Part I Tools We Need
1 Of Fields and Forces ......................................... 3
1.1 Electromagnetic Fields of Charged Particles ............. 3
1.1.1 Vector and Scalar Potential ...................... 5
1.1.2 Wave Equation .................................... 5
1.1.3 Induction ........................................ 7
1.1.4 The Lorentz Force ................................ 7
1.1.5 Equation of Motion ............................... 8
1.1.6 Energy Conservation ............................. 10
1.2 Primer in Special Relativity ........................... 11
1.2.1 Lorentz Transformation .......................... 11
1.2.2 4-Vectors ....................................... 13
1.2.3 Spatial and Spectral Distribution of
Radiation ....................................... 17
1.3 Elements of Classical Mechanics ........................ 18
1.3.1 How to Formulate a Lagrangian? .................. 20
1.3.2 The Lorentz Force ............................... 21
1.3.3 Frenet Serret Coordinates ...................... 22
1.4 Hamiltonian Formulation ................................ 23
1.4.1 Cyclic Variables ................................ 25
1.4.2 Canonical Transformations ....................... 25
1.4.3 Curvilinear Coordinates ......................... 28
1.4.4 Extended Hamiltonian ............................ 30
1.4.5 Change of Independent Variable .................. 30
2 Particle Dynamics in Electromagnetic Fields ................. 37
2.1 The Lorentz Force ...................................... 37
2.2 Fundamentals of Charged Particle Beam Optics ........... 38
2.2.1 Particle Beam Guidance .......................... 38
2.2.2 Particle Beam Focusing .......................... 42
2.3 Equation of Motion ..................................... 46
2.4 Equations of Motion from the Lagrangian and
Hamiltonian ............................................ 49
2.4.1 Equations of Motion from Lagrangian ............. 49
2.4.2 Canonical Momenta ............................... 51
2.4.3 Equation of Motion from Hamiltonian ............. 51
2.4.4 Harmonic Oscillator ............................. 53
2.4.5 Action-Angle Variables .......................... 54
2.5 Solutions of the Linear Equations of Motion ............ 55
2.5.1 Linear Unperturbed Equation of Motion ........... 55
2.5.2 Matrix Formulation .............................. 57
2.5.3 Wronskian ....................................... 57
2.5.4 Perturbation Terms .............................. 59
3 Electromagnetic Fields ...................................... 63
3.1 Pure Multipole Field Expansion ......................... 63
3.1.1 The Laplace Equation ............................ 63
3.1.2 Deflecting Magnets .............................. 65
3.1.3 Focusing Device ................................. 66
3.1.4 Multipole Magnets ............................... 74
3.1.5 Multipole Fields for Beam Transport Systems ..... 77
3.2 General Transverse Magnetic-Field Expansion ............ 80
3.3 Third-Order Differential Equation of Motion ............ 87
3.4 Longitudinal Field Devices ............................. 92
3.5 Air Coil Magnets ....................................... 94
3.6 Periodic Wiggler Magnets ............................... 99
3.6.1 Wiggler Field Configuration .................. 100
3.7 Electric Field Components ............................. 104
3.7.1 Electrostatic Deflectors ....................... 104
3.7.2 Electrostatic Focusing Devices ................. 105
3.7.3 Iris Doublet ................................... 107
3.7.4 Einzellens ..................................... 108
3.7.5 Electrostatic Quadrupolc ....................... 109
Part II Beam Dynamics
4 Single Particle Dynamics ................................... 115
4.1 Linear Beam Transport Systems ......................... 116
4.1.1 Nomenclature ................................. 117
4.2 Matrix Formalism in Linear Beam Dynamics .............. 118
4.2.1 Drift Space .................................... 120
4.2.2 Quadrupolc Magnet .............................. 120
4.2.3 Thin Lens Approximation ........................ 122
4.2.4 Quadrupole End Field Effects ................... 125
4.3 Focusing in Bending Magnets ........................... 129
4.3.1 Sector Magnets ............................... 130
4.3.2 Fringe Field Effects ........................... 131
4.3.3 Finite Pole Gap ................................ 133
4.3.4 Wedge Magnets .................................. 135
4.3.5 Rectangular Magnet ............................. 137
4.3.6 Focusing in a Wiggler Magnet ................... 138
4.3.7 Hard Edge Model of Wiggler Magnets ............. 141
4.4 Elements of Beam Dynamics ............................. 143
4.4.1 Building Blocks for Beam Transport Lines ....... 143
4.4.2 Isochronous Systems ............................ 146
5 Particle Beams and Phase Space ............................. 153
5.1 Beam Emittance ........................................ 154
5.1.1 Liouville's Theorem ............................ 155
5.1.2 Transformation in Phase Space .................. 158
5.1.3 Beam Matrix .................................... 161
5.2 Betatron Functions .................................... 166
5.2.1 Beam Envelope .................................. 169
5.3 Beam Dynamics in Terms of Betatron Functions .......... 169
5.3.1 Beam Dynamics in Normalized Coordinates ........ 172
5.4 Dispersive Systems .................................... 175
5.4.1 Analytical Solution ............................ 175
5.4.2 (3 x 3)-Transformation Matrices ................ 177
5.4.3 Linear Achromat ................................ 178
5.4.4 Spectrometer ................................... 183
5.4.5 Measurement of Beam Energy Spectrum ............ 184
5.4.6 Path Length and Momentum Compaction ............ 187
6 Longitudinal Beam Dynamics ................................. 191
6.1 Longitudinal Particle Motion .......................... 192
6.1.1 Longitudinal Phase Space Dynamics ............ 194
6.2 Equation of Motion in Phase Space ..................... 197
6.2.1 Small Oscillation Amplitudes ................... 199
6.2.2 Phase Stability ................................ 203
6.2.3 Acceleration of Charged Particles .............. 207
6.3 Longitudinal Phase Space Parameters ................... 211
6.3.1 Separatrix Parameters .......................... 211
6.3.2 Momentum Acceptance ............................ 213
6.3.3 Bunch Length ................................... 216
6.3.4 Longitudinal Beam Emittance .................... 218
6.3.5 Phase Space Matching ........................... 219
6.4 Higher Order Phase Focusing ........................... 224
6.4.1 Path Length in Higher Order .................... 224
6.4.2 Higher Order Phase Space Motion ................ 226
6.4.3 Stability Criteria ............................. 231
7 Periodic Focusing Systems .................................. 237
7.1 FODO Lattice .......................................... 238
7.1.1 Scaling of FODO Parameters ..................... 239
7.1.2 Betatron Motion in Periodic Structures ......... 243
7.1.3 General FODO Lattice ........................... 245
7.2 Beam Dynamics in Periodic Closed Lattices ............. 249
7.2.1 Hill's Equation ................................ 249
7.2.2 Periodic Betatron Functions .................... 252
7.2.3 Periodic Dispersion Function ................... 255
7.2.4 Periodic Lattices in Circular Accelerators ..... 263
7.3 FODO Lattice and Acceleration ......................... 275
7.3.1 Lattice Structure .............................. 275
7.3.2 Transverse Beam Dynamics and Acceleration ...... 276
7.3.3 Adiabatic Damping .............................. 280
Part III Beam Parameters
8 Particle Beam Parameters ................................... 289
8.1 Definition of Beam Parameters ......................... 289
8.1.1 Beam Energy .................................... 289
8.1.2 Time Structure ................................. 290
8.1.3 Beam Current ................................... 290
8.1.4 Beam Dimensions ................................ 292
8.2 Damping ............................................... 293
8.2.1 Robinson Criterion ........................... 294
8.3 Particle Distribution in Longitudinal Phase Space ..... 301
8.3.1 Energy Spread .................................. 301
8.3.2 Bunch Length ................................... 303
8.4 Transverse Beam Ernittance ............................ 303
8.4.1 Equilibrium Beam Ernittance .................... 304
8.4.2 Ernittance Increase in a Beam Transport Line ... 305
8.4.3 Vertical Beam Ernittance ....................... 306
8.4.4 Beam Sizes ..................................... 307
8.4.5 Beam Divergence ................................ 310
8.5 Variation of the Damping Distribution ................. 310
8.5.1 Damping Partition and rf-Frequency ........... 310
8.6 Variation of the Equilibrium Beam Ernittance .......... 312
8.6.1 Beam Ernittance and Wiggler Magnets ............ 312
8.6.2 Damping Wigglers ............................... 315
8.7 Robinson Wiggler ...................................... 317
8.7.1 Damping Partition and Synchrotron
Oscillation .................................... 317
8.7.2 Can we Eliminate the Beam Energy Spread? ....... 318
8.8 Beam Life Time ........................................ 319
8.8.1 Beam Lifetime and Vacuum ....................... 321
8.8.2 Ultra High Vacuum System ....................... 329
9 Vlasov and Fokker—Planck Equations ......................... 335
9.1 The Vlasov Equation ................................... 336
9.1.1 Betatron Oscillations and Perturbations ........ 341
9.1.2 Damping ........................................ 343
9.2 Damping of Oscillations in Electron Accelerators ...... 345
9.2.1 Damping of Synchrotron Oscillations ............ 345
9.2.2 Damping of Vertical Betatron Oscillations ...... 349
9.2.3 Robinson's Damping Criterion ................... 352
9.2.4 Damping of Horizontal Betatron Oscillations .... 355
9.3 The Fokker Planck Equation ............................ 355
9.3.1 Stationary Solution of the Fokker Planck
Equation ....................................... 358
9.3.2 Particle Distribution within a Finite
Aperture ....................................... 362
9.3.3 Particle Distribution in the Absence of
Damping ........................................ 364
10 Equilibrium Particle Distribution .......................... 369
10.1 Particle Distribution in Phase Space .................. 369
10.1.1 Diffusion Coefficient and Synchrotron
Radiation ...................................... 369
10.1.2 Quantum Excitation of Beam Emittance ........... 372
10.2 Equilibrium Beam Emittance ............................ 373
10.2.1 Horizontal Equilibrium Beam Emittance .......... 373
10.2.2 Vertical Equilibrium Beam Emittance ............ 374
10.3 Equilibrium Energy Spread and Bunch Length ............ 375
10.3.1 Equilibrium Beam Energy Spread ................. 375
10.3.2 Equilibrium Bunch Length ....................... 376
10.4 Phase-Space Manipulation ......................... 377
10.4.1 Exchange of Transverse Phase-Space
Parameters ..................................... 377
10.4.2 Bunch Compression .............................. 378
10.4.3 Alpha Magnet ................................... 380
10.5 Polarization of a Particle Beam ....................... 383
11 Beam Emittance and Lattice Design .......................... 389
11.1 Equilibrium Beam Emittance in Storage Rings ........... 391
11.1.1 FODO Lattice ................................... 391
11.1.2 Minimum Beam Emittance ......................... 392
11.2 Beam Emittance in Periodic Lattices ................... 396
11.2.1 The Double Bend Achromat Lattice (DBA) ......... 397
11.2.2 The Triple Bend Achromat Lattice (TBA) ......... 399
11.2.3 The Triplet Achromat Lattice (TAL) ............. 400
11.2.4 Limiting Effects ............................... 402
11.2.5 The FODO Lattice ............................... 404
11.2.6 Optimum Emittance for Colliding Beam
Storage Rings .................................. 407
Part IV Perturbations
12 Perturbations in Beam Dynamics ............................. 411
12.1 Magnet Field and Alignment Errors ..................... 412
12.1.1 Dipole Field Perturbations ..................... 414
12.1.2 Existence of Equilibrium Orbits ................ 415
12.1.3 Closed Orbit Distortion ........................ 418
12.1.4 Quadrupole Field Perturbations ................. 426
12.2 Chromatic Effects in a Circular Accelerator ........... 435
12.2.1 Chromaticity ................................... 436
12.2.2 Chromaticity Correction ........................ 439
12.3 Kinematic Perturbation Terms .......................... 441
12.4 Control of the Central Beam Path ...................... 443
12.4.1 Launching Error ................................ 444
12.4.2 Statistical Alignment and Field Error .......... 445
12.5 Dipole Field Errors and Dispersion Function ........... 450
12.5.1 Self Compensation of Perturbations ............. 451
12.5.2 Perturbations in Open Transport Lines .......... 452
12.6 Dispersion Function in Higher Order .............. 454
12.6.1 Chromaticity in Higher Approximation ........... 456
12.7 Nonlinear Chromaticity ........................... 458
12.8 Perturbation Methods in Beam Dynamics ................. 463
12.8.1 Periodic Distribution of Statistical
Perturbations .................................. 464
12.8.2 Periodic Perturbations in Circular
Accelerators ................................... 467
12.8.3 Statistical Methods to Evaluate
Perturbations .................................. 469
13 Hamiltonian Resonance Theory ....................... 479
13.1 Resonances ............................................ 479
13.1.1 Resonance Conditions ........................... 480
13.1.2 Coupling Resonances ............................ 484
13.1.3 Resonance Diagram .............................. 485
13.2 Nonlinear Hamiltonian ................................. 487
13.3 Resonant Terms ........................................ 490
13.4 Resonance Patterns and Stop-Band Width ................ 492
13.4.1 Half integer stop band ......................... 493
13.4.2 Separatrices ................................... 495
13.5 General Stop-Band Width .......................... 497
13.6 Third-Order Resonance ................................. 498
13.6.1 Particle Motion in Phase Space ................. 501
14 Hamiltonian Nonlinear Beam Dynamics ........................ 503
14.1 Higher Order Beam Dynamics ............................ 503
14.1.1 Multipole Errors ............................... 503
14.1.2 Nonlinear Matrix Formalism ..................... 507
14.2 Aberrations ........................................... 512
14.2.1 Geometric Aberrations .......................... 514
14.2.2 Filamentation of Phase Space ................... 520
14.2.3 Chromatic Aberrations .......................... 523
14.2.4 Particle Tracking .............................. 526
14.3 Hamiltonian Perturbation Theory ....................... 528
14.3.1 Tune Shift in Higher Order 534
Part V Acceleration
15 Charged Particle Acceleration .............................. 541
15.1 Preinjector and Beam Preparation ...................... 541
15.1.1 Prebuncher ..................................... 541
15.1.2 Beam Chopper ................................... 544
15.2 rf-Waveguides and Cavities ............................ 545
15.2.1 Wave Equation .................................. 545
15.2.2 Rectangular Waveguide Modes .................... 547
15.2.3 Cylindrical Waveguide Modes .................... 551
15.3 Linear Accelerator .................................... 554
15.3.1 Basic Waveguide Parameters ..................... 555
15.3.2 Particle Capture in a Linear Accelerator
Field .......................................... 560
15.4 rf-Cavities ........................................... 563
15.4.1 Energy Gain .................................... 565
15.4.2 rf-Cavity as an Oscillator ..................... 566
15.4.3 Cavity Losses and Shunt Impedance .............. 568
15.5 rf-Parameters ......................................... 572
15.5.1 Synchronous Phase and rf-voltage ............... 573
16 Beam—Cavity Interaction .................................... 577
16.1 Coupling between rf-Field and Particles ............... 577
16.1.1 Network Modelling of an Accelerating Cavity .... 578
16.2 Beam Loading and rf-System ............................ 581
16.3 Higher Order Mode Losses in an rf-Cavity .............. 587
16.3.1 Efficiency of Energy Transfer from Cavity
to Beam ........................................ 590
16.4 Beam Loading .......................................... 591
16.5 Phase Oscillation and Stability ....................... 593
16.5.1 Robinson Damping ............................... 594
16.5.2 Potential Well Distortion ...................... 598
Part VI Coupled Motion
17 Dynamics of Coupled Motion ................................. 605
17.1 Equations of Motion in Coupled Systems ................ 605
17.1.1 Coupled Beam Dynamics in Skew Quadnipoles ...... 606
17.1.2 Particle Motion in a Solenoidal Field .......... 608
17.1.3 Transformation Matrix for a Solenoid Magnet .... 611
17.2 Betatron Functions for Coupled Motion ................. 614
17.3 Conjugate Trajectories ................................ 614
17.4 Hamiltonian and Coupling .............................. 621
17.4.1 Linearly Coupled Motion ........................ 621
17.4.2 Higher Order Coupling Resonances ............... 630
17.4.3 Multiple Resonances ............................ 630
Part VII Intense Beams
18 Statistical and Collective Effects ......................... 635
18.1 Statistical Effects ................................... 636
18.1.1 Schottky Noise ................................. 636
18.1.2 Stochastic Cooling ............................. 638
18.1.3 Touschek Effect ................................ 638
18.1.4 Intrabeam Scattering ........................... 640
18.2 Collective Self-Fields ................................ 642
18.2.1 Stability of a Charged-Particle Beam ........... 642
18.2.2 Self-Field for Particle Beams .................. 644
18.2.3 Beam Beam Effect ............................... 647
18.2.4 Transverse Self-Fields ......................... 649
18.2.5 Fields from Image Charges ...................... 650
18.2.6 Space-Charge Effects ........................... 655
18.2.7 Longitudinal Space-Charge Field ................ 660
18.3 Beam-Current Spectrum ................................. 662
18.3.1 Longitudinal Beam Spectrum ..................... 662
18.3.2 Transverse Beam Spectrum ....................... 665
19 Wake Fields and Instabilities .............................. 671
19.1 Definitions of Wake Field and Impedance ............... 672
19.1.1 Longitudinal Wake Fields ....................... 678
19.1.2 Transverse Wake Fields ......................... 683
19.1.3 Panofsky-Wcnzel Theorem ........................ 684
19.2 Impedances in an Accelerator Environment .............. 685
19.2.1 Space-Charge Impedance ......................... 686
19.2.2 Resistive Wall Impedance ....................... 686
19.2.3 Cavity-Like Structure Impedance ................ 687
19.2.4 Overall Accelerator Impedance .................. 688
19.2.5 Broad-Band wake Fields in a Linear
Accelerator .................................... 691
19.3 Coasting-Beam Instabilities ........................... 692
19.3.1 Negative-Mass Instability ...................... 692
19.3.2 Dispersion Relation ............................ 695
19.3.3 Landau Damping ................................. 701
19.3.4 Transverse Coasting-Beam Instability ........... 703
19.4 Longitudinal Single-Bunch Effects ..................... 705
19.4.1 Potential Well Distortion ...................... 705
19.5 Transverse Single-Bunch Instabilities ................. 713
19.5.1 Beam Break-up in Linear Accelerators ........... 713
19.5.2 Fast Head-Tail Effect .......................... 715
19.5.3 Head-Tail Instability .......................... 719
19.6 Multibunch Instabilities .............................. 722
Part VIII Synchrotron Radiation
20 Fundamental Processes ...................................... 731
20.1 Radiation from Moving Charges ......................... 731
20.1.1 Why Do Charged Particles Radiate? .............. 732
20.1.2 Spontaneous Synchrotron Radiation .............. 733
20.1.3 Stimulated Radiation ........................... 734
20.1.4 Electron Beam .................................. 735
20.2 Conservation Laws and Radiation ....................... 736
20.2.1 Cherenkov Radiation ............................ 737
20.2.2 Compton Radiation .............................. 738
20.3 Electromagnetic Radiation ............................. 739
20.3.1 Coulomb Regime ................................. 740
20.3.2 Radiation Regime ............................... 741
21 Overview of Synchrotron Radiation .......................... 749
21.1 Radiation Sources ..................................... 750
21.1.1 Bending Magnet Radiation ....................... 750
21.1.2 Superbends ..................................... 751
21.1.3 Wavelength Shifter ............................. 752
21.1.4 Wiggler Magnet Radiation ....................... 753
21.1.5 Undulator Radiation ............................ 757
21.1.6 Back Scattered Photons ......................... 763
21.2 Radiation Power ....................................... 765
21.3 Spectrum .............................................. 768
21.4 Spatial Photon Distribution ........................... 773
21.5 Fraunhofer Diffraction ................................ 775
21.6 Spatial Coherence ..................................... 778
21.7 Temporal Coherence .................................... 780
21.8 Spectral Brightness ................................... 782
21.8.1 Matching ....................................... 783
21.9 Photon Source Parameters .............................. 785
22 Theory of Synchrotron Radiation ............................ 789
22.1 Radiation Field ....................................... 789
22.2 Total Radiation Power and Energy Loss ................. 796
22.2.1 Transition Radiation ........................... 796
22.2.2 Synchrotron Radiation Power .................... 799
22.3 Spatial and Spectral Radiation Distribution ........... 802
22.3.1 Radiation Lobes ................................ 802
22.3.2 Synchrotron Radiation Spectrum ................. 807
22.4 Radiation Field in the Frequency Domain ............... 807
22.4.1 Spectral Distribution in Space and
Polarization ................................... 812
22.4.2 Spectral and Spatial Photon Flux ............... 814
22.4.3 Harmonic Representation ........................ 815
22.4.4 Spatial Radiation Power Distribution ........... 816
22.5 Asymptotic Solutions .................................. 818
22.5.1 Low Frequencies and Small Observation Angles ... 818
22.5.2 High Frequencies or Large Observation Angles ... 818
22.6 Angle-Integrated Spectrum ............................. 819
22.7 Statistical Radiation Parameters ...................... 825
23 Insertion Device Radiation ................................. 829
23.1 Particle Dynamics in a Periodic Field Magnet .......... 831
23.2 Undulator Radiation ................................... 833
23.2.1 Fundamental Wavelength ......................... 833
23.2.2 Radiation Power ................................ 834
23.2.3 Spatial and Spectral Distribution .............. 835
23.2.4 Line Spectrum .................................. 847
23.2.5 Spectral Undulator Brightness .................. 851
23.3 Elliptical Polarization ............................... 852
23.3.1 Elliptical Polarization from Bending Magnet
Radiation ...................................... 853
23.3.2 Elliptical Polarization from Periodic
Insertion Devices .............................. 855
24 Free Electron Lasers ....................................... 861
24.1 Small Gain Regime ..................................... 862
24.1.1 Energy Transfer ................................ 864
24.1.2 Equation of Motion ............................. 866
24.1.3 FEL-Gain ....................................... 868
Solutions ..................................................... 875
Part IX Appendices
Useful Mathematical Formulae .................................. 907
A.l Vector Algebra ........................................ 907
A.1.1 Differential Vector Expressions ................ 907
A.1.2 Algebraic Relations ............................ 908
A.1.3 Differential Relations ......................... 909
A.1.4 Integral Relations ............................. 909
A.1.5 Series Expansions .............................. 909
A.1.6 Fourier Transform .............................. 909
A.1.7 Parceval's Theorem ............................. 910
A.1.8 Coordinate Transformations ..................... 910
Physical Formulae and Parameters .............................. 913
B.l Physical Constants .................................... 913
B.2 Relations of Fundamental Parameters ................... 914
B.3 Unit Conversion ....................................... 914
B.4 Maxwell's Equations ................................... 914
B.5 Wave and Field Equations .............................. 915
B.6 Relativistic Relations ................................ 916
B.6.1 Lorentz Transformation ......................... 916
B.6.2 Four-Vectors ................................... 917
B.6.3 Square of the 4-acceleration ................... 918
B.6.4 Miscellaneous 4-Vectors and Lorentz Invariant
Properties ..................................... 918
Transformation Matrices in Beam Dynamics ...................... 919
C.l General Transformation Matrix ......................... 920
C.l.l Symmetric Magnet Arrangement ................... 920
C.l.2 Inverse Transformation Matrix .................. 920
C.2 Specific Transformation Matrices ...................... 921
C.2.1 Drift Space .................................... 921
C.2.2 Bending Magnets ................................ 921
C.2.3 Quadrupol ...................................... 923
References .................................................... 925
Index ......................................................... 917
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