Tables for
Volume B
Reciprocal space
Edited by U. Shmueli

International Tables for Crystallography (2006). 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:

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(] 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(] 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(]

Expression ([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, ([link] implies that [{\bf S} = {\bf H}] with [{\bf H} = h{\bf a}^{*} + k{\bf b}^{*} + l{\bf c}^{*}].

The first factor in ([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(]

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