1. The design of elliptical cavities Gabriele Costanza

2. Introduction Design = optimization of the shape of the cavity with respect to a set of parameters – RF parameters – Mechanical parameters Manufacturing, cleaning, testing – Chemical polishing: Buffered Chemical Polishing or Electrop-polishing. Removes a damaged surface layer (due to the manufacturing process) and reduces roughness. – Heat treatment: removes H from the – Rinsing with high pressure, ultrapure water To design a cavity we need to characterize it from an electromagnetic and mechanical point of view

3. Introduction

4. Multicell structures: Less expensive/m !! Fewer couplers, easier phasing….. Advantages of single cell structures: No field flattness problem Easier to damp HOMS The input coupler transfers less power Easier to manufacture and clean

5. Example: pillbox

6. RF parameters THE DESIGN OF ELLIPTICAL CAVITIES

7. RF parameters port

8. RF parameters

12. RF parameters: summary

13. Mechanical parameters THE DESIGN OF ELLIPTICAL CAVITIES

14. Mechanical parameters Assume a wall thickness of 3.6 mm Cavity Stiffness [KN/mm]: 1 KN is applied at one end, the other end is grounded. The displacement is calculated Tuning Sensitivity Δf/Δz [KHz/mm]: a displacement of 1 mm is imposed at one end, the other end is grounded. The new frequency of the π mode is calculated. 1 KN

15. Mechanical parameters Pressure Sensitivity [Hz/mbar]: vibrations coming from various sources cause the detuning of the cavity. The major contributor is the variation of the helium pressure. In this simulation a uniform pressure of 1 mbar is applied to the external boundary. The frequency shift is calculated. Both ends are grounded

16. Mechanical parameters

17. Design THE DESIGN OF ELLIPTICAL CAVITIES

18. Design The radius of the iris is a very powerful variable to trim the RF parameters All the other parameters have a ”second order” influcence Too many parameters to design an entire cavity all at once Design flow: All the cells are designed with COMSOL. I wrote a code to explore one section of the parameter space at a time. The code launches COMSOL to simulate the structure, tunes the cell to 704 MHz and calculates the RF parameters. The mechanical simulations are performed only on the full cavity. There are 5 RF parameters, the optimal choice is not obvious! (tradeoffs) Inner cell RF Parameter calculation & selection of the best geometry end cell RF Parameter calculation & selection of the best geometry cavity

19. Parameter trends All the parmeters are connected between each other and it’s not clear what the ”best solution” is For example: Riris Kcc Peak Fields R/Q G

20. parameter increases Bpk/EaccEpk/Eacc A-~+ B~~ a~~- b - More on parameter trends - A ”tall” minor ellipse leads to a lower electric peak field (α increases). - A ”large” major ellipse leads to a lower magnetic peak field - B has little influence on the RF properties. - The same applies to the outer cells but it’s harder to achieve the same performance due to the beam tube High peak fields can limit the maximum achievable gradient

21. The code The optimizing code…

22. The code The optimizing code…

23. Results THE DESIGN OF ELLIPTICAL CAVITIES

24. RF parameters R/Q[Ohm]302.30308.29309.81 G[Ohm]198.7204.5203.58 G R/Q [Ohm 2 ]600776306963071 Epk/Eacc2.5082.60522.5578 Bpk/Eacc [mT/MV/m]4.9364.80974.816 Field flattness [%]99.9899.96799.93 Kcc [%]1.321.431.36 Freq. distance between 4π/5 and π mode [KHz] 840908.7861.8 Mechanical parameters (no stiffening rings) Cavity Stiffness [KN/mm] 0.9560.7140.659 Tuning Sensitivity Δf/Δz [KHz/mm] 244.9239.4244.2 KL [Hz/(MV/m) 2 ] Both ends fixed 1.7391.4991.53 Pressure Sensitivity [Hz/mbar] 28.735.634 63+257_2+20 63_2+31 Found in ”Medium β Elliptical Cavity – Cyromodule Technology Demonstrator”. S. Molloy Can we use higher gradients? larger dome ellipse=>higher Kcc

25. Results Courtesy of Paolo Pierini, HPSL Workshop SPL CDR II 4.5 cm Riris to increase The R/Q but a lower beta Leads to higher Kcc Lower beta => lower R/Q => Smaller Riris

26. Results 63+2

27. Results 63+2

28. Results 63+2 The cavities tend to have better performances for β> β g

29. Results 63+2

30. Results The cavities must be tuned to obtain a high field flattness

31. Results 63+2

32. Results 57_2+20

33. Results 57_2+20

34. Results 57_2+20

35. Results

36. 57_2+20

37. Results 63_2+31

38. Results 63_2+31

39. Results 63_2+31

40. Results 63_2+31

41. Results 63_2+31

42. Bonus Section (if you’re not too bored….) THE DESIGN OF ELLIPTICAL CAVITIES SLUT, TACK

43. Results: HOM 1pole list All HOMs with their R/Q’s are calculated up to 3 GHz. Study of the HOMs started Two modes close to 6f 0 : f 0 = 352.21 MHz 2.111337 GHz 2.11135 GHz Does this mode really exist?

44. On the number of cells per cavity 1.The lowe the number of cells, the higher the maximum Eacc. The maximum is not obtained at the geometric beta 2.The higher the number of cells, the lower the energy / velocity acceptance but 4 cell cavities lead to longer accelerator & more € βgβg

45. On the number of cells per cavity 4 cavities per cryo 5 cavities per cryo 6 cavities per cryo β g =0.65 β g =0.67 β g =0.69 1 m 15 cm 10 cm 2 m Is a higher β g better?

46. On the number of cells per cavity Higher β g => wider energy/velocity acceptance, higher injection energy => more spokes. Are they more efficient / less expensive than elliptical cavities? If not it’s possible to use ”few” β g = 0.65 ell. cavities (lower injection energy) and more high β cavities which are more efficient than β g = 0.67 cavities Lower β g => lower performances (but it’s possible to find a good compromise). Cavities for β g higher peak fields βgβg

48. RF parameters R/Q[Ohm]309.81 G[Ohm]203.58 G R/Q [Ohm 2 ]63071 Epk/Eacc2.5578 Bpk/Eacc [mT/MV/m]4.816 Field flattness [%]99.93 Kcc [%]1.36 Freq. distance between 4π/5 and π mode [KHz] 861.8 Mechanical parameters (w stiffening rings) Cavity Stiffness [KN/mm]1.65 Tuning Sensitivity Δf/Δz [KHz/mm] 254.4 KL [Hz/(MV/m) 2 ] Both ends fixed 0.93 Pressure Sensitivity [Hz/mbar] 0.67 63_2+31 Simulations of stiffened cavities

49. RF parameters R/Q[Ohm]302.30464.671559.7 G[Ohm]198.7338196.637201.87 Epk/Eacc2.5082.4522.4725 Bpk/Eacc [mT/MV/m]4.9364.83894.8646 Field flattness [%]99.98 Kcc [%]1.321.302 Freq. distance between 4π/5 and π mode [ KHz] 840 Mechanical parameters Cavity Stiffness [KN/mm]0.956 Tuning Sensitivity Δf/Δz [KHz/mm] 244.5 KL [Hz/(MV/m) 2 ] Both ends fixed 1.739 Pressure Sensitivity [Hz/mbar] 28.68 63+2 63 2 Some results

50. RF parameters R/Q[Ohm]308.2966.264560.625 G[Ohm]204.5778202.92207.05 Epk/Eacc2.60522.52842.5546 Bpk/Eacc [mT/MV/m]4.80974.68754.7496 Field flattness [%]99.967 Kcc [%]1.431.4 Freq. distance between 4π/5 and π mode [ KHz] 908.7 Mechanical parameters Cavity Stiffness [KN/mm]0.714 Tuning Sensitivity Δf/Δz [KHz/mm] 239.4 KL [Hz/(MV/m) 2 ] Both ends fixed 1.499 Pressure Sensitivity [Hz/mbar] 84 57_2+20 57_220 Some results

51. RF parameters R/Q[Ohm]309.8166.5960.91 G[Ohm]203.58201.69206.39 Epk/Eacc2.55782.49962.5192 Bpk/Eacc [mT/MV/m]4.8164.69914.7511 Field flattness [%]99.93 Kcc [%]1.361.339 Freq. distance between 4π/5 and π mode [ KHz] 861.8 Mechanical parameters Cavity Stiffness [KN/mm]0.659 Tuning Sensitivity Δf/Δz [KHz/mm] 244 KL [Hz/(MV/m) 2 ] Both ends fixed 1.5 Pressure Sensitivity [Hz/mbar] 43 63_2+31 63_231 Some results