1. NEWS ABOUT COLLIMATOR IMPEDANCE
FOR PHASE 1 & 2
Elias Métral
Some more info about the transverse impedance of the LHC
1st part of the answer to the question (from R. Assmann) about the EM fields near the collimator (For SPS MD in 2004)  Deduced from Zotter2005 formalism (circular geometry + classical “thick-wall” regime)
Preliminary results for a ceramic collimator or a collimator with slots or rods
Elias Métral, Phase 2 Specification and Implementation Meeting, 04/04/2008

2. ZOTTER2005’S THEORY FOR 1 GRAPHITE COLLIMATOR
3) “High-frequency” regime  Not relevant for LHC!
1) “Inductive-bypass” regime
2) Classical “thick-wall” regime
Interesting frequency range for LHC  From few kHz to few GHz
Elias Métral, Phase 2 Specification and Implementation Meeting, 04/04/2008

3. SIMPLEST FORMULA FOR THE LHC COLLIMATOR TRANSVERSE IMPEDANCE (round case) (1/2)
Valid for any relatively good conductor with r real and r = 1
with
Modified Bessel function  
There are Yokoya’s factors to go from round to flat (2 / 12 and 2 / 24 )
Classical “thick-wall” regime
Elias Métral, Phase 2 Specification and Implementation Meeting, 04/04/2008

4. The maximum of the real part is reached when
SIMPLEST FORMULA FOR THE LHC COLLIMATOR TRANSVERSE IMPEDANCE (round case) (2/2)
The maximum of the real part is reached when
It is a broad maximum  
This scaling was also found analytically using the approximated model of L. Vos (as said in the LHC Design Report, p. 100)
N.A.:
Elias Métral, Phase 2 Specification and Implementation Meeting, 04/04/2008

5. ZOOM (between 8 kHz and 20 MHz) OF THE LHC TRANSVERSE IMPEDANCE AT TOP ENERGY (AFTER THE SQUEEZE)
The value of the real part of the impedance at 8 kHz (1st unstable betatron line) is ~ 141 M/m
The value of the real part of the impedance at 20 MHz (frequency limit of the transverse damper) is ~ 55 M/m
The ratio between the two values is only ~ 2.6 (it would have been 50 in the case of the classical resistive-wall theory!)
And, as the gain of the power amplifier rolls off rapidly when approaching 20 MHz, there might be some problems there…
Elias Métral, Phase 2 Specification and Implementation Meeting, 04/04/2008

6. SOURCE CHARGE DENSITY USED FOR THE COMPUTATIONS
A macro-particle of charge is assumed to move along the pipe (in the s - direction) with an offset in the direction and with velocity
A cos m beam generates EM fields in cos m and sin m  Different multipoles are decoupled (consequence of the axial symmetry)
 The charge density can be written
where is the mth multipole moment
The cylindrical coordinate system is used
Numerical values are given for the 2004 SPS experiment
Elias Métral, Phase 2 Specification and Implementation Meeting, 04/04/2008

7. REMINDER: Perfectly Conducting wall (1/3) m = 0 m = 1
Used to compute the longitudinal impedance
Used to compute the transverse impedance
Elias Métral, Phase 2 Specification and Implementation Meeting, 04/04/2008

8. REMINDER: Perfectly Conducting wall (2/3) m = 0 m = 1
Force on a particle with charge q
m = 0
m = 1
Elias Métral, Phase 2 Specification and Implementation Meeting, 04/04/2008

9. REMINDER: Perfectly Conducting wall (3/3)
For L = 2  R
For L = 2  R
Behind
the bunch
Elias Métral, Phase 2 Specification and Implementation Meeting, 04/04/2008

10. CASE OF A RESISTIVE OBJECT (1/4) m = 0 m = 1
  
 z > 0 is ahead of the beam and z < 0 is behind the beam
Different from Chao page 54 (see next slide)!
Elias Métral, Phase 2 Specification and Implementation Meeting, 04/04/2008

11. CASE OF A RESISTIVE OBJECT (2/4) I have (at the moment…) for m = 1
Chao finds the same result for r = b and  = 0
r = 1 for Chao
Chao finds a result 2 times bigger
Chao has (r 2 – b2) instead of r 2
Chao has (r 2 + b2) instead of r2
Elias Métral, Phase 2 Specification and Implementation Meeting, 04/04/2008

12. CASE OF A RESISTIVE OBJECT (3/4)
m = 0
m = 1
Elias Métral, Phase 2 Specification and Implementation Meeting, 04/04/2008

13. CASE OF A RESISTIVE OBJECT (4/4)
m = 0
m = 1
Elias Métral, Phase 2 Specification and Implementation Meeting, 04/04/2008

14.  Scan in resistivity  from 10-6 to 1020 m and
PRELIMINARY ANALYTICAL COMPUTATIONS OF THE TRANSVERSE IMPEDANCE OF A CERAMIC COLLIMATOR (1/2)
 Scan in resistivity  from 10-6 to 1020 m and
Many thanks to Fritz and Alexej for many interesting discussions!
Elias Métral, Phase 2 Specification and Implementation Meeting, 04/04/2008

15. PRELIMINARY ANALYTICAL COMPUTATIONS OF THE TRANSVERSE IMPEDANCE OF A CERAMIC COLLIMATOR (2/2)
1 layer of thickness 2.5 cm and then Perfect Conductor
1 layer of infinite thickness
1 layer of thickness 2.5 cm and then Perfect Conductor +
Lossy dielectric
Elias Métral, Phase 2 Specification and Implementation Meeting, 04/04/2008

16. PRELIMINARY SIMULATIONS OF THE TRANSVERSE IMPEDANCE OF A CERAMIC COLLIMATOR BY TOM KROYER
Typo  “below”
Typo  “above”
In conclusion: Only preliminary results for the moment!!!  We still need to carefully check them!
Elias Métral, Phase 2 Specification and Implementation Meeting, 04/04/2008

17. NEWS ABOUT COLLIMATOR WITH SLOTS & RODS (1/2)
Full Cu
Slots
Rods
NEWS ABOUT COLLIMATOR WITH SLOTS & RODS (1/2)
Current density at 10 kHz
Width = 1 mm
Depth = 5 mm
22 mm2
Length = 5 mm
Elias Métral, Phase 2 Specification and Implementation Meeting, 04/04/2008

18. NEWS ABOUT COLLIMATOR WITH SLOTS & RODS (2/2)
Measurements were also started by Fritz, Benoit and Federico  More news soon…
Elias Métral, Phase 2 Specification and Implementation Meeting, 04/04/2008