1. Professor Philip Burrows John Adams Institute for Accelerator Science Oxford University ACAS School for Accelerator Physics January 2014 Longitudinal Dynamics I 1

2. Acknowledgements 2 Ted Wilson Ken Peach Emmanuel Tsesmelis

3. Lecture 4 outline 3 Longitudinal motion Momentum compaction Transition Dispersion Radio frequency (RF) systems Longitudinal phase space Chromaticity

4. Longitudinal motion (1) 4 So far assumed motion in accelerator with constant speed (momentum) in the direction of the orbit, s Neglected: differences in momentum between particles acceleration (change in speed/momentum) as particles traverse accelerator

5. 5 If particle momentum increases: Particles travel faster, until ultra-relativistic when v ≈ c In a fixed dipole bend field the orbit will get longer Longitudinal motion (2)

6. 6 How does revolution frequency, f, change with momentum? df / f=dv / v-dr / r frequencyspeedradius If p increases, the length of the orbit will change (for a given magnetic lattice) Can define dr / r = α p dp / p α p is the momentum compaction factor Change in radius of closed orbit for a change in momentum So df / f=dv / v-α p dp / p Longitudinal motion (3)

7. 7 So df / f=dv / v-α p dp / p Longitudinal motion (4)

8. 8 So df / f=dv / v-α p dp / p Recall β = v / c, so dv / v = dβ / β Longitudinal motion (4)

9. 9 So df / f=dv / v-α p dp / p Recall β = v / c, so dv / v = dβ / β p = γ m 0 v = γ m 0 β c= m 0 c β / √ (1 - β 2 ) Longitudinal motion (4)

10. 10 So df / f=dv / v-α p dp / p Recall β = v / c, so dv / v = dβ / β p = γ m 0 v = γ m 0 β c= m 0 c β / √ (1 - β 2 ) Exercise: show that dp / dβ = m 0 c γ 3 = γ 2 p / β Longitudinal motion (4)

11. 11 So df / f=dv / v-α p dp / p Recall β = v / c, so dv / v = dβ / β p = γ m 0 v = γ m 0 β c= m 0 c β / √ (1 - β 2 ) Exercise: show that dp / dβ = m 0 c γ 3 = γ 2 p / β dβ / β= 1 / γ 2 dp / p Longitudinal motion (4)

12. 12 So df / f=dv / v-α p dp / p Recall β = v / c, so dv / v = dβ / β p = γ m 0 v = γ m 0 β c= m 0 c β / √ (1 - β 2 ) Exercise: show that dp / dβ = m 0 c γ 3 = γ 2 p / β dβ / β= 1 / γ 2 dp / p df / f= ( 1 / γ 2 - α p ) dp / p Longitudinal motion (4)

13. 13 df / f= ( 1 / γ 2 - α p ) dp / p momentum lattice Transition

14. 14 df / f= ( 1 / γ 2 - α p ) dp / p momentum lattice For a particle in a constant magnetic bend field: Low p: β α p f ↑ as p ↑ Transition

15. 15 df / f= ( 1 / γ 2 - α p ) dp / p momentum lattice For a particle in a constant magnetic bend field: Low p: β α p f ↑ as p ↑ High p:β  1, γ >> 1: 1 / γ 2 < α p f ↓ as p ↑ Transition

16. 16 df / f= ( 1 / γ 2 - α p ) dp / p momentum lattice For a particle in a constant magnetic bend field: Low p: β α p f ↑ as p ↑ High p:β  1, γ >> 1: 1 / γ 2 < α p f ↓ as p ↑ Transition energy: 1 / γ 2 = α p Transition

17. 17 df / f= ( 1 / γ 2 - α p ) dp / p = η dp / p frequency slip factor Frequency slip factor

18. 18 df / f= ( 1 / γ 2 - α p ) dp / p = η dp / p frequency slip factor Below transition (low p): η > 0 Transition:η = 0 Above transition (high p):η < 0 Frequency slip factor

19. 19 df / f= ( 1 / γ 2 - α p ) dp / p = η dp / p frequency slip factor Below transition (low p): η > 0 Transition:η = 0 Above transition (high p):η < 0 Proton Synchrotron (CERN): transition energy ≈ 6 GeV Frequency slip factor

20. 20 df / f= ( 1 / γ 2 - α p ) dp / p = η dp / p frequency slip factor Below transition (low p): η > 0 Transition:η = 0 Above transition (high p):η < 0 Proton Synchrotron (CERN): transition energy ≈ 6 GeV Not applicable to electron accelerators – discuss! Frequency slip factor

21. 21 So far assumed beam has no energy/momentum spread: Δ E / E = 0 = Δ p / p Different momentum  different radius of curvature r in main dipole magnets Particles pass through quadrupoles at different positions Quadrupoles bend differently, depending on position Closed orbits different, depend on momentum Horizontal displacement characterised by dispersion function D(s): depends on s Dispersion (1)

22. 22 Local radial displacement at position s caused by momentum spread: Δ x (s) = D (s) Δ p / p D (s) is a property of the magnetic lattice, units meters Non-zero Δ p / p  non-zero Δ x Momentum spread causes finite beam size Normally no vertical dipoles: D = 0 in vertical plane Dispersion (2)

23. 23 Magnetic fields deflect, but do not accelerate, charged particles For acceleration need electric field Early days: V fixed Modern accelerators: use an oscillating voltage of ‘radio frequency’ (RF) Acceleration

24. 24 Consider single particle in oscillating RF longitudinal electric field produced by a RF cavity Set oscillation frequency = period of revolution of the particle RF electric field (1)

25. 25 Add a second particle of lower energy, which arrives later in time, with respect to the first particle, at the RF cavity On first turn: B arrives late w.r.t. A  sees a higher voltage and gets accelerated w.r.t. A Second turn: still late, but not quite as late, accelerated again RF electric field (2)

26. 26 Turn 1

27. 27 Turn 100

28. 28 Turn 200

29. 29 Turn 400

30. 30 Turn 500

31. 31 Turn 600

32. 32 Turn 700

33. 33 Turn 800

34. 34 Turn 900

35. Synchrotron oscillations 35 In this toy example, after 900 turns particle B has made one full oscillation around particle A Motion due to energy difference is ‘synchrotron oscillation’ Amplitude of oscillation depends on initial energy difference, i.e. initial ‘phase difference’

36. 36 Potential well (1) A B Cavity voltage Potential

37. 37 Potential well (2) A B Cavity voltage Potential

38. 38 Potential well (3) A B Cavity voltage Potential

39. 39 Potential well (4) A B Cavity voltage Potential

40. 40 Potential well (5) A B Cavity voltage Potential

41. 41 Potential well (6) A B Cavity voltage Potential

42. 42 Potential well (7) A B Cavity voltage Potential

43. 43 Potential well (8) A B Cavity voltage Potential

44. 44 Potential well (9) A B Cavity voltage Potential

45. 45 Potential well (10) Cavity voltage Potential A B

46. 46 Potential well (11) Cavity voltage Potential A B

47. 47 Potential well (12) Cavity voltage Potential A B

48. 48 Potential well (13) Cavity voltage Potential A B

49. 49 Potential well (14) Cavity voltage Potential A B

50. 50 Potential well (15) Cavity voltage Potential A B

51. Longitudinal phase space 51 By analogy with transverse phase space (x, x’) and (y, y’) define longitudinal phase space in terms of conjugate variables E and t, or the relative variables ΔE and Δt:

52. Motion in phase space (1) 52 Define zero to be at phase space position of particle A Particle B oscillates around particle A:

53. Motion in phase space (2) 53 Define zero to be at phase space position of particle A Particle B oscillates around particle A:

54. Motion in phase space (3) 54 Define zero to be at phase space position of particle A Particle B oscillates around particle A:

55. Motion in phase space (4) 55 Define zero to be at phase space position of particle A Particle B oscillates around particle A:

56. Motion in phase space (5) 56 Define zero to be at phase space position of particle A Particle B oscillates around particle A: synchrotron oscillations

57. Chromaticity 57 Recall normalised focussing strength of quadrupoles: k=1/Bρ dB y /dx

58. Chromaticity 58 Recall normalised focussing strength of quadrupoles: k=1/Bρ dB y /dx andB ρ~p

59. Chromaticity 59 Recall normalised focussing strength of quadrupoles: k=1/Bρ dB y /dx andB ρ~p hence Δk / k=- Δp / p

60. Chromaticity 60 Recall normalised focussing strength of quadrupoles: k=1/Bρ dB y /dx andB ρ~p hence Δk / k=- Δp / p but tune shift ΔQ / Q~ Δk / k

61. Chromaticity 61 Recall normalised focussing strength of quadrupoles: k=1/Bρ dB y /dx andB ρ~p hence Δk / k=- Δp / p but tune shift ΔQ / Q~ Δk / k so ΔQ / Q= ξ Δp / p

62. Chromaticity 62 Recall normalised focussing strength of quadrupoles: k=1/Bρ dB y /dx andB ρ~p hence Δk / k=- Δp / p but tune shift ΔQ / Q~ Δk / k so ΔQ / Q= ξ Δp / p ξ is chromaticity

63. Interpretation of chromaticity 63 Chromaticity relates tune spread of transverse motion to the momentum spread of the beam: ΔQ / Q= ξ Δp / p

64. Calculation of chromaticity 64 RecallΔQ = 1/4π β Δk ds and Δk / k = - Δp/p HenceΔQ / Q = - 1/4π β k/Q ds Δp/p comparing ΔQ / Q = ξ Δp/p ξ = - 1/4π β k/Q ds To correct the tune spread ΔQ need to: increase k for higher momentum particles decrease k for lower momentum particles

65. This is physically easy to see 65 Chromaticity relates tune spread of transverse motion to the momentum spread of the beam: ΔQ / Q= ξ Δp / p

66. Sextupole magnets 66 Used to compensate for chromatic aberrations in strongly- focussing magnetic structures Particle motion in horizontal plane is coupled to vertical plane Field along x-axis:

67. Sextupole magnets: examples 67 LEP sextupole: electromagnet, 1m long, few 100 kg LHC correction sextupole: SC magnet, 11cm long, 10 kg 500A @ 2K  1630 T/m 2

68. Summary 68 RF accelerating system forms a ‘potential well’ within which particles oscillate: synchrotron oscillation Describe motion using longitudinal phase space: energy vs. time (or phase) Works for particles below ‘transition’ Oscillation is non-linear motion Chromaticity and sextupole corrections