1. Collective Effect II Giuliano Franchetti, GSI CERN Accelerator – School Prague 11/9/14G. Franchetti1

2. Type of fields 11/9/14G. Franchetti2 Collective Effects Collective Effects ?

3. Robinson Instability 11/9/14G. Franchetti3

4. Robinson Instability 11/9/14G. Franchetti4 It is an instability arising from the coupling of the impendence and longitudinal motion real part control the energy loss in a bunch the revolution frequency controls the impendence R0R0 RF slower particle (if below transition) change particle energy change revolution frequency

5. The coupling of two effects 11/9/14G. Franchetti5 Energy loss due to impedance via the longitudinal dynamics Change of revolution frequency because of the impedance Z(ω)

6. Below transition 11/9/14G. Franchetti6

7. Below transition 11/9/14G. Franchetti7 Energy lost in one turn

8. Below transition 11/9/14G. Franchetti8 Where is given by the impedance

9. Below transition 11/9/14G. Franchetti9 energy lost per particle for non oscillating bunch

10. Below transition 11/9/14G. Franchetti10 In one turn energy is lost but compensated by the RF cavity frequency

11. Below transition 11/9/14G. Franchetti11 cavity frequency revolution frequency changes larger smaller

12. Below transition 11/9/14G. Franchetti12 Energy lost  increase ω  increase Z r  increase energy loss !!!

13. Below transition 11/9/14G. Franchetti13

14. Below transition 11/9/14G. Franchetti14 Stable

15. Below transition 11/9/14G. Franchetti15 Stable

16. Below transition 11/9/14G. Franchetti16 Unstable

17. Below transition 11/9/14G. Franchetti17 Unstable

18. Summary of the reasoning 11/9/14G. Franchetti18 below transition above transition stable unstable stable

19. More complicated 11/9/14G. Franchetti19 E < E T No Energy Loss: RF give the same energy lost by the impedance

20. 11/9/14G. Franchetti20 set the cavity frequency here Remember that energy lost is V*I

21. Source of difficulty 11/9/14G. Franchetti21 E < E T Energy gain Energy lost Impedance effect 

22. 11/9/14G. Franchetti22

23. Still we neglect something 11/9/14G. Franchetti23 E < E T position of the bunch at turn “k” Q s is the synchrotron tune

24. 11/9/14G. Franchetti24

25. Current 11/9/14G. Franchetti25 The bunch current can be described by 3 components with frequency very close side band this component is out of phase (because is a sin) side band

26. 11/9/14G. Franchetti26 That means that the energy loss due to the impedance has to be computed on the 3 currents… Voltage created by the resistive impedance Main component 1 st sideband 2 nd sideband

27. 11/9/14G. Franchetti27 But Prosthaphaeresis formulae

28. 11/9/14G. Franchetti28 Main component 2 nd sideband 1 st sideband Therefore the induced Voltage depends on Voltage created by the resistive impedance

29. Energy lost in one turn 11/9/14G. Franchetti29 energy lost per particle per turn this term can give rise to a constant loss, or a constant gain of energy

30. In terms of the energy of a particle 11/9/14G. Franchetti30 This is a slope in the energy, and the sign of the slope depends on and

31. The longitudinal motion now! 11/9/14G. Franchetti31 Ifthere is a damping Ifthere is an instability Robinson Instability

32. 11/9/14G. Franchetti32 below transition above transition stable unstable stable Robinson Instability

33. Longitudinal space charge and resistive wall impedance 11/9/14G. Franchetti33

34. Space charge longitudinal field 11/9/14G. Franchetti34 dz RwRw ErEr ErEr EzEz EwEw

35. 11/9/14G. Franchetti35 For a KV beam Electric Field

36. 11/9/14G. Franchetti36 Therefore

37. 11/9/14G. Franchetti37 Magnetic Field

38. 11/9/14G. Franchetti38 from the equation of continuity again we find the factor! Maxwell-Faraday Law

39. Space charge impedance 11/9/14G. Franchetti39 Local density

40. 11/9/14G. Franchetti40 Perfect vacuum chamber E zw = 0

41. Resistive Wall impedance 11/9/14G. Franchetti41 dz RwRw ErEr ErEr EzEz EwEw Do not take into account B E w = E z

42. 11/9/14G. Franchetti42 pipe wall currents skin depth Beam on axis Wall currents are related to the electric field by Ohm’ law The thickness of the wall currents is called skin depth

43. 11/9/14G. Franchetti43 Impedance of the surface (pipe) Longitudinal impedance (beam) J z (0) x y z μ,σ

44. Transverse impedance 11/9/14G. Franchetti44

45. Origin 11/9/14G. Franchetti45 v t Beam passing through a cavity on axis

46. Origin 11/9/14G. Franchetti46 t Beam passing through a cavity off-axis v

47. But the field transform it-self ! 11/9/14G. Franchetti47 E

48. 11/9/14G. Franchetti48 B

49. 11/9/14G. Franchetti49 E

50. 11/9/14G. Franchetti50 B

51. Effect on the dynamics 11/9/14G. Franchetti51 The dynamics is much more affected by B, than E because this speed is high

52. The beam creates its own dipolar magnetic field ! 11/9/14G. Franchetti52 (dipolar errors create integer resonances…. we expect the same…)

53. Transverse impedance 11/9/14G. Franchetti53 Definition of longitudinal impedance (classical) Impedance System

54. 11/9/14G. Franchetti54 source of the effect For a displaced beam It means that in the equation of motion we have to add this effect Effect this field acts on a single particle

55. 11/9/14G. Franchetti55 In the time domain therefore for a weak effect or distributed we find

56. 11/9/14G. Franchetti56 Now the situation is the following: System But is like a “strange” voltage it depends on frequency

57. Transverse beam coupling impedance 11/9/14G. Franchetti57 now the question is what is ω ?

58. What is it ω ? 11/9/14G. Franchetti58 It is given by the fractional tune, as this is the frequency seen in a cavity Example:Q = 2.23fractional tuneq = 0.23 beam position seen by the cavity turns

59. B-field induced by beam displacement 11/9/14G. Franchetti59 From electric field at the position of beam x 0 is Longitudinal impedance

60. 11/9/14G. Franchetti60 Transverse impedance The magnetic field comes from Maxwell taking

61. 11/9/14G. Franchetti61 pipe wall currents skin depth Beam x off axis

62. 11/9/14G. Franchetti62 xbxb image charge Therefore the field on the beam is (for small x b /r p )

63. 11/9/14G. Franchetti63 More charges here Less charges here beam

64. 11/9/14G. Franchetti64 Transverse resistive Wall impedance

65. Transverse instability 11/9/14G. Franchetti65

66. Coasting beam instability 11/9/14G. Franchetti66 Force due to the impedance (in the complex notation) Equation of motion of one particle for a beam on axis Equation of motion of a beam particle when the beam is off-axis

67. Collective motion 11/9/14G. Franchetti67 On the other hand the beam center is If all particles have the same frequency, i.e. each particle experience a force with therefore then

68. 11/9/14G. Franchetti68 We can define a coherent “detuning” because this is a linear equation

69. 11/9/14G. Franchetti69 that is But now is a complex number !! Solution

70. 11/9/14G. Franchetti70 Is the growth rate of the transverse resistive wall instability This instability always take place  Landau damping Instability suppression

71. An important assumption 11/9/14G. Franchetti71 We assumed that all particles have the same frequency so that This assumption means that each particle of the beam respond in the same way to a change of particle amplitude Coherent motion drive particle motion, which is again coherent

72. Chromaticity ? 11/9/14G. Franchetti72 What happened if the incoherent force created by the accelerator do not allow a coherent build up Momentum spread dp/p

73. 11/9/14G. Franchetti73 chromaticity one particle with off-momentum dp/p has tune If each particle of the beam has different dp/p then the force that the lattice exert on a particle depends on the particle !

74. Incoherent motion damps x b 11/9/14G. Franchetti74 Equation of motion without impedances Motion of center of mass as an effect of the spread of the frequencies of oscillation The momentum compaction also provides a spread of the betatron oscillations

75. 11/9/14G. Franchetti75 Example: N. particles = 5 dq/q = 0

76. 11/9/14G. Franchetti76 Example: N. particles = 5 dq/q = 5E-3

77. 11/9/14G. Franchetti77 Example: N. particles = 5 dq/q = 1E-2

78. 11/9/14G. Franchetti78 Example: N. particles = 5 dq/q = 2.5E-2 damping of x b

79. But incoherent motion reduces x b 11/9/14G. Franchetti79 Example: these are 5 sinusoid with amplitude linearly growth

80. 11/9/14G. Franchetti80 Example: now a spread dq/q of 1E-2 is added to the 5 curves The center of mass growth slower

81. 11/9/14G. Franchetti81 Example: now a spread dq/q of 1E-2 is added to the 5 curves the spread of the particles damps the oscillations of the center of mass  the instability cannot develop

82. Situation 11/9/14G. Franchetti82 Coherent effect Incoherent effect Growth rateDamping rate The faster wins

83. instability of a single bunch 11/9/14G. Franchetti83 Q = 2.23 Q = 2.00 Q = 1.77 No oscillations  Example beam position at the cavity

84. 11/9/14G. Franchetti84 Q = 2.23 Q = 2.00 Q = 1.77 Slow side band Fast

85. behavior of the field in the cavity 11/9/14G. Franchetti85 T r = time of oscillation of the field in the cavity

86. Cavity tuned upper sideband 11/9/14G. Franchetti86

87. Cavity tuned upper sideband 11/9/14G. Franchetti87

88. 11/9/14G. Franchetti88 As for the Robinson Instability

89. Negative mass instability 11/9/14G. Franchetti89 x Above transition z Example with 2 particles ! Repulsion forces from Coulomb Repulsion forces from Coulomb reference frame of synchronous particle

90. Negative mass instability 11/9/14G. Franchetti90 x Above transition z Example with 2 particles ! Repulsion forces from Coulomb Repulsion forces from Coulomb reference frame of synchronous particle gain speed revolution time longer lose speed revolution time shorter

91. Negative mass instability 11/9/14G. Franchetti91 x Above transition z Example with 2 particles ! reference frame of synchronous particle

92. Negative mass instability 11/9/14G. Franchetti92 x Above transition z repulsive forces attract particles as if their mass were negative perturbation x z growth

93. Summary 11/9/14G. Franchetti93 Robinson instability Longitudinal space charge and resistive wall impedance Transverse impedance Transverse instability Landau damping Single bunch instability Negative mass instability