Tables for
Volume B
Reciprocal space
Edited by U. Shmueli

International Tables for Crystallography (2006). Vol. B. ch. 2.5, p. 312   | 1 | 2 |

Section Thick crystals

B. K. Vainshteinc Thick crystals

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When the specimen thickness exceeds a certain critical value ([\sim]50–100 Å), the kinematic approximation does not hold true and the scattering is dynamic. This means that on the exit surface of a specimen the wave is not defined as yet by the projection of potential [\varphi (xy) = {\textstyle\int} \varphi ({\bf r}) \hbox{ d}z] ([link], but one has to take into account the interaction of the incident wave [\psi_{0}] and of all the secondary waves arising in the whole volume of a specimen.

The dynamic scattering calculation can be made by various methods. One is the multi-slice (or phase-grating) method based on a recurrent application of formulae ([link] for n thin layers [\Delta z_{i}] thick, and successive construction of the transmission functions [q_{i}] ([link], phase functions [ Q_{i} = {\scr F}q_{i}], and propagation function [p_{k} = [k / 2\pi i \Delta z] \exp [ik (x^{2} + y^{2}) / 2 \Delta z]] (Cowley & Moodie, 1957[link]).

Another method – the scattering matrix method – is based on the solution of equations of the dynamic theory (Chapter 5.2[link] ). The emerging wave on the exit surface of a crystal is then found to diffract and experience the transfer function action [([link], ([link],b[link])].

The dynamic scattering in crystals may be interpreted using Bloch waves: [\Psi\hskip 2pt^{j}({\bf r}) = {\textstyle\sum\limits_{H}} C^{\;j}_{H} \exp (-2\pi i{\bf k}^{\;j}_{H} \cdot {\bf r}). \eqno(] It turns out that only a few (bound and valence Bloch waves) have strong excitation amplitudes. Depending on the thickness of a crystal, only one of these waves or their linear combinations (Kambe, 1982[link]) emerges on the exit surface. An electron-microscopic image can be interpreted, at certain thicknesses, as an image of one of these waves [with a correction for the transfer function action ([link], ([link],b[link])]; in this case, the identical images repeat with increasing thickness, while, at a certain thickness, the contrast reversal can be observed. Only the first Bloch wave which arises at small thickness, and also repeats with increasing thickness, corresponds to the projection of potential [\varphi (xy)], i.e. the atom projection distribution in a thin crystal layer.

An image of other Bloch waves is defined by the function [\varphi ({\bf r})], but their maxima or minima do not coincide, in the general case, with the atomic positions and cannot be interpreted as the projection of potential. It is difficult to reconstruct [\varphi (xy)] from these images, especially when the crystal is not ideal and contains imperfections. In these cases one resorts to computer modelling of images at different thicknesses and defocus values, and to comparison with an experimentally observed pattern.

The imaging can be performed directly in an electron microscope not by a photo plate, but using fast-response detectors with digitized intensity output on line. The computer contains the necessary algorithms for Fourier transformation, image calculation, transfer function computing, averaging, and correction for the observed and calculated data. This makes possible the interpretation of the pattern observed directly in experiment (Herrmann et al., 1980[link]).


First citation Cowley, J. M. & Moodie, A. F. (1957). The scattering of electrons by atoms and crystals. I. A new theoretical approach. Acta Cryst. 10, 609–619.Google Scholar
First citation Herrmann, K. H., Krahl, D. & Rust, H.-P. (1980). Low-dose image recording by TV techniques. In Electron microscopy at molecular dimensions, edited by W. Baumeister & W. Vogell, pp. 186–193. Berlin: Springer-Verlag.Google Scholar
First citation Kambe, K. (1982). Visualization of Bloch waves of high energy electrons in high resolution electron microscopy. Ultramicroscopy, 10, 223–228.Google Scholar

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