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

487 Pages·1999·14.25 MB·English

by  Professor Dr. Helmut Wiedemann (auth.)

Particle Accelerator Physics II
Springer
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Helmut Wiedemann
Particle
Accelerator Physics II
Nonlinear and Higher-Order
Beam Dynamics
Second Edition
With 118 Figures
, Springer
Professor Dr. Helmut Wiedemann
Applied Physics ~partment
and Synchrotron Rfodiation Laboratory
Stanford Univenity, Stanford, CA 94309-0110, USA
ISBN-13: 978-3-642-64177-0 Springer-Verlag Berlin Heidelberg New York
Library ofCongreu Cataloging-in-Publication DatL
Wiedemann, Helmut, 19)8-Particle acuJerator phys.ics II : nonlinear and higher-order beam dynamia
I Helmut Wiedemann. -lnd ed. p.em. Includes bibliographical references and indexes. ISBN-13: 978
-3-642-64In-O (alk. paper) I. Bcam dynamia. :I.. Lincar accelcrators. I. Title. IN PROCBSS 539.7'
3-dc1198-34118 CIP
ISBN-13: 978-3-642-64171-0 e-ISBN-13: 978-3-642-59908-8
001: 10.1007/978-3-642-59908-8
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e Springer-Verlag Berlin Heidelberg 1995, 1999
Soft cover reprint ofthe hardcover 1st edition 1999
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Preface
This second edition of "Particle Accelerator Physics II" does not contain
major changes in content. Primarily, errors have been eliminated as far as
they have been detected. Progress made in the field of accelerator design
since the publication of the first edition made it necessary to udate the bib
liography. The author appreciates the many suggestions made by observant
readers to reduce errors and misprints.
Paolo Alto, October 1998 Helmut Wiedemann
Preface to the First Edition
This text is a continuation of the first volume of "Particle Accelerator
Physics I" 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 detailed discussions 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 detailed 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,
VIII Preface to the First Edition
e.g. whenever electrical or magnetic units are used. These conversion factors
are enclosed in square brackets like [J 1a nd should be ignored by those
41TEo
who use formulas in the cgs system. The conversion factors are easy to iden
tify since they include only the constants c, 1T, Eo ,J.Lo 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
X 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 of the 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 Thne 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 XI
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 Power 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