1. Topic One: Acceleration UW Spring 2008 Accelerator Physics J. J. Bisognano 1 Accelerator Physics Topic I Acceleration Joseph Bisognano Synchrotron Radiation Center University of Wisconsin

2. Topic One: Acceleration UW Spring 2008 Accelerator Physics J. J. Bisognano 2 Relativity

3. Topic One: Acceleration UW Spring 2008 Accelerator Physics J. J. Bisognano 3 Maxwell’s Equations

4. Topic One: Acceleration UW Spring 2008 Accelerator Physics J. J. Bisognano 4 Vector Identity Games Poynting Vector Electromagnetic Energy

5. Topic One: Acceleration UW Spring 2008 Accelerator Physics J. J. Bisognano 5 Propagation in Conductors

6. Topic One: Acceleration UW Spring 2008 Accelerator Physics J. J. Bisognano 6 Free Space Propagation

7. Topic One: Acceleration UW Spring 2008 Accelerator Physics J. J. Bisognano 7 Conductive Propagation

8. Topic One: Acceleration UW Spring 2008 Accelerator Physics J. J. Bisognano 8 Boundary Conditions

9. Topic One: Acceleration UW Spring 2008 Accelerator Physics J. J. Bisognano 9 AC Resistance

10. Topic One: Acceleration UW Spring 2008 Accelerator Physics J. J. Bisognano 10 Cylindrical Waveguides Assume a cylindrical system with axis z For the electric field we have And likewise for the magnetic field

11. Topic One: Acceleration UW Spring 2008 Accelerator Physics J. J. Bisognano 11 Solving for E tangential

12. Topic One: Acceleration UW Spring 2008 Accelerator Physics J. J. Bisognano 12 Maxwell’s equations then imply (k=  /c)

13. Topic One: Acceleration UW Spring 2008 Accelerator Physics J. J. Bisognano 13 All this implies that E 0z and B 0z tell it all with their equations For simple waveguides, there are separate solutions with one or other zero (TM or TE) For complicated geometries (periodic structures, dielectric boundaries), can be hybrid modes

14. Topic One: Acceleration UW Spring 2008 Accelerator Physics J. J. Bisognano 14 TE Rectangular Waveguide Mode b a x y

15. Topic One: Acceleration UW Spring 2008 Accelerator Physics J. J. Bisognano 15 a TE mode Example

16. Topic One: Acceleration UW Spring 2008 Accelerator Physics J. J. Bisognano 16 Circular Waveguide TE m,n Modes

17. Topic One: Acceleration UW Spring 2008 Accelerator Physics J. J. Bisognano 17 Circular Waveguide TE m,n Modes

18. Topic One: Acceleration UW Spring 2008 Accelerator Physics J. J. Bisognano 18 Circular Waveguide TM m,n Modes

19. Topic One: Acceleration UW Spring 2008 Accelerator Physics J. J. Bisognano 19 Circular Waveguide Modes

20. Topic One: Acceleration UW Spring 2008 Accelerator Physics J. J. Bisognano 20 Cavities d

21. Topic One: Acceleration UW Spring 2008 Accelerator Physics J. J. Bisognano 21 Cavity Perturbations Now following C.C. Johnson, Field and Wave Dynamics

22. Topic One: Acceleration UW Spring 2008 Accelerator Physics J. J. Bisognano 22 Cavity Energy and Frequency ++++ - - Attracts I -I B E Repels

23. Topic One: Acceleration UW Spring 2008 Accelerator Physics J. J. Bisognano 23 Energy Change of Wall Movement

24. Topic One: Acceleration UW Spring 2008 Accelerator Physics J. J. Bisognano 24 Bead Pull J. Byrd

25. Topic One: Acceleration UW Spring 2008 Accelerator Physics J. J. Bisognano 25 Lorentz Theorem Let and be two distinct solutions to Maxwell’s equations, but at the same frequency Start with the expression

26. Topic One: Acceleration UW Spring 2008 Accelerator Physics J. J. Bisognano 26 Vector Arithmetic

27. Topic One: Acceleration UW Spring 2008 Accelerator Physics J. J. Bisognano 27 Using curl relations for non-tensor  one can show that expression is zero So, in particular, for waveguide junctions with an isotropic medium we have S1S1 S2S2 S3S3

28. Topic One: Acceleration UW Spring 2008 Accelerator Physics J. J. Bisognano 28 Scattering Matrix Consider a multiport device Discussion follows Altman S1S1 S2S2 SpSp

29. Topic One: Acceleration UW Spring 2008 Accelerator Physics J. J. Bisognano 29 S-matrix Let a p amplitude of incident electric field normalize so that  a p  2 = 2(incident power) and  b p  2 = 2(scattered power)

30. Topic One: Acceleration UW Spring 2008 Accelerator Physics J. J. Bisognano 30 Two-Port Junction Port XPort Y a1a1 b1b1 a2a2 b2b2

31. Topic One: Acceleration UW Spring 2008 Accelerator Physics J. J. Bisognano 31 Implication of Lorentz Theorem

32. Topic One: Acceleration UW Spring 2008 Accelerator Physics J. J. Bisognano 32 Lorentz/cont. Lorentz theorem implies or

33. Topic One: Acceleration UW Spring 2008 Accelerator Physics J. J. Bisognano 33 Unitarity of S-matrix Dissipated power P is given by For a lossless junction and arbitrary this implies

34. Topic One: Acceleration UW Spring 2008 Accelerator Physics J. J. Bisognano 34 Symmetrical Two-Port Junction

35. Topic One: Acceleration UW Spring 2008 Accelerator Physics J. J. Bisognano 35 Powering a Cavity b1b1 a1a1 b2b2 a2a2 

36. Topic One: Acceleration UW Spring 2008 Accelerator Physics J. J. Bisognano 36 Power Flow

37. Topic One: Acceleration UW Spring 2008 Accelerator Physics J. J. Bisognano 37 Power Flow/cont.

38. Topic One: Acceleration UW Spring 2008 Accelerator Physics J. J. Bisognano 38 Optimization With no beam, best circumstance is ; I.e., no reflected power

39. Topic One: Acceleration UW Spring 2008 Accelerator Physics J. J. Bisognano 39 At Resonance

40. Topic One: Acceleration UW Spring 2008 Accelerator Physics J. J. Bisognano 40 Shunt Impedance Consider a cavity with a longitudinal electric field along the particle trajectory Following P. Wilson z1z1 z2z2

41. Topic One: Acceleration UW Spring 2008 Accelerator Physics J. J. Bisognano 41 Shunt Impedance/cont

42. Topic One: Acceleration UW Spring 2008 Accelerator Physics J. J. Bisognano 42 Shunt Impedance/cont. Define where P is the power dissipated in the wall (the term) From the analysis of the coupling “  ” where is the generator power

43. Topic One: Acceleration UW Spring 2008 Accelerator Physics J. J. Bisognano 43 Beam Loading When a point charge is accelerated by a cavity, the loss of cavity field energy can be described by a charge induced field partially canceling the existing field By superposition, when a point charge crosses an empty cavity, a beam induced voltage appears To fully describe acceleration, we need to include this voltage Consider a cavity with an excitation V and a stored energy What is ?

44. Topic One: Acceleration UW Spring 2008 Accelerator Physics J. J. Bisognano 44 Beam Loading/cont. Let a charge pass through the cavity inducing and experiencing on itself. Assume a relative phase Let charge be bend around for another pass after a phase delay of 

45. Topic One: Acceleration UW Spring 2008 Accelerator Physics J. J. Bisognano 45 Beam Loading/cont. VeVe V2V2 V1V1 V 1 +V 2  

46. Topic One: Acceleration UW Spring 2008 Accelerator Physics J. J. Bisognano 46 Beam Loading/cont. With negligible loss But particle loses Since  is arbitrary,  and

47. Topic One: Acceleration UW Spring 2008 Accelerator Physics J. J. Bisognano 47 Beam Loading/cont. Note: we have same constant (R/Q) determining both required power and charge-cavity coupling

48. Topic One: Acceleration UW Spring 2008 Accelerator Physics J. J. Bisognano 48 Beam Induced Voltage Consider a sequence of particles at

49. Topic One: Acceleration UW Spring 2008 Accelerator Physics J. J. Bisognano 49 Summary of Beam Loading References: Microwave Circuits (Altman); HE Electron Linacs (Wilson, 1981 Fermilab Summer School )

50. Topic One: Acceleration UW Spring 2008 Accelerator Physics J. J. Bisognano 50 Vector Addition of RF Voltages    VbVb VcVc VbVb V br VgVg V gr

51. Topic One: Acceleration UW Spring 2008 Accelerator Physics J. J. Bisognano 51 Vector Algebra

52. Topic One: Acceleration UW Spring 2008 Accelerator Physics J. J. Bisognano 52 Required Generator Power Trig yields

53. Topic One: Acceleration UW Spring 2008 Accelerator Physics J. J. Bisognano 53 E.g, assume 

54. Topic One: Acceleration UW Spring 2008 Accelerator Physics J. J. Bisognano 54 Scaling of Shunt Impedance Consider a pillbox cavity of radius b & length L

55. Topic One: Acceleration UW Spring 2008 Accelerator Physics J. J. Bisognano 55 Pillbox Cavity The energy stored and power loss are given by

56. Topic One: Acceleration UW Spring 2008 Accelerator Physics J. J. Bisognano 56 Summary of Scaling

57. Topic One: Acceleration UW Spring 2008 Accelerator Physics J. J. Bisognano 57 More Multicell Cavities Given a solution of a single cell cavity, one can consider the coupling of multiple cells in that mode by an expansion where –the expansion coefficients give the strength of excitation of each cell in that mode –coupling comes from perturbation of purely conductive boundary by holes communicating field between cells This is a recondite subject, with all sorts of dangers from “conditional” covergence of Fourier series; see Slater (and Gluckstern) for a complete picture of this

58. Topic One: Acceleration UW Spring 2008 Accelerator Physics J. J. Bisognano 58 Periodic Structures

59. Topic One: Acceleration UW Spring 2008 Accelerator Physics J. J. Bisognano 59 Expansion Equations

60. Topic One: Acceleration UW Spring 2008 Accelerator Physics J. J. Bisognano 60 Holes But if there are holes in the cavity talking to neighboring cells, we have E.g., Bethe says Tangential electric field

61. Topic One: Acceleration UW Spring 2008 Accelerator Physics J. J. Bisognano 61 Coupled Equations

62. Topic One: Acceleration UW Spring 2008 Accelerator Physics J. J. Bisognano 62

63. Topic One: Acceleration UW Spring 2008 Accelerator Physics J. J. Bisognano 63 Result of Floquet Theorem that solutions of differential equations with periodic coefficients have form of periodic function times exp(j   z) Phase velocities less than c  particle acceleration possible From Slater, RMP 20,473

64. Topic One: Acceleration UW Spring 2008 Accelerator Physics J. J. Bisognano 64 Pi Mode Tesla Pi-mode

65. Topic One: Acceleration UW Spring 2008 Accelerator Physics J. J. Bisognano 65 Floquet Theorem For example, for a disk loaded circular cylindrical structure, the TM 01 is of the form

66. Topic One: Acceleration UW Spring 2008 Accelerator Physics J. J. Bisognano 66 Field Relations for Cylindrical Systems

67. Topic One: Acceleration UW Spring 2008 Accelerator Physics J. J. Bisognano 67 Field Relations for Cylindrical Systems

68. Topic One: Acceleration UW Spring 2008 Accelerator Physics J. J. Bisognano 68 Integrated Force at v=c Let be longitudinal force seen by a particle. Consider a trajectory z=vt, r=r 0. The integrated force is then= Only q=  /v contributes;i.e.  n =  /v

69. Topic One: Acceleration UW Spring 2008 Accelerator Physics J. J. Bisognano 69 E.g., a TM Mode

70. Topic One: Acceleration UW Spring 2008 Accelerator Physics J. J. Bisognano 70 Force on Relativistic Particle

71. Topic One: Acceleration UW Spring 2008 Accelerator Physics J. J. Bisognano 71 Panofsky Wenzel Theorem Pure TE mode doesn’t kick; pure TM mode, as in previous example, behaves to cancel denominator, so falls off as  -2 ; hybrid modes don’t cancel denominator, so finite kick may obtain even when v=c

72. Topic One: Acceleration UW Spring 2008 Accelerator Physics J. J. Bisognano 72 Superconductivity Basic mechanism –Condensation of charge carriers into Cooper pairs, coupled by lattice vibrations –Bandgap arises, limiting response to small perturbations (e.g., scattering) –No DC resistance At temperatures above 0 K, some of the Cooper pairs are “ionized” But for DC, these ionized pairs are “shorted out” and bulk resistance remains zero

73. Topic One: Acceleration UW Spring 2008 Accelerator Physics J. J. Bisognano 73 RF Superconductivity But pairs exhibit some inertia to changing electromagnetic fields, and there are some residual AC fields (sort of a reactance) These residual fields can act on the ionized, normal conducting carriers and cause dissipation But it’s very small at microwave frequencies (getting worse as f 2 ) At 1.3 GHz, copper has R s ~10 milli-ohm At 1.3 GHz, niobium has R s ~800 nano-ohm at 4.2 K At 1.3 GHz, niobium has R s ~15 nano-ohm at 2 K Q’s of 10 10 vs. 10 4

74. Topic One: Acceleration UW Spring 2008 Accelerator Physics J. J. Bisognano 74 CEBAF RF Parameters Superconducting 5-cell cavity in CEBAF

75. Topic One: Acceleration UW Spring 2008 Accelerator Physics J. J. Bisognano 75 CEBAF Cavity Assembly

76. Topic One: Acceleration UW Spring 2008 Accelerator Physics J. J. Bisognano 76 Cavity Specifications Frequency1497 MHz Nominal length0.5 meters Gradient>5 MeV/m Accel Current200  A  5 passes Number cells5 R/Q480 ohms Nominal Q 0 2.4·10 9 Loaded Q L 6.6 ·10 6

77. Topic One: Acceleration UW Spring 2008 Accelerator Physics J. J. Bisognano 77 What’s Q L ? Run in linac mode, essentially on crest,  = 0 In storage rings,  0 for longitudinal focusing Wall losses V 2 /R=(2.5 · 10 6 volts) 2 /(480  ·  2.4·10 9 ) are 5.4 watts Power to beam: ( 200  A  5 passes) ·(2.5 · 10 6 volts) is 2500 watts

78. Topic One: Acceleration UW Spring 2008 Accelerator Physics J. J. Bisognano 78 Would Copper Work Typical Q is now  10 4 Wall losses (2.5 · 10 6 volts) 2 /(480  ·  10 4 ) are now 1.3 MW vs. beam power of 2500 watts Some optimizations could yields “2’s” of improvement More importantly, SRF losses are at 2 K, which requires cryogenic refrigeration. Efficiencies are order 10 -3 So, 5 watts at 2 K is 5 kW at room temperature, but still factor of 100 to the good

79. Topic One: Acceleration UW Spring 2008 Accelerator Physics J. J. Bisognano 79 Higher Order Modes There are higher order modes, which can be excited by the beam These can generate wall losses, and fields can act on beam to generate destructive collective effects First question is whether wall losses are large or small compared to fundamental wall losses

80. Topic One: Acceleration UW Spring 2008 Accelerator Physics J. J. Bisognano 80 Loss Factors When a bunch passes through a cavity, it loses energy of

81. Topic One: Acceleration UW Spring 2008 Accelerator Physics J. J. Bisognano 81 Loss Factor for HOMs Bunch spectrum extends out to For a typical 1 ps linac bunch, For for a 1.5 GHz fundamental, there are many tens of longitudinal HOMs for the beam to couple; coupling is weaker than to fundamental because of more rapid temporal and spatial variation From codes,

82. Topic One: Acceleration UW Spring 2008 Accelerator Physics J. J. Bisognano 82 Power Estimates For 1 mA CEBAF 5-pass beam, with 0.5 pC/superbunch, we have only10mW of loss But in, say FEL application, with 100 pC bunches at 5mA, we have 10 watts in the wall, more than the power dissipated by fundamental! So extraction of HOM power is issue for high current applications with short bunches

83. Topic One: Acceleration UW Spring 2008 Accelerator Physics J. J. Bisognano 83 Couplers and Kicks Waveguide couplers break cylindrical symmetry Result is that the nominal TM 01 mode now has TE content and m  0; hybridized By Panofsky-Wenzel, introduces steering and skew quad fields that require compensation

84. Topic One: Acceleration UW Spring 2008 Accelerator Physics J. J. Bisognano 84 Homework: Topic I From Accelerator Physics, S.Y. Lee –Reading: Chapter 3, VIII, p. 352 & onward –Problems:3.8.1 (really table 3.10), 3.8.2, 3.8.4 The next generation CEBAF cavity can achieve 20 MV/m gradients with a 7 cell structure. a) Assuming R/Q is 7/5 higher, what would be the heat load generated per cavity if the Q is unchanged? b) How much higher a Q is necessary to main the heat load at the levels from the first generation cavity discussed in the lecture? Consider a single cell RF system operating at 500 MHz with an effective length of 0.3 meters, which is to be operated at a gradient of 2 MV/m with beam current of 100 mA. Assume the R/Q of the cavity is 100 ohms and that the Q from resistive wall losses is 40,000. A) Calculate the optimal coupling coefficient for powering the cavity with the 100 a mA beam passing through it. B) Calculate the power necessary with 100 mA beam to power the cavity. C) Calculate the power received by the beam and the power dissipated in wall losses. D) Describe what will happen to the power requirements and reflected power from the cavity if the beam is lost, but the feedback system attempts to maintain the 2 MV/meter gradient. A CEBAF problem: Use the nominal specs given in lecture a) For a 1 mA at 5 MV/m calculate total power and reflected power on resonance and 10 degrees off resonance with bunch on crest. b) If this beam is allowed to pass for a second time through the cavity for energy recovery at 170 degrees off accelerating crest, simultaneously with the first pass beam on crest, calculate the total and reflected power when the cavity is tuned on resonance.