Kinematical theory of electron diffraction

We now consider the diffraction of electrons by a single crystal in terms of the so-called kinematical theory. Although this theory has serious limitations, it is useful in practice under certain conditions, and it also provides an introduction to the more satisfactory dynamical theory, which we develop in Chapter 4. 

In the kinematical theory, we consider the diffraction of a plane wave (of wavelength X) incident upon a three-dimensional lattice array of identical scattering points, each of which consists of a group of atoms and acts as the center of a spherical scattered wave. Our problem is to find the combined effect of the scattered waves at a point outside the crystal, at a distance from the crystal that is large compared with its linear dimensions. In developing the theory, we make several important assumptions  

1 There is no attenuation of the incident wave in the crystal so that the incident wave has the same amplitude at each scattering point. This is equivalent to neglecting any interaction between the incident wave in the crystal and the scattered waves. 

2 Each scattered wave travels through the crystal without being rescattered by other scattering points. 

3 There is no absorption of either the incident or the scattered waves in the crystal. 

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In developing the kinematical theory of diffraction in Chapter 3, we assumed that a beam of electrons has wavelike characteristics and can be described by an equation of the form... 

Hirsch, P. B., Howie, A., Whelan, M. J. (1960). A kinematical theory of diffraction contrast of electron transmission microscope images of dislocations and other defects. Phil. Trans. Roy. Soc. (London), A252, 499-529. 

Diffraction by an ideal mosaic crystal is best described by a kinematical theory of diffraction, whereas diffraction by an ideal crystal is dynamical and can be described by a much more complex theory of dynamical diffraction. The latter is used in electron diffraction, where kinematical theory does not apply. X-ray diffraction by an ideal mosaic crystal is kinematical, and therefore, this relatively simple theory is used in this book. The word "mosaie" describes a crystal that consists of many small, ideally ordered blocks, which are slightly misaligned with respect to one another. "Ideal mosaic" means that all blocks have the same size and degree of misalignment with respect to other mosaic blocks. Most of this chapter is dedicated to conventional crystallographic symmetry, where three-dimensional periodicity is implicitly assumed. 

When investigating polycrystalline materials with disordered or partially ordered orientation, ring diagrams or texture diagrams respectively, are obtained. In these cases, there is sufficient conformity with the prediction of the kinematical theory. Therefore, the texture diagram is the most important type for structure analysis with electrons. Vainshtein, Pinsker and co-workers [2-7], as well as Cowley [8, 9] evaluated the basis, about 50 years ago, of the determination of crystal structure by means of electron diffraction. 

The reason lies in the non-applicability of the kinematical approximation (theory of single scattering) in the case of diffraction of electrons by single crystals. This, in comparison with X-ray quanta, is due to the much stronger scattering of the electrons at the lattice atoms and the consequent very high intensity of the singly and multiply diffracted electrons. 

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