Abstract 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.

中文翻译：

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环形离子阱轴对称环形谐波的幂级数展开

摘要 本研究提出了一种根据距诱捕圆的径向和轴向距离获得环形谐波的幂级数展开的方法。为了得到单个环谐波的幂级数展开式，推导出三项递推关系，涉及n - 1 , n, n + 1 阶环谐波和n 阶环谐波的导数。使用这些三项递推关系，提出了一个系统程序，以获得任意阶次的环形谐波的幂级数展开，直至达到所需的项数。通过这个程序，可以得到 5 阶环形谐波的幂级数展开。该理论的验证是在任意环形离子阱上进行的。将通过幂级数获得的 78 Th 单电荷离子的电位和轨迹与使用边界元方法 (BEM) 计算的那些进行比较。比赛被认为非常好。