Transcript AC Resistive Wall Wake Field Measurement, Theory

Resistive Wall Wakes in the Undulator
Beam Pipe – Theory, Measurement
Karl Bane
FAC Meeting
April 7, 2005
• Work done with G. Stupakov, Jiufeng Tu
see:
SLAC-PUB-10707/LCLS-TN-04-11 Revised, Oct. 2004
SLAC-LCLS-TN-05-6, Feb. 2005
April 7-8, 2005
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Outline
• Review ac resistive wall wake calculations given last time
•Analyze reflectivity measurements performed by J. Tu at Brookhaven
•Discuss implications
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Introduction
• in the LCLS, the relative energy variation (within the bunch) induced
within the undulator must be kept to a few times the Pierce parameter;
if it becomes larger, part of the beam will not reach saturation
• the largest contributor to energy change in the undulator is the
(longitudinal) resistive wall wakefield
• earlier calculations included only the so-called dc conductivity of the
metal (beam pipe) wall; we showed last time that:
--if the ac conductivity is included the wake effect is larger
--that the anomalous skin effect is negligible, and can be ignored
--that the wake effect can be ameliorated by going to aluminum,
and to a flat chamber.
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•the free-electron model of conductivity is expected to be valid at
low frequencies; it has two parameters: dc conductivity , and
relaxation time 
• at 295K: Cu--= 5.261017/s, = 2.5210-14 s;
Al --= 3.351017/s, = 0.7510-14 s
•if it is valid, the wake in a round beam pipe can be approximated:
with the plasma frequency
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Point charge wake
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charge—1 nC,
energy—14 GeV,
tube radius—2.5 mm,
tube length—130 m
induced energy deviation
for round chamber
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induced energy deviation
for flat chamber
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Table I: Figure of merit, E (minimum total energy variation over 30
m stretch of beam), for different assumptions of beam pipe
shape and material. Nominally, vertical aperture is 2a= 5 mm.
Note that Cu-dc results are not physically realizable.
--figure of merit only gives rough idea of effect in LCLS; better to use
analytic model of Z. Huang and G. Stupakov or simulations
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FTIR Reflectivity Measurements
(J. Tu talk)
~  ~( )  n  ik
n
4~ ( )
41
~
 ( ) 
i  1 
i


~
Eref
n~  1
i ( )
~
~   ( )e
n 1
Einc
Kramers-Kronig relation: causality
0
 ( 0 ) 



ln[ R( )]  ln[ R( 0 )]
0
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 02   2
8
We can measure
~ 2 ~ 2
Eref
n 1
R ~
 ~
n 1
Einc
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Measured conductivity compared with calculation
Kramers Kronig analysis of
measurements
Calculation using Ashcroft-Mermin , 
--1(0) is dc conductivity
--Kramer’s Kronig analysis of measurements gives dc conductivity factor
1/2.5 of expected value
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Behavior of R for free-electron model
Frequency k vs. reflectivity R for a metallic conductor, assuming the free
electron model (solid line). For this example =1.21016 /s and =5.410-15
s. Analytic guideposts are also given (dashes).
--for LCLS bunch interested in : [10,100] m, or k: [0.06,0.6]m-1
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Brookhaven reflectivity data for Cu and evaporated Al
range of interest
Measurement results from Brookhaven: reflectivity R vs.
frequency k for copper and evaporated aluminum.
--Cu absorption band at 10 m-1 also seen in plot in Ashcroft-Mermin
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Al survey of measurements, Shiles, et al, 1980
range of interest
1evk= 5 m-1
--also slope in R(), but only ½ as steep
--clear absorption resonance, only hinted at in Brookhaven results
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Al fit
region of interest
Aluminum reflectivity: comparison of measurements (blue) with
calculations using nominal ,  (green); and fitted values: 0.63 nominal ,
0.78 nominal  (red). The position of k= 1/c for the fit is also shown.
--in literature no data below 0.2 m-1
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Cu fit
region of interest
Copper reflectivity: comparison of measurements (blue) with calculations
using nominal ,  (green); and fitted values: 0.66 nominal , 0.67 nominal
 (red). The position of k= 1/c for the fit is also shown.
--in literature no data below 0.3 m-1
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For Al, how does the fit taken affect
E for the LCLS?
Consider (,):
nom.– (3.351017/s, 0.7510-14 s)
fit– (2.121017/s, 0.5810-14 s)
model2– (1.681017/s, 0.4810-14 s)
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Conclusion and Discussion
--The resistive wall wakefield in the beam pipe of the LCLS undulator will
induce a significant energy variation within the bunch; it seems that this will
inhibit much of the beam from lasing.
--This effect can be ameliorated by going from a pipe made of Cu to one of
Al (by ~2), and from one with a round to a flat cross-section (by ~30%).
--To really quantify this, one should go to analytical calculation of Z. Huang
and G. Stupakov, or simulations (S. Reichle and W. Fawley)
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Brookhaven Measurements
--fits (/nom, /nom) are approx. (0.65, 0.80) for Al, and (0.65, 0.65) for
Cu; fit is reasonably good, but does not cover desired frequency (k) range
for Cu (only 40%)
--At higher frequencies the measured R varies linearly with frequency,
unlike the constant dependence expected by the free-electron model.
This point should be understood before giving too much credence to our
results
--The deviation in R for Cu above 0.3 m-1 is small and may not have
much effect. However, it is difficult to know precisely the implication for
the wakefield of a copper pipe: reflectivity at normal incidence alone,
without a model, is not enough information to make such a calculation.
--Measurements of more Al samples are in the works.
April 7-8, 2005
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Karl Bane
[email protected]