Influence of uniform electron clouds on the coupling impedance

The contribution to the longitudinal coupling impedance from an electron cloud in the form of a uniformly distributed non-neutral plasma of electrons is investigated analytically. The beam-pipe is assumed to be of circular cross section with a thick resistive wall. The beam charge distribution is uniform in the transverse direction. The electron contribution to the charge and current densities are obtained from the collective electron response to the beam passage through the pipe. We obtain the radial differential equation governing the field variation in the presence of the electron background and a general closed formula for the longitudinal coupling impedance is derived. The depletion of the coupling impedance with the density of the electron cloud is discussed for the examples of GSI SIS-18 and SIS- 100, CERN SPS and PS, and the KEKB LER, and conditions for the minimum excitation frequency as a function of the electron density are derived. Furthermore, the case of over-dense plasmas is also studied.