A theorem of higher-order spheric aberrations in Glaser's bell-shaped magnetic lens

The author (J. Ximen) has developed canonical aberration theory, performed a generalized integration transformation on Hamiltonian eikonals, derived electron optical aberrations for up to the ninth-order approximation in rotationally symmetric magnetic and electrostatic lenses. In the present article, this powerful technique of integration transformation has been applied to Glaser's bell-shaped magnetic lens. Third-, fifth-, seventh-, ninth-order spheric aberrations have been analytically derived and numerically calculated. Based on numerical computations of spheric aberrations, it has been shown that third-and seventh-order spheric aberration coefficients always have positive signs, while fifth-and ninth-order spheric aberration coefficients may be positive or negative. Nevertheless, the total spheric aberration is still positive. This theoretical conclusion would be a development to O. Scherzer's theorem, and would lead to a break through in theoretical electron optics. It is to be expected that this theoretical research may be useful for estimating variations and effects of higher-order spheric aberrations in relationally symmetric magnetic lenses.