Third-rank chromatic aberrations of electron lenses

In this paper the third-rank chromatic aberration coefficients of round electron lenses are analytically derived and numerically calculated by Mathematica. Furthermore, the numerical results are cross-checked by the differential algebraic (DA) method, which verifies that all the formulas for the third-rank chromatic aberration coefficients are completely correct. It is hoped that this work would be helpful for further chromatic aberration correction in electron microscopy.