A semi-analytical aberration calculation method using polynomials and Lie algebra

To investigate aberration coefficients up to order 11 or higher in rotationally symmetric magnetic fields, a semi-analytical method is described. The field distribution and the trajectory of a charged particle are expressed by several polynomials to differentiate and integrate them analytically. The approximation error of the field can be reduced to less than 10−13. These polynomials are substituted in “interaction” Hamiltonian, and the aberration coefficients are obtained by Lie algebraic theory. An exponential model and Glaserʹs bell-shaped model are used to compare the aberration coefficients above order 4.