On the density oscillations of a warm particle bunch

Abstract

The density oscillations of warm particle bunches is investigated theoretically. Two different mathematical approaches are employed to derive the basic equation describing density oscillations; one is a fluid approach and the second is a more general Green’s function formulation. The motion is analyzed in first-order perturbation theory where it is shown, under the assumption of no degeneracy, that there are only stable oscillations. Second-order perturbation theory gives damping of the motion. The perturbation theory is examined, and a criterion is exhibited for its proper use. Thus, when the resistivity is small enough (but nonzero) the motion is stable, but when the resistivity is large the motion is essentially unstable with a growth rate which is that of an unbunched beam. The criterion is approximately evaluated using a model for a bunched beam.