Analytic and numerical analysis of the longitudinal coupling impedance of an annular cut in a coaxial liner

Beam pipes of high-energy superconducting colliders require a shielding tube (liner) with pumping slots to screen cold chamber walls from synchrotron radiation. Pumping slots in the liner walls are required to keep high vacuum inside the beam pipe and provide for a long beam lifetime. As previously discussed [Fedotov and Gluckstern, Phys. Rev. E 54, 1930 (1996)], for a long narrow slot whose length may be comparable with the wavelength, the usual static approximation for the polarizability and susceptibility which enter into the impedance is a poor one. Our objective is eventually to analyze and obtain numerical values for a rectangular slot of arbitrary dimensions. In this paper we present an analysis, based on a variational formulation, for the impedance of an annular cut in the inner conductor, including both the realistic coaxial structure of the beam-pipe and the effect of finite wavelength. For low frequencies, the numerical results are checked against analytical results, with which they agree.