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(~