Twofold Astigmatism
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Under axial illumination conditions it is not possible to measure all aberration coefficients. In this case, only the defocus and the twofold astigmatism can be determined by the defocus-based techniques. The other aberration coefficients, e.g. spherical aberration, need to be provided by the microscope manufacturer or to be determined by different methods. However, those aberration coefficients, such as the axial coma and the threefold astigmatism, cannot be ignored at high spatial resolutions approaching 0.1 nm. Therefore, the measurements of the other aberration coefficients should be done by other methods such as tilting the illumination. Note that we need to realize that tilting illumination also changes the defocus, astigmatism, and the original aberrations. Table 3678 gives the aberration coefficient nomenclature. The two-fold astigmatisms are marked in blue.

Table 3678. 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 [3]; 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

Note that a single aberration violating the π/4 criterion may also cancel out the other one with the same symmetry if the have different sign, such as a small positive third-order spherical aberration and a small underfocus, or a small positive twofold astigmatism and a small negative third-order star aberration with the same azimuth.

Later on, Haider et al. [1] corrected the aberrations in TEM imaging mode successfully. Figure 3678 shows an example of diffractogram tableaus of the aligned TEM under both uncorrected and corrected conditions. The figures taken under the uncorrected condition demonstrates that the electron beam tilt about the coma-free pivot point introduces primarily a defocus and an axial twofold astigmatism. The strength of these aberrations does not depend on the azimuthal direction of incident beam due to the rotational symmetry of the aligned TEM. After the aberration correction all diffractograms in the tableau exhibit approximately the same appearance revealing the properties of an aplanatic lens. In this case the illumination tilt does neither introduce defocus nor two-fold astigmatism confirming the correction of spherical aberration.

Figure 3678. Diffractograms obtained in an aligned microscope without (a) and with (b) correction of the spherical aberration. τ stands for tilt angles. [1]

[1] Maximilian Haider, Harald Rose, Stephan Uhlemann, Bernd Kabius and Knut Urban, Towards 0.1 nm resolution with the first spherically corrected transmission electron microscope, Journal ofElectron Microscopy 47(5): 395-405 (1998).

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