Longitudinal coupling impedance for a smooth resistive cylindrical pipe of two conducting layers of finite thicknesses

The longitudinal coupling impedance and the corresponding longitudinal transmission coefficients have been investigated analytically in a smooth resistive cylindrical beam pipe. The particle beam has a finite transverse size and the beam-pipe wall is made up of two conducting layers of arbitrary thickness. We obtain closed form expressions for the longitudinal coupling impedance and its resistive-wall part, and the corresponding transmission coefficient. We also derive simplified approximate expressions for the resistive-wall impedance and the transmission coefficient which reproduce the exact calculations rather well without any visible differences, and recover our previous results of a one-layered beam-pipe [A. Al-Khateeb, O. Boine-Frankenheim, R. Hasse, I. Hofmann, Phys. Rev. E 71 (2005) 026501]. Finally the model is applied to a coated beam tube of SIS18 of GSI-Darmstadt. For the parameters used, no effect of the inner side coated with NEG material TiZrV on the longitudinal impedance is found.