beam pipe - - - - - -- chao (1993) collective instabilities in wakefield coupled bunches objective -...
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Beam pipe - - - - - --
- - - - - --
Chao (1993)
Collective Instabilities in Wakefield Coupled Bunches
Objective - OCS6 Damping Ring
- Transverse Growth Rates
Kai Hock
Liverpool Accelerator Group Meeting, Cockcroft 14 February 2007
Uniform Resistive Wall
Transverse Force
zzW
1)(1
No wakefieldthis side
Chao (1993)
yzWF )(1
Wake potential
v
Equation of motiony
...)2()2()()()()( 21110
02
tysWtysWT
cNrtyty nnnn
s = c
0)()( 2 tyty nn
No wakefield
)()( 110
0
tysWT
cNrn
Wakefield from bunch ahead
n n+1
n+2
Damping Ring
y0
y1
ym
yM-1
- yn = transverse displacement- periodic nature modes
...)2()2()()()()( 21110
02
tysWtysWT
cNrtyty nnnn
tinn eyty )0()(Trial solution
Modes
Eigenvector / Mode
1
0
2
)()(~M
m
N
mi
m etyty
yy
cccc
cccc
cccc
nn
n
)(...)()()(
......
......
)(...)()()(
)(...)()()(
0321
2001
1210Circulant Matrix
(Gray 2006)
Characteristic Equation
nin
nin
ii ebebebeba 22
)12(12
221
2 ...
e.g. 2 bunches, Mode 0
Multiple solutions:
If assume dominated by betatron oscillation …
a = 1b1 = 0.1b2 … = 0tau = 1
|Left hand side – right hand side|
Growth Rate
Im1
)(
… derive analytic expression for small wakefield
Chao (1993)
tieyty
)0(~)(~Mode
yn(0)FFT
Simulation Method
...)3()2()()()( 33221102
0 tywtywtywtyty
sincos)( 00
vyy
qvyy cossin)( 00
- Integrate over one time interval between slices- Repeat for next interval
SHM Kick
OCS6 Damping Ring
Mode Amplitudes
High frequency oscillations?
- Mode amplitude ~ exp(t/)
- Growth rate 1/ ~ initial gradient
OCS6 Growth Rate
Assume constant beta for analytic curve.
Problems
tie not complete. May also be
Non-exponential behaviour?
tinet
not general. Could be .......)0(~ )3()2()1(
321 tititi ecececy
Growth rate ?
(Wright 1948)
tiey
)0(~