Accurate calculation of geometric aberrations in electrostatic lenses

Using accurate numerical methods, electric fields and meridional electron trajectories have been computed for the three-cylinder einzel lens with sufficient precision to determine values for all the third- and fifth-order geometric aberration coefficients. The third-order coefficients agree very well with those obtained by integration of paraxial trajectories and the fifth-order coefficients are given with estimated accuracy ranging from 5% to 15%. Finally, a study of the applicability of the Renau and Heddle approximation [J. Phys. E 19 (1986) 288] is reported.