New Results on the Impedance of Resistive Metal Walls of Finite Thickness

The resistive wall impedance of cylindrical vacuum chambers was first calculated over 40 years ago. The original results were valid for metal vacuum chamber walls which are thick compared to the skin depth at the frequencies of interest. Recently the subject has again become important for beam stability in the LHC where in particular the transverse impedance of the large number of graphite collimators to be installed could severely limit its performance, if the “thick wall” formulae were correct. The frequencies of the slow betatron waves in such a large machine are very low and thus the transverse impedance, originally found to be proportional to the inverse square root of frequency, could lead to instabilities. However, when the skin depth exceeds the wall thickness, the transverse resistive wall impedance is strongly reduced and a number of papers have recently been published to estimate this reduction. However, all of these had restrictions of validity. Here we give a consistent derivation of the general expression for the transverse impedance of walls made of arbitrary materials which are valid at all frequencies. The results are compared with previous ones in the graphs for the LHC collimators - where they agree quite well, but also for the SPS-MKE kickers which are required for injection into the LHC and where the agreement is less perfect.