1. Tune Shift Induced by Flat-Chamber Resistive Wall Impedance
LHC Collimator Experiment
in the SPS
Frank Zimmermann
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2. Frank Zimmermann, GSI Meeting 31.03.2006
introduction
collimators are largest impedance in LHC
~1-m long graphite blocks (for survival), half gap ~1.5 mm
2004 experiment aimed at validating our impedance model
best measured quantity: coherent tune shift
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3. Frank Zimmermann, GSI Meeting 31.03.2006
prototype LHC collimator installed in the SPS
(R. Assmann)
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4. Frank Zimmermann, GSI Meeting 31.03.2006
parameters
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5. Frank Zimmermann, GSI Meeting 31.03.2006
measurement
data from Marek Gasior
(BBQ monitor)
For smallest gaps, intensity was reduced & transverse emittance increased
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6. Frank Zimmermann, GSI Meeting 31.03.2006
comparing measurement with classical theory
applies if
with
Chao, Physics of Collective Instabilities in High
Energy Accelerators, J. Wiley, New York 1993
c~2x10-5 for s~105 W-1m-1
and half gap b~1.5 mm
→OK from 1 MHz to 1 THz
classical
factor
2.5 difference
at small
gaps
data
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7. Frank Zimmermann, GSI Meeting 31.03.2006
A.Burov,
V. Lebedev,
EPAC’02
comparing with Burov-Lebedev theory
includes effect of finite chamber thickness and so-called ‘inductive bypass
effect’ (correct dependence at low frequency)
applies if
with
and
Burov-Lebedev
factor
2 difference
at small
gaps
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8. Frank Zimmermann, GSI Meeting 31.03.2006
consideration
rms beam size ~1/4 half gap
→ nonlinear component of the wake field could be important
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9. Frank Zimmermann, GSI Meeting 31.03.2006
wake potential & nonlinear deflection
potential for nonlinear resistive-wall impedance between two parallel
plates was derived by Piwinski (DESY , Eq. (52)) and re-written
by Bane, Irwin, and Raubenheimer (NLC ZDR p. 594).
2b: full gap
nonlinear kick to test particle:
Piwinski formula applies if
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10. Frank Zimmermann, GSI Meeting 31.03.2006
time dependence of kick along
the bunch is described by fR
tail
head
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11. introduce new coordinates and perform 2 integrations
coherent tune shift:
introduce new coordinates
and perform 2 integrations
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12. Frank Zimmermann, GSI Meeting 31.03.2006
the function G(X,Y)
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13. Frank Zimmermann, GSI Meeting 31.03.2006
the function G(X,Y) )
note: change of sign for large X
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14. Frank Zimmermann, GSI Meeting 31.03.2006
the function G(X,Y)
note: divergence for Y→2b
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15. Frank Zimmermann, GSI Meeting 31.03.2006
variation of coherent tune shift with emittance g
example parameters
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16. Frank Zimmermann, GSI Meeting 31.03.2006
comparing SPS measurement with tune shift
expected from nonlinear wake field
nonlinear wake field
data
20%
difference
for small
gaps
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17. Frank Zimmermann, GSI Meeting 31.03.2006
comparing SPS measurement with tune shift
expected from nonlinear wake field for a
50-mm closed orbit offset at the collimator
c2 of the agreement increases from 0.80 to 0.83
nonlinear wake field
with 50 mm c.o. offset
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18. Frank Zimmermann, GSI Meeting 31.03.2006
generalized formula: combine correct frequency dependence
of Burov-Lebedev with complete nonlinear on transverse
coordinates from Piwinski, assuming that the two dependencies
remain factorized
generalized formula
nearly perfect
agreement
measurement
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19. Frank Zimmermann, GSI Meeting 31.03.2006
incoherent tune shift
single-particle tune nonlinearly depends on transverse
coordinates & on position along the bunch t
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20. Frank Zimmermann, GSI Meeting 31.03.2006
incoherent tune spread
b=1.0 mm
b=1.5 mm
Monte-Carlo evaluation of analytical formula
for example parameters
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21. Frank Zimmermann, GSI Meeting 31.03.2006
centroid motion from multi-particle tracking with nonlinear wake
field (2000 particles over turns)
no collimator
b=1.5 mm,
no synchr.osc.
b=1.0 mm,
no synchr.osc.
b=1.5 mm,
with synchr.osc.
b=1.0 mm,
no synchr.osc.,
1000 particles
b=1.0 mm,
with synchr.osc.
w/o synchrotron motion
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22. Frank Zimmermann, GSI Meeting 31.03.2006
FFT of centroid motion for the same 6 cases
no collimator
b=1.5 mm,
no synchr.osc.
b=1.0 mm,
no synchr.osc.
b=1.5 mm,
with synchr.osc.
b=1.0 mm,
no synchr.osc.,
1000 particles
b=1.0 mm,
with synchr.osc.
increased tune spread for small gaps
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23. Frank Zimmermann, GSI Meeting 31.03.2006
conclusions
nonlinear terms of resistive-wall wake field become important if aperture comparable to rms beam size
generalized formula combining Burov-Lebedev (dependence on w) & Piwinski (dependence on x and y) in perfect agreement with SPS measurement
for small gaps, incoherent tune spread from nonlinear wake field increases beam stability via enhanced Landau damping
More details in CERN-AB-Note
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