Models of longitudinal space-charge impedance for microbunching instability

A 1D model of space-charge impedance, assuming a transversely uniform beam with circular cross section, has been proposed and is being extensively used in the modeling of the microbunching instability of relevance for the beam delivery systems of x-ray free-electron lasers. In this paper we investigate the limitation of the model when applied to studying the effect of shot noise—one of the sources of the microbunching instability. We make comparison with a fully 3D calculation and identify the upper end of the frequency spectrum for applicability of the 1D model. Relaxation of the assumptions regarding axis symmetry and uniformity of the transverse density is also reviewed.