Effects of External Fields on RF Cavity Operation

Diktys StratakisAdvanced Accelerator Group

Brookhaven National Laboratory

Muon Collider Design Workshop – Jefferson LabDecember 09, 2008 1

Thanks to: J. S. Berg, R. C. Fernow, J. C. Gallardo, H. Kirk, R. B. Palmer (PO-BNL), X. Chang (AD-BNL), S. Kahn (Muons Inc.)

Outline

• Motivation• Introduction/ Previous Work• Model Description • Results• Summary

2

Motivation

• Maximum gradients were found to depend strongly on the magnetic field

• Consequently the efficiency of the RF cavity is reduced

805 MHz

Moretti et al. PRST - AB (2005) 3

Introduction and Previous Work (1)

• Dark currents electrons were observed in a multi-cell 805 MHz cavity

• They arise most likely from field enhanced regions on the cavity iris.

• Measured local field gradients where up to 10 GV/m. Such enhancement is mainly due: (i) the cavity geometry (iris), and (ii) material imperfections

• Those electron emitters are estimated to be around 1000, each with an average surface field enhancement βe=184.

Norem et al. PRST - AB (2003) 4

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Objectives of this Study

• Model the propagation of emitted electrons from field enhanced regions (asperities) through an RF cavity. In the simulation we include:– RF and externally applied magnetic fields– The field enhancement from those asperities– The self-field forces due space-charge

• Estimate the surface temperature rise after impact with the wall.

• Can we explain the dependence of the maximum gradient on the strength of the magnetic field (measured in Moretti's and Norem's experiment)?

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Model Description

• Use a surface field such that to produce the observed high local gradient on Norem’s experiment

(i.e. )

• Define the geometry of the asperity such that to produce a local field enhancement consistent with that on Norem’s experiment (i.e )

c

b

• Model each individual emitter (asperity) as a prolate spheroid. Then, the field enhancement at the tip is:

16μm, 0.7μm, 184ec b β

ln(2 ) 1TIP surf e surf

crE E β Ebr

2br

c

Eyring et al. PR (1928)

TIP (184) 50MV/m 10GV/mE 6

Electron Emission Model

• Current Density:

9 1.5

0.5

6.53 102 26 4.521.54 10 10 e surf

φ

β Ee surf φβ EJ e

φ

2

2 ( ) 1dR

ds πR z dzdz

• Average Current density for an RF period:

I Jds

9 1.5

0.50,

6.53 102.5 2.50,12 4.52

1.750

1( ) 6.0 10 10 e surf

φTβ Ee surf φβ E

J J t dt eT φ

J. W. Wang, PhD Dis. (1989)R.H. Fowler and L.W. Nordheim, Proc. Roy. Soc. (1928)

0, sin( )surf surfE E ωt• In RF:

• Assume that the electron emission is described by Fowler-Nordheim model (quantum tunneling through a barrier).

• Average Current: where

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Asperity

Current

Enhancement

Conditions at the Beam Source

8

rad. curvature = 0.03 μm

Simulation Details

9

0 19 /E MV m

Asperity

B

• SUPERFISH/MATLAB is used to create a grid that is a superposition of the RF fields and the asperity enhanced fields and PARMELA is used for electron tracking

Initial Distribution (t=0)

Initial Energy = 1eV

Current and RF Phase

• Oscillation will finally wash-out

• Particles emitted between 20-110 deg. will reach the other side of the wall

• Current will vary with phase

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Energy and Power at Impact Point

• Final kinetic energies up to 1 MeV• Particles emitted between 90-95 deg. are those with the

highest impact

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Dependence of Beamlet Radial Size with Field

• How rms beamlet radius scales with current at impact point?

At z=8.1cm

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Dependence of Beamlet Size with Current • Beam Envelope Equation:

2 2 2

2 2 3 3

10

2 2θpγ R γ R qB ε K

Rβ γ β γ mcβγ mcβγ R R R

: Canonical angular momentum

: Beam emittance

: Generalized perveance

: RMS beamlet radius

θp

ε

K

R

• Assume:– Conditions:– "Matched Beam" – Flat emitter (No radial fields)

0, 0θp ε 0, 0R R

• Then:0.5

0

2 ( )

2

βγR I

qIB

mc

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Dependence of Beamlet Size with Current in Asperities

• The asperity radial fields will reduce the effect of space-charge

• Therefore, less beam defocusing is expected

jIR

B

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Dependence of Beamlet Size with Current and Field

• Higher fields reduce the final beamlet size but increase the energy density

• How much damage it creates to the cavity wall has to be estimated

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jIR

B

Summary

• Electrons were tracked inside an 805 MHz RF cavity with external magnetic fields

• Electrons, get focused by the external magnetic field and hit the cavity wall with large energies (1MeV).

• Space-charge is deflecting the electrons leading to larger beams.

• Depending on the field the final beamlet rms size is found to be 200-700 μm.

• We found that the beamlet radial size scales as 0.38 with the current, and is independent from the external field.

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