This book (Practical Electron Microscopy and Database) is a reference for TEM and SEM students, operators, engineers, technicians, managers, and researchers.

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It is necessary to eliminate all the secondorder aberrations (highlighted in light orange in Table 3616) before correcting the thirdorder aberrations (highlighted in green in Table 3616), since the secondorder aberrations are larger than the thirdorder aberrations caused by the rotationally symmetric fields.
Table 3616. 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 [3]; 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 
Sextupole elements can be conveniently used to produce threefold symmetry of electromagnetic fields. The course of the deviation of the second order path along the optic axis depends on the arrangements and locations of both the round lenses and the sextupoles. These sextupole elements are normally employed in chargedparticle optics to compensate for the primary secondorder aberrations arising in the systems with a curved optical axis, such as spectrometers or imaging energy filters.
Sextupoles cannot be used to correct spherical aberrations if both the signs of the aberrations generated by sextupoles and round lenses are the same. Unfortunately, this is exactly the case if the secondorder aberrations are eliminated. However, it is possible to reverse the sign of the spherical aberration by employing a round lens doublet. Figure 3616 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 [1].
Figure 3616. Schematic illustration of a simplest sphericalaberration corrector. Adapted from [2].
[1] Born M. and Wolf E., 1975, Principles of Optics (Oxford:
Pergamon Press).
[2] Rose H. H., Optics of highperformance electron microscopes, Sci. Technol. Adv. Mater. 9 (2008) 014107.
[3] O. Scherzer, J. Appl. Phys. 20 (1949) 20.
