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Thirdorder spherical aberration is one of the dominant geometric aberrations that limit the spatial resolution of EMs (electron microscopes). Figure 3654 shows the octupole design, normally used for spherical aberration (C_{s}) correction. The optical axis, along which the electrons travel, is into the page. indicates the direction of the electron beam. The blue arrowed curves represent the magnetic field lines, while the red and plum arrowed lines represent the Lorentz force on the electrons. From Equation 4313, we can know the field in the octupole system can be given by,
 [3654]
Namely, it varies by r^{3}, which acts directly as third order aberrations, increasing with distance from the optical axis. Note that multipoles of higherorder than 3 do not produce 3rdorder spherical aberrations at all.
Figure 3654. The octupole design.
The plum
arrows show focusing
function, while the red arrows show defocusing function.
Table 3654 lists the aberration coefficient nomenclature, their order (of both the ray deviation (N) and the wavefront), and radial (azimuthal) symmetry, as well as highlights the thirdorder spherical aberration in blue. This table also compares both commonly used notations [1, 2].
Table 3654. 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 
The 6fold astigmatism (A_{6}) generated by the first and second hexapoles in a C_{s} corrector can be cancelled out by the one from the combination of thirdorder aberrations of the transfer lenses and the 3fold astigmatism introduced by the hexapole elements. [4]
[1] Krivanek, O. L., Dellby, N., and Lupini, A. R. (1999). Towards subÅ electron beams.
Ultramicroscopy 78, 1.
[2] Typke, D., and Dierksen, K. (1995). Determination of image aberrations in highresolution
electron microscopy using diffractogram and cross correlation methods. Optik 99, 155.
[3] O. Scherzer, J. Appl. Phys. 20 (1949) 20.
[4] 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.
