Power series expansion of axially symmetric toroidal harmonics for toroidal ion trap

This study presents a method to obtain power series expansions of toroidal harmonics in terms of radial and axial distances from the trapping circle. In order to obtain the power series expansion of individual toroidal harmonics, three-term recurrence relations are derived, which involve toroidal harmonics of order n - 1, n, n + 1 and derivative of toroidal harmonic of order n. Using these three-term recurrence relations a systematic procedure is presented to obtain the power series expansion for a toroidal harmonic of arbitrary order, up to the desired number of terms. With this procedure, the power series expansions of toroidal harmonics till order 5 are presented. Verification of this theory was carried out on an arbitrary toroidal ion trap. The potential and the trajectory of a singly charged ion of 78 Th obtained by the power series were compared with those computed using the Boundary Element Method (BEM). The match was found to be very good. (C) 2019 Elsevier B.V. All rights reserved.