the design of elliptical cavities

The design of elliptical cavities

Gabriele Costanza

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

Introduction• The medium β cavity has

5 cells and operates in the TM01π mode.

• The longitudinal E-field has a 180 phase shift from one cell to the next so that the particles experience always an accelerating field. The length of each cell is then:

IntroductionMulticell 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 simplest model of an accelerating cavity = pillbox• Let’s consider a pillbox of radius a and length h.• To find the fields of the accelerating mode (TM010) we need to solve the transverse

problem:• • and the longitudinal problem:

• The solution consists in the eigenmodes and eigenvalues.• The accelerating (fundamental) mode is the (TM010):

• The dispersion relation is:

• For the TM010 mode to resonate at 704.42 MHz, a=16.29 cm

RF parameters

THE DESIGN OF ELLIPTICAL CAVITIES

RF parameters• With the fields we can calculate

several quantities:• Stored Energy:

• Power Dissipated:– part of the energy stored in the

cavity is dissipated on the walls

• Power exchanged with the external circuit:

– Power extracted by the HOM coupler or injected by the FPC

port

RF parameters• Intrinsic quality factor Q0: • Measures of how quickly the energy stored in the cavity is lost by dissipation in the cavity

walls. • External quality factor Qext: • Measures how quickly the energy stored in the cavity is radiated through the ports .• Geometric factor: • Measures of the energy lost by dissipation in the cavity walls considering a Rsurf of 1 ohm. • The surface resistance of SC structures can be modeled with:

• The residual resistance is almost constant with temperature and is a measure of the quality of the material. The clearner the surface, and the purer the metal, the lower is the residual resistance.

• The BCS resitance grows very quickly with the frequency and decreases exponentially with the temperature.

RF parameters• We define the R/Q as:

• Where:

is a measure of how efficient the cavity accelerates the beam, • a large R/Q implies that little energy is required to

produce a large acceleration, therefore the R/Q is a measure on how efficient the energy exchange between a mode and the beam is (beam coupling impedance)

• R/Q does not depend on the material of the cavity.

RF parameters• The higher the parameter: the higher the accelerating voltage with respect to the power dissipated• Peak Fields:– Epk/Eacc , where Epk is the peak electric

field on the surface of the cavity and – Bpk/Eacc [mT/(MV/m)] , where Bpk is the

peak magnetic field on the surface of the cavity.

RF parameters• Cell to Cell Coupling Kcc: • It’s a measure of the width of a

band. It’s usually calculated only for the fundamental passband.

• It’s important to have a high cell-to-cell coupling because:– It’s easier to obtain a high field

flattness, that is, field is more even among cells

– enhanced frequency separation between the 4π/5 and the π modes

– HOMs are better coupled to the outer cells and possibly extracted by an antenna

RF parameters: summary• Rf parameters summary:

,these are not the only parameters to take into account…

• The end cells and the inner cells are different because the outer cells are connected to the beam tubes, so I consider them separately

• Let’s take a look at geometry of the inner cell:– 6 geometric parameters:

• A,B = radiuses of the major ellipse• a,b = radiuses of the smaller ellipse• Riris = the radius of the iris• D = the diameter of the cell is a tuning

parameter• The end cells add other 5 parameters (for

symmetric cavities)

Mechanical parameters

THE DESIGN OF ELLIPTICAL CAVITIES

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

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• Lorentz Detuning Coefficient [Hz/(MV/m) 2]: The Lorentz Detuning Coefficient is defined as

• The frequency detuning is caused by the EM pressure on the cavity walls. The pressure is

• Both ends are grounded

Design

THE DESIGN OF ELLIPTICAL CAVITIES

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

Parameter trends• All the parmeters are connected between each other and it’s not clear what

the ”best solution” is• For example:

Riris

KccPeak Fields

R/QG

parameter increases

Bpk/Eacc Epk/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

The code• The optimizing code…

The code• The optimizing code…

Results

THE DESIGN OF ELLIPTICAL CAVITIES

RF parametersR/Q[Ohm] 302.30 308.29 309.81

G[Ohm] 198.7 204.5 203.58

G R/Q [Ohm2] 60077 63069 63071

Epk/Eacc 2.508 2.6052 2.5578

Bpk/Eacc [mT/MV/m] 4.936 4.8097 4.816

Field flattness [%] 99.98 99.967 99.93

Kcc [%] 1.32 1.43 1.36

Freq. distance between 4π/5 and π mode [KHz]

840 908.7 861.8

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

63+2 57_2+20 63_2+31Found in ”Medium β Elliptical Cavity – Cyromodule Technology Demonstrator”. S. Molloy

Epk/Eacc < 2.63 Epk/Eacc < 2.66

Bpk/Eacc< 5.26 mT/MV/m

Bpk/Eacc < 5.33 mT/MV/m

Can we use higher gradients?

larger dome ellipse=>higher Kcc

ResultsCourtesy 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

Results63+2

Results63+2

Results63+2

• The cavities tend to have better performances for β> βg

Results63+2

Results

The cavities must be tuned to obtain a high field flattness

Results63+2

Results57_2+20

Results57_2+20

Results57_2+20

Results

Results57_2+20

Results63_2+31

Results63_2+31

Results63_2+31

Results63_2+31

Results63_2+31

Bonus Section (if you’re not too bored….)

THE DESIGN OF ELLIPTICAL CAVITIES

SLUT, TACK

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 6f0 :f0 = 352.21 MHz

2.111337 GHz

2.11135 GHz

Does this mode really exist?

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 €

βg

On the number of cells per cavity

Cryostat Filling Factor = Cryostat accelerating efficiency

=

4 cavities per cryo5 cavities per cryo

6 cavities per cryo

βg =0.65

βg =0.67

βg =0.69

1 m

15 cm

10 cm

2 m

Is a higher βg better?

On the number of cells per cavity

• 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

βg

RF parametersR/Q[Ohm] 309.81

G[Ohm] 203.58

G R/Q [Ohm2] 63071

Epk/Eacc 2.5578

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

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 [mT/MV/m] 4.936 4.8389 4.8646

Field flattness [%] 99.98

Kcc [%] 1.32 1.302

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

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 [mT/MV/m] 4.8097 4.6875 4.7496

Field flattness [%] 99.967

Kcc [%] 1.43 1.4

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_2 20Some results

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 [mT/MV/m] 4.816 4.6991 4.7511

Field flattness [%] 99.93

Kcc [%] 1.36 1.339

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_2 31

Some results