Models of Longitudinal Space-Charge Impedance for the Study of the 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 FELs. 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