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

International Tables for Crystallography (2010). Vol. B, ch. 1.2, p. 10   | 1 | 2 |

Section 1.2.3. Scattering by a crystal: definition of a structure factor

P. Coppensa*

aDepartment of Chemistry, Natural Sciences & Mathematics Complex, State University of New York at Buffalo, Buffalo, New York 14260–3000, USA
Correspondence e-mail: coppens@buffalo.edu

1.2.3. Scattering by a crystal: definition of a structure factor

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In a crystal of infinite size, [\rho ({\bf r})] is a three-dimensional periodic function, as expressed by the convolution [\rho_{\rm crystal} ({\bf r}) = {\textstyle\sum\limits_{n}} {\textstyle\sum\limits_{m}} {\textstyle\sum\limits_{p}} \rho_{\rm unit\ cell} ({\bf r})\ast \delta ({\bf r} - n{\bf a} - m{\bf b} - p{\bf c}), \eqno(1.2.3.1)]where n, m and p are integers, and δ is the Dirac delta function.

Thus, according to the Fourier convolution theorem, [A({\bf S}) = \hat{F} \{\rho ({\bf r})\} = {\textstyle\sum\limits_{n}} {\textstyle\sum\limits_{m}} {\textstyle\sum\limits_{p}} \hat{F} \{\rho_{\rm unit\ cell} ({\bf r})\} \hat{F} \{\delta ({\bf r} - n{\bf a} - m{\bf b} - p{\bf c})\}, \eqno(1.2.3.2)]which gives [A({\bf S}) = \hat{F} \{\rho_{\rm unit\ cell} ({\bf r})\} {\textstyle\sum\limits_{h}} {\textstyle\sum\limits_{k}} {\textstyle\sum\limits_{l}} \delta ({\bf S} - h{\bf a}^{*} - k{\bf b}^{*} - l{\bf c}^{*}). \eqno(1.2.3.3)]

Expression (1.2.3.3)[link] is valid for a crystal with a very large number of unit cells, in which particle-size broadening is negligible. Furthermore, it does not account for multiple scattering of the beam within the crystal. Because of the appearance of the delta function, (1.2.3.3)[link] implies that S = H with [{\bf H} =h{\bf a}^{*} \ +] [k{\bf b}^{*} + l{\bf c}^{*}].

The first factor in (1.2.3.3)[link], the scattering amplitude of one unit cell, is defined as the structure factor F: [F({\bf H}) = \hat{F} \{\rho_{\rm unit\ cell} ({\bf r})\} = {\textstyle\int_{\rm unit\ cell}} \rho ({\bf r}) \exp (2\pi i{\bf H}\cdot {\bf r}) \hbox{ d}{\bf r}. \eqno(1.2.3.4)]








































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