1. Accelerator Fundamentals Brief history of accelerator Accelerator building blocks Transverse beam dynamics coordinate system Mei Bai BND School, Sept. 1-6, 2015

2. Mei Bai BND School, Sept. 1-6, 2015 Why Accelerator? high energy particles are excellent probes in studying the micro-structure of matters – Wave length of a high energy particle Discover new particles Generate secondary beams for research in physics as well as other scientific fields, material science, biology, etc – Neutron source – Synchrotron radiation Also widely used in industry as well as medical research and treatment such as hadron therapy

3. Mei Bai BND School, Sept. 1-6, 2015 Accelerator Timeline

4. Mei Bai BND School, Sept. 1-6, 2015 Legends of Accelerator Development John D. Cockcroft and Ernest Walton – Invented Cockcroft-Walton Accelerator Rolf Wideroe: a nuclear physicist – Invented concept of LINAC when he was a Ph. D student at RWTH – Invented the principle of Betatron Ernest Orlando Lawrence: a nuclear physicist – Invented/implemented cyclotron – Nobel Laureate in 1939 Ernest Courant: Accelerator physicist – Invented strong focusing principle Bruno Touschek: particle physicist – father of the 1 st e+e- collider (AdA) Simon van der Meer: particle physicist – Invented/implemented Stochastic Cooling, and Nobel Laureate in 1984

5. Mei Bai BND School, Sept. 1-6, 2015 Both electric field and magnetic field can be used to guide the particles path. Magnetic field is more effective for high energy particles, i.e. particles with higher velocity. -For a relativistic particle, what kind of the electric field one needs to match the Lorentz force from a 1 Telsla magnetic field?

6. Mei Bai BND School, Sept. 1-6, 2015 Dipoles: uniform magnetic field in the gap -Bending dipoles -Orbit steering Quadrupoles -Providing focusing field to keep beam from being diverged Sextupoles: -Provide corrections of chromatic effect of beam dynamics Higher order multipoles

7. Mei Bai BND School, Sept. 1-6, 2015 Two magnetic poles separated by a gap homogeneous magnetic field between the gap Bending, steering, injection, extraction g

8. Mei Bai BND School, Sept. 1-6, 2015 For synchrotron, bending field is proportional to the beam energy where p is the momentum of the particle and q is the charge of the particle ρ

9. Mei Bai BND School, Sept. 1-6, 2015 Quadrupole Magnetic field is proportional to the distance from the center of the magnet Produced by 4 poles which are shaped as Providing focusing/defoucsing to the particle –Particle going through the center: F=0 – Particle going off center x y

10. Mei Bai BND School, Sept. 1-6, 2015 Quadrupole magnet Theorem Pick the loop for integral For the gap is filled with air,

11. Mei Bai BND School, Sept. 1-6, 2015 Sextupole Focusing strength in horizontal plane:  Place sextupole after a bending dipole where dispersion function is non zero

12. Mei Bai BND School, Sept. 1-6, 2015 Focusing from quadrupole Required by Maxwell equation, a single quadrupole has to provide focusing in one plane and defocusing in the other plane x Δx’ f s

13. Mei Bai BND School, Sept. 1-6, 2015 Transfer matrix of a qudruploe Thin lens: length of quadrupole is negligible to the displacement relative to the center of the magnet

14. Mei Bai BND School, Sept. 1-6, 2015 Transfer matrix of a drift space x Δx’ L s

15. Mei Bai BND School, Sept. 1-6, 2015 Lattice Arrangement of magnets: structure of beam line – Bending dipoles, Quadrupoles, Steering dipoles, Drift space and Other insertion elements Example: – FODO cell: alternating arrangement between focusing and defocusing quadrupoles L One FODO cell f-f L

16. Mei Bai BND School, Sept. 1-6, 2015 FODO lattice Net effect is focusing!

17. Mei Bai BND School, Sept. 1-6, 2015 FODO lattice Net effect is focusing Provide focusing in both planes!

18. Mei Bai BND School, Sept. 1-6, 2015 Curvilinear coordinate system Coordinate system to describe particle motion in an accelerator Moves with the particle Set of unit vectors: Reference orbit

19. Mei Bai BND School, Sept. 1-6, 2015 Equation of motion Equation of motion in transverse plane Δθ=Δs/ρ x x s

20. Mei Bai BND School, Sept. 1-6, 2015 Equation of motion

21. Mei Bai BND School, Sept. 1-6, 2015 Equation of motion

22. Mei Bai BND School, Sept. 1-6, 2015 Solution of equation of motion Comparison with harmonic oscillator: A system with a restoring force which is proportional to the distance from its equilibrium position, i.e. Hooker’s Law: Where k is the spring constant Equation of motion: Amplitude of the sinusoidal oscillation Frequency of the oscillation

23. Mei Bai BND School, Sept. 1-6, 2015 transverse motion: betatron oscillation The general case of equation of motion in an accelerator  For k > 0 Where k can also be negative  For k < 0

24. Mei Bai BND School, Sept. 1-6, 2015 Transfer matrix of a quadrupole For a focusing quadrupole For a defocusing quadrupole

25. Mei Bai BND School, Sept. 1-6, 2015 Hill's equation In an accelerator which consists individual magnets, the equation of motion can be expressed as, Here, k(s) is an periodic function of Lp, which is the length of the periodicity of the lattice, i.e. the magnet arrangement. It can be the circumference of machine or part of it. Similar to harmonic oscillator, expect solution as or:

26. Mei Bai BND School, Sept. 1-6, 2015 Hill’s equation: cont’d with  Hill’s equation is satisfied

27. Mei Bai BND School, Sept. 1-6, 2015 Betatron oscillation Beta function : – Describes the envelope of the betatron oscillation in an accelerator Phase advance: Betatron tune: # of betatron oscillations in one orbital turn

28. Mei Bai BND School, Sept. 1-6, 2015 Phase space In a space of x-x’, the betatron oscillation projects an ellipse where  The are of the ellipse is X’ X

29. Mei Bai BND School, Sept. 1-6, 2015 Courant-Snyder parameters The set of parameter ( β x, α x and γ x ) which describe the phase space ellipse Courant-Snyder invariant: the area of the ellipse in unit of

30. Mei Bai BND School, Sept. 1-6, 2015 Transfer Matrix of beam transport Proof the transport matrix from point 1 to point 2 is  Hint:

31. Mei Bai BND School, Sept. 1-6, 2015 One Turn Map Transfer matrix of one orbital turn  Closed orbit: Stable condition

32. Mei Bai BND School, Sept. 1-6, 2015 Stability of transverse motion Matrix from point 1 to point 2  Stable motion requires each transfer matrix to be stable, i.e. its eigen values are in form of oscillation With and

33. Mei Bai BND School, Sept. 1-6, 2015 Dispersion function Transverse trajectory is function of particle momentum ρ ρ+Δρ Define Dispersion function Momentum spread

34. Mei Bai BND School, Sept. 1-6, 2015 Dispersion function Transverse trajectory is function of particle momentum.

35. Mei Bai BND School, Sept. 1-6, 2015 Dispersion function: cont’d In drift space dispersion function has a constant slope and  In dipoles, and

36. Mei Bai BND School, Sept. 1-6, 2015 Dispersion function: cont’d  For a focusing quad, dispersion function oscillates sinusoidally and  For a defocusing quad, dispersion function evolves exponentially and

37. Mei Bai BND School, Sept. 1-6, 2015 Compaction factor  The difference of the length of closed orbit between off- momentum particle and on momentum particle, i.e.

38. Mei Bai BND School, Sept. 1-6, 2015 Path length and velocity  For a particle with velocity v,  Transition energy : when particles with different energies spend the same time for each orbital turn Below transition energy: higher energy particle travels faster Above transition energy: higher energy particle travels slower

39. Mei Bai BND School, Sept. 1-6, 2015 Chromatic effect Comes from the fact the the focusing effect of an quadrupole is momentum dependent -Higher energy particle has less focusing  Chromaticity: tune spread due to momentum spread momentum spread Particles with different momentum have different betatron tune Tune spread

40. Mei Bai BND School, Sept. 1-6, 2015 Chromaticity Transfer matrix AB  Transfer matrix of a thin quadrupole

41. Mei Bai BND School, Sept. 1-6, 2015 Chromaticity

42. Mei Bai BND School, Sept. 1-6, 2015 Chromaticity Assuming the tune change due to momentum difference is small

43. Mei Bai BND School, Sept. 1-6, 2015 Chromaticity of a FODO cell L One FODO cell L β βfβf βdβd

44. Mei Bai BND School, Sept. 1-6, 2015 Chromaticity correction Nature chromaticity is always negative and can be large and can result to large tune spread and get close to resonance condition Solution: -A special magnet which provides stronger focusing for particles with higher energy: sextupole

45. Mei Bai BND School, Sept. 1-6, 2015 Sextupole Focusing strength in horizontal plane: where and, l is the magnet length Tune change due to a sextupole:

46. Mei Bai BND School, Sept. 1-6, 2015 Chromaticity Correction Sextupole produces a chromaticity with the opposite sign of the quadrupole! It prefers to be placed after a bending dipole where dispersion function is non zero Chromaticity after correction

47. Mei Bai BND School, Sept. 1-6, 2015 Chromaticity correction ξ=20 ξ=1

48. Mei Bai BND School, Sept. 1-6, 2015 How to measure betatron oscillation?  Excite a coherent betatron motion with a pulsed kicker  Record turn – by – turn beam position X’ X

49. Mei Bai BND School, Sept. 1-6, 2015 How to measure betatron oscillation? Turn-by-turn beam position monitor data betatron tune is obtained by Fourier transform TbT beam position data Kicker

50. Mei Bai BND School, Sept. 1-6, 2015 Beam Position Monitor (BPM) A strip line bpm response – Right electrode response – And left electrode response Hence, Let x=rcosθ b

51. Mei Bai BND School, Sept. 1-6, 2015 Coherent betatron oscillation at RHIC

52. Mei Bai BND School, Sept. 1-6, 2015 How to measure betatron functions and phase advance?

53. Effects of Errors - dipole errors - quadrupole errors

54. Mei Bai BND School, Sept. 1-6, 2015 Closed orbit distortion Dipole kicks can cause particle’s trajectory deviate away from the designed orbit -Dipole error -Quadrupole misalignment  Assuming a circular ring with a single dipole error, closed orbit then becomes: s0s0 BPM s

55. Mei Bai BND School, Sept. 1-6, 2015 Closed orbit: single dipole error  Let’s first solve the closed orbit at the location where the dipole error is  The closed orbit distortion reaches its maximum at the opposite side of the dipole error location

56. Mei Bai BND School, Sept. 1-6, 2015 Closed orbit distortion  In the case of multiple dipole errors distributed around the ring. The closed orbit is  Amplitude of the closed orbit distortion is inversely proportion to sinπQ x,y -No stable orbit if tune is integer!

57. Mei Bai BND School, Sept. 1-6, 2015 Measure closed orbit  Distribute beam position monitors around ring.

58. Mei Bai BND School, Sept. 1-6, 2015 Control closed orbit  minimized the closed orbit distortion.  Large closed orbit distortions cause limitation on the physical aperture  Need dipole correctors and beam position monitors distributed around the ring  Assuming we have m beam position monitors and n dipole correctors, the response at each beam position monitor from the n correctors is:

59. Mei Bai BND School, Sept. 1-6, 2015 Control closed orbit  Or,  To cancel the closed orbit measured at all the bpms, the correctors are then

60. Mei Bai BND School, Sept. 1-6, 2015 Quadrupole errors Misalignment of quadrupoles -dipole-like error: kx -results in closed orbit distortion  Gradient error: -Cause betatron tune shift -induce beta function deviation: beta beat

61. Mei Bai BND School, Sept. 1-6, 2015 Tune change due to a single gradient error Suppose a quadrupole has an error in its gradient, i.e.

62. Mei Bai BND School, Sept. 1-6, 2015 Tune shift due to multiple gradient errors In a circular ring with a multipole gradient errors, the tune shift is

63. Mei Bai BND School, Sept. 1-6, 2015 Beta beat In a circular ring with a gradient error at s0, the tune shift is s0s0 s Unstable betatron motion if tune is half integer!

64. Mei Bai BND School, Sept. 1-6, 2015 Beta beat In a circular ring with multiple gradient errors, Unstable betatron motion if tune is half integer! Beta beat wave varies twice of betatron tune around the ring

65. Mei Bai BND School, Sept. 1-6, 2015 Resonance condition Tune change due to a single quadrupole error If, the above equation becomes and Q x can become a complex number which means the betatron motion can become unstable

66. Mei Bai BND School, Sept. 1-6, 2015 Resonance X’ x Integer resonanceHalf Integer resonance x