1. Impedance analysis for collimator and beam screen in LHC and Resistive Wall Instability
Liu Yu Dong

2. Wall impedance for multilayer chamber
Beam screen: stainless steel (0.6mm) with coating copper (50um)
Injection protection collimator: hBN (hexagonal boron nitride, 六方氮化硼) coating with Ti（5um）
Primary collimator: CFC (Carbon-Carbon)
Secondary collimator: CFC
Absorber: W

3. Axisymmetiric multilayer chamber
According to reference : Nicolas Mounet, Doctoral thesis, 2012
The impedance for multilayer axisymmetiric chamber can be expressed as:
(1)
Here, the parameter α0TM and α0TM are decided by the properties of multilayer
Above formula are general expression without any approximation

4. Flat multilayer chamber (collimator structure)
the analysis process can be found in the same reference
The impedance can be expressed as :
(2)
Here, the parameters αare decided by the properties of multilayer.

5. The impedance of flat multilayer chamber derived from equation (1) with constant Yokoyal’s(Laslett’s) form factors.
(3)
The above Yokoyal’s form factors expressed as:
For single jaw collimator, the Yokoyal’s form factors expressed as
The detail analytic process for formula (3) and Yokoyal’s form factors are in references:
R. L. Gluckstern, J. van Zeijts, and B. Zotter. Coupling impedance of beam pipes of general cross section. Phys. Rev. E, 47:656–663, 1993.
K. Yokoya. Resistive wall impedance of beam pipes of general cross section. Part. Acc., 41:221, Also published as KEK Preprint
A. Burov and V. Danilov. Suppression of transverse bunch instabilities by asymmetries in the chamber geometry. Phys. Rev. Lett., 82:2286–2289, 1999

6. Reference* give another formula to calculate the resistive wall impedance for multilayer flat chamber.
(4)
a is pipe radius and d is thickness of the layer
The Yokoyal’s form factors have been considered in formula (4).
*Alexey Burov and Valeri Lebedev, TRANSVERSE RESISTIVE WALL IMPEDANCE FOR MULTI-LAYER FLAT CHAMBERS, Proceedings of EPAC 2002, Paris, France
*Alexey Burov and Valeri Lebedev, TRANSVERSE RESISTIVE WALL IMPEDANCE FOR MULTI-LAYER ROUND CHAMBERS, Proceedings of EPAC 2002, Paris, France

7. The single layer axisymmetric classic thick wall impedance formula:
(5)
Above four methods, formla (3)、 (2)、(4)、(5) were used to calculate the collimators in LHC by Nicolas Mounet in his thesis.
Titanium-coated hBN collimator

8. Wall impedance for beam screen
One layer of copper with thickness 50um and then infinite layer of stainless steel
In the presence of high fields and low temperatures, copper is magnetoresistant and suffers from the anomalous skin effect.
Equivalent elliptical shape for beam screen
Magnetoresistance was taken into account with empirical law*:
Yokoya factors for an elliptical shape** are applied in formula (3) to get the impedances of an elliptic structure approximating the beam screen cross-section
*C. Rathjen. Electrical measurements of beam screen wall samples in magnetic fields, Investigation report, LHC, EDMS Document No , CERN
**K. Yokoya. Resistive wall impedance of beam pipes of general cross section. Part. Acc., 41:221, 1993.

9. Nicolas Mounet’s calculation results for LHC beam screen
The above calculation with formula (3)、 (2)、(4)、(5) have been implemented by Nicolas’s Mounet’s code, and he has sent me his source code.

10. Resistive Wall Instability
The growth rate of the resistive wall instability described by rigid particle model is give by:
is the averaged beta function over the ring the real part of the resistive wall impedance, the beam curren, the beam energy, the betatron tune , the mode number of the coupled bunch oscillation and M the number of bunches.

11. Electron Cloud Instability for some super pp colliders
LHC
FCC-hh
SPPC
Bunch particles (1011)
1.15
1.0
0.4/0.8/2.0
Bunch spacing (ns) 25
5/10/25
Beam energy (TeV) 7 50
31.7
Pipe radius (mm) 20 13
Parameter n
0.165
0.189
1.001/0.249/0.040
Neutralization line density (1010/m)
1.53
1.33
2.66
Neutralization volume density (1013/m3)
1.22
2.51
5.01
Wake field W/L (103/m2)
3.15
3.149
Betatron tune
43.3 -
60.3
Synchrotron tune
0.006
0.002
0.005
Growth time (ms)
4.31
1.752
Circumference (km)
26.7
100
Threshold electron density (1013/m3)
0.66
0.147
0.468

12. Simulation parameters for electron cloud density
Value
Bunch particles nb(1011)
2.0
Bunch spacing (ns) 25
Pipe radius (mm) 12
Circumference(km)
54.7
Tune
60.3
rms bunch length(mm)
75.5
Bunch size (rm)
1/1
Dipole magnetic field (T) 20
Quadrupole(T/m2)
1000

13. Electron Cloud in Drift Region

14. Electron Cloud in Dipole Magnetic Field

15. Thank you for attention!