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Particle Accelerator Physics I
Springer-Verlag Berlin Heidelberg GmbH
Helmut Wiedemann
Particle Accelerator Physics I Basic Principles and Linear Beam Dynamics
Second Edition With 160 Figures
Springer
Professor Dr. Helmut Wiedemann Applied Physics Department and Synchrotron Radiation Laboratory Stanford University, Stanford, CA 94309.0210, USA e-mail: [email protected]
Library of Congress Cataloging-in-Publication Data. Wiedemann, Helmut, 1938- Particle accelerator physics I : basic principles and linear beam dynamics I Helmut Wiedemann. - 2nd ed. p.cm. Includes bibliographical references and index. ISBN 978-3-662-03829-1 1. Beam dynamics. 2. Linear accelerators. I. Title. QC793.3.B4 W54 1998 539.7'3-ddc2l 98-46048 CIP
This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable for prosecution under the German Copyright Law.
© Springer-Verlag Berlin Heidelberg 1993, 1999 Originally published by Springer-Verlag Berlin Heidelberg New York in 1999
Softcover reprint of the hardcover 2nd edition 1999 The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use.
Typesetting: Camera-ready copy from the author using a Springer TJ3X macro package Cover design: design & production GmbH, Heidelberg
SPIN: 10683486 55/3144/mf - 5 43210 - Printed on acid-free paper
ISBN 978-3-662-03829-1 ISBN 978-3-662-03827-7 (eBook) DOI 10.1007/978-3-662-03827-7
To my sons Frank and Martin
Preface
In this second edition of Particle Accelerator Physics, Vol. 1, is mainly a reprint of the first edition without significant changes in content. The bibliography has been updated to include more recent progress in the field of particle accelerators.
With the help of many observant readers a number of misprints and errors could be eliminated. The author would like to express his sincere appreciation to all those who have pointed out such shortcomings and welcomes such information and any other relevant information in the future.
The author would also like to express his special thanks to the editor Dr. Helmut Lotsch and his staff for editorial as well as technical advice and support which contributed greatly to the broad acceptance of this text and made a second edition of both volumes necessary.
Palo Alto, California November 1998
Helmut Wiedemann
VII
Preface to the First Edition
The purpose of this textbook is to provide a comprehensive introduction into the physics of particle accelerators and particle beam dynamics. Particle accelerators have become important research tools in high energy physics as well as sources of incoherent and coherent radiation from the far infra red to hard x-rays for basic and applied research. During years of teaching accelerator physics it became clear that the single most annoying obstacle to get introduced into the field is the absence of a suitable textbook. Indeed most information about modern accelerator physics is contained in numerous internal notes from scientists working mostly in high energy physics laboratories all over the world.
This text intends to provide a broad introduction and reference book into the field of accelerators for graduate students, engineers and scientists summarizing many ideas and findings expressed in such internal notes and elsewhere. In doing so theories are formulated in a general way to become applicable for any kind of charged particles. Writing such a text, however, poses the problem of correct referencing of original ideas. I have tried to find the earliest references among more or less accessible notes and publications and have listed those although the reader may have difficulty to obtain the original paper. In spite of great effort to be historically correct I apologize for possible omissions and misquotes. This situation made it necessary to rederive again some of such ideas rather than quote the results and refer the interested reader to the original publication. I hope this approach will not offend the original researchers, but rather provide a broader distribution of their original ideas, which have become important to the field of accelerator physics.
This text is split into two volumes. The first volume is designed to be self contained and is aimed at newcomers into the field of accelerator physics, but also to those who work in related fields and desire some background on basic principles of accelerator physics. The first volume therefore gives an introductory survey of fundamental principles of particle acceleration followed by the theory of linear beam dynamics in the transverse as well as longitudinal phase space including a detailed discussion of basic magnetic focusing units. Concepts of single and multi particle beam dynamics are introduced. Synchrotron radiation, its properties and effect on beam dynamics and electron beam parameters is described in considerable detail followed
IX
by a discussion of beam instabilities on an introductory level, beam lifetime and basic lattice design concepts.
The second volume is aimed specifically to those students, engineers and scientists who desire to immerse themselves deeper into the physics of particle accelerators. It introduces the reader to higher order beam dynamics, Hamiltonian particle dynamics, general perturbation theory, nonlinear beam optics, chromatic and geometric aberrations and resonance theory. The interaction of particle beams wi th rf fields of the accelerating system and beam loading effects are described in some detail relevant to accelerator physics. Following a detailed derivation of the theory of synchrotron radiation, particle beam phenomena are discussed while utilizing the Vlasov and Fokker Planck equations leading to the discussion of beam parameters and their manipulation and collective beam instabilities. Finally design concepts and new developments of particle accelerators as synchrotron radiation sources or research tools in high energy physics are discussed in some detail.
This text grew out of a number of lecture notes for accelerator physics courses at Stanford University, the Synchrotron Radiation Research Laboratory in Taiwan, the University of Sao Paulo in Brazil, the International Center for Theoretical Physics in Trieste and the US Particle Accelerator School as well as from interaction with students attending those classes and my own graduate students.
During almost thirty years in this field I had the opportunity to work with numerous individuals and accelerators in laboratories around the world. Having learned greatly from these interactions I like to take this opportunity to thank all those who interacted with me and have had the patience to explain their ideas, share their results or collaborate with me. The design and construction of new particle accelerators provides a specifically interesting period to develop and test theoretically new ideas, to work with engineers and designers, to see theoretical concepts become hardware and to participate in the excitement of commissioning and optimization. I have had a number of opportunities for such participation at the Deutsches Elektronen Synchrotron DESY in Hamburg, Germany and at the Stanford University at Stanford, California and am grateful to all colleagues who hosted and collaborated with me. I wished I could mention them individually and apologize for not doing so.
A special thanks goes to the operators of the electron storage rings SPEAR and PEP at the Stanford Linear Accelerator Center, specifically to T. Taylor, W. Graham, E. Guerra and M. Maddox, for their dedicated and able efforts to provide me during numerous shifts over many years with a working storage ring ready for machine physics experimentation.
I thank Mrs. Joanne Kwong, who typed the initial draft of this texts and introduced me into the intricacies of TEX typesetting. The partial support by the Department of Energy through the Stanford Synchrotron Radiation Laboratory in preparing this text is gratefully acknowledged. Special thanks
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to Dr. C. Maldonado for painstakingly reading the manuscript. Last but not least I would like to thank my family for their patience in dealing with an "absent" husband and father.
Palo Alto, California April 1993
Helmut Wiedemann
XI
Contents
1. Introduction .......................................... . 1.1 Short Historical Overview .......................... . 1.2 Particle Accelerator Systems ....................... .
1.2.1 Basic Components of Accelerator Facilities .... . 1.2.2 Applications of Particle Accelerators ......... .
1.3 Basic Definitions and Formulas ..................... . 1.3.1 Units and Dimensions ...................... . 1.3.2 Basic Relativistic Formalism ................. . 1.3.3 Particle Collisions at High Energies .......... .
1.4 Basic Principles of Particle-Beam Dynamics .......... . 1.4.1 Stability of a Charged-Particle Beam ......... .
Problems
2. Linear Accelerators .................................... . 2.1 Principles of Linear Accelerators .................... .
2.1.1 Charged Particles in Electric Fields .......... . 2.1.2 Electrostatic Accelerators ................... . 2.1.3 Induction Linear Accelerator ................ .
2.2 Acceleration by rf Fields ........................... . 2.2.1 Basic Principle of Linear Accelerators ........ . 2.2.2 Waveguides for High Frequency EM Waves .... .
2.3 Preinjector Beam Preparation ...................... . 2.3.1 Prebuncher ................................ . 2.3.2 Beam Chopper ............................ .
Problems
3. Circular Accelerators .................................. . 3.1 Betatron ......................................... . 3.2 Weak Focusing ................................... . 3.3 Adiabatic Damping ............................... . 3.4 Acceleration by rf Fields ........................... .
3.4.1 Microtron ................................. . 3.4.2 Cyclotron ................................. . 3.4.3 Synchro Cyclotron ......................... . 3.4.4 Isochron Cyclotron ......................... .
1 1 5 5 8 9 9
12 15 17 20 21
25 25 25 27 30 31 31 33 47 47 49 50
53 54 58 60 62 63 65 67 68
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3.5 Synchrotron. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 3.5.1 Storage Ring ............................... 71
3.6 Summary of Characteristic Parameters ............... 72 Problems .............................................. 73
4. Charged Particles in Electromagnetic Fields . . . . . . . . . . . . . . . 75 4.1 The Lorentz Force ................................. 75 4.2 Coordinate System ................................. 76 4.3 Fundamentals of Charged Particle Beam Optics ...... . 81
4.3.1 Particle Beam Guidance ..................... 81 4.3.2 Particle Beam Focusing ...................... 84
4.4 Multipole Field Expansion .......................... 88 4.4.1 Laplace Equation ........................... 88 4.4.2 Magnetic Field Equations .................... 91
4.5 Multipole Fields for Beam Transport Systems ......... 92 4.6 Multipole Field Patterns and Pole Profiles ............ 97 4.7 Equations of Motion in Charged Particle Beam Dynamics 100 4.8 General Solution of the Equations of Motion .......... 103
4.8.1 Linear Unperturbed Equation of Motion ....... 104 4.8.2 Wronskian ................................. 106 4.8.3 Perturbation Terms ......................... 107 4.8.4 Dispersion Function ......................... 109
4.9 Building Blocks for Beam Transport Lines ............ 110 4.9.1 General Focusing Properties .................. 110 4.9.2 Chromatic Properties ........................ 112 4.9.3 Achromatic Lattices ......................... 112 4.9.4
Problems Isochronous Systems 113
116
5. Linear Beam Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 5.1 Linear Beam Transport Systems ..................... 119
5.1.1 Nomenclature .............................. 120 5.2 Matrix Formalism in Linear Beam Dynamics .......... 121
5.2.1 Driftspace. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 5.2.2 Quadrupole Magnet ......................... 123 5.2.3 Thin Lens Approximation .................... 125 5.2.4 Quadrupole End Field Effects ..... . . . . . . . . . . . 127 5.2.5 Quadrupole Design Concepts ................. 131
5.3 Focusing in Bending Magnets ............ . . . . . . . . . . . 137 5.3.1 Sector Magnets ............................. 138 5.3.2 Wedge Magnets ............................. 143 5.3.3 Rectangular Magnet ......................... 145
5.4 Particle Beams and Phase Space ..................... 147 5.4.1 Beam Emittance ............................ 147 5.4.2 Liouville's Theorem ......................... 149
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5.4.3 Transformation in Phase Space ............... 152 5.4.4 Measurement of the Beam Emittance .......... 157
5.5 Betatron Functions ................................ 159 5.5.1 Beam Envelope ............................. 162 5.5.2 Beam Dynamics in Terms of Betatron Functions 162 5.5.3 Beam Dynamics in Normalized Coordinates .... 164
5.6 Dispersive Systems ................................. 168 5.6.1 Analytical Solution .......................... 168 5.6.2 (3 x 3)-Transformation Matrices .............. 170 5.6.3 Linear Achromat ............................ 171 5.6.4 Spectrometer. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174
5.7 Path Length and Momentum Compaction ............. 178 Problems .............................................. 180
6. Periodic Focusing Systems ............................... 184 6.1 FODO Lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185
6.1.1 Scaling of FODO Parameters ................. 186 6.2 Betatron Motion in Periodic Structures ............... 190
6.2.1 Stability Criterion ........................... 190 6.2.2 General FODO Lattice ...................... 192
6.3 Beam Dynamics in Periodic Closed Lattices ........... 196 6.3.1 Hill's Equation ............................. 196 6.3.2 Periodic Betatron Functions .................. 199
6.4 Periodic Dispersion Function ........................ 202 6.4.1 Scaling of the Dispersion in a FODO Lattice 202 6.4.2 General Solution for the Periodic Dispersion .... 205
6.5 Periodic Lattices in Circular Accelerators ............. 210 6.5.1 Synchrotron Lattice ......................... 210 6.5.2 Phase Space Matching ....................... 212 6.5.3 Dispersion Matching .. . . . . . . . . . . . . . . . . . . . . . . 213 6.5.4 Magnet Free Insertions ...................... 215 6.5.5 Low Beta Insertions ......................... 218 6.5.6 Example of a Colliding Beam Storage Ring ..... 219
Problems .............................................. 221
7. Perturbations in Beam Dynamics . .. . . . . . . . . . . . . . . . . . . . .. 225 7.1 Magnet Alignment Errors ........................... 226 7.2 Dipole Field Perturbations .......................... 228
7.2.1 Existence of Equilibrium Orbits ............... 229 7.2.2 Closed Orbit Distortion ...................... 231 7.2.3 Closed Orbit Correction ..................... 238
7.3 Quadrupole Field Perturbations ..................... 240 7.3.1 Betatron Tune Shift ......................... 240 7.3.2 Resonances and Stop Band Width ........... . 243 7.3.3 Perturbation of Betatron Functions ........... 246
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8.
9.
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7.4 Resonance Theory ................................. 248 7.4.1 Resonance Conditions ....................... 249 7.4.2 Coupling Resonances ........................ 253 7.4.3 Resonance Diagram ......................... 254
7.5 Chromatic Effects in a Circular Accelerator ........... 255 7.5.1 Chromaticity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256 7.5.2 Chromaticity Correction ..................... 260
Problems
Charged Particle Acceleration .......................... . 8.1 Longitudinal Particle Motion ....................... .
8.1.1 Longitudinal Phase Space Dynamics .......... . 8.1.2 Equation of Motion in Phase Space ........... . 8.1.3 Phase Stability ............................ . 8.1.4 Acceleration of Charged Particles ............ .
8.2 Longitudinal Phase Space Parameters ............... . 8.2.1 Separatrix Parameters ...................... . 8.2.2 Momentum Acceptance ..................... . 8.2.3 Bunch Length ............................. . 8.2.4 Longitudinal Beam Emittance ............... . 8.2.5 Phase Space Matching ...................... .
Problems
Synchrotron Radiation ................................. . 9.1 Physics of Synchrotron Radiation ................... .
9.1.1 Coulomb Regime ........................... . 9.1.2 Radiation Regime .......................... . 9.1.3 Spatial Distribution of Synchrotron Radiation .. 9.1.4 Radiation Power ........................... . 9.1.5 Synchrotron Radiation Spectrum ............. . 9.1.6 Photon Beam Divergence ................... .
9.2 Coherent Radiation ............................... . 9.2.1 Temporal Coherent Synchrotron Radiation .... . 9.2.2 Spatially Coherent Synchrotron Radiation ..... . 9.2.3 Spectral Brightness ......................... . 9.2.4 Matching ................................. .
9.3 Insertion Devices ................................. . 9.3.1 Bending Magnet Radiation .................. . 9.3.2 Wave Length Shifter ........................ . 9.3.3 Wiggler Magnet Radiation .................. . 9.3.4 Undulator Radiation ....................... .
9.4 Back Scattered Photons ........................... . 9.4.1 Radiation Intensity ......................... .
Problems
262
265 266 267 271 277 282 285 285 287 290 292 293 297
300 302 302 303 306 307 311 315 317 318 320 324 325 326 326 327 327 330 333 333 335
10. Particle Beam Parameters .............................. . 10.1 Definition of Beam Parameters ..................... .
10.1.1 Beam Energy .............................. . 10.1.2 Time Structure ............................ . 10.1.3 Beam Current ............................. . 10.1.4 Beam Dimensions .......................... .
10.2 Damping ........................................ . 10.2.1 Robinson Criterion ......................... .
10.3 Particle Distribution in Phase Space ................ . 10.3.1 Equilibrium Phase Space .................... . 10.3.2 Transverse Beam Parameters ................ .
10.4 Variation of the Equilibrium Beam Emittance ........ . 10.4.1 Beam Emittance and Wiggler Magnets ....... . 10.4.2 Damping Wigglers ......................... .
10.5 Variation of the Damping Distribution ............... . 10.5.1 Damping Partition and rf Frequency ......... . 10.5.2 Robinson Wiggler .......................... . 10.5.3 Damping Partition and Synchrotron Oscillation . 10.5.4 Can We Eliminate the Beam Energy Spread? ...
Problems
11. Beam Life Time ....................................... . 11.1 Beam Lifetime and Vacuum ........................ .
11.1.1 Elastic Scattering .......................... . 11.1.2 Inelastic Scattering ......................... .
11.2 Ultra High Vacuum System ........................ . 11.2.1 Thermal Gas Desorption .................... . 11.2.2 Synchrotron Radiation Induced Desorption .... .
Problems ............................................. .
12. Collective Phenomena ................................. . 12.1 Linear Space-Charge Effects ........................ .
12.1.1 Self Field for Particle Beams ................ . 12.1.2 Forces from Space-Charge Fields ............. .
12.2 Beam-Beam Effect ................................ . 12.3 Wake Fields ...................................... .
12.3.1 Parasitic Mode Losses and Impedances ....... . 12.4 Beam Instabilities ................................ . Problems ............................................. .
337 337 337 338 338 339 341 342 348 349 355 358 358 360 363 363 365 365 366 368
370 371 372 378 380 381 381 383
384 384 385 387 388 390 391 396 401
13. Beam Emittance and Lattice Design ...................... 402 13.1 Equilibrium Beam Emittance in Storage Rings ........ 403 13.2 Beam Emittance in Periodic Lattices ................. 407
13.2.1 The Double Bend Achromat Lattice (DBA) 407 13.2.2 The Triple Bend Achromat Lattice (TBA) 410
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13.2.3 The Triplet Achromat Lattice (TAL) .......... 410 13.2.4 The FODO Lattice .......................... 413
13.3 Optimum Emittance for Colliding Beam Storage Rings . 416 Problems .............................................. 417
Appendices
A. Suggested Reading 419
B. Bibliography 424
References 427
Author Index 437
Subject Index 441
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