1. TRANSVERSE RESISTIVE-WALL IMPEDANCE FROM ZOTTER2005’S THEORY
Elias Métral ( min, 25 slides)
“Low-frequency” regime
Introduction and motivation
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
High-frequency regime
Elias Métral, CERN-GSI bi-lateral working meeting on Collective Effects – Coordination of Theory and Experiments, GSI, 30-31/03/06

2. THE MOTIVATION: THE LHC GRAPHITE COLLIMATORS
INTRODUCTION AND MOTIVATION (1/2)
THE MOTIVATION: THE LHC GRAPHITE COLLIMATORS
First unstable betatron line
Skin depth for graphite (ρ = 10 μΩm)
Collimator thickness 
 One could think that the classical “thick-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

3. 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 :
New regime :
Elias Métral, CERN-GSI bi-lateral working meeting on Collective Effects – Coordination of Theory and Experiments, GSI, 30-31/03/06

4. N.A. FOR A LHC COLLIMATOR (1/4)
COMPARISON ZOTTER2005-BUROV&LEBEDEV2002
Classical thick-wall
BL’s results (real and imag. parts) in black: dots without and lines with copper coating
Elias Métral, CERN-GSI bi-lateral working meeting on Collective Effects – Coordination of Theory and Experiments, GSI, 30-31/03/06

5. N.A. FOR A LHC COLLIMATOR (2/4)
GLOBAL PLOT FROM ZOTTER2005
Low beam velocity case (e.g. PSB : , )
Same as Bane1991
Negative
AC conductivity
Elias Métral, CERN-GSI bi-lateral working meeting on Collective Effects – Coordination of Theory and Experiments, GSI, 30-31/03/06

6. N.A. FOR A LHC COLLIMATOR (4/4)
Elias Métral, CERN-GSI bi-lateral working meeting on Collective Effects – Coordination of Theory and Experiments, GSI, 30-31/03/06

7. N.A. FOR A SPS MKE KICKER
Elias Métral, CERN-GSI bi-lateral working meeting on Collective Effects – Coordination of Theory and Experiments, GSI, 30-31/03/06

8. REVIEW OF ZOTTER’S THEORY (1/17)
1) Maxwell equations
In the frequency domain, all the field quantities are taken to be proportional to
Combining the conduction and displacement current terms yields
with
Elias Métral, CERN-GSI bi-lateral working meeting on Collective Effects – Coordination of Theory and Experiments, GSI, 30-31/03/06

9. REVIEW OF ZOTTER’S THEORY (2/17)
2) Scalar Helmholtz equations for the longitudinal field components
Using , one obtains (using the circular cylindrical coordinates r, θ, z)
The homogeneous equation can be solved by separation of variables
Elias Métral, CERN-GSI bi-lateral working meeting on Collective Effects – Coordination of Theory and Experiments, GSI, 30-31/03/06

10. REVIEW OF ZOTTER’S THEORY (3/17)
m is called the azimuthal mode number
(m=1 for pure dipole oscillations) 
and
k is called the wave number
Reinserting the time dependence ( ), the axial motion is seen to be a wave proportional to , with phase velocity
which may in general differ from the beam velocity
R (r) is given by with
Radial propagation constant
The solutions of this differential equation are the modified Bessel functions and
Elias Métral, CERN-GSI bi-lateral working meeting on Collective Effects – Coordination of Theory and Experiments, GSI, 30-31/03/06

11. 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
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 )  See slide 14
C1,2 and D1,2 are constants to be determined
Elias Métral, CERN-GSI bi-lateral working meeting on Collective Effects – Coordination of Theory and Experiments, GSI, 30-31/03/06

12. REVIEW OF ZOTTER’S THEORY (5/17)
3) Transverse field components deduced from the longitudinal ones using Maxwell equations (in a source-free region)
Elias Métral, CERN-GSI bi-lateral working meeting on Collective Effects – Coordination of Theory and Experiments, GSI, 30-31/03/06

13. REVIEW OF ZOTTER’S THEORY (6/17)
4) Source of the fields: Ring-beam distribution  Infinitesimally short, annular beam of charge and radius traveling with velocity along the axis
Charge density in the frequency domain
where 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

14. REVIEW OF ZOTTER’S THEORY (7/17)
Longitudinal source terms (from Gluckstern)  Valid for , i.e. in the vacuum between the beam and the pipe = region (1)
with
and 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

15. REVIEW OF ZOTTER’S THEORY (8/17)
5) The total (i.e. resistive-wall + space charge) horizontal impedance 
with L the length of the resistive pipe and
Elias Métral, CERN-GSI bi-lateral working meeting on Collective Effects – Coordination of Theory and Experiments, GSI, 30-31/03/06

16. REVIEW OF ZOTTER’S THEORY (9/17)
The space-charge impedance is obtained with a perfect conductor at r = b, i.e. when and , with  If 
and 
Elias Métral, CERN-GSI bi-lateral working meeting on Collective Effects – Coordination of Theory and Experiments, GSI, 30-31/03/06

17. REVIEW OF ZOTTER’S THEORY (10/17)
The resistive-wall impedance is obtained by subtracting the space-charge impedance from the total impedance 
Only 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

18. 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 and have to be continuous
Then the radial components of the displacement and of the induction are also continuous, i.e. matching of the radial components is redundant
At a Perfect Conductor (PC) : 
At a Perfect Magnet (PM) : 
At r  Infinity  Only is permitted as diverges
Elias Métral, CERN-GSI bi-lateral working meeting on Collective Effects – Coordination of Theory and Experiments, GSI, 30-31/03/06

19. REVIEW OF ZOTTER’S THEORY (12/17)
General form of the field strengths in region (p)
(Ep, Gp, αp and ηp) are constants to be determined …
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20. REVIEW OF ZOTTER’S THEORY (13/17)
7) General 1-layer formula
α2 and η2 are determined by the boundary conditions at the outer chamber wall r = d
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21. REVIEW OF ZOTTER’S THEORY (14/17)
d  ,
or PC or PM
(2) Layer 1
d  
Perfect Conductor (PC)
(1) Vacuum
Beam a b
Perfect Magnet (PM) d
Elias Métral, CERN-GSI bi-lateral working meeting on Collective Effects – Coordination of Theory and Experiments, GSI, 30-31/03/06

22. REVIEW OF ZOTTER’S THEORY (15/17)
8) General 2-layer formula
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
Elias Métral, CERN-GSI bi-lateral working meeting on Collective Effects – Coordination of Theory and Experiments, GSI, 30-31/03/06

23. REVIEW OF ZOTTER’S THEORY (16/17)
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

24. REVIEW OF ZOTTER’S THEORY (17/17)
(3) Layer 2
e  ,
or PC or PM
(2) Layer 1
(1) Vacuum
e  
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

25. 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] B. Zotter, CERN-AB (2005)
[2] R.L. Gluckstern, CERN yellow report (2000)
[3] E. Métral, CERN-AB (2005)
[4] A. Burov and V. Lebedev, EPAC’02 (2002)
[5] K. L.F. Bane, SLAC/AP-87 (1991)
[6] F. Caspers et al., EPAC’04 (2004)
[7] H. Tsutsui, LHC-PROJECT-NOTE-318 (2003)
[8] L. Vos, CERN-AB ABP (2003) and CERN-AB ABP (2003)
Elias Métral, CERN-GSI bi-lateral working meeting on Collective Effects – Coordination of Theory and Experiments, GSI, 30-31/03/06