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In aberration correctors, multipole aberration corrections produce electron beam distortions of welldefined angular exponent and azimuthal symmetry because the inaccuracies of EM (electron microscope) construction practically prevent from ideal performance, and even a perfect EM system may experience changes of mechanical, electrical and materials properties after the construction. For instance, Nion quadrupoleoctupole 3^{rd} order corrector [1] introduces beam distortions consisting of 1^{st} and 3^{rd} order, 2fold and 4fold symmetry.[2, 3] Therefore, parasitic aberrations, not depending on the fundamental symmetries of the manufactured EM systems can essentially limit the corrector performance at any order or azimuthal symmetry. In the quadrupoleoctupole systems, parasitic aberrations normally include sextupolelike 4^{th} order types, having 1, 3, and 5fold symmetries (aberration coefficients C_{4,1,a}, C_{4,1,b}, C_{4,3,a}, C_{4,3,b}, C_{4,5,a}, and C_{4,5,b}), which are highlighted in green in Table 3650 and are indicated by the examples in Figure 3650.
Table 3650. 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 
Figure 3650. Aberration coefficients of C_{4,1,a}, C_{4,3,a}, and C_{4,5,a}.
[1] N. Dellby, O.L. Krivanek, P.D. Nellist, P.E. Batson, and A.R. Lupini, J. Electron Microscopy, 50 (2001)177  185.
[2] P.W. Hawkes and E. Kasper, Principles of Electron Optics (Academic Press, London, 1994).
[3] A.R. Lupini, Dissertation (Cambridge University, Cambridge, U.K., 2001).
[4] O. Scherzer, J. Appl. Phys. 20 (1949) 20.
