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The design of elliptical cavities

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The design ofellipticalcavities

Gabriele Costanza

Introduction

- To design a cavityweneedtocharacterize it from an electromagnetic and mechanicalpointofview

- Manufacturing, cleaning, testing
- Chemicalpolishing: Buffered ChemicalPolishing or Electrop-polishing. Removes a damagedsurfacelayer (dueto the manufacturing process) and reducesroughness.
- Heat treatment: removes H from the
- Rinsingwithhighpressure, ultrapure water

- Design = optimizationof the shapeof the cavitywithrespectto a set of parameters
- RF parameters
- Mechanical parameters

Introduction

- The medium βcavity has 5 cells and operates in the TM01πmode.
- The longitudinal E-field has a 180 phaseshift from one cell to the next so that the particlesexperiencealways an acceleratingfield. The lengthofeach cell is then:

Introduction

Multicell structures:

- Less expensive/m !!
- Fewer couplers, easierphasing…..

- Advantagesofsingle cell structures:
- No fieldflattness problem
- Easierto damp HOMS
- The input coupler transfers less power
- Easiertomanufacture and clean

Example: pillbox

- The simplestmodelof an acceleratingcavity = pillbox
- Let’sconsider a pillboxofradius a and length h.
- To find the fieldsof the accelerating mode (TM010) weneedtosolve 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 toresonate
at 704.42 MHz, a=16.29 cm

The design of elliptical cavities

RF parameters

RF parameters

- With the fieldswecancalculateseveralquantities:
- Stored Energy:
- Power Dissipated:
- part of the energystored in the cavity is dissipated on the walls

- Power exchangedwith the
external circuit:

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

- Power extracted by the HOM

port

RF parameters

- Intrinsicqualityfactor Q0:
- Measuresofhowquickly the energystored in the cavity is lost by dissipation in the cavitywalls.
- ExternalqualityfactorQext:
- Measureshowquickly the energystored in the cavity is radiatedthrough the ports
.

- Geometricfactor:
- Measuresofthe energylost by dissipation in the cavitywallsconsidering a Rsurfof 1 ohm.
- The surfaceresistanceof SC structurescan be modeledwith:
- The residualresistance is almostconstantwithtemperature and is a measureof the qualityof the material. The clearner the surface, and the purer the metal, the lower is the residualresistance.
- The BCS resitancegrowsveryquicklywith the frequency and decreasesexponentiallywith the temperature.

RF parameters

- We define the R/Q as:
- Where:
is a measureofhowefficient the cavityacceleratesthebeam,

- a large R/Q impliesthatlittleenergy is requiredtoproduce a large acceleration, therefore the R/Q is a measure on howefficient the energyexchangebetween a mode and the beam is (beamcouplingimpedance)
- R/Qdoes not depend on the material of the cavity.

RF parameters

- The higher the parameter:
the higher the acceleratingvoltagewithrespectto the powerdissipated

- Peak Fields:
- Epk/Eacc , whereEpkis the peakelectricfield on the surfaceof the cavityand
- Bpk/Eacc [mT/(MV/m)], whereBpk is the peakmagneticfield on the surfaceof the cavity.

RF parameters

- Cell to Cell CouplingKcc:
- It’s a measureof the widthof a band. It’susuallycalculatedonly for the fundamental passband.
- It’simportanttohave a high cell-to-cell couplingbecause:
- It’seasiertoobtain a highfieldflattness, that is, field is moreevenamong cells
- enhancedfrequency separation between the 4π/5 and the π modes
- HOMsarebettercoupledto the outer cells and possiblyextracted by an antenna

RF parameters: summary

- Rf parameters summary:
,

theseare not the only parameters totakeintoaccount…

- The end cells and the inner cells are different because the outer cells areconnectedto the beamtubes, so I considerthemseparately
- Let’stake a look at geometryof the inner cell:
- 6geometric parameters:
- A,B = radiusesof the major ellipse
- a,b = radiusesof the smallerellipse
- Riris = the radiusof the iris
- D = the diameter of the cell is a tuning parameter

- 6geometric parameters:
- The end cells addother 5 parameters (for symmetriccavities)

The design of elliptical cavities

Mechanical parameters

Mechanical parameters

- Assume a wallthicknessof 3.6 mm
- CavityStiffness [KN/mm]: 1 KN is applied at one end, the other end is grounded. The displacement is calculated
- TuningSensitivityΔf/Δz [KHz/mm]: a displacementof 1 mm is imposed at one end, the other end is grounded. The new frequencyof the π mode is calculated.

1 KN

Mechanical parameters

- PressureSensitivity [Hz/mbar]: vibrations coming from varioussources cause the detuningof the cavity. The major contributor is the variation of the helium pressure. In this simulation a uniform pressureof 1 mbar is appliedto the external boundary. The frequencyshift is calculated. Bothendsaregrounded

Mechanical parameters

- Lorentz DetuningCoefficient [Hz/(MV/m) 2]: The Lorentz Detuning Coefficient is defined as
- The frequency detuning is caused by the EM pressure on the cavitywalls. The pressure is
- Bothendsaregrounded

Design

- The radiusof the iris is a verypowerfulvariableto trim the RF parameters
- All the other parameters have a ”second order” influcence
- Toomany parameters to design an entirecavity all at once
- Design flow:
- All the cells are designed with COMSOL. I wrote a codetoexploreonesectionof the parameter space at a time. The codelaunches COMSOL tosimulate the structure,tunes the cell to 704 MHz and calculates the RF parameters. The mechanical simulations areperformedonly on the full cavity.
- Thereare 5 RF parameters, the optimal choice is not obvious! (tradeoffs)

RF Parameter calculation & selectionof the best geometry

RF Parameter calculation & selectionof the best geometry

Inner cell

cavity

end cell

Parameter trends

- All the parmeters areconnectedbetweeneachother and it’s not clearwhat the ”best solution” is
- For example:

Kcc

Peak Fields

Riris

R/Q

G

More on parameter trends

Highpeakfieldscan limit the maximum achievable gradient

- A ”tall” minor ellipseleadsto a lowerelectricpeakfield (αincreases).

- A ”large” major ellipseleadsto a lowermagneticpeakfield

- B has littleinfluence on the RF properties.

- The same appliesto the outer cells butit’shardertoachieve the same performancedueto the beamtube

The code

- The optimizingcode…

The code

- The optimizingcode…

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largerdomeellipse=>higherKcc

Found in ”Medium βEllipticalCavity – CyromoduleTechnologyDemonstrator”. S. Molloy

Canweusehigher gradients?

Courtesyof Paolo Pierini, HPSL Workshop

ResultsLower beta => lower R/Q

=> SmallerRiris

SPL CDR II

4.5 cm Riristoincrease

The R/Q but a lower beta

LeadstohigherKcc

Results

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Results

The cavities must be tunedtoobtain a highfieldflattness

Results

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Results

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Results

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Results

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Results

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Results: HOM 1pole list

All HOMswiththeir R/Q’sare

calculatedupto 3 GHz.

Studyof the HOMsstarted

2.111337 GHz

Two modes closeto6f0 :

f0 = 352.21 MHz

2.11135 GHz

Does this mode reallyexist?

On the numberof cells per cavity

βg

The lowe the numberof cells, the higher the maximum Eacc. The maximum is not obtained at the geometric beta

The higher the numberof cells, the lower the energy / velocityacceptancebut 4 cell cavitiesleadtolonger accelerator & more€

On the numberof cells per cavity

Cryostat FillingFactor = Cryostatacceleratingefficiency

=

βg =0.69

Is a higherβg better?

6 cavitiesper cryo

βg =0.67

5cavities per cryo

4 cavities per cryo

βg =0.65

2 m

1 m

10 cm

15 cm

On the numberof cells per cavity

βg

- Higherβg => widerenergy/velocityacceptance, higherinjectionenergy => morespokes. Aretheymoreefficient / less expensivethanellipticalcavities?
- If not it’spossibletouse ”few” βg = 0.65 ell. cavities (lowerinjectionenergy) and morehighβcavitieswhicharemoreefficientthanβg = 0.67 cavities
- Lowerβg => lowerperformances (butit’spossibletofind a goodcompromise). Cavities for βg<1 have a smallervolume, for the same frequency, w.r.t βg=1 cavities, and lowerEaccbecauseof the reducedlength => higherpeakfields

Simulations ofstiffenedcavities

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