The Short-Range Resistive Wall Wakefields

Abstract

In an accelerator when the bunch length becomes comparable to a characteristic distance s0, one which depends on the radius and the conductivity of the beam tube and in typical structures is on the order of tens of microns, the usual formulas for the resistive wall wakefield do not apply. In this report we derive the short‐range resistive wall wakefields of an ultra‐relativistic point particle in a metallic, cylindrical tube, both for a model in which the wall conductivity is taken to be independent of frequency and for one in which a frequency dependence is included. On this scale the wakefield is found to be dominated by a damped, high frequency resonator component. For the case of constant conductivity the resonant frequency is given by ω=∛c/s0 and the Q‐factor equals ∛/2. We provide a physical model to explain these results. For the case of a frequency dependent conductivity the resonator parameters depend also on the relaxation time of the metal τ. For cτ/s0≳0.5 the frequency ω∼√2ωpc/b, with ωp the plasma frequency of the free electrons in the metal and b the tube radius, and the 1/e damping time becomes 4τ. Finally, we calculate the wakefield and loss factor of a short Gaussian bunch.