Table of Contents

Preface page xi
1 Introduction 1
1.1 Differences or Differentials? 1
1.2 Phase Space Co-ordinates 3
1.3 Iterations, Ancient and Modern 5
1.4 Accelerator History: The Two Golden Ages 6
Exercises 10
2 Linear Motion 13
2.1 Stable Oscillations 13
2.2 Transverse Motion through Magnets 16
2.3 Matrix Equations of Motion 19
Exercises 22
3 Strong Focusing Transverse Optics 24
3.1 Linear Stability and Twiss Functions 25
3.2 Turn-by-Turn Motion in Phase Space 27
3.3 Propagation across a Fraction of a Turn 29
3.4 Continuous Propagation 30
3.5 FODO Cell Optics 32
Exercises 35
4 Longitudinal and Off-Momentum Motion 37
4.1 Constant Momentum Offset: Transverse Motion 37
4.2 The Dispersion Function 39
4.3 Oscillating Momentum: Longitudinal Motion 41
4.4 The Standard Map 45
Exercises 48
5 Action and Emittance – One Particle or Many? 50
5.1 Transverse Action-Angle Co-ordinates 50
5.2 Unnormalised Emittances and Beam Sizes 52
5.3 Tune Spread and Filamentation 54
5.4 Linac (Phase Space Area) Emittances 56
5.5 Normalised Emittance and Adiabatic Damping 58
5.6 Longitudinal Phase Space Parameters 60
Exercises 61
6 Magnets 63
6.1 Normal and Skew Multipole Magnets 63
6.2 Iron-Dominated Magnets 65
6.3 Conductor-Dominated Magnets 66
6.4 Field Quality and Errors 67
Exercises 69
7 RF Cavities 73
7.1 Waveguides 73
7.2 Transverse Modes 74
7.3 Cylindrical Resonant Cavities – Pill-Boxes 76
7.4 Cavity Performance Limits 79
Exercises 81
8 Linear Errors and Their Correction 83
8.1 Trajectory and Closed Orbit Errors 83
8.2 Linear Coupling 88
8.3 Tune Shifts and β-Waves 89
Exercises 92
9 Sextupoles, Chromaticity and the Hénon Map 94
9.1 Chromaticity in a FODO Lattice 94
9.2 Chromaticity Correction 96
9.3 The Hénon Map – A Unit Strength Sextupole in 1-D 98
9.4 A Taxonomy of 1-D Motion 100
9.5 Dynamic Aperture 103
Exercises 104
10 Octupoles, Detuning and Slow Extraction 107
10.1 Single Octupole Lattice 107
10.2 Discrete Motion in Action-Angle Space, (J, φ) 109
10.3 Two-Turn Motion with Q ≈ 1/2 110
10.4 Slow Extraction near the Half-Integer 111
Exercises 113
11 Synchrotron Radiation – Classical Damping 116
11.1 Spectrum and Distribution Pattern 116
11.2 Energy Loss Per Turn and Longitudinal Damping 119
11.3 Continuous Acceleration 124
11.4 Transverse Damping and Partition Numbers 126
Exercises 128
12 Synchrotron Radiation – Quantum Excitation 131
12.1 Energy Spread 132
12.2 Horizontal Emittance 134
12.3 Vertical Emittance 138
Exercises 139
13 Linacs – Protons and Ions 141
13.1 Time Structures 142
13.2 Multi-Cell Synchronism 145
13.3 Linear Motion 147
13.4 Radio Frequency Quadrupoles 153
13.5 Beam Losses and Haloes 157
Exercises 158
14 Linacs – Electrons 159
14.1 Longitudinal and Transverse Focusing 159
14.2 RF Capture 161
14.3 Bunch Compression 162
14.4 Recirculating and Energy Recovery Linacs 164
14.5 Beam Breakup 166
Exercises 169
15 The Beam–Beam Interaction and 1-D Resonances 170
15.1 Round Beam-Beam Interaction 170
15.2 First-Order Theory of 1-D Resonances 174
15.3 Resonance Island Tunes and Widths 176
Exercises 180
16 Routes to Chaos 181
16.1 Resonance Overlap 182
16.2 Tune Modulation 184
16.3 Dynamical Zones in Tune Modulation Space 187
Exercises 190
Appendix A Selected Formulae for Accelerator Design 191
A.1 Matrices for Linear Motion through Accelerator Elements 191
A.2 Propagation of Twiss Functions 196
References 198
Index 201