Impedance aspects on beam chamber specifications

Impedance aspects on beam chamber specifications

C. Zannini

E. Métral, G. Rumolo, B. Salvant

17/10/2019 ALERT 2019 workshop, Ioannina, Greece

Overview

• Introduction

• The impedance of vacuum chambers– The chamber aperture

– The surface impedance• Examples

– Coated ceramic chamber

– aC coating on good conductor

– Shape and asymmetries• Example of the LHC beamscreen

• Summary

2

Overview

• Introduction






• Summary

3

4

Beam environment interaction

5

Evolution of the electromagnetic fields (Es) in the cavity while and after the beam has passed


Evolution of the electromagnetic fields (Es) in the kicker while and after the beam has passed


Longitudinal Wake function Longitudinal Impedance

6



Transverse Wake

7



For structure with top/bottom and left/right symmetries

8



9

Why do we need impedance models

• Undesired effect of impedance (intensity limitations)

– Local effects• Heating and sparking

– Outgassing and/or damage

– Beam Instabilities• Driven by single elements• Machine impedance

• Impedance optimization

– Existing machines• Ensure completeness of impedance model (beam-based measurements)• Make impedance checks of any layout modification or new device installation• Identify impedance sources causing limitations and implement mitigations

– New machines• Optimize the impedance in the design phase

10

Component design flow process

New devices

(BI, RF, Vacuum,

Magnets etc.)

Functional

design

Impedance check

(All impedance

induced effects)

Detailed design

Modification needed

In design Impedance

check

Modification needed

Impedance

compliant

Impedance perspective

11

Analytical models

Simulations

Bench or beammeasurements

Estimate impedances of individual elements

Wall(IW2D, TLwall etc.)

Bench: wire method, beadpull, RF antennasBeam: tune shift, impedance localization, heating

(ABCI, ACE3P, Ansys, CST, ECHO2D/3D, GdfidL etc.)

Impedances: e.g. wall, cavities, beam diagnostic devices, kickers

and septa, steps etc.

Electromagnetic model of materials versus frequency

If possible estimate impedances in two ways

Need for correct simplificationsof the simulation models

Bench measurements are used to validate the model

12

Overview

• Introduction






• Summary

13

The impedance of vacuum chambers

• The machine impedance needs to be keptbelow a certain budget to allow operation atthe desired intensity

– The beam chamber is usually a very importantcontributor to the global machine impedance

• Special attention must be devoted to the impact ofimpedance effects on the beam chamber specifications

14


𝑍∥ Ω/𝑚 ≈𝐹∥ 𝜁 𝜔

2 𝜋 𝑏

𝑍𝑥,𝑦 Ω/𝑚2 ≈𝐹𝑥,𝑦 𝜁 𝜔 𝛽 𝑐

𝜔 𝜋 𝑏3

Longitudinal impedance

Transverse impedance

Radius or half aperture of the chamber

Form factor with respectto circular shape

Surface impedance(material properties)

15

The impedance of vacuum chambers: the chamber aperture

• Choice of the aperture

– Large aperture is aimed to have smaller impedance (especially important for the transverse impedance 𝑍𝑥,𝑦 ∝ 1/b3)

– Constraints, e.g.• Magnet gaps

– Small gaps allow for large gradients and short magnets which results in strong focusing, low emittance and a compact lattice

– In the design phase of the accelerator one should investigate that the chamber aperture is reasonable from the impedance point of view

– Estimation of impedance budget for TMCI

16

The impedance of vacuum chambers: the chamber aperture

Elias Métral, workshop "Beam Dynamics meets Vacuum, Collimation and Surfaces", Karlsruhe, Germany, 08-10/03/2017

𝐼𝑚 𝑍𝑦𝑒𝑓𝑓

𝑏, 𝜁𝐶𝑢 < 𝐼𝑚 𝑍𝑦𝑒𝑓𝑓

𝑚𝑎𝑥Is the chamber aperture reasonable?

17

Frequency range of interest

fi =(n-Q)*frev

𝑓𝑐𝑇𝐸 = 1.841

𝑐

2 𝜋 𝑏𝑓𝑐𝑇𝑀 = 2.405

𝑐

2 𝜋 𝑏

Mimimum frequency of interest

Cut-off frequency of the pipe

The chamber aperture defines at which frequency electromagnetic modes propagates

Frequency width of the spectrum ∝ 1/σ𝑡

18

The impedance of vacuum chambers:the surface impedance


2 𝜋 𝑏


𝜔 𝜋 𝑏3






𝜁 𝜔 ≈1+ j

𝜎 𝜔 𝛿 𝜔

19

• Choice of the material

– Low surface impedance

• High conductive material

• Shielded low conductive material

– Good conductors coated on bad conductors

– Good conductors coated on ceramics

• Transparent coatings (aC or NEG) on high conductive material

• Surface roughness degrades electrical conductivity

– 20% reduction in electrical conductivity at 3 GHz for 0.4 μm roughness on Cu

– Constraints

• Vacuum properties

• Mechanical and thermal properties

– Induced heating and mechanical stress from Impedance

– Keeping in mind the constraints find the best trade-off for impedance effects (impedance budget for TMCI, beam induced heating, coupled bunch instability)


20

g=10000

Pipe radius=18.4 mm

LHC beamscreen


Thick wall

Thin wall

ModelFirst layer metal with σ=1.82GS/m, thickness=0.05 mmSecond layer metal with σ=1.67MS/m, thickness=1 mm


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g=10000

Pipe radius=18.4 mm

LHC beamscreen

ModelFirst layer metal with σ=1.82GS/m, thickness=0.05 mmSecond layer metal with σ=1.67MS/m, thickness=1 mm


Thick wall

Thin wall 𝑓𝑚𝑎𝑥𝑅𝐸 ∝ 𝑅

Beam induced heatingTransverse mode Coupling InstabilityCoupled bunch Instability


22

coating

coating

tr

LR

2

Uniform Titanium coating

Resistance of the coating

The beam coupling impedancestrongly depends on the resistance of the coating

Example: Good conductors coated on ceramics

Ferrite

Ceramic

23

mtcoating 12.0

Titanium coating

The shielding (coated ceramic chamber) strongly reduces the longitudinal impedance


24

Real part of the longitudinal impedance

kHzf 500


25

Real part of the longitudinal impedance

kHzf 110

mtcoating 2

Titanium coating

Minimum R defined by the required kicker

rise time


Minimum R gives the best shielding 26

The coating shifts the impedance spectrum to lower frequencies


27

The frequency shift is proportional to the coating resistance


28

The frequency fr is proportional to the coating resistance

Real part of the transverse impedance

fr


29

Ideally R small enough to shield the full frequency range of interest. If not possible choice of R to be based on machine impedance effects criticalities (not always the minimum R!!!!)

Real part of the transverse impedance

fr


30

• Case without coating– 1 layer structure

• 1st layer (B)

• Case with coating• 2 layer structure

• 1st layer (F)

• 2st layer (B)

Bm Felr

FF

FB

s

'

0

FBm sj

From transmission line theory one can derive the surface impedance seen by the beam

The coating introduces an additional contribution to the imaginary impedance

Coating thickness

Example: coating on good conductor

31


Effect of aC coating on the LHC beamscreen

Strong effect of the coating on the imaginary part above a certain frequency

Coating consists of 500 nm of amorphous Carbon on 100 nm of Titanium

32


Strong effect of the coating on the imaginary part above a certain frequency

Coherent tune shift Effect on the TMCI thresholds

Effect of aC coating on the LHC beamscreen

Coating consists of 500 nm of amorphous Carbon on 100 nm of Titanium

33

The impedance of vacuum chambers:the chamber shape


2 𝜋 𝑏


𝜔 𝜋 𝑏3






𝐹∥ = 𝐹𝑥,𝑦 = 1

For circular pipe

34

2 b2 b

Vacuum

Conductive material

Analytical form factors of the wake functions for rectangular pipes

K. Yokoya. Resistive Wall Impedance of Beam Pipes of

General Cross Section. Number 41. Part. Acc., 1993.

2 a


35

2 b2 b

Vacuum

Conductive material

2 a


N. Mounet, E. Métral. Generalized form factors for the

beam coupling impedances in a flat chamber, IPAC10

Advanced calculations of complex chambers can be made with 3D

simulation tools36


Tune shift from vacuum chamber impedance

Sligtly higher tune shift compared

to round 𝜋2

8form factor

Zero tune shift: perfect compensation of driving and detuning impedances in the horizontal plane

37

The LHC beam screen

weld

38


𝑍∥ Ω/𝑚 ≈𝐺∥ 𝐹∥ 𝜁 𝜔

2 𝜋 𝑏

𝑍𝑥,𝑦 Ω/𝑚2 ≈𝐺𝑥,𝑦 𝐹𝑥,𝑦 𝜁 𝜔 𝛽 𝑐

𝜔 𝜋 𝑏3



Weld factor

39

The LHC beam screen: effect of the weld

40

Computed with CST Particle Studio

Overview

• Introduction

• The resistive wall impedance design– The chamber aperture





• Summary

41

Summary

• The impact on impedance effects of beam chamber parameters (aperture andsurface impedance) must be deeply investigated in the design phase.

• Due to constraints coming from costs, mechanical properties or devicefunctionality (e.g. kickers) the optimal choice can be complex.– Characterization of material properties including roughness is fundamental to

make correct assessment.

• Effect of chamber shape can be accounted using the Yokoya form factors.– Dependence on frequency of the form factors can be obtained with a more

advanced theory (N. Mounet and E. Métral).• Analytical studies are ongoing to have a generalized theory also for the

elliptical shape (N. Biancacci, M. Migliorati et al.).

• Advanced calculations including wall asymmetries (e.g. weld), complex chamber shapes and pumping holes can be performed with 3D simulation tools.

42

Thank you very much for your attention

43

Effect of the ceramic chamber on the KSW transverse impedance

Rfr 44

Silicon-Steel model for the SPSSilicon-Steel electromagnetic model

jf21

μ1μμμμ i

0r0

33

500

1021

)Extraction(

)Injection(

KHzf

i

i

r

0

0r0πfε2

jε'εεεε

1

105

'

m

At low frequency the real part ofthe longitudinal impedance is muchlarger with silicon steel boundary

45

• 2 layer structure• 1st layer (F)

• 2st layer (B)

𝜁𝑚=𝜁𝐹𝜁𝐵+𝑗𝜁𝐹 tan 𝑘𝐹𝑠𝐹

𝜁𝐹+𝑗𝜁𝐵 tan 𝑘𝐹𝑠𝐹𝜁𝑚=𝜁𝐹

𝜁𝐵+𝑗𝜁𝐹𝑘𝐹𝑠𝐹

𝜁𝐹+𝑗𝜁𝐵𝑘𝐹𝑠𝐹

𝜁𝑚=𝜁𝐹𝜁𝐹𝜁𝐵(1+𝑘𝐹

2𝑠𝐹2)+𝑗(𝜁𝐹

2−𝜁𝐵2)𝑘𝐹𝑠𝐹

𝜁𝐹2+𝜁𝐵

2𝑘𝐹2𝑠𝐹

2

𝜁𝑚=𝜁𝐵 + 𝑗𝜁𝐹𝑘𝐹𝑠𝐹

bLZ m

2||

• 1 layer structure• 1st layer (B)

𝜁𝑚=𝜁𝐵

𝛿𝐹 ≫ 𝑠𝐹

𝜁𝐵 ≪ 𝜁𝐹All hypothesis applies to the

aC coating or dielectrics (Al2O3) on metals

𝜁𝑚=𝜁𝐵 + 𝑗𝜔𝜇𝐹𝑠𝐹

Example: aC coating on good conductor

46

Verification of the formula𝛿𝐹 ≫ 𝑠𝐹

𝜁𝐵 ≪ 𝜁𝐹

𝜁𝑚=𝜁𝐵 + 𝑗𝜔𝜇𝐹𝑠𝐹F layer aC: σ=400 S/m, εr =5.4, s=0.5 μmB layer Cu: σ=1 GS/m

bLZ m

2||

47

Verification of the formula𝛿𝐹 ≫ 𝑠𝐹

𝜁𝐵 ≪ 𝜁𝐹

𝜁𝑚=𝜁𝐵 + 𝑗𝜔𝜇𝐹𝑠𝐹F layer aC: σ=400 S/m, εr =5.4, s=0.5 μmB layer Cu: σ=1 GS/m

bLZ m

2||

48

Impedance model for analytical calculation

• Case with aC coating– 5 layer structure

• 1st layer (aC)• 2st layer (Ti)• 3st layer (Cu)• 4st layer (StSt)• 5st layer (Vacuum)

• Case without aC coating– 3 layer structure

• 1st layer (Cu)• 2st layer (StSt)• 3st layer (Vacuum)

Material σel [S/m] εr Thickness [µm]

aC coating 400 5.4 0.5

Titanium coating 106 1 0.1

Copper 109 1 50

Stainless steel 1.35 106 1 1000

Vacuum 0 1 Infinity49

50


Longitudinal Impedance

Transverse Impedances

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source particle (x0, y0)

test particle (x, y)

Longitudinal Wake function

Longitudinal Impedance

000 yyxx

Nominal orbit

The EM problem

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Beam induced heating

Beam spectrum

bunch spacing

Beam profile Beam spectrum

1/(bunch spacing)

Fourier Transform

t f

Revolution frequency of particlesin the ring of circumference C

Longitudinal Impedance means heating of the components

The EM problem

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0yx AA

source particle (x0, y0)

test particle (x, y)

Transverse Wake functionTransverse Impedance

The EM problem

0yxaa

)()()( det

,,, fZfZfZ yx

driv

yx

gen

yx

The generalized term has an impact on the transverse tune shifts with beam

intensity

The driving term has an impact on the instability of the beam centroid

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eff

yxyx ZQ ,,

0 5000 10000 15000

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

Turn #

Ho

r. p

os

itio

n. [a

.u.]

Bunch 52

Driving and detuning impedances

Frequency range of interest

Elias Métral, workshop "Beam Dynamics meets Vacuum, Collimation and Surfaces", Karlsruhe, Germany, 08-10/03/2017

f

Power spectrum Bunch spectrum

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Impedance aspects on beam chamber specifications

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