International
Tables for Crystallography Volume B Reciprocal space Edited by U. Shmueli © International Union of Crystallography 2006 |
International Tables for Crystallography (2006). Vol. B. ch. 1.3, pp. 58-59
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The central role of the Fourier transformation in X-ray crystallography is a consequence of the kinematic approximation used in the description of the scattering of X-rays by a distribution of electrons (Bragg, 1915; Duane, 1925; Havighurst, 1925a,b; Zachariasen, 1945; James, 1948a, Chapters 1 and 2; Lipson & Cochran, 1953, Chapter 1; Bragg, 1975).
Let be the density of electrons in a sample of matter contained in a finite region V which is being illuminated by a parallel monochromatic X-ray beam with wavevector . Then the far-field amplitude scattered in a direction corresponding to wavevector is proportional to
In certain model calculations, the `sample' may contain not only volume charges, but also point, line and surface charges. These singularities may be accommodated by letting ρ be a distribution, and writing F is still a well behaved function (analytic, by Section 1.3.2.4.2.10) because ρ has been assumed to have compact support.
If the sample is assumed to be an infinite crystal, so that ρ is now a periodic distribution, the customary limiting process by which it is shown that F becomes a discrete series of peaks at reciprocal-lattice points (see e.g. von Laue, 1936; Ewald, 1940; James, 1948a p. 9; Lipson & Taylor, 1958, pp. 14–27; Ewald, 1962, pp. 82–101; Warren, 1969, pp. 27–30) is already subsumed under the treatment of Section 1.3.2.6.
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