Abstract
Advances in electron optics have made it possible to form electron beams of sub-nanometer diameters, and these beams have enabled high-resolution imaging methods with incoherent scattering at high angles. The origin of this incoherence is discussed. Some practical issues of high-angle annular dark field imaging are presented, and the fortunate effect of electron channeling along ion cores is explained. One of the enabling technologies for this work is an aberration correction system. Methods of Cs correction are described, one in detail. The various higher-order aberrations are classified and enumerated. Some examples of atomic-scale imaging and chemical analysis of individual atoms are shown, with comments on future directions.
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Notes
- 1.
Another viewpoint is that the electron is coherently imaged into a small object onto the specimen, and the smallness of this point in real space depends on having a large range of Δk passed coherently by the objective lens.
- 2.
Recall the argument of Sect. 2.4.2—different electrons cannot mutually interfere because they are fermions. Furthermore, rarely are two electrons present simultaneously at the sample.
- 3.
This is analogous to how the index of refraction in a glass fiber is larger than in the surrounding air, allowing a light ray to be trapped in the optical fiber by total internal reflection.
- 4.
- 5.
The analysis is known as “first-order degenerate perturbation theory.” It is used to find the Bloch waves of Chap. 13, which are combinations of diffracted beams. It is also used in physical chemistry to find bonding and antibonding orbitals that differ in energy by ±E, where E is the matrix element coupling the two atomic states.
- 6.
Substituting the elementary result for a harmonic oscillator, \(\omega = \sqrt{k/M}\), gives the potential energy of a fully compressed spring, E therm=1/2k〈u 2〉.
- 7.
Recall that the coherent (Bragg) scattering is diminished by the factor D(Δk), whereas the thermal diffuse scattering grows as 1−D(Δk).
- 8.
Here is a more physical argument. Recall that the electron wavefunction interferes coherently with itself in making a HRTEM image, especially between the forward beam and the diffracted beams, requiring accurate phase relationships between all these beams. The image is made up of many such interferences between different electrons, so all electrons should have the same phase contrast conditions if the image is to have sharp detail. For HAADF imaging, the incoherent scattering depends on the presence of one real electron at a point in space, and the electron density is less sensitive to the fluctuations of electron phase caused by microscope instabilities.
- 9.
A more powerful reciprocity theorem exists, however. It equates the amplitudes (and hence phases) of the waves between interchanged source and detector, not just their intensities. This originates with the symmetry of incident and scattered wavefunctions in (4.73), for example.
- 10.
This can be derived by recognizing that α=Δk/k, and substituting Δk=2πα/λ into lΔk=π of (10.159).
- 11.
Recall that distance, d=1/2at 2 and a=F/m, where a is acceleration and t is time allowed for acceleration. The time, t, is assumed the same for paths through the lens, or at least for all positions along a line like the one selected in Fig. 12.7b.
- 12.
With a bare minimum of 6 lenses, the multiple degrees of freedom of the hexapole lens currents, and a high degree of accuracy needed for compensating mechanical misalignments, selecting the optimal lens currents for the C s corrector of Fig. 12.8 is not simple.
- 13.
For example, a second-order spherical aberration coefficient C 20 would be circularly-symmetric by m=0, but this is inconsistent with a cubic dependence on Δk, where inversion across the optic axis requires a change in sign.
- 14.
This causes some loss of intensity, but electron monochromatization is of special interest for improving the energy resolution of EELS spectrometry.
- 15.
An “interband transition” refers to an electron excitation where the initial state is in the valence band, and the final state is in the conduction band.
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Fultz, B., Howe, J. (2013). High-Resolution STEM and Related Imaging Techniques. In: Transmission Electron Microscopy and Diffractometry of Materials. Graduate Texts in Physics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-29761-8_12
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DOI: https://doi.org/10.1007/978-3-642-29761-8_12
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