Particle Studio simulations of the
resistive wall impedance of copper
cylindrical and rectangular beam pipes
C. Zannini
E. Metral, G. Rumolo, B. Salvant
(CERN – BE-ABP-LIS)
Special acknowledgement:
O. Sebastia (AB desktop)
GSI/CERN collaboration meeting - Feb 19th 2009 – GSI Darmstadt
1
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
2
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
3
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
4
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
5
Wall impedance in the Headtail code?
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
6
Detuning and driving terms of the
transverse wake
Wxgeneral ( s )  Wxdriving ( s )  W detuning ( s )
simulated
y
x1  x2  0
x
calculated
y1  y2  0
simulated
x1  0, x2  0
y
y1  y2  0
x
Wygeneral (s)  Wydriving (s)  W detuning(s)
y
simulated
x1  x2  0
x
y1  y2  0
calculated
simulated
y
x1  x2  0
y1  0, y2  0
x1 , y1
transv erse position of the source bunch
x2 , y2
transv erse position of the test particle
x
7
Why do we want to separate the
dipolar and quadrupolar contribution?
Wxgeneral ( s )  Wxdriving ( s )  W detuning ( s )
The general wake has
an impact on the transverse
betatron tune shift
measured 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
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
9
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-9
β=1
10
11
Horizontal wake in a rectangular shape
0.2
driving (calc.)
detuning
general
W[V/pCm]
0.1
0.0
-0.1
-0.2
0.0
0.2
0.4
0.6
0.8
1.0
t[ns]
 In the horizontal plane, Wgeneral=0, and Wdriving=- Wdetuning
12
Vertical wake in a rectangular shape
nydet
Detuning
nygen
General
nydriv
Driving (calc.)
W[V/pCm]
0.4
0.2
0.0
-0.2
0.0
0.2
0.4
0.6
0.8
1.0
t[ns]
 In the vertical plane, Wgeneral=3*Wdetuning, and Wdriving= 2* Wdetuning
13
Summary plot for the rectangular shape: Vertical and horizontal wakes
0.5
0.4
Wx driving
driving (calc.)
Wx
(calc.)
Wx detuning
detuning
Wx
Wy driving
driving (calc.)
Wy
(calc.)
Wy detuning
general
Wy
Wx general
general
Wx
Wy general
detuning
Wy
W[V/pCm]
0.3
0.2
0.1
0.0
-0.1
-1
0
1
2
3
4
5
6
7
t[ns]
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)
14
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
15
Simulation Parameters
Geometric parameters
Thickness Copper = 0.2cm 1cm
Length = 1m 0.2m
Vacuum Chamber:
Cylindrical shape : radius=2cm
Particle Beam Parameters
σbunch = 1cm 0.8cm
0.5cm
Charge = 1e-9
β=1
16
Cylindrical shape
0.025
W[V/pCm]
0.020
nxdet
Wy drivingnxdriv
= Wxdriving
nydriv
Wy detuning
= Wxdetuning
nydet
0.015
0.010
0.005
0.000
-1
0
1
2
3
4
5
6
7
t[ns]
Detuning terms are nonexistent, as expected.
However, unphysical ripple observed for the cylindrical shape
17
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
18
Modelling a lossy metal
without the conducting wall
condition in CST 2009
The lossy metal is explicitly
modelled
around the vacuum
New boundary
condition in CST 2009
The lossy metal is only modelled
through a boundary condition
(background material has
to be changed to loss metal too)19
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.
20
Comparison of the simulated wake potential with the
theoretical wake potential of a point charge
Theory: from Palumbo, Vaccaro, Zobov, INFN, 1994
theory (cylindrical)
simulation (cylindrical)*4.44
theory (rectangular)
simulation (rectangular)*4.44
Wake[V/pCm]
2
Number of mesh ~ 106
Device length = 20 cm
b=1cm
Rms bunch length = 1 cm
Displacement =0.1*b
Boundary conditions: conducting
wall in x and y open in z
Normalization at device of 1m
0
0.0
0.4
t[ns]
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
21
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 simulations
Also the theory is not valid at high frequencies
22
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
23
Simulations with MWS 2008
form factor studies
Form factor q:
2b
hb
q
hb
2h
24
Simulation parameters
• Number of mesh
 3 *106
• 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
25
Rectangular shape with form factor q=0.5
All the results simulated are normalized by the factor  2
0.40
0.35
q=0.5
0.30
xdet
xdriv
xgen
ydet
ydriv
ygen
0.25
W[V/pCm]
0.20
0.15
2b
2h
0.10
q=0.5  h=3b
0.05
0.00
-0.05
-0.10
-0.15
-0.3
0.0
0.3
0.6
0.9
1.2
1.5
1.8
t[ns]
26
Rectangular shape with form factor q=0.33
All the results simulated are normalized by the factor  2
0.45
0.40
xdet
xdriv
ydet
ydriv
ygen
xgen
0.35
0.30
q=0.33
W[V/pCm]
0.25
0.20
2b
2h
0.15
q=0.33  h=2b
0.10
0.05
0.00
-0.05
-0.10
-0.15
-0.3
0.0
0.3
0.6
0.9
1.2
1.5
1.8
t[ns]
27
Rectangular shape with form factor q=0.1
All the results simulated are normalized by the factor  2
0.025
xdet
xdriv
xgen
ydet
ydriv
ygen
0.020
q=0.1
W[V/pCm]
0.015
0.010
2b
2h
0.005
q=0.1  h ~ 1.22 b
0.000
-0.005
-0.3
0.0
0.3
0.6
0.9
1.2
1.5
1.8
t[ns]
28
Comparison of the theoretical and simulated wake form factor
Theory: from Gluckstern, Ziejts, Zotter, Phys. Rev., 1992
29
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
30
Conclusion
• A factor 4.4 (probably  2 ) 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.
31
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)?
32
Future plans: coupling terms and non linear terms
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?
33
First results of simultaneously moving x and y location of the source beam
Test beam
1
Wake[V/pCm]
Source beam
ydriv
ygen
ydet
0.1
y
x
0.01
δx
bx
1E-3
0.0
x
bx

0.1
y
by
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
/b
the displacement is along the diagonal of the rectangular shape
and the wake is normalized to the displacement
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)
34
Thank you for your attention!
35
6
Number of mesh ~ 10
Device length = 2.5cm
Displacement =0.0333*b,h
Boundary conditions: electric in x and y open in z
Normalization at device of 1m
2b
2h
q=0.5  h=3b
36
Different boundary conditions
37
38