Interpolation Functions for Describing the Boundary Trajectories of Electron Beams Propagated in Ionised Gas

The linear and square interpolation functions for simulation the trajectories of electron beams, propagated in ionized gas, are considered in the paper. The basic physic-mathematical model for propagation of electron beam with compensation the space charge of electrons by the positive ions of residual gas, described by the set of algebra-differential equations, is considered. It has been proved theoretically, that for the beams with high convergence angle on the most part of its propagation in such conditions the boundary trajectory is closed to linear. In such circumstance's combination of linear and square functions is usually enough for accurate describing of beam boundary trajectories. Possibility of using the square root from square polynomic as suitable interpolation function for solving this task is also considered. Examples of interpolation are given and analyzed. The maximal relative error of interpolation was 0.2% by the focus position and 7.7% by the focal beam radius, but in the most cases relative error was smaller, then 1%.