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The 6fold astigmatism A_{6} in double hexapole (or dodecapole) systems is given by [2],
 [2239]
where,
f_{0}  The focal length of the objective lens;
M  The (de)magnification for (S)TEM between
the corrector and the objective lens;
C_{s}  The original spherical aberration
coefficient of the objective lens;
z  The multipole length.
The effect of A_{6} astigmatism is more critical at low accelerating voltages than at high voltages because the focal length at low voltages is much smaller than that at high voltages.
Two main methods for compensating for the A_{6} have been proposed. Müller et al. optimized the hexapole length for the compensation [1]. The A_{6} astigmatism generated by the first and second hexapoles is successfully cancelled out by the one from the combination of thirdorder aberrations of the transfer lenses and the 3fold astigmatism introduced by the hexapole elements. Sasaki et al. proposed a delta corrector [2  4], consisting of a triple dodecapole system, to compensate for the positive A_{6} (generated by the first and second dodecapoles) with a negative A_{6} (from the second and third dodecapoles).
[1] Müller H, Uhlemann S, Hartel P, and Haider M (2006) Advancing the
hexapole Cscorrector for the scanning transmission electron microscope.
Microsc. Microanal. 12: 442–455.
[2]
Sawada H, Sasaki T, Hosokawa F, Yuasa S, Terao M, Kawazoe M,
Nakamichi T, Kaneyama T, Kondo Y, Kimoto K, and Suenaga K
(2009) Correction of higher order geometrical aberration by triple
3fold astigmatism field. J. Electron Microsc. 58: 341–347.
[3]
Sawada H, Sasaki T, Hosokawa F, Yuasa S, Terao M, Kawazoe M,
Nakamichi T, Kaneyama T, Kondo Y, Kimoto K, and Suenaga K
(2010) Higherorder aberration corrector for an imageforming system
in a transmission electron microscope. Ultramicroscopy, doi:10.1016/j.
ultramic.2010.01.010.
[4]
Takeo Sasaki, Hidetaka Sawada, Fumio Hosokawa, Yuji Kohno, Takeshi Tomita, Toshikatsu Kaneyama, Yukihito Kondo, Koji Kimoto, Yuta Sato, and Kazu Suenaga, Performance of lowvoltage STEM/TEM with delta corrector and cold field emission gun, Journal of Electron Microscopy 59(Supplement): S7–S13 (2010).
