On the residual six-fold astigmatism in DCOR/ASCOR

After the introduction of the hexapole C_s-correctors for scanning transmission electron microscopes (STEM), the next big step forward was the strong reduction of the six-fold astigmatism A₅ by means of an advanced hexapole design (DCOR/ASCOR). As a result all axial aberrations up to fifth order are sufficiently small to allow for large semi-aperture angles beyond 40 mrad for electron energies in the range of 30 to 300 kV without deterioration of the STEM resolution. In this paper we derive simple expressions for the optimum hexapole strength for minimum A₅ and the size of the residual A₅. Both quantities are intrinsic properties of the hexapoles and the transfer lens doublet in between. The optimum hexapole strength scales with the inverse of the electron wavelength, while the residual A₅ does not depend on the electron energy directly, but on the spherical aberration C_s of the pole piece. With the given properties of the DCOR/ASCOR and typical values of C_s in the range of 0.5 to 2.7 mm, at all acceleration voltages A₅ remains in the range from 0.03 to 0.4 mm, the latter even for a large-gap pole piece.