particle accelerator physics1.5.7 chargedparticles in anelectromagneticfield 33 1.5.8...
TRANSCRIPT
Helmut Wiedemann
Particle Accelerator Physics
Fourth Edition
^ Springer
Contents
Part I Introduction
1 Introduction to Accelerator Physics 3
1.1 Short Historical Overview 3
1.2 Particle Accelerator Systems 7
1.2.1 Main Components of Accelerator Facilities 7
1.2.2 Applications of Particle Accelerators 10
1.3 Definitions and Formulas 11
1.3.1 Units and Dimensions 11
1.3.2 Maxwell's Equations 13
1.4 Primer in Special Relativity 14
1.4.1 Lorentz Transformation 15
1.4.2 Lorentz Invariance 18
1.4.3 Spatial and Spectral Distribution of Radiation 22
1.4.4 Particle Collisions at High Energies 24
1.5 Principles of Particle-Beam Dynamics 26
1.5.1 Electromagnetic Fields of Charged Particles 26
1.5.2 Vector and Scalar Potential 27
1.5.3 Wave Equation 28
1.5.4 Induction 30
1.5.5 Lorentz Force 30
1.5.6 Equation of Motion 31
1.5.7 Charged Particles in an Electromagnetic Field 33
1.5.8 Linear Equation of Motion 34
1.5.9 Energy Conservation 35
1.5.10 Stability of a Charged-Particle Beam 37
References 41
2 Linear Accelerators 43
2.1 Principles of Linear Accelerators 43
2.1.1 Charged Particles in Electric Fields 44
2.1.2 Electrostatic Accelerators 45
xvii
xviii Contents
2.2 Electric Field Components 48
2.2.1 Electrostatic Deflectors 48
2.2.2 Electrostatic Focusing Devices 49
2.2.3 Iris Doublet 51
2.2.4 Einzellens 52
2.3 Acceleration by rf Fields 54
2.3.1 Basic Principle of Microwave Linear Accelerators .... 54
References 57
3 Circular Accelerators 59
3.1 Betatron 60
3.2 Weak Focusing 63
3.3 Adiabatic Damping 66
3.4 Acceleration by rf Fields 68
3.4.1 Microtron 68
3.4.2 Cyclotron 70
3.4.3 Synchro-Cyclotron 73
3.4.4 Isochron Cyclotron 74
3.4.5 Synchrotron 75
3.4.6 Storage Ring 77
3.4.7 Summary of Characteristic Parameters 77
References 79
Part II Tools We Need
4 Elements of Classical Mechanics 83
4.1 How to Formulate a Lagrangian? 85
4.1.1 The Lagrangian for a Charged Particle
in an EM-Field 85
4.2 Lorentz Force 86
4.3 Frenet-Serret Coordinates 87
4.4 Hamiltonian Formulation 88
4.4.1 Cyclic Variables 90
4.4.2 Canonical Transformations 90
4.4.3 Curvilinear Coordinates 93
4.4.4 Extended Hamiltonian 95
4.4.5 Change of Independent Variable 96
References 98
5 Particle Dynamics in Electro-Magnetic Fields 99
5.1 The Lorentz Force 99
5.2 Fundamentals of Charged Particle Beam Optics 100
5.2.1 Particle Beam Guidance 100
5.2.2 Particle Beam Focusing 102
5.3 Equation of Motion 106
Contents xix
5.4 Equations of Motion from the Lagrangian and Hamiltonian 109
5.4.1 Equations of Motion from Lagrangian 110
5.4.2 Canonical Momenta 112
5.4.3 Equation of Motion from Hamiltonian 112
5.4.4 Harmonic Oscillator 114
5.4.5 Action-Angle Variables 115
5.5 Solutions of the Linear Equations of Motion 116
5.5.1 Linear Unperturbed Equation of Motion 117
5.5.2 Matrix Formulation 118
5.5.3 Wronskian 119
5.5.4 Perturbation Terms 120References 123
6 Electromagnetic Fields 125
6.1 Pure Multipole Field Expansion 125
6.1.1 Electromagnetic Potentials and Fields
for Beam Dynamics 126
6.1.2 Fields, Gradients and Multipole Strength Parameter... 128
6.1.3 Main Magnets for Beam Dynamics 131
6.1.4 Multipole Misalignment and "Spill-down" 1376.2 Main Magnet Design Criteria 138
6.2.1 Design Characteristics of Dipole Magnets 138
6.2.2 Quadrupole Design Concepts 140
6.3 Magnetic Field Measurement 145
6.3.1 Hall Probe 147
6.3.2 Rotating Coil 148
6.4 General Transverse Magnetic-Field Expansion 152
6.4.1 Pure Multipole Magnets 153
6.4.2 Kinematic Terms 155
6.5 Third-Order Differential Equation of Motion 160
6.6 Longitudinal Field Devices 165
6.7 Periodic Wiggler Magnets 167
6.7.1 Wiggler Field Configuration 168
6.8 Electrostatic Quadrupole 172
References 174
Part III Beam Dynamics
7 Single Particle Dynamics 177
7.1 Linear Beam Transport Systems 178
7.1.1 Nomenclature 179
7.2 Matrix Formalism in Linear Beam Dynamics 180
7.2.1 Driftspace 182
7.2.2 Quadrupole Magnet 182
7.2.3 Thin Lens Approximation 184
7.2.4 Quadrupole End Field Effects 187
xx Contents
7.3 Focusing in Bending Magnets 190
7.3.1 Sector Magnets 191
7.3.2 Fringe Field Effects 193
7.3.3 Finite Pole Gap 195
7.3.4 Wedge Magnets 196
7.3.5 Rectangular Magnet 198
7.3.6 Focusing in a Wiggler Magnet 200
7.3.7 Hard-Edge Model of Wiggler Magnets 203
7.4 Elements of Beam Dynamics 205
7.4.1 Building Blocks for Beam Transport Lines 205
7.4.2 Isochronous Systems 208
References 211
8 Particle Beams and Phase Space 213
8.1 Beam Emittance 214
8.1.1 Liouville's Theorem 215
8.1.2 Transformation in Phase Space 218
8.1.3 Beam Matrix 222
8.2 Betatron Functions 227
8.2.1 Beam Envelope 230
8.3 Beam Dynamics in Terms of Betatron Functions 231
8.3.1 Beam Dynamics in Normalized Coordinates 233
8.4 Dispersive Systems 236
8.4.1 Analytical Solution 237
8.4.2 3 x 3-Transformation Matrices 238
8.4.3 Linear Achromat 240
8.4.4 Spectrometer 244
8.4.5 Measurement of Beam Energy Spectrum 245
8.4.6 Path Length and Momentum Compaction 248
References 251
9 Longitudinal Beam Dynamics 253
9.1 Longitudinal Particle Motion 254
9.1.1 Longitudinal Phase Space Dynamics 256
9.2 Equation of Motion in Phase Space 259
9.2.1 Small Oscillation Amplitudes 262
9.2.2 Phase Stability 266
9.2.3 Acceleration of Charged Particles 270
9.3 Longitudinal Phase Space Parameters 274
9.3.1 Separatri x Parameters 274
9.3.2 Momentum Acceptance 275
9.3.3 Bunch Length 278
9.3.4 Longitudinal Beam Emittance 280
9.3.5 Phase Space Matching 282
Contents xxi
9.4 Higher-Order Phase Focusing 286
9.4.1 Dispersion Function in Higher Order 287
9.4.2 Path Length in Higher Order 289
9.4.3 Higher Order Momentum Compaction Factor 291
9.4.4 Higher-Order Phase Space Motion 292
9.4.5 Stability Criteria 296
References 302
10 Periodic Focusing Systems 303
10.1 FODO Lattice 304
10.1.1 Scaling of FODO Parameters 305
10.1.2 Betatron Motion in Periodic Structures 309
10.1.3 General FODO Lattice 311
10.2 Beam Dynamics in Periodic Closed Lattices 315
10.2.1 Hill's Equation 315
10.2.2 Periodic Betatron Functions 318
10.2.3 Periodic Dispersion Function 321
10.2.4 Periodic Lattices in Circular Accelerators 329
10.3 FODO Lattice and Acceleration 339
10.3.1 Lattice Structure 339
10.3.2 Transverse Beam Dynamics and Acceleration 341
References 349
Part IV Beam Parameters
11 Particle Beam Parameters 353
11.1 Definition of Beam Parameters 353
11.1.1 Beam Energy 353
11.1.2 Time Structure 354
11.1.3 Beam Current 354
11.1.4 Beam Dimensions 356
11.2 Damping 358
11.2.1 Robinson Criterion 358
11.3 Particle Distribution in Longitudinal Phase Space 365
11.3.1 Energy Spread 366
11.3.2 Bunch Length 368
11.4 Transverse Beam Emittance 368
11.4.1 Equilibrium Beam Emittance 369
11.4.2 Emittance Increase in a Beam Transport Line 371
11.4.3 Vertical Beam Emittance 371
11.4.4 Beam Sizes 373
11.4.5 Beam Divergence 375
11.5 Variation of the Damping Distribution 375
11.5.1 Damping Partition and Rf-Frequency 375
xxii Contents
11.6 Variation of the Equilibrium Beam Emittance 377
11.6.1 Beam Emittance and Wiggler Magnets 377
11.6.2 Damping Wigglers 380
11.7 Robinson Wiggler 382
11.7.1 Damping Partition and Synchrotron Oscillation 382
11.7.2 Can We Eliminate the Beam Energy Spread? 384
11.8 Beam Life Time 385
11.8.1 Beam Lifetime and Vacuum 386
11.8.2 Ultra High Vacuum System 395
References 399
12 Vlasov and Fokker-Planck Equations 401
12.1 The Vlasov Equation 402
12.1.1 Betatron Oscillations and Perturbations 408
12.1.2 Damping 410
12.2 Damping of Oscillations in Electron Accelerators 411
12.2.1 Damping of Synchrotron Oscillations 412
12.2.2 Damping of Vertical Betatron Oscillations 416
12.2.3 Robinson's Damping Criterion 419
12.2.4 Damping of Horizontal Betatron Oscillations 422
12.3 The Fokker-Planck Equation 422
12.3.1 Stationary Solution of the Fokker-Planck Equation ...425
12.3.2 Particle Distribution within a Finite Aperture 430
12.3.3 Particle Distribution in the Absence of Damping 432
References 435
13 Equilibrium Particle Distribution 437
13.1 Particle Distribution in Phase Space 437
13.1.1 Diffusion Coefficient and Synchrotron Radiation 438
13.1.2 Quantum Excitation of Beam Emittance 440
13.2 Equilibrium Beam Emittance 441
13.2.1 Horizontal Equilibrium Beam Emittance 441
13.2.2 Vertical Equilibrium Beam Emittance 442
13.3 Equilibrium Energy Spread and Bunch Length 444
13.3.1 Equilibrium Beam Energy Spread 444
13.3.2 Equilibrium Bunch Length 444
13.4 Phase-Space Manipulation 446
13.4.1 Exchange of Transverse Phase-Space Parameters 446
13.4.2 Bunch Compression 446
13.4.3 Alpha Magnet 449
13.5 Polarization of a Particle Beam 453
References 457
Contents xxiii
14 Beam Emittance and Lattice Design 459
14.1 Equilibrium Beam Emittance in Storage Rings 461
14.1.1 FODO Lattice 461
14.1.2 Minimum Beam Emittance 462
14.2 Absolute Minimum Emittance 465
14.3 Beam Emittance in Periodic Lattices 468
14.3.1 The Double Bend Achromat Lattice (DBA) 469
14.3.2 The FODO Lattice 470
14.3.3 Optimum Emittance for Colliding Beam
Storage Rings 472
References 472
Part V Perturbations
15 Perturbations in Beam Dynamics 477
15.1 Magnet Field and Alignment Errors 478
15.1.1 Self Compensation of Perturbations 479
15.2 Dipole Field Perturbations 480
15.2.1 Dipole Field Errors and Dispersion Function 482
15.2.2 Perturbations in Open Transport Lines 482
15.2.3 Existence of Equilibrium Orbits 484
15.2.4 Closed Orbit Distortion 486
15.2.5 Statistical Distribution of Dipole Field
and Alignment Errors 490
15.2.6 Dipole Field Errors in Insertion Devices 492
15.2.7 Closed Orbit Correction 494
15.2.8 Response Matrix 496
15.2.9 Orbit Correction with Single Value
Decomposition (SVD) 497
15.3 Quadrupole Field Perturbations 499
15.3.1 Betatron Tune Shift 500
15.3.2 Optics Perturbation Due to Insertion Devices 502
15.3.3 Resonances and Stop Band Width 503
15.3.4 Perturbation of Betatron Function 506
15.4 Chromatic Effects in a Circular Accelerator 509
15.4.1 Chromaticity 509
15.4.2 Chromaticity Correction 513
15.4.3 Chromaticity in Higher Approximation 514
15.4.4 Non-linear Chromaticity 517
15.5 Kinematic Perturbation Terms 522
15.6 Perturbation Methods in Beam Dynamics 524
15.6.1 Periodic Distribution of Statistical Perturbations 525
15.6.2 Periodic Perturbations in Circular Accelerators 528
15.6.3 Statistical Methods to Evaluate Perturbations 530
xxjv Contents
15.7 Control of Beam Size in Transport Lines 531
References 538
16 Resonances 539
16.1 Lattice Resonances 539
16.1.1 Resonance Conditions 540
16.1.2 Coupling Resonances 544
16.1.3 Resonance Diagram 545
16.2 Hamiltonian Resonance Theory 547
16.2.1 Non-linear Hamiltonian 547
16.2.2 Resonant Terms 550
16.2.3 Resonance Patterns and Stop-Band Width 553
16.2.4 Half-Integer Stop-Band 555
16.2.5 Separatrices 556
16.2.6 General Stop-Band Width 558
16.3 Third-Order Resonance 560
16.3.1 Particle Motion in Phase Space 563
References 564
17 Hamiltonian Nonlinear Beam Dynamics 565
17.1 Higher-Order Beam Dynamics 565
17.1.1 Multipole Errors 565
17.1.2 Non-linear Matrix Formalism 569
17.2 Aberrations 573
17.2.1 Geometric Aberrations 575
17.2.2 Filamentation of Phase Space 581
17.2.3 Chromatic Aberrations 584
17.2.4 Particle Tracking 587
17.3 Hamiltonian Perturbation Theory 588
17.3.1 Tune Shift in Higher Order 595
References 599
Part VI Acceleration
18 Charged Particle Acceleration 603
18.1 Rf-Waveguides and Cavities 603
18.1.1 Wave Equation 604
18.1.2 Rectangular Waveguide Modes 605
18.1.3 Cylindrical Waveguide Modes 610
18.2 Rf-Cavities 614
18.2.1 Square Cavities 614
18.2.2 Cylindrical Cavity 614
18.2.3 Energy Gain 616
18.2.4 Rf-Cavity as an Oscillator 617
18.2.5 Cavity Losses and Shunt Impedance 619
Contents xxv
18.3 Rf-Parameters 623
18.3.1 Synchronous Phase and Rf-voltage 62518.4 Linear Accelerator 625
18.4.1 Basic Waveguide Parameters 626
18.4.2 Particle Capture in a Linear Accelerator Field 632
18.5 Preinjector and Beam Preparation 634
18.5.1 Prebuncher 634
18.5.2 Beam Chopper 636
18.5.3 Buncher Section 638
References 640
19 Beam-Cavity Interaction 641
19.1 Coupling Between rf-Field and Particles 641
19.1.1 Network Modelling of an Accelerating Cavity 642
19.2 Beam Loading and Rf-System 645
19.3 Higher-Order Mode Losses in an Rf-Cavity 650
19.3.1 Efficiency of Energy Transfer from Cavity to Beam...
653
19.4 Beam Loading 654
19.5 Phase Oscillation and Stability 656
19.5.1 Robinson Damping 657
19-5.2 Potential Well Distortion 662
References 665
Part VII Coupled Motion
20 Dynamics of Coupled Motion 669
20.1 Equations of Motion in Coupled Systems 669
20.1.1 Coupled Beam Dynamics in Skew Quadrupoles 670
20.1.2 Particle Motion in a Solenoidal Field 672
20.1.3 Transformation Matrix for a Solenoid Magnet 675
20.2 Betatron Functions for Coupled Motion 678
20.3 Conjugate Trajectories 679
20.4 Hamiltonian and Coupling 685
20.4.1 Linearly Coupled Motion 686
20.4.2 Higher-Order Coupling Resonances 695
20.4.3 Multiple Resonances 695
References 697
Part VIII Intense Beams
21 Statistical and Collective Effects 701
21.1 Statistical Effects 702
21.1.1 Schottky Noise 702
21.1.2 Stochastic Cooling 704
21.1.3 Touschek Effect 705
21.1.4 Intra-Beam Scattering 706
xxvi Contents
21.2 Collective Self Fields 708
21.2.1 Self Field for Elliptical Particle Beams 709
21.2.2 Beam-Beam Effect 712
21.2.3 Transverse Self Fields 715
21.2.4 Fields from Image Charges 715
21.2.5 Space-Charge Effects 720
21.2.6 Longitudinal Space-Charge Field 725
21.3 Beam-Current Spectrum 727
21.3.1 Longitudinal Beam Spectrum 727
21.3.2 Transverse Beam Spectrum 730
References 734
22 Wake Fields and Instabilities 737
22.1 Definitions of Wake Field and Impedance 738
22.1.1 Parasitic Mode Losses and Impedances 739
22.1.2 Longitudinal Wake Fields 743
22.1.3 Transverse Wake Fields 749
22.1.4 Panofsky-Wenzel Theorem 750
22.2 Impedances in an Accelerator Environment 751
22.2.1 Space-Charge Impedance 751
22.2.2 Resistive-Wall Impedance 752
22.2.3 Cavity-Like Structure Impedance 753
22.2.4 Overall Accelerator Impedance 754
22.2.5 Broad-Band Wake Fields in a Linear Accelerator 756
22.3 Coasting-Beam Instabilities 756
22.3.1 Negative-Mass Instability 757
22.3.2 Dispersion Relation 760
22.3.3 Landau Damping 767
22.3.4 Transverse Coasting-Beam Instability 769
22.4 Longitudinal Single-Bunch Effects 771
22.4.1 Potential-Well Distortion 771
22.5 Transverse Single-Bunch Instabilities 779
22.5.1 Beam Break-Up in Linear Accelerators 779
22.5.2 Fast Head-Tail Effect 781
22.5.3 Head-Tail Instability 786
22.6 Multi-Bunch Instabilities 789
References 795
Part IX Synchrotron Radiation
23 Fundamental Processes 799
23.1 Radiation from Moving Charges 799
23.1.1 Why Do Charged Particles Radiate? 800
23.1.2 Spontaneous Synchrotron Radiation 801
23.1.3 Stimulated Radiation 802
23.1.4 Electron Beam 803
Contents xxvii
23.2 Conservation Laws and Radiation 804
23.2.1 Cherenkov Radiation 805
23.2.2 Compton Radiation 806
23.3 Electromagnetic Radiation 807
23.3.1 Coulomb Regime 808
23.3.2 Radiation Regime 809
References 813
24 Overview of Synchrotron Radiation 815
24.1 Radiation Sources 816
24.1.1 Bending Magnet Radiation 816
24.1.2 Superbends 817
24.1.3 Wavelength Shifter 818
24.1.4 Wiggler Magnet Radiation 819
24.1.5 Undulator Radiation 822
24.2 Radiation Power 830
24.3 Spectrum 834
24.4 Spatial Photon Distribution 839
24.5 Fraunhofer Diffraction 840
24.6 Spatial Coherence 843
24.7 Temporal Coherence 846
24.8 Spectral Brightness 848
24.8.1 Matching 849
24.9 Photon Source Parameters 851
References 854
25 Theory of Synchrotron Radiation 857
25.1 Radiation Field 857
25.2 Total Radiation Power and Energy Loss 864
25.2.1 Transition Radiation 865
25.3 Spatial Radiation Distribution 868
25.3.1 Radiation Lobes 868
25.4 Radiation Field in the Frequency Domain 873
25.4.1 Spectral Distribution in Space and Polarization 877
25.4.2 Spectral and Spatial Photon Flux 879
25.4.3 Harmonic Representation 880
25.4.4 Spatial Radiation Power Distribution 881
25.5 Asymptotic Solutions 883
25.5.1 Low Frequencies and Small Observation Angles 884
25.5.2 High Frequencies or Large Observation Angles 884
25.6 Angle-Integrated Spectrum 885
25.7 Statistical Radiation Parameters 891
References 893
xxviii Contents
26 Insertion Device Radiation 895
26.1 Particle Dynamics in a Periodic Field Magnet 896
26.2 Undulator Radiation 899
26.2.1 Fundamental Wavelength 899
26.2.2 Radiation Power 900
26.2.3 Spatial and Spectral Distribution 901
26.2.4 Line Spectrum 914
26.2.5 Spectral Undulator Brightness 917
26.3 Elliptical Polarization 918
26.3.1 Elliptical Polarization from Bending Magnet
Radiation 918
26.3.2 Elliptical Polarization from Periodic Insertion
Devices 921
References 927
27 Free Electron Lasers 929
27.1 Small Gain Regime 930
27.1.1 Energy Transfer 932
27.1.2 Equation of Motion 934
27.1.3 FEL-Gain 937
27.2 High Gain Free Electron Laser 942
27.2.1 Electron Dynamics in a SASE FEL 942
27.2.2 Electron Source 945
27.2.3 Beam Dynamics 945
27.2.4 Undulator 947
References 947
Solutions 949
A Useful Mathematical Formulae 983
A.l Vector Algebra 983
A.1.1 Differential Vector Expressions 984
A.1.2 Algebraic Relations 984
A. 1.3 Differential Relations 985
A. 1.4 Partial Integration 985
A. 1.5 Trigonometric and Exponential Functions 985
A. 1.6 Integral Relations 986
A.l .7 Dirac's Delta Function 986
A. 1.8 Bessel's Functions 986
A. 1.9 Series Expansions 987
A. 1.10 Fourier Series 987
A. 1.11 Coordinate Transformations 988
Contents xx'x
B Physical Formulae and Parameters 993
B.l Physical Constants 993
B.2 Relations ofFundamental Parameters 994
B.3 Unit Conversions 994
B.4 Maxwell's Equations 995
B.5 Wave and Field Equations 995
B.6 Relativistic Relations 996
B.6.1 Lorentz Transformation 996
B.6.2 Four-Vectors 997
B.6.3 Square of the 4-Acceleration 998
B.6.4 Miscellaneous 4-Vectors and Lorentz
Invariant Properties 998
B.7 Transformation Matrices in Beam Dynamics 998
B.8 General Transformation Matrix 999
B.8.1 Symmetric Magnet Arrangement 999
B.8.2 Inverse Transformation Matrix 1000
B.9 Specific Transformation Matrices 1000
B.9.1 Drift Space 1000
B.9.2 Bending Magnets 1000
B.9.3 Quadrupole 1003
Index 1005