The design of elliptical cavities Gabriele Costanza Introduction

The design of elliptical cavities Gabriele Costanza

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

Introduction •

Introduction 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

Example: pillbox •

THE DESIGN OF ELLIPTICAL CAVITIES RF parameters

RF parameters • port

• RF parameters

RF parameters •

RF parameters: summary •

THE DESIGN OF ELLIPTICAL CAVITIES Mechanical parameters

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 1 KN • 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.

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

Mechanical parameters •

THE DESIGN OF ELLIPTICAL CAVITIES Design

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: RF Parameter calculation & selection of the best geometry Inner cell • • RF Parameter calculation & selection of the best geometry end cell cavity 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)

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

More on parameter trends High peak fields can limit the maximum achievable gradient parameter increases A Bpk/Eacc Epk/Eacc - ~+ B ~ ~ a ~ ~- b ~- - - 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

The code • The optimizing code…

THE DESIGN OF ELLIPTICAL CAVITIES Results

larger dome ellipse=>higher Kcc 63+2 57_2+20 63_2+31 Found in ”Medium β Elliptical Cavity – Cyromodule Technology Demonstrator”. S. Molloy RF parameters R/Q[Ohm] 302. 30 308. 29 309. 81 G[Ohm] 198. 7 204. 5 203. 58 G R/Q [Ohm 2] 60077 63069 63071 Epk/Eacc 2. 508 2. 6052 2. 5578 Bpk/Eacc [m. T/MV/m] 4. 936 4. 8097 4. 816 Field flattness [%] 99. 98 99. 967 99. 93 Kcc [%] 1. 32 1. 43 1. 36 908. 7 861. 8 Freq. distance between 840 4π/5 and π mode [KHz] Can we use higher gradients? Mechanical parameters (no stiffening rings) Cavity Stiffness [KN/mm] 0. 956 0. 714 0. 659 Tuning Sensitivity Δf/Δz [KHz/mm] 244. 9 239. 4 244. 2 KL [Hz/(MV/m) 2] Both ends fixed 1. 739 1. 499 1. 53 Pressure Sensitivity [Hz/mbar] 28. 7 35. 6 34 Epk/Eacc < 2. 63 Epk/Eacc < 2. 66 Bpk/Eacc< 5. 26 Bpk/Eacc < 5. 33 m. T/MV/m

Results Courtesy of Paolo Pierini, HPSL Workshop SPL CDR II Lower beta => lower R/Q => Smaller Riris 4. 5 cm Riris to increase The R/Q but a lower beta Leads to higher Kcc

Results 63+2

Results 63+2 • The cavities tend to have better performances for β> βg

Results 63+2

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

Results 63+2

Results 57_2+20

Results

Results 57_2+20

Results 63_2+31

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

Results: HOM 1 pole list All HOMs with their R/Q’s are calculated up to 3 GHz. Study of the HOMs started 2. 111337 GHz Two modes close to 6 f 0 : f 0 = 352. 21 MHz 2. 11135 GHz Does this mode really exist?

On the number of cells per cavity βg 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 €

On the number of cells per cavity βg =0. 69 Is a higher βg better? 6 cavities per cryo 5 cavities per cryo 4 cavities per cryo 2 m 10 cm 15 cm βg =0. 67 βg =0. 65

On the number of cells per cavity βg • • • Higher βg => 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” βg = 0. 65 ell. cavities (lower injection energy) and more high β cavities which are more efficient than βg = 0. 67 cavities Lower βg => lower performances (but it’s possible to find a good compromise). Cavities for βg <1 have a smaller volume, for the same frequency, w. r. t βg =1 cavities, and lower Eacc because of the reduced length => higher peak fields

Simulations of stiffened cavities RF parameters 63_2+31 R/Q[Ohm] 309. 81 G[Ohm] 203. 58 G R/Q [Ohm 2] 63071 Epk/Eacc 2. 5578 Bpk/Eacc [m. T/MV/m] 4. 816 Field flattness [%] 99. 93 Kcc [%] 1. 36 Freq. distance between 4π/5 861. 8 and π mode [KHz] 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

Some results 63+2 2 63 RF parameters R/Q[Ohm] 302. 304 64. 6715 59. 7 G[Ohm] 198. 7338 196. 637 201. 87 Epk/Eacc 2. 508 2. 452 2. 4725 Bpk/Eacc [m. T/MV/m] 4. 936 4. 8389 4. 8646 Field flattness [%] 99. 98 Kcc [%] 1. 32 Freq. distance between 4π/5 and π mode [ KHz] 840 1. 302 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

Some results 57_2+20 57_2 20 RF parameters R/Q[Ohm] 308. 29 66. 2645 60. 625 G[Ohm] 204. 5778 202. 92 207. 05 Epk/Eacc 2. 6052 2. 5284 2. 5546 Bpk/Eacc [m. T/MV/m] 4. 8097 4. 6875 4. 7496 Field flattness [%] 99. 967 Kcc [%] 1. 43 Freq. distance between 4π/5 and π mode [ KHz] 908. 7 1. 4 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

Some results 63_2+31 63_2 31 RF parameters R/Q[Ohm] 309. 81 66. 59 60. 91 G[Ohm] 203. 58 201. 69 206. 39 Epk/Eacc 2. 5578 2. 4996 2. 5192 Bpk/Eacc [m. T/MV/m] 4. 816 4. 6991 4. 7511 Field flattness [%] 99. 93 Kcc [%] 1. 36 Freq. distance between 4π/5 and π mode [ KHz] 861. 8 1. 339 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