Coherent and Incoherent Tune Shift Induced by the Nonlinear Flat-Chamber Resistive-WallWake Field with an Application to the LHC Collimator Experiment in the SPS

We derive formulae for the coherent and incoherent tune shift due to the non-linear resistive-wall wake £eld for a single beam travelling between two parallel plates. The theoretical predictions are compared with measurements performed on an LHC prototype collimator during the 2004 SPS run. We show that the tune shift derived from the nonlinear wake £eld yields a better agreement with the measurements than two conventional tune-shift expressions which consider only the linear wake £eld. In particular, we demonstrate that the nonlinear terms of the resistive-wall wake £eld become important, if, as in the SPS experiment, the collimator gap is comparable to the rms beam size. In addition, for small gap sizes, the incoherent tune spread induced by the nonlinear wake £eld increases beam stability via enhanced Landau damping.