leading to longitudinal instabilities. Furtunately all these phenomena are suppressed by the bunching.

5 Beam break up Apart from this longitudinal effect, there might be also transverse instabilities; these are hardly affected by longitudinal bunching. Fortunately, the structures used so far exhibit always a very weak coupling between a beam circulating on their axis and their transverse modes; no instability of this type has been observed yet in proton linear accelerators. It remains however a potential risk.

PLA

THEORY

C.l


1.3b I Transients and Beam Loading Effect T. NISHI KAWA Department of Physics University of Tokyo, Tokyo, Japan

1

Various methods of analysis

In the energy range of interest, the velocity of a proton changes significantly during acceleration, so that the proper control of the accelerating field under the presence of beam loading is of great importance. For this reason, the standing-wave type is usually adopted for proton linacs. In the traveling-wave case the correct distribution of the axial accelerating field can be set up only for a particular value of beam current, while, for the standing-wave type, it may be reproduced for any beam currents by changing the input power. In order to find a proper design, however, analyses of the energy flow and transients in such a standing-wave cavity are important, for considerable amount of extra power will be necessary to compensate beam loading. Various methods of transients analysis have been developed in recent years: they are 1 . An equivalent circuit analysis; 2. An analysis using waveguide modes with the field given by waves traveling in opposite directions; 3. An analysis based on the microwave cavity theory in which the field is expanded in terms of normal modes. A simple equivalent circuit analog for standing-wave cavity-chains, or the "coupled resonator model", has been given by Nagle [1964], Nagle and Knapp [1964], Nagle et al. [1967], B. C . Knapp et al. [1965a,b]* in the course of the design study for the

* See also Swain [1966]. 8 North-Holland Publ. Co., Anrsterdam

1969

Refermces p. 824

810

T. NISHIKAWA


mesonfactory project at Los Alamos. Jameson (see Jameson [1964a, b, c] and Jameson et al. [1965]) has made a transients analysis of such a circuit leading to agreement with the model experiments. An alternative of the equivalent circuit in transmission line mode has also been proposed by Hereward (see Hereward [1965], Hereward and Lapostolle [1966]) concerning energy propagation in the drift-tube type structure. Equivalent circuit analyses permit one to investigate the nature of energy flow and phase gradient in an accelerating structure without detailed knowledge about the field configuration. It is, however, not straightforward to evaluate beam-cavity interactions from circuit elements which represents integrated field effects; only a simple analog with beam loading effect has been given by Gluckstern [1964, 19651. On the other hand, in a microwave treatment, the components of the field excited by the power source and induced by the beam can be derived separately and later added to obtain the total field with beam loading. In this line, Lapostolle [1965a, b] has developed a theory using a cylindrical waveguide model, and Nishikawa [1965b, 1966a, b] with the normal mode analysis of the TM,,,-type cavity *.

2 Transients calculation by cylindrical waveguide model In a cylindrical waveguide of radius a, the longitudinal electric field of TM,, mode will be written as E,

= Jo

(kcr) f (z, t )

2

where kc = xtoc/c = 2.405/a, oc and x being the cut-off frequency and the dielectric constant, respectively. Including wall-losses, Lapostolle derives the wave-equation for f (z, t) as

In this expression, z = a/(8oc),

o:

= oc(1 - 36/a),

(2.4) (2.5)

and 6 is the usual skin depth. Eq. (2.3) is of the telegraphy type and f (z, t) is given by

* Abraham [I9661 attempted to apply the S-matrix approach in waveguide theory to a transients analysis of a linac cavity.

TRANSIENTS AND BEAM LOADING

C.1.3b

811

the form t

x Jo [a: {t" - ( X Z ~ / C ~ ) ) ' ]dt'

, for t > IiI J x / c ,

At the input end (z=O), f ( 0 , t ) is zero for negative t and can be applied to any input signal for t > O . Assuming an exponential build-up input signal with the frequency wi, Lapostolle calculates amplitude and phase responses along an infinite waveguide. For a resonant cavity (at a cut-off mode) of length L, he gets an approximate formula,

with

and

Effect of drift-tubes loaded in the Alvarez-type cavity can be replaced by an

RELATIVE TIME

t/r

Fig. 1. Amplitude response at the end of a cut-off mode cavity calculated from the cylindrical waveguide model (Lapostolle (1965a1).

812

T. NISHIKAWA


effective change in the dielectric constant x . By applying a dispersion relation such as given by Leiss (see Leiss [1965], Leiss et al. [1962]), the periodic configuration of the loaded structure can also be taken into account. The calculated results for the usual drift-tube cavity are shown in figs. 1 and 2. These figures show the general response time proportional to LZ with phase shift and delay along the cavity, in agreement with measurements (Giordano [1964], E. A. Knapp and B. C . Knapp

-

RELATIVE

TIME

t /r

Fig. 2. Phase response at the end of a cut-off mode cavity calculated from the cylindrical waveguide model (Lapostolle [1965a]).

3 Normal mode analysis For an actual cavity, measurements also show some wiggles or scallops in transient amplitude and phase responses (Keane [1964]). Normal mode analysis based on microwave cavity theory gives a better representation of the field including these further effects. From a general cavity theory, it is possible to expand the actual field in terms of the normal modes which form a complete orthogonal set of functions for the field in an ideal cavity (Slater [1950]). If such a normal mode is well defined and separated from its nearby modes, then normal mode analysis is a useful method by which the various effects caused by interactions or perturbations within the cavity may be r, O), the actual field can be expressed examined. Taking the normal electric field, En(z, as E (z, r, 8, t) = C Vn(t) En(z, r, 6). (3.1) n

C.1.3b


813

From Maxwell's equations and orthogonal relations, the expansion coefficient of V n ( t )satisfies the equation of forced oscillations,

The first term on the right hand side is integrated over the non-ideally conducting surfaces, S , such as lossy metallic walls. By means of the standard evaluation of Q, from wall losses, this can be rewritten as a damping term

The surface integral in the second term is performed over the open surfaces, S ' , and gives the effect of coupling the cavity to an outside system. One part of this term gives forced oscillations excited by an external source and the other additional damping due to circuit losses (cc l/Q,,,). Effects of the beam within the cavity are represented by the last term where the current density J at a point is multiplied by the normal electric field a t that point and integrated over the cavity volume. If the cavity is excited near one of the normal mode frequencies, o,(n=l), then the modes near the lth mode will predominate in normal mode analysis. For a long cavity, such as used in linacs, these nearby modes will have a different z-dependence from the resonant mode but the axial field will have the same radial and azimuthal dependencies (e.g., TM,,, modes for most cases of linac cavities). Considering the periodic structure of normal mode fields, we may write the axial component as* m

E,(z, r, 0)

=

(8, (r, 0) a,')

n = O , 1 , 2 ,..., N ,

1

T,, cos {(n

+ 2 s N ) nzILJ,

\=-w

(3.4)

where L is the total length, N is the number of cells, Lo = LIN being the cell length, T,, is the so-called transit-time factor, and a, is a normalization constant,

a,

=

1,

for other modes

Here, we neglect the asymmetric distribution of loaded obstacles, which corresponds to the velocity change of accelerated particles. For a drift-tube cavity, assuming that the field is constant over a gap and zero inside the region occupied by drift-tubes, we

* The derivation of normal-mode fields is given in appendix 1 in Nishikawa [1965b]. See also Beringer and Gluckstern [I 9641.

814

T.NISHIKAWA


get a transit time factor as

where g is the gap length and kn=nn/L. Similarly, for a loosely-coupled multicell cavity, in which the field may be assumed to be constant in each cell, T,, is given by eq. (3.5) with g = Lo. Consider a cavity, with no beam, excited by a well-padded generator, then eqs. (3.2) and (3.3) give (Nishikawa [1965a, b])

where Q,, Q,,,,, and K, are given by

and

Zo is the characteristic impedance of the waveguide coupled to the cavity, only one

propagating mode being assumed in the guide. At the surface S' the cavity normal mode field, En, is related to the normal tangential component of the waveguide mode, e,

En = Une (U, is a real constant). With a good approximation, we can rewrite eq. (3.6) for forced oscillations of the resonant mode (n = 1) at resonance as

and for nonresonant modes (n # I)

where the lth mode resonant frequency is to be given by

Eq. (3.9) corresponds to eq. (2.5) given by Lapostolle. These equations can be solved In addition, in order to by assuming proper initial conditions for A, Vn and determine the field inside a cavity it is necessary to assume a proper boundary condition at S' which leads to an evaluation of Un.If the power is fed from one end of the cavity (z = 0 ) then the nearby modes have nearly the same normal mode field patterns at S' as for the resonant mode, so all modes are excited in phase at z = 0 . Thus, except

K.


C.1.3b

815

for high n's, (a,,)' U,, = (a,)+ U, . For a cavity coupled at any other point (2,) in the cavity the value of U,, will depend on the z-dependence of the mode in question and we may write (Batchelor et al. [19671) (c# U,, cos (klzf) = (aI)' UI cos(k,zf). It should be noted that this also may be in error for a loaded cavity due to the asymmetric distribution of obstacles along the cavity axis.

4 Results and comparison with measurements Results of normal mode analysis particularly for the drift-tube cavity can immediately be compared with measurements, such as made at the AGS injector linac cavity (Batchelor et al. [1966, 19671; Keane [1964]; Nishikawa [1965b, 1966a, b]).

4.1

Steady state field

For a drift-tube cavity with zero mode operation, the steady state solution is given by E ( z , t)

= E,

expjobt 1 + 2j(ob/Qoo)

cos (k,z,).cos ( k , z )

where

with the assumptions of

lor

w b 4 / Q n< - wo I , T,, = 1 , for s = 0 and = 0 , for s f 0 . 121

The first term in the bracket of eq. (4.1) arises from the resonant zero mode, and the others from higher modes. It is noted that the first-order effect is the phase shift which is given by a linear combination of higher mode components, each proportional to (LC);-o$)-'. Excitations of these modes are necessary for compensating wall losses along the length since the velocity of energy transport by means of the cut-off mode field is zero corresponding t o the vanishing group velocity at that mode. The calculated and measured phase shift for the injector linac of AGS at Brookhaven is given in fig. 3. The present AGS linac is driven at about the center of the cavity (z,=+L). In practice, however, the mechanical center is not at the electrical

816

T.NISHIKAWA


center of the cavity, so there is some coupling to odd-numbered modes resulting in asymmetric effects along the length. For the computation, nearby modes up to n = 4 are included by using their measured frequencies and Q-values with an assumed value of z, slightly shifted from +L.

- CALCULATED 0 X

MEASURED

I

I

X

0.5

-4.01

I

I

I

1

.

1

~

1

I

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0 9 AXIAL DISTANCE / LINAC LENGTH

I

1.0

Fig. 3. Steady state phase shift along the length of AGS linac cavity. Following values are assumed in the calculation: zp =0.565L, 5201 =29.5 kc/s, 5202 = 121.8 kc/s, 5203= 301.6 kc/s, 5204 = 512.3 kc/s, Ql = 36600, QZ= 21 900, Q3 = 32900 and Q4 = 29200 (Batchelor et al. [1966, 19671 and Nishikawa [1966b]).

4.2 Transient field Assume a step function incident wave ( A = 0 for t GO and A =constant for t > 0 ) and initial conditions Vn= v,,= 0 at t = 0. In the first-order approximation for transients, the build-up curve of the resonant mode (n=O) is, from the solution of eq. (3.7),

Eo exp jobt [I - exp ( - t/z0)], whereas for a nonresonant mode (n # 0, n 4 N) the term

2jEo Qext,

I

o

[1 - exp(- t l ~ ~-) l

- exp [ - (tlz,) + j (w: - wb) t ] cos ( k n z f .cos ) (knz), is obtained from the solution of eq. (3.8). The decay time T , is now given by 2Q,/w,. The terms having the angular frequency w: (n#O) represent free oscillations and terms involving wb are forced oscillations. Eq. (4.3) shows wiggles which arise from the beat between the free oscillations of that mode and the driving force, the angular


C.1.3b

817

frequency of the beat being

Figs. 4 and 5 show calculated amplitude and phase responses for the AGS linac a t a near-center-fed case. The measured curves and values show good agreement with

calculations. Calculated delay time at the end of the cavity is about 2 psec and agrees with the experiment, too *.

Fig. 4a. Amplitude response at the center of AGS linac cavity. (a) Calculated from the normal mode analysis.

Fig. 4b. Observed by a pick-up electrode (20 psldiv). (Batchelor ct al. [I9671 and Nishikawa [1965b, 1966b1).

*

Detailed discussions on the delay time are given by Lapostolle [1965a, b] and Nishikawa [1965b].

818

5


T. NISHIKAWA

Field induced by the beam

Both in the microwave waveguide and cavity theory, beam loading effect can be taken into account by calculating induced field. Usually the time taken for the beam bunches to fill the length of the cavity is short enough to be neglected compared with the time constant or the filling time of the cavity. The negative transient, induced by beam loading, has therefore a build-up of pure exponential type with no phase nor amplitude variations along the cavity, provided parameters of beam bunches are uniform all the way along. In terms of the normal mode analysis, this corresponds to the argument that in a long cavity the beam gives very small excitation to nonsynchronous mode since the fields induced by the bunches, which are distributed along the length, are superimposed with the proper phase relations only on a synchronized mode. -220

-220

-180 k

-180 t (a) L O W E N E R G Y E N D

- CALCULATED 0 x MEASURED

(b) H I G H E N E R G Y E N D

t -60 z ; -20 2 L

20

Fig. 5. Transient phase shifts between center and both ends of AGS linac cavity. (Batchelor et al. [1966, 19671 and Nishikawa [1966b]).

It is shown that the resulting beam-cavity coupling integral in eq. (3.2) for the synchronous Ath mode is

s

J.E, dv = +Z,b,LT,f,a~

expj (wit + %) ,

(5.1)

where I,, = beam current averaged over bunches; cp,=phase angle of the center of beam bunches; TA = transit time factor for the synchronized space harmonic; we assume that the fundamental harmonic (s=O) synchronizes with the beam; fn =form factor of bunches. The synchronized condition is given by 0; = 2rcVVb/Lb )

.

‘;.I if the particle velocity is vr,, the spacing between bunches is L,, and v is a positive integer (we consider the case of v= 1). Assuming a step function beam (I, =0 for

C.1.3b

TRANSIENTS

AND

BEAM

LOADING

819

t < t, and 1, = constant for t> to) uniformIy distributed along the cavity, we get an equation of forced oscillations similar to that of eq. (3.7), leading to the induced field on the axis, E,, (z, r = 0, t) = - r,Jofn& ewj (dt - vb)

x [l - exp { - (t - t,)/r,}] cos (kAz) .

(5.2) *

In this expression, the effective shunt impedance reA is given by reijQOl = a; (r = 0) J!z,2/(2&4).

(5.3)

The form factor f’ can be calculated for some simple shapes of bunches and, if the beam spread is less than 60”, we may assume thatf, = 1. Corresponding to the minus sign of eq. (5.2), the induced field is in the opposite phase to that of the beam bunches. The synchronous phase angle (Pi is usually chosen to be a few tens of degrees behind the resultant field within the cavity. Then the real and the imaginary part of E,, give the resistive and the reactive loading, respectively.

6

Efiects of beam loading

If interactions between the beam and the external power source are neglected, then each component of the field excited by the power source and that induced by the beam are added to obtain the cavity field with beam loading. For many cases of standing wave linacs, the time constant tl is so long compared with the beam pulse length that beam loading effect is of a transient nature.

6.1

Phase shift and energy spread due to field variation

First of all, in order to satisfy the required change in particle velocities from one cell to the next, the field decrease due to transient beam loading will shift the synchronous phase angle towards the crest of the wave as the induced field builds up. If the field decrease is small enough, such a phase shift will approximately be given by a relation (L. Smith [1959]), cos % (t> - cos (Pb (43)

z Hcp,‘(h,) - d(t)1 = I-G,(~)l/l~‘(4,)l.

For a typical value of (Pi, the phase shift will amount to one degree per one percent change in the field strength E. The time dependent phase shift will also cause an extra energy spread which results from the smaller phase stability, and consequently, with a 10% decrease in the field the particles will be lost. In addition, the dynamic change in the longitudinal * A factor of l/r~ is is because the different of the reference. See also Lee [1968].

820

T. NISHIKAWA


phase motion will affect the transverse motion of particles resulting in dynamic emittance variation.

6.2 Effect of reactive loading The reactive beam loading will cause a phase shift between the resultant cavity field and the driving field from the power source. This phase shift is given by

where IEI and IEbI are the amplitudes of the resultant and the induced field, respectively (fig. 6). In particular, if an accelerator is composed of a number of cavityINDUCED FIELD

-----__------_

'.

----_

RESULTANT FIELD

FIELD

Fig. 6. Composition of the cavity field from the driving field and the induced field.

tanks (or it is associated with a separate bunching system), then phases between separate tanks should be adjusted corresponding to the different value of A q in each tank. The value of A q will reach the order of ten degrees for a 100 mA current (Carter [1967a]; Nishikawa [1964, 1966bl).

6.3 Beam loading compensation In general, beam loading effect will be compensated by feeding extra power to the cavity. To achieve a good compensation, it is customary to increase the input power during the beam pulse by an amount equal to the beam power. This extra energy may be added as a step function or a ramp function both of which give rise to a transient response in the cavity due to excitation of nonresonant modes (n#l=A). The beam itself, however, gives very small excitation to these nonsynchronized modes. Consequently, amplitude and phase errors will remain after compensation, which vary with position along the cavity length due to the normal field distribution of higher order modes. Table 1 gives a measurement at the AGS linac to show the effect of beam loading for the case with and without compensation. Even without particular compensation technique, almost the maximum capture efficiency of the linac is achieved for 20 mA loading by increasing the input power level and adjusting the phase between the beam and the wave. As discussed in § 6.1, however, transient beam loading causes a larger


C.1.3b

821

TABLE 1 A measurement of beam loading effects in the AGS linac (From Nishikawa [1965b]) -

-

-

-

Compensation

No

Yes

Yes

-

10

Current output (mA) Current input (mA) Maximum field change at center Obs. (MV/rn)

-

0.02 0.015

Calc. (MV/m) Energy spread of output beam (half width, MeV) Number of particles accelerated by AGS ( x 1012 ppp)

0.18

0.57

CALCULATED FIELD VARIATION DUE TO 30rnA BEAM LOADING WlTH OPTIMUM COMPENSATION PULSE. (COMPENSATION PULSE IS ASSUMED TO BE 5psec IN AWANCE OF BEAM PULSE AND BUILD-UP LINEARLY WlTH A RISE TlME OF 15rsec) BEAM ON

I

I

0

l

10 TlME

l

I

I

20

30

40

I

I

(YS~C)

Fig. 7. Field variation due to 30 mA beam loading after an optimum compensation (Batchelor et al. [I 966, 19671). /

I

l

l

EXTRA PHASE SHIFT DUE TO COMPENSATION PULSE : CALCULATED AT FAR ENDS (6% COMP. WITHOUT BEAM)

TlME

(p Sec)

Fig. 8. Extra phase shift due to the beam compensation pulse (Batchelor et a1 [1966,1967]).

822

T. NISHIKAWA


energy spread of output beam and a smaller percentage of particles accepted in the AGS. Such effects can greatly be improved by adding extra compensation power during the beam pulse, but there still remain phase and amplitude errors due to nearby modes effects. Figs. 7 and 8 show calculated and measured phase and amplitude errors, as functions of time and length, after an optimum compensation for 30 mA beam loading in the AGS linac. Similar measurements have also been made by Taylor and Dupuis [1964] at the CERN PS linac and by Livdahl [1964], Perry and Abraham [I9671 at the ZGS linac; their results can also be explained at least qualitatively by the analysis described above.

7 Methods for improvements 7.1 Use of multiple feeds If a cavity is operated at a cut-off mode such as a zero- or z-mode (n = 0 or N), then the transient phase and amplitude delays are roughly quadratic in total length L. In terms of normal mode analysis, this is because the mode separation near a cut-off mode satisfies an approximate relation,

(ul2+, - 0:)cc m 2 / ~ 2 .

(7.1)

Thus a shorter cavity has an advantage of achieving a perfect beam loading compensation. Another reduction in phase and amplitude delays, however, is also possible by feeding power into the cavity at more than one point. For example, if the cavity is excited at two points, z=+L and $L, then the nearest higher mode excited by the generator can be raised to m = 4, so a considerable reduction in amplitude and phase variation is expected. An experimental study of such a multi-drive excitation has been carried out by Giordano et al. [I9671 who obtained promising results. Voekler [1964a, b] has also proposed to use a distributed power feed system with an rf manifold along the cavity.

7.2 Use of middle of pass-band or alternating periodic structure If a uniform periodic multicell cavity is operated in the middle of a passband or in the +c-mode (n = $ N ) , the field distribution becomes different from that of a cut-off mode. The anti-symmetric frequency distribution of nearby modes around that mode cancels out the most part of nonresonant-mode effects. Thus, both for a steady and a transient state, any additional phase shift does not appear as a first order effect. The only disadvantage of using this mode is a relatively lower shunt impedance than at the cut-off mode; this predominantly comes from the normalization factor a, [biac U,E; in eq. (5.3)]. Therefore, some new accelerating structures having an alternating periodicity have been proposed and developed by Giordano, Knapp and others. For instance, the multi-stem drift-tube cavity proposed by Giordano (see Giordano

C.1.3b


823

[1966], Giordano and Hannwacker [1966, 19671 has a set of modes associated with resonance in the circumferential field in the stem system, which couples to the usual TM,,, modes. The coupled resonance system forms an alternating periodic structure. Essentially the same type of drift-tube cavity with resonant coupler systems is being studied by E. A. Knapp et al. [1967], Swenson et al. [1967]. It should also be noted that similar modes are discussed by Carne et al. [I9661 in the treatment of the crossbar structure. For the iris-loaded type cavity, the Los Alamos group explores the sidecoupled structure (E. A. Knapp [1964, 19651, E. A. Knapp et al. [1966]), while Brookhaven (Giordano [1965a, b], Wheeler and Giordano [1965]) and other superconducting linac groups (Backenstoss et al. [1967], Smith et a/. [1966], Weaver et al. [1967]) investigate the structure in which short and long cells alternate. Theoretical aspects of the dispersion relation and field stabilization for such an alternating periodic structure (APS) have been investigated by Blewett (Blewett [1965]), Nagle (Nagle [1966], Nagle et a/. [1964, 19671) and Nishikawa (Nishikawa [1965a, b, 1966~1,Nishikawa et al. [1966])*. As shown in fig. 9, the dispersion curve splits into two pass bands corresponding to double periodicity, and the APS mode is essentially a cut-off mode having two different frequencies o,and w, for two field configurations under different end-conditions. There is a stop band, in general, between w , and a,.By means of a slight change in the boundary of one of the two cell-types, however, one can make w , close enough to w,, so that two passbands may merge, as the limit, into a single passband having a finite group velocity at the APS mode. A measure of insensitivity to beam loading and any other perturbations can be given by v,(w, Lo) for the FREQUENCY -MEASURED CALCULATED CURVE BY NORMAL 0

MODE ANALYSIS flMc)

6"+2" APS (WITHOUT NOSE CONE)

6"+2' APS WITH NOSE CONE IN LONGER CELLS

840 -

820

820

780 O

$

+

~

MODE

T

0

I6 r 7 3 r2 Z y F T MODE

Fig. 9. Calculated and measured dispersion curves for alternating periodic structures (Nishikawa et a/. [1966]).

*

See also Carter [1967a, b] and Claus [1967].

824

T. NISHIKAWA


matched condition (o, = 01.), and by lo, - o, l/o, for the nearly matched case (o, z 4, where v,, Lo and o, are the group velocity, the periodic length and the total passband width, respectively.

7.3 Timing and shape control of beam pulse Other possibilities of reducing nearby mode effects will be given from considerations on transients control. In this respect, Carne and Batchelor [1964] have proposed use of the transient build-up of the main pulse for beam loading compensation by adjusting the timing of beam pulse. Alternatively, if the beam pulse has the shape of a ramp function as well as the compensation pulse, errors of beam loading compensation due to high-order modes will be much reduced compared with the case of a step function type (Nishikawa [1966b]).

8 Conclusive remarks In conclusion, if the beam current is much less than 100 mA, then the beam loading effect in proton linacs will be moderate and the practical design will be a compromise with some other factors. For a high current, however, beam effects will predominate in the practical design of linacs. It should also be noted that the higher the current, the more the stability of the ion source and other parameters associated with the beam is essential. If the beam current fluctuates by 10% at 100 mA, the resulting cavity field will vary by about the same percentage, due to the change in the induced field. Thus an automatic control system for the cavity field and the beam current will be necessary for stable operation. In addition, although we give no detailed discussions here, the coherent beam instability due to the excitation of a synchronized higher-order mode will be important as the beam current becomes much higher.

References ABRAHAM, J., 1966, Transient wave propagation in linear accelerators, Linear accelerator Conf. 1966, LA Rept. 3609, p. 337. BACKENSTOSS, G., et al., 1967, Studie iiber einen supraleitenden Protonen Linearbeschleuniger im GeV Bereich. Inst. Experimentelle Kernphysik, Kernforschungszentrum, Karlsruhe. K., R. CHASMAN and T. WERNTZ, 1966, Measurements and calculations on the 50 MeV BATCHELOR, Brookhaven linac, Linear accelerator Conf. 1966, LA Rept. 3609, p. 28. and T. WERNTZ,1967, Numerical analysis of the rf field in a drift BATCHELOR, K., T. NISHIKAWA tube loaded cavity, IEEE Trans. Nucl. Sci. N.S. 14, no. 3, p. 295. BERINGER, R. and R. L. GLUCKSTERN, 1964, On the normal modes in linac structures, Yale Rept. Y 9. J. P., 1965, Network representation of 7212 mode accelerating waveguide system, BNL BLEWETT, Rept. AADD 65. CARNE, A. and K. BATCHELOR, 1964, Design of a high current 200 MeV proton linear accelerator, RHEL Rept. NIRL/R 55. CARNE,A., G. D ~ M E N., FEWELL and W. JUNGST, 1966, Development of the cross-bar structure for a proton linear accelerator, 5th Conf. on high energy accelerators, Frascati, 1965, p. 642.

C.1.3b


825

CARTER, D., 1967a, Linac phase error calculations, BNL Rept. AGSCD 21. CARTER, D., 1967b, Field stabilization in periodic systems, BNL Rept. AGSCD 24. J., 1967, On the influence of stem impedances upon the dispersion curves of proton linac CLAWS, cells, BNL Rept. AGSCD 16. GIORDANO, S., 1964, Model measurements and correction of beam loading effects in proton linacs, Linear accelerator Conf. 1964, MURA Rept. 714, p. 252. GIORDANO, S., 1965a. Multiply periodic rf structures for proton acceleration, BNL Rept. AADD 64. GIORDANO, S., 1965b, Some new rf accelerating structures, lEEE Trans. Nucl. Sci. N.S. 12, no. 3, p. 213. GIORDANO, S., 1966, Measurements of the multistem drift tube structure for proton accelerators, BNL Rept. AGSCD 11. GIORDANO, S. and J. P. HANNWACKER, 1966, Measurement on a multistem drift tube structure, Linear accelerator Conf. 1966, LA Rept. 3609, p. 88. GIORDANO, S. and J. P. HANNWACKER, 1967, Studies of multistem drift tube accelerator structures, IEEE Trans. Nucl. Sci. N.S. 14, no. 3, p. 290. GIORDANO, S., J. P. HANNWACKER and J. T. KEAN,1967, Studies of multidrive excitation for Alvarez structures, lEEE Trans. Nucl. Sci. N.S. 14, no. 3, p. 303. GLUCKSTERN, R. L., 1964, Beam loading effects in proton linear accelerator, Proton linear accelerator Conf. 1963, Yale Rept., p. 95. GLUCKSTERN, R. L., 1965, Equivalent circuits for cavity chains, LASL Rept. MP 4/RLG 4. H. G . , 1965, Some examples of energy flow in the Alvarez structure, CERN Rept. MPS/ HEREWARD, DL/int. 65-1. HEREWARD, H. G . and P. LAPOSTOLLE, 1966, Energy flow and transients in the Alvarez structure, 5th Conf. on high energy accelerators 1965, Frascati, p. 742. JAMESON, R. A,, 1964a, Rf phase and amplitude control, Linear accelerator Conf. 1964, MURA Rept. 714, p. 505. R. A,, 1964b, Envelope response of a chain of coupled circuits, LASL Tech. Memo P 1I/ JAMESON, RAJ 3. JAMESON, R.A., 1964c, Transient response of a chain of coupled circuits, LASL Tech. Memo P 111 RAJ 4. JAMESON, R.A., T. F. TURNER and N. A. LINDSAY, 1965, Design of the rf phase and amplitude control system for a proton linear accelerator, IEEE Trans. Nucl. Sci. N.S. 12, no. 3, p. 138. J. T., 1964, Experiments on rf field patterns in the BNL linac, Linear accelerator Conf. 1964, KEANE, MURA Rept. 714, p. 398. KNAPP,B. C., E. A. KNAPP,G . J. LUCASand J. M. POTTER,1965, Resonantly coupled accelerating structures for high current proton linacs, IEEE Trans. Nucl. Sci. N.S. 12, no. 3, p. 159. KNAPP,B. C. et a/.,1965, Electrical behavior of long linac tanks and a new tank coupling scheme, IEEE Trans. Nucl. Sci. N.S. 12, no. 3, p. 623. KNAPP,E. A,, 1964, 800 MeV rf structures, Linear accelerator Conf. 1964, MURA Rept. 714, p. 31. KNAPP,E. A,, and B. C. KNAPP1964, Experimental checks on the validity of the coupled circuit approximation for a chain of coupled resonators, LASI, Tech. Memo P ll/EAK/BCK 1. KNAPP,E. A., P. W. ALLISON,C. R. EMIGH,L. N. ENGEL,J. M. POTTERand W. J. SHLAER,1966, Acceleratingstructure research at Los Alamos, Linear accelerator Conf. 1966, LA Rept. 3609, p. 83. KNAPP,E. A,, J. M. POTTER,E. J. SCHNEIDER and D. A. SWENSON, 1967, Stabilization of thedrift tube linac by operation in the in cavity mode, LASL Rept. MP 3IEAK/JMP/EJS/DAS I. LAPOSTOLLE, P., 1965a, Phenomtnes transitoires dans un guide d'ondes electromagnttiques au voisinage de la frequence de coupure; application a la structure Alvarez des accelerateurs lineaires a protons, CERN Rept. AR/int. SG/65-9. LAPOSTOLLE, P., 1965b, Phenomenes transitoires dans une structure periodique, comparaison des modes 0 et n ; application a des structures d'accelerateurs linkaires a protons, CERN Rept. AR/int. SG/65-12. LEE,M. J., 1968, Transient beam loading in standing wave linacs, Linear accelerator Conf. 1968, BNL 50120, p. 108. LEISS,J. E., 1965, Beam loading in linear accelerators, IEEE Trans. Nucl. Sci. N.S. 12, no. 3, p. 566. LEISS,J. E., and R . A. SCHRACK, 1962, Transient and beam loading in linear electron accelerators, NBS internal Rept.

826

T. NISHIKAWA


LIVDAHL, P. V., 1964, ZGS injector observations, Linear accelerator Conf. 1964, MURA Rept. 714, p. 384. NAGLE, D. E., 1964, Coupled resonator model of linear accelerator tanks, Linear accelerator Conf. 1964, MURA Rept. 714, p. 21. NAGLE,D. E., 1966, High energy proton linear accelerators., 5th Conf. on high energy accelerators, Frascati, 1965, p. 403. NAGLE,D. E. and E. A. KNAPP,1964, Steady state behaviour of a ring or of a chain of coupled circuits, Proton linear accelerators Conf. 1963, Yale Univ., p. 171. NAGLE,D. E., B. C. KNAPPand E. A. KNAPP,1964, Perturbation theory for the doubly periodic chain, LASL Rept. P ll/DN/BK/EK 1. NAGLE,D. E., E. A. KNAPPand B. C. KNAPP,1967, A coupled resonator model for standing wave accelerator tanks, Rev. Sci. Instr. 38 1583. NISHIKAWA, T., 1964, An approach to the study of beam loading for the linear accelerator, Linear accelerator Conf. 1964, MURA Rept. 714, p. 214. NISHIKAWA, T., 1965a, Study of field distribution and beam loading in proton linacs at high energy, IEEE Trans. Nucl. Sci. N.S. 12, no. 3, p. 630. NISHIKAWA, T., 1965b, Normal mode analysis of standing wave linacs; field excitation and beam loading in linac cavities, BNL Rept. AADD 87. NISHIKAWA, T., 1966a, Normal mode analysis of standing wave linacs, 5th Conf. on high energy accelerators, Frascati, 1965, p. 651. NISHIKAWA, T., 1966b, Beam loading effects in standing wave linacs, Linear accelerator Conf. 1966, LA Rept. 3609, p. 294. T., 1966c, Equivalent circuit and dispersion relation for the multistem drift tube structure, NISHIKAWA, BNL Rept. AADD 125. T., S. GIORDANO and D. CARTER, 1966, Dispersion relation and frequency characteristics NISHIKAWA, of alternating periodic structure for linear accelerator, Rev. Sci. Instr. 37 652. 1967, Measurement of phase shift due to beam loading in an AlvarezPERRY, R. and J. ABRAHAM, type linac, 6th Conf. on high energy accelerators, Cambridge, 1967, p. 270. J. C., 1950, Microwave electronics (D. Van Nostrand, New York), p. 57-67. SLATER, SMITH,L., 1959, Linear accelerators, in Handbuch der Physik Band XLIV, eds. S. Fliigge and E. Creutz (Springer, Berlin) p. 380. W. M. FAIRBANK and P. B. WILSON,1966, Stanford's superSMITH,T. I., H. A. SCHWETTMAN, conducting accelerator program, Linear accelerator Conf. 1966, LA Rept. 3609, p. 491. SWAIN, G. R., 1966, Circuit analog techniques for analysis of resonantly coupled linear accelerator structures, Linear accelerator Conf. 1966, LA Rept. 3609, p. 125. D. A., E. A. KNAPP,J. M. POTTERand E. J. SCHNEIDER, 1967, Stabilization of the driftSWENSON, tube linac by operation in the n/2 cavity mode, 6th Conf. on high energy accelerators, Cambridge, 1967, p. 167. C. S. and Y. DUPUIS,1964, The CERN PS linac beam loading and rf studies, Linear acTAYLOR, celerator Conf. 1961, MURA Rept. 714, p. 239. VOEKLER, F., 1964a, Multiple excitation of rf cavities with an rf manifold, Proton linear accelerator Conf. 1963, Yale Univ., p. 321. F., 1964b, Thoughts on mode distribution of rf manifold, Linear accelerator Conf. 1964, VOEKLER, MURA Rept. 714, p. 497. J. N., T. I. SMITH and P. B. WILSON,1967, Accelerating structures for superconducting elecWEAVER, tron linacs, IEEE Trans. Nucl. Sci. N.S. 14, no. 3, p. 345. 1965, Rf structures for linear accelerators, IEEE Trans. Nucl. WHEELER, G. W. and S. GIORDANO, Sci. N.S. 12, no. 3, p. 110.

PLA THEORY

C.1


1.3~

I

Space Charge Effects R. L. GLUCKSTERN University of Massachusetts, U.S.A.

I

Introduction

In previous sections the stability of a charged particle about its synchronous equilibrium position has been treated. In particular longitudinal stability has been achieved by having the particles travel through the accelerating region as the rf field increases. The resulting transverse defocussing* is usually overcomet by providing magnetic quadrupole fields which to lowest order provide transverse stability without affecting longitudinal stability. The presence of space charge forces alters these considerations. It is true that the bunches as a whole will move subject only to the external focussing fields (except for effects of imaging in the conducting walls of the linac). The motion of the individual particles, however, now takes place in the combination of the external focussing fields and internal defocussing fields due to space charge repulsion. The sum of the "spring constants" of the longitudinal and two transverse motions is decreased by a quantity depending only on the charge density (via the Poisson equation) in the bunch. To overcome this defocussing in all three directions one must increase the strzngth of the magnetic quadrupoles. Some of the added transverse focussing is then transferred to longitudinal focussing by operating the linac at larger synchronous phases. The above simple considerations are primarily qualitative. We shall attempt to make them somewhat quantitative in the sections that follow. It might, however, be well to point out several factors which make all of these simple analyses suspect. These include:

* A result of Earnshaw's theorem which applies to first order in the fields, in the absence of both acceleration and space charge. t Transverse focussing obtained with grids or solenoidal magnetic fields is also used. Ccr ~Vorth-HollunilPubl. Co., Amsterdam

Rrferentrs p. 836

1969

827

828

R.L. GLUCKSTERN


a. b. c. d. e.

The asymmetry of the longitudinal focussing force; The non-uniform longitudinal charge distribution; The non-uniform transverse charge distribution; The effects of acceleration in changing the focussing and defocussing forces; The non-circular transverse beam cross-section in an alternating gradient focussing system; f. The rapidly changing image forces in drift tubes, gaps, bunchers, beam pipes. As a result of these and similar considerations the only reliable method of investigating space charge effects is a truly particle by particle numerical computation in the presence of both the external focussing fields and the repulsion between each pair of particles. Present efforts are clearly heading in this direction as the capacity and speed of digital computers increase.

2 Longitudinal space charge efect The most attractive model for the space charge forces is that which assumes the beam bunch to be a three-dimensional ellipsoid of uniform density (Lapostolle [1965]), centered at the synchronous position. The solution of the appropriate Poisson equation then corresponds to a quadratic potential function within the ellipsoid and to continued separation of the equations of motion in the usual three directions. The equation for the longitudinal motion can then be written, in the non-relativistic region (Gluckstern [I9661; Nishikawa [1967]), as

where k, is the wave number of the longitudinal oscillation in the absence of space charge, X =$I - 4, is the phase deviation of the particle from its synchronous value d,, and p, is a longitudinal space charge parameter which can be written in terms of the average beam current, I, as

Here

-

where b, a, a' are the longitudinal and two transverse semi-axes of the ellipsoid. The expression (3) is an approximate one (Lapostolle [1965]), valid for a and a' not greatly different from one another. Tf one uses the approximate form f (5) (35)- ', valid in

C.1.3~

SPACE CHARGE EFFECTS

829

the range 0.8 < 5 < 5, one can write

It is clear that eq. (1) can be obtained from the corresponding equation in the absence of space charge [eq. (1) with p,=O] by the replacements

This implies a reduction of the phase stable region from -14,1 1 MV In the case where the total energy of certain future accelerating tubes (Huguenin et al. 119671; Vienet [1967]) will be above 1 MV, they will probably have to be mounted inside pressurized tanks. Since cables are not available for voltages above 1 MV, the ht supply should be mounted in the same vessel and, if the tube gradient lies much above the generator gradient, should probably be arranged at right angles to the tube or in line with it. This solution gives greater independence between the design of the tube and the generator and allows a damping resistor to be connected between them. Voltage hold-off problems will concern the accelerating gradient, the gradient on the insulating wall in the vacuum and in the gas. Electron loading will have to be solved (4 4.3.9). To maintain high reliability of the system it might be necessary to have a spare pre-injector available for immediate operation, a powerful gas treatment plant and a suitable mechanical layout of the pressure tank for quick access to the source.

Note added in proof Since May 1968 several developments have taken place. 1) The AGS high gradient tube in Brookhaven has been operational in early 1969. There was clearly seen the importance of vacuum cleanliness. A new version of the tube has been made, inspired from CERN technique (with an indium joint between cement and vacuum). 2 ) Amalthe'e 11 tube has been installed on Saturne in August 1969 (Faure [1968]). 3) The NAL tube in Batavia, inspired from the ZGS high gradient tube (fig. 14) has been running since late spring 1969. 4) Information exists on the ITEP tube in Moscow (Batalin et al. [1967]). The Serpukhov tube is similar to it.

PRE-INJECTORS

C.2.1

873

Bibliography

KEMKE, D., 1956, Elektronen und Ionenquellen. In: Handbuch der Physik. Ed. S. Fliigge (Springer Verlag, Berlin) pp. 36-122. VON ARDENNE, M., 1962, Tabellen zur angewandten Physik, vol. 1 (V.E.B. Deutscher Verlag der Wissenschaften, Berlin), pp. 640-670. KERETNIKOV, D . V., SLIVKOV, I. N., TEPLYAKOV, V. A,, FLDOTOV, A. P., SHEMBEL, B. K., 1962, Linear Ion Accelerators, U.S. Atomic Energy Commission, Div. of Technical Information, pp. 110-138. GABOVIE, M . D., 1964, Plasma Ion Sources (Naukova Dumka, Kiev).

2.2

Ionization

NEWHALL, H. F., 1942, Phys. Rev. 62, 11. BOHM,D., BURHOP,E. H . S., and MASSEY, H . S. W., 1949, The use of probes for plasma exploration in strong magnetic fields. In: The characteristics of electrical discharges in magnetic fields. Ed. Guthrie, A,, and Wakerling, R . K . (McGraw-Hill, New York) p. 14.

2.3 PIC sources PINNING, F. M., 1937, Physica 4, 71. K. K., 1965, Fifth International Conf. IAPITSKY,Yu. YA., KHOROSHKOV, V. S. and ONOSOVSKY, High Energy Accelerators, Frascati, p. 92. B F N N ~G T ., W. and TURNER, C. M., 1967, IEEE Trans. Nucl. Sci. NS-14, No. 3, p. 65. C. M., 1953, Rev. Sci. Instr. 24, 807. ROGERS,E. J . and TURNER, v. STEENBERGEN, A,, 1965, IEEE Trans. Nucl. Sci. NS-12, No. 3, p. 746. Gow, J. and FOSTER, R., 1953, Rev. Sci. Instr. 24, 606, and UCRL 1698.

E. R., 1955, AERE Report No. GP/R 1190. THONEMANN, P. C., and HARRISON, L. E., GOBBET, R . H., STROUD, P. T., 1965, lEEE Trans. Nucl. Sci. NS-12, No. 3, p. 247. COLLINS, L. C. W.,HARRISON, E. R. and WHITBY, H . C., 1960, J. Nucl. Energy, Part C, Plasma Phys. HOBBIS, 1, 130. TALLGREN, U., 1962, Recent experiments and measurements on the CPS-RF ion source, CERN, Geneva, Internal Report: MPS/Int. LIN 62-3, 2.8.1962. TALLGREN, U., 1966, Progress report on the CPS-RF ion source work, CERN, Geneva, Internal Report: MPS/lnt. LLN 66-8, 23.12.1966.

FROHLICH, H., 1959, Nukleonik 1, 183. U. V., GUTKIN, T. I., International Conference on DEMIRKHANOV, R. A,, FROHLICH, H., KURSANOV, high energy accelerators, Brookhaven 1961 (Papers from USSR) BNL 767 (C-36), p. 224. M . A., GERASIMOV, V. P., ZHELEZNIKOV, F. G., ZABLOTSKAYA, G . R., IVANOV, N . F., ABKOYAN, IVLEV, A. V., KOMAROV, V. L., KUZNETSOV, V. S., LATMANIZOVA, G . M., ROYFE,I. M., and SOLNYSHKOV, A. I . , 1964, Proc. Intern. Conf. High Energy Acc., Dubna (USSR) 1963 (Atoniizdat, Moscow), p. 507. WROE,H., 1967, Nucl. Instr. Methods 52, 67. R., BERNARD, P., and FAURE, J., 1966, Centre d'Etudes Nucleaires de Saclay (CEN) Rept. VIENET, CEA-R-3068.

J. HUGUENIN, B. VOSICKI

874

PROTON LINACS: TECHNOLOGY

VOSICKI,B., BUZIC,M. and CHERETAKIS, A., 1966, The duoplasmatron ion source for the CERN PS linac. Linear Acc. Conference, Los Alamos, Rept. LA 3609, p. 344. J. A., CURTIS,C. D., and LEE,G. M., 1966, Proc. Linear Acc. Conference, Los Alamos, FASOLO, p. 371. ALLISON, Jr., R. W., CORK,B., RICHTER, R. M., SMITH,J. F., WHITE,G . E. and ZAJEC,E., 1961, Rev. Sci. Instr. 32, 1331. EMIGH,C. R., 1967, Los Alamos Scientific Laboratory Rept. LA-3772-MS, p. 35. GAUTHERIN, G., 1967, Doctoral thesis, Faculte des Sciences d'Orsay, France. 3 Optics GALEJS, A. and ROSE,P. H., 1967, Optics of electrostatic accelerator tubes. In: SEPTIER, A,, Focusing of charged particles, vol. 2 (Academic Press, New York) pp. 297-326. ELKIND, M. M., 1953, Rev. Sci. Instr. 24, 129. PIERCE, J. R., 1940, J. Appl. Phys. 11, 548. FAURE,J., 1964, CEN Saclay, Internal Rept. SEFS-TD 64-2. SLUYTERS, TH. J. M., 1966, Proc. Linear Acc. Conference, Los Alamos, p. 383. CURTIS,C. D., LEE,G . M., FASOLO, J. A., 1966, Proc. Linear Acc. Conference, Los Alamos, p. 365. EMIGH,C. R., 1966, Proc. Linear Acc. Conference, Los Alamos, p. 398. SEPTIER, A., PRANGERE, F., ISMAIL, H., and GAUTHERIN, G., 1965, Nucl. Instr. Methods 38, 41. FAURE, J., 1966, Centre d'Etudes NuclCaires de Saclay (CEN), Rept. CEA-R 3002, p. 46. HUGUENIN, J., and DUBOIS, R., 1965, Measurements on a high gradient acceleration tube model Investigation of the properties of titanium electrodes. Rept. CERN 65-23, 10.6.1965, European Organisation for Nuclear Research, Geneva. PIERCE, J. R., 1954, Theory and design of electron beams, 2nd ed. (Van Nostrand, Toronto). SEPTIER, A., 1967, Focusing of charged particles (Academic Press, New York) Chapters 3 and 5. KIRSTEIN, P. T., and HORNSBY, J. S., 1964, IEEE Trans. Electron Devices ED-11, No. 5, p. 196. HORNSBY, J. S., 1963, A computer program for the solution of elliptic partial differential equations (Potential problems), Rept. CERN 63-7, Geneva 1963. 4.2 The Alvarez-linac preinjector Name

Paper

Year

C. M. TURNER A. J. HUDGINS F. L. FILLMORE M. R . JEPPSON J. D. Gow

The Berkeley Electrostatic Generator, UCRL-352

1949

The Berkeley Proton Linear Accelerator, UCRL-236

4.3 Second generation. Low gradient accelerating rubes for voltages 4.3.1

Inside electrodes

I

CEA no. 189

5 Gain of 516> 10). Linac Conf. 1966, p. 183. NIRL/R/44.

OSC.

Bevatron 11

3W 10,000

ZGS

RCA 7835

Itep

GI-27 A

Serpukhov

GI-27 A

--

-

Triode OSC. Triode amp. Triode amp. Triode amp.

-

Note. Other relevant information will be found in table 1 chapter C.2 2a.

BNL Conf. 1962, p. 1. Dubna Conf. 1963, p. 499. Prib. i Tekh. Eksp. No. 5, 1967, p. 34 or Instr. Exp. Techn. 1967-2, p. 1004. Dubna Conf. 1963, p. 462.

*

9 "

3

C] ---

--

-

3

9

C.2.3

RADIOFREQUENCY PROBLEMS

927

For triodes, the anode high tension supply, typically of the order of 40 kV, may consist of pulses from a delay line modulator and pulse transformer with or without a hard-tube regulator, or from a capacitor bank and hard-tube modulator, or the capacitor bank may supply long anode pulses through a hard-tube modulator with the amplifier biassed beyond cut-off. The output pulse is then formed by a shorter grid-drive pulse (as at Saclay). The delay line modulator is the simplest of these solutions and contains the least stored energy, but is not in itself capable of fast, closed-loop regulation of the amplifier power output. Fast regulation has been applied successfully to the AGS and ZGS injectors and to the PLA (Gallagher-Daggitt [I 9651).

4 The coupled system

-

ampliJier and accelerating cavity

We shall now examine the interactions between the power source and the accelerating cavity. The complete system is shown schematically in fig. 14 where the triode amplifier is represented by a voltage generator in series with an internal impedance. In view of the non-linearity of the tube characteristics for the wide voltage swings encountered in class B or class C operation, the internal impedance has to be given the meaning of an average value r, over the rf cycle. Similarly the voltage generator is described by -Ce,, where C is averaged over the cycle, being indeterminate for the period when no plate current is flowing. In general both r, and -Ge, will vary with the amplitude of the voltage swing across the load. The tube load in this case is everything to the right of AB, transferred to this plane. For simplicity we consider first the steady state at the end of build-up. When the system is on-tune and perfectly matched, the loop of the accelerating cavity transforms the shunt resistance R, to an input impedance Z , , which is made equal to the Z , of the feeder line. This line presents to the amplifier terminals an impedance Z,, which is transferred by the amplifier coupling circuit to a value equal to the tube load impedance ZLrequired for agiven power output. 2, is the normal tube load impedance.

For maximum power transfer from the tube to the load, Z , will be equal to the average internal impedance r,. In this steady-state condition power will flow from the

Fig. 14. Equivalent circuit of the complete system.

928

C.S.TAYLOR


amplifier to the accelerating cavity where it will be completely absorbed by cavity losses in setting up the required accelerating field. If now the steady-state condition is interrupted by switching off the anode supply and the drive signal, the stored energy in the cavity will start to be dissipated in the cavity itself and in the external circuit, i.e. in the feeder and amplifier, and will decay at a rate determined by the loaded Q, which will depend on the value of the real impedance presented to the cavity terminals by everything to the left of the terminals E, F. This will include the diode action of the amplifier tube, which will absorb some power from the cavity and contribute a real component at the terminals E, F, depending upon the electrical length L/1 (For the CERN Tank I, 2Q,/oO=40 ps, whereas the measured decay time constant is z 95 ps). The behaviour of the systemduring build-up is more complicated. At some moment during build-up when the cavity input impedance is very low, the amplifier will be presented with a complex impedance which will vary with L / i as shown in the complex impedance plot of fig. 15, being a low real impedance for L/A=+n and a high real

Fig. 15. Impedance plot in the plane of the amplifier terminals CD.

impedance for L/A=f(2n - 1). Intermediate values of L will produce complex impedances as shown. Variations in the real component of this impedance will change the load seen by the tube and will therefore affect the voltage swing applied to the cavity, while changes in the reactive component will in principle modify the anode cavity tuning. As a secondary effect the internal impedance 7, and generator voltage z.7-e, may change with L as a result of changes in power output, and this will affect the loaded Q and the rate of rise. Ignoring this latter effect, one expects from the above arguments a maximum voltage at L/1= f (2n - 1 ) and a minimum at L/A =)n at any given moment during build-up, the differences diminishing as the matched condition is approached. This effects are shown diagrammatically in fig. 16a and b as a function of time and of the electrical length L/1. We now ask how the behaviour of this tuned and matched system will change if it is de-tuned or mis-matched at some point. In a qualitative way, one can understand the effect of de-tuning the amplifier output circuit by considering the admittance chart in the plane of the amplifier terminals C , D. De-tuning the output circuit will add a susceptance to the load seen by the tube, and so the final steady-state admittance

C.2.3


Fig. 16. (a) Cavity voltage rise as a function of time for L/A--+n and i(2tz-1). Vtl as a function of L1.l.

929

(b) Cavity voltage

load will be moved from the matched condition at the center to a point A as in fig. 17, at which point the amplifier will deliver less than its full power. We can now choose a line length L l i such that the load admittance will move along the dashed trajectory (1) during build-up, passing through the center and the full-power condition before the end of build-up. This will encourage a faster rate of rise than is achieved in the tuned state. Conversely, we could choose a line length shorter or longer than this by /1/1/4so that the tube load admittance would move along trajectory (2). In this case the amplifier would never develop full power and the rate of rise would be decreased. For these line lengths then, the effect of detuning the amplifier is to modify the rate of rise given by fig. 16 for the tuned output circuit. Intermediate values of L / i will have less efrect. By a similar argument, onecan show the effect on the rate of rise of overcoupling or undercoupling the cavity loop. In the case of an amplifier which is not well matched to the load (which is often the case in practice) feeding a cavity which is "long" by our previous definition, so that the trajectory during build-up is not a straight line (Hagerman [1967]), this argument stands in principle and one can still affect the rate of build-up by the choice of L, although the problem is made more complicated. These build-up phenomena can be treated quantitatively by transferring the equivalent circuit of the generator and feeder into the secondary circuit of the accelerating cavity and solving the second-order differential equations for the voltage rise across the cavity as a function of time. In the simple case of fig. 18, corresponding to a feeder length L/A=+n, the solution is given by the following expression.

Fig. 17. Admittance in the plane of the amplifier terminals CI) looking towards the cavity.

Fig. 18. Simplified equivalent circuit of complete system for LIE. pn.

-

930

C.S.TAYLOR


where w is the drive angular frequency, wo is the resonant angular frequency of the cavity, y = w/wo; furthermore

q2 = tan-'

1

-y2

( 4 ~ : - 1)'-

For large values of QL and for y close to 1, eq. (21) simplifies to

v = Vy (RIR,) [sin wt - exp (- wot/2QL)sin mot].

(22)

From eq. (22) one sees that detuning the cavity will produce a damped modulation of the exponential rise. Batchelor et a/. [I9651 have analysed the amplitude and phase changes in the accelerating cavity in this fashion, introducing additional susceptance at the generator as a perturbing factor, and have made a comparison with phase measurements on the PLA. The results suggest that the assumption of constant generator characteristics in the analysis is valid for the latter 300 ys of the 400 ,us rf pulse. The criteria adopted for the choice of L/A in the PLA is that the phase change produced by generator susceptance should be zero during the latter part of build-up in order to minimize the beam energy spread, and that the rate of rise should be a maximum in order to permit the longest possible beam pulse. It has been possible to satisfy these criteria simultaneously on the PLA Tank I by adjustment of L. When intense proton beam currents are accelerated through the cavity and the additional power is supplied by the same amplifier which excites the cavity, the amplifier is presented with a change of load during the passage of the beam. It has been suggested (Murin [1963]) that where the amplifier has power in reserve it should be possible to choose L so that the amplifier automatically delivers more power when the beam load appears. For example, one can see that if L/A= $, the passage of the beam will produce a higher resistance load at the amplifier, and this could be arranged to move the power output from somewhere below the maximum during build-up to the maximum during the beam pulse. The interaction between the beam and the amplifier has been further analyzed in subsequent papers (Murin and Kvasha [1965, Murin et al. [1967], Lee [1968]). In the course of this chapter, the significance of the electrical length LII has been referred to in connection with several different phenomena, viz. build-up time, multipactoring, amplifier breakdown, and phase and amplitude changes during the

C.2.3


931

beam pulse, with and without beam-loading. There is also the practical matter of convenience of physical layout to be considered. It is apparent that it would not be easy to formulate a general rule for the optimum length L/A, but it should be clear what factors need to be taken into account in a complete study of this problem for a particular case. We conclude this section with two topics of a more general nature. The first concerns the generally assumed principle of one or more amplifiers feeding one tank to form a generator-load unit ~ h i c hmay be varied in phase and amplitude independently of the rest of the system. At Berkeley an alternative approach has been proposed of feeding all the power amplifiers into a common manifold, from which power can be tapped off t o individual tanks (Voelker [1966]). This scheme requires the distance from the rf manifold to each tank to be $1. in order that the tank levels and phases be locked together. An attractive aspect of this scheme is the availability of the total installed rf power for driving only one or two tanks through multipactoring during running-in. The second concerns the uses to which reserve power may be put in the conventional generator-load unit. In the case of heavy beam-loading, it has been suggested that the maximum available rf power be used a t the beginning of the pulse in order to obtain a fast build-up to accelerating level, at which point the power is cut back and then re-applied as needed t o keep the field constant during the passage of the proton beam (Carne and Batchelor 119641). A method similar in principle has been used on the Itep machine in order to obtain a fast rise. There, rf negative feedback from the tank was employed to make the full power available at the beginning of the pulse (Bobylev et al. [1967]). The use of positive feedback in combating multipactoring has been described by Polyakov et al. [1961], Bobylev et al. [1967], Mints et al. [1967].

5

Controls and instrumentation

A complete schematic diagram for the rf system of a multi-tank linac looks very complicated at first glance. In operation however, complications arise not so much from the proliferation of amplifiers, modulators, phase shifters, servo loops and rf lines as from inter-action between elements which are not supposed to inter-act. A good example of this was seen in the CERN-PS linac when pollution from some 50 MeV experimental apparatus was allowed to communicate with the debuncher vacuum. The rf window of the debuncher became contaminated and then started to break down, which produced standing waves on its feeder and eventually caused breakdown of a cable connector supplying drive power to the second tank of the linac through a power divider. Unexpectedly, the servo loop of the third tank then started drifting and the tank went off tune, due to a surge from the rf break-down which had blown a diode in a low level circuit. The probability of these minor disasters occurring can be considerably reduced by interposing isolators at suitable points, and adequate

932

C.S. TAYLOR


isolation between sections of the rf system can also bring one a step nearer having controls which change only one variable at a time, i.e. a line lengthener which changes only the phase of one tank, and not its level nor the level of other tanks as well.

Acknowledgements A number of people have contributed to this chapter through helpful discussions. In particular the author wishes to acknowledge the contributions of R. P. Featherstone of the Minnesota 68 MeV machine.

References AITKEN, D. K., 1968, Long Transit-time Multipactoring at Ultra-high Frequencies and the Effect of Surface Emitting Layers. Proc. lEEE International Convention on Microwave Valves. Paper 2584 R. May 1958. ALVAREZ, L. W., et a/., 1955, Berkeley Proton Linear Accelerator. Rev. Sci. Instr. 26 111. K. and G. E. GALLAGHER-DAGGITT, 1965, Accurate Field Level and Phase Control and BATCHELOR, Monitoring for a Proton Linear Accelerator. IEEE Trans. Nucl. Sci. NS-12 No. 3 (Washington Conference) p. 195. BOBYLEV, V. I., et al., 1967, High-frequency Supply System for the I-2-Linear Proton Accelerator. Prib. i Tekh. Eksp. No. 5, p. 34-39; Instr. Exp. Techn. 1967-2, p. 1004-1008. BOYD,T. J. and R. JAMESON, 1967, Optimum Generator Characteristics of R.F. Amplifier for Heavily Beam-loaded Accelerators. IEEE Trans. Nucl. Sci. NS-14 No. 3 (Washington Conference) p. 213-216. 1964, Design of a High Current 200 MeV Proton Linear Accelerator. CARNE, A. and K. BATCHELOR, Rutherford High Energy Laboratory, P.L.A. Accelerator Physics Group NIRL/R/55. DAVIES, A. J., 1969, Some thoughts on Radio Frequency Spark Breakdown in Linear Accelerators. CERN Internal Report MPS/LIN 69-13. DAVIES,D E., 1959, Influence of the Cathode Work Function on the Sparking Potential in Hydrogen. British J. Appl. Phys. 10 502. DUPUIS,Y., 1963, Etude de la charge introduite par le faisceau dans les cavitCs du Linac du C.P.S. CERN Internal Report MPS/Int. LIN 63-8. DWORK,L. E., 1950, Maximum Tank Voltage in Class C Amplifiers. Proc. I.R.E. 38 No. 6 p. 637-644. GALLAGHER-DAGGITT, G. E., 1965, Automatic Control of the Electromagnetic Field in Linac Resonant Cavities. Accelerator Physics Group P.L.A. Rutherford High Energy Laboratory NIRL/R/76. GINZTON,E. L., 1957, Microwave Measurements (McGraw-Hill) p. 435 et seq. HAGERMAN, D. C., 1967, High Duty Factor R.F. Sources at 800 MHz. IEEE Trans. Nucl. Sci. NS-14 No. 3 (Washington Conference) p. 197. HATCH,A. J. and H. B. WILLIAMS, 1954, The Secondary Electron Resonance Mechanism of Low Pressure High-Frequency Gas Breakdown. J. Appl. Phys. 25 No. 4 p. 417-423. 1958, Multipactoring Modes at High Frequency Gaseous BreakHATCH,A. J. and H. B. WILLIAMS, down. Phys. Rev. 112 681. JAMESON, R. A,, 1965, Analysis of a Proton Linear Accelerator R.F. System and Application to R.F. Phase Control. Los Alamos Scientific Laboratory, Report LA-3372. Section V-C-3. KEANE, J., 1964, Experiments on R.F. Field pattern in the BNL Linac. Linear Accelerator Conference, Mura D. 398-406. KILPATRICK, W. D., 1953, A Criterion for Vacuum Sparking Designed to Include both R.F. and D.C., UCRL-2321. LASL, 1967, Quarterly Status Report o n the Medium-Energy Physics Program for the Period ending July 31, 1967. LA-3772-MS. p. 21.

C.2.3

RADIOFREQUENCY P R O B L E M S

933

LIBACQZ,J. V., 1965, High Power Klystrons. IEEE Trans. Nucl. Sci. NS-12 No. 3 (Washington Conference) p. 86-95. L ~ EM. , S., 1968, Steady State Beam Loading of the Linac R F System. BNL Int. Report A G S C D 28, April 1968. LIVINGSTON, M. S. and J. P. BLEWETT, 1962, Particle Accelerators (McGraw-Hill) p. 345-348. MINTS,A. L., ct 01.. 1967, Injector for the Serpukhov PS: Start-up Works. Vlth Intern. Conf. o n High Energy Accelerators, Cambridge, CEAL 2000, p. 255-259. M O R G AG ~ ,R E YC., 1965, Fundamentals of Electric Discharges in Gases. Handbook of Vacuum Physics 2, Physical Electronics Part I. (Pergamon Press). MORRIS,A. J . and C. A. MARTIN-VEGUE, 1965, A Comparison of Triode and Klystrons for Particle Accelerator Applications. IEEE Trans. Nucl. Sci. NS-12 No. 3 (Washington Ccnference) p. 96109. MULLETT, L. M., R. E. CLAY,and R. J . B. HADDEN,1953, Multipactor Effect in Linear Accelerators and other evacuated R.F. Systems, and a New Cold Cathode Valve. AERE G P / R . 1076. MURA,1964, Informal Discussion of Sparking Phenomena. Linear Accelerator Conference. Mura, p. 606-61 5. MURIN,B. P., 1963, (Radio Technical Institute) Moscow. Prelinlinary Notes o n the Organization of the R.F. Field in Linear Accelerators of 50-100 mA Pulsed Proton Beams. C E R N Internal Report MPS/lnt. LIN 63-7. M U R I N B. , P. and A. I. KVASKA,1965, Accelerating Field Stabilization in Beamloaded Proton Linac by Choosing R F Supply Parameters and Operating Conditions. Vth lntcrnational Conference o n High Energy Accelerators, Frascati. MURIN,B. P. et ul., 1967, K F Field Stabilization in Strong Current Linacs. Vlth International Conference o n High Energy Accelerators, Cambridge, CEAL 2000, p. A54-55. N A G L ~D., E., E. A . K K A P Pand B. C. KNAPP,1967, Coupled Resonator Modcl for Standing Wave Accelerator Tanks. Rev. Sci. Instr. 38 No. 1 1 p. 1583-1587. NISHIKAWA, T., 1966, Beam Loading Effects in Standing Wave Linacs. Proceedings of the 1966 Linear Accelerator Conference, Los Alamos, p. 294. OWEN,C. W., C . A. RADNEKand D. E. YOUNG,1966, High Field Measurements at 200 M H z in Conventional Proton Linear Accelerator Geometries at 5, 50 and 130 MeV. Proceedings of the 1966 Linear Accelerator Conference. Los Alamos, p. 176-1 82. V. I., 1963, Some Problems in the Theory of Low-voltage Electron Discharges (MultiPLYRUNIN, patting). Electric Instrumentation in High-Energy Physics, Moscow. Collection of papers No. I. Edited E. G . Komar, translation published for A.E.G. and N.S.F. P ~ L Y A K OB. V ,I., F. BLOCK and U. KRACHT,1961, About a Method of Eliminating Multipactor Instability in the Radio-frequency Impulse of the CEKN Proton Synchrotron Linac. C E R N Internal Report MPS/LIN 61-5. PRCIST,D. H., and A. C. TALCOTT,1961, O n the Heating of Output W i n d o w of Microwave Tubes by Electron Bombardment. IRE Trans. Electron Devices ED-8 No. 4, p. 243-251. SCHMELZER, CH., 0 1 a/., 1966, Sparking Experiments at 60 Mc/s. Part I. Unilac Project. University of Heidel berg. J. C., 1950, Microwave Electronics (Van Nostrand) p. 95-101. SLATER, TALCOTT, A. C., 1962, The Effects of Titanium Films o n Secondary Electron Emission Phenomena in Resonant Cavities and at Dielectric Surfaces. IRE Trans. Electron Devices ED-9 No. 5 p. 405410. TAYLOR C. A. and Y . DUPULS,1964, The Cern-PS Linac-Beam Loading and R.F. Studies. Linear Accelerator Conference, Mura, p. 239. VOELKER, F., 1966, Measurements on a Model R.F. Manifold System. Proceedings of the 1966 Linear Accelerator Conference, Los Alamos, LA-3609, p. 146-1 52. YOUNG,D . E., 1963, Drift-tube Structures. Yale Conference o n Proton Linear Accelerators, p. 204. ZACCHERONI, E., 1958, The 2.5 M W H.F. Amplifier of the C E R N Linear Accelerator. C E R N Report 58-22. ZAGER,B. A. and V. G . TISHIN, 1962, High Frequency Resonant Discharges and the Possibility of their Elimination. Dubna Int. Report 1058.

PLA

TECHNOLOGY

C.2

Beam Monitoring, Accelerator Shielding and Services A. VAN STEENBERGEN Brookhaven National Laboratory Upton, IY. Y., U.S.A.

1

Beam monitoring

1.1 Introduction Related to the desired performance of the linear accelerator it is essential that certain beam characteristics such as beam momentum, momentum spread, transverse phase space characteristics and beam intensities are known at various locations along the length of the accelerator. Also, essential linac parameters such as the average accelerating field in each cavity, the synchronous phase angle and the relative phase between the various cavities of a multi-cavity linear accelerator, need to be known for proper performance. Especially during the commissioning period, accurate measurements of some of the mentioned parameters are essential in order to "converge" to an optimum tuned accelerator. Typically, transverse phase space matching of the preinjector beam to linear accelerator acceptance depends on a detailed knowledge, not only of the pertinent phase space area but also of its density distribution. Further, if the linear accelerator is to serve as a medium energy physics facility or an injector into an adjunct synchrotron ring, the beam momentum and momentum spread from the accelerator need to be known precisely. Knowledge of these parameters, especially of momentum distribution, is essential to tuning of the linac cavities. In the following, some of the approaches used in the measurements of the beam 0 North-Holland Publ. Co., Amstrrdam

1969

References p . 969

936

A. VAN STEENBERGEN


dynamical parameters are further indicated. Most of the techniques, as described here, have been extensively developed at BNL, however, the description also includes novel techniques used for example at CERN, LRL, etc. No pretense is made that this would form a complete summary. Indeed, further development of beam monitoring techniques is a continuous process, and is seemingly only limited by the inventiveness of the accelerator builder.

1.2 Beam intensity monitoring Arbitrarily, the beam intensity monitors to be discussed here are divided in "nondestructive" monitors and beam "destructive" monitors. Examples of the first type are the current transformer and electrostatic pick-up electrode. Where beam momentum or emittance analysis is being utilized, in which case the beam would be lost for physics utilization or injection into an adjunct synchrotron, the "destructive" technique is acceptable. Examples are a total beam intercepting target or a Faraday cup. Further included in the latter category are also the secondary emission chamber and air ionization chamber. The current transformer is essentially a single turn primary multiturn secondary transformer, whereby the particle beam forms the primary loop. Related to the response requirements of the current monitoring system, such as pulse rise time and acceptable pulse "droop", the following expressions, valid for a somewhat oversimplified system of a beam current transformer coil, shunted by a resistor R,,,serve to define the relationship between the various parameters:

where n equals the number of turns. With z = LIR and T=pulse length, it follows for z $ T that e ( t ) = -(RSh/n)lheam( t ) where e(t) is measured across R,,. Similarly, for z~ T

It is obviously desirable to make the ratio e(t)/I(t) optimum, in order to achieve maximum current transformer sensitivity. With more sophisticated arrangements, use is made of feedback amplifiers (Sharp [1962]), whereby, related to the response of the overall system, the amplifier characteristics are specifically matched to the parameters of the pulse transformer. Typically, by sacrificing band width and rise time a higher loop gain can be obtained, resulting in a greater current transformer sensitivity, as defined above. Since it would be difficult to measure individual linac beam bunches, normally a current transformerpreamplifier combination with a relatively long time constant, z, is used. The second non-destructive beam intensity monitor to be mentioned here is the

BEAM MONITORING.

C.2.4

SHIELDING AND SERVICES

937

electrostatic pick-up electrode. In its simplest form, this would be a metallic cylinder of length L, with capacitance versus the surrounding of C, which would be traversed by the particle beam, with average intensity, I,,,,. In this case, the following simple relatianships are valid: L

L

Here it may be desirable to resonate the electrode-preamplifier combination with the beam bunch traversal rate, as determined by the linac cavity frequency, in order to obtain greater sensitivity. With pick-up electrodes it is essential that no spurious charges reach the electrodes. This may be achieved partly by preceding the electrodes with a defining diaphram, which is slightly smaller in diameter than the clearance aperture of the pick-up electrode proper. Further, rest gas ionization within the pick-up electrode cylinder could result in a sufficient large charge collection to make the signal to noise ratio unacceptable. Consequently, a good vacuum pressure in the region of interest is important. The intensity monitors as described here, normally serve as "secondary" beam intensity standards and may have been calibrated initially against other beam measurement techniques, some of which will be mentioned in the following. Operationally, the "destructive" linac beam intensity monitor is normally used in colijunction with beam emittance analysis or beam momentum determination. A typical example is the Farads)) cup. The execution of this device for measurement of beam intensities at low beam energies is well known. At higher beam energies, typically for proton beams of 50 MeV and higher, the design of the Faraday cup is sufficiently more complex so that its use at these energies has been more limited. At these energies the range value of the protons is such that a thick absorber is required to stop and absorb the protons completely. Also, secondaries produced are more difficult to contain and espxially the back scattered electrons may have sufficient energy that trapping by electrostatic fields become some-vhat difficult. It should also be kept in mind, that energetic secondary y-rays are produced. Allowance must be made in the absorber thickness for this, in order to minimize ejection of electrons from the back of the absorber. Related to backstreaming secondary electrons, the normal approach is to reduce the magnitude of the solid angle, viecz,ed in the backward direction, by means of a defining cylinder, in addition to the use of (the classical approach) a sequence of biased beam diaphrams and a transverse magnetic field. T o illustrate a typical design approach, the Faraday cup developed by Burr (Burr [1964]) for a 6 GeV electron beam intensity measurement, is given in fig. 1. This approach and similar work is equally useful for the construction of a Faraday cup for proton beam intensity measurement. At BNL use has been made of total targets to obtain monitoring signals proportional to the beam intensity. The beam is not stopped by the target but traverses with

938

A. VAN STEENBERGEN


only a minor energy loss. The free charges which are produced as a result of interactions and ionization are integrated to serve as a measure for the beam intensity. Special precautions are taken to shield against collection of free charges from the surrounding media. This technique is used only as a relative monitor and does need calibration for absolute measurements. In addition, ionization chambers and secondary emission chambers are used for beam intensity measurements. Both devices are in essence low beam intensity devices, but by means of special construction their linear range has been extended. The basic Stainless Steel Vacuum Enclosure

11

d

To Insulator Guard Ring Bias To Electrometer

I & Fore Pump To Vacion Pump Fig. 1. Faraday cup for higher energy beam intensity measurement.

concept of the parallel plate ionization chamber is simple, but because of ionic recombination "saturation" occurs with higher beam intensities. By reducing the chamber gas density and interelectrode distance, it has been possible to obtain a linear device for up to 1015 particles/cm2.sec, and further improvements seem possible. As a result of the development work with ionization chambers the concept of the secondary electron emission chamber has evolved. Further development, to extend the useful range for higher beam intensity measurements, has been done by Agoritsas (Agoritsas [1966]). An example of this work, given in fig. 2, illustrates best the basic

C.2.4

BEAM MONITORING. SHIELDING AND SERVICES

939

principle involved in the secondary emission chamber design. Again, calibration against a "primary" standard, for example the Faraday cup, is necessary.

1.3 Beam position monitoring Following again the classification of non-destructive (non-interfering) beam position monitors and destructive monitors, special execution of the design of the electrostatic

B i a s Voltage

Power Supply

Vacuum Chamber Walls I

4

Collecting Foil! Thin Window

Thin w i n d o t

-

Beam

d

l ~ m i t t i n gF o i l

-secondary To Vacuum Pump

electron

)orbits Charge Measuring Device

Fig. 2.

Secondary emission chamber.

pick-up electrode and of the current transformer are useful as beam position monitors. An example of the first device is shown in fig. 3, which was developed for the 20 MeV linear accelerator injector for the Bevatron (Allison et al. [1966]). The electrodes are taken up in a resonant rf bridge, tuned to the linac frequency (200 MHz) o r beam bunch traversal frequency for greater sensitivity. The beam position resolution with this particular system is of the order of k0.1 mm and a linear response with beam

940

A. VAN STEENBERGEN


displacement has been obtained. The longitudinal asymmetry of these particular electrodes eliminate phase shifts due to beam transit time effects. For position determination, knowledge of the total beam intensity is necessary. This can be measured with the same position electrode system by summing the subelectrode signals. The current transformer has also been adapted to beam position measurement.

-Ceramic Cylinder

--

"Vacuum Sputteredrf copper e l e c t r o d e

Beam

200 MHz

Comparator

C

Phase Detector

Set Negative

-

Sign Sample

TotalIntensity

n

I

Normalizer

Sample & Hold

Fig. 3. Electrostatic position electrode.

This is done by arranging four coils with their axes in a plane perpendicular to the beam and separated 90" apart circumferentially. By connecting two of the coils (typically in the horizontal plane) in series opposition, the net voltage will be proportional to the beam displacement from the beam axis. The interference ("destructive") beam position and size measurement makes use of well-known techniques utilizing quartz plates, scintillation screens, etc. Also slit

C.2.4

BEAM MONITORING.


941

systems or fixed diaphrams are used to define beam size and position. Especially with a scanning slit the real space density distribution may be obtained and from this the beam centerline position may be accurately determined. Note added i r ~proof "Nondestructive" beam profile monitors of the residual gas ionization type have recently been further developed and are proving to be extremely useful diagnostic devices at the ZGS (DeLuca [1969]) and at the PS (Johnson [1969]).

1.4 Beam monitoring, transverse phase-space The concept of beam emittance and transverse phase-space has been used extensively in the frame work of accelerator technology. It has been especially useful whenever it is required t o "match" the beam from one component of a n accelerator (with known beam "emittance") to the so called "acceptance" of a sequential component of the accelerator. I n essence, the mentioned concepts refer to the description of the particle beam, whereby the state of the ensemble of beam particles is described by the canonical conjugate variables E, t , x, p,, y, p, in the six dimensional phase-space. Especially the density distribution of the phase space points is of significant interest, also related to Liouville's density conversation principle. In practice, of course, it is somewhat (!) difficult to visualize the six dimensional distribution. Disregarding here the influence of space charge coupling phenomena or classical longitudinal-transverse coupling phenomena, the longitudinal phase space projection (E, t ) , is normally treated separately from the four dimensional transverse phase space representation. Further, related to the description of the transverse beam properties, it is actually the projections of the four dimensional phase space on the two dimensional coordinate planes, which is being used normally to describe the beam at any particular location in the accelerator, i.e., the distribution of particles in the .r. p, projection is used to define the emittance of a particle beam. The simplest method of transverse phase measurement is well known. This is done by means of two slits, separated in the longitudinal direction and followed by a beam intensity monitor o r beam observation screen. By means of the first slit, a beam sample is defined at a transverse location y. The angular extent, or transverse momentum range, of this beam sample is determined by means of the second slit-beam monitor combination. Repeating this procedure for various first slit locations y will provide the two dimensional phase-space domain F ( y , y') representing the particle beam, where y ' = (dyldz) = (p,/pZ) and p,, the longitudinal momentum, is a constant of the measurement. Early measurements of beam transverse phase space did not in particular deal with the beam phase space density distribution. Moreover, the assumption was normally made that the projection in two dimensional phase space was a homo-

942

A. VAN STEENBERGEN


geneously filled ellipse. This demands that the four dimensional distribution is a homogeneous particle distribution on the surface of a hyper-ellipsoid, which, related to actual proton beams, is rather unreal. With these simplifying assumptions, however, particularly simple methods, which have proven quite useful in practice, are possible. In these cases use is made of only a few beam size determinations (Walsh [1960]) rather than a complete sequential scanning of a combination of slits. For the case of a non-elliptical phase space projection and arbitrary density distribution, it is useful to state that all particles in the beam are represented by the four dimensional density function p(x, x', y, y'). In order to obtain now a two dimensional density distribution, typically, in the (y, y') phase plane projection, a slit method may be used whereby the slits extend "infinitely" in the x transverse direction. Consequently, integration over all x, x' components is obtained implicitly and the integrated partial density function D(y, y') is obtained (Teng [1960]), given by D(Y, Y') = JJ 4% x',Y, Y') dx dx'. The procedure to determine D(y, y') is as follows: For a value y=i, the first slit location, a current distribution I,,,(y') is obtained by scanning the second slit and simultaneously determining the transmitted current with a beam intensity measuring device. Zero points, (yb), and (yb,),, of this function define a maximum angular extent of this beam sample. Zero points, (y;), and (y;,),, for a set of functions IY(y1)define a boundary in the two dimensional phase-space projection. Cutting off the IY(y1)functions at a fixed (although arbitrary) value, unequal to zero, provide a new set of values, (y,), and (y;,), and consequently a new equidensity contour in the y, y' projection. A family of contours may be obtained in this fashion, each of which would "enclose" a certain fraction of the total number of beam particles. By integration, this fractional beam content may be determined resulting in the quantitative density distribution, i.e., it is necessary to evaluate for the various equidensity contours,

whereby C, a constant, relates beam intensity to particle density (Van Steenbergen [1967]). The particular density contour, which includes 90% of the phase points or particles in the total beam defines now an area, which, when divided by 71, yields the value of the phase space "emittance" of the beam. The usage of this definition of beam emittance has grown recently. It is particularly relevant that no assumption need be made as to the boundary shape in the two dimensional phase space or its density distribution. An alternating method of transverse phase space measurement involves the use of two pairs of crossed slits. Here the beam sample is further defined at the first location by an additional stationary slit, at x=O, extended in they coordinate direction. This is done similarly at the second location. In this case a specific partial density function


Beam

943

w-

P h a s e Space Measurement Schematic

x = R cos 0 y = R sin 0 = R' cos 0 y ' = R' s i n 8

-

+

t sin 8 t cos 8

Distribution of Beam Density in Relation to Angles in Transverse Phase Space

Fig. 4. "Four dimensional" phase-space measurement.

D,,o,,.=o(y, y') may be obtained. This result is sometimes of interest, when comparing experimental observations with theoretical acceptance calculations for typically a linear accelerator, whereby in general the intercept of the four dimensional phase space with the two dimensional phase plane has been determined.

944

A.VAN STEENBERGEN


The method whereby beam samples are defined by means of pin holes (pepperpot techniques) fall in this class, i.e., in this case also specific partial density functions are determined in the two dimensional phase space. An additional method to determine the transverse phase space density distribution should be mentioned here. In this case use is made again of crossed slits for beam sampling, but the resultant angular distribution ,(xl,y') is completely determined for each set of x, y values by means of a densitometer analysis of the beam sample image on a photographic film (Collins and Stroud [1964]). This method has been analyzed in detail by Ivanov et al. [1965]. With this particular method, which is schematically illustrated in fig. 4 it is possible to determine the four dimensional phase-space "volume" and its density distribution. From this, with certain limiting assumptions related to the four dimensional boundary shape, the two dimensional phase space projection may be obtained. This method is more elaborate than those mentioned before, but provides more information describing the particle density distribution in the four dimensional phase space. A comparison of the various methods has been made by Van Steenbergen (Van Steenbergen [1967]), whereby the limiting conditions were examined for which the foregoing methods would yield similar results for the beam emittance, as defined above. Related to the importance of continuous linac beam monitoring various methods have evolved to measure the beam emittance with automated equipment. An example of this is shown (Otis et al. [1967]) in fig. 5, which illustrates the BNL 50 MeV "slow" emittance plotter, whereby a linac beam emittance diagram, averaged over approxi3 is directly plotted by means of an X,Y mately 200 linac beam pulses ( ~ min) recorder. A faster technique has been developed by Allison et al. [I9661 based on the earlier work of VoSicki at CERN. In this particular case the beam is sampled by a scanning slit and following this sampling slit a repetitively pulsed magnet deflects the beam sample across a Faraday cup slit combination for transverse momentum or angular distribution analysis. All components are electronically coupled for direct readout on an oscilloscope of the two dimensional phase space diagram. Using a similar technique, Sluyters et al. [I9671 developed a "fast" emittance analyzer for the BNL 50 MeV linac beam, whereby only a single 40 p e c beam pulse is sufficient to obtain a complete emittance plot. In this case two ferrite magnets, which are sinusoidally driven at a frequency of approximately 25 kHz, displace the beam across a sample slit. Following this, an analyzer ferrite magnet, driven at 500 kHz, scans the beam sample across a second slit followed by a Faraday cup. The schematic arrangement is shown in fig. 6. Both sampling magnets and analyzer magnet are coupled to the horizontal and vertical deflection plates, respectively, of an oscilloscope, in addition the signal from the Faraday cup is being fed via a discriminator to the cathode ray tube grid for Z-axis modulation. The technique of fast emittance analysis has already proved to be very useful in optimum tuning of the linear accelerator.

+,,

C.2.4


945

946

A.VAN STEENBERGEN


I

I

I

I

FARADAY

ANALYZER (500 Kc/s)

Fig. 6. Single pulse emittance device; schematic.

1.5 Beam monitoring, longitudinal phase-space parameters Equally important for the establishment of good performance of a linear accelerator is the measurement of the longitudinal phase space parameters of the linac output beam, i.e., beam momentum, momentum distribution and bunch width. After the preinjector, the beam in the longitudinal phase-space representation AE, A t , or AE, Arp is a horizontal band of small energy width having a uniform line density. The energy width is determined by the random variation of the CockcroftWalton final voltage, normally kept below 0.2%, and superimposed on this an organized voltage variation related to the beam loading of the C-W generator. With use of a beam buncher structure this phase space pattern is distorted into an S-shape pattern. Traversal through the linac, with the addition of certain nonlinearities of the longitudinal motion, leads to a spiral like particle distribution in the longitudinal phase space projection, of which the orientation is determined by the total number of phase oscillations in the linear accelerator. Its projection on the energy axis in the phase space diagram determines the particle distribution as a function of energy, similarly its projection on the phase coordinate axis determines the particle distribution as a function of time. Related to particle physics experiments to be performed with the linac beam or its utilization in an adjunct accelerator (see chapter C.3.1), it is normally desired to obtain a small energy spread*. Also, the absolute value of the particle energy is required, not only related to experimental usage, but also from the point of view of comparing experimental linac performance with theoretical expectation. This is particularly relevant for optimum tuning of individual cavities in a multi-cavity linac.

*

Defined here as the full width at half maximum amplitude of the energy spectrum.

C.2.4


947

The beam energy measurements may be typically done by one of three of the following methods: time of flight method, momentum determination with a momentum analyzer magnet of which the B d l along the particle orbit is precisely known as a function of magnet current, or by means of range measurements of the protons in a variety of materials for which the range energy relationships are known. A time-ofijight method has been developed by Batty (Batty and Warner [1961]) for precise determination of the energy spectrum of the proton beam from the PLA (Rutherford Laboratory, see chapter C.3.2). In this particular case the proton energy was measured in the range from 10 MeV to 50 MeV. It was shown that the precision of energy determination was of the order of +_0.16'x which compares favorably with some of the other methods mentioned below. Experimental arrangements for the time-of-flight energy determination involved two scattering chambers, containing a thin aluminum foil, which defined the flight path. With scintillation counters only the elastically scattered protons were detected. In order to "tag" now a certain bunch or group of' bunches produced by the linac a special sequence of linac output bunches was obtained by means of an rf beam deflector which was located between preinjector and linear accelerator. The normal time structure of the proton beam involved bunches of 0.5 nsec long, separated by time spacings of approximately 5 nsec. By means of the rf beam deflector, and a defining aperture in front of the linac, bursts of beam bunches could be obtained separated by time spacings of the order of 90 nsec. These "trains" of proton bunches were usable with the time-of-flight technique. In practice the flight path length was kept at a value determined by the length traversed by the protons in approximately 90 nsec, and the small difference between the flight time and bunch "train" spacing was measured. Impressive results were obtained with this method, however, careful experimental techniques are required in order t o maintain the accuracy of energy determination. Typically, measurements of time differences of the order of 0.1 nsec are involved. If a somewhat lesser accuracy of absolute energy determination would be sufficient, the use of a standard spectrometer magnet might be preferred. Moreover, related t o the energy distribution determination a resolution of the order of O.l:/, can still be obtained. It should be noted here, that, if the linac serves as a n injector for a n adjunct accelerator, especially for a previously commissioned facility, operationally lt is only the energy spread which needs to be determined with high accuracy. With the BNL 50 MeV linac this technique is used continuously during operation by deflecting one beam pulse of three via a 5" pulsed deflect~onmagnet, into a 25" spectrometer magnet, which is followed by a flight path of the order of 10 m. At the end of this drift space a 12 channel Faraday cup is located at a transverse waist location of the beam. Using a vertical slit, located immediately following the 5" pulsed bending magnet, an energy resolution of approximately 30 keV could be obtained. The histogram response from the Faraday cup is converted into a (semi) smooth curve by means of sophisticated electronic circuits. Further, the energy spread is

948

A. VAN STEENBERGEN

PROTON LINACS : TECHNOLOGY

determined automatically and provided in digital form (Otis et al. [1967]). The general layout and representative results are illustrated in fig. 7. The final method to be mentioned here for energy determination is the use of range measurements. This has also been used for energy determination of the PLA proton beam (Hannan and Hodges [1963]). Here a monitor consisting of a differential ion chamber and a variable wedge degrader was used. This is illustrated in fig. 8, which is further self-explanatory. The thickness of the thin degrader, as shown, is equivalent to two standard deviations of the (Gaussian) range distribution curve. The relative length of the ion chambers A and B is such that the larger number of protons passing through A, is compensated by the greater ionization loss per proton in B, in order to produce equal currents in A and B, when the mean range of the protons is coincident with the thin degrader. By using available range energy relationships this method may provide a simple technique for relatively accurate (approximately 0.5%) energy determination of the linac proton beam. It is obvious of course that energy spread cannot be measured by this technique and as indicated before, this is a necessary parameter for linac operational beam monitoring. Further information, related to the longitudinal phase space properties, may be obtained by the measurement of the (relative) phase distribution of the linac output

Fig. 7a. Arrangement of pulsed momentum analyzer and emittance monitoring systems.


C.2.4

949

beam. For a 200 MHz linac frequency, this involves beam bunch measurements of bunches less than 0.5 nsec "wide", obviously somewhat difficult with present techniques. Nevertheless, this has been done by Warner [1965], using specially matched beam probes. A desirable extension would be, to combine this approach with the energy spectrum measurements and to obtain correlated beam bunch distributions for a sequence of beam momentum "bites" with a momentum dispersed beam. The results would permit a more complete plotting of the particle distribution in the longitudinal phase space and assist in comparing linac experimental performance with theoretical prediction. Although somewhat out of context here, a discussion of accelerator beam monitoring would not be complete without mentioning the particle mass analysis of the 50 MeV PROTONS

12-channel Faraday cup.

- 4 Aa(=m) 120 K ~ V I C .

or

69.85 KeV1f.r.d.y

-1.2

-0.8

c u p channel

-

-0.4

-2

Y\

Cenrcr cnersy

-

a, cam

50.8 W ~ V

/

Energy Distribution Function Approximation. Dlsperaion of the Momentum Analyzer.

Fig. 7b. Faraday cup, optics and experimen~alresults of pulsed momentum analyzer.

950

A.VAN STEENBERGEN


preinjector beam. With present ion sources the preinjector beam proton content may vary typically between 50% and 90%. The obvious method, of course, is to use standard mass spectrometer techniques. Other methods whereby an approximate estimate of the major mass components of the preinjector beam can be obtained, involve the use of the linac structure as a mass analyzer. This may be done either by the time-of-flight technique (especially applicable with long linacs) or by using the sequence of magnetic focusing quadrupoles (in an Alvarez linac) as the particle mass selector. Entrance Pup i 1

Thick

Thin

DegPer

Proton Beam

\/ Ion

VariablH Wedge Degrader

Chambers

Fig. 8. Layout of beam energy monitor.

In conclusion it should be noted, that the combination of "instantaneous" measurement of beam intensity, beam transverse phase space and beam momentum and momentum distribution, as developed to a high degree for the BNL 50 MeV linac is considered essential monitoring equipment for tuning a linear accelerator to optimum performance, not only during the early commissioning period, but more specifically, during regular continuous operation.

2 Proton linac shielding 2.1 Introduction In the following shielding, activation and radiation damage will be discussed related to proton linear accelerators only. Specific requirements related to target stations for typically a meson factory will not be dealt with. An extensive treatment of this specifically has been provided by the Yale University High Intensity Accelerator Study Group (Yale Design Study Staff [1964]). The logical approach in the evaluation of the shield wall thickness is to estimate

C.2.4


951

first the anticipated proton losses, secondly to define the source of radiation and finally to evaluate the shield thickness required in order to reduce environment radiation levels to acceptable values. This treatment is based in part on the shielding studies (Wheeler and Moore [1966]) done for the Brookhaven National Laboratory, 200 MeV linear accelerator design for the A G S Conversion program.

2.2 Beam loss in the linear accelerator The anticipated proton losses in the linac are distinguishable as continuous proton losses, occurring even in a perfectly tuned machine and proton "fault" losses related specifically to excessive linac parameter variations and definite performance faults. Under ideal conditions then, the continuous loss is related t o the optimum longitudinal trapping efficiency and transverse phase space beam matching. For a convenlional Alvarez linac a longitudinal trapping efficiency of approximately 25% may be obtained without a beam bunching system, with a simple single gap buncher, the trapping efficiency might be as high as 70%. Even higher values may be obtained with special "harmonic" bunchers or possibly a special linac section for "adiabatic" capture. Nevertheless, not all of the injected protons will be trapped in the rf "bucket". These protons are essentially all lost in the energy region from injection to approximately 10 MeV, if typically, injection takes place below energies of the order of 1 MeV. In addition to continuous losses related to longitudinal capture, beam losses related to transverse miss-matching of the injected beam to the transverse phase space acceptance of the linac should be considered. It is assumed here that this matching has been done properly and that, as defined, for an ideal tuned linear accelerator no loss takes place related to transverse acceptance specifically. A definite continuous loss does occur due to the heavy ion content frcm the pre-accelerator beam. This uill be normally in the range from 20-30%. These particles will not be kept within the transverse stability domain of the linac becauze of the miss-match of the rf defocusing forces and the transverze correction focusing forces, which are adjusted cortectly for r r c t o n s cnly. Theoretical calculations a r d experimental tvidence indicate that the particle losses as described above, occLr ncrmally below a prctcn energy in tke 1il;ac of approximately 10 MeV. As mentioned above, deperding on uhether or r,ot a prebuncher cavity is uscd, the total loss below 10 MeV might therefore be in tke range of 25%--757; of the preinjector beam current. Since in principle it should be possible with a well designed and properly tuned linac to reduce proton losses above 10 MeV to approximately 1"; (or even less) of the total accelerated beam intensity, all particle losses significantly above this value will therefore be classified as f a ~ ~losses. lt Again a distinction may be made between loss of longitudinal stability o r loss of transverse stability. Fault losses related to loss of longitudinal stability may be due to a low accelerating

952

A.VAN STEENBERGEN


field in a particular cavity, or a sequence of rf cavities whereby a single cavity is operating of resonance, or an incorrect phase relation between cavities or transient phenomena due to transient field variations related to beam "loading". In the last case it may be possible to eliminate this loss by shaping the beam pulse from the ion source in order to reduce the transient variations. Normally the tunable parameters are maintained between acceptable limits by means of servo control systems. If the pertinent parameters vary outside set limits, it has been estimated for practical cases that the proton loss is widely distributed along the length of the linac starting somewhat downstream from the point of loss of phase stability. Fault losses related to incorrect transverse focusing parameters, would similarly result in a beam spill, spread again widely along the axis of the linac, following immediately downstream from the failure location. Summarizing the criteria mentioned, above 10 MeV a continuous loss along the length of the accelerator will occur due to transient phenomena related to beam loading. Percentage wise this loss depends on the pulse length employed and should be estimated in relevant cases. Conservatively estimated, continuous beam losses due to all possible causes would normally be of the order of a few percent of the accelerated total beam. Significant total proton losses, especially above 10 MeV, must be prevented by means of fast acting devices. Considering further the proton losses below 10 MeV, it is useful to treat this separately. This is enforced rather naturally because of the threshold levels for the ) 10.1 MeV ( 6 5 C ~ )res, (p, n) reactions in copper, which are 10.6 MeV ( 6 3 C ~and pectively. Therefore, no neutron production does occur in the first cavity below 10 MeV. The lost protons and heavy ions will be stopped on the copper surfaces of the drift tubes and will produce soft Rontgen rays, but this radiation is readily absorbed by the cavity wall. This source will be supplemented to some extent by X-rays due to accelerated free electrons in the cavity fields. The peak of this spectrum will, however, be below 1 MeV. Based on experience with the present BNL 50 MeV linear accelerator and preinjector, minimum thickness structural walls provide adequate shielding around the pre-accelerator and in the vicinity of the first cavity. Therefore, in the following the significant beam losses below 10 MeV will be neglected, since from the point of view of shielding and radiation hazards it is easily dealt with.

2.3 Definition of source The combined loss of protons per unit length and per unit time (above 10 MeV) may be expressed as follows:

C.2.4

BEAM MONITORING.


953

where f =fractional continuous loss (normal operation); typically, 0.1 %

(More precise empiric expressions are given in Nicolaev's review [1965], by Leischner [I9661 and by Betz et a/. [1967].) The efficiency of a stripper, i.e. the relative intensity of the most probable charge state, is therefore equal to o r slightly less than

1036

CH. SCHMELZER

HEAVY ION LINACS

Typical values derived from Leischner's [I9661 measurements are given in table 2 (Betz et al. [1966]). TABLE 2 N m a x for various ions and 82 0.015 --

Ion Nmax

N 0.55

S 0.36

--

A 0.24

s

.

-

I 0.19

U 0.15

- -

The mean equilibrium charge [ depends mainly on velocity and nuclear charge of the ions, on the volume density of the stripper and, to a lesser degree, on the stripper material. Measurements with heavy ions at particle energies up to about 100 MeV fit for Pk0.01 and within the range 7 5 2 5 9 2 a simple empiric relation

1 - ( Q Z ) = C exp { - 6Pla)

(3)

(Heckman et al. [1963], Betz et al. [1966]) with a=e2/Ac (see fig. 5). The constants 1.05 C 5 1.1 and 6 = Z - ? depend on Zionas well as on stripper properties. Typical values are given in table 3 (Betz et al. [1966]). It is seen that for stripping in rarefied

Fig. 5. Straight line plot of eq. (3) for stripping in air (L) and formvar foils (F).

gases ( p s 1 torr) 6 g Z - * which is in agreement with theories by Knipp and Teller [I9411 and by Bohr [1948]. A convenient empiric expression for 6 derived by Leischner [I9661 is

with a=0.067 for gas- and a=0.053 for foil-stripping. Combining eq. (3) and (4) leads to a useful interpolation formula

D.l

SPECIAL PROBLEMS IN HEAVY ION ACCELERATION

1037

TABLE 3 Experimental results for C , 6 and y of eq. (3) -

Ion

-

p

p

6

C

7

Alr stripper

S As

I

U*

1.135 i0.049 1.117tO007 1.065 0.004 (1.01)

+

+

0.159 0.007 OIlltO001 0.078 + 0 001 (0.042)

0.663 i 0.016 O628tO003 0.641 10.003 (0.70)

Formvar-foil stripper

S As 1

U

1.083 ir 0.020 1.098 t 0.006 1.030 0.002 I .030 ir 0.002

+

0.187 0.004 0.153 &0.001 0.128 &0.001 0.100 0.001

+

0.604 3 0.007 0.538 $ 0.002 0.518 % 0.002 0.510 $ 0.002

* The high magnetic rigidity of the lower charge U ions prevented the measurement of complete charge distributions. In this case C, 6 and y were determined from values extrapolated under the assumption of a gaussian charge distribution.

i

which yields values for 5 within a relative error of a few percent (P/c(=4.57 x uion, if uion is measured in cm sec-I). It must be emphasized that these empiric formulae rest mainly on measurements below 100 MeV. Since [(b, Z ) is a key parameter in the design and optimization of heavy ion accelerators, it is important to know how far they can be extrapolated. It can be shown that for 7 5 2 5 2 5 the results become unreliable outside the range 0 . 2 s [ / Z s 0 . 8 . Above Z=30, the simple exponential relation eq. (3) begins to break down earlier. This has been demonstrated in the case of iodine by Grodzins et al. [1967], where eq. (3) is shown to be valid only up to 5 ~ 2 (see 5 fig. 6). This limit corresponds to particle energies of about 0.6 A MeV in case of foil-, and about

Fig. 6. Same plot as fig. 5 for iodine. 0 after Leischner [1966], x after Grodzins et al. [1967]. Note change of slope at 1 ~ 2 5 .

1038

CH.SCHMELZER

HEAVY ION LINACS

2 A MeV in case of gas-stripping. At higher energies, [-values calculated by eqs. (3) and(5) have to be reduced by 20percent in order toget agreement with the experimental data. Consequently, not only eqs. (3) and (4), but also all other %@,Z) relations proposed so far should not be extrapolated for 2 2 4 0 much beyond 0.6 and 2 A MeV in case of foil- and gas-stripping, respectively. The deviations observed by Grodzins et al. are almost certainly caused by a shell-effect. At [=25, all electrons in the two outer (0, n) shells of the iodine atom have been stripped away. Since the ionisation potentials of the electrons in the following m-shell are considerably higher than those in the n-shell, the cross sections for electron loss are smaller, causing 1to increase less rapidly with increasing particle velocity. All stripping experiments, especially those with the heavier ions, show that at given In the case of iodine below 0.6 A MeV, for instance, [foil/[gasZ 2. velocity This so called foil or density effect has been attributed to the fact that, in sufficiently dense strippers, the mean collision time and the life times of some excited electronic states of the ion become comparable. Since the cross sections for electron loss are larger for excited than for ground state ions, lfoi, > [,,, (Bohr [I 9481). It seems, however, that this explanation does not yield a complete, quantitative description of the foil effect. The obvious superiority of foil strippers is unfortunately impaired by their inability to withstand high beam currents. Even foils made out of such resistant materials as beryllium oxide or carbon show only a mean life time of a few hours if bombarded with several times 1012 argon ions per second at 1 A MeV. Moreover, irregular thickness variations tend to change the charge distribution during foil life. Rarefied gas strippers, on the other hand, consume much space and need elaborate differential pumping systems in order to separate the gas cell from the vacuum system of the accelerator. The best solution for reliable stripping of high intensity beams seems to be the supersonic vapour jet perpendicularly crossing the particle beam, as it was used in the Berkeley HILAC (Hubbard et al. [1961]). It is able to handle high beam currents and can be separated rather easily from the vacuum system of the accelerator. Stripper media like water vapour (Roos [1965]) or carbon dioxide (Borovik et al. [1964]) are preferable to mercury vapour originally used at Berkeley. Strippers allow to economize considerably on total accelerating voltage. The total potential drop necessary to accelerate, for instance, uranium 11 ions to 7 A MeV is 151 MV. By using one (gas-) stripper in the correct position, this value already is reduced by 45 per cent to 83 MV. Because of the low stripping efficiency (see table 2), multiple stripping leads to prohibitive intensity losses, in particular for the very heavy ions, and should therefore be avoided.

c,,.

+

5 The pressure transparency Collisions between ions and residual gas atoms in the vacuum chamber of the acceler-

D.l


1039

ator give rise to undesired charge change processes, which lead to particle loss or at least to increased energy spread, depending on the type of accelerator used. The relative number of ions accelerated without suffering charge change processes outside localized strippers will be called pressure transparency, T,. It is an important parameter in the design of vacuum systems for heavy ion accelerators, and given by

where L measures the total beam path, A = 3 . 3 n : ~1016 t o n - ' cm-"or n-atomic residual gas molecules, a , is the total charge change cross section per atom and p the residual gas pressure. In more simple terms, the pressure transparency can be written T, r exp { - E L )

with the 'coefficient of extinction'

where 5,represents a mean, velocity independent cross-section and p is assumed to be constant along the beam path. Unfortunately, relatively little is known about the dependence on /3 and ( of o,, especially for the very heavy ions. The results on [(b, Z) and T,however, combined with the theoretically sound assumption (Bell [1953]) that electron capture cross sections of ions of sufficiently high charge are independent on Zion,allowed a semiempiric estimate of a , (Betz et al. [1967]). Typical results for sulfur, arsenic, iodine and uranium ions colliding with nitrogen molecules are shown in fig. 7a-d. At low energies electron capture dominates a,, which varies almost proportional to /3-5. With increasing energy loss processes begin to contribute and finally completely govern the velocity dependence of o,. In fig. 8 the result of this semi-empiric estimate is compared with measurements by Angert [1968], Angert ef al. [I9681 and by Moller [1968], Moller et al. [I9681 for 8' iodine ions in nitrogen. Above 0.1 A MeV, the agreement is rather good. It seems therefore, that the a,-values plotted in fig. 7 give sufficiently correct information for designing vacuum systems of heavy ion accelerators. One additional point deserves special mention. If the particle energies decrease below approximately 0.1 A MeV, a , no longer increases, but tends to become constant until below about 10 A keV a , begins to drop. The existence of this plateau has been demonstrated by Nikolaev [I9651 and although there are no measurements in this energy region available for heavy ions, its existence has to be accepted. It is furthermore highly probable that the height of the plateau increases with (, which points towards rather severe vacuum problems in the initial stage of acceleration of very highly charged ions. A typical example for the pressure dependence of T,, which also illustrates the

1040

CH. SCHMELZER

HEAVY ION LINACS

Fig. 7. Calculated total charge change cross sections vs. velocity for a) sulfur, b) arsenic, c) iodine, d) uranium in nitrogen.

D.l


1041

Fig. 8. Measured (full line) and calculated (dashed line) total charge change cross sections for iodine 8+ in nitrogen.

vacuum problems during the early part of the acceleration process, is given in table 4. A linear accelerator is considered, producing 7 A MeV particles in the mass range 1005A, 0.5. Thin foil f o c ~ ~ s i nisgabove all of theoretical interest. A single attempt was made to use it in the initial Alvarez accelerator (the foils were of beryllium) but this solution quickly had to be abandoned because the foils caused unacceptable scattering of the beam and did not resist the least breakdown between the drift tubes. Moreover, their thermal dissipation is very low and the beam intensity is therefore extremely limited.

1076

2.2

D. IZOUSSARD

HEAVY ION LINACS

Grids

The thin foils have very quickly been replaced by metal grids which close only a part of the aperture of the drift tube. The field distribution and the shape of the lines of force in the gap are therefore more complex, but it is still possible to bring out the focusing effect of a gap fitted with a grid. By means of the Laplace equation, which gives rise to the incompatibility theorem (the potential does not show any extremes outside the charges, but only saddle points), it is possible to write, in axial symmetry:

In the case of a gap fitted with a grid, the Poisson equation must be used because of the presence of electric charges on the wires of the grid. As a result of these electric charges, eq. (7) becomes:

q being the charge density.

When the gap is an accelerating gap the righthand tube carries negative charges (acceleration of positive ions). It is, therefore, conceivable that if the grid is connected electrically to this tube (fig. 2a), the second term of eq. (8) will become strongly negative and ensure convergence. Considered from this angle, grid focusing is not unlike the method of space charge neutralization of an ion beam by an electron beam. To simplify matters, let us imagine a grid reduced to a single ring with radius Q connected electrically to the righthand tube (fig. 2a). According to the value of r which is supposed to remain constant during passage through the gap (approximation as for thin lenses), two regions are distinguishable:

Fig. 2. (a) Drift tube with a circular grid. (b) Limit of stability of a gridded gap (circular grid).

RF FOCUSING IN HEAVY ION LINACS

D.3

1077

The inner region (r < Q ) . The incompatibility theorem is applicable in the whole of this region and the gap is defocusing when 4 is negative. The outer region (r>e). Because of the charges carried by the ring, the gap is focusing even when 4 < O (fig. 2b). In the vicinity of the ring, the stability region is extensive as is shown by the shape of the equipotentials, whereas at the edge of the tube ( r = a ) the field distribution is only slightly changed by the presence of the grid (4 1.0). However, the limit of stability does not reach 4=O because the electric centre of the gap is shifted by the grid. Unlike the case of thin foil focusing, the convergence of a gap fitted with a grid is very strongly non-linear (fig. 2b). The complete calculation of the trajectories across the accelerator (succession of lenses with very strong aberrations) is extremely complicated*. The non-linearity which causes an increase in the beam diameter, can be reduced by using several concentric rings, but this impairs the transparency of the grid. If, on the contrary, the ring is reduced to a single stem on the axis (Karetnikov et al. [1962]) the aberrations become very great. The grids used in linear accelerators are generally of the shape shown in fig. 3. This type of grid has been studied in detail at Berkeley (Smith and Gluckstern [1955]). It is not possible to calculate the radial impulses directly, therefore an experimental

-

Fig. 3. Berkeley grids. Limit of stability for different values of 0 (gla 1.625, g / L = 0.25). - - - - - - - Average limit of stability, - - - - Theoretical limit of stability for a thin foil.

...

* The calculation of the trajectories in a thin foil machine was carried out by King [I9541 and Bell [1954a].

1078

D. BOUSSARD

HEAVY ION LINACS

determination is used based on potential measurements carried out in an electrolytic tank. The radial impulse:

(Alvarez type machine) is written in the form:

with F ( r , 8) = -

S

V ( r , 8, 2 ) - V ( 0 ,0, z ) cos 2n'd vm

(T)

L

in these expressions V ( r , 0, z) is the potential measured in the electrolytic tank, E is the mean electric field along the accelerator, and Vm is the maximum voltage in the gap. The stability condition, rewritten:

is obtained graphically (according to measurements by Smith and Gluckstern) for three values of 8: 8=0°, 22.5", 45" (fig. 3). As the grids are placed in any azimuthal position along the accelerator, the focusing efficiency can be represented by a mean curve (fig. 3). It is interesting to compare this curve with the limit of theoretical stability for a thin foil focusing machine. The imperfect closing of the drift tube aperture leads to a reduced phase acceptance. The wires of the grid are made of tungsten or molybdenum strip and are arranged parallel to the axis of the machine. This guarantees a high degree of transparency for the beam (98%) and good thermal conductance. The latter is indispensable for removing the thermal power received by the grids (which may easily reach 10 W), and to avoid the formation of hot points liable to cause a disturbing electron emission (an emission which would be accentuated by the field effect). Berkeley type grids are still in use in the initial sections of heavy ion linear accelerators (Yale, Berkeley, Manchester). The total current performance (phase acceptance and transparency) of such machines always remains very low. For the Manchester machine (with 40 drift tubes and Sloan-Lawrence structure) it scarcely exceeds 2 to 3% (Nassibian et al. [1961]).


D.3

1079

3 Alternating focusing 3.1 Alternating phase focusing This method of focusing was proposed shortly after the invention of the alternating gradient synchrotron (Good [1953], Mullett [1953], Fainberg [1956]). Its principle, copied from that of the new synchrotrons, is as follows: with a carefully chosen machine lay-out the synchronous phase can be made to be positive and negative alternately. The accelerator, therefore, has a succession of focusing gaps (axially instable) and defocusing gaps (phase stable). By analogy with alternating gradient focusing it can, therefore, be expected that an axially symmetrical machine will have overall stability of motion, i.e. both axial and radial stability. Various systems have been proposed for alternating the synchronous phase. For a travelling wave accelerator, it is possible to excite the successive sections of the machine by waves which are suitably dephased (Adlam [1953]). Also for a SloanLawrence structure. it is conceivable to feed each drift tube with a suitably phased rf voltage. If the synchronous phase is well chosen (4,= -45"), an electrically simple structure is obtained which requires only one rf generator with symmetrical outputs (Boussard [1963]). Another possibility is to have drift tubes which are alternately longer and shorter than the normal lengths (Boussard [1963], Teissier et a/. 119661). This tempting method of focusing has, however, not found any practical application and the reasons for this lack of success are t o be found in the order of magnitude of the parameters. In axial symmetry, the axial and radial convergences of a gap are expressed by:

c,= Ai: = -

E

wV, -

2 U

Tsin4,;

C,=

At

r

=

o V, -

4 U

T sin 4,

in which expression c represents the abscissa deviation in relation to the synchronous particle, T is the transit time factor and U is the mean particle energy (expressed in electron-volts). The numerical values of these convergences are always relatively weak even if 4, is given large values (which is inconvenient for acceleration). On the other hand the absolute values of C , and C, decrease along the machine (term V,/U). The characteristic exponent ki of the transfer matrix of a lattice. which can be expressed by: cosp

=

2

+ 2k ( n / w )( C , + C,) + k 2 ( n 2 / w 2 C) , C 2

(1 3)

( C , , C, are convergences of two successive gaps; k = w L / n u , e q ~ ~ a l l i nI gfor a conven-

tional machine), very quickly tends to zero. As a result, the radial acceptance and more especially the phase acceptance are extremely weak. Even if the best parameters are chosen, the phase acceptance obtained is a few degrees (Teissier et al. [1966]). The success of alternating focusing ('strong' focusing) for synchrotrons has been

1080

D.BOUSSARD

HEAVY ION LINACS

due to the considerable increase in the convergence of each element. In the case of a linear accelerator using alternating phase focusing, on the contrary, the convergences of the elements are always weak and this explains the failure of such a simple focusing method.

3.2 Rectangular drift tubes To obtain a suitable acceptance while still using the alternating focusing mechanism, the successive convergences and divergences of the gaps must be increased. This can be done by using drift tubes with a rectangular cross-section (fig. 4), which were proposed at the same time at CERN and in the USSR (Lapostolle [1962], Anisimov and Teplyakov [1963]).

*

z

Fig. 4. Rectangular drift tubes.

In this structure the synchronous phase is kept constant (6, < 0) and this ensures longitudinal stability. But the shape of the lines of force of the electric field in a gap is such that there is focusing in one direction and defocusing in the direction perpendicular to it. The field distribution is illustrated in fig. 5 and, as in the case of thin foil focusing, it can be assumed as a first approximation that the impulse is localized at the input or at the output end of the gap. This approximation has moreover been confirmed by

-

Fig. 5. Field distribution in a gap between rectangular drift tubes.


D.3

1081

more detailed studies (Fer et al. [1963], Marcou [1965]). The usual convergence calculation (for a Sloan-Lawrence type machine) is, therefore, as follows (Ox direction) :

(x = constant, thin lens approximation). In the region where the impulse is concentrated, the transverse component E, depends very little o n y (fig. 5). In the electrostatic equation:

the second term is, therefore, cancelled and we get:

Em being the electric field in the gap (assumed constant). This expression can be developed to obtain:

A i -- wLVm - cos 4 x 2ngU

-

+ 04 vUm sin 4 -

The first term, which is negative, corresponds to the focusing effect of the gap while the second corresponds to the conventional divergence (in axial symmetry). When the real field distribution is taken into account the first term is subject to a correction factor (asymmetry factor) with a value of the order of 0.9 (Fer et al. [1963]). The field configuration can be determined in an electrolytic tank or analytically (Marcou [1965]). With the usual values for the parameters (g/L=0.25, 4,= -3O0), the alternating type convergence (first term) is found to be four times greater than the conventional divergence (second term). The focusing is, therefore, more efficient than that produced by the alternating phase focusing, since the successive convergences and divergences of the gaps are clearly higher, as is shown by the relative amplitude of the two terms in eq. (16). However, their rate of decrease along the machine is Vm/U and the acceptance of such a system remains relatively low. Fer et al. [I9631 give a calculation which leads to a radial acceptance of 70 mm mrad for a machine having the following characteristics : injection voltage output energy

150 kV 1 MeV (singly charged ions)

1082

D.BOUSSARD

HEAVY ION LINACS

frequency

35 MHz

g =constant = 5 mm

2a = 5 mm (input) asymmetry factor

2a = 8 mm (output) R=0.9

Focusing with rectangular drift tubes has been studied in the USSR both theoretically (Mal'tsev, Teplyakov [1965]) and experimentally (Teplyakov et al. [1965], Mal'tsev et al. [1964]). The first accelerating section (50 kV injection, 5 drift tubes) has been constructed on this principle and also a low energy electron model (2.1 keV injection, 3.6 keV output). The current performance with this model is more than 30%, but it seems that in view of the rather particular operating conditions, this remarkable result is due to the influence of phase oscillations rather than to true alternating focusing (Mal'tsev et al. [1966]).

3.3 Drift tubes fitted with "fingers" These were first proposed by Vladimirsky [1956], and recently studied in more detail (Tcplyakov [1964], Boussard [1965a, b], Mal'tsev et al. [1967]). The drift tubes are extended by diametrically opposed fingers (fig. 6), which overlap one another and produce a quadrupole type field distribution inside the gap. In a gap the potential V ( r , 0, z ) is expressed by the Fourier expansion:

1, is the modified Bessel function of the first kind and of the nth order. The boundary

conditions,on the cylinder r = a can be expressed simply and exactly (Boussard [1965a]). With them the coefficients A,, can be calculated and so the convergences of' the

L

J

Fig. 6. Drift tubes with "fingers".

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1083

equivalent thin lenses can be determined. As in the case of the rectangular tubes. the convergence is obtained as two terms: the conventional defocusing term due to transit time and the desired quadrupole term: Ai I.

=

V,

U

sin4

w sin(ng/2L) cos (7rhlL) + ... 4 7rg/2L lo (TOIL) -

-

-

g , h and a are defined in fig. 6. The coupling terms between the radial and axial motions have been disregarded in this formula. For the usual values of the parameters ( g / L =

0.25, L / a = 4 ; 6 , = - 30°, g/h = 2), the quadrupole term is already five times higher than the defocusing term even at the beginning of the accelerator (Vm/U=0.7). But unlike the case of a rectangular tube structure, the quadrupole term can be kept constant all along the machine (by keeping for example a, V, and h/L constant), as is the case for magnetic quadrupole focusing. This remarkable property guarantees excellent optical qualities for this structure. The asymmetry factor, which is limited to unity for tubes with flat surfaces, can have much higher values in the case of tubes fitted with "fingers". The focusing properties of this structure have been confirmed experimentally on a low energy model (Boussard and Septier [1965]). The main parameters of the machine were: ion, He' injection, 15 keV output, 200 keV

g/L=0.25 frequency, 20 M H z 1m g/h=2 total length, 2a= I 0 mm number of tubes, 21 beam diameter at injection: 6 mm

An optimum synchronous phase of -26" was measured (Boussard [1965a]) which gave a theoretical current performance (limited only by the phase acceptance) of 3 40/2~[=227& The value obtained experimentally amounted to 20%. The fairly considerable coupling between transverse and axial movements does not seem to have any particular effect on the optical qualities of the machine (it must be noted that the structures considered are "short" and the resonances are not, therefore, very disturbing). The most serious obstacles encountered when putting this method into operation are of a technological nature: the machining of the drift tubes, the rf power c o n s ~ ~ m p t i oand n breakdown. However, it must be remembered that the rf power required by the first drift tubes is only a small fraction of that required for the whole accelerator. T o demonstrate the possibilities of the focusing method sing drift tubes fitted with "fingers", it has been proposed to apply this method to a first section of a heavy ion linear accelerator with characteristics which might be as follows (Boussard [I 9661) :

1084

D. BOUSSARD

ion, Ne++ injection, 150 keV output, 5 MeV frequency, 20 MHz total length, 1.80 m number of tubes, 17

HEAVY ION LINACS

rf voltage, 100-200 kV maximum electric field at any point, 100 kV/cm mean accelerating field, 1.1 MV/m

inner diameter of drift tubes, 25-40 mm g / L = 0.25 g/h = 2 synchronous phase, - 26"

Fig. 7 shows the shape of the trajectories calculated for this machine. The radial acceptance can be deduced approximately from them by adopting the value 2 for the oscillation form factor. This gives very nearly 1100 mm mrad.

A4

20"

10"

o0

____c

tube number

-lo0

-20"

Fig. 7. Typical trajectories in a self focused linac (drift tubes with "fingers"). radial motion (plane xOz), ------- radial motion (plane yOz), - - - - phase oscillation.

...

At the output end of this section magnetic quadrupole focusing can be used (the required field gradient is no more than 0.61 kG/cm, the drift tubes are longer than 16 cm) and the adaptation of the two quadrupole systems can be arranged without any particle loss.

3.4

New types of grids

Following the work on rectangular tube focusing a new type of grid was suggested,

D.3


1085

consisting of two parallel wires (fig. 8). From one drift tube to the next the direction of the grid wires is altered by 90" on analogy with the layout of the rectangular tubes. Bieth et al. [1966a] have made a study of this type of grid. The curves in fig. 9 were obtained from measurements in an electrolytic tank using the method indicated above in section 2.2.

Fig. 8. New type of grid (two parallel wires).

For r < Q (Q is the distance of the grid wires to the axis) the focusing or defocusing character of the gap depends mainly on the symmetry plane chosen (fig. 9) as in the

Fig. 9. Limit of stability for grids (two parallel wires). - - - - - - - plane xOz, - .- .- .- plane yOz.

case of rectangular drift tubes. In the central channel the focusing can, therefore, be considered to be of the alternating type with, however, less efficiency than in the case of rectangular tubes (because of the circular shape of the upstream tube). Outside this channel ( r > Q ) , on the other hand, the curves obtained are comparable with those from conventional grids. Another interpretation of these curves consists in bringing the stability condition in the vertical plane (i.e. parallel to the grid wires) closer to that obtained for a thin

1086

D. BOUSSARD

HEAVY ION LINACS

foil (fig. 3), since in the vertical plane close to the grid wire, the electric field distribution is given approximately by fig. lb. In the other symmetry plane, the stability condition is the same as for a conventional grid. The overall improvement in performance compared with a Berkeley type grid is thus connected with the superiority of the thin foils over the grids. A further conceivable development along these lines would be to keep only one single diametrical grid wire but there is a danger that considerable aberrations would spoil this solution (as in the case of grids reduced to a single axial stem). An experimental study has been made under the same conditions as that made on the "fingers7' tubes (Bieth et al. [1966b]). Measurements showed a total current performance of 14% (without a buncher) which confirms the expected improvement compared with conventional grids. This improvement also includes an increase in the transparency of the grids (individual transparency of each gap and alignment of the grids). The practical advantages of this means of focusing are obvious: it can be applied to an existing machine without major alterations to the rf structure, and the construction of the grids, although delicate, does not raise any particular problems. For these reasons, parallel wire grids have been chosen for the heavy ion linear accelerator at Orsay (cyclotron injector). The grids are made of tungsten strips which become wider from the centre of the tube towards the edges (in order to improve the thermal dissipation). They are cut in the middle to avoid deformation through expansion. Through very careful manufacture of these grids (with regard to thickness, flatness, alignment), it was possible to reduce to 30% the particle loss on the tungsten strips fitted in the 56 drift tubes.

References J. H., 1953, A method of simultaneously focusing and accelerating a beam of protons, ADLAM, AERE GP/M 146 - At. Energy Res. Estab. Harwell, England. G. M. and V. A. TEPLYAKOV, 1963, Pribory i Tekhn. Eksp. 1,21; Inst. Exp. Techn. 1, 15. ANISIMOV, BELL,J. S., 1954a, Grid focused linear accelerator dynamics. AERE TM/95, At. Energy Res. Estab. Harwell, England. BELL,J. S., 1954b, A divided drift tube for focusing in proton linear accelerators, AERE TR/1072 At. Energy Res. Estab. Harwell, England. BERNARD, M. Y., 1953, J. Phys. Le Radium 14, 381. BERNARD, M. Y., 1954, Compt. Rend. Acad. Sci. Paris 238, 675. BIETH, C., A. CABRESPINE, C. H. GOLDSTEIN, 1966a, IEEE Trans. Nucl. Sci. NS-13, no. 4, p. 182. BIETH,C., A. CABRESPINE, C. H. GOLDSTEIN, J. BOSSER, J. ARIANER, 1966b, A one MeV/nucleon heavy ion linear accelerator. J. W. McDonald, ed., Proc. 1966 Lin. Acc. Conf. Los Alamos LA 3609, p. 508. BOUSSARD, D., 1963, Doctoral thesis, Orsay (France). BOUSSARD, D., 1965a, IEEE Trans. Nucl. Sci. NS-12, no. 3, p. 648. BOUSSARD, D., 1965b, Compt. Rend. Ac. Sci. 261, 4665. BOUSSARD, D., 1966, Self focusing in heavy ion linacs. J. W. McDonald, ed., Proc. 1966 Lin. Acc. Conf. Los Alamos LA 3609, p. 96. BOUSSARD, D., and A. SEPTIER, 1965, IEEE Trans. Nucl. Sci. NS-12, no. 3, p. 652.

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R F FOCUSING I N HEAVY ION LINACS

1087

FAINBERG, I. B., 1956, Alternating phase focusing. Proc. C E R N Symp. High Energy Accelerators Geneva, Switzerland, p. 91. C. BIETH,A. CABRESPINE, 1963, Cross gradient focusing in linear accelerFER,F., P. LAPOSTOLLE, ators, MSC. 0. 181999 CERN Geneva, Switzerland. GOOD,M. L., 1953, Phys. Rev. 92, 538. A. P. FEDOTOV,and B. K. SHEMBEL, 1962, D. V., I. N. SLIVKOV,V. A. TEPLYAKOV, KARETNIKOV, Linear ion accelerators (Gosatomizdat Moscow) 67-80. KING,N. M., 1954, Proton dynamics in the linear accelerator AERE TM/107. At. Energy Res. Estab. Harwell, England. LAPOSTOLLE, P., 1962, Dispositif de focalisation pour accelerateurs lidaires B ions lourds, MSC-62-4 CERN, Geneva, Switzerland. R. B. MEUSER,1966, High gradient drift tube quadrupole magnets. MAIN,R . M., J. M. HAUGHIAN, Proc. 1966 Lin. Acc. Conf. Los Alamos LA 3609, p. 153. A . P., E. A. ZOVOTA,V. A. TEPLYAKOV, 1964, Pribory i Tekhn. Eksp. 4, 20; Inst. Exp. MAL'TSEV, Techn. 4,739. A . P., V. A . TEPLYAKOV, 1965, Pribory i Tekhn. Eksp. 4, 29; Inst. Exp. Techn. 4, 763. MAL'TSEV, MAL'TSEV, A. P., E. A. ZOVOTA,V. A. TEPLYAKOV, 1966, Pribory i Tekhn. Eksp. 6, 17; Inst. Exp. Techn. 6, 1300. MAL'TSEV, A. P., S. M. ERMANOV, V. A. TEPLYAKOV, 1967, Atomnaya Energiya 23, 3. J., 1965, Thesis, Orsay (France). MARCOU, MCMILLAN, E. M., 1950, Phys. Rev. 80, 493. MULLETT,L. B., 1953, Linear accelerators focusing by periodic changing of the synchronous phase position. AERE GP/M 147. At. Energy Res. Estab. Harwell, England. S. DEVONS,R . W. R. HOISINGTON, V. E. MILLER, NASSIBIAN, G., J. BENNETT, JR., D. BROADBENT, 1961, Rev. Sci. Instr. 32, 12, 1316. PANOFSKY, W. K. H., 1951, Linear accelerator beam dynamics, UCRL 1216 Univ. Calif. Radiation Lab. Berkeley, California. 1955, Rev. Sci. Instr. 26, 2, 220. SMITH,L., R. L. GLUCKSTERN, STADLER, B., 1966, Univ. Heidelberg, Germany, private communication. B. VEYRON,1966, Etude numerique du principe d'acceleration et de TEISSIER, G., A. CHABERT, focalisation par phase alternee. LYCEN/6528 (Rev.) Insl. de Phys. Nucl. Lyon (France). TEPLYAKOV, V. A,, 1964, Pribory i Tekhn. Eksp. 6, 24; Inst. Exp. Techn. 6, 1162. TEPLYAKOV, V. A., S. M. ERMANOV, A. I. MAKAROV, Y. G . GENDEL,V. 1. KRASNOVSKII, B. K. SHEMBEL, 1965, Pribory i Tekhn. Eksp. 4, 26; Inst. Exp. Techn. 4, 759. V. V., 1956, Pribory i Tekhn. Eksp. 3, 35. VLADIMIRSKII,

PART E

Superconducting

Linear Accelerators A. CITRON H. SCHOPPER

A. L. SEPTIER P. B. WILSON

SULA

E

The Surface Impedance of Superconducting Metals at High Frequencies A. SEPTIER Znstitut d'Electronique Fondamentale Laboratoire Associe au CNRS Faculte des Sciences, 91 Orsay, France

1 The historical background Since 1911 (Kamerlingh Onnes [1911]), it has been known that the electrical conductivity a of certain metals, cooled to very low temperatures, suddenly becomes infinite beneath a particular critical temperature, Tc (which lies between 0 and 20°K). A second fundamental property of this superconducting state was discovered by Meissner [1933]: a static magnetic field H cannot penetrate into a superconductor (which is thus perfectly diamagnetic), provided that the magnitude of this field is not greater than a critical value, H,. For H > Hc, the conductivity a again becomes finite and the metal returns to the normal state. Finally, London showed in 1940 (London and Ochsenfeld [1940]) that the high-frequency surface conductivity Z, of a superconducting metal increases abruptly beneath Tc, but does not become infinite even though a is infinite. A variety of phenomenological theories were developed in an attempt to explain these different properties (for example, Maxwell [1964]), but it was not until 1957 that a satisfactory microscopic theory of the superconducting state appeared (Bardeen et al. [1957]). With the aid of this BCS theory, which has been reconsidered and expanded by many research groups, very general expressions have been established giving the uhf surface impedance (Mattis and Bardeen [1958]; Miller [1960]; Abrikosov et al. [1959, 19601; Khalatnikov and Abrikosov [1959]. It is difficult to extract 0 North-Holland Publ. Co., Amsterdam

1969

References p. llO5

1089

1090

A. SEPTIER

SUPERCONDUCTING LINACS

exact numerical results from these expressions which have therefore been simplified and used primarily in the two important limiting cases: the penetration depth A of the electromagnetic fields in the metals is either very small or very large, compared with the mean free path I of the electrons and the coherence length to,which will be defined below. In the superconductors that are most commonly used in high-frequency work, we usually have A$~,(0), when the mean free path I has a finite value; the coherence length is given by an expression similar to eq. (3.6) In a pure superconductor, variations with T (T0.3TC. If T0.5 (MV/m)-' in fig. 11, the curve is rather erratic, indicating that the loss enhancement is due to several causes. In the region above 30 kV/cm, however, field emission appears to be the dominant loss mechanism in these two cavities. When the field emission loading is heavy, substantial radiation can be also measured in the vicinity of the cavity.

A

950 MHz DISK LOADED STRUCTURE 2855 MHz TMolo

MODE CAVITY

Fig. I 1 . Fowler-Nordheim plot of the loading observed in two superconducting cavities at high electric field strengths. In this figure Ec is the maximum electric field on the axis of the cavity. For the disk-loaded structure Ep = 1 .6Er =3.2Ea.

There can be considerable variation in this type of loading from cavity to cavity depending on the plating technique and on the subsequent treatment of the surface. In addition it is possible to reduce the effective enhancement factor and the resulting field emission current by a period of "processing" in which the cavity is operated at power levels sufficiently high to destroy the sharper projections (Turneaure [1968b]). In initial tests on the Stanford 5-ft superconducting accelerator (9 5.1), a gradient of about 4 MeV/m was reached before the field emission loading became substantial. Dc measurements have been made (McAshan [1968b]) comparing the field emission current from polished niobium and electroplated lead electrodes. These measurements indicate that a substantially higher gradient should be possible in a niobium structure as compared to a structure with an "as electroplated" lead surface. In measurements at 8.5 GHz using a solid niobium TM,,, modecavity, fields equivalent to a gradient of about 10 MeV/m were attained without appreciable loss enhancement due to field emission (Weissman and Turneaure [1968]). The fields again appeared to be limited by flux penetration. It should be noted that field emission loading is not best described as an abrupt breakdown, but will normally limit the gradient indirectly. The effect provides a mechanism for converting rf power into heat dissipated at low temperature. At some

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field level the loading will equal the cavity wall losses, and will then increase rapidly with increasing field until a limit set by the capacity of the refrigerator is reached. A field strength of 42 MV/m has been reported in a superconducting cavity having a re-entrant coaxial geometry (Passow [1968]). The power dissipation at this field level due to field emission is not reported, but may be substantial.

4.3 Beam break-up The possibility must be considered that either regenerative or cumulative beam breakup might limit the current in a superconducting linac. Regenerative beam break-up, in which the dc beam excites an oscillation by interacting with the transverse T M l l like deflection mode within a single standing wave accelerating cavity, is discussed in chapter B.1.4. If a coupled cavity model is used to approximate the accelerating structure, then it can be shown that {Z,/(LQ)) x 100/1,, where Z,/L is the transverse shunt impedance per unit length and 1,is the wavelength at the break-up frequency. Using this approximation, the starting current for regenerative break-up in the steady state case can be written io

= 0.025 ( V ~ ~ ; ) / ( Q , L ~ ) .

(4.3)

Here Vo is the electron energy (assumed constant for simplicity), and 1,for the T M l l like mode is given approximately by i1,x$i1. The effect will be most severe near the beginning of an accelerator where the energy is lowest. As an example, consider a structure operating at 1 GHz (1, x 20 cm) with L = 3 m and Vo = 30 MeV. Then from eq. (4.3) we have i0Q,x3 x lo3. If a current of 100 pA is desired, the loaded Q for the deflection mode must be on the order of 3 x lo7 or less to avoid oscillations of this type. If the electron energy is very low, as is the case if electrons are injected from a gun into a section with the rf off, oscillation can also be observed due to the interaction of the beam with the longitudinal TMol-like modes (Wilson [1967a]). Cumulative beam break-up can also limit the current in a multisection superconductinglinac. In this type of break-up, each section acts like a stage of amplification for the transverse modulation on the beam in much the same manner as the center cavities in a multi-cavity klystron enhance the longitudinal modulation of the dc beam. Cumulative beam break-up in the steady state limit is analyzed in chapter B. 1.4 (see also Wilson [1967b]). If the transverse shunt impedance is approximated as before using {Z,/(LQ)}x 100/At, the total gain can be written as exp (F) where F x 30 [ ( l z i , ~ , ) / ( ~v1' )~] ~ .

(4.4) Here z is the distance along the accelerator, V' is the energy gradient, and I is the length over which fields are excited in a section with total length L. Take as a n example z = 150 m, V' = 10 MV/m, 1,= 20 cm, and I/L = 0.5. Then from eq. (4.4) we calculate ~ , x 0 . 4 ( i , Q , ) ~ .If break-up occurs for F x 2 0 and if a beam current of 100 pA is desired, then the loaded Q must be held to about 2 x lo7 or less. Thus the

E.2

ELECTRON LINAC THEORY AND DESIGN

1135

loading needed to prevent both types of beam break-up is comparable for the examples chosen. One method for loading the Q of the deflection mode is to make the diameter of the beam coupling holes in the ends of the structure just large enough to propagate at the break-up frequency, but yet be below cut-off for the fundamental mode. This scheme has the advantage of simplicity, but the disadvantage that the length of the cut-off tubes needed to contain the accelerating mode can be quite long at low frequencies (e.g. about 40 cm at 1 GHz). In addition the threshold for cumulative break-up can be raised by staggering the resonant frequencies for the transverse mode in successive sections along the accelerator. This will take place to some extent without a special effort in a superconducting linac, since the bandwidth of the resonance is so narrow.

5

Application to accelerators

5.1 Superconducting test accelerators Tests in which a low energy beam from an electron gun is accelerated by a short section of superconducting disk-loaded structure have been important in demonstrating the feasibility of superconducting linacs and in obtaining information of use in the design of full-scale accelerators. In 1965 an electron beam of about 1 pA was accelerated at Stanford to 0.5 MeV using a three cell, disk-loaded cavity 10 cm in length (Schwettman [1965]). The cavity, resonant in the 571-mode at 2.85 GHz, is described by Schwettman et al. [1966]. A much larger test accelerator, using a bi-periodic 3n-mode structure 1.5 m in length with a resonant frequency of 950 MHz, has been built at Stanford and was tested initially in 1967. The structure and rf system for this accelerator are described by Weaver et al. [1967]. A description of the injector, cryogenic system, and beam analyzing system, and some initial test results are given by Chambers [1967]. The unloaded Q measured for the lead-plated structure was about 2.5 x lo8 (an improvement factor of 1 x lo4) near 2°K. A maximum (but unstable) electron energy of 5.6 MeV was observed, corresponding to an accelerating field of 5.5 MV/m. Stable accelerating fields on the order of 4 MV/m and accelerated currents on the order of several microamperes were obtained. It should be emphasized that these results are preliminary. It is expected that higher values for the Q, gradient, and accelerated current can be obtained in further test using an improved rf structure and variable coupling system (fig. 10).

5.2 Large superconducting electron linacs Since 1961, when measurements were begun at Stanford University on superconducting

1136

P.B. WILSON


cavities, the possibility has been seriously considered that the present Mark 111 linac at the High Energy Physics Laboratory might eventually be replaced by a superconducting accelerator. As an ultimate goal, the conversion of the two-mile SLAC accelerator to a superconducting machine has been envisioned. In order to point out the potential advantages of large superconducting accelerators, tentative parameters for superconducting linacs comparable in length to the Mark 111 and SLAC accelerators have been presented from time to time (Wilson et al. [1964], Schwettman et al. [1965], Wilson and Schwettman [1965], Wilson et al. [1966]). As the results being obtained from cavity measurements have improved, such "paper accelerators" have looked increasingly more attractive. Recently, plans for a large superconducting accelerator at the High Energy Physics Laboratory have become more definitive. It has been proposed that such an accelerator, 500 ft in length, be constructed in a tunnel located directly below the existing Mark 111 linac. Some tentative design parameters have been suggested for the proposed accelerator: unity duty factor at an energy of 2.0 GeV (although pulsed operation at higher gradients may be possible, depending on the limitations discussed and a power dissipation of 2.3 W/ft at a in §4), a relative energy spread of temperature of 1.85"K and a frequency near 1 GHz (Smith [1968a], Turneaure [1968b], McAshan [1968a]). Although the preceding figure for power dissipation per unit length is based on the theoretical surface resistance of superconducting lead, it may well prove feasible to build a niobium or niobium-plated accelerating structure. In this case the theoretical loss would be lower by about a factor of two. The average current and loaded Q for the proposed accelerator will depend on the available rf power per unit length. If, for example, a power of 1 kW/ft is supplied, the average current at the design energy will be 250 pA and the loaded Q at optimum efficiency will be about 5 x 10' (assuming the structure is similar to that shown in fig. 10). Since the design of the proposed accelerator is still in the preliminary stage, further details will not be presented here. A more comprehensive account can be expected in future publications.

5.3 Application to multi-turn accelerators The use of superconducting linac sections has been proposed as a means of increasing the maximum energy of an electron synchrotron (e.g. Wilson and Schwettman [1965]). In such an accelerator the required energy gain per turn at maximum energy is determined by synchrotron radiation, which varies as the fourth power of energy. At the limiting energy the rf power requirement is usually dominated by cavity wall losses, which then vary as the eighth power of energy. For example, in the Cornell 10 GeV synchrotron the energy loss per turn at 10 GeV due to synchrotron radiation is 9 MeV, while the peak and average rf power required for acceleration to this energy are about 400 kW and 100 kW respectively (Tigner [1967]). This power requirement

E.2


1137

is calculated for a circulating current of 7 mA, corresponding to I x 10" electrons per pulse, but the figures would not be substantially lower even for a vanishingly small accelerated current. At 20 GeV (assuming energy 1s not limited by other machine parameters) the energy loss per turn due to synchrotron radiation increases to 140 MeV. The cavity losses increase by a factor of 256, resulting in an average power requirement on the order of 25 MW. If, however, superconducting cavities with an improvement factor of 10' are used, the average power dissipated in the cavities would be on the order of 250 watts. A refrigerator with an input power of about 400 kW would be required. For acceleration to 20 GeV, the average power loss to synchrotron radiation would be about 200 kW for the same circulating current. The power requirements for both the rf system and the refrigerator are seen to be reasonable. Superconducting linac sections have also been proposed for use as the accelerating cavities in a racetrack microtron (Wiik et a/. [1966], Hanson et a/. [1966]). An accelerator of this type seems well adapted for producing a high current cw beam in the 200-600 MeV energy range. Only a single rf accelerating cavity about 3 m in length is required, although in some designs a second cavity is used as an injector. The machine is therefore quite compact, and in addition has the inherent high energy resolution associated with a phase-stable, multi-turn accelerator.

Acknowledgments

The foregoing article is largely a summary of work carried out at the High Energy Physics Laboratory, Stanford University, by the author and others concerning the theory and design of superconducting electron linacs. An attempt has been made to acknowledge by appropriate references the specific contributions made by various individuals. The formal list of references, however. does not adequately reflect the contributions made by Professors W. M. Fairbank and H. A. Schwettman in the effort at Stanford University to develop superconducting accelerators. In writing this summary, conversations with G. A. Loew, R. Miller, T. I. Smith, and J. P. Turneaure have been particularly helpful.

Notes added in proof During the short time that has elapsed since the preceding text was written, the field of superconducting linac technology has been advancing at a particularly rapid rate. Some of the most significant new developments are summarized in the following notes. The results obtained from rf measurements on niobium cavities continue to be encouraging. Recently peak fields of 77 MV/m and 1050 Oe, at a Q improvement factor of 5 x I@, have been measured at Stanford University (J. P. Turneaure, private communication). The cavity, resonant in the TMolo mode at 8.5 GHz, was machined from solid niobium in two pieces, which were then joined by electron beam welding. Processing at high temperature and high vacuum is still found to be

1138

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necessary. The possibility of producing a structure by hydroforming assemblies from niobium sheet, and then electron beam welding these assemblies together to form the completed structure, is also being investigated at Stanford (E. Jones, private communication). Several papers related to the development of superconducting electron linacs were given at 1968 Summer Study on Superconducting Devices and Accelerators (Brookhaven National Laboratory, June 1CJuly 19,1968; see: Report BNL 50155). L. R. Suelzle (p. 67) reports on the use of feedback stabilization techniques to control the rf amplitude and phase in a superconducting cavity to the and k0.2 degrees respectively. E. E. Chambers (p. 79) presents detailed calcuorder of * 7 x lations on particle motion in a standing-wave accelerator. Special attention is given to the problem of capture and bunching. R. B. Neal (p. 101) gives sample parameters for a 100 GeV superconducting version of the two-mile SLAC accelerator. The same author, in another paper (p. 11I), discusses the use of a traveling-wave section with feedback (traveling-wave resonator) as a structure for a superconducting accelerator, taking into account the effect of beam loading. Finally, the possibility of producing a superconducting structure by electroplating technetium on copper is pointed out in a paper by M. A. Allen and H. A. Hogg (p. 34). A number of papers concerning superconducting linac development were also presented at the 1969 Particle Accelerator Conference (Washington, D.C., March 5-7; proceedings are: published in IEEE Trans. Nucl. Sci. NS-16, No. 3). E. Jones et al. (p. 1000) report on the performance of a superconducting injector for the Stanford 500 ft linac. Initial test results are very encouraging: an energy at an energy of 1.3 MeV. spectrum of 2%, a bunch length less than 2O, and a stability of about W. B. Herrmannsfeldt et al. (p. 1004) give some results of a continuing study at SLAC to investigate the feasibility of a two-mile superconducting linac. D. C. Sutton et a[. (p. 985) present calculations of the phase and spatial stabihty of the electron trajectories in a 600 MeV microtron. This accelerator, which uses a 30 MeV superconducting linac between two uniform field bending magnets, has been proposed for the University of Illinois. It is expected to produce a cw beam with excellent energy homogeneity ( w Papers related to the design of structures for superconducting linacs were presented respectively by E. A. Knapp (p. 329) and P. B. Wilson (p. 1092). In the former paper the advantages of resonant coupling is emphasized, and the possibility of extending this concept beyond bi-periodic structures to higher order "superperiodic" structures (e.g. tri-periodic, quadru-periodic, etc.) is pointed out. In the latter paper the effect of a mismatch in a superconducting traveling-wave resonator is given particular attention. It is shown that such a resonator must be matched to a high degree of precision if power flow in the reverse direction is to be avoided. Work directed toward an understanding of the behavior of superconductors at high rf electric and magnetic field strengths is continuing. H. Schopper has pointed out (Proceedings of the 1968 Proton Linear Accelerator Conference, Brookhaven National Laboratory, BNL 50120, p. 755) that the field emission process, as outlined in 8 4.2, may not be the principal source of loss at high electric field strengths in lead-plated cavities. The apparent enhancement factors, as obtained from the slopes of Fowler-Nordheim plots, especially at low and moderate field levels, are often unreasonably high in comparison with the values obtained in dc measurements. A number of alternative loss mechanisms have been suggested by J. Halbritter (The Dependence of the Surface Impedance on the R F Field Amplitude, External Report 3168-8, Institut fur Experimentelle Kernphysik, Karlsruhe, Germany, October 1968). It should also be noted in this regard that at very high field strengths, above about 30 MV/m, recent measurements have given reasonable values for the enhancement factor, consistent with a field emission loss mechanism (W. Kuhn, private communication). On the whole, however, it is fair to state that at the present time the electric field dependent Q degradation which occurs in TM-mode lead-plated cavities, often at relatively low field strengths, is not well understood. It is encouraging to note that properly prepared niobium cavities do not show similar losses, even at very high field levels.

References ABRIKOSOV, A. A., L. P. GOR'KOV and I. M. KHALATNIKOV, 1959, Sov. Phys. JETP 8,182. ALPERT,D., D. A. LEE,E. M. LYMAN and H. E. TOMASCHKE, 1964, J. Vac. Sci. Technol. 1, 35. ALTMAN, J. L., 1964, Microwave Circuits (D. Van Nostrand Co., Inc., New York) 8 5.3 and 5 5.4.

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BARDEEN, J., L. N. COOPERand J. R. SCHRIEFFER, 1957, Phys. Rev. 108, 11 75. CHAMBERS, E. E., 1967, Superconducting Accelerator Project at Stanford: 11. The Five and Five Hundred Foot Accelerators. I n : MACK,R. A,, ed., Proc. 6th Intern. Conf. High Energy Accelerators, Cambridge, Mass., 1967 (Cambridge Electron Accelerator, Cambridge, Mass. Also available as Report CEAL-2000, Clearinghouse for Federal Scientific and Technical Information. U.S. Department of Commerce, Springfield, Va.) p. 381. CHAMBERS, E. E., 1968, private communication. and M. H. STREETER, 1967, Rev. Sci. Instr. 38, 1654. COLLINS, S. C., R. W. STUART FAIRBANK, W. M., J. M. PIERCE and P. B. WILSON,1963, High Power Superconducting Cavities for Accelerators. In: DAVIES, R. O., ed., Proc. 8th Intern. Conf. Low Temperature Physics, London, 1962 (Butterworths, Washington, D.C.) p. 324. FINNEMORE, D. K., T. F. STROMBERG and C. A. SWENSON, 1966, Phys. Rev. 149, 231. FONTANA, J. R . and H. J. SHAW,1958, Proc. IRE 46, 1424. GINZTON, E. L., 1957, Microwave Measurements (McGraw-Hill, New York) 5 9.1. GOLDE,H., 1959, Extended Interaction Klystrons with Traveling Wave Cavities, M.L. Report No. 582, Microwave Laboratory, Stanford University (April, 1959) 5 3F. HAIMSON, J., 1965, IEEE Trans. Nucl. Sci. NS-12, no. 3, p. 499. A. O., D. JAMNIK, C. S. ROBINSON and C. L. RODGERS, 1966, Plans for a superconducting HANSON, cw Electron Linac at the University of Illinois. In: Proc. 1966 Linear Accelerator Conference, Los Alamos, New Mexico (Report LA-3609, Clearinghouse for Federal Scientific and Technical Information, U.S. Department of Commerce, Springfield, Va.) p. 494. HOYT,H. C., D . D. SIMMONDS and W. F. RICH, 1966, Rev. Sci. Instr. 37, 755. MCASHAN, M. S., 1968a, Low Temperatures and Linear Accelerators: 111. Heat Transport and RefrigeK. D., ed., Advances in Cryogenic Engineering 13 (Plenum Press, New ration. I n : TIMMERHAUS, York) p. 428. MCASHAN, M. S., l968b, private communication. MILLER,R., 1968, private communication. NAGLE,D. E., E. A. KNAPPand B. C. KNAPP,1967, Rev. Sci. Instr. 38, 1583. PASSOW, C., 1968, Results of Design Study of 7 GeV Superconducting Proton Linear Accelerator. In: Prcc. 6th Intern. Conf. High Energy Accelerators, Cambridge, Mass., 1967, p. 383. J. M., H . A. SCHWETTMAN, W. M. FAIRBANK and P. B. WILSON,1965, Rf Losses in SuperPIERCE, conducting Lead and Niobium. In: DAUNT,J. G. et a / . ,eds., Proc. 9th Intern. Conf. Low Temperature Physics, Columbus, Ohio, 1964 (Plenum Press, New York) p. 396. J. M., 1967, The Microwave Surface Resistance of Superconducting Lead, Trapped Magnetic PIERCE, Flux and a New Magnetometer Using Superconductivity. Ph.D. thesis, Department of Physics, Stanford University. Also available as Technical Report No. 514, High Energy Physics Laboratory, Stanford University (June 1967). SCHWETTMAN, H . A,, 1965, Results presented at 5th Intern. Conf. High Energy Accelerators, Frascati, Italy, 1965. 1965, The App!ication of SCHWETTMAN, H. A., P. B. WILSON,J. M. PIERCEand W. M. FAIRBANK, K. D., ed., International Superconductivity to Electron Linear Accelerators. In: TIMMERHAUS, Advances in Cryogenic Engineering 10 (Plenum Press, New York) p. 88. SCHWETTMAN, H . A,, P. B. WILSONand G . Y. CHURILOV, 1966, Measurements at High Field Strengths on Superconducting Accelerator Cavities. In: GRILLI,M., ed., Proc. 5th Intern. Conf. High Energy Accelerators, Frascati, Italy, 1965 (Comitato Nazionale Energia Nucleare, Rome) p. 690. H . A,, J. P. TURNEAURE, W. M. FAIRBANK, T. I. SMITH, M. S. MCASHAN, P. B.WILSON SCHWETTMAN, and E. E. CHAMBERS, 1967, IEEE Trans. Nucl. Sci. NS-14, no. 3, p. 336. SMITH,T. I., 1966, Standing Wave Modes in a Superconducting Linear Accelerator, HEPL Report No. 437, High Energy Physics Laboratory, Stanford University (April 1966), p. 23. SMITH,T. I., 1968a, Low Temperatures and Linear Accelerators: I. Superconducting Linear AccelK. D., ed., Advances in Cryogenic Engineering 13 (Plenum Press, erators. In: TIMMERHAUS, New York) p. 102. SMITH,T . I., 1968b, private communication. TIGNER, M., 1967, IEEE Trans. Nucl. Sci. NS-14, no. 3, p. 241. H. E. and D. ALPERT,1967a, J. Vac. Sci. Technol. 4, 192. TOMASCHKE, TOMASCHKE, H. E. and D. ALPERT,1967b, J . Appl. Phys. 38, 881.

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TURNEAURE, J. P., 1968a, private communication. TURNEAURE, J. P., 1968b, Low Temperatures and Linear Accelerators: 11. Microwave Properties of Superconductors. In: TIMMERHAUS, K. D., ed., Advances in Cryogenic Engineering 13 (Plenum Press, New York) p. 109. J. P., 1968c, Microwave Measurements of the Surface Impedances of Superconducting Tin TURNEARE, and Lead. Ph.D. thesis, Department of Physics, Stanford University. Also available as Technical Report No. 507, High Energy Physics Laboratory, Stanford University. TURNEAURE, J. P. and I. WEISSMAN, 1968, J. Appl. Phys. 39, 4417. WEISSMAN, I. and J. P. TURNEAURE, 1968, Appl. Phys. Lett. 13, 390. WEAVER, J. N., T. I. SMITHand P. B. WILSON,1967, IEEE Trans. Nucl. Sci. NS-14, no. 3, p. 345. and P. B. WILSON,1966, A 200 MeV Superconducting Racetrack WIIK,B. H., H. A. SCHWETTMAN Microtron. In: GRILLI,M., ed., Proc. 5th Intern. Conf. High Energy Accelerators, Frascati, Italy, 1965 (Comitato Nazionale Energia Nucleare, Rome) p. 686. WILSON, P. B., 1967a, Excitation of Longitudinal and Deflection Mode Oscillations in a Superconducting Accelerator Structure by a Low Energy dc Beam. TN-67-6 (Technical Note) High Energy Physics Laboratory, Stanford University (July 1967). WILSON, P. B., 1967b, A Simple Analysis of Cumulative Beam Break-up for the Steady-state Case. TN-67-8 (Technical Note) High Energy Physics Laboratory, Stanford University (September 1967). WILSON, P. B., 1967c, Superconducting Accelerator Project at Stanford University: I. Design Problems. In: MACK,R. A., ed., Proc. 6th Intern. Conf. High Energy Accelerators, Cambridge, Mass., 1967 (Cambridge Electron Accelerator, Cambridge, Mass. Also available as Report CEAL-2000, Clearinghouse for Federal Technical and Scientific Information, U.S. Department of Commerce, Springfield, Va.) p. 378. 1965, IEEE Trans. Nucl. Sci. NS-12, no. 3, p. 1045. WILSON,P. B. and H. A. SCHWETTMAN, and W. M. FAIRBANK, 1964, Status of Research at Stanford WILSON,P. B., H. A. SCHWETTMAN University on Superconducting Electron Linacs. In: KOLOMENSKY, A. A., A. B. KUSNETSOV and A. N. LEBEDEV, eds., Proc. Intern. Conf. High Energy Accelerators, Dubna, U.S.S.R., 1963 (Atomizdat, Moscow) p. 535. WILSON, P. B., W. M. FAIRBANK, H. A. SCHWETTMAN, T. I. SMITH and J. P. TURNEAURE, 1966, Applications of rf Superconductivity to Accelerators and Instrumentation. In: Proc. 1966 Intern. Conf. Instrumentation for High Energy Physics, Stanford Linear Accelerator Center, Stanford, California (Report No. C0NF.-660918, Clearinghouse for Federal Scientific and Technical Information, U.S. Department of Commerce, Springfield, Va.) p. 236.

SULA

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Superconducting Proton Linear Accelerators and Particle Separators A. CITRON H. SCHOPPER Znstitur fur Experinzentelle Kernphysik der Universiriit ( T H ) und des Kernforschungszentrums Karlsruhe, Germany

1 Introduction The main advantages that can be gained by applying the superconducting technique to proton linear accelerators are the same as for electron linear accelerators. The drastic reduction of rf losses in the walls of the structure makes it possible to produce a continuous wave beam (duty cycle= 1) and to obtain comparatively high energy gains per unit length (310 MeV/m). Transient effects should be suppressed considerably by the continuous beam operation which might result in a very good energy homogeneity. On the other hand a superconducting linear accelerator is also a very adequate device for time-of-flight measurements. This stems from the fact that because of the small losses the characteristic times for a field change in the structure are of the order of several msec. Therefore the large rf energy stored in the cavities acts as a buffer between the pulsed beam on one side and the rf source and the cooling system on the other side. The latter two have only to be designed for the average power of the beam, whereas they do not "notice" its actual time structure. Conventional proton linear accelerators differ from electron accelerators in the operation mode, since protons are usually accelerated in standing waves whereas electron linacs use travelling waves. In comparing these two modes of operation, it is 0 North-Holland Publ. Co., Amsterdam

References p. 1 1 79

1969

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often stated that travelling wave acceleration is twice as efficient as standing wave operation for the same peak field. It is argued that the component of the standing wave which travels in the direction opposite to that of the particle does not contribute to the acceleration, but doubles the peak fields. As already mentioned in chapter E.2, tj 2.1, this is not generally true. It holds only when the particle is subject to the action of the field most of the way; i.e. in the absence of drift tubes. Proton accelerators usually have drift tubes. In the limit of long drift tubes and short gaps the particle is exposed to the field only for a short moment and does not notice in which direction the field travels. Quantitatively, the energy gain in the travelling wave case (W,,) and in the standing wave case (W,,) are, for optimum phasing and 7-c mode operation, in the ratio (see also chapter C .1. le) R = -W W W,,

(29124 sin (g12A) cos (gI2A)

= -

-

+ (g12iZ)

( A = 427~).

A structure without any drift tubes corresponds to g =+A (g =gap length, A =wave R length of the mode in the structure) which gives R = 2. On the other hand for g Teissier G . 1087 CHADWICK C. +Ballam J. 412 CHAMBERS E.E. 1 130, 1 135, 1139 t s c h w e t t m a n H.A. 113, 1106, I139 CHAMBERS R.G. 1092, 1105 CHASMAN R. 800,801,804,824,836 -+Batchelor K. 824 -+Cluckstern R. 745 CHEHABR . 257,263, 378,383 +Borie H. 382 CHEKHLOV K.V. +Vaghin V.A. 583 CHENCG . H . H . 185,218,219 CHERETAKIS A. +VoSicki B. 874, 1009 CHERNYAK L.L. +Bolotin L.I. 1070 CHIAO-MIN CHU 725, 726, 734 CHICK D.R. 728, 735 CHIDLEY B.G. 1015, I023 CHMARA F. +Grodzins L. 1044

CHOCHLOV V.K. +Val'ter A.K. 221 -tKhokhlov V.K. CHODOROW M. 8, 10, 17, 39,50,91, 100, 110, 111,120, 127,145,266,276,312,320, 335, 354, 385,412,459, 469, 667, 668, 674, 676,681, 682, 685, 689, 690, 726, 735 CHRISTOPHILOS N.C. 591,599 C H UE.L. 7 , 17, 22, 37, 39,44, 87, I l l , 116, 145, 185, 188,217, 219,266, 267,269,312, 726, 735 +Ayers W.R. 469 CHURCHT.G. 101 1 , 1018, I023 CHURILOV G.Y. +Schwettman H.A. 113, 1139 CITRONA. 1 105, 1105, 1 141 CLARKA.F. 101 1, 1023 CLARKH.H. 362,383 CLAUSJ. 823,825 CLAYR.E. +Mullett L.M. 933 CLELAND M.R. 860,876 COATESW.M. 4,17 COCHRANE D.R.F. 1019,1023 COHNC.1. 47, 111 COLLINSL.E. 847,873,944, 969 COLLINS S.C. 1 130, 1139 CONVISER S.R. 369,383 COPENHAGEN D.R. 690, 735 CORKB. +Allison R.W. 874 COOPERL.N. +Bardeen J. 1105, 1139 COURANT E.D. 134, 145,218,219, 786, 792, 795,800,801,802,804, 1025, 1028 COWARDD. 401,412 C o x J . 393,412 CRACGJ.D. 861,876 CRAIGR.A. +Chodorow M. 735 CRANBERG L. 864, 865,876,877 CRANDALL K.R. 466,469,835,836,983,985 CRAWFORD D.F. 528, 529,544,551 CROUZETJ. 720,723, 735 CROWLEY-MILLING M.C. 173, 176,219, 220, 495,499, 514, 522, 574,582 +Jarvis T.R. 220,583 CURNOWH.J. 687, 735 CURTISC.D. 852, 870,874,878, 1002, 1009 +Fasolo J.A. 874 CUSICKJ.H. 484,485,493 D A H L H .21,37 +Johnsen K. 736 DAHLP.F. 88, 111 DAMMR. 859,875 +Sluyters Th.J.M. 969 DANZERM. 726,728, 735, 1066,1071 DARDENNE G . 362,383 DATZS. +Moak C.D. 1045 DAUGHERTY J.D. 1033, 1044 DAVIERM. 409,412 DAVIESA.J. 922, 932

AUTHOR INDEX DAVIES D.E. 915,932 DAVIN J.M. +Brooks F.P. 469 DAVISR.C. -+Morgan O.B. 1023 DAVYDOV M.S. 581,582 DAYE.A. 12,17 DEANN.W. +Batchelor K. 996 DEAVER B.S. +Taimuty S.I. 235 DEBRAY P. 1057,1071 DEDRICK K.G. 362,383 DEFOREST T. 388,413 DE GENNES P.G. 1094,1105 DELEZENNE E. +Berg&re R. 234 DELUCA W.H. 941,969 DELMON M. +Leboutet H. 112,469 DEMIRKHANOV R.A. 847,873 DEMOS P.T. 110,111,274,312,423,469,499, 522 DEMYANENKO G.K. +Artemov V.J. 219 DE RAADB. 134,146,578,582 DERADO I. +Davier M. 412 DESCHAMPS G.A. 100,111 DE STAEBLER H. 362,383,524,528,529,551 +Brechna H. 145 DEVINN.R. +Metropolis 969 DEVITOB. +Otis A. 969 DEVONS S. +Nassibian G. 17, 737, 1071, 1087 DE VRIESC. 52, 111 DICKSON J.M. 920,987,992,993,996 +Batchelor K. 996 +Came A. 9% DOBROLYBOV J.I. +Artemov V.J. 219 +Val'ter A.K. 221 DOBSON J. +Williams C.B. 354 DOEHLER 0. 340,354 DOLBILOV G.V. +Veksler V.I. 1024 D ~ MG. E 118,145,246,263, 597,599,634, 635, 635, 637, 644, 645, 648, 650, 652, 663, 670, 671, 674,693, 694,696, 697,699, 700,704, 709,725, 726,728, 729, 731,733, 735, 742, 745,882,894 +Carrie A. 599, 734,824 DONETS E.D. 1033,1044 DORFAN D. 410,412 DORFMAN L.G. +Davydov M.S. 582 DOVBUSH A S . +Artemov V.J. 219 DRELLS. 386, 393,412 DRICKEY D. +Davier M. 412 DUBOIS R. 852, 859, 866,874,875,877 +Huguenin J. 878 DUNNP.D. 596,600, 667, 668, 673-675, 677, 681, 735 +Adlam J.H. 733 DUPUISY. 822,826,9 12,9 1 3,932,933,98 1 , 982,985 DWORK L.E. 925,932

EDWARDS T.W. 620,627,635 EHLERS K.W. +Anderson C.E. 1044,1070 +Hubbard E.L. 17,1044, 1071 EKIMOV V.V. +Davydov M S . 582 EL BEZR. -+Huguenin J. 878 ELDREDGE A.L. 265 +Borghi R.P. 111,312 +Brown K.L. 219,353,582 ELKIND M.M. 852,874 EMIGHC.R. 849,851,874,983,985 +Brolley J.E. 1179 +Knapp E.A. 469,616, 736,825 ENGELL.N. -+Knapp E.A. 469,616, 736,825 ENSTROM J. +Dorfan D. 412 EPPLINGF.J. +Michael I. 875 EPSZTEIN B. 169, 170,172, 674,680, 735 EREMENKO E.V. -+Val'ter A.K. 113,221,413 ERMANOV S.M. +Mal'tsev A.P. 1087 +Teplyakov V.A. 737,1087 ESCHELBACHER H. 606,612,616,687, 735, 1143,1179 -+Bauer W. 734 EVANS D.M. +Allison R.W. 969,976 EVERHART T.E. +Birdsall C.K. 734 FABER T.E. 1095, I 105 FAINBERG I.B. 1079, 1087 FAIRBANK W.M. 1109, 1139 +Pierce J.M. 1139 +Schwettman H.A. 113,1106,1139 +Smith T.I. 113,616,826,1106 +Wilson P.B. 113,1140 FALNES J. 576,582 FARINHOLT E.V. 201,220 +Altenmueller O.H. 219,469 FARKAS Z.D. 444,469 +Altenmueller O.H. 219,469 FARRELL P. 860,876 FASOLO J.A. 849,874 +Curtis C.D. 874,878, 1009 FAURE J. 851, 852,872,874,878 +Bernard P. 878,985 +Vienet R. 873 FEATHERSTONE R.P. 858,875 +Day E.A. 17 FECHTER H.R. +Hofstadter R. 412 FEDOTOV A.P. +Karetnikov D.V. 873, 1087 +Murin B.P. 736 FELDB.T. +Janes G.S. 1044 FELICIN.J. 860,876 FERF. 1081, 1087, 1150, 1179 FERLENGHI E. 531,537, 538,551 FESTER H.R. +Tautfest G.W. 235

AUTHOR INDEX FEWELL N. 592,599 +Came A. 734,824 FILIPPOV E.I. +Kulikov O.F. 235 FILLMORE F.L. +Turner C.M. 874,876 FINKELSTEIN 1.1. +Davydov M.S. 582 FINNEMORE D.K. 1 131,1139 FLECHER P. 1101,1 105,1105 FLESHER G.T. 47,111 FOELSCHE H.W.J. 570,581,582 FOMINL.P. +Levin V.M. 220 FONTANA J.R. 1132,1139 FORSYTHE G.E. 621,628,635 FORTESCUE R.L. 864,876 FORTUNE R.D. -.Bell M. 582 ->Bramham P. 582 FOSTERE. H. 1180 -+Hahn H. 1105 FOSTERR. 842,873 FOWLER R.G. 866,878 Fox A.G. 343,354 FRANCK J.V. -.Alvarez L.W. 17,783 FRANK N.H. 31,37 FRANKEL R. 975,976 FRANZKE B. 1035,1044 -+Anger1 N. 1044 -.Moiler A. 1045 FRASER J.S. 1016,1023 FRIEDLANDER G. -.Metropolis 969 FRIEHMELT R. +Blasche K. 1070 FRIESD. +Davier M. 412 FROHLICH H . 554,555,566,847,873 eDemirkhanov R.A. 873 F R YD.W. 30,37,39,111,273,312 FULMER C.B. 958,969 -.Martin J.A. 1023 -Toth K.S. 969 GABILLARD R. 576,582 GABOVIC M.D. 839,873 GALEJS A. 852,874 GALLAGHER W.J. 91,100,105,111,293,312 -Ayers W.R. 469 GALLAGHER-DAGGITT G.E. 927,932,990,997 -Batchelor K. 932,996 +Webb J.A. 895 GALLIR. 864,877 GARAULT Y. 178,220,575,582 GARDNER J.W. 134,145 GARFUNKEL M.P. 1/05 GAROLLA G. 536,551 GAUTHERIN G . 850,874 tSeptier A. 874 GAVET J.C. +Bres M. 1070 GAVIN B. 1053,1071 GENDEL Y.G. +Teplyakov V.A. 1087

GENTHER W. 1033,1044 GERASIMOV V.P. +Abroyan M.A. 873 GHIROSO A. 1053,1054,1071 GIEBLER R.H. 1021,1023 GILBERT E.V. 633,635 GINZBURG V.L. 1101,1/05 GINZTON E.L. 7,10,17,91,100,112,266,274, 1112,11 19,1139 312,421,469,907,932, +Chodorow M. 17,111,145,312,354,412,469 G~ORDANO S. 36,37,71, 112,597,600,606,612,

616,672,674,678,679,680,687,691,695, 701,717,735,736,812,822,823,825,826, 1143,1179 +Dahl P.F. I11 +Nishikawa T. 112,220,616, 737,826,1180 GLEICHLAUF P.H. 857,875 CLUCKSTERN R.L. 191-193,196,216,220,591, 600,742,745,789,792,796,797-801,803, 828,836,1020, 804,s10,81 3,824,825,827, 1023,1077,1087,1 1 59,1160,1 1 63,1180 +Hubbard E.L. 17,1044,1071 GOBBET R.H. +Collins L.E. 873 GOEJ.D. +Alvarez L.W. 783 GOLDEH. lllI,lI39 GOLDSTEIN CH. +Bieth C. 734,1070,1086 GOODM.L. 1079,1087 GORBACHEV S.K. +Pasyuk A S . 1045 GORDON H.S. 879,894 -tAlvarez L.W. 17,783,874 +Hubbard E.L. 17,1044,1071 GORKOV L.P. -.Abrikosov A.A. 1105,1138 GOROHOVATSKY P.S. +Artemov V.J. 219 GOROSHOVATSKY F.A. -+Va13terA.K. 221 GOUTTEFANGEAS M. -*Bernard P. 985 Gow J.D. 842,873 +Alvarez L.W. 17 +Turner C.M. 874,876 GOWARD F.K. 26,37 GRAND P. 886,893,894,956,969 GREEN G.801,804 GRIEMM.L. +Carpender J.W.J. 493 GRIFFY T.A. +Janssens T. 412 GRINDBERG A.P. 554,566 GRISHAEV I.A. +Artemov V.J. 219 +Val'ter A.K. 113,221,413 +Vishnyakov V.A. 221 GRIVET P. -.Biquard F. 1105 GRIZHKO V.M. +Vishnyakov V.A. 221 GRODZINS L. 1035,1037,1044 +Daugherty J.D. 1044 GRUNDINGER U. 1021,1044 GU~NAR P.D25 1,264 GUIRAGOSSIAN Z. +Ballam J. 412 GUMOWSKI 1. 24,37 GUSTAVSON D. +Anderson R. 412

AUTHOR INDEX GUTKIN T.I. +Demirkhanov R.A. 873

HENDERSON C. 556,567 HENDRICHS R.H. 858,875 HADDEN R.J.B. +Mullett L.M. 933 HENOCH B.T. 574,575,583 HADINGER G. +Chabert A. 734,1070 HENRY-BEZY G. +Doehler 0. 354 HADJOUT J.P. +Chabert A. 1071 HENSHALL 865,877 HAEBERLI W. 993,997 HERBR.G. 864,877 HAGERMAN D.C. 914,929,932 +Michael I. 875 HAHN,H. 173,178,180,220,570,575,581,582, HEREWARD H.G. 19, 26, 34, 37, 576, 583, 672,

1099,1103,1104,1105, ll45,ll72,ll74,ll77, 736,794,795,810,825,977,981,985 +Bell M. 582 1178,1180 +Foekche H.W.J. 582 HERM~NGHAUS H. +Danzer M. 735,1071 HAHNW.C. 40,112 HAIMSONJ. 110,112,119,145,158,172,173,220, HERRMANNSFELDT W.B. 141, 143,146,312,312, 371,383,412,413 244,246,250,251,253,256,257,263,277, 312, 330,354,415,417,421,422,440,453,460,462, +Altenmueller O.H. 219,469 F.F. +Henderson C. 567 464,466,469,484,485,493,524,528, 529,531, HEYMANN HIETSCHOLD R. 537,551,1130,1139 +Flecher P. 1105 +Amman F. 550 +Bertozzi W. Ill,219,469 +Halbritter J. 1105 HAISSINSKI J. 524,528,529,531,537,551 HILAIRE A. +Bernard P. 985 HALAMA H.J. 173,178,180,220,570,575,581, HILL 864,877 582,1099,1105, 1172, 1174,1177,1178,1180 HINEM.G.N. 11 67,1180 +Foekche H.W.J. 582 H~RAKAWA H . 190,220 +Hahn H. 1105 HIRATA S. +Tanaka J. 313 HALBACH K. +Main R.M. 1071 HIRELR. 246,264,576,583 HALBRITTER J. 1100-1 102,1105,1146 HOARD C.T. +Brechna H. 145 +Flecher P. 1105 HOBBIS L.C.W. 846,873 HANFORD R.N. 134,145 HODGER.I. 1016,1023 HANNA R.C. 948,969,990,997 HODGES T.A. 948,969,990,997 HANNWACKER J.P. 674,691,695,735,823,825 HOFMAN K. +Flecher P. 1105 +Dahl P.F. I11 HOFSTADTER R. 385-387,412 +Giordano S. 600,736,825 +Janssens T. 412 HANSEN J. +Chodorow M. 412 H0ccH.A. 315 H A N S E N W .7,10,17,39,87,111 ~. +Altenmueller O.H. 219,469 +Chodorow M. 17,111,145,312,412,469 +Farinholt E.V. 220 +Ginzton E.L. 312 +Williams C.B. 354 HANSON A.O. 1 137,1139 HO~SINGTON R.W.R. 1056,1071 HARDER D. +Kretschko J. 235 +Nassibian G. 17,737,1071,1087 HARDT W. 985,985 HORNSBY J.A. 627,635, 852,874 HARRIS N.W. 677,689, 736 HORTIGG. 1029,1044 HARRISON E.R. 843,873 +Betz H.-D. 1044 HARTWIG E. +Allison R.W. 976 HOWELL L.N. +Martin J.A. 1023 HARVEY A.F. 673,736 HOWRY S.K. 975,976 HATCHA.J.916,932 +Moore C.H. 146,567 HAUGHIAN J.M. 1054,1071 HOYTH.C. 88,112,610,616, 621,622,623, +Main R.M. 1071,1087 635,717,736,1003,1009,1127,1139, 1143. HECKMAN H.H. 1036,1044 1180 H E W F. ll63,lI80 Hu M. +Howry S.K. 976 +Mittag K. 1180 HUBBARD E.L. 14,17,588,600,1038,1044, HELMR.H.87,112,115,132,134,137,143,145, 1053,1071 146,163,164,166,169,171,172,173,197,199, +Ghiroso A. 1071 200,201,203,220,390,412,454,469,526,531, +Heckman H.H. 1044 532,535,539,542,547,551 HUBER P. +Galli R. 877 +Brechna H. 145 HUDG~NS A.J. +Turner C.M. 874,876 +Brown K.L. 219,353,5Y2 HUDSON E.D. +Martin J.A. 1023 +Farinholt E.V. 220 HUGHES E.B. +Janssens T. 412

AUTHOR INDEX

HUGUENIN J. 837, 852, 866, 867, 872,874,877, 8 78 HYNEMAN R.F. 190, 196,197,219 I I N OY. +lkeda S. 312 S. 281,312 IKEDA -Tanaka J. 313 ILLGEN J. 1032, 1044 ILYNSCHENKO V.I. +Donets E.D. 1044 INAL E.K. 864, 865,877 ISABELLE D.B. 224, 228,235, 377,383, 388,412 ISING G . 1, 2, 17, 880,894 ISMAIL H. +Septier A. 874 ISRAEL H.I. +Cochrane D.R.F. 1023 IVANOV G.A. +Veksler V.1. 1024 IVANOV (3.1. +Artemov V.J. 219 IVANOV I.F. 944,969 IVANOV I.N. +Veksler V.I. 1024 IVANOV N.F. +Abroyan M.A. 873 IVLEV A.V. +Abroyan M.A. 873 JACOBS K. +Cluckstern R.L. 220 JAKOBSON M. 795, 795 JAMES C.R. 594,600 JAMESON R.A. 665, 736, 810,825,913,925,932 --Putnam T. 976 JAMNIK D. +Hanson A.O. 1139 JANES G.S. 1029, 1044 +Daugherty J.D. 1044 T. 387.412 JANSSENS JARVIS T.R. 178; 179, 188, 189,211,220, 575, 583 Xrowley-Milling M.C. 219,522,582 -Saxon G. 221,583 JASBERG J.H. +Brown K.L. 219,353,582 JEANR. 279,290, 306,312,313,423,469 JELLETTK.+Dahl P.F. I l l R.E. +Henderson C. 567 JI:NNINGS JEONGD.H. 269,312 JEWSON M.R. -+Turner C.M. 874,876

pHereward H.G. 985 JOHNSON C.D. 941,969 JOHNSON J. +Anderson R. 412 JOHNSON K. 228,235 JOHNSON L.H. +Day E.A. 17 JOHNSTON L.H. 381,383 JONESL.W. +Symon K.R. 146 JONES W. +Anderson R. 412 JOPSONR.C. +Clark A.F. 1023 J~~NGSTW 1171, . 1172, 1174, 1177, 1179,1180 -+Carrie A. 734,824

KAMERLINCH ONNESH. 1089, 1106 KANEJ.V. 865,877 KAPCHINSKY I.M. 743, 745, 786,789,790, 795, 831, 832,836,982,985 KAPITZA S.P. 554,567 KARETNIKOV D.V. 1077, 1087 +Keretnikov D.V. 873 KARZMARK C.J. 330,354 KASPAR K. +Blasche K. 1070 KATZA. 617,627,635 +Bernard P. 985 KATZL. 440,469 KAWASHIMA M. +Tanaka J. 313 KEANE J.T. 812, 815,825,914, 932 +Giordano S. 825 KEARB.K. 1 34, 145 KEILE. 578, 582, 1180 +Bell M. 582 +Bramham P. 582 KEITH-WALKER D.G. +Chick D.R. 735 KELLEY G.G. +Morgan O.B. 1023 KELLIHER M.G. 58, 111, 176,220,273,312, 574, 583 KELLOG E.M. 864,877 KEMKE D. 839,873 P. +Main R.M. 1071 KENNEDY KENNEDY W.R. 7, 10, 17 +Ginzton E.L. 312 KERETNIKOV D.V. 839, 852,873 -.Karetnikov D.V. 1087 KERSTD.W. +Symon K.R. 146 K.N. 1032, 1045 KERVALIDZE KHALATNIKOV I.M. 1089,1105 +Abrikosov A.A. 1105, 1138 KHOKLOV V.K. +Va13ter A.K. 113,413 -.Chochlov V.K. KHOROSHKOV V.S. -.Lapitsky Yu.Ya. 873 J.D. 962, 969 KIESLING KILERT A. -*Ballam J. 412 KILPATRICK W.D. 598,600,920,932 KINGN.M. 1077, 1087 KINGR.C.M. 677, 736 KINOG.S. 674, 681-684, 687, 733, 734 K I PA.F. +Demos P.T. 111,312,469 KIRSTEIN P.T. 852, 874 KITCHEN S.W. 592, 600 KITCHING P. 628,636 KLEINH. 726, 728, 736, 1066, 1071 i D a n z e r M. 735, 1071 KLEINP. +Ballam J. 412 KLIPPERT T. +Johnson K . 235 KNAPPB.C. 313,679,680,681, 736, 808,825 +Knapp E.A. 616, 736 +Nagle D.E. 112,616. 737,826, 933, 1009, 1139 KNAPP E.A. 35,37,71,112,313,423,424,469,

AUTHOR INDEX 597,600, 601,606, 608,609,616, 672-674, 678,680,690,717, 736, 809, 812, 823, 825, 826, 897, 1013,1023 +Copenhagen D.R. 735 +Knapp B.C. 736,825 +Nagle D.E. 112,616, 737,826,933,1009, 1139 +Swenson D.A. 616, 737,826,1009,1023 KNECHT D.J. +Michael I. 875 KNEISEL P. +Halbritter J. 1105 KNIPPJ. 1036,1044 KNOTTM. 974,976 KNOXW.J. +Hubbard E.L. 17,1044,1071 KOBAYASHI K. +Tanaka J. 313 KOBEZSKY V.M. +Artemov V.J. 219 KOFOID856,875 KOLODY D. +Main R.M. 1071 KOLOMENSKII A.A. 556,557,567 K o ~ o s o vV.I. +Va13ter A.K. 221 KOMAR E.G. +Val'ter A.K. 221 KOMAROV V.L. +Abroyan M.A. 873 KONDO K. +Tanaka J. 313 KONDRATENKO V.V. +Artemov V.J. 219 +Val'ter A.K. 113, 221, 413 KOONTZ R.F. +Altenmueller O.H. 219,469 +Farinholt E.V. 220 +Miller R.H. 146,567 KOROZA V.I. 190,221 +Voskresensky G.V. 221 KOSHKAREV D. 978,985 KOVALENKO N.A. +Val'ter A.K. 221 KOVARICK V.J. 868,878 +Sluyters Th. J.M. 1023 KOZHUKHOV I.V. +Veksler V.I. I024 U. +Polyakov B.I. 933 KRACHT KRSNOVSKII V.I. +Teplyakov V.A. 1087 KREINICK D. +Anderson R. 412 KRETSCHKO J. 224,235 KRIEGLER F. 627,635 KROLLN.M. 407,412 KRONROD A. 832,836 KRUPPH. 1032,1044 KRUSEC.J. 355, 357, 358, 363, 383 +Altenmueller O.H. 219,469 +Farkas Z.D. 469 K ~ ~ W. H N1149,1180 KULIKOV O.F. 225,235 KULMAN V.G. +Andreev V.G. 734 +Murin B.P. 736 KULYGIN Y.F. +Bolotin L.I. 1070 KUNTZE M. 11 62,1180 +Bauer W. 734 +Mittag K. 1180

KUROKAWA K. 665, 736 KURSANOV U.V. +Demirkhanov R.A. 873 KUSNETSOV S.B. +Veksler V.I. 1024 KUZNETSOV A.B. +Vaghin V.A. 583 KUZNETSOV G.F. +Val'ter A.K. 113,221,413 KUZNETSOV V.S. +Abroyan M.A. 873 KVARTSKAVA 1.F. 1032,1045 KVASKA A.I. 930,933 KYHLR.L. 105,112 +Chodorow M. 17,111,145,312,412,469 LACHJ. +Foekche H.W.J. 582 LAKINW. +COX J. 412 LAMB W.A.S. +Clark A.F. 1023 LAMPIE.E. +Day E.A. 17 LANCASTER H. 975,976 +Allison R.W. 976 LANDAU L.D. 1101,1105 LANZL L.H. +Carpender J.W.J. 493 +Skaggs L.S. 493 LAPITSKY Yu.YA. 842,873 LAPOSTOLLE P.M. 34,37, 246,247,264, 574, 583, 585,588,599, 635,635,720, 736,739, 742, 744, 745, 747, 756, 763, 769, 771, 773, 783, 789, 796,805, 810-812,817,825,828, 834,836,977,984,985, 1065,1071, 1080, 1087 +Came A. 37,783 +Dame G. 735 +Fer F. 1087,1179 +Hereward H.G. 736,985 LARSEN R.R. 576,581,583 +Altenmueller O.H. 219 +Ballam J. 412 LARSEN R.S. 229,235 LARSON R. +Otis A. 969 LASLETT L.J. +Symon K.R. 146 LATMAN~ZOVA G.M. +Abroyan M.A. 873 LAWRENCE, E.O. 4,17,709, 737, 880,895 LAWTON, E.J. 40,112 LAZAREV, N. +Kapchinsky I. 745 P. +Bernard P. 582,1179 LAZEYRAS, LEBACQZ J.V. 3 I5,346,354,918,933,102l, 1023 +Brown K.L. 219,353,582 LEBOUTET K. 39,52,112,203,216,220,279, 313,416,441,469,471 LEED.A. +Alpert D. 1138 LEEG.M. 870,878 +Curtis C.D. 874,878,1009 +Fasolo J.A. 874 LEEM. 819,825,913,933 +Gluckstern R.L. 745 +Herrmannsfeldt W.B. 312 +Williams C.B. 354

AUTHOR INDEX LEE-WHITING G.E. 1020,1023 LEFEBVRE J.M. 891,894 -Bernard P. 985 LEHMANN P. 387,412 LEISCHNER E. 1035,1036,1037,1044 -Bet2 H.-D. 1044 LEISSJ.E. 110, 112, 147, 152, 155, 158, 163, 164,169, 170, 171,172,247,264,355,357, 358, 363,383,422,469,812,825 LEITHD.W.G.S. +Ballam J. 412 LENGELER H. 1167, 1179,1179,1180 +Bernard P. 582, 1179 -tBramham P. 582 LEVIN V.M. 216,220 -Davydov M S . 582 -tVal'ter A.K. 113, 221,413 LEVYA. +Ballam J. 412 LEVYR.H. +Daugherty J.D. 1044 +Janes G.S. 1044 LEWISL. 974,976 LICHTENBERC A.J. 5 12,522 LINDENBAUM S.J. 955,969 LINDSAY N.A. +Jameson R.A. 736,825 LIOTR. 367, 369,383 LISINA.V. 265, 312,313 LIVDAHL P.V. 822,826, 881,886, 889,894,975, 976 LIVINGOOD J.J. 789, 796, 889,894 LIVINGSTON M.S. 889,894,916,919,933,957, 969 R.S. +Martin J.A. 1023 LIVINGSTON LLEWELLYN F.B. 320,354 LOACHB.G. 273,313 -Shersby-Harvie R.B.R. 737 LOBBD.E. 134, 146,531, 537,551 LOCKEY, R. +Otis A. 969 LOEWG.A. 39, 173, 199,220,355,357,358, 363,383,412,454,469, 576,583 +Altenmueller O.H. 219, 469 -+Borghi R.P. 111,312 -*Brown K.L. 219,353,582 -+Farinholt E.V. 220 +Farkas Z.D. 469 ->Helm R.H. 146,220 +Herrmannsfeldt W.B. 412 +Larsen R.R. 583 -+Williams C.B. 354 LONDONF. 1093, 1106 LONDON H. 1089, 1091, 1093,1106 H . +Chick D.R. 735 LONGLEY LORDR.S. +Martin J.A. 1023 LUCASG.J. +Knapp B.C. 736,825 Luccr F. 531,551 LUDLAM T. +Foelsche H.W.J. 582 LUNDQUIST D. +Prepost R. 413

LUTZH.O. +Moak C.D. 1045 LYMAN E.M. -+Alpert D. 1138 LYNTON E.A. lO96,lIO6 MACGREGOR M.H. 472,483,484,493 +Sund R.E. 551 MACHEN D. 975,976 +Allison R.W. 976 MACMASTER G.H. +Skowron J.F. 1023 MAINR.M. 1053, 1054, 1065, 1069,1071, 1073, 1087 +Ghiroso A. 1071 +Hubbard E.L. 17, 1044, 1071 A.I. +Teplyakov V.A. 1087 MAKAROV MAKHANKOV V.G. +Veksler V.I. 1024 MAKHNENKO L.A. 130, 132, 146 +Artemov V.J. 219 +Vishnyakov V.A. 221 MALKIN M.S. +Hubbard E.L. 17,1044, 1071 MALLORY K.B. 100,112,276,313,381,383 MALTHOUSE H.F. +Huguenin J. 877 +Warner D.J. 895 MAL'TSEV A.P. 1082, 1087 -Teplyakov V.A. 737 MALWITZ E. +Bohne D. 1070 I.F. MALYSHEV +Davydov M S . 582 +Va13ter A.K. 113, 221, 413 MANGIN J.P. 247,254,264 +Epsztein B. 172 +Leboutet H. 112,469 MANGO,L. 531, 537, 538,551 MANNJ.E. +Martin J.A. I023 MARCOU J. 1081,1087 MARKH. 1011, 1023 MARSHALL L.C. +Alvarez L.W. 17, 783 MARTINF. +Cox J. 412 MARTINJ.A. 101 1, 1020,1023 MARTINIM. 629, 636, 753, 783 MARTIN-VEGUE C.A. 925,933 +Morris A.J. 1023 MASSEY H.S.W. +Bohm D. 873 MATTISD.C. 1089, 1097,1106 MAVROGENES G.S. 1032, 1033, 1044 MAXWELL E. 1089, 1106 MCASHANM S . 1103, 1106, 1123, 1124, 1130, 1133, 1136, 1139 +Schwettman H.A. 113,1106, 1139 MCCONUELL R.A. +Farkas Z.D. 469 MCDIARMID D.R. 674,675,684, 736 MCINTOSH L.R. +Buechner W.W. I l l +Robinson C.S. 877 MCINTYRE J.A. +Hofstadter R. 412 MCKIBBEN J.L. 857, 864, 865,875,877

AUTHOR INDEX MCKINNEY T.O. +Miller R.H. 146,469 MCMILLAN E.M. 19,21,37, 1074,1087 MECKLENBURG B. 119,145,466,469 MEEKJ.M. 861,876 MEISSNER K. 1089,1106 MELARD L. 110,112,180,220 MELEKHIN V.N. +Kapitza S.P. 567 MELLORS G.W. 1103,1106 MENKE M. +Ballam J. 412 MESSEL H. 528, 529, 544,551 METROPOLIS 953,969 MEUSER R.B. +Main R.M. 1071,1087 MICHAEL I. 857, 865,875,877 MIERSCH E. 1042, I044 MILLERC.W. 173,219,274,313 +Crowley-Milling M.C. 219,522,582 MILLERD. +Dorfan D. 412 MILLERJ. +Bernardini M. 551 +Metropolis 969 MILLERP.B. 1089, 1094, 1097,1106 MILLER R. 115, 119,124,146,249,250,258, 264,373,383,459,469,561,567, 1130,1139 +Altenmueller O.H. 219, 469 +Farinholt E.V. 220 MILLERV.E. +Nassibian G. 17, 737, 1071, 1087 MILTONJ.C.D. +Bartholomew G.A. 1023 MINKNER R. 876 MINTEN A. 578,582 MINTSA.L. 931,933 MITTAGK. ll63,ll8O +Kuntze M. 1180 MIYAHARA A. +Tanaka J. 313 MOAKC.D. 1034,1035,1045 MOLLER A. 1039,1045 +Angert N. 1044 +Franzke B. 1044 MONTAGUE B.W. 569, 571,575,578,579,583, 728, 736, 789, 796, 984,985,985 +Bell M. 582 MOOREC.H. 134,146,560,567 MOOREW.H. +Wheeler G.W. 969 MORGAN G.C. 920,933 MORGAN O.B. 1022,1023 M o ~ o zE.M. 554,567 MORRIS A.J. 925, 933, 101 8, 1023 MORTON P. 21 8,220, 832,836 M o s ~ S.W. o 1020, 1023 +Martin J.A. I023 MOTCHESHNIKOV N.I. +Val'ter A.K. 221 MOTZH. 225,235 MOZLEY R. +Brown K.L. 219,353,582 +Davier M. 412 MUELLER D.W.

+Brolley J.E. 1179 +Cochrane D.R.F. 1023 MUFELV.B. +Val'ter A.K. 221 MUHLSCHLECEL B. 1098, I106 MUKHIN S.V. +Vaghin V.A. 583 M ~ ~ L LW. E R1066,1071 MULLETT L.B. 56, 113, 132, 146,273,313 + R y e D.W. 312 +Shersby-Harvie R.B.R. 737 MULLETT L.M. 91 6, 933 MURDOCH J.C. 703,704, 737 MURINB.P. 714, 736,913,930, 933 +Andreev V.G. 734 MURNICK D. +Grodzins L. 1044 MURPHY F. +Anderson R. 412 MURRAY J.J. 397,412 MYAKOTA V.I. +Artemov V.J. 219 MYFELV.B. +Artemov V.J. 219 NAGLED.E.70, 88, 112, 199,220,601, 604,616, 662, 665, 667, 670, 681,717, 736, 809, 823, 826, 891,894,914,933,999, 1002, 1009, 1126, 1139, 1143, 1180 +Cluckstern R.L. 220,1023 +Knapp B.C. 825 NAKAMURA M. 421,469 NALOSE.J. +Chodorow M. 735 NASSIBIAN G. 5, 17,710, 737, 1056, 1071, 1078, 1087 NEALR.B.39,63,78, 80, 110, 112, 154,172, 173, 208, 220, 266, 267, 276, 313, 316, 317, 322, 324,354, 378, 383, 385, 388,400,412, 418,447,469, 1021, 1023 +Borghi R.P. I l l , 312 +Brown K.L. 219,353,582 +Chodorow M. 17, I l l , 145,312,412,469 +Herrmannsfeldt W.B. 412 NEETD. +Howry S. 976 NEILSEN I.R. +Chodorow M. 17,354 NEVINS JR. J.E. 727, 737 NEVYAZHSKY 1.c~. +Andreev V.G. 734 +Murin B.P. 736 NEWHALL H.F. 838,873 NEWTON T. 832,836,836 NGUYEN TUONGVIET 1102-1 104,1106 NIELV.K. +Morton P.L. 220 NIKOLAEV V.S. 1035, 1039, 1041,1045 NILSSON 0. 723, 737 NISHIKAWA T. 88,112, 194, 199,220,280,313, 606,616, 662,665, 674, 681, 704, 737,798, 803,804, 809, 810,813-821, 823, 824,826, 828-830,836,913,914,933, 1148, 1180 +Batchelor K. 734,824 +Carter D. 734 +Tanaka J. 313

AUTHOR INDEX

NORCLIFFE S. 992, 997 -+Batchelor K. 996 +Carrie A. 996 NORDBY D. +Knott M. 976 NORRISN.I. +Hubbard E.L. 1044 +Sund R.E. 551 NORTHCLIFFE L.C. +Michael I. 875 +Moak C.D. 1045 NUNANC.S. 110, 112 +Amman F. 550 NYCARD J.C. 176,220, 574,583 +Robinson C.S. 877 OCHSENFELD R. 1089, 1091-1093, I106 ODIANA. +Davier M. 412 OHNUMA S. 795, 796, 799, 800,804, 831,832, 836 OKABE S. 224,235 OKUMURA S. 803,804,829,836 OLEKSUIK L. +Taylor C.S. 986 O'MEARAJ.E. 891, 892,895 ONOSOVSKY K.K. +Lapitsky Yu.Ya. 873 F. +Alvarez L.W. 17, 783 OPPENHEIMER OTISA. 944,948, 969 +Sluyters Th.J.M. 969 OTSUBO Y. +lkeda S. 312 OUTRAM S. +Walkinshaw W. 600, 738 O V A D I1.A -tBess L. 235 OVSYAMIKOV V.M. +Bolotin L.1. 1070 OWENC.W. 920,933 M. +Dorfan D. 412 PACIOTTI PACKK. +Kuntze M. 1180 PAKHOMOV V.1. +Makhnenko L.A. 146 PALLUEL P. +Doehler 0. 354 PANOFSKY W.K.H. 131, 132,146,173, 195,201, 202, 203, 208, 220, 378, 379, 383, 404,407, 412,447,469, 569, 570, 572-574,583, 747, 756, 783, 1087, 1168, 1180 +Alvarez L.W. 17, 783,874 -Brown K.L. 219,353,582 4Chodorow M. 17, 111,145,312,412,469 -.Helm R.H. 146,220 PAPASC.H. 665, 734 PARAIN J. 795, 796 PARZEN G. 593,600,634,636,691, 737 +Dahl, P.F. 111 PASYUK A.S. 1032, 1033, 1045 Passow C. 1 134, 1139 P A U L IA. N 554,567 PEARCE A.F. 674,677, 737 PENNING F.M. 841,873 PERELSTEIN E.A. +Veksler V.I. 1024 PERLM. +COX J. 412

PERRYR. 822,826, 889, 890,895 PETELIN I.G. +Davydov M.S. 582 PETERW. 859,875 R. +Hubbard E.L. 17, 1044,1071 PETERS PETERSON H.K. +Gluckstern R.L. 745 PETRENKO V.V. +Artemov V.J. 219 +Val'ter A.K. 221 PETRIED.P.R. +Chick D.R. 735 V.I. 91 6, 9 17, 933 PETRUNIN +Davydov M.S. 582 PETUKHOV V.A. +Vaghin V.A. 583 PICKUPE. +Ballam J. 412 PIERCEJ.M. llOl,1106, 1109, 1122,1139, 1145, 1180 +Fairbank W.M. 1139 +Schwettman H.A. 1106, 1139 PIERCE J.R. 127, 146, 720, 722, 737, 850, 853, 8 74 PINEJ. 388, 390,413, 523,551 PIPPARD A.B. 1093-1095, 1105, 1/06 PLACESM. +Borie H. 382 PLOTNIKOV V. +Kapchinsky 1. 745 PLYUTTO A.A. 1032, 1045 POHLIT W. +Kretschko J. 235 POLKL. 271,313, 883,895 POLYAKOV B.1. 931,933 PONOMARENKO B.A. -tArtemov V.J. 219 POPEJ.A. 277,313 P o ~ o vV.A. +Davydov M.S. 582 i V a g h i n V.A. 583 V.1. +Artemov V.J. 219 POPENKO POSTR.F. 7, 17, 176, 220, 266,275,313, 574, 583 POTTERJ.M. 313 +Copenhagen, D.R. 735 +Knapp B.C. 736,825 +Knapp E.A. 469,600,616, 736,825 +Swenson D.A. 616, 737,826, 1009, 1023 POTTIER J . 712, 737 POULAIN M. 884,895 F. +Septier A. 874 PRANGERE PREPOSTR. 387,413 +Anderson R. 412 PRESITD.H. 91 8,933 F. 856, 857,874,875 PRFVOT +Bruck H . 875 PRICEV.G. 265 PROMEM. 619,636,741,743, 745,747, 769, 771, 773,774, 783, 785,880,882,890,891,894,895, 979, 980, 985 +Bernard P. 985 +Came A. 37, 783 T. 971,976, 976 PUTNAM

AUTHOR INDEX

QUINND. -+Prepost R. 413 QUINTON A.R. +Hubbard E.L. 17,1044,1071 RADNER C.A. +Owen C.W. 933 RALMER W.J. +Johnson K. 235 +Mavrogenes G.S. 1044 RAMOS. 43,86,112 RASHEVSKY V.P. +Veksler V.I. 1024 RASTOIX G. +Bernard P. 985 RAYMOND D. +Dorfan D. 412 REDHEAD P.A. 1033,1045 REEDER T.M. 667,674,677,737 REGENSTREIF E. 33,37,789, 790,796,801,804, 886,888,895,977,985 REGINATO L.L. 860,876 REGLER M. 1025 REMBSER J. +Miiller W. 1071 RESHETNIKOVA K.A. +Veksler V.I. 1024 REUTER G.E.H. 1091,1106 REVUTSKY E.J. +Bolotin L.I. 894,1070 +Bomko A.V. 734,1070 RHEINHOLD G. 860,876 RICHM. 747,783 RICHW.F. +Hoyt H.C. 616,635,736,1139, I180 RICHARDSON Jr. E.G. +Martin J.A. 1023 RICHMAN C. +Alvarez L.W. 17,783,874 RICHTER B. 404,407,413 RICHTER R.M. +Allison R.W. 874,969 RITSOND. +Anderson R. 412 ROBERTS A. 554,555,557,567 ROBINSON C.S. 562,567,865,877 +Hanson A.O. 1139 ROBSON P.N. 91,112,421,470 RODGERS C.L. +Hanson A.O. 1139 ROGERS B.T. +Boyd T.J. 17,493 ROGERS E.J. 842,873 ROLAND S. +Berg&re R. 235 ROMMEL G. +Bernard P. 985 Roos M. 1038,1045 ROSEP.H. 852,874 +Grodzins L. 1044 +Roes M. 1045 ROSENL. 1000,1009 ROSENBLUTH M.N. 386,413 Ross M. +Walkinshaw W. 738 Ross1 B.B. 144,146 ROUBINE E. 719,737 Roux J. +Chabert A. 734,I070 ROYFEI.M. +Abroyan M.A. 873 RUBINN.B. +Vaghin V.A. 583 +Veksler V.I. 1024

RUBINS.B. +Veksler V.I. 1024 Ruesov K.S. +Artemov V.J. 219 RUHLAC. +Chabert A. 734,1070 RUMYANTSEV V.V. +Val'ter A.K. 113,221,413 RYLTSEV P.I. +Veksler V.I. 1024 SABEL C.S. +Dunn P.D. 735 +Walkinshaw W. 600,738 SALSBURG B.L. 143,146 SAMOILOV A. +Bernard P. 582 SANDERHOFF S. 1103, 1106 SANDWEISS J. +Foekche H.W.J. 582 SANGBOONAM1097,1100,1101,1106 SARANTSEV V.P. +Veksler V.I. 1024,1045 SARAZIN A. 11,17,73,113 SARCENT C.P. +Bertozzi W. 111,219,469 SAXON G.178,221,423,575,583 , +Crowley-Milling M.C. 219,522,582 +Jarvis T.R. 220,583 SCHELBERC A.D. +Kitchen S.W. 600 SCHIFFL. 385,413,554,567 SCHINZEL, D. 1148,1180 SCHMELZER CH. 915,921,922,933,1029,1047, 1062,1071 +Angert N. 1044 +Bet2 H.-D. 1044 +Blasche K. 1070 +Franzke B. 1044 +Moiler A. 1045 SCHNEIDER E.J. 703,737 +Knapp E.A. 825 +Swenson D.A. 616,737,826,1009,1023 SCHNEIDER H.R. 703,704,737 SCHNELL W. 570,572,583 SCHNIZER B. 747,751,758,761,764, 783 SCHOLL R. +Howry S.K. 976 SCHOPPER H. 606,612,616, 1141,1143,1167, 1171-1175,1179,I180 +Eschelbacher H. 735 +Halbritter J. 1105 SCHRACK R.A. +Leis J.E. 172,825 SCHRIBER S.O. 1020,1023 SCHRIEFFER J.R. 1095,1105 +Bardeen J. 1105,1139 SCHUHL C. +Bernardini M. 551 SCHULTHEIS T. +Putnam T. 976 SCHULTZ H.L. 10,17 SCHULZE D. ll52,Il53,ll80 SCHWARTZ M. +Dorfan D. 412 SCHWARZ L. +Hubbard E.L. 17,1044,1071 SCHWETTMAN H.A. 76,77,82,113,1097, 1098, 1101-1103,1106,1109, 1123-1125,1130,1132, 1134,1136,1139,1140 +Pierce J.M. 1139 +Smith T.I. 113,616,826,1106

AUTHOR INDEX -+Wiik B.H. 567,1140 -+Wilson P.B. 113,1140 SEDLAEEK M. 561,566 SEMENOV A.N. +Va17ter A.K. 113 SEMENYUSHKIN 1.N. +Vaghin V.A. 583 SENSIPER S. 720,723,727,737 SEPPIE. +Howry S.K. 976 SEPTIER A.L. 134,146,850,851,853,874, 1089,

1092,1/05 -+Biquard F. 1105 SEREBRYAKOV Y.N. 190,221 -+Voskresensky G.V. 221 SEVRANCKX R. +DBme G. 735 SESSLER A.M. 218,219,221 -+Morton P.L. 220 SEYBOTH P. +Ballam J. 412 SHARPJ.B. 936,969 SHAWH.J. 1132,1/39 SHEMBEL B.K. +Karetnikov D.V. 1087 -+Keretnikov D.V. 873 -+Teplyakov V.A. 737,1087 SHERSBY-HARVIE R.B.R. 47,56,113,266,273,

313,674,675, 737 -+Fry D.W. 312 SHERWOOD A.J. +Allison R.W. 969 SHIREN N.S. 7,17,275,313 SFILAER W.J. +Knapp E.A. 469,616,736,825 SIMMONDS D.D. -Hoyt H.C. 616,635,736,

1/39,1/80

SMITHT.I. 40,71,76,88,113,606,616,823,826,

llO5,llO6,1126-1128,1136,1139 i s c h w e t t m a n H.A. 113,1106,1139 +Weaver J.N. 113,826,1\40 SMITSR.G. +Kitchen S.W. 600 SMIRNOV V.L. +Levin V.M. 220 SNODDY L.B. 4,17 SNYDER H.S. 134, 145,786,792,795,800, 801,

802,804 SOFFER J. +Leboutet H. 112,469 SOKOLOVSKY A. 52,113,416,470 SOLINAS L. +Huguenin J. 877 SOLNYSHKOV A.I. 860,876 i A b r o y a n M.A. 873 -1vanov I.F. 969 SONDHEIMER E.H. 1091,1106 SONKIN S. +Chodorow M. 17 SPENCE D.A. +Ghiroso A. 1071 SPERDUTO A. +Buechner W.W. 111 SPRANZA J. iHerrmannsfeldt W.B. 312 STADLER B. 1073,1078 +Bet2 H.-D. 1044 STAFFORD H.G. 76,I11 STARKL. 720,722,723, 737 STEELE C.W. 243,264,421,470 STEFFEN K.G. 134,146 STEPANOV K.N. 143,146 +Makhnenko L.A. 146 STEPANOV S.B. +Teplyakov V.A. 737 STEPANYUK V.L. 4Vaghin V.A. 583 STEVENS R. +Gluckstern R. 804 STORMM. +Metropolis 969 STRATTON J.A. 749,783 STREETER M.H. +Collins S.C. 1139 STROMBERG T.F. +Finnemore D.K. 1139 STROUD P.T. 944,969

SIMON W.G. +Heckman H.H. 1044 SINELNIKOV K.D. 888,895 SIVKOV U.P. +lvanov I.F. 969 SKACCS L.S. 489,493 +Carpender J.W.J. 493 SKARPAAS K. 310, 313 +Callings L.E.873 SKOWRON J.F. 1021,1023 STUART R.W. +Collins S.C. 1139 SKUBKO V.A. +Artemov V.J. 219 L.R. 1159,1180 SLATER, J.C. 10,17,19,28,30,37, 39,71-73,87, SUELZLE 93,113,116,118,146,159,171,172,193,194, SUNDR.E. 524,537,551 V.A. +Bolotin L.I. 894,1070 221,266,274,313,421,470,665, 737,812,826, SUPRUNENKO SURAJ. 789,796 908,933 SUSSKIND C. 127,145 +Demos P.T. 111,312,469 -&hodorow M. 354 SLIVKOV 1.N. SWAING.R. 606,616,826 +Karetnikov D.V. 1087 SWALLOW A.J. 474,475,493 +Keretnikov D.V. 873 SWENSON D.A. 616,616,662, 667,674,701, 703, SLOAND.H. 4,17,709,737,880,895 704,737,747,756,758,771,772, 783,823,826, SLUYTERS TH. J.M.849,851,852, 861,868,874,

876,878,944,969,1022,1023,1025,1028

1002,1009,1020,1023

+Finnemore D.K. 1/39 SMITH B.H. 860,876,1053,1071 SMITHJ. + h t n a m T. 976 +Knapp E.A. 825 SWOBODAC.+Knott M. 976 SMITHJ.F. +Allison R.W. 874 SYMON K.R. 132,146 SMITHL. 22,37,789,792,796,801,804,819,826, SYMONDS J.L. 423,469 889,895,1025,1028,1077,1087,1150,1/80 SZECSIL. 1145,1146,1180 +Clark A.F. 1023 +Hubbard E.L. 17,1044,1071

AUTHOR INDEX

TABATA T. +Okabe S. 235 TAIMUTY S.I. 224,235 TALCOTT A.C. 918,933 TALLGREN U. 846,873 +Huguenin J . 878 TAMAS G. +Bernardini M. 551 TANT. +Ballam J. 412 +Cox J. 412 TANAKA J . 279,313 TANGUY P. +Regenstreif E. 804 TAUTFEST G.W. 224,235 +Brown K.L. 235 TAYLOR C.S. 822,826,879,905,913,933, 981-983,986 TAYLOR R. 27,37,628,636,747,753, 772, 783 798,804 +Lehmann P. 412 TAYLOR R.E. 404,407,413 TEICHMANN T. 683, 737 TEISSIER G. 1079, 1087 TELLER E. 1036,1044 TELNOV Y.V. +Kulikov O.F. 235 TENGL C . 789, 796,942,969 TEPLYAKOV V.A. 714,733, 738,1080, 1082, 1086,1087 +Karetnikov D.V. 1087 +Keretnikov D.V. 873 +Mal'tsev A.P. 1087 TERVELDL.K. 864,865,877 TERWILLIGER K.M. +Symon K.R. 146 TESCHE F.R. +Boyd T.J. 17,493 THOMPSON D.J. 677, 735 +Dunn P.D. 735 THONEMANN P.C. 843,873 THORNDAHL L. +Johnson C.D. 969 THORSON I.M. 1019,1023 TIENP.K. 723, 738 +Pierce J.R. 737 TIGNER M. 1 1 36,1139 TISHINV.G. +Zager B.A. 933 TOLMAN R.C. 26,37 TOMASCHKE H.E. 1132,1139 +Alpert D. 1138 TONERW. +COX J. 412 TOTHK.S. 958,969 +Martin J.A. 1023 TRAND.T. 789, 796 +Bres M. 1070 +Chabert A. 1071 +Epsztein B. 735 TRETIAKOV Yu.P. +Pasyuk A.S. 1045 TRIGGER K. +Herrmannsfeldt W.B. 312 TRONCD. 247,248,264 TROTTER H. 4,17 TRUEMPY K. 876

TRUMP J.G. +Van de Graaff R.J. 875 TRUSHIN N.F. +Davydov M.S. 582 TSANG D.D. +Miller R.H. 146,567 TUCKER E.B. +Day E.A. 12, 17 TUNNICLIFFE P. 101 1 , 1023 TURCHINETZ W . +Bertozzi W. 111, 219,469 TURKEVICH A. +Metropolis 969 TURKIN F.F. +Val'ter A.K. 113,221,413 TURNEAURE J.P. 1094,1097, 1101,1102,1104, 1106, 1109, 1121, 1127, 1131, 1133, 1136, 1140, 1149, I I80 +Schwettman H.A. Il3,IIO6,ll39 TURNER C.B. +Mavrogenes G.S. 1044 TURNER C.M. 856,860,864,865,874,876,877 +Bennet G.W. 873 +Rogers E.J. 873 TURNER T.F. +Brown K.L. 219,353,582 +Jameson R.A. 736,825 TWISTLETON J.R.J. 1021,1023 TZARA C. +Bernardini M. 551 UMANSKY 1.G. +Davydov M.S. 582 VAGHIN V.A. 581,583, 1179 +Bernard P. 582, 1/79 VALASSIS J. +Bess L. 235 VAL'TER A.K. 110,113,173,221, 363,383, 385,413 VANATTAC.M. +Clark A.F. 1023 +Hubbard E.L. 17, 1044, 1071 VANDE GKAAFFR.J. 857, 865,875,877 +Buechner W.W. 111 +Grodzins L. 1044 VANDERBEKEN H. 975,976 VANDE VIJVERR.E. +Vanhuyse V.I. 235 VANHUYSE v.1. 224,235 VANPRAET G.I. +Vanhuyse V.I. 235 VANSTEENBERGEN A. 842,848,873,935,942, 944,969 +Otis A. 969 +Taylor C.S. 986 VARIAN ASSOCIATES 250,264,537,551 VARIAN R.H. 6,17 VARIAN S. 6 , 17 VASTEL J . 1 1 , 17,73,113 VEKSLERV.~. 19,37,553,567,569,583, 1016, 1024,1029, 1045 +Vaghin V.A. 583 VENABLE D. +Boyd T.J. 17,493 VERMEULEN F. 983,985 VESSCHER W.M. 1020,1024 VETTER J. 1161,1180

AUTHOR INDEX

--Kuntze M. 1180 - M i t t a g K. 1180 VEYRON B. ~ B r e M. s 1070 FChabert A. 734, 1070, 1071 +Teissier G. I087 VEYSSIERE A. 368,382,428,469 Bergkre R. 111,234,235 VIENET R. 849,872,873,878 -.Bernard P. 878, 985 VILLAF . +Davier M. 412 G. +Huguenin J. 878 VISCONTI VISHNYAKOV V.A. 21 1,216,221 t A r t e m o v V.J. 219 VISSCHER W.M. 795, 795, 796 ->Cluckstern R.L. 220, 1023 VISWANATHAN R.V. 625,636 VLADIMJRSKY V. 83 1,836,982,985, 1082, 1087 VLASSOV A.D. 789, 796,832,836 F . 822,826, 86l,876,93 1, 933 VOELKER + G h i r o s o A. 1071 + H u b b a r d E.L. 17, 1044, 1071 VOGTE.W. -.-Bartholomew G.A. 1023 VOISING. -+Bres M. 1070 -+Chabert A. 734, 1070, 1071 VONARDENNE M. 839, 847,873 VOSICKIB. 837, 849, 85 1,874, 1002, 1009 +Huguenin J. 877 VOSKRESENSKY G.V. 190,201,221 Voss 507 +

WADEY W.G. 10, I7 WALECKA J.D. 388.413 WALKER G.B. 40,47, 113, 667, 673-675, 738 +James C.R. 600 +McDiarmid D.R. 736 WALKINSHAW W. 30, 37, 39, 111,574, 583, 590, 600, 629, 636, 718, 729, 730, 738, 1066, 1071 + D u n n P.D. 600 + F r y D.W. 312 +Shersby-Harvie R.B.R. 737 WALLACE R.W. 955,969 WALSHT.R. 942,969 J. 742, 745 WALTERS WALTON R.B. +Sund R.E. 551 WARNECKE R. 251,264 WARNER D.J. 592,599, 629, 636, 753, 783, 891, 895,947, 949,969,982,986,990,994,996,997 WASOWW.R. 621,628,635 WATANABE A. +Main R.M. 1071 E.D. iVanhuyse V.I. 235 WATTECAMPS WATKINS D.A. 44,87, 113 WEAVER J.N. 76, 113, 344, 354, 431, 444, 470, 823,826, 1 127, 1135, 1140 WEBBG.S. 893,895

+Carrie A. 996 WEIHRAUCH J. +Bet2 H.-D. 1044 WEINSTEIN R. +Anderson R. 412 WEISSM. +Huguenin J. 878 WE ISM AN^. 1101, 1103, 1104, 1106, 1109, 1131, 1133, 1140 +Turneaure J.P. 1180 1. 963, 969 WEITMAN WENTWORTH F.L. 100, 105, 113 WENZEL W.A. 195,220,573,583 T. +Batchelor K. 734,824 WERNTZ WESTN . 40,47, 113, 592, 600, 798,804 +Walker G.B. 738 WESTBROOK E. 100, 105, 113, 293, 313 WESTCOTT C.H. 101 3,1024 WHEELER G.W. 823,826,951,955, 969 +Hubbard E.L. 17, 1044, 1071 WHINNERY J.R. 43,86, 112 WHITBYH.C. 873 WHITEG.E. +Allison R.W. 874 WHITE1. 423,634,635,693,697, 699,709, 735 +Dbme G . 599, 735,894 +Saxon G. 221,583 WHITESIDE D. 134, 145 WIDEROE, R. 2, 17, 709, 738, 880,895 H . 134,146,521,522,524,527,53 1, WIEDEMANN, 536, 546,551 WJGNEK E.P. +Teichmann T. 737 WIIKB.H. 553, 554, 558,560,567, 1 137, 1140 WILKINS J.J. 592, 600 WILLIAMS C.B. 329, 354, 378,382 WILLIAMS H.B. 916,932 WILLIAMS I.R. ~ F u l m e C.B. r 969 WILLIAMS J.H. +Day E.A. 12, 17 WILLIAMS S.H. +Ballam J. 412 WILMUNDER A.R. +Williams C.B. 354 WILSON,P.B. 76, 77, 82, 113, 177, 185, 187, 188, 191, 203, 218,221, 355, 357, 358, 363,383, 553, 558, 560, 567, 1107, 1132, 1134, 1136, 1140, 1159, 1160, 1162, 1163, 1177, 1178, 1180 -.Fairbank W.M. 1139 +Pierce J.M. 1139 +Schwettman H.A. 113, 1106, 1139 +Smith T.1. 113,616, 826, 1106 +Weaver J.N. 113,826, 1140 +Wiik B.H. 567, 1140 WILSONR. 320, 354, 407,413, 89 1 , 895 +Lehmann P. 412 WITTKOWER A.B. + R o w M. 1045 WOJCICKI S. -+Dorfan D. 412 WOLFG . +Ballam J. 412 WOLFANGEL U . +Blasche K. 1070 WOODSH.A. 229,235 J.R. +Alvarez L.W. 17, 783, 874 WOODYARD WORSHAM R.E. +Martin J.A. 1023 WORSTELL H.G. 897, 903

AUTHOR INDEX

YAKIMENKO M.N. +Kulikov O.F. 235 YAMAZAKI E. +Tanaka J. 313 YAMNITSKII V.A. +Zeidlits P.M. 738 YARKOVOY 0.1. +Veksler V.I. 1024 YEARIAN M.R. +Janssens T. 412 YOUNGD.E. 924,933 +Owen C.W. 933 +Swenson D.A. 783 YOUNGD.M. 623,636 YOUNTD. 388,413,523,551 +Davier M. 412 YOURDR. +Main R.M. 1071 YOVNOVITCH M.L. +Veksler V.I. 1024 YUFEROV V.I. +Borovik E.A. 1044

ZABLOTSKAYA G.R. +Abroyan M.A. 873 ZACCHERONI E. 925,933 ZAGER,B.A. 919,933 ZAJECE. +Allison R.W. 874,969 ZALMANZON V.B. +Davydov M.S. 582 ZDANISR. -+Davier M. 412 ZEIDLITS P.M. 710, 71 1, 712, 730, 738 ZEITLENOK G.A. +Davydov M.S. 582 +Val'ter A.K. 113,221,413 ZHELEZNIKOV F.C. +Abroyan M.A. 873 ZIEGLER N.F. +Martin J.A. 1023 ZIPFT. +COX J. 412 ZOVOTA E.A. +Mal'tsev A.P. 1087 ZUBAREV V.N. +Varin V.A. 583 ZUCKER A. 1011,1024 ZYKOV A.I. +Arternov V.J. 219 +Val'ter A.K. 221 +Vishnyakov V.A. 221

Subject Index Accelerating gap 747 Accelerating structures alignment 3 11, 892 Alvarez 588, 879, 888, 1053 analytic techniques of calculation 86 cloverleaf 689 cold tests on 91 computer techniques of calculation 86, 617, 630, 767 constant gradient 60, 64 constant impedance 42, 64 cooling and temperature control 301, 304 coupler problems 96 cross-bar 674, 706 dielectric loaded 675 design parameters 47, 741, 767 different types of 30, 32, 46, 88, 637 empirical methods of design 88 helix 718, 1067 high power tests 299 history 6, 1 1 H-type 713, 1065 interdigital 71 1, 1065 iris 20, 419, 716 iris loaded with drift tubes 679, 717, 1144 modes in ELA structures 90 multistem 693 post coupled 674, 701, 1004 side coupled 609, 1005 single gap 1064 Sloan-Lawrence 709, 1056, 1144 slotted iris 674, 683, 688 superconducting 76, 1 126, 1146 supports 306, 885 technology 265, 271, 886, 890, 897, 1 137 tuning 99, 897 washer loaded 7 14 with resonant coupling 665, 674 Activation 1019 Adjustment methods 375 Admittance non-linear effects 797 of a synchrotron 497

of an optical system 526 of a PLA Alvarez linac 776 of rf circuits 929 phase space diagram 27, 123, 374, 776 Alignment 3 10 Alternating gradient focusing by static lenses 134, 787, 1054, 1 151 by rf fields 1079 Alvarez accelerators 587, 1053, 1069 structure 588, 881 A-matrix 626 Amplitron 319, 1020 Attenuation constant 50, 660, 672 measurements 300 waveguide 417 Beam alignment 370 characteristics 225, 935, 990 detectors 224, 377, 938 dynamics 1 15 emittance 941, 981 excited modes 171, 1159 failure 1157 losses 951, 1019 measurements on 223, 936, 981 monitoring 373, 935 monitors 224, 936 optics 369, 785 polarized 993 shielding 371, 950 switchyard 399 transport 373, 398, 785 Beam break-up ( B . B . U ) backward wave oscillator 185 coupled resonator formulation 192 cumulative 198 effect of focusing on 203, 217 experimental evidence 175, 204, 1 162 growth rate 188, 192 interaction mechanisms 183, 192

SUBJECT INDEX regenerative 176, 183, 196, 1159 remedies 215, 217, 450 standing wave 190, 1161 starting conditions 185, 191, 210, 1159 survey on 173 transient 201 Beam loading compensation 820 energy gain 151, 164, 1116 energy spectrum 427, 819 in a buncher 242 in a preinjector tube 860, 865 in a superconducting ELA 1118; (PLA) 1152 including dispersive effects 158 nature of 148 parameter 1115 reactive 820 steady state 151, 1118, 1156 theory 151, 809 transient solutions 153, 810 Biperiodic structures electrons 35, 1125 protons 602, 662, 679, 1145 Boundary conditions 623 Breakdown in preinjector tubes 865 radiofrequency 368, 915, 919 Brillouin diagram accelerating mode 32, 43, 603, 612, 671, 699, 704, 722, 1147 deflecting mode 180, 448, 577 Bubble chamber 404, 1167 Building 356, 400, 401, 406, 989, 994, 1000, 1012 Build-up time in accelerators with feedback 58 in SW accelerators 72, 908, 929 Bunch change of dimension 833 compression 467 effect of space charge 833 model 465, 828, 834 narrow width 458, 1130, 1138 phase 436 shape 467 width 432, 435 Buncher ELA buncher design 239 energy and phase variations 468,497 prebuncher 251,462 various types 245 PLA buncher 744, 893, 1057 Cavity deflecting 253, 509

dimensions 268, 286, 420, 589, 610 Cell of Alvarez structure 589, 741, 768 Chain (of resonators) 41, 88, 601, 662 Chopper low frequency 507 prebuncher system 466 rf deflecting cavity 253, 507 Compensation (of a structure) 671, 703 Computation bunch shape 467 cavity parameters 740 electrical parameters 630 fields in a gap 749 linac 767 methods 617, 747 programs 623 Confluence (of pass-bands) 666, 674, 684, 697, 1148 Controls of dimensions 89 of rf power supplies 931 Convergence (relaxation methods) 628 Converters electron-positron 390, 544 electron X-rays 479 Cooling influence on the phase 365 of superconducting linac 1129 technique 301, 959, 962, 965 Cost accelerator 355 energy 477 optimization 597, 1018, 1152, 1166, 1177 refrigerators 1129 superconducting PLA 1165 Coupled resonators 41, 88, 193, 601, 662, 685 Coupler asymmetry 98, 143 design 97, 615, 1 128 effects 462 matching 293 problems 96 Coupling between longitudinal and transverse motion3 799 coefficient 1 1 15 elements 674 loops 676, 911 non-resonant 709 resonant 665 Cross-bar structure 674, 706 Cryogeny 1128, 1164 Debuncher 51 1, 744, 893 Deflecting cavity 253, 507

SUBJECT INDEX

mode 176, 448, 453, 572 structures 572 systems 994, 1 168 Defocusing 20, 761 Degaussing 372 Detectors 224, 327, 402 Deuteron (acceleration) 1025 Dimensional requirements 266, 419, 881 Disc-loaded structure different types 89, 674 for superconducting ELA 1126 properties 30, 42, 716 technology 274 Dispersion curves accelerating mode (ELA) 32, 43; (PLA) 603, 61 2 biperiodic structures 602, 1147 deflecting mode 180, 448, 577 helix 722 multistem structures 699, 704 resonantly coupled cavities 671 I h t y cycle 357, 41 5, 426, 1003, 1018, 1055, 1107, 1 I67 Dynan~ics asynchronism wave-particle 657 beam dynamics 22, 115, 129 coefficients 756 in a buncher 240 proton across a gap 757, 763, 1081 Efficiency (of a SW structure) 1 1 15, 1154 Electron gun properties 124, 254, 462 injectors 237 linacs characteristics (summary) 107 model experiments 1l6l Electric field calculation 749 configuration 30, 90, 703, 729, 749 limitations in superconducting linacs 1132, 1138 measurements 92, 700 of deflecting mode 18 1 repartition along axis 60, 128, 700 Electrostatic preaccelerators beam loading in 860, 865 description '856, 867 high gradient 854, 865, 872 low gradient 853, 856 Emittance buncher 238, 258, 503, 51 7 emittance-current curve 982 ion sources 842, 846, 848 matching 536 measurements 941, 945, 981 positron source 525

Energy analysis 379, 948 at vanishing beam current 164 at high current 155, 157, 167, 1 1 16 gain 28, 53, 73, 84, 423, 11 16 stability in a superconducting linac 1123 stored 55 variable energy machines 992, 1060, 1068 variations due to transients 819, 983 Energy spectrum calculations 440 effect of parameter variations 501, 1156 in a microtron 561 in a superconducting linac 1123, 1 138 in high current ELA 427, 437 measurement 947 production of a narrow 497, 1123 Energy spread due to bunch width 435 due to bunch position 436 due t o transients 819 in a buncher 239, 459, 467 reduction by debunching 513 Equivalent circuit chain of cavities 668 coupled resonators 664, 685 of an Alvarez tank 9 1 1, 927 post-coupled structure 705 SW accelerator 11 14 Errors effects on motion 801 in computation 626 of phasing 1 156 of tuning 593, 605, 609, 672, 1128, 1I56 power and current 1157 Experimental facilities 355, 363, 399, 401, 995, 1005, 1012 Experiments at high energy 41 1, 987 Fabrication methods ELA structures 271 PLA structures 886, 897 superconducting structures 1 126, 1137 Feedback (of rf power) 59, 1138, 1142 Field amplitude in a cell 643 computation in a cavity 617 distribution 30 electron emission 1 132 expressions TMol mode 43 deflecting mode 575 in a gap 749, 754, 1082 in Alvarez structure 590 induced by the beam 432, 8 18 in post-coupled structure 703

SUBJECT INDEX limitations in superconducting structures 1131 measurements 93 series expansion of 572 stability 673 symmetry properties (in periodic structures) 641 uniformity 101, 700 Filling time of SW linacs 71 of TW linacs 48, 55, 62, 65 of SW superconducting linacs 83 of TW superconducting linacs 80 Focusing alternating gradient 134, 535, 785, 1051, 1150 external 133 magnetic 127 of positrons 530, 535 periodic 525, 533 radiofrequency 130, 1073, 1150 matching systems 536 methods 785 Frequency choice 49, 1171 stability 322, 456, 1124 spectrum of the beam 225 Gap acceleration 747 field in 749 motion of protons across a gap 761 Grid focusing principle 1074 technology 1053, 1077, 1085 Group velocity 32, 43, 48, 62, 703 Harmonics (space) 31, 44, 180, 590, 641 Heavy ions acceleration 1029 accelerators 1042, 1052, 1061 collisions with gas 1038 production 1031 Helix accelerator 1066 different types 725 properties 718 shunt impedance 725 High duty-cycle accelerators attenuation 417 design parameters 438 energy spectrum 427 general design 416 length 418 optimization 425 shunt impedance 423

High energy physics electrons 385, 399 protons 987 High energy PLA structures 601 High voltage accelerator tubes 853, 867 breakdown 864 general layout 861 source supply 859 voltage and current measurements 861 History 1 H-type structure 713, 1065 Hybrid mode (HE111 dispersion diagram 180, 448 effects 175 field distribution 575 properties 178, 572 Injector characteristics 960 electron 237, 258 for a microtron 559 for cyclic electron accelerators 495, 514 for cyclic proton accelerators 977 for a PLA 853, 865, 872, 1021 in high duty cycle accelerator ELA 458 PLA 1021 measurements on 256, 935 particle dynamics in 119, 239 Instabilities different type of 174, 218, 1134, 1159 growth rate 188 Installations 355, 363, 399, 401 Interdigital structure 711, 1065 Ion sources duoplasmatron 847 extraction systems 850 heavy ions 1031 H- ions 1013 ionization 838 multicharged ions 1031, 1057 PIG sources 841 properties 842, 846, 848 rf sources 843 Iris dimensions 268, 286, 421 loaded with drift tubes 679, 717 slotted 674, 683, 688 waveguide!20, 419, 716 Irradiation (industrial applications) 471 Klystron 319, 334, 338, 347, 441 Liouville theorem 25, 121, 373 Longitudinal motion

SUBJECT INDEX at injection 116, 239 equations of 22, 116, 129 L.oop coupled structure 675 coupling 9 1 1 Low energy structures 587 Magnetic field critical (in superconductors) 1090, 1101 focusing 126 limitations 1101, 1301 repartition along a buncher 262 shielding 371 stray 142, 371 Magnetron 318, 331 Matching accelerating structures 99, 109, 292, 929 adiabatic 539 dephasing in matching sections 541 errors 1156 optical systems 536 Measurements of beam parameters 937, 941 of injector characteristics 256 on ELA structures 91 Merit factor (see Quality factor) Meson factory 999 muon beam 391 pion spectrum 1001 production 391, 393 Microtron design 559 injection in 558 phase stability 556 race-track 553, 559 superconducting 1 137, 1 I38 trajectory stability 557 Misalignment effects in accelerator guides 141, 801 in focusing lenses 137 Mode analysis (in a cavity) 812 rr mode (ELA) 32, 90; (PLA) 606 n/2 mode (ELA) 34, 42, 90; (PLA) 606 2a/3 mode 90, 93, 419, 423, 449 excitation of 171 deflecting 176, 448, 572 nomenclature (in periodic structures) 661 Momentum analysis 948 Motion across a gap 747 longitudinal 20, 1 16, 128, 239 non-linear effects (coupling) 799 transverse 121, 124, 131 Multipactor

effects 91 7, 1055 theory 915 Neutrons 954, 1011, 1016 Non-linear effects due t o space charge 803 longitudinal phase acceptance 797 longitudinal-transverse coupling 799 Nuclear Physics (applications to) 385, 987 Optimization cost 597, 1018, 1152, 1166, 1177 coupling and tuning 1 1 16, 1120, 1155 parameters 598, 725, 1142, 1152 superconducting linacs 1127, 1 155 Particle detectors 224, 327, 402 dynamics (ELA) 115; (PLA) 747 monitors 937 Passband accelerating mode 45, 160 coupled resonators chain 603, 1147 deflecting mode 179 forward and backward 663 Periodic optical systems 134, 524, 1151 Periodic structures different types (ELA) 46; (PLA) 603 doubly periodic 35, 603, 1146 fields in 3 1, 42 modes 32, 34, 42, 603 principle 30, 40 superconducting 1 I46 Perturbation method 93 of motion (misalignments) 137 theory (motion) 757 Phase acceptance calculation (PLA) 776 acceptance measurement 226, 943 control 445 diagram 25, 118, 579, 642 distortion 432 errors 1 156 focusing 1079 in a cell 642 law 742, 761 oscillations 498, 5 1 1, 562 space diagrams 118, 121, 525, 941, 943, 981, 1043 spread (in ELA bunchers) 256, 458, 461, 1138 stability 19, 323, 556, 1151 transients 8 18, 930 velocity 23, 43, 1 19, 239, 721 Phase shift

SUBJECT INDEX errors 105 in cells 34, 160, 418 reactive 433 Phasing system 325, 339, 345 errors 1124, 1156 Polarized beams 993 Positron acceleration 388, 523, 546 angular distribution 529 current distribution 548 focusing 535 production 390, 528, 544 Post-coupled structure 661, 1002 Power build-up time 56, 908 electrical sources 363 rf power 28, 315, 905 stability 325 Prebuncher drift-space parameters 463 phase curves 505 principle 251 Preinjector 837, 1053 Prestripper 1051 Protection 358, 950, 1005 Pulse control of beam pulse shape 824 length 51 loss of pulses (multipactor) 918 shortening 175 transmission in a guide 162 Quadrupole focusing 134, 785, 1050, 1073 Quality factor definition 47 measurement method 95 of deflecting mode 453 values (ELA) 95.453: (PLA) 613, 908; Radiation chemistry 471 damages 959 doses 360,474, 493, 954 nature 360 protection 358, 950, 1005 Radiofrequency distribution systems 337, 3 focusing 1073 generators 318, 926, 1055 losses in a long cavity 963 multiple feed of 63 vower 28. 1130 bower soirees 318, 363,441, 925, 1004, 1020

problems 315, 329, 331, 333, 882, 905, 914 separators 397, 569, 1167, 1178 Radiography 482, 486 Radioisotope production 472 Radiotherapy 489 Refrigerator (helium) 1129, 1165 Repetition rate 51 Resonant cavity chain 607, 1064 coupled structures 667 Resistive-wall effect 21 8 Scattering by residual gas 144, 368 elastic, inelastic 387, 407 electron-nucleon 386, 407 Separators (rf) aberrations 1175 applications 397, 1167 design 580, 1177 fields in 575 long pulses 1167 optics 578 optimization 1169 principle 569, 1168 structures 572, 1177 superconducting 1167, 1178 Shielding magnetic 371 radiation 362, 950, 955 Shunt impedance comparison between PLA structures 728, 730, 731 definition 47, 423, 611, 653, 907 measured values 61 1, 653, 701, 709 of SW structures 69, 653, 907, 1144, 1148 of TW structures 655 of superconducting structures 1122, 1129, 1146 optimal 612 theoretical values 28 Side-coupled cavity structure 607, 1003 Sloan-Lawrence accelerator 5, 1056, 1058 structure 709 Smith chart 102, 294,928 Sources electrons 124 ions 841, 1031 mesons 1000, 1017 negative hydrogen ions 1013 neutrons 1015 polarized particles 993 positrons 390, 544 radiofrequency 319, 334, 338, 347, 441, 926, 1004

SUBJECT INDEX

X-rays 479 Space charge effects o n bunching 242, 509, 828 effects on energy dispersion 436 fields in a bunch 465, 831 limitation of beam density 851 longitudinal fields 435 non-linear effects 803 self-consistent models 83 1 tranverse effects 830 Space harmonics definition 31, 44, 160, 590, 641, 719 excitation of 171 influence on axial field 107 in injectors 243 measurements in ELA 92 of deflecting mode 180 Sparking damages 922, 1055 in preinjector tube 864 Stability diagrams (focusing systems) 788, 1077, 1084 frequency 322 mechanical 883 phase 323, 1138, 1151, 1156 power 325, 1 156 stabilization techniques 1 138 tuning 1 156 Standing wave operation (periodic structure) 646, 1109 power dissipation 652 Standing wave accelerators principle 65, 1 1 12 pulsed operation 71 superconducting 82, 1107, 1141 Stem as coupling element 691 effect of stems 634 multistem structure 692 Stripping 1034, 1048, 1054 Superconducting accelerators advantages 1 141 SW 76, 82, 1111, 1127, 1135, 1138, 1146 TW (with feedback) 78, 1138 theory and design 1107, 1 163 Superconductors rf applications 1104 rf properties 1092, 1 121 Superposition theorem (beam loading) 148, 432, 820,912, 1118, 1152 Support structures 306 Surface rf resistance at low temperature 1090 of superconductors 1092, 1 121 residual 1 100

values 1097, 1102, 1145 Symmetry properties (of fields) 641 Temperature control 301, 902 influence of 365, 882, 1125 operating (superconducting linac) 1 122 phase control 445 repartition (in cavity wall) 964, 1125 Tests high power 299 on structures 91 Tolerances frequency 593, 609, 1124 injection energy 460 intensity or power 429 mechanical 268, 270, 419, 881 temperature 882, 1125 Transient effects beam break-up 201 beam loading 153, 431, 809 energy spread 8 19 improvements 822 in periodic structures 672 in superconducting linacs 1155 phase 818, 930 theory 153, 810 Transit time coefficient (in a gap) 754 factor 33, 638 Transverse motion defocusing 20, 747 in injectors 244 in rf fields 124, 131, 1074, 1082 phase space 12 1 Traveling wave accelerators 52, 1 109, 1 138, 1142 field configuration 90, 642 in a periodic structure 641 properties 21, 639 Triode 319, 925, 1003 Tuners (in a linac tank) 595 Tuning considerations 454 errors 593, 1 124, 1 156 machine 295 of structures (ELA) 99, 290, 454; (PLA) 595 requirements 885, 9 15, 959, 1156 Vacuum in a preinjector 859 installations 367, 960, 1055 Vector diagram (of fields) 148, 432, 820, 913, 1118, 1152

1204

Wave standing 67, 646, 652 traveling 20, 52, 639 Waveguide attenuation 411, 421 choice of length 418

SUBJECT

Water jacket 304 Wideriie structure 2. 1063 X-rays production 330,479,483 radiography 482, 484

INDEX

Linear Accelerators

RF Linear Accelerators

RF Linear Accelerators (Physics Textbook)

Principles of RF linear accelerators

Particle Accelerators

Proton Radiotherapy Accelerators

Medical Electron Accelerators

RF superconductivity for accelerators

Electrostatic Accelerators: Fundamentals and Applications

Principles of Cyclic Particle Accelerators

Electromagnetic radiation: variational methods, waveguides and accelerators

Scientific Computing with Multicore and Accelerators

A Primer on Theory and Operation of Linear Accelerators in Radiation Therapy

RF Linear Accelerators (Wiley Series in Beam Physics and Accelerator Technology)

Electromagnetic Radiation Variational Methods Waveguides and Accelerators

Linear spaces and linear operators

Linear Algebra and Linear Models

Linear Algebra and Linear Models

Linear Algebra and Linear Models

Linear Algebra and Linear Models

Linear Algebra and Linear Models

Linear Operators and Linear Systems)

Linear spaces and linear operators

Linear Operators and Linear Systems)

Electromagnetic Radiation: Variational Methods, Waveguides and Accelerators

The Physics of Particle Accelerators: An Introduction

Electromagnetic radiation variational methods waveguides and accelerators

Induction Accelerators (Particle Acceleration and Detection)

Linear Algebras

Linear Algebra

Linear Algebra

Linear Accelerators

RF Linear Accelerators

RF Linear Accelerators (Physics Textbook)

Principles of RF linear accelerators

Particle Accelerators

Proton Radiotherapy Accelerators

Medical Electron Accelerators

RF superconductivity for accelerators

Electrostatic Accelerators: Fundamentals and Applications

Principles of Cyclic Particle Accelerators

Electromagnetic radiation: variational methods, waveguides and accelerators

Scientific Computing with Multicore and Accelerators

A Primer on Theory and Operation of Linear Accelerators in Radiation Therapy

RF Linear Accelerators (Wiley Series in Beam Physics and Accelerator Technology)

Electromagnetic Radiation Variational Methods Waveguides and Accelerators

Linear spaces and linear operators

Linear Algebra and Linear Models

Linear Algebra and Linear Models

Linear Algebra and Linear Models

Linear Algebra and Linear Models

Linear Algebra and Linear Models

Linear Operators and Linear Systems)

Linear spaces and linear operators

Linear Operators and Linear Systems)

Electromagnetic Radiation: Variational Methods, Waveguides and Accelerators

The Physics of Particle Accelerators: An Introduction

Electromagnetic radiation variational methods waveguides and accelerators

Induction Accelerators (Particle Acceleration and Detection)

Linear Algebras

Linear Algebra

Linear Algebra

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