Solutions Manual for

Solutions Manual for Hands-On Accelerator Physics Using MATLAB ⃝R

• Ziemann

Contents

• Introduction 4
• For Chapter 2 4

2.1 FODO rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.2 Racetrack . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.3 Dog-leg . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.4 Underpass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.8 Equations of motion . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.9 Moments of distributions . . . . . . . . . . . . . . . . . . . . . . . . 8 2.10 Projections are also Gaussians . . . . . . . . . . . . . . . . . . . . . . 9 2.11 27183 random numbers . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.12 31416 random numbers . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.13 Fitting distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.14 Adding Gaussian random variables . . . . . . . . . . . . . . . . . . . 11 2.15 Adding uniformly distributed random variables . . . . . . . . . . . . 11 2.16 Adding Lorentz-distributed random variables . . . . . . . . . . . . . 12

• For Chapter 3 12

3.1 Phase-advance of FODO cell with thin quadrupoles . . . . . . . . . . 12 3.2 Phase-advance of FODO cell with thick quadrupoles . . . . . . . . . 13 3.3 Phase-space plots and trajectories . . . . . . . . . . . . . . . . . . . 13 3.4 Matching section between FODO cells, 2D . . . . . . . . . . . . . . . 14 3.5 Limit of stability of FODO cell . . . . . . . . . . . . . . . . . . . . . 15 3.6 Make small spots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 3.7 Dispersion in doublet lattice . . . . . . . . . . . . . . . . . . . . . . . 17 3.8 Matching section between FODO cells, 4D . . . . . . . . . . . . . . . 18 3.9 Necktie diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 3.10 Analyze FODO ring . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 3.11 Analyze doublet ring . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 3.12 Achromat . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 3.13 Bunch compressor . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 3.14 Ring with rolled quadrupole . . . . . . . . . . . . . . . . . . . . . . . 23

• For Chapter 4 26

4.1 Ampere-turns for dipole . . . . . . . . . . . . . . . . . . . . . . . . . 26 4.2 Ampere-turns for a quadrupole . . . . . . . . . . . . . . . . . . . . . 26 4.3 Dipole with bad iron near gap . . . . . . . . . . . . . . . . . . . . . . 26 4.4 Dipole with bad iron near top . . . . . . . . . . . . . . . . . . . . . . 28 4.5 Bumped dipole . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 1 NOTE: For Complete File, Download link at the at the end of this File 1 / 4

4.6 The other bumped dipole . . . . . . . . . . . . . . . . . . . . . . . . 30 4.7 Narrower quadrupole . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 4.8 Sextupole . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 4.9 Magnets with super-conducting laments . . . . . . . . . . . . . . . 34 4.10 Bad coils in super-conducting dipole . . . . . . . . . . . . . . . . . . 35 4.11 Try to x the problem with the bad coils . . . . . . . . . . . . . . . 36 4.12 Halbach dipole . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 4.13 Field in undulator . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 4.14 Compensate poor permanent magnets in undulator . . . . . . . . . . 37 4.15 Rotating radial coil . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 4.16 Rotating tangential coil . . . . . . . . . . . . . . . . . . . . . . . . . 38

• For Chapter 539

5.1 Why accelerate with TM-modes? . . . . . . . . . . . . . . . . . . . . 39 5.2 Why do TE-modes normally not work? . . . . . . . . . . . . . . . . . 39 5.3 Example of a TEM{mode . . . . . . . . . . . . . . . . . . . . . . . . 39 5.4 Plot elds for TM010{mode . . . . . . . . . . . . . . . . . . . . . . . 39 5.5 Plot elds for a few other TMmnp{mode . . . . . . . . . . . . . . . . 40 5.6 100 MHz pill-box cavity . . . . . . . . . . . . . . . . . . . . . . . . . 41 5.7 Synchrotron frequency in CELSIUS . . . . . . . . . . . . . . . . . . . 41 5.8 Bucket half-height in electron ring . . . . . . . . . . . . . . . . . . . 42 5.9 Bunch-rotation simulation . . . . . . . . . . . . . . . . . . . . . . . . 42 5.10 De-bunching simulation . . . . . . . . . . . . . . . . . . . . . . . . . 43 5.11 Re-bunching simulation . . . . . . . . . . . . . . . . . . . . . . . . . 44 5.12 Re-bunching at higher harmonic . . . . . . . . . . . . . . . . . . . . 45

• For Chapter 645

6.1 Cutoff frequency for rectangular wave guide . . . . . . . . . . . . . . 45 6.2 Why are square waveguides unsuitable . . . . . . . . . . . . . . . . . 45 6.3Hzin several TE{modes . . . . . . . . . . . . . . . . . . . . . . . . . 46 6.4 Transverse elds of several TE{modes . . . . . . . . . . . . . . . . . 46 6.5 Dented WR-340 waveguide . . . . . . . . . . . . . . . . . . . . . . . 46 6.6 Triangular waveguide . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 6.7 Transmission line with transformers . . . . . . . . . . . . . . . . . . 48 6.8 Reection coefficient . . . . . . . . . . . . . . . . . . . . . . . . . . 48 6.9 Buckled pillbox cavity . . . . . . . . . . . . . . . . . . . . . . . . . . 49 6.10 Generator current and phase . . . . . . . . . . . . . . . . . . . . . . 50 6.11 P-controller in transient beam-loading simulation . . . . . . . . . . . 51

• For Chapter 752

7.1 Quadrupole scan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 7.2 Three wire scanners . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 7.3 Four wire scanners . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 7.4 Optimum phase advance . . . . . . . . . . . . . . . . . . . . . . . . . 55 7.5 Tunes from turn-by-turn data . . . . . . . . . . . . . . . . . . . . . . 55

• For Chapter 856

8.1 IP-steering knob . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 8.2 Closed IP-knob . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 8.3 Orbit correction in doublet lattice . . . . . . . . . . . . . . . . . . . 57 8.4 Orbit correction in FODO ring . . . . . . . . . . . . . . . . . . . . . 60 8.5 Orbit correction with an additional steering magnet . . . . . . . . . 63 8.6 Displacement of long quadrupoles . . . . . . . . . . . . . . . . . . . . 64 8.7 Random quadrupole errors . . . . . . . . . . . . . . . . . . . . . . . . 65

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8.8 Correct chromaticity . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 8.9 Tune diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 8.10 LOCO in a toy-ring . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 8.11 Dispersion term in response coefficient . . . . . . . . . . . . . . . . . 73

• For Chapter 973

9.1 Target thickness to get data over a weekend . . . . . . . . . . . . . . 73 9.2 Energy loss in a lithium target . . . . . . . . . . . . . . . . . . . . . 73 9.3 Luminosity for elliptic beams . . . . . . . . . . . . . . . . . . . . . . 74 9.4 Beam-beam deections in MATLAB . . . . . . . . . . . . . . . . . . 74 9.5 Weak-strong beam-beam simulation . . . . . . . . . . . . . . . . . . 75 9.6 Add displacement todisruption.m. . . . . . . . . . . . . . . . . . 76 10 For Chapter 1078 10.1 Radiation loss in LEP . . . . . . . . . . . . . . . . . . . . . . . . . . 78 10.2 Numerically evaluate synchrotron radiation integrals in FODO ring . 78 10.3 Numerically evaluate synchrotron radiation integrals in doublet ring 80 10.4 Undulator parameters . . . . . . . . . . . . . . . . . . . . . . . . . . 80 10.5 Phase shifter tolerance . . . . . . . . . . . . . . . . . . . . . . . . . . 80 10.6 Effect of Momentum Spread in FEL oscillator . . . . . . . . . . . . . 81 11 For Chapter 1183 11.1 Amplitude-dependent tune shift from tracking data . . . . . . . . . . 83 11.2 Tracking with octupole and decapole . . . . . . . . . . . . . . . . . . 84 11.3 1000-turn dynamic aperture . . . . . . . . . . . . . . . . . . . . . . . 85 11.4 Survival plots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 11.5 Hamiltonian of thin corrector and other magnets . . . . . . . . . . . 87 11.6 Kicks from the Hamiltonians in the previous exercise . . . . . . . . . 87 11.7 Beam line with two steering magnets . . . . . . . . . . . . . . . . . . 87 11.8 Beam line with two other magnets . . . . . . . . . . . . . . . . . . . 88 11.9 Simulation with two sextupoles . . . . . . . . . . . . . . . . . . . . . 89 11.10Transverse displacements . . . . . . . . . . . . . . . . . . . . . . . . 90 11.11Beam line with three octupoles . . . . . . . . . . . . . . . . . . . . . 90 11.12Amplitude-dependent tune shift from normal forms . . . . . . . . . . 90 12 For Chapter 1291 12.1 Space-charge tune shift in CELSIUS . . . . . . . . . . . . . . . . . . 91 12.2 Space-charge tune shift for generalAandZ. . . . . . . . . . . . . . 91 12.3 Space-charge in doublet lattice . . . . . . . . . . . . . . . . . . . . . 92 12.4 Space-charge in a ring . . . . . . . . . . . . . . . . . . . . . . . . . . 95 12.5 Touschek effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 12.6 Wakes from box distribution . . . . . . . . . . . . . . . . . . . . . . . 97 12.7 Wakes from parabolic distribution . . . . . . . . . . . . . . . . . . . 99 12.8 Stability diagram for parabolic distribution . . . . . . . . . . . . . . 100 12.9 Keil-Schnell criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 12.10Phase-shift due to beam current . . . . . . . . . . . . . . . . . . . . 100 12.11Current dependence of the synchrotron tune . . . . . . . . . . . . . . 101 12.12Coupled-bunch instability . . . . . . . . . . . . . . . . . . . . . . . . 101 13 For Chapter 13103 13.1 Parameters in electron gun . . . . . . . . . . . . . . . . . . . . . . . 103 13.2 Pump-down time scale . . . . . . . . . . . . . . . . . . . . . . . . . . 104 13.3 Differential pumping . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 13.4 Average dose . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

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• Introduction

The material com-prises of all MATLAB code and the online appendices from the book. Note that Appendix B.5 contains discussions of the code examples. Since the solutions, dis-cussed in this manual, are based on the code examples from the book, it is strongly recommended to download the additional material, which is freely available from the web site mentioned above.

• For Chapter 2

For the design of the beam lines in the rst few exercises, it is convenient to encap- sulate most of the script layout.m in a separate function, such that it can be used repeatedly without excessive copying.% layout_function.m function vvpos=layout_function(beamline) hold on nlines=size(beamline,1); % number of lines

nele=sum(beamline(:,2))+1; % number of elements

vvpos=zeros(3,nele); % element positions f=fopen('layout.scad','w'); % open output file for 3D view vv=[0;0;0];% x,y,z or origin ww=eye(3);% orientation of tripod ic=1;% element counter

for line=1:nlines% loop over input elements

for seg=1:beamline(line,2) % loop over repeat-count

v0=vv; w0=ww;% remember previous point ic=ic+1; switch beamline(line,1) case {1,2,5,7} % drift, quadrupole, solenoid dv=[0;0;beamline(line,3)]; dw=eye(3); case 4 % sector dipole phi=beamline(line,4)*pi/180; % convert to radians if abs(phi)>1e-7 rho=beamline(line,3)/phi;% bending radius dv=[rho*(cos(phi)-1);0.0;rho*sin(phi)]; % sagitta dw=wmake(0,-phi,0); dw2=wmake(0,-phi/2,0); % for 3D renderer only else dv=[0;0;beamline(line,3)]; dw=eye(3); end

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