Relaxation calculation of the electrostatic properties of compensated Penning traps with hyperbolic electrodes

Spectacular accuracies have been achieved with single elementary particles trapped in hyperbolic Penning traps, but only after anharmonicities have been tuned out of these traps by varying the potential of extra compensation electrodes introduced into the traps. The relaxation calculation reported here is the first theoretical study of the electrostatic properties of such compensated Penning traps. Enough computations have been completed to clarify the basic physics involved in anharmonicity compensation and to provide useful, quantitative information for experiments in progress (especially for testing of anharmonicity systematics). The clearer picture of anharmonicity compensation which emerges suggests that the design of existing Penning traps could be significantly improved. An optimal electrode configuration is proposed which, in principle, makes the harmonic oscillation frequency of a trapped particle independent of changes in the compensation potential.