TRANSVERSE RESISTIVE-WALL
IMPEDANCE FROM ZOTTER2005’S THEORY
Elias Métral (25 + 5 min, 25 slides)


Introduction and motivation
“Low-frequency” regime
High-frequency regime
Numerical applications for a
 LHC collimator (vs. Burov-Lebedev2002 and Bane1991)
 SPS MKE kicker (vs. Burov-Lebedev and 2-wire measurements)

Review of Zotter’s theory ( For a circular beam pipe)
 Any number of layers
 Any beam velocity
 Any frequency  Unification of 3 regimes (BL, “thick-wall” and Bane)
 Any σ (conductivity), ε (permittivity) and μ (permeability)

Conclusion and work in progress
Elias Métral, CERN-GSI bi-lateral working meeting on Collective Effects – Coordination of Theory and Experiments, GSI, 30-31/03/06
1/25
INTRODUCTION AND MOTIVATION (1/2)
THE MOTIVATION: THE LHC GRAPHITE COLLIMATORS
f 1  8 kHz

First unstable betatron line

Skin depth for graphite (ρ = 10 μΩm)

Collimator thickness dth  2.5 cm

  f  
  8 kHz   1.8 cm

 d th
  f
 One could think that the classical “thick- Z thick wall  f

wall” formula would be about right

Elias Métral, CERN-GSI bi-lateral working meeting on Collective Effects – Coordination of Theory and Experiments, GSI, 30-31/03/06
1
b3
2/25
f
INTRODUCTION AND MOTIVATION (2/2)

In fact it is not  The resistive impedance is ~ 2 orders of
magnitude lower at ~ 8 kHz !
 A new physical regime was revealed by the LHC collimators
Usual regime : d th ,   b
dth
New regime : dth  b ,   d th
beam
beam
dth
b
 beff  b
b
 beff  b when δ  d th
Induced
Induced
current
current
Elias Métral, CERN-GSI bi-lateral working meeting on Collective Effects – Coordination of Theory and Experiments, GSI, 30-31/03/06
3/25
N.A. FOR A LHC COLLIMATOR (1/4)
Zy
1. 10
m
COMPARISON ZOTTER2005-BUROV&LEBEDEV2002
1 meter long round LHC collimator
10
b  2 mm
dC  
Classical thick-wall
1.
10
8
C  10 μm
Im
1.
106
10000
f 1  8 kHz
Re
100
d Cu  5 μm
BL’s results (real and imag. parts) in black: dots
without and lines with copper coating
100
10000
1.
10
6
1. 10
8
Cu  17 nm
1. 10
Elias Métral, CERN-GSI bi-lateral working meeting on Collective Effects – Coordination of Theory and Experiments, GSI, 30-31/03/06
10
4/25
f Hz
N.A. FOR A LHC COLLIMATOR (2/4)
GLOBAL PLOT FROM ZOTTER2005
Zy
m
1. 10
8
1. 10
6
1 meter long round LHC collimator
Low
beam velocity case
(e.g. PSB :   1.05 ,   0.3)
Same as
Bane1991
10000
Negative
100
 AC   DC /  1  j  
  0.8 ps
AC conductivity
1000
1. 10

6
1.
10
9
1.
f Hz
12
10
Elias Métral, CERN-GSI bi-lateral working meeting on Collective Effects – Coordination of Theory and Experiments, GSI, 30-31/03/06
5/25
N.A. FOR A LHC COLLIMATOR (4/4)
Ratio
2
1.75
Re Zy, Zotter
Re Zy, BL
1.5
Im Z y, Zotter
Im Z y,BL
 
Im Z y
1.25
@ fc 
1
0
Z
0

0.75
If  2  f c 
0.5
0.25
1
1000
1.
106
1.
109
1. 10 12
Zy M
0.4
Re Zy, Zotter
Re Zy, Bane
1.5
Im Zy, Zotter
Im Z y, Bane

2

  DC2
b



1/ 4
 0.94 THz
 2  1
f Hz
Ratio
2
1.75
c
1/ 4
3
m
0.2
1.25
0
1
0.75
0.5
0.2
0.25
1
1000
1.
106
1.
109
1. 10 12
f Hz
f THz
0.25
0.5
0.75
1
1.25
1.5
1.75
Elias Métral, CERN-GSI bi-lateral working meeting on Collective Effects – Coordination of Theory and Experiments, GSI, 30-31/03/06
2
6/25
N.A. FOR A SPS MKE KICKER
Zy M
0.8
m
Zotter
Burov Lebedev
0.6
2 wire meas.
0.4
0.2
f GHz
0.5
1
1.5
2
2.5
3
3.5
4
0.2
Elias Métral, CERN-GSI bi-lateral working meeting on Collective Effects – Coordination of Theory and Experiments, GSI, 30-31/03/06
7/25
REVIEW OF ZOTTER’S THEORY (1/17)

1) Maxwell equations
 In the frequency domain, all the field quantities are taken to be
proportional to e j  t
 Combining the conduction and displacement current terms yields



curl H     j   c E
with


B H


curl E   j   H
  0    0  r  1  j tanM

div H  0
 
div E 


 c 0   0  r 
j 2 f
 0 r
Elias Métral, CERN-GSI bi-lateral working meeting on Collective Effects – Coordination of Theory and Experiments, GSI, 30-31/03/06
8/25
REVIEW OF ZOTTER’S THEORY (2/17)

2) Scalar Helmholtz equations for the longitudinal field components
Using curl curl  grad div   , one obtains (using the circular
cylindrical coordinates r, θ, z)
1   
 r

r r r

 1 2
 1   r   r   
2
2
  2

   c  H z  


2
2
z
r 
 r 
 r 

1   
 r

r r r

 1 2
2
1 
2
  2
 2     c  Ez 
 j    z
2
z
 z
 r 

 The homogeneous equation can be solved by separation of
variables
H z or Ez      Z  z  R  r 
Elias Métral, CERN-GSI bi-lateral working meeting on Collective Effects – Coordination of Theory and Experiments, GSI, 30-31/03/06
9/25
REVIEW OF ZOTTER’S THEORY (3/17)

     e  j m
m is called the azimuthal mode number
(m=1 for pure dipole oscillations)
and Z  z   e  j k z
k is called the wave number
Reinserting the time dependence ( e j  t ), the axial motion is seen to
be a wave proportional to e j   t  k z  , with phase velocity    c   / k
which may in general differ from the beam velocity b   b c
1 d  d R   m2
 r
   2   2
R (r) is given by
r d r d r   r

 R  0 with   k 1   2    

Radial propagation constant
The solutions of this differential equation are the modified Bessel
functions I  r  and K  r 
m
m
Elias Métral, CERN-GSI bi-lateral working meeting on Collective Effects – Coordination of Theory and Experiments, GSI, 30-31/03/06
10/25
REVIEW OF ZOTTER’S THEORY (4/17)
 Conclusion for the homogeneous scalar Helmholtz equations
• For pure dipole oscillations excited by a horizontal cosine
modulation propagating along the particle beam, one can
write the solutions for Hz and Ez as
H z  sin  e j   t  k z   C1 I1  r   C2 K1   r

Ez  cos  e j   t  k z   D1 I1  r   D2 K1  r  
C1,2 and D1,2 are
constants to be
determined
• Sine and cosine are interchanged for a purely vertical
excitation (see source fields)
• Only the solutions of the homogeneous Helmholtz
equations are needed since all the regions considered are
source free except the one containing the beam where the
source terms have been determined separately by
Gluckstern (CERN yellow report 2000-011)  See slide 14
Elias Métral, CERN-GSI bi-lateral working meeting on Collective Effects – Coordination of Theory and Experiments, GSI, 30-31/03/06
11/25
REVIEW OF ZOTTER’S THEORY (5/17)

3) Transverse field components deduced from the longitudinal ones
using Maxwell equations (in a source-free region)


G  Z0 H
Ez  Ez 0 cos 
Er  Er 0 cos 
G  G 0 cos 
Gz  Gz 0 sin 
Gr  Gr 0 sin 
E  E 0 sin 
Gz 0 d E z 0
jk 
Er 0  2    

 
r
dr



d Gz 0
j k  Ez 0
E 0   2 
  
  r
dr



E z 0 d Gz 0
jk 
Gr 0  2    

 
r
dr



d Ez 0
j k  Gz 0
G 0  2 
  
  r
dr



Elias Métral, CERN-GSI bi-lateral working meeting on Collective Effects – Coordination of Theory and Experiments, GSI, 30-31/03/06
12/25
REVIEW OF ZOTTER’S THEORY (6/17)

4) Source of the fields: Ring-beam distribution  Infinitesimally short,
annular beam of charge Q  Nb e and radius a traveling with
velocity    c along the z axis
 Charge density in the frequency domain
P
 jk z


  r ,  , z;   

r

a
cos

e
 a2 
where P  Q f rev  x is the horizontal dipole moment
Elias Métral, CERN-GSI bi-lateral working meeting on Collective Effects – Coordination of Theory and Experiments, GSI, 30-31/03/06
13/25
REVIEW OF ZOTTER’S THEORY (7/17)
 Longitudinal source terms (from Gluckstern)  Valid for a  r  b , i.e.
in the vacuum between the beam and the pipe = region (1)
(2) Layer 1
Ez( s )  r ,  , z   j C cos  F1  u 
(1) Vacuum
Gz( s )  r ,  , z   j C sin   TE I1  u 
Beam
with
a
P
 jk z


C
I
s
e
1
 a 0  2  2
F1  u   K1  u   TM I1  u 
b
u 
d
 TE
kr

s 
ka


1
1  2
and  TM will be determined by the boundary conditions at b and d
Elias Métral, CERN-GSI bi-lateral working meeting on Collective Effects – Coordination of Theory and Experiments, GSI, 30-31/03/06
14/25
REVIEW OF ZOTTER’S THEORY (8/17)

5) The total (i.e. resistive-wall + space charge) horizontal impedance
Zx 
j
f 
P
j

P


 

dz E x  b B y e j k z




Zx 
 (s) 

dz  E  a,  , z
2




(s) 
  b Br  a,  , z
2


  jk z
e

j L Z 0 I1  s 
 K1  s    TM I1  s  
f  
2
2
a 
with L the length of the resistive pipe and Z 0  120 
Elias Métral, CERN-GSI bi-lateral working meeting on Collective Effects – Coordination of Theory and Experiments, GSI, 30-31/03/06
15/25
REVIEW OF ZOTTER’S THEORY (9/17)
 The space-charge impedance is obtained with a perfect conductor at
r = b, i.e. when   1 and   0 , with
1
1
1   TM
 Z
SC
x

I1  x1 
K1  x1 
x1 
kb

j L Z 0 I12  s   K1  s  K1  x1  
f  


2
2
 a    I1  s  I1  x1  
• If x  1 

1   TE
I1  x1 
K1  x1 
Z
SC
x

x
I1  x  
2
and
1
K1  x  
x
j L Z0  1
1 
f  
 2
2  2
2    a
b 
Elias Métral, CERN-GSI bi-lateral working meeting on Collective Effects – Coordination of Theory and Experiments, GSI, 30-31/03/06
16/25
REVIEW OF ZOTTER’S THEORY (10/17)
 The resistive-wall impedance is obtained by subtracting the spacecharge impedance from the total impedance

Z xRW 
j L Z 0 I12  s  K1  x1 
 1  1 
f  
2
2
 a   I1  x1 
Only 1 remains to be
determined (by field matching)
Elias Métral, CERN-GSI bi-lateral working meeting on Collective Effects – Coordination of Theory and Experiments, GSI, 30-31/03/06
17/25
REVIEW OF ZOTTER’S THEORY (11/17)

6) Field matching
 At the interfaces of 2 layers (r = constant) all field strength
components have to be matched, i.e. in the absence of surface
charges and currents the tangential field strengths E
and
z ,
have to be continuous
H
z ,
 Then the radial components of the displacement Dr and of the
induction B are also continuous, i.e. matching of the radial
r
components is redundant
 At a Perfect Conductor (PC) : E z  E  0  d G / d r  0
z
 At a Perfect Magnet (PM) : Gz  G  0  d E / d r  0
z
 At r  Infinity  Only
K1  x  is permitted as I1  x 
diverges
Elias Métral, CERN-GSI bi-lateral working meeting on Collective Effects – Coordination of Theory and Experiments, GSI, 30-31/03/06
18/25
REVIEW OF ZOTTER’S THEORY (12/17)
 General form of the field strengths in region (p)
 K u 
I1  u p 
1
p
( p)

Ez 0  u p   E p
p
 K x 
I1  x p 
 1 p




 K u 
I1  u p 
1
p
( p)

Gz 0  u p   G p
p
 K x 
I1  x p 
 1 p




up  p r
( p)
( p)



E
u
d
G
j
k
z
0
p
z0

E( 0p )  u p   
   p
 p 
up
d up
( p)
( p)



G
u
d
E
j
k
p
z0
 z0
G( 0p )  u p  
   p
 p 
up
d up
(Ep, Gp, αp and ηp)
are constants to
be determined








x p   p bp
b1  b
b2  d
Elias Métral, CERN-GSI bi-lateral working meeting on Collective Effects – Coordination of Theory and Experiments, GSI, 30-31/03/06
…
19/25

Z
REVIEW OF ZOTTER’S THEORY (13/17)
7) General 1-layer formula
RW
x
j L Z 0  I12  s  K1  x1  x12 x22    P1  Q1     P1  k   Q


 a 2  2 I1  x1     x2  k x1 2    x1 x2
a  Beam radius

b  Inner radius of the pipe
L  Length of the object
  0    0  r  1  j tanM
c 0   0  r 
x1 
kb

s
ka


j 2 f
k
2 f
c
  k 1   2     x2   b
1

P1 
 f 
2    P1  k   Q    P1  k   Q  
Q 
Q2   2 P2
1  2
Q 
I1  x1 
K  x 
I  x
Q1  1 1 P2  1 2
I1  x1 
I1  x 2
K1  x1 
Q2 
K1  x2
K1  x 2




1
Q2  2 P2
1  2


Z 0  120 
1  2
α2 and η2 are determined by the
boundary conditions at the outer
chamber wall r = d
Elias Métral, CERN-GSI bi-lateral working meeting on Collective Effects – Coordination of Theory and Experiments, GSI, 30-31/03/06
20/25
REVIEW OF ZOTTER’S THEORY (14/17)
d  ,
or PC or PM
(2) Layer 1
 d
 2INF  2INF  0
 Perfect Conductor (PC)
y  d
(1) Vacuum



K1  y  I1  x2 
PC
2 
I1  y  K1  x2 
Beam
PC
2
a
b
K1  y  I 1  x 2

I1  y  K1  x2
 Perfect Magnet (PM)
 2PM 2PC 2PM   2PC
d
Elias Métral, CERN-GSI bi-lateral working meeting on Collective Effects – Coordination of Theory and Experiments, GSI, 30-31/03/06
21/25

REVIEW OF ZOTTER’S THEORY (15/17)
8) General 2-layer formula
Z
RW
x
j L Z 0 I12  s  K1  x1 

E2   2  1 
2
2
a 
I1  x1 
where the parameters (E2, α2) are 2 parameters out of 4 (α2, η2, E2 and G2)
which have to be found by solving the matching equations at each layer
boundary. The 4 unknowns are given by the following system of 4 linear
equations
  2 x 2 E 2  1   2     2 x1 x 2  G 2  1   2  P1  k x1 E 2  1   2   k x1 x 2   2 G 2  Q2   2 P2 
  2 x 2 G 2  1   2     2 x1 x 2   Q1  P1  P1 E 2  1   2    k x1 G 2  1   2   k x1 x 2   2 E 2  Q2   2 P2
 3 x 4 E 2  K 32   2 I 32    3 x 3 x 4   2 G 2  Q32   2 P32    2 x 3 E 2  K 32   2 I 32    2 x 3 x 4   3 G 2  K 32   2 I 32

 3 x 4 G 2  K 32   2 I 32    3 x 3 x 4   2 E 2  Q32   2 P32    2 x 3 G 2  K 32   2 I 32    2 x 3 x 4   3 E 2  K 32   2 I 32

Elias Métral, CERN-GSI bi-lateral working meeting on Collective Effects – Coordination of Theory and Experiments, GSI, 30-31/03/06

Q4   3 P4
1  3
Q4   3 P4
1 3
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REVIEW OF ZOTTER’S THEORY (16/17)
P1, 2 
Q4 
I1  x1, 2
I1  x1, 2


Q1, 2 
K1  x4
K1  x 4


P4 
K1  x1, 2
K1  x1, 2
I1  x4
I 1  x4




K 32 
x1, 2   1, 2 b
 1, 2,3  k 1   1, 2,3 1, 2,3
2
K1  x3
K1  x2


I 32 
x3   2 d
I1  x3
I1  x2


Q32 
K1  x3
K1  x 2


P32 
I1  x3
I1  x2
x4   3 d
1,2,3 refer to the vacuum (between
the beam and the first layer), the first
and second layer respectively
Elias Métral, CERN-GSI bi-lateral working meeting on Collective Effects – Coordination of Theory and Experiments, GSI, 30-31/03/06
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

REVIEW OF ZOTTER’S THEORY (17/17)
(3) Layer 2
e  ,
or PC or PM
(2) Layer 1
(1) Vacuum
 e
 3INF  3INF  0
Beam
a
b
d
e
Elias Métral, CERN-GSI bi-lateral working meeting on Collective Effects – Coordination of Theory and Experiments, GSI, 30-31/03/06
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CONCLUSION AND WORK IN PROGRESS

Zotter2005’s formula has been compared to other approaches from
Burov-Lebedev2002, Tsutsui2003 (theory and HFSS simulations) and
Vos2003 (see Ref. 6 below)  Similar results obtained in the
new (Burov-Lebedev2002) low-frequency regime

Work in Progress
 Multi-bunch or “long” bunch  Wave velocity  Beam velocity
 Finite length of the resistive beam pipe
REFERENCES
[1]
[2]
[3]
[4]
[5]
[6]
[7]
[8]
B. Zotter, CERN-AB-2005-043 (2005)
R.L. Gluckstern, CERN yellow report 2000-011 (2000)
E. Métral, CERN-AB-2005-084 (2005)
A. Burov and V. Lebedev, EPAC’02 (2002)
K. L.F. Bane, SLAC/AP-87 (1991)
F. Caspers et al., EPAC’04 (2004)
H. Tsutsui, LHC-PROJECT-NOTE-318 (2003)
L. Vos, CERN-AB-2003-005-ABP (2003) and CERN-AB-2003-093-ABP (2003)
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