Ellipsoid Space Charge Model for Electron Beam Dynamics Simulations

The pulsed beam current in linear accelerators can approach significant values up to tens of Amperes. Such high currents cause many specific adverse effects in the accelerators. One of such effects is the repelling forces of the space charge as they become comparable to the forces of the electromagnetic accelerating fields, and can influence the stability of phase and radial particle motion. In the numerical analysis of the beam dynamics in linear accelerators, it is necessary to choose one of the different space charge models depending on the desired accuracy, complexity, speed and computer resources availability. The most accurate results are achieved by the numerical solution of the Poisson equation. However, this method may require significant resources and time and may not be suitable when fast analysis is required in the linac design stages. Analytical methods, on the other hands, base on the analytical solution of Poisson equations for the pre-defined shape of the particles distribution inside the beam. One of the most popular analytical space charge models is the ellipsoidal beam approximations. Despite being a well-developed model, many published approaches lack some important features as fully three-dimensional ellipsoid asymmetry and multi-bunch model. In this paper, we will derive the equations of the space charge field for the full-3D non-relativistic ellipsoid bunch step by step, starting from the Poisson equation, and compare this model with the other known models.