A new method for numerical calculation of potentials and trajectories in systems of cylindrical symmetry

Abstract

The charge density method for solving Laplace's equation is treated for a system with cylindrical symmetry, where the electrodes can be represented by smooth curves of arbitrary shape. It is assumed that each electrode has two separate sharp edges, but this is no severe restriction. The singularity in the charge density at the edges is eliminated by approximating the integral equation with Gauss-Chebyshev quadrature. A second singularity, occurring if the potential is evaluated on an electrode, is split off and integrated analytically. The resulting set of linear equations is characterised by a symmetric, positive definite matrix and is solved with the Choleskii method. The potential and the electric field can be calculated anywhere outside the electrodes directly from the charge densities. The equations of motion are solved with the method of rational extrapolation. This allows the tracing of electron rays in the non-paraxial region. Some results are given and their accuracy is discussed. In comparison with existing methods either the accuracy is significantly improved or the computer requirements are considerably reduced.