Theory and Design of Charged Particle Beams

Martin Reiser0-47-1-306-16-9, 978-0-47-1-306-16-0

Although particle accelerators are the book’s main thrust, it offers a broad synoptic description of beams which applies to a wide range of other devices such as low-energy focusing and transport systems and high-power microwave sources. Develops material from first principles, basic equations and theorems in a systematic way. Assumptions and approximations are clearly indicated. Discusses underlying physics and validity of theoretical relationships, design formulas and scaling laws. Features a significant amount of recent work including image effects and the Boltzmann line charge density profiles in bunched beams.

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Table of contents :
Theory and Design of Charged Particle Beams……Page 2
Contents……Page 10
Preface……Page 16
Acknowledgments……Page 20
1.1 Exposition……Page 24
1.2 Historical Developments and Applications……Page 27
1.3 Sources of Charged Particles……Page 30
References……Page 37
2.1 The Lorentz Force and the Equation of Motion……Page 38
2.2 The Energy Integral and Some General Formulas……Page 42
2.3.1 Hamilton’s Principle and Lagrange’s Equations……Page 46
2.3.2 Generalized Potential and Lagrangian for Charged Particle Motion in an Electromagnetic Field……Page 48
2.3.3 Hamilton’s Equations of Motion……Page 51
2.3.4 The Hamiltonian for Charged Particles and Some Conservation Theorems……Page 53
2.4.1 The Principle of Least Action and the Euler Equations……Page 60
2.4.2 Relativistic Euler Equations in Axially Symmetric Fields……Page 64
2.5.1 Planar Diode without Space Charge……Page 66
2.5.2 Planar Diode with Space Charge (Child-Langmuir Law)……Page 68
2.5.3 Charged-Particle Motion in a Uniform Magnetic Field……Page 69
2.5.4 Charged Particle Motion in a Radial Electric Field……Page 70
2.5.5 The Harmonic Oscillator……Page 72
Problems……Page 75
3.1 Beam Emittance and Brightness……Page 79
3.2 Liouville’s Theorem……Page 85
3.3.1 Series Representation of Axisymmetric Electric and Magnetic Fields……Page 89
3.3.2 Derivation of the Paraxial Ray Equation……Page 92
3.3.3 General Properties of the Solutions of the Paraxial Ray Equations……Page 99
3.4.1 General Parameters and Transfer Matrix of a Lens……Page 103
3.4.2 Image Formation and Magnification……Page 106
3.4.3 Electrostatic Lenses……Page 109
3.4.4 Solenoidal Magnetic Lenses……Page 121
3.4.5 Effects of a Lens on the Trace-Space Ellipse and Beam Envelope……Page 126
3.4.6 Aberrations in Axially Symmetric Lenses……Page 129
3.5 Focusing by Quadrupole Lenses……Page 134
3.6.1 Betatron Oscillations……Page 139
3.6.2 The Trace-Space Ellipse and Beam Envelope in a Betatron-type Field……Page 144
3.6.3 Focusing in Axisymmetric ExB Fields……Page 149
3.6.4 Energy Spread, Momentum Compaction, and Effective Mass……Page 153
3.7 Sector Magnets and Edge Focusing……Page 158
3.8.1 Periodic Focusing with Thin Lenses……Page 162
3.8.2 General Theory of Courant and Snyder……Page 168
3.8.3 The FODO Quadrupole Channel……Page 176
3.8.4 Sector-Focusing Cyclotrons……Page 180
3.8.5 Strong-Focusing Synchrotrons……Page 185
3.8.6 Resonances in Circular Accelerators……Page 189
3.9 Adiabatic Damping of the Betatron Oscillation Amplitudes……Page 194
References……Page 197
Problems……Page 198
4.1 Theoretical Models of Beams with Space Charge……Page 206
4.2.1 Laminar Beam with Uniform Density Profile……Page 214
4.2.2 Beam Envelope with Self Fields and Finite Emittance……Page 225
4.2.3 Limitations of the Uniform Beam Model and Limiting Currents……Page 226
4.2.4 Self-Focusing of a Charge-Neutralized Beam (Bennett Pinch)……Page 231
4.3.1 The Paraxial Ray Equation with Self Fields……Page 233
4.3.2 Beam Transport in a Uniform Focusing Channel……Page 235
4.4.1 Beam Transport in a Periodic Solenoid Channel……Page 244
4.4.2 Beam Transport in a Quadrupole (FODO) Channel……Page 257
4.4.3 Envelope Oscillations and Instabilities of Mismatched Beams……Page 263
4.4.4 Coherent Beam Oscillations due to Injection Errors and Misalignments……Page 275
4.5.1 Betatron Tune Shift due to Self Fields……Page 283
4.5.2 Current Limits in Weak- and Strong-Focusing Systems……Page 288
4.5.3 Effects of Image Forces on Coherent and Incoherent Betatron Tune……Page 290
4.6.1 Ionization Cross Sections for Electrons and Proton Beams in Various Gases……Page 296
4.6.2 Linear Beam Model with Charge Neutralization……Page 301
4.6.3 Gas Focusing in Low-Energy Proton and H- Beams……Page 304
4.6.4 Charge-Neutralization Effects in Intense Relativistic Electron Beams……Page 308
4.6.5 Charge-Neutralization Effects in High-Energy Synchrotrons and Storage Rings……Page 312
4.6.6 Plasma Lenses……Page 317
References……Page 319
Problems……Page 321
5.1 Introduction……Page 327
5.2.1 A Cylindrical Beam in an Infinitely Strong Magnetic Field……Page 329
5.2.2 Nonrelativistic Laminar Beam Equilibria……Page 334
5.2.3 Relativistic Laminar Beam Equilibria……Page 346
5.2.4 Paraxial Analysis of Mismatched Laminar Beams in Uniform Magnetic Fields……Page 354
5.3.1 The Vlasov Equation……Page 358
5.3.2 The Kapchinsky-Vladimirsky (K-V) Distribution……Page 364
5.3.3 Stationary Distributions in a Uniform Focusing Channel……Page 370
5.3.4 RMS Emittance and the Concept of Equivalent Beams……Page 381
5.4.1 Coulomb Collisions between Particles and Debye Shielding……Page 387
5.4.2 The Fokker-Planck Equation……Page 391
5.4.3 The Maxwell-Boltzmann Distribution for a Relativistic Beam……Page 395
5.4.4 The Stationary Transverse Distribution in a Uniform or Smooth Focusing Channel……Page 402
5.4.5 Transverse Temperature and Beam-Size Variations in Nonuniform Focusing Channels……Page 413
5.4.6 The Longitudinal Distribution and Beam Cooling due to Acceleration……Page 416
5.4.7 Stationary Line-Charge Density Profiles in Bunched Beams……Page 425
5.4.8 Longitudinal Motion in rf Fields and the Parabolic Bunch Model……Page 437
5.4.9 Longitudinal Beam Dynamics in Circular Machines……Page 451
5.4.10 Effects of Momentum Spread on the Transverse Distribution……Page 460
5.4.11 Coupled Envelope Equations for a Bunched Beam……Page 470
5.4.12 Matching, Focusing, and Imaging……Page 476
References……Page 484
Problems……Page 485
6.1 Causes of Emittance Change……Page 490
6.2.1 Analytical Theory……Page 493
6.2.2 Comparison of Theory, Simulation, and Experiment……Page 502
6.3.1 Transverse Beam Modes and Instabilities in Periodic Focusing Channels……Page 514
6.3.2 Longitudinal Space-Charge Waves and the Resistive-Wall Instability……Page 521
6.3.3 Longitudinal Instability in Circular Machines and Landau Damping……Page 538
6.4.1 The Boersch Effect……Page 548
6.4.2 Intrabeam Scattering in Circular Machines……Page 553
6.4.3 Multiple Scattering in a Background Gas……Page 560
6.5.1 The Need for Emittance Reduction……Page 564
6.5.2 Electron Cooling……Page 565
6.5.3 Stochastic Cooling……Page 567
6.5.4 Radiation Cooling……Page 569
6.6 Concluding Remarks……Page 577
References……Page 580
Problems……Page 583
Appendix 1 : Example of a Pierce-Type Electron Gun with Shielded Cathode……Page 587
References……Page 589
Appendix 2: Example of a Magnetron Injection Gun……Page 590
References……Page 592
Appendix 3: Four-Vectors and Covariant Lorentz Tranrsformations……Page 593
Appendix 4: Equipartitioning in High-Current rf Linacs……Page 596
References……Page 603
Appendix 5: Radial Defocusing and Emittance Growth in High-Gradient rf Structures (Example: The rf Photocathado Electron Gun)……Page 604
References……Page 611
List of Frequently Used Symbols……Page 613
Bibliography (Selected list of Books)……Page 621
Index……Page 624