An introduction to the physics of high energy accelerators

Accelerator X-Ray Sources......Page 4
Contents......Page 8
Preface......Page 16
1.1 Preview......Page 22
1.2 Coordinate Definitions......Page 23
1.3 One-dimensional Transverse Propagation Equations......Page 25
1.4.2 Thin Lens......Page 28
1.4.3 Thick Lens......Page 30
1.4.4 Erect Quadrupole Lens......Page 32
1.5 Elliptical (in Phase Space) Beams......Page 34
1.6 Beam Envelope E(s)......Page 36
1.7 Gaussian Beams: their Variances and Covariances......Page 38
1.8 Pseudoharmonic Trajectory Description......Page 39
1.9 Transfer Matrix Parametrization......Page 42
1.10.1 Beam Evolution Through a Drift Section......Page 44
1.10.2 Beam Evolution Through a Thin Lens......Page 45
References......Page 46
2.1 Preview......Page 48
2.2 Scalar Wave Equation......Page 49
2.3 The Short Wavelength, Geometric Optics Limit......Page 51
2.3.1 Determination of Rays from Wavefronts......Page 52
2.3.2 The Ray Equation in Geometric Optics......Page 53
2.3.3 Obtaining Phase Information from Intensity Measurement......Page 55
2.4 Wave Description of Gaussian Beams......Page 57
2.4.1 Gaussian Beam in a Focusing Medium......Page 58
2.4.2 Spatial Dependence of a Wave Near a Free Space Focus......Page 60
2.4.3 The ABCD Law......Page 61
2.4.4 Optics Using Mirrors......Page 63
2.4.5 Wave Particle Duality for Electrons......Page 65
2.5 Synchrotron Radiation: Waves or Particles?......Page 66
2.6 X-ray Holography and Phase Contrast and Lens-free Imaging......Page 67
References......Page 69
3.1 Capsule History of Synchrotron Light Sources......Page 70
3.2 Generalities......Page 72
3.3 Potentials and Fields......Page 74
3.4 Relations Between Observation Time and Retarded Time......Page 75
3.5 Evaluation of Electric and Magnetic Fields......Page 79
3.5.1 Radial Field Approximation......Page 81
3.6 Total Power Radiated and its Angular Distribution......Page 83
3.7.1 Estimate of Frequency Spectrum from Pulse Duration......Page 86
3.7.2 Radial Approximation......Page 88
3.7.3 Accurate Formula for Spectral Power Density......Page 89
3.8 Radiation from Multiple Charges......Page 91
3.9 The Terminology of “Intensity” Measures......Page 92
3.10 Photon Beam Features “Inherited from” the Electron Beam......Page 95
3.11 Intensity Estimates for Bending Magnet beams......Page 96
References......Page 102
4.1 Preview......Page 104
4.2 The Uniform Field Ring......Page 105
4.3 Horizontal Stability......Page 106
4.4 Vertical Stability......Page 108
4.5 Simultaneous Horizontal and Vertical Stability......Page 109
4.6 Dispersion......Page 110
4.7 Momentum Compaction......Page 111
4.8 Chromaticity......Page 112
4.9 Strong Focusing......Page 113
5.1 Preview......Page 116
5.2.1 Total Energy Radiated......Page 117
5.2.2 The Distribution of Photon Energies; “Regularized Treatment”......Page 120
5.2.3 Randomness of the Radiation......Page 124
5.3 The Damping Rate Sum Rule: Robinson’s Theorem......Page 126
5.3.1 Vertical Damping......Page 131
5.3.2 Longitudinal Damping......Page 132
5.3.3 Horizontal Damping and Partition Numbers......Page 140
5.4 Equilibrium between Damping and Fluctuation.......Page 141
5.5 Horizontal Equilibrium and Beam Width......Page 142
5.6.1 Energy Spread......Page 149
5.6.2 Bunch Length......Page 152
5.7.1 Emittance of Pure Wiggler Lattice......Page 153
5.7.2 Thermodynamic Analogy......Page 158
References......Page 161
6.1 Acknowledgement and Preview......Page 162
6.2.1 Transit Time Factor......Page 163
6.2.2 Shunt Impedance......Page 164
6.2.3 Cavity Q, R/Q, and Decay Time......Page 165
6.2.4 Phase Stability and Adiabatic Damping......Page 166
6.2.5 Transverse Defocusing......Page 172
6.3.1 Introduction......Page 174
6.3.2 Particle Acceleration by a Wave......Page 175
6.3.3 Wave Confined by Parallel Planes......Page 176
6.3.4 Circular Waveguide......Page 182
6.3.5 Cylindrical “Pill-box” Resonator......Page 184
6.3.6 Lumped Constant Model for One Cavity Resonance......Page 185
6.3.7 Cavity Excitation......Page 187
6.3.8 Wave Propagation in Coupled Resonator Chain......Page 189
6.3.9 Periodically Loaded Structures......Page 193
6.3.10 Space Harmonics......Page 195
7.1 Preview......Page 200
7.2 Introduction......Page 201
7.3 Electron Orbit in a Wiggler or Undulator......Page 205
7.4 Energy Radiated From one Wiggler Pole......Page 208
7.5 Spectral Analysis for Arbitrary Longitudinal Field Profile......Page 209
7.6.1 Orbit Treated as Arc of Circle......Page 212
7.6.2 Radiation from a Single, Short, Isolated, Magnet, K « 1......Page 213
7.7 Coherence from Multiple Deflections......Page 217
7.8 Phasor Summation for K « 1......Page 220
7.9.1 Energy Distribution from the n = 1 Undulator Fundamental......Page 224
7.10.1 Analytic Formulation......Page 225
7.10.2 Diffraction Grating Analogy......Page 233
7.10.3 Numerical/Graphical Representation of Undulator Radiation......Page 234
7.10.4 Approximation of the Integrals by Special Functions......Page 241
7.10.5 Practical Evaluation of the Series......Page 243
7.11 Post-monochromator Profile......Page 244
7.11.1 Monochromatic Annular Rings......Page 246
7.11.3 Is the Forward Undulator Peak Subject to Angular Narrowing?......Page 248
References......Page 251
8.1 Preview......Page 252
8.2 Considerations Governing Undulator Parameters......Page 253
8.3 Simplified Radiation Formulas......Page 255
8.4 A Hybrid, Electo-permanent, Asymmetric Undulator......Page 258
8.4.1 Electromagnet Design......Page 260
8.4.2 Permanent Magnet Design—Small Gap Limit......Page 265
8.4.3 Combined Electro-/Permanent- magnet Design......Page 267
8.4.4 Estimated X-ray Flux......Page 268
References......Page 269
9.1 Preview......Page 270
9.2 Beam Line Generalities......Page 271
9.3 Accelerator Parameters......Page 273
9.4 Bragg Scattering and Darwin Width......Page 275
9.5.1 Undulator Radiation, n = 1, Negligible Electron Divergence......Page 278
9.5.2 Effect of Electron Beam Emittances on Flux and Brilliance......Page 282
9.5.3 Brilliance with K > 1 and n > 1......Page 284
9.6.2 Elliptical Mirrors......Page 287
9.6.3 Hyperbolic Mirrors......Page 288
9.7.2 Focusing a Monochromatic Undulator Radiation Ring......Page 291
9.7.3 Undulator-specific X-ray Lens......Page 293
9.7.4 Lens Quality......Page 299
9.8.1 The Pin-hole Camera......Page 300
9.8.2 Imaging the Beam with Visible Light......Page 301
9.8.3 Practicality of Lens-based, X-ray Beam Camera?......Page 304
9.9.1 Aperture-free Rationale......Page 307
9.9.2 Aperture-free Microbeam Line Based on Lenses......Page 308
9.9.3 Effective Lens Stop Caused by Absorption......Page 312
9.9.4 Choice of Undulator Parameter K......Page 313
9.9.5 Estimated Flux......Page 314
9.9.6 Estimated Brilliance and Qualifying Comments......Page 316
References......Page 318
10.1 Preview......Page 320
10.2 Introduction......Page 321
10.3 Emittance Evolution in a DC Electron Gun......Page 322
10.4 Qualitative Description of Lattice Design Issues......Page 327
10.5 Isochronous Arc Design......Page 330
10.6.1 Deceleration through Linac Section to Dump......Page 335
10.6.2 Triplet Design......Page 339
10.6.3 Acceleration through the High Energy Linac Section......Page 340
10.7 Emittance Growth Due to CSR and Space Charge......Page 342
References......Page 346
11.1 Preview......Page 348
11.2 Low Emittance Lattices......Page 349
11.3.1 A Formula for Brilliance B......Page 354
11.3.2 A Strategy to Maximize B......Page 355
11.3.3 Hypothetical Utilization of an Existing Large Ring......Page 356
11.4 Acceleration Scenario......Page 359
11.5.1 Average Power......Page 363
11.5.2 Instantaneous Power......Page 364
11.6 Critical Components and Parameter Dependencies......Page 365
11.7.1 Basic Formulas......Page 367
11.7.2 Thin Lens Treatment......Page 369
11.7.3 Thick Lens Treatment......Page 374
11.7.4 Zero Dispersion Straight Sections......Page 376
11.7.5 Nonlinearity and Dynamic Aperture......Page 378
11.8 Emittance Evolution During Acceleration......Page 380
11.9 Touschek Lifetime Estimate......Page 383
11.10.1 Brilliance from Short Undulator......Page 384
11.10.2 Refinement of Brilliance Calculation......Page 386
References......Page 389
12.1 Preview......Page 390
12.2 Compton Scattering Kinematics......Page 391
12.3.1 Back-scattered Photons......Page 394
12.3.2 Orthogonal Photon Incidence in the Laboratory......Page 395
12.3.3 Orthogonal Electron Frame Incidence......Page 396
12.4 Total Compton Cross Section......Page 401
12.5 The Photon Beam Treated as an Electromagnetic Wave......Page 402
12.5.1 Determination of the Electron’s Velocity Modulation......Page 403
12.5.2 Undulator Parametrization of Electron Motion in a Wave......Page 407
12.6 Undulator Fields in Electron Rest Frame......Page 409
12.6.2 Treatment of an Undulator Magnet as an Electromagnetic Wave......Page 410
12.7.1 Introduction......Page 412
12.7.3 Electric Dipole Radiation [6]......Page 413
12.7.4 Scattering Rate Expressed as a Total Cross Section......Page 415
12.8.1 Solid Angle Transformation......Page 416
12.8.2 Photon Energy Distribution in the Laboratory......Page 417
12.8.3 A Theorem Applicable to Isotropic Distributions......Page 419
12.8.4 Energy Distribution of Undulator Radiation......Page 420
12.9 Rate Estimates for a Laser Wire Diagnostic Apparatus......Page 422
12.9.2 Laser Wire Treated as Undulator......Page 424
12.9.3 Laser Wire Treated via Electron Rest Frame......Page 426
12.9.4 Invariant Cross Section Applied to Laser Wire......Page 427
12.9.5 Bunched Beam Rates......Page 429
References......Page 432
13.1 Acknowledgement and Preview......Page 434
13.2 Introduction to the String Space Charge Formalism......Page 435
13.3 Self-force of Moving Straight Charged String......Page 439
13.4 Self-force of Moving Straight Charged Ribbon......Page 444
13.5 Curve End Point Determination......Page 451
13.6 Field Calculation......Page 455
13.7 “Regularization” of the Longitudinal Force......Page 459
13.8 Coherent Synchrotron Radiation......Page 461
13.9 Evaluation of Integrals......Page 463
13.10 Calculational Practicalities......Page 464
13.11 Suppression of CSR by Wall Shielding......Page 465
13.12 Effects of Entering and Leaving Magnets......Page 466
13.13 Space Charge Calculations Using Unified Accelerator Libraries......Page 468
13.13.1 Numerical Procedures Used by UAL......Page 469
13.13.2 Program Architecture......Page 471
13.13.3 Numerical Procedures......Page 473
13.13.4 Comparison with TRAFIC4 [24]......Page 474
References......Page 477
14.1 Absorption and Spontaneous and Stimulated Emission......Page 478
14.2 Closed and Open FELs......Page 479
14.3 Interpretation of Undulator Radiation as Compton Scattering......Page 480
14.4 Applicability Condition for Semi-Classical Treatment......Page 482
14.5 Comparison of Storage Ring, ERL, and FEL......Page 483
References......Page 486
Index......Page 488