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