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In conventional highresolution TEM, the fixed large value of the thirdorder spherical aberration imposes limit to the contrast transfer of thin objects. However, it is necessary to eliminate all the secondorder aberrations (highlighted in light orange in Table 3618) before correcting the thirdorder aberrations (highlighted in green in Table 3618), since the secondorder aberrations are larger than the thirdorder aberrations caused by the rotationally symmetric fields.
Table 3618. Aberration Coefficient Nomenclature. The aberration coefficients have two
main types of notations, namely Krivanek notation, and Typke and Dierksen notation.
Krivanek notation 
Typke and Dierksen notation 
Radial Order 
Azimuthal Symmetry 
Nomenclature 
Ray 
Wave (k) 
C_{0,1} 
A_{0} 
0 
1 
1 
Image Shift 
C_{1,2} 
A_{1} 
1 
2 
2 
Twofold axial astigmatism (or axial astigmatism of the 1st order) 
C_{1,0} 
C_{1} 
1 
2 
0, ∞ 
Defocus (overfocus positive, or spherical aberration of the 1st order; Real numbers and describing rotationally symmetric contributions to the wave aberration) (alt: Δf) 
C_{2,3} 
A_{2} 
2 
3 
3 
Threefold axial astigmatism (or axial astigmatism of the 2nd order)

_{C2,1} 
B_{2} 
2 
3 
1 
Axial coma 
_{C3,4} 
A_{3} 

4 
4 
Fourfold axial astigmatism or axial astigmatism of the 3rd order C_{s} 
C_{3,2} 
B_{3} 

4 
2 
Twofold astigmatism of C_{s} (or Third order twofold astigmatism, or Axial star aberration of the 3rd order) 
C_{3,0} 
C_{3} 

4 
0, ∞ 
Thirdorder spherical aberration (always positive for round lenses [4]; Real numbers and describing rotationally symmetric contributions to the wave aberration) (alt: C_{s} ) 
C_{4,5} 
A_{4} 

5 
5 
Fivefold axial astigmatism or axial astigmatism of the 4th order 
C_{4,1} 
B_{4} 

5 
1 
Fourthorder axial coma 
C_{4,3} 
D_{4} 
4 
5 
3 
Fourth order threefold astigmatism (or Three lobe aberration) 
C_{5,6} 
A_{5} 

6 
6 
Sixfold axial astigmatism or sixfold axial astigmatism of the 5th order 
C_{5,4} 
R_{5} 
5 
6 
4 
Fourfold astigmatism of C_{5 }(or Fifth order rosette aberration) 
C_{5,2} 
S_{5} 
5 
6 
2 
Twofold astigmatism of C_{5} (or Fifthorder axial star aberration) 
C_{5,0} 
C_{5} 

6 
0, ∞ 
Fifthorder spherical aberration 

D_{5} 



Four lobe aberration of the 5th
order 
The secondary effects of sextupoles produce spherically symmetric 3rdorder aberrations if paraxial optics is rotationally symmetric. These secondary aberrations depend quadratically on the sextupole strength. Therefore, magnetic hexapoles combined with transfer round lenses [1] are used for thirdorder spherical aberration correction at high accelerating voltages in TEM and STEM. However, the image quality can be improved by this correction only if the primary 2ndorder aberrations of the sextupoles are eliminated as well and if the 4thorder aberrations can be kept adequately small.
Figure 3618 shows the simplest system which can eliminate all secondorder fundamental rays and thus all the secondorder aberrations outside of the corrector. It consists of a round lens doublet and two identical sextupoles. The outer focal points of the corrector are the same as the nodal points N_{1} and N_{2} of the round lens doublet. The coils of the round lenses are connected oppositely in order to avoid an image rotation from the first sextupole, which is independent of the current strength. Therefore, the doublet images the front sextupole with magnification of −1 exactly onto the second sextupole centered about the nodal point N_{2} without introducing an offaxial thirdorder coma. This system can correct the thirdorder spherical aberrations of electron microscopes [2].
Figure 3618. Schematic illustration of a simplest sphericalaberration corrector. Adapted from [3].
Note that a single aberration violating the π/4 criterion may also cancel out the other one with the same symmetry if the have different sign, such as a small positive thirdorder spherical aberration and a small underfocus, or a small positive twofold astigmatism and a small negative thirdorder star aberration with the same azimuth.
[1] Haider, M., Braunshausen, G. and Schwan, E. 1995. Correction of the spherical aberration of a 200 kV TEM
by means of a hexapolecorrector, Optik, 99, 167–179.
[2] Born M. and Wolf E., 1975, Principles of Optics (Oxford:
Pergamon Press).
[3] Rose H. H., Optics of highperformance electron microscopes, Sci. Technol. Adv. Mater. 9 (2008) 014107.
[4] O. Scherzer, J. Appl. Phys. 20 (1949) 20.
