1

Impedance and its link to vacuum chamber geometry

T.F. GünzelVacuum systems for synchrotron light sources

12th september 2005

2

Outline– Motivation

– Basic properties of wakefields

– Resistive wall impedance

– Impedance budget

– Comparison to measured single bunch thresholds

– Incoherent tune shifts

– Effective impedance under different aspects

– Head-Tail instability

– Conclusion

3

Motivation

• ESRF particularly concerned by impedance-related instabilities, the single bunch transverse thresholds are low <0.7mA vertically, <1.7mA horizontally

ESRF is an excellent case to study the link between vacuum chamber geometry and transverse impedance

• Verification of the impedance model by comparison of the calculated TMCI-thresholds to the measured ones

• Establishment of the impedance budget, identification of the elements with the largest detrimental effect

4

Basic properties of wakefields

),,(),,( tsctzEdtq

csW zlong

rrr 0

long. wakefield excited by a particle at r0 and observed by a test (witness) particle at r

)),,(),,((),,( tsctzctsctzdt

q

cs z rBerErrW 0

r0

origin

test particle

r0

transverse wakefield (2-dimensional vector field) : ),( s0rr,W

5

wakefield in a circular geometry with tapers

axial-symmetric taper geometry in the (z,r)-planesource particle at offset r0 to the origin

r0

origin

source particle

0rW )(),,( 0 sFsrr

all arrows point in the direction of the offset

no dependence of the test particle positionr0

if zero offset, the wake is also zero (everywhere!)

this dipolar field contributes to the impedance

6

Without offset W only depends on the test particle position, it is locally quadrupolar

However, this field does not contribute to the impedance

wakefield in a rectangular chamber with vertical tapers and offset r0=0

magnification

7

wakefield in a rectangular chamber with a vertical tapers and horizontal offset r0 0

the wakefield is a sum of a quadrupolar field (depending on the test particle position) and a dipole field (depending on the offset of the exciting particle)

source particle with a 1mm offset in x-direction

r0

a vertical taper produces horizontal impedance !

magnification

8

The role of the vacuum chamber cross section

• 2D calculation possible• wakefield only depends on

source particle position• resistive wall impedance given

by

• 3D calculation necessary

• wakefield depends on source as well as on test particle position

• resistive wall impedance (chamber is approximated by 2 parallel plates)

a

3

1~

aZZ HV

3

1~2

aZZ HV

a

b

all quantities only depend on one geometric parameter: the vertical extension of the chamber a

9

Resistive wall budget

form-factors as well -functions of vertical and horizontal plane taken into account

standard ESRF vacuum chamber (33mm x 79mm)makes up two third of the vertical budget

the horizontal RW-impedance budget even larger than the vertical one

chamber type V[m] H[m] ( ZVV)eff [MW] (ZHH)eff [MW]

all low-gap sections (invacuum open) 3.5 24 0.73 2.65ESRF standard beampipe + other elements 24.6 16.6 1.78 0.78

2.51 3.43

Vhorizontal values are high in the straight sections,vertical values are high in the dipoles.

10

Geometrical Impedance budget

33 (39) different elements in the budget, 1775 elements in total (status of november 2004)

• 26 taper pairs (without in-vacuum)

• 8 invacuum undulators (all open)

• 293 RF-fingers

• 569 flanges

• 134 vertical pumps, 448 horizontal pumps

• 6 cavities and 3 cavity tapers

• 2 scrapers in operation position

• 277 BPM’s

• 7 kicker like chambers

• 1 septum

Calculation of impedance with GdfidL (W.Bruns)

11

Broadband Impedance budget (vertical and horizontal plane)

12

Effective dipolar impedance budget (geometric and resistive wall)

Vertical impedance distributed smoothly around the ring

Horizontal impedance concentrated in the low-gap sections

13

• Coherent tune shift : dipolar impedance

Thresholds and tune shifts in single bunch

the thresholds only depend on the dipolar impedance• vertical measured 0.65mA calculated 1.05mA

• horizontal measured 1.7mA (nov. 2004) calculated 1.10mA

probably the horizontal (resistive wall) impedance is overestimated

for tune shift : dipolar impedance + quadrupolar wakefield

Incoherent tune shift deteriorates the operation in single bunch

• Incoherent tune shift : quadrupolar wakefield

14

Coherent and incoherent tune shifts in single bunch

Synonym with calculation of kick factors (effective impedance)of dipolar and quadrupolar wakefields

This is what can be measured by the

bump method

• Vertical incoherent and coherent kick factors add up positively

• Horizontal incoherent and coherent kick factors cancel out each other

15

Incoherent tune shift in multi-bunch

-0.02

-0.015

-0.01

-0.005

0

0.005

0.01

0.015

0.02

0.025

0.03

0 50 100 150 200

QH_close

QV_close

QH_open

QV_open

Explained by R. Nagoaka, PAC 2001 (Chigaco) 3531

horizontal

verticalMeasured in november 2004

160.00 5.00 10.00 15.00 20.00 25.00 30.00 35.00

CV2000x19m mCV5000x20m mCV5000x19m m

APS-cham ber 12m m ins ideCV5000 SS 11m m gap

m inigap openm inigap closed 10m m

CV5000 Al 8m m gapWiggler cham berCV5000 SS 8m m

1.6m in-vacuum open1.6m in-vacuum closed 6m m2m in-vacuum s tandard open

2m in-vacuum s tandard closed 6m m

6 ESRF cavities3 cavity taper pairs

conical scrapertapered scraper

bellow-fingertaperfingerCV4-fingerCV7-fingerCV9-fingerCV2-finger

CV16-finger

long vertical pum pshort vertical pum p with absorber

short vertical pum p without absorberhorizontal pum p

kicker cham ber with absorbergeom etry around the septum

flange perim eter 150m m , s lit 0.15m mflange perim eter 150m m , s lit 0.1m m

BPM 33m m gapBPM 16m m gapBPM 8m m gap

piece in front of in-vacuum prototype

dipolar cham ber with absorber

Zeff(kOhm/m) dipolar+quad.

0.00 0.50 1.00 1.50 2.00 2.50 3.00

BPM 33mm gap

BPM 16mm gap

BPM 8mm gap

17

Effective impedance map (dipolar + quadrupolar) (σ=40ps)

Calculated values about 2/3 of the measured values apart from invacuum undulators + 10mm SS-chambers (larger deviations)

effective Impedance (kOhm/m) dipolar+quad.

0

20

40

60

80

100

120

140

160

180Calculation

Measurement Measurements from Th. Perron

PhD thesis

18

Head-Tail Instability (only general remarks)impedance not only has bad effects

• Head-Tail instability is driven by transverse impedance.

• P. Kernel (PhD thesis) showed that at the ESRF the vertical

head-tail instability is damped by the incoherent synchrotron

tune spread caused by longitudinal impedance.

• On the horizontal plane this finding has still to be checked

• The limit of stability is given by the Post-Head-Tail instability(P. Kernel, R. Nagaoka, JL Revol, EPAC 2000, Vienne)

19

Conclusions

• The vertical impedance is determined by many different elements

• The horizontal impedance is mainly determined by the low gap chambers.

• The model explains 2/3 of the vertical mode detuning,• but in the horizontal plane it predicts a too small threshold compared

to the measured one

• Flat vacuum chambers give rise to an incoherent tune shift tune shift affects the beam in multibunch as well as in single bunch

• The Horizontal as well as vertical impedance are essentially created by the vertical walls : its geometrical variation and its

resistivity

20

Conclusions

High impedance budget of the ESRF is mainly due to

• modularity of the vacuum system

• alternating vertical and horizontal β-function distribution

• stainless steel vacuum chambers (is evolving towards more aluminium chambers)

• flatness of the vacuum chambers

21

Acknowledgements

– discussions and explanations of R. Nagaoka

– good user support from W. Bruns (GdfidL)

– all participants to this work, in particular

P Elleaume, L. Farvacque, T.Perron, JL Revol