1. Transverse Beam Dynamics Part 2 Piotr Skowronski In large majority based on slides of O.Brüning http://bruening.web.cern.ch/bruening/ and http://cds.cern.ch/record/941313?ln=en B.Holzer https://indico.cern.ch/event/173359/contribution/9/material/0/0.pdf W.Herr http://zwe.home.cern.ch/zwe/ Y.Papaphilippou http://yannis.web.cern.ch/yannis/teaching/ B. Goddard

2. Overview Next in transverse beam dynamics to follow – Errors from different kinds of elements – Orbit correction – Decoherence – Filamentation – Amplitude detuning – Resonances – Tune diagram – Dynamic aperture – Coupling 2

3. Exercise 1 Simulate motion of a particle with some initial x and p x around a ring – Observe particles at injection where β x = 10 m, α x =0, ring phase advance 10.29×2π – Plot the coordinates every turn for 50k turns Assume different phase advance and observe how the particle moves. In particular when tune is close to –1–1 – 0.5 – 0.333 Put 10 particles with different amplitudes up to 10mm Add a small dipole error. Normally magnet errors are within 1e-4. Bring the tune towards integer 3

4. Integer resonance Dipole kicks add up A smallest dipole error make the particles drift away 4

5. Move away from integer to 0.5 Add quadrupole error and choose a random (not a round) tune Slowly, step by step, bring tune close to 0.5 5 Exercise 2

6. Half-integer resonance Dipole errors are nicely damped But quadrupole errors add-up 6

7. Exercise 3 Add sextupole kick and see what happens around 1/3 tune Octupole and close to 1/4 – Why the tune needs to be little bit less to observe the features? Amplitude detuning 7

8. Exercise 4 Add the second plane, implement a sextupole kick Keep some sextupole and choose tunes 0.18 and 0.62 Bring the vertical tune towards 0.59 Why do you see the “features”? What is the condition to observe these features? 8

9. Saturn The gaps between rings are due to resonances 9

10. Resonance condition 10

11. MAD-X Methodological Accelerator Design version X http://mad.web.cern.ch/mad/ Tutorials https://indico.cern.ch/category/5177/ 11

12. Error Sources Linear field errors – Magnetic field imperfections from design values – Powering errors – Feed-down errors due to transverse alignment errors or bad orbit (regular precision: ± 0.1 mm) – Roll positioning errors (creates coupling) (± 0.5 mrad) – Beam energy errors (change of normalized strength) – Calibration Non-linear field errors – Naturally it is impossible to build a perfect magnet 12

13. Accelerator sensitivity Off-center magnets change the beam optics – Dispersion and focusing Usually not possible to correct entirely – It spoils emittance and luminosity Alignment Vibrations and ground motion 13

14. LEP: When people sleep the machine is stable 14 Accelerator sensitivity

15. LEP: Trains changed the bending magnet field 15

16. LEP sensitivity LEP energy (green dots) and expected effect due to two-body motion of the moon and the earth Below LEP energy and orbit change with the lake water level 16

17. Accelerator sensitivity Magnet motion in HERA, Hamburg 17

18. Accelerator Sensitivity LEP beam felt moon position (tides) and the lake water level Linear Colliders with their nano-meter emittance care about nano-meter ground motion – For example, Atlantic haul is sensible in Geneva at this level – Need active stabilization – Beam based alignment 18

19. Feed-down errors 19

20. Skew Multipoles Example: Skew Quadrupole Quadrupole rotated by 45° gives a skew quadrupole Horizontal kick depends on vertical offset 20

21. Errors Treatment Perturbed Hills equation The force term is the Lorenz force of a charged particle in a magnetic field: Where q is x or y  0 : constant  -function and phase advance along the storage ring F is arbitrary function of all coordinates Generally this equations are not solvable – Need a numerical integration But some special cases can be solved, or using some approximations – They eventually give some insight into specific effects 21

22. Errors Treatment Perturbed Hills equation 22

23. Errors to linear motion Perturbation from a dipole filed error Perturbation from a quadrupole filed error Reminder, we use the normalized multipole strengths 23

24. Multipole Expansion of Magnetic Fields skew multipoles a n : Taylor expansion of the magnetic field: with multipole dipole quadrupole sextupole octupole order 0 1 2 3 BxBx ByBy rotation of the magnetic field 90 o for dipole magnets by half of the coil symmetry: 45 o for quadrupole magnets 30 o for sextupole magnets 24

25. Perturbed Hills Equation normalized multipole gradients : with: perturbed equations of motion: ByBy BxBx 25

26. Perturbed Hills Equation General solution of this type of equations is linear combination of – Solution of homogenous equation where F is set to 0 – Particular u p solution that we usually guess, for example F=cos(x)  u p =a p cos(x) F=x3  u p =a p3 x 3 + a p2 x 2 + a p1 x + a p0 Perturbation theory – Calculate recursively corrections to higher and higher orders hoping that few terms will be sufficient to find an accurate solution – Does not really work For example breaks in vicinity of resonances Does not reveal all the details of the dynamics – Still, some features can be studied – The proper way to proceed is the normal form analysis It is quite vast and heavy topic, I spare it for a separate lecture 26

27. Perturbation theory vs tracking 27 1 st order perturbation theory

28. 28 Perturbed Hills Equation

29. Dipole error: orbit change The orbit can be treated as a single particle It obeys At s 0, after one turn, it receives additional kick x’(s 0 ) = Δk 0 The kicked orbit must close on itself after one turn 29

30. Dipole error: perturbed orbit Orbit ‘kink’ for single perturbation (SPS with 90 o Q = n.62): 30 These coordinates can be propagated to arbitrary point using transfer matrix It is symmetric around the error position Nota bene: if Q is an integer then closed orbit does not exist!

31. Orbit Correction Deflection angle: Trajectory response: Closed orbit bump – compensate the trajectory response with additional dipole fields further down-stream  ‘closure’ of the perturbation within one oscillation 31

32. 2 corrector bump In case of an orbit error it is the best to correct it at the place – Requires a corrector at every element Eventually, we can try to correct after π phase advance Limits: – Must be lucky to find a corrector exactly after π – Sensitive to BPM errors – Large number of correctors 32

33. 3 corrector bump Closure Works for any type of lattice Limits: – Sensitive to BPM errors; – Need large number of correctors 33

34. 4 corrector bump Of course one can involve any number of correctors 34

35. Global correction A is orbit response matrix on correctors excitation – n x m matrix n BPMs m corrector magnets – Can be obtained from the machine model – Or measured Excite each corrector and observe orbit change We can write the orbit as Correction is Problem: A is normally not invertible – It is normally not even a square matrix! Solution: minimize the norm 35

36. SVD Algorithm Find a matrix B such that Attains a minimum with B being a m x n matrix and: Singular value decomposition (SVD): – Any matrix can be written as: where O 1 and O 2 are orthogonal matrices and D is diagonal D has diagonal form – Define a pseudo inverse matrix 1 k being the k x k unit matrix 36

37. SVD Algorithm Correction matrix defined as Main properties – SVD allows you to adjust k corrector magnets – If k = m = n one obtains a zero orbit (using all correctors) – For m = n, SVD minimizes the norm (using all correctors) – The algorithm is not stable if D has small Eigenvalues Can be used to find redundant correctors! – Nota bene, SVD is extensively used to clean the measured data and hence reduce errors of the measurements 37

38. Most effective corrector the orbit error is dominated by a few large perturbations:  minimize the norm: using only a small set of corrector magnets brut force: select all possible corrector combinations –  time consuming but god result selective: use one corrector at the time + keep most effective  much faster but has a finite chance to miss best solution and can generate  bumps MICADO: selective + cross correlation between orbit residues and remaining correcotr magnets 38

39. LEP See the phase advance… 39

40. Harmonic Filtering Unperturbed solution (smooth approximation): In case of orbit perturbation F and Closed Orbit are periodic  Fourier decomposition Closed Orbit Fourier peak around tune – Distribute correctors to filter out the main harmonics wich create the main orbit waves 40

41. Quad error A focusing error can be represented as additional thin lens quadrupole strength – Where is unperturbed map Modified one turn map is 41

42. Quad error: tune change For distributed errors we have to integrate around ring – As bigger the beta function as more the error prenounced Biggest errors from final focus quadrupoles 42

43. Quad error: β-function change 43 =

44. Quad error: β-function change Arithmetics for reference 44

45. Quad error: β-function change Arithmetic for reference 45 Inserting We get Useand =

46. Beta-function error Keeping 1 st order in And for the all errors in the ring Beta function error is oscillatory (Beta beat) of twice the betatron frequency Errors around the ring average out due to different phases Use all the quads for tune correction, it will affect beta less Or use 2 quads with the same beta but 90 degree apart – Tune will change, but beta changes will cancel 46

47. Natural Chromaticity Offset in particle energy can be seen as offset in focusing strength The resulting tune change is natural chromaticity – Chroma without sextupole correction 47

48. Chromaticity Off centre sextupole Using dispersion orbit we get focusing change And resulting chromaticity Chroma correction strength is proportional to – Beta inside the sextupole – Dispersion – Strength of the sextupole 48

49. Matching ring injection Ring has a unique closed solution in terms of orbit and Twiss parameters What happens if beam injected not onto this solution? If beam was perfectly monochromatic or there was no chromaticity or amplitude detuning the beam was beat around the closed solution But real bunches and real accelerators are never perfect: decoherence and filamentation 49

50. Decoherence Effect of the orbit error

51. Decoherence

60. Damping of injection oscillations Residual transverse oscillations lead to an emittance blow-up through filamentation “Transverse damper” systems used to damp injection oscillations bunch position measured by a pick-up that is linked to a kicker Damper measures offset of bunch on one turn, then kicks the bunch on a subsequent turn to reduce the oscillation amplitude Injection oscillation

61. Optical Mismatch at Injection x x’ Matched phase-space ellipse Mismatched injected beam Can also have an emittance blow-up through optical mismatch

62. Optical Mismatch at Injection Beam filaments and fills closed solution ellipse, but with bigger area: Emittance increase x x’ Turn 0 Turn N Time

63. Amplitude detuning In a perfectly linear machine tune does not depend on amplitude of oscillations Non linear fields are used – Sextupoles to correct chromaticity – Octupoles for Landau damping Even if they are not used intentionally they are present due to imperfect magnet construction or other unwanted phenomena These lead to tune change with the oscillation amplitude: amplitude detuning Tune foot-print is obtained to see it 63

64. Coupling 64

65. Coupling I: Identical Coupled Oscillators  mode:  mode: fundamental modes for identical coupled oscillators: 65 weak coupling (k << k 0 ):  description of motion in unperturbed ‘x’ and ‘y’ coordinates  degenerate mode frequencies

66. solution by decomposition into ‘Eigenmodes’: distributed coupling: 66 Coupling II: Equation of Motion in Accelerator With orthogonal condition: 

67. take second derivative of q 1 and q 2 : 67 Coupling II: Equation of Motion in Accelerator  use Orthogonal condition for calculating a,b,c,d (set b=1=d)  expressions for  1 and  2 as functions of a, b, c, d,  x,  y with: yields:

68. very different unperturbed frequencies: 68 Coupled Oscillators Case Study: Case 1 expansion of the square root:  ‘nearly’ uncoupled oscillators 

69. almost equal frequencies: 69 Coupled Oscillators Case Study: Case 2   keep only linear terms in  expansion of the square root for small coupling and  :  with: 

70. measurement of coupling strength: 70 Coupled Oscillators Case Study: Case 2 measure the difference in the Eigenmode frequencies while bringing the unperturbed tunes together:  the minimum separation yields the coupling strength!!  1,2 xx yy

71. Coupled Oscillators Case Study: Case 2 with initial oscillation only in horizontal plane: 71 and    modulation of the amplitudes sum rules for sin and cos functions:

72. Beating of the Transverse Motion: Case 1 modulation of the oscillation amplitude: frequencies can not be distinguished and energy can be exchanged between the two oscillators two almost identical harmonic oscillators with weak coupling:  -mode and  =mode frequencies are approximately identical! x s 72

73. Driven Oscillators  large number of driving frequencies! equation of motion  driven un-damped oscillators: 73 Perturbation treatment: substitute the solutions of the homogeneous equation of motion: into the right-hand side of the perturbed Hills equation and express the ‘s’ dependence of the multipole terms by their Fourier series (the perturbations must be periodic with one revolution!)

74. Driven Oscillators general solution: 74 single resonance approximation: consider only one perturbation frequency (choose ): without damping the transient solution is just the HO solution

75. Driven Oscillators resonance condition: 75 stationary solution: where ‘  ’ is the driving angular frequency! and W(  ) can become large for certain frequencies!  justification for single resonance approximation:  all perturbation terms with: de-phase with the transient  no net energy transfer from perturbation to oscillation (averaging)!

76. solution by decomposition into ‘Eigenmodes’: different oscillation frequencies: 76 Coupling IV: Orthogonal Coupled Oscillators yields: with: