Penning trap with an inclined magnetic field

A modified Penning trap with a spatially uniform magnetic field B inclined with respect to the axis of rotational symmetry of the electrodes is considered. The inclination angle can be arbitrary. Canonical transformation of phase variables transforming the Hamiltonian of the considered system into a sum of three uncoupled harmonic oscillators is found. We determine the region of stability in space of two parameters controlling the dynamics: the trapping parameter κ and the squared sine of the inclination angle ϑ0. If the angle ϑ0 is smaller than 54°, a charge occupies a finite spatial volume within the processing chamber. A rigid hierarchy of trapping frequencies is broken if B is inclined at the critical angle: the magnetron frequency reaches the modified cyclotron frequency while the axial frequency exceeds them. Apart from this resonance, we reveal the family of resonant curves in the region of stability. In the relativistic regime, the system is not linear. We show that it is not integrable in the Liouville sense. The averaging over the fast variable allows to reduce the system to two degrees of freedom. An analysis of the Poincaré cross-sections of the averaged systems shows the regions of effective stability of the trap.