Space charge algorithm for the multi ensemble model

Derived from the Vlasov equation the Ensemble Model [A. Novokhatski, T. Weiland, PAC’99, New York, March 1999] has been elaborated for fast and efficient beam dynamics simulations. The Model represents a particle beam as a set of sub-beams or Ensembles, described by coordinates of the centroid and 6D phase space correlations. Whereas a space charge routine for the Single Ensemble Model (SEM) has been developed and tested [M. Krassilnkov, et al., ICAP 2000, Darmstadt, September 2000], implementation of the space charge algorithm for the Multi Ensemble Model (MEM) needs more efforts. A space charge model based on the Multi-Centered Gaussian Expansion (MCGE) [M. Krassilnikov, T. Weiland, ICAP’02, East Lansing, USA, October 2002] implies a smooth particle density distribution within an Ensemble but it requires rather large computational efforts. This paper presents another space charge algorithm, based on the analytical solution for the electromagnetic field of an ellipsoidal 3D charge distribution [M. Comunian, et al., Phys. Rev. Spec. Topics—Acc. Beams 4 (2001) 124201]. Using this algorithm one can calculate the space charge force and its gradient inside and outside the driving Ensemble. Features of the implementation and simplifying approximations are discussed in this paper.