Transcript Effects of External Magnetic Fields on the operation of an RF Cavity D.

Effects of External Magnetic Fields
on the operation of an RF Cavity
D. Stratakis, J. C. Gallardo, and R. B. Palmer
Brookhaven National Laboratory
RF Workshop III, FNAL, July 7-8 2009
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Outline
•
•
•
•
Motivation
Introduction and previous work
Breakdown model description
Particle tracking results within the pillbox cavity under
external magnetic fields
• Discussion of the effects of different cavity materials
• Summary
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Motivation/ Introduction
B
Moretti et al. PRST - AB (2005)
805 MHz
• Maximum gradients were found to depend strongly on
the external magnetic field
• Consequently the efficiency of the RF cavity is reduced
• Available data suggest that such problems are
associated with the unwanted emission from locally
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enhanced regions.
Model Description
B=1 T
Start
End
• Step 1: Emitted electrons from an asperity
are getting focused by the magnetic field
and reach the far cavity side or window.
Vacuum Metal
• Step 2: Those high power electrons strike P
R
the cavity surface and penetrate within the
d
metal up to a distance d.
δ
• Step 3: Surface temperature rises. The rise
induces thermal fatigue leading to surface
P : Incident P ower
damage and most likely to breakdown
R : BeamletRadius
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Pulsed Heating Experiments at SLAC
• Mushroom-type cavity used with no surface electric
fields at the bottom flange
• Sample bottom surface is pulsed heated from magnetic
fields (eddy currents) due the high powered rf pulse.
• Different sample materials were tested. For Cu
considerable degree of damage at 50 degrees
Pritzkau, PRST AB (2003)
L. Laurent HG Meeting(2009)
S. Tantawi HG Meeting (2009)
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Thermal Fatigue
• The metal surface heats up during each pulse and cools
down between pulses
• The temperature rise is enough so that expansion and
contraction of the surface causes internal stresses which
create deformation
• As the number of pulses increases, finally microcracks
appear. Damage occurs; likely breakdown starts.
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Objectives of this Study
• As in the SLAC case it is likely that the pillbox cavity
surface may experience thermal fatigue by the repeated
bombardment of dark current electrons as they are
focused by the B-fields
• ΔΤ~ Deposited Energy, beamlet geometry/ size, material
properties, # pulses
• To estimate ΔΤ we simulate the transport of emitted
electrons from field enhanced regions (asperities). In the
simulation we include:
– RF and externally applied magnetic fields
– The field enhancement from those asperities
– The self-field forces due space-charge
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Simulation Details
b
b2
r
c
c
• Model each individual emitter (asperity) as a prolate spheroid.
Then, field enhancement at the tip:
c


ETIP


r

 Esurf  βe Esurf
b
 ln(2 )  1 
r


• Electron emission is described by Fowler-Nordheim model
β E
1
12
J   J (t )dt  6.0 10
T 0
φ
T
2.5
e
2.5
0, surf
1.75
6.5310 φ


0.5
β E
104.52 φ  e e 0,surf


9 1.5
R.H. Fowler and L.W. Nordheim, Proc. Roy. Soc. (1928)
J. W. Wang, PhD Dis. (1989)




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Particle Tracking inside Cavity
B
electrons
2R
400 nm
z=0.0 cm
z=0.02 cm
R
z=8.1 cm
400 μm
END
400 μm
START
• Asperity was placed on
axis
• Particles launched
normal to surface at 1
eV energies
• Cavity identical to the
805 MHz pillbox (PB)
• Es, B similar to the PB
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Scale of Final Beamlet Size with B
METAL
R
1
• For any gradient, final beamlet radius scales as: R 
B
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Beam Transport in a Uniform Focusing
Channel
• Beam Envelope Equation:
2
2
γR γR  qB   pθ  1 ε 2 K
R  2  2  
 
 3  3  0
β γ 2 β γ  2mcβγ   mcβγ  R
R
R
pθ : Canonical angular momentum
ε : Beam emittance
K : Generalized perveance
R : RMS beamlet radius
Flat emitter (No radial fields)
• Assume:
– Conditions: pθ  0, ε  0
– "Matched Beam" R  0, R  0
• Then:
R
2 (  ) 0.5 0.5
I 0.5
I C
I0  q 
B

B
 2m c 
805 MHz RF
I 0.5
R~
B
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Scale of Final Beamlet Size with
Current and B for Asperities
For Asperities
• In real experiments we have asperities with radial fields
that reduce the space-charge effect. So:
I [ A]0.330.01
R[ m]  22.6 
B[T ]
• This result is independent from the magnetic field
strength
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Off-Axis Beamlet Sweep
• Sweeping was ignored in my
previous talks.
• Variation of the RF Phase:
– Spot on-Axis:
R
– Spot off Axis: Sweeping
on the x-y plane
 [m m]  aB[T ]b
a  0.26, b  1.0
• As Bob showed in previous talks the sweeping is B-field
dependent.
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CASINO Simulation of Electron Penetration
Cu
Be
1 MeV
• Simulation assumed zero magnetic fields on metal
CASINO, Scanning 29, p. 92 (2007)
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Energy Deposition on Wall
Copper, Cu
Aluminum, Al
Beryllium, Be
• Note that electrons penetrate deep in Beryllium
• Thus, less surface temperature rise would be expected.
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Average Energy Deposition at Surface
dE
[ MeV / cm]  aE[ MeV ]b
dx
a  4.98, b  0.49 Aluminum
a  25.7, b  0.40 Copper
a  2.5,
b  0.32 Beryllium
• Note that electrons penetrate deep in Beryllium
• Thus, less surface temperature rise would be expected.
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Summary
• Electrons were tracked inside an 805 MHz RF cavity with
external magnetic fields
• Necessary parameters to predict the outcomes of future
experiments such as spot geometry, beamlet size,
energy deposition were determined. Also scaling laws
with external magnetic fields were offered.
• Current data can provide an estimate of ΔT (Palmer).
Precise calculation of the temperature rise requires a
detailed simulation of the surface physics (ANSYS?,
PENELOPE?, other?).
• Beryllium appears to be a better material candidate
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