Longitudinal plasma oscillations are becoming a subject of great interest for XFEL physics in connection with LSC microbunching instability [1] and certain pump-probe synchronization schemes [2]. In the present paper we developed the first exact analytical treatment for longitudinal oscillations within an axis-symmetric, (relativistic) electron beam, which can be used as a primary standard for benchmarking space-charge simulation codes. Also, this result is per se of obvious theoretical relevance as it constitutes one of the few exact solutions for the evolution of charged particles under the action of self-interactions.

During the design of X-FELs, space-charge codes are required to simulate the evolution of longitudinal plasma oscillation within an electron beam in connection with LSC microbunching instability [1] and certain pump-probe synchronization schemes [2]. In the paper [3] we presented an analytical solution to the initial value problem for longitudinal plasma oscillation in an electron beam. Such a result, besides its theoretical importance, allows one to benchmark space-charge simulation programs against a self-consistent solution of the evolution problem. In this paper we present a comparison between our results [3] and the outcomes of the simulation code ASTRA.