First-Order Theory of Electrical Focusing in Cyclotron-Type Two-Dimensional Lenses with Static and Time-Varying Potentials

Abstract

The potential distribution in cyclotron-type gaps with liner (outer electrodes at ground potential) is determined via Schwarz-Christoffel transformation, as well as by computer analysis (relaxation method). First-order focusing formulas for both static, as well as time-varying, potentials are derived. In addition exact calculations were carried out by direct numerical integration of the equations of motion with a computer program. The numerical data permitted an accurate evaluation of the validity of the analytical approximation, as well as further improvement of the theoretical formulas. Focusing relations are presented in a generalized form which shows the scaling laws and is readily applicable to different types of particles, energies, or lens geometries. As an example, the theory is applied to the axial motion of ions in a cyclotron.