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

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

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

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.

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, 1993. Also published as KEK Preprint 92-196. A. Burov and V. Danilov. Suppression of transverse bunch instabilities by asymmetries in the chamber geometry. Phys. Rev. Lett., 82:2286–2289, 1999

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

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

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, 2007. Investigation report, LHC, EDMS Document No. 329882, CERN **K. Yokoya. Resistive wall impedance of beam pipes of general cross section. Part. Acc., 41:221, 1993.

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.

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.

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

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

Electron Cloud in Drift Region

Electron Cloud in Dipole Magnetic Field

Thank you for attention!