Tables for
Volume B
Reciprocal space
Edited by U. Shmueli

International Tables for Crystallography (2010). Vol. B, ch. 1.2, p. 22   | 1 | 2 |

Section 1.2.13. The generalized 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:

1.2.13. The generalized structure factor

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In the generalized structure-factor formalism developed by Dawson (1975)[link], the complex nature of both the atomic scattering factor and the generalized temperature factor are taken into account. We write for the atomic scattering factor: [\eqalignno{f_{j} ({\bf H}) &= f_{j, \, {\rm c}} ({\bf H}) + if_{j, \, {\rm a}} ({\bf H}) + f^{'}_{j} + if^{''}_{j} &(\cr T_{j} ({\bf H}) &= T_{j, \, {\rm c}} ({\bf H}) + iT_{j, \,{\rm a}} ({\bf H}) &(}]and [F({\bf H}) = A({\bf H}) + iB({\bf H}), \eqno(]where the subscripts c and a refer to the centrosymmetric and noncentrosymmetric components of the underlying electron distribution, respectively. Substitution in ([link][link][link][link] gives for the real and imaginary components A and B of [F({\bf H})] [\eqalignno{A ({\bf H}) &= {\textstyle\sum\limits_{j}} (f_{j, \, {\rm c}} + f^{'}_{j}) [\cos (2\pi {\bf H} \cdot {\bf r}_{j}) T_{\rm c} - \sin (2\pi {\bf H} \cdot {\bf r}_{j}) T_{\rm a}]\cr &\quad - (f_{j, \, {\rm a}} + f^{''}_{j}) [\cos (2\pi {\bf H} \cdot {\bf r}_{j}) T_{\rm a} + \sin (2\pi {\bf H} \cdot {\bf r}_{j}) T_{\rm c}]\cr&&(}]and [\eqalignno{B ({\bf H}) &= {\textstyle\sum\limits_{j}} (f_{j, \, {\rm c}} + f^{'}_{j}) [\cos (2\pi {\bf H} \cdot {\bf r}_{j}) T_{\rm a} + \sin (2\pi {\bf H} \cdot {\bf r}_{j}) T_{\rm c}]\cr &\quad + (f_{j, \, {\rm a}} + f^{''}_{j}) [\cos (2\pi {\bf H} \cdot {\bf r}_{j}) T_{\rm c} - \sin (2\pi {\bf H} \cdot {\bf r}_{j}) T_{\rm a}]\cr& &(}](McIntyre et al., 1980[link]; Dawson, 1967[link]).

Expressions ([link][link] illustrate the relation between valence-density anisotropy and anisotropy of thermal motion.


Dawson, B. (1967). A general structure factor formalism for interpreting accurate X-ray and neutron diffraction data. Proc. R. Soc. London Ser. A, 248, 235–288.
Dawson, B. (1975). Studies of atomic charge density by X-ray and neutron diffraction – a perspective. In Advances in Structure Research by Diffraction Methods. Vol. 6, edited by W. Hoppe & R. Mason. Oxford: Pergamon Press.
McIntyre, G. J., Moss, G. & Barnea, Z. (1980). Anharmonic temperature factors of zinc selenide determined by X-ray diffraction from an extended-face crystal. Acta Cryst. A36, 482–490.

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