Single particle motion in a Penning trap: description in the classical canonical formalism

Abstract

This paper aims at the development of methods for the calculation of the characteristic frequencies of a Penning trap, taking into account deviations of the actual geometry from the ideal one, anharmonicities of the electric potential, misalignments and inhomogeneities of the magnetic field, additional time dependent electromagnetic fields, and so on. The paper starts by describing the motion of a single charged particle in an ideal hyperbolic Penning trap using the formalism of classical hamiltonian mechanics. The usefulness of rotating coordinates is pointed out, and the importance of conservation of canonical angular momentum is stressed. After transformation to action-angle variables the ideal hyperbolic Penning trap can be considered as a 0-th approximation for a systematic classical hamiltonian perturbation approach to "real" Penning traps, as they are used in the laboratory. It is explained how a given problem has to be set up and prepared for the perturbation calculation, then the general principles of the method are outlined. Finally the application of the method is illustrated on hand of a number of worked out examples.