Three-fold Symmetry of Electromagnetic Fields
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This book (Practical Electron Microscopy and Database) is a reference for TEM and SEM students, operators, engineers, technicians, managers, and researchers.


The primary aberrations of the TEM systems with three-fold symmetry are of second order (highlighted in light orange in Table 3617). It is necessary to eliminate all the second-order aberrations before correcting the third-order aberrations (highlighted in green in Table 3617), since the second-order aberrations are larger than the third-order aberrations caused by the rotationally symmetric fields.

Table 3617. 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)
C0,1 A0 0 1 1 Image Shift
C1,2 A1 1 2 2 Two-fold axial astigmatism (or axial astigmatism of the 1st order)
C1,0 C1 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)
C2,3 A2 2 3 3 Three-fold axial astigmatism (or axial astigmatism of the 2nd order)
C2,1 B2 2 3 1 Axial coma
C3,4 A3   4 4 Four-fold axial astigmatism or axial astigmatism of the 3rd order Cs
C3,2 B3 4 2

Twofold astigmatism of Cs (or Third order twofold astigmatism, or Axial star aberration of the 3rd order)

C3,0 C3 4 0, ∞ Third-order spherical aberration (always positive for round lenses [1]; Real numbers and describing rotationally symmetric contributions to the wave aberration) (alt: Cs )
C4,5 A4   5 5 Five-fold axial astigmatism or axial astigmatism of the 4th order
C4,1 B4   5 1 Fourth-order axial coma
C4,3 D4 4 5 3

Fourth order threefold astigmatism (or Three lobe aberration)

C5,6 A5   6 6 Six-fold axial astigmatism or sixfold axial astigmatism of the 5th order
C5,4 R5 5 6 4 Fourfold astigmatism of C5 (or Fifth order rosette aberration)
C5,2 S5 5 6 2

Twofold astigmatism of C5 (or Fifth-order axial star aberration)

C5,0 C5   6 0, ∞ Fifth-order spherical aberration
  D5       Four lobe aberration of the 5th order

Sextupole elements can be conveniently used to produce three-fold symmetry of electromagnetic fields. These elements are normally employed in charged-particle optics to compensate for the primary second-order aberrations arising in the systems with a curved optical axis, such as spectrometers or imaging energy filters.





[1] O. Scherzer, J. Appl. Phys. 20 (1949) 20.




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