The Resonator Impedance Model of Surface Roughness Applied to the LCLS Parameters

The resonator impedance model of surface roughness in a cylindrical beam tube, derived in Ref. 1, is compared to the inductive impedance model of Ref. 2. It is shown that for long, smooth bunches the two models both give an inductive response, that the effective inductance per length is proportional to the corrugation depth over the beam pipe radius, and that the absolute results also are comparable. For a non-smooth bunch shape, such as is found in the undulator region of the LCLS, however, the inductive impedance model is no longer valid; and the resonator model gives a non-inductive response, with the induced energy spread decreasing much more slowly with increasing bunch length than for a smooth distribution. When applied to the actual bunch shape and parameters in the LCLS, the resonator model predicts that, to remain within tolerances for induced energy spread, the beam tube roughness must be kept to ~10 nm. Further calculations suggest, however, that if the period-to-depth aspect ratio of the surface features is large, (as has been found in recent measurements of polished beam tube surfaces), then the wake field effect may be greatly suppressed, and the roughness tolerance greatly increased.