Numerical Calculation of the Potential Distribution in Ion Slit Lens Systems I

Abstract

The potential distribution is calculated in an ion lens, consisting of three parallel collinear slits in three parallel electrodes. The slit system is supposed to be infinite in the direction of the slits, so the problem becomes two dimensional in a plane perpendicular on the direction of the slits. In this plane the potential distribution is calculated by the method of conformal transformation. The SCHWARZ—CHRISTOFFEL transformation is used to map conformally the region between the projections of the electrodes of the slit system. It proves to be very simple to perform this transformation. Formulae are given for the case of an ion lens consisting of slits in three parallel plates. A series expansion and an iteration method are developed to find the necessary parameters. Both methods prove to be satisfactory if the slit widths are smaller than the distance to the neighbouring electrodes. Symmetrical lenses, not satisfying this condition will be treated in a second paper. In a third paper slit system will be treated with an arbitrary number of electrodes. In the transformed region LAPLACE'S equation is solved, having as boundary conditions the potentials on the electrodes. In this way the exact potential distribution in the lens system is found. In a typical example the potential distributions are calculated along the axis for several potentials on the electrodes, together with the corresponding fields.