coherent tune shift due to collimator impedance

- its dependence on gap size, bunch

length, chromaticity,beta function, conductivity,beam energy, #bunches

-thanks to Elias Metral & Javier Resta Lopez

Questions by Francesco 27.04.2006 to Elias, Giovanni and me“Understand/clarify the scaling of the effective LHC collimator impedance as a function of the collimator gap. Does a simple scaling law exist?”→ Impact of Running LHC with non-nominal settings→ Rating of ILC IR upgrade options

Questions by Jean-Pierre 23.05.2006 to Ralph and me1) Is the Cu collimator an option for the upgrade? It was the nominal system, rejected for lack of robustness. Can it be considered again for the upgrade with much higher beam power? If yes, impedance problem disappears. 2) What is dominant and should be used for scaling: single bunch or coupled bunch? Shall I take the tune shift to be proportional to the bunch charge or to the product of the bunch charge by the number of bunches?

C

Cu

C

Cu

C=105 -1m-1

Cu=5.9x107 -1m-1

=70 mb=6 md=3 cm7 TeVB-L theory

impedance of single flat collimator

carbon

factor 23

factor 23

factor 22

factor 21

factor 22

impedance increase for 2x smaller gap

222 /2

cm

zm

zec

h

m=0

m=1

Q’=10

Q’=0

Q’=10

Gaussian weight functions

coherent coupled-bunch head-tail tune shifts

F. Sacherer, 1974A. Chao, 1993

222 /|2|

cm

zm

zec

h

psmsllm mlpMhmlpMZi

T

MNr

lQ

000010

0, !2

1

4

1

coh.tune shift – =70 m, b=67TeV

coh.tune shift – =70 m, b=67 TeV

half of the modesunstable

all modes damped

all modes unstable

“Ruggiero graph”

coh.tune shift – =70 m, 7 TeV

opening the collimators from 6 to 8reduces tune shift more than 2 times

critical modes:

largest Q or ImQ

z

(cm)

mat. b m Q’ mode#

max(Q)

mode#

max(ImQ)

7.55 C 0 0 0 3377

7.55 C 0 10 3564 3380

7.55 C 1 10 0 3369

7.55 C 0 0 0 3254

7.55 Cu 0 0 3564 3563

3.77 C 0 0 0 3374always 0 or 3564 changes by a few 100

carbonQ’=0,

m=0=70 m,

7 TeV

injection

maximum growth rate ImQ vs. gap for m=0, Q’=0

fitted curve

where b is the half gap size

nearly inversely linear dependence on gap size!

7 TeV

s

1 061.11

1

s

1 9907.2

1

1.83092

log08932.0

0.92305

log219332.0

2

2

b

e

b

e

b

Cu

b

C

~1/b!

~1/b2!

variation with conductivity

maximum growth rate ImQ vs. gap for m=0, Q’=0

fitted curve

where b is the half gap size

computed growth rates

fit result

nearly inversely quadratic dependence on gap size!

carbon

s

1 1317.9

11.76325

log05786.0 2

b

e b

injection

~1/b2!

maximum tune shift |Q| vs. gap, m=0, Q’=0, =70 m

Q’=0, m=0

carbon

copper

fitted curves

variation with conductivity

3130.2

log059376.0

,0',0

7405.2

log020997.0

,0',0

2

2

001184.0

009573.0

b

eQ

b

eQ

b

CuQm

b

CQm

~1/b2.7~1/b3

~1/b2.3~1/b2

7 TeV

maximum tune shift |Q| vs. gap for m=0, Q’=0 & 10

carbonm=0

Q’=0

Q’=10

fitted curves almost~1/b3!

variation with chromaticity

7548.2

log011834.0

10',0

7405.2

log020997.0

0',0

2

2

009025.0

009573.0

b

eQ

b

eQ

b

Qm

b

Qm

7 TeV

maximum tune shift |Q| vs. gap for m=0, Q’=0

computed tune shifts

fitted curves

carbonm=0

Q’=0

fitted curves almost~1/b3!

67203.2

log06977.0

0',0

2

002945.0

b

eQ

b

Qm

injection

maximum tune shift |Q| vs. gap for m=0 & 1, Q’=5

carbonQ’=5

m=0

m=1

fitted curves almost~1/b3!

variation with head-tail mode

8623.2

log007386.0

5',1

7448.2

log019049.0

5',0

2

2

001679.0

009428.0

b

eQ

b

eQ

b

Qm

b

Qm

maximum tune shift |Q| vs. gap, m=0, Q’=0, varying

computed tune shifts

fitted curves

carbonQ’=0, m=0

=70 m

=700 m

fitted curves almost~1/b3!

variation with beta function

7766.2

log076547.0

700,0',0

7405.2

log020997.0

70,0',0

2

2

004267.0

009573.0

b

eQ

b

eQ

b

mQm

b

mQm

tune shift decrease for 10x larger about factor2

carbon, Q’=0, m=0

maximum tune shift |Q| vs. gap, m=0, Q’=0, varying

carbonQ’=0, m=0gap=6

variation with beta function cont’d

23662.0σ6gap,0',0m

0001883.0

QmQ

computed tune shifts

fitted curves

maximum tune shift |Q| vs. gap, m=0, Q’=0, =70 m

computed tune shifts

fitted curves

Q’=0, m=0carbon

z=3.77 cm

fitted curves

z=7.55 cm

variation with bunch length

~1/b3!

8051.2

log017125.0

cm77.3,0',0

7405.2

log020997.0

cm55.7,0',0

2

2

014598.0

009573.0

b

eQ

b

eQ

b

Qm

b

Qm

z

z

maximum tune shift |Q| vs. nb, m=0, Q’=0, =70 m

fitted curves

variation with # bunches

~nb!2149

Cu,0',0

2139C,0',0

1045.91095.30000028.0

1013.41096.10000625.0

bbQm

bbQm

nnQ

nnQ

~const.

maximum growth rate vs. nb, m=0, Q’=0, =70 m

fitted curves

variation with # bunches

~nb1.2

s

11011.5

1

s

11056.4

1

19.1log01509.05

Cu,0',0

42.1log00746.06

C,0',0

2

2

bn

Qm

bn

Qm

ne

ne

b

b

~nb1.4

2452.2

log071275.0

Cubunches, 5616 cm,1.7e11,77.3,0',0

7405.2

log020997.0

Cbunches, 2808 ,cm,1.15e1155.7,0',0

2

2

003092.0

009573.0

b

eQ

b

eQ

b

Qm

b

Qm

z

z

maximum tune shift |Q| vs. gap, m=0, Q’=0, =70 m

computed tune shifts

fitted curves

Q’=0, m=0

carbon, 2808 bunches,z=7.55 cm, Nb=1.15x1011

fitted curves

variation with bunch length, #bunches,conductivity, bunch charge

~1/b2.2

copper, 5616 bunches,z=3.77 cm, Nb=1.7x1011

~1/b2.7

b

b yx

XY

y

ydYdXYXG

b

b

b xy2

2

44

22

2

4,

2erf

erf2

NLC

2

2

2

2

correction factor from nonlinear wake components

b

X

b

YX

b

X

b

YX

b

Yb

b

YY

b

X

b

YYb

b

Yb

b

X

b

Yb

b

YY

b

X

b

Y

bYXG

sinh2

cos2

sinh3cos2

cos8

2sin4

2coshsin12cos4

2cosh

2cos8

2sin2

2cosh

2cos

8,

2

3

3

2

derived from A. Piwinski’s wake field, in“Impedance of Elliptical Vacuum Chambers,”DESY 94-068, Eq. (52)

nonlinear correction vs. gap

b/

NLC

round beam

can we detect the inductive bypass effect with single bunches in the SPS?

for larger opening nonlinear correction is small, but tune shift is small too if we go very close to integer resonance, classical formula diverges

b=4m270 GeV1 bunch

conclusions

• for carbon jaw Q ~1/b2.75 , for Cu jaw ~1/b2.5

• value for Cu almost 10 times smaller• weak dependence on : Q ~1/0.25

• halving bunch length increases Q by ~50%

• LHC upgrade with half z, 1.7x1011 ppb, 5616 bunches, and Cu collimators → 1/3 tune shift of nominal LHC with C jaws

• correction from nonlinear wake field a few percent for half gaps of 6or larger