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Particle Studio simulations of the resistive wall impedance of copper

cylindrical and rectangular beam pipes

C. ZanniniE. Metral, G. Rumolo, B. Salvant

(CERN – BE-ABP-LIS)

GSI/CERN collaboration meeting - Feb 19th 2009 – GSI Darmstadt

Special acknowledgement:O. Sebastia (AB desktop)

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Overview• Context and Objectives

• Definition of the detuning, driving and general wake

• First simulations– Rectangular shape– Cylindrical shape

• New boundary condition in CST 2009

• Form factor studies

• Conclusions

• Open questions

• Future Plans

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Context• High intensity in the CERN complex for nominal LHC operation, and foreseen LHC

upgrade

• Need for a good knowledge of the machines beam impedance and their main contributors

• To obtain the total machine impedance, one can:– Measure the quadrupolar oscillation frequency shift (longitudinal) or the tune shift

(transverse) with the SPS beam

– obtain the impedance of each equipment separately and sum their contributions:• Analytical calculation (Burov/Lebedev, Zotter/Metral or Tsutsui formulae) for simple geometries• Simulations for more complicated geometries• RF Measurements on the equipment

available impedance and wake data compiled in the impedance database ZBASE

In this talk, we focus on the benchmark of theory and time domain simulations of the wakes of simple structures with finite conductivity

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Objectives

• Separation of the dipolar and quadrupolar terms of the rectangular shape with Particle Studio simulations, and comparison with theory.

• Simulation of the wake form factor in a rectangular shape

• Analysis of the nonlinear term in the wake of the rectangular shape

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Broader objectives for the “impedance team”:1) Which code should we trust to obtain the wakes for Headtail? (Headtail needs the dipolar and quadrupolar terms disentangled)3) Should we include coupled or higher order terms of the Resistive Wall impedance in the Headtail code?

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Overview• Context and Objectives

• Definition of the detuning, driving and general wake

• First simulations– Rectangular shape– Cylindrical shape

• New boundary condition in CST 2009

• Form factor studies

• Conclusions

• Open questions

• Future Plans

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)()()( sWsWsW detuningdrivingx

generalx

)()()( sWsWsW detuningdrivingy

generaly

simulated

00

21

21

yyxx

00

21

21

yyxx

00 ,0

21

21

yyxx

0 ,00

21

21

yy

xx

particle test theofposition erse transv,bunch source theofposition erse transv,

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yxyx

Detuning and driving terms of the transverse wake

x

y

x

y

x

y

x

y calculated simulated

simulated calculated simulated

)()()( sWsWsW detuningdrivingx

generalx

Why do we want to separate the dipolar and quadrupolar contribution?

The general wake has an impact on the transverse betatron tune shiftmeasured in the machine

The driving wake has an impact on the transverse instability threshold

Therefore, in machines with flat chambers:- no negative horizontal tune shift (or even positive one)- but existence of a horizontal instability threshold

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Overview• Context and Objectives

• Definition of the detuning, driving and general wake

• First simulations– Rectangular shape– Cylindrical shape

• New boundary condition in CST 2009

• Form factor studies

• Conclusions

• Open questions

• Future Plans

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Simulation Parameters

Geometric parameters Thickness Copper = 0.2cm Length = 1m Vacuum Chamber:Rectangular shape : height=2cm; width= 6cm

Particle Beam Parameters

σbunch = 1cm Charge = 1e-9β=1

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-0.2 0.0 0.2 0.4 0.6 0.8 1.0

-0.1

0.0

0.1

0.2

W[V/pCm]

t[ns]

driving (calc.) detuning general

Horizontal wake in a rectangular shape

In the horizontal plane, Wgeneral=0, and Wdriving=- Wdetuning

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-0.2 0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.2

0.4

W[V/pCm]

t[ns]

nydet nygen nydriv

Vertical wake in a rectangular shape

DetuningGeneralDriving (calc.)

In the vertical plane, Wgeneral=3*Wdetuning, and Wdriving= 2* Wdetuning

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-1 0 1 2 3 4 5 6 7

-0.1

0.0

0.1

0.2

0.3

0.4

0.5

W[V/pCm]

t[ns]

Wx driving (calc.) Wx detuning Wy driving (calc.) Wy detuning Wx general Wy general

Finally, Wy detuning=Wx driving, and all relative values of these wakes are consistent with the theory Yokoya (Part. Acc. 1993) and Gluckstern, Zotter, Zeijts (Phys Rev 1992)

Summary plot for the rectangular shape: Vertical and horizontal wakes

Wx driving (calc.)Wx detuningWy driving (calc.)Wy generalWx generalWy detuning

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Overview• Context and Objectives

• Definition of the detuning, driving and general wake

• First simulations– Rectangular shape– Cylindrical shape

• New boundary condition in CST 2009

• Form factor studies

• Conclusions

• Open questions

• Future Plans

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Simulation Parameters

Geometric parameters Thickness Copper = 0.2cm 1cm Length = 1m 0.2mVacuum Chamber:Cylindrical shape : radius=2cm

Particle Beam Parameters

σbunch = 1cm 0.8cm 0.5cmCharge = 1e-9β=1

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Cylindrical shape

-1 0 1 2 3 4 5 6 7

0.000

0.005

0.010

0.015

0.020

0.025

W[V/pCm]

t[ns]

nxdet nxdriv nydriv nydet

Detuning terms are nonexistent, as expected.However, unphysical ripple observed for the cylindrical shape

Wy driving = Wxdriving

Wy detuning = Wxdetuning

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Overview• Context and Objectives

• Definition of the detuning, driving and general wake

• First simulations– Rectangular shape– Cylindrical shape

• New boundary condition in CST 2009 and comparison with theory

• Form factor studies

• Conclusions

• Open questions

• Future Plans

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New boundary condition in CST 2009

Modelling a lossy metalwithout the conducting wall

condition in CST 2009

The lossy metal is explicitly modelled

around the vacuum

The lossy metal is only modelled through a boundary condition

(background material has to be changed to loss metal too)

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Boundary condition conducting wall

The conducting wall boundary condition allows to simulate easily also the cylindrical shape.To simulate explicitly the cylindrical copper layer without ripple, an unmanageable number of mesh cells has to be used.

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0.0 0.4

0

2

Wake[V/pCm]

t[ns]

theory (cylindrical) simulation (cylindrical)*4.44 theory (rectangular) simulation (rectangular)*4.44

Number of mesh ~ 106

Device length = 20 cmb=1cmRms bunch length = 1 cmDisplacement =0.1*bBoundary conditions: conducting wall in x and y open in zNormalization at device of 1m

Comparison of the simulated wake potential with the theoretical wake potential of a point charge

Theory: from Palumbo, Vaccaro, Zobov, INFN, 1994

But we are comparing the simulated wake of a gaussian bunch with the theoretical wake of a point charge. We need to convolute the theoretical wake with the source bunch

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Comparison of the simulated wake potential with the theoretical wake potential of a Gaussian bunch

Theoretical and simulated wake potential are very similar Short range wakes are subject to more noise in simulationsAlso the theory is not valid at high frequencies

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Overview• Context and Objectives

• Definition of the detuning, driving and general wake

• First simulations– Rectangular shape– Cylindrical shape

• New boundary condition in CST 2009

• Form factor studies

• Conclusions

• Open questions

• Future Plans

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Simulations with MWS 2008

form factor studies

2 b

2 h

bhbhq

Form factor q:

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Simulation parameters• Number of mesh

• Device length = 20cm

• b=1cm

• Displacement = 0.1*b,h

• Boundary conditions: electric in x and y open in z

• Normalization at device of 1m

• All wakes (including the driving term) are now simulated

610*3

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-0.3 0.0 0.3 0.6 0.9 1.2 1.5 1.8-0.15

-0.10

-0.05

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

q=0.5

W[V

/pC

m]

t[ns]

xdet xdriv xgen ydet ydriv ygen

Rectangular shape with form factor q=0.5

2 b

2 h

q=0.5 h=3b

All the results simulated are normalized by the factor 2

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-0.3 0.0 0.3 0.6 0.9 1.2 1.5 1.8-0.15

-0.10

-0.05

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

0.45

q=0.33

W[V/pCm]

t[ns]

xdet xdriv ydet ydriv ygen xgen

2 b

2 h

q=0.33 h=2b

Rectangular shape with form factor q=0.33 All the results simulated are normalized by the factor 2

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-0.3 0.0 0.3 0.6 0.9 1.2 1.5 1.8

-0.005

0.000

0.005

0.010

0.015

0.020

0.025

q=0.1

W[V

/pC

m]

t[ns]

xdet xdriv xgen ydet ydriv ygen

2 b

2 h

q=0.1 h ~ 1.22 b

Rectangular shape with form factor q=0.1 All the results simulated are normalized by the factor 2

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Comparison of the theoretical and simulated wake form factorTheory: from Gluckstern, Ziejts, Zotter, Phys. Rev., 1992

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Overview• Context and Objectives

• Definition of the detuning, driving and general wake

• First simulations– Rectangular shape– Cylindrical shape

• New boundary condition in CST 2009

• Form factor studies

• Conclusions

• Open questions

• Future Plans

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Conclusion

• A factor 4.4 (probably ) is observed between the amplitude of simulated wakes and theoretical wakes.

• This amplitude factor aside, we have separated the dipolar and quadrupolar terms in the rectangular shape, and they agree with the theory.

• The simulated wakes obtained for several rectangular shape form factors also agree with the theoretical curve.

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Open questions• Factor 4.4 between theory and simulations

most likely a difference of convention.

• Issues with cylindrical shape

• Particle Studio outputs the wake potential (gaussian bunch source), but Headtail expects the wake function (point charge source).

should we simulate short bunches for high frequency applications (e.g. multi bunch effects), and long bunches for

low frequency applications (single bunch effects)?

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In Headtail, the wake is assumed to have linear uncoupled dependance on the source particle and the test particle.

This linear approximation should be valid for small particle amplitudes.

If the amplitude grows, do we have to include higher order terms? At what displacement?

Besides, are there coupled terms between planes?

Future plans: coupling terms and non linear terms

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0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.91E-3

0.01

0.1

1

Wake[V/pCm]

/b

ydriv ygen ydet

bx

δx

Test beam

Source beam

xy

y

y

x

x

bb

the displacement is along the diagonal of the rectangular shapeand the wake is normalized to the displacement

First results of simultaneously moving x and y location of the source beam

These first results are difficult to explain without involving non linear higher order dependance of the wake on the transverse location.

The threshold for the onset of a nonlinear dependance seems very low (~0.1 b)

Thank you for your attention!

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Number of mesh ~ Device length = 2.5cmDisplacement =0.0333*b,hBoundary conditions: electric in x and y open in zNormalization at device of 1m

610

2 b

2 h

q=0.5 h=3b

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Different boundary conditions

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