An analytic Solution for the Potential Distribution within an Asymmetrically Excited Electrostatic Quadrupole Lens

Abstract

This paper describes the solution of Laplace's equation for an asymmetrically excited electrostatic quadrupole comprising four spherical electrodes. The introduction explains the advantages of such a lens and the failure of numerical relaxation techniques to solve its potential distribution. The mathematical techniques employed to arrive at an analytic solution are then developed. A description of the various coordinate systems employed, and the connexions between them, is presented first. This is followed by the solution of Laplace's equation in the various systems by the method of separation of variables. At this stage the geometry of the chosen system is described and a comparison is made with the usual cylindrical polar form near the axis. The process of obtaining the full solution by superposing two bispherical systems is then developed, with details of the method employed to fit the boundary conditions; and finally the transformation of this solution into the standard form is described.