Detection, damping, and translating the center of the axial oscillation of a charged particle in a Penning trap with hyperbolic electrodes

The axial oscillation frequency of a trapped particle in a Penning trap with hyperbolic electrodes has proven to be a very important observable for precision experiments. Shifts in this frequency have been used to measure magnetron frequencies, cyclotron frequencies, and $g-2\mathrm{}$ transitions for single trapped particles. Detection and damping of the axial oscillation were thought to be well understood except that a key constant which depends upon trap geometry had not been calculated. A relaxation calculation has thus been performed to provide a calculated value for this constant and several others, and to determine their dependence upon trap geometry. A surprising result is a discrepancy between the calculated value of the key constant and that deduced from measured values of the width of the axial oscillation frequency in the several cases where it has been measured.