Longitudinal impedance and shielding effectiveness of a resistive beam pipe for arbitrary energy and frequency.

The longitudinal coupling impedance of a cylindrical beam pipe for arbitrary relativistic gamma(0) and mode frequency is obtained analytically for finite wall conductivity and finite wall thickness. Closed form expressions for the electromagnetic fields excited by a beam perturbation are derived analytically. General expressions for the resistive-wall impedance in the presence of a metallic shield and for the rf shielding effectiveness of the beam pipe have been obtained and then compared with approximate expressions. The results are applied to the GSI synchrotron SIS, where the thickness of the vacuum chamber in the dipole magnets is much smaller than the skin depth at injection energy.