Power loss in the LHC beam screen at 7 TeV due to the multi-layer longitudinal impedance N. Mounet and E. Métral Goal: Check the effect of the multi-layer formula on the power loss of the currently installed beam screen, and the effect of the Cu coating thickness on the beam screen power loss in one of the LHC upgrade scenario. Assumptions: Circular geometry, Copper conductivity takes into account magneto-resistance and anomalous skin effect, Neglecting anomalous skin effect on the power loss itself, No weld. 3rd Impedance meeting – 25/02/2010 – N. Mounet and E. Métral - CERN/BE-ABP-LIS

Current beam screen power loss: parameters used Beam screen geometry: 2D (infinitely long), circular shape with radius b=18.4mm (not exactly the case, see design report, chapter 5). Beam screen materials: 75 mm copper, with r (high B field at 7 TeV, 20K)=5.5 10-10 W.m (no relaxation time, eb=mr=1 ), 1 mm Stainless steel 304L with r (20K)=6.10-7 W.m (no relaxation time, eb=mr=1), LHC parameters: Bunch length (RMS): sz = 74.95 mm (or st = 0.25 ns) (gaussian bunch), Number of particles per bunch: Nb=1.15 1011, Number of bunches: M=2808, LHC total circumference: L=26,658.883 m, Revolution frequency: f0=w0 /(2p) = c/L=11.2455 kHz, Relativistic mass factor: g = 7,462.69. Impedance meeting - N. Mounet and E. Métral - CERN/BE-ABP-LIS 2

Beam screen longitudinal impedance Resistive-wall impedance: comparison between classic thick wall and multilayer Zotter’s formalism (per unit length) Revolution frequency For the classic thick wall formula the crosses are at each harmonic pw0 of the revolution frequency, up to the maximum used in the power loss calculation (see next slide). Impedance meeting - N. Mounet and E. Métral - CERN/BE-ABP-LIS 3

Power loss computation Formula used (see G. Rumolo USPAS 2009 course, slide 37, http://impedance.web.cern.ch/impedance/USPAS/LongPlane.pdf): where is the longitudinal bunch distribution in frequency domain. For the classic thick wall case, we can also apply an analytic formula: (power per unit meter) where G is the gamma function and Z0=m0 c is the free space impedance. Impedance meeting - N. Mounet and E. Métral - CERN/BE-ABP-LIS 4

Current beam screen power loss: conclusion We have computed the power loss per unit length of beam screen using 5 different approaches: Using the analytic formula for the classic thick wall case (with copper), Using the infinite sum formula with a longitudinal impedance computed from either: The classic thick wall formula for copper, Zotter’s formalism for one infinite layer of copper, Zotter’s formalism for copper coated stainless steel (2 layers), Zotter’s formalism for copper coated stainless steel with vacuum behind (3 layers). All those computations end up with the same result: Ploss=85.0 mW / m (with a maximum difference of 0.01 %) Impedance meeting - N. Mounet and E. Métral - CERN/BE-ABP-LIS 5

“Upgraded” beam screen power loss: parameters used Beam screen geometry: 2D (infinitely long), circular shape with radius b=39.35mm (not exactly the case; this is the radius of the largest inscribable circle). Beam screen materials: Copper coating, with r (high B field at 7 TeV, 20K)=5.5 10-10 W.m (no relaxation time, eb=mr=1 ), Infinite Stainless steel 304L with r (20K)=6.10-7 W.m (no relaxation time, eb=mr=1), LHC parameters: same as previously. 3rd Impedance meeting – 25/02/2010 – N. Mounet and E. Métral - CERN/BE-ABP-LIS

“Upgraded” Beam screen longitudinal impedance: no coating Resistive-wall impedance: comparison between classic thick wall and multilayer Zotter’s formalism with different thickness of stainless steel (no coating) Revolution frequency 3rd Impedance meeting – 25/02/2010 – N. Mounet and E. Métral - CERN/BE-ABP-LIS

“Upgraded” beam screen longitudinal impedance: coating Resistive-wall impedance: comparison between different coating thicknesses using the multilayer formalism Revolution frequency 3rd Impedance meeting – 25/02/2010 – N. Mounet and E. Métral - CERN/BE-ABP-LIS 8

Power loss vs. copper coating Power loss with respect to the thickness of the Cu coating: Above 1mm, the power loss is the same as for infinite copper 3rd Impedance meeting – 25/02/2010 – N. Mounet and E. Métral - CERN/BE-ABP-LIS 9

“Upgraded” beam screen power loss: Conclusion For a pure stainless steel pipe, we have computed the power loss per unit length of beam screen (upgrade scenario) in 4 different cases: Using the analytic formula for the classic thick wall case, Using the infinite sum formula with our multilayer longitudinal impedance for 1mm, 13 mm, or infinite stainless steel. All those computations end up with the same result: Ploss=1313 mW/m (the maximum difference being 0.02%) (Using the multilayer formalism): for a copper coated pipe, 1mm is enough to get (within 2%) the power loss obtained with one infinite layer of copper in the beam screen, the latter being:  Ploss=39.8 mW/m. 3rd Impedance meeting – 25/02/2010 – N. Mounet and E. Métral - CERN/BE-ABP-LIS