Hamiltonian description of ion motion in crossed electric and magnetic fields with cylindrical symmetry

Abstract

A Hamiltonian description is given of charged particle motion around a circular design trajectory in static electromagnetic systems possessing cylindrical symmetry. In particular the ion motion in a constant magnetic field in the z-direction and a hyperboloid electric potential, V(x,y,z) = 1 2 k 2z 2 − 1 4 k 2(x 2 + y 2) , where k 2 is the potential strength parameter, is considered in Cartesian coordinates, to provide an exact solution to the Hamiltonian problem. The Wien filter can be considered as a special application of the system described. For a general electric field shape an expansion of the potential around the design orbit is required and a treatment in cylindrical coordinates seems more appropriate. Applications of these systems can be found in inflectors and deflectors in circular ion accelerators as well as in Wien filters or ion separators in various ion optical devices.