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Three-fold Symmetry of Electromagnetic Fields
- Practical Electron Microscopy and Database -
- An Online Book -
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https://www.globalsino.com/EM/
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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.
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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 |
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D5 |
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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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