Particle Accelerator Physics II: Nonlinear and Higher-Order Beam Dynamics

477 Pages·1995·12.07 MB·English

by  Professor Dr. Helmut Wiedemann (auth.)

Particle Accelerator Physics II
Helmut Wiedemann
Particle
Accelerator Physics II
Nonlinear and Higher-Order Beam Dynamics
With 118 Figures
Springer
Professor Dr. Helmut Wiedemann
Applied Physics Department and Stanford Synchrotron Radiation Laboratory
Stanford University, Stanford, CA 94309-0210, USA
ISBN-13: 978-3-642-97552-3 e-ISBN-13: 978-3-642-97550-9
001: 10.1007/978-3-642-97550-9
Library of Congress Cataloging-in-Publication Data. Wiedemann, Helmut, 1938-Particle accelerator physics II:
nonlinear and higher-order beam dynamics 1 Helmut Wiedemann. p. cm. Includes bibliographical references and
indexes.ISBN-13: 978-3-642-97552-3 I.
Beam dynamics. 2. Particle accelerators-Design and construction. I. Title. QC793.3.B4W55 1995 539.7'3-
dc20 94-35447
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© Springer-Verlag Berlin Heidelberg 1995
Softcover repriot of the hardcover 1st edition 1995
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Typesetting: Camera-ready copy from the author using a Springer TEX macro package
SPIN 10070489 54/3144 -5 4 3 2 I 0 -Printed on acid-free paper
Preface
This text is a continuation of the first volume of "Particle Accelerator
Physics" on "Basic Principles and Linear Beam Dynamics". While the first
volume was written as an introductory overview into beam dynamics, it
does not include more detailled discussion of nonlinear and higher-order
beam dynamics or the full theory of synchrotron radiation from relativistic
electron beams. Both issues are, however, of fundamental importance for
the design of modern particle accelerators.
In this volume, beam dynamics is formulated within the realm of Hamil
tonian dynamics, leading to the description of multiparticle beam dynamics
with the Vlasov equation and including statistical processes with the Fokker
Planck equation. Higher-order perturbations and aberrations are discussed
in detail, including Hamiltonian resonance theory and higher-order beam
dynamics. The discussion of linear beam dynamics in Vol. I is completed
here with the derivation of the general equation of motion, including kine
matic terms and coupled motion. To build on the theory of longitudinal
motion in Vol. I, the interaction of a particle beam with the rf system, in
cluding beam loading, higher-order phase focusing, and the combination
of acceleration and transverse focusing, is discussed. The emission of syn
chrotron radiation greatly affects the beam quality of electron or positron
beams and we therefore derive the detailled theory of synchrotron radiation,
including spatial and spectral distribution as well as properties of polariza
tion. The results of this derivation are then applied to insertion devices such
as undulator and wiggler magnets. Beam stability in linear and circular ac
celerators is compromized by the interaction of the electrical charge in the
beam with its environment, leading to instabilities. Theoretical models of
such instabilities are discussed and scaling laws for the onset and rise time
of instabilities are derived.
Although this text builds upon Vol. I, it relates to it only as a refer
ence for basic issues of accelerator physics, which could be obtained as well
elsewhere. This volume is aimed specifically at those students, engineers,
and scientists who desire to aqcuire a deeper knowledge of particle beam
dynamics in accelerators. To facilitate the use of this text as a reference,
many of the more important results are emphazised by a frame for quick
detection. Consistent with Vol. I we use the cgs system of units. However,
for the convenience of the reader used to the system of international units,
conversion factors have been added whenever such conversion is necessary,
VI Preface
e.g. whenever electrical or magnetic units are used. These conversion factors
are enclosed in square brackets like [v'471"f 1 and should be ignored by those
o
who use formulas in the cgs system. The conversion factors are easy to iden
tify since they include only the constants c, 71", 10o, fLo and should therefore
not be mixed up with other factors in square brackets. For the convenience
of the reader, the sources of these conversion factors are compiled in the
Appendix together with other useful tools.
I would like to thank Joanne Kwong, who typed the initial draft of this
text and introduced me to the intricacies of 'lEX typesetting, and to my
students who guided me through numerous inquisitive questions. Partial
support by the Division of Basic Energy Sciences in the Department of En
ergy through the Stanford Synchrotron Radiation Laboratory in preparing
this text is gratefully acknowledged. Special thanks to Dr. C. Maldonado for
painstakingly reading the manuscript and to the editorial staff of Springer
Verlag for support during the preparation of this text.
Palo Alto, California Helmut Wiedemann
March 1994
Contents
1. Hamiltonian Formulation of Beam Dynamics . . . . . . . . . . . . . . . 1
1.1 Hamiltonian Formalism ............................. 1
1.1.1 Lagrange Equations .......................... 1
1.1.2 Hamiltonian Equations ....................... 4
1.1.3 Canonical Transformations .................... 6
1.1.4 Action-Angle Variables ....................... 10
1.2 Hamiltonian Resonance Theory ...................... 12
1.2.1 Nonlinear Hamiltonian ....................... 12
1.2.2 Resonant Terms ............................. 16
1.2.3 Resonance Patterns and Stop-Band Width ...... 18
1.2.4 Third-Order Resonance ....................... 25
1.3 Hamiltonian and Coupling ........................... 29
1.3.1 Linearly Coupled Motion ..................... 29
1.3.2 Higher-Order Coupling Resonances ............ 38
1.3.3 Multiple Resonances ......................... 39
1.4 Symplectic Transformation .......................... 39
Problems ............................................... 41
2. General Electromagnetic Fields ... . . . . . . . . . . . . . . . . . . . . . . . . 43
2.1 General Transverse Magnetic-Field Expansion .......... 43
2.2 Third-Order Differential Equation of Motion ........... 51
2.3 Periodic Wiggler Magnets ........................... 57
2.3.1 Wiggler Field Configuration ................... 57
2.3.2 Focusing in a Wiggler Magnet ................. 61
2.3.3 Hard-Edge Model of Wiggler Magnets .......... 64
2.4 Superconducting Magnet ............................ 66
Problems ............................................... 71
3. Dynamics of Coupled Motion ............................. 73
3.1 Conjugate Trajectories .............................. 73
3.2 Particle Motion in a Solenoidal Field . . . . . . . . . . . . . . . . . 75
3.3 Transverse Coupled Oscillations ...................... 80
3.3.1 Equations of Motion in Coupling Systems ....... 80
3.3.2 Coupled Beam Dynamics in Skew Quadrupoles .. 80
3.3.3 Equations of Motion in a Solenoid Magnet ...... 82
3.3.4 Transformation Matrix for a Solenoid Magnet .. , 83
VIII Contents
3.3.5 Betatron FUnctions for Coupled Motion ........ 86
Problems ............................................... 92
4. Higher-Order Perturbations .............................. 93
4.1 Kinematic Perturbation Terms ....................... 93
4.2 Control ofthe Central Beam Path .................... 95
4.3 Dipole Field Errors and Dispersion FUnction ........... 102
4.4 Dispersion FUnction in Higher Order .................. 105
4.4.1 Chromaticity in Higher Approximation ......... 107
4.4.2 Nonlinear Chromaticity ....................... 110
4.5 Perturbation Methods in Beam Dynamics ............. 114
4.5.1 Periodic Distribution of Statistical Perturbations. 115
4.5.2 Statistical Methods to Evaluate Perturbations ... 121
Problems ............................................... 126
5. Hamiltonian Nonlinear Beam Dynamics .................... 127
5.1 Higher-Order Beam Dynamics.. .... . .... .. .. .. . .. . . . . 127
5.1.1 Multipole Errors ............................. 127
5.1.2 Nonlinear Matrix Formalism .................. 131
5.2 Aberrations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
5.2.1 Geometric Aberrations ....................... 137
5.2.2 Filamentation of Phase Space ................. 143
5.2.3 Chromatic Aberrations ....................... 147
5.2.4 Particle Tracking ............................ 149
5.3 Hamiltonian Perturbation Theory .................... 152
5.3.1 Tune Shift in Higher Order ................... 158
Problems ............................................... 160
6. Charged Particle Acceleration .. . . . . . . . . . . . . . . . . . . . . . . . . . . 163
6.1 Accelerating Fields in Resonant rf Cavities ............. 163
6.1.1 Wave Equation .............................. 164
6.1.2 Waveguide Modes. . . .. .. . . . .. . . . . . . . . . .. ... . . 165
6.1.3 rf Cavities .................................. 170
6.1.4 Cavity Losses and Shunt Impedance ............ 175
6.1.5 Determination of rf Parameters ................ 179
6.2 Beam-Cavity Interaction ............................ 181
6.2.1 Coupling Between rf Field and Particles ........ 181
6.2.2 Beam Loading and rf System .................. 187
6.2.3 Higher-Order Mode Losses in an rf Cavity 192
6.2.4 Beam Loading in Circular Accelerators ......... 197
6.3 Higher-Order Phase Focusing ........................ 208
6.3.1 Path Length in Higher Order .................. 208
6.3.2 Higher-Order Phase Space Motion ............. 210
6.3.3 Stability Criteria ............................ 214
6.4 FODO Lattice and Acceleration ...................... 220
Contents IX
6.4.1 Transverse Beam Dynamics and Acceleration .... 222
6.4.2 Adiabatic Damping .......................... 225
Problems ............................................... 227
7. Synchrotron Radiation ................................... 229
7.1 Theory of Synchrotron Radiation ..................... 229
7.1.1 Radiation Field .............................. 229
7.2 Synchrotron Radiation Pow:er and Energy Loss ......... 236
7.3 Spatial Distribution of Synchrotron Radiation .......... 241
7.4 Synchrotron Radiation Spectrum ..................... 245
7.4.1 Radiation Field in the Frequency Domain ....... 246
7.4.2 Spectral Distribution in Space and Polarization .. 251
7.4.3 Angle-Integrated Spectrum ................... 260
Problems ............................................... 267
8. Hamiltonian Many-Particle Systems ....................... 269
8.1 The Vlasov Equation ............................... 269
8.1.1 Betatron Oscillations and Perturbations ........ 275
8.1.2 Damping ................................... 277
8.2 Damping of Oscillations in Electron Accelerators ....... 279
8.2.1 Damping of Synchrotron Oscillations ........... 279
8.2.2 Damping of Vertical Betatron Oscillations ...... 285
8.2.3 Robinson's Damping Criterion ................. 287
8.2.4 Damping of Horizontal Betatron Oscillations .... 290
8.3 The Fokker-Planck Equation ......................... 291
8.3.1 Stationary Solution of the Fokker-Planck Equation 294
8.3.2 Particle Distribution Within a Finite Aperture .. 298
8.3.3 Particle Distribution in the Absence of Damping. 301
Problems ............................................... 302
9. Particle Beam Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305
9.1 Particle Distribution in Phase Space .................. 305
9.1.1 Diffusion Coefficient and Synchrotron Radiation . 305
9.1.2 Quantum Excitation of Beam Emittance ........ 308
9.1.3 Horizontal Equilibrium Beam Emittance ........ 308
9.1.4 Vertical Equilibrium Beam Emittance .......... 309
9.2 Equilibrium Energy Spread and Bunch Length ......... 311
9.3 Phase-Space Manipulation ........................... 313
9.3.1 Exchange of Transverse Phase-Space Parameters. 313
9.3.2 Exchange of Longitudinal Phase-Space Parameters 314
9.4 Polarization of Particle Beam ........................ 320
Problems ............................................... 323
10. Collective Phenomena 325
10.1 Statistical Effects 325
X Contents
10.1.1 Schottky Noise .............................. 326
10.1.2 Stochastic Cooling ........................... 328
10.1.3 Touschek Effect ............................. 328
10.1.4 Intra-Beam Scattering ........................ 330
10.2 Collective Self Fields ................................ 332
10.2.1 Transverse Self Fields ........................ 332
10.2.2 Fields from Image Charges .................... 334
10.2.3 Space-Charge Effects ......................... 338
10.2.4 Longitudinal Space-Charge Field .............. 343
10.3 Beam-Current Spectrum ............................. 345
10.4 Wake Fields and Impedance ......................... 350
10.4.1 Definitions of Wake Field and Impedance ....... 353
10.4.2 Impedances in an Accelerator Environment ..... 361
10.5 Coasting-Beam Instabilities .......................... 368
10.5.1 Negative-Mass Instability ..................... 368
10.5.2 Dispersion Relation .......................... 372
10.5.3 Landau Damping ............................ 378
10.5.4 Transverse Coasting-Beam Instability .......... 380
10.6 Longitudinal Single-Bunch Effects .................... 382
10.6.1 Potential-Well Distortion ..................... 382
10.7 Transverse Single-Bunch Instabilities .................. 390
10.7.1 Beam Break-Up in Linear Accelerators ......... 390
10.7.2 Fast Head-Tail Effect ......................... 392
10.7.3 Head-Tail Instability ......................... 397
10.8 Multi-Bunch Instabilities ............................ 399
Problems ............................................... 404
11. Insertion Device Radiation ............................... 406
11.1 Particle Dynamics in an Undulator ................... 407
11.2 Undulator Radiation ................................ 409
11.3 Undulator Radiation Distribution ..................... 412
11.4 Elliptical Polarization ............................... 429
Problems ............................................... 434
Appendix .................................................. 435
References ................................................. 445
Author Index 453
Subject Index 457
Contents to Volume I
1. Introduction
1.1 Short Historical Overview
1.2 Particle Accelerator Systems
1.2.1 Basic COJIlPonents 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
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