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Axial coma is an image aberration that is introduced when the
illumination is not parallel to the optical axis of the objective lens and is found mostly at high magnifications and high resolutions in TEM observations. The effect of coma aberration was initially demonstrated by Zemlin using a diffractogram tableau method that is a set of the diffractograms obtained by rotating the azimuth of the beam tilt direction [2]. Unlike the effects of defocus and spherical aberration, the effect of coma generally cannot be represented in terms of contrast transfer function.
A specific tilt angle of the electron beam can be found such that the image coma becomes negligible for a chosen point in the TEM specimen. In this case, the position of the unscattered (transmitted) electron beam, focused in the back focal plane of the objective lens, lies on the optical axis. More accurately speaking, one refers to this axis as the comafree axis for the given point in the specimen, but one normally refers to the axis defined by comafree alignment as being optical axis.
In the comafree aligned case, where the unscattered electrons are along the comafree axis as shown in Figure 4326a (a), the phasedistortion function (or called phase shift) has the same value on the opposite sides of the scattered (diffracted) waves when the incident illumination is tilted relative to the optical axis of the objective lens. In Figure 4326a (b), the origin of the phase distortion function is shifted relative to the origin of the structure factor when the illumination is tilted, and thus the respective phase shifts at spatial frequencies s and s are no longer the same for the pair of interference fringes that is generated by Friedel mates F(s) and F(s).
Figure 4326a. Phasedistortion functions in comafree aligned (a) and unaligned (b) cases.
The spatial resolution is reduced in one direction if axial coma exists. If the intensity distribution in the pattern is not symmetric, a coma component in the defocus is probably present. In coma case, most of its intensity distributes on one side and a “comet tail” on the other as shown in Figure 4326b.
Figure 4326b. Axial coma. Adapted from [1]
Table 4326 gives the aberration coefficient nomenclature. The coma aberrations are marked in blue.
Table 4326. 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)

C_{2,1} 
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 
It is interesting to mention that ideally the aplanatic lenses, which can in some cases be condenser or objective lenses, are free of spherical aberration and offaxial coma simultaneously.
[1] Joachim Zach and Maximilian Haider, Aberration correction in a low voltage SEM by a multipole corrector, Nuclear Instruments and Methods in Physics Research A 363 (19953 316325.
[2] F. Zemlin, K. Weiss, P. Schiske, W. Kunath, K.H.
Herrmann, Ultramicroscopy 3 (1978) 49.
