“spherical aberration” has been improved in the Glossary of TEM Terms (Figures have been added).

The spherical aberration is the third-order geometrical aberration with axial symmetry. Among the geometrical aberrations of the objective lens, the spherical aberration is the most significant aberration to degrade the resolution of high-resolution images. Although the fifth-order and higher-order (odd-order) spherical aberrations exist, only the third-order spherical aberration, which has a significant effect on the image, has been discussed in electron microscopy.

Spread (Blur) of the electron beam due to spherical aberration

When spherical aberration is present, electron beams exiting from the point on the optical axis at the object plane with an angle to the optical axis pass through the lens, do not meet at the point on the optical axis at the ideal image plane, but intersect the optical axis at an upper plane than the ideal image plane (at the lens side). As a result, these electron beams produce a circularly spread image (circular blur) on the ideal image plane. The amount of the spread (blur) is given by the product of the spherical aberration coefficient Cs and cube of the angle α, Csα³ (α is the angle between an electron beam and the optical axis). The value of Cs is always positive for a rotationally symmetric lens with respect to the optical axis.

In the case of TEM

In TEM observation, parallel electron beams are illuminated on a specimen placed near the center of the objective lens. The angular range over which an image is formed with no aberration is limited by the spherical aberration of the post-magnetic field of the objective lens (below the specimen), and the electron beams running at larger angles than the angle for aberration-free imaging produce a spread image (an image blur) on the ideal image plane (Fig.(a)). The angular range with no effect of spherical aberration is given by the position (angle) of the First Zero in the Fourier transformed pattern obtained from a high-magnification TEM image of an amorphous thin-film taken at the Scherzer focus condition. In the example of Fig. (b), the angle of the First Zero is 13 mrad at an accelerating voltage of 200 kV, and the spatial resolution corresponding to the image space is 0.19 nm.

Measurement of spherical aberration

Because of the spherical aberration in the post-field of the objective lens, when the incident electron beam onto the specimen is tilted against the optical axis, two-fold astigmatism and focal shift appear. The larger the tilt angle, the larger the focal shift and two-fold astigmatism. The spherical aberration coefficient Cs can be obtained by measuring the tilt angle dependence on the amounts of the focal shift and two-fold astigmatism. The amount of these aberrations is practically measured using a set of diffractograms (Diffractogram Tableau) (in Fig.(c)), which are taken by varying the tilt and azimuth angles of the electron beam.

In the case of STEM

In STEM observation, a focused electron beam is illuminated onto a specimen placed near the center of the objective lens. Due to the spherical aberration of the pre-field of the objective lens (in front of the specimen), the angular range of the electron beams which meet at a single point on the specimen is limited, and the electron beams with angles larger than the angle for aberration-free probe formation, produce a spread probe (Fig.(d)). The angular region where spherical aberration does not contribute is given as a region of uniform intensity in the electron wave interference pattern (Ronchigram). When acquiring a STEM image, the electron beam is illuminated onto the specimen by selecting the beams of the aberration-free angular region using the condenser aperture. In the example shown in Fig. (e), the angle with no spherical aberration is 11 mrad at an accelerating voltage of 200 kV, and the corresponding probe size in real space is 0.14 nm.

Corrections of the third-order spherical aberration and fifth-order spherical aberration

Nowadays, the (third-order) spherical aberration has been successfully corrected mainly by the use of Cs correctors that combine hexapoles and transfer lenses. After correction of the third-order spherical aberration, the fifth-order spherical aberration is subjected to be corrected at high angles. The Cs correctors mentioned can also correct the fifth-order spherical aberration.

(a) Ray diagram for TEM, where spherical aberration below a specimen placed in the objective lens is taken into account. Electron beams exiting from the specimen and entering into the post-field of the objective lens at an angle α with respect to the optical axis, are more strongly deflected due to the spherical aberration than the electron beams with no aberration. The image magnified by the objective lens is further magnified by the intermediate lens and the projection lens to form the final image on the screen.
The electron beams that exit from a single point on the specimen do not meet at the single point on the screen but create a spread image (image blur) due to spherical aberration. The amount of the image spread is given by Csα³ (Cs: spherical aberration coefficient) in terms of the specimen plane.

(b) Fourier transformed (diffraction) pattern (diffractogram) of an amorphous thin-film image obtained using an objective lens with a (third-order) spherical aberration coefficient Cs of 0.5 mm at an accelerating voltage of 200 kV under the Scherzer focus condition (condition that gives optimum spatial resolution in the presence of spherical aberration). The angle of the First Zero that determines the resolution is indicated by a dotted circular arc.

(c) Diffractogram Tableau (a set of diffractograms) obtained using an imaging system with spherical aberration. The incident electron beam is tilted up to 10 mrad against the specimen. As the tilt angle increases, each pattern becomes more elliptical due to spherical aberration. For spherical aberration, the Diffractogram Tableau is point-symmetric with respect to the center of the Tableau.

(d) Ray diagram for STEM, where spherical aberration above a specimen placed in the objective lens is taken into account. In STEM, the electron source is demagnified by the condenser lens and the objective lens to create a small probe with a size less than 1 nm, and by scanning the probe over the specimen with a scan coil the image of the specimen is formed. Electron beams that exit from the electron source and enter into the pre-field of the objective lens at an angle α with respect to the optical axis, are more strongly deflected due to the spherical aberration than the electron beams with no aberration. That is, the electron beams that exit from a point on the electron source do not meet at a single point on the specimen but create a spread probe. The size of the probe is given by Csα³, which determines the resolution of the STEM image.

(e) Ronchigram pattern of an amorphous thin-film obtained using an objective lens with the (third-order) spherical aberration coefficient Cs of 0.5 mm at an accelerating voltage of 200 kV. The angular region of uniform intensity in the figure indicates the region where the electron waves are in phase (shown by an arrow). The electron beams within this angular range do not spread on the specimen and meet at a single point. On the other hand, the electron beams outside this angular range cause a probe spread on the specimen.

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