Introduction

Bricogne, G.

doi:10.1107/97809553602060000551

International
Tables for
Crystallography
Volume B
Reciprocal space
Edited by U. Shmueli

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International Tables for Crystallography (2006). Vol. B. ch. 1.3, pp. 58-59 | 1 | 2 |

Section 1.3.4.1. Introduction

G. Bricogne^a

^a MRC Laboratory of Molecular Biology, Hills Road, Cambridge CB2 2QH, England, and LURE, Bâtiment 209D, Université Paris-Sud, 91405 Orsay, France

1.3.4.1. Introduction

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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 $[\rho ({\bf X})]$ 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 $[{\bf K}_{0}]$ . Then the far-field amplitude scattered in a direction corresponding to wavevector $[{\bf K} = {\bf K}_{0} + {\bf H}]$ is proportional to $[\eqalign{F({\bf H}) &= {\textstyle\int\limits_{V}} \rho ({\bf X}) \exp (2\pi i{\bf H} \cdot {\bf X}) \;\hbox{d}^{3}{\bf X}\cr &= \bar{\scr F}[\rho]({\bf H})\cr &= \langle \rho_{\bf x}, \exp (2\pi i{\bf H} \cdot {\bf X})\rangle.}]$

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 $[F({\bf H}) = \bar{\scr F}[\rho]({\bf H}) = \langle \rho_{\bf x}, \exp (2\pi i{\bf H} \cdot {\bf X})\rangle.]$ 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.

References

Bragg, W. H. (1915). X-rays and crystal structure. (Bakerian Lecture.) Philos. Trans. R. Soc. London Ser. A, 215, 253–274.Google Scholar

Bragg, W. L. (1975). The development of X-ray analysis, edited by D. C. Phillips & H. Lipson. London: Bell.Google Scholar

Duane, W. (1925). The calculation of the X-ray diffracting power at points in a crystal. Proc. Natl Acad. Sci. USA, 11, 489–493.Google Scholar

Ewald, P. P. (1940). X-ray diffraction by finite and imperfect crystal lattices. Proc. Phys. Soc. London, 52, 167–174.Google Scholar

Ewald, P. P. (1962). Fifty years of X-ray diffraction. Dordrecht: Kluwer Academic Publishers.Google Scholar

Havighurst, R. J. (1925a). The distribution of diffracting power in sodium chloride. Proc. Natl Acad. Sci. USA, 11, 502–507.Google Scholar

Havighurst, R. J. (1925b). The distribution of diffracting power in certain crystals. Proc. Natl Acad. Sci. USA, 11, 507–512.Google Scholar

James, R. W. (1948a). The optical principles of the diffraction of X-rays. London: Bell.Google Scholar

Laue, M. von (1936). Die aüßere Form der Kristalle in ihrem Einfluß auf die Interferenzerscheinungen an Raumgittern. Ann. Phys. (Leipzig), 26, 55–68.Google Scholar

Lipson, H. & Cochran, W. (1953). The determination of crystal structures. London: Bell.Google Scholar

Lipson, H. & Taylor, C. A. (1958). Fourier transforms and X-ray diffraction. London: Bell.Google Scholar

Warren, B. E. (1969). X-ray diffraction. Reading: Addison-Wesley.Google Scholar

Zachariasen, W. H. (1945). Theory of X-ray diffraction in crystals. New York: John Wiley.Google Scholar

International Tables for Crystallography (2006). Vol. B. ch. 1.3, pp. 58-59