Analytical analysis of longitudinal space charge effects for a bunched beam with radial dependence

The longitudinal space-charge (LSC) force can be a major cause of the microbunching instability in the linac for an x-ray free-electron laser. In this paper, the LSC-induced beam modulation is studied using an integral equation approach that takes into account the transverse (radial) variation of the LSC field for both the coasting-beam limit and a bunched beam. Variation of the beam energy and the transverse beam size is also incorporated. We discuss the validity of this approach and compare it with other analytical analyses as well as numerical simulations.