Beam Dynamics in High Energy Particle Accelerators

606 Pages·2014·4.58 MB·English

by  Andrzej Wolski

BEAM DYNAMICS
IN HIGH ENERGY PARTICLE
ACCELERATORS
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BEAM DYNAMICS
IN HIGH ENERGY PARTICLE
ACCELERATORS
Andrzej Wolski
University of Liverpool, UK
Imperial College Press
ICP
P899_9781783262779_tp.indd 2 19/12/13 11:11 am
Published by
Imperial College Press
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British Library Cataloguing-in-Publication Data
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BEAM DYNAMICS IN HIGH ENERGY PARTICLE ACCELERATORS
Copyright © 2014 by Imperial College Press
All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means,
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ISBN 978-1-78326-277-9
Printed in Singapore
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To my parents, for all the many years of patience and support.
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Contents
Preface xi
I Electromagnetism and Classical Mechanics 1
1 Electromagnetic Fields in Accelerator Components 3
1.1 Boundary Conditions on Electromagnetic Fields 7
1.1.1 Surface of an infinite permeability material 8
1.1.2 Surface of an ideal conductor 11
1.2 Two-Dimensional Multipole Fields 12
1.2.1 Current distribution for a pure multipole 18
1.2.2 Geometry of iron-dominated multipole magnets 22
1.2.3 Multipole decomposition 26
1.3 Three-Dimensional Fields 33
1.3.1 Cartesian and cylindrical modes 34
1.3.2 Generalised gradients 39
1.4 Fields in Radiofrequency Cavities 43
1.4.1 Rectangular cavities 46
1.4.2 Cylindrical cavities 53
2 Hamiltonian for a Particle in an Accelerator Beam Line 59
2.1 The Hamiltonian for a Straight Beam Line 61
2.2 Dynamical Variables for Beam Dynamics 68
2.3 The Hamiltonian in a Curved Co-ordinate System 72
2.4 Symplectic Transfer Maps and Liouville’s Theorem 76
vii
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viii Beam Dynamics in High Energy Particle Accelerators
II Single-Particle Linear Dynamics 81
3 Linear Transfer Maps for Common Components 83
3.1 Drift Space 85
3.2 Dipole Magnet 88
3.3 Dipole Fringe Fields and Edge Focusing 96
3.4 Quadrupole Magnet 101
3.5 Solenoid 106
3.6 Radiofrequency Cavity 109
3.7 Spin Dynamics 116
4 Linear Optics in Uncoupled Beam Lines 127
4.1 A FODO Lattice 128
4.2 The Courant–Snyder Parameters 132
4.3 Action–Angle Variables 136
4.4 Courant–Snyder Parameters in a FODO Beam Line 143
4.5 Hill’s Equation 147
4.6 Courant–Snyder Parameters and Particle Distribution 150
5 Coupled Optics 157
5.1 Transverse–Longitudinal Coupling 158
5.1.1 Dispersion 158
5.1.2 Momentum compaction and phase slip 162
5.1.3 Synchrotron motion 168
5.2 Fully Coupled Motion 176
5.3 Dispersion Revisited 188
5.4 Examples of Coupled Optics 190
5.4.1 Uniform solenoid field 190
5.4.2 Flat-beam electron source 191
6 Linear Imperfections in Storage Rings 195
6.1 The Closed Orbit 196
6.2 Dipole Field Errors 198
6.3 Quadrupole Alignment Errors 203
6.4 Focusing Errors 204
6.5 Beam-Based Alignment of Quadrupoles 207
6.6 Coupling Errors 210
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Contents ix
7 Effects of Synchrotron Radiation 215
7.1 Classical Radiation: Radiation Damping 217
7.2 Quantum Radiation: Quantum Excitation 230
7.3 Equilibrium Emittance and Lattice Design 237
7.3.1 Natural emittance in a FODO storage ring 239
7.3.2 Double-bend achromat 243
7.3.3 TME lattices and multibend achromats 246
7.4 Computation of Equilibrium Emittances 250
7.5 Synchrotron Radiation and Spin Polarisation 258
III Single-Particle Nonlinear Dynamics 263
8 Examples of Nonlinear Effects in Accelerator Beam Lines 265
8.1 Longitudinal Dynamics in a Bunch Compressor 265
8.2 Chromaticity in a Linear FODO Beam Line 274
8.3 Chromaticity in Storage Rings 279
9 Representations of Transfer Maps 287
9.1 Lie Transformations 289
9.2 Power Series Map for a Sextupole 296
9.3 Mixed-Variable Generating Functions 301
10 Symplectic Integrators 309
10.1 Splitting Methods 310
10.2 Explicit Symplectic Integrator for s-dependent Fields 319
10.3 Symplectic Runge–Kutta Integrators 325
11 Methods for Analysis of Single-Particle Dynamics 331
11.1 A Lie Transformation Example: the −I Transformer 332
11.2 Canonical Perturbation Theory 335
11.2.1 Dipole perturbations: closed orbit distortion 343
11.2.2 Quadrupole perturbations: focusing errors 346
11.2.3 Skew quadrupole perturbations: coupling 348
11.2.4 Sextupole perturbations 357
11.3 Resonances and Dynamic Aperture 360
11.4 Normal Form Analysis 373
11.5 A Numerical Method: Frequency Map Analysis 384