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

International Tables for Crystallography (2010). Vol. B, ch. 4.6, p. 603   | 1 | 2 |

Section 4.6.3.2.3. Structure factor

W. Steurera* and T. Haibacha

aLaboratory of Crystallography, Department of Materials, ETH Hönggerberg, HCI G 511, Wolfgang-Pauli-Strasse 10, CH-8093 Zurich, Switzerland
Correspondence e-mail:  walter.steurer@mat.ethz.ch

4.6.3.2.3. Structure factor

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The structure factor [F({\bf H})] of a composite structure consists of the weighted contributions of the subsystem structure factors [F_{\nu}({\bf H}_{\nu})]:[\eqalign{F({\bf H}) &= \textstyle\sum\limits_{\nu}| J_{\nu} |F_{\nu}({\bf H}_{\nu})\semi\cr F_{\nu}({\bf H}) &= \textstyle\sum\limits_{(R^{\nu}, \, {\bf t}^{\nu})}^{\infty} \textstyle\sum\limits_{k = 1}^{N'} \textstyle\int\limits_{0}^{1} \hbox{d}\bar{x}_{4, \, k}^{\nu}\ldots \textstyle\int\limits_{0}^{1} \hbox{d}\bar{x}_{3 + d, \, k}^{\nu} f_{k}^{\nu} ({\bf H}^{\parallel})p_{k}^{\nu}\cr &\quad \times \exp {\left(- \textstyle\sum\limits_{i, \, j = 1}^{3 + d} h_{i}^{\nu} [R^{\nu} B_{ijk}^{\nu}R^{\nu T}]h_{j}^{\nu}+\,2\pi i \textstyle\sum\limits_{j = 1}^{3 + d} h_{j}^{\nu} R^{\nu} x_{jk}^{\nu} + h_{j}^{\nu} t_{j}^{\nu}\right),}}]with coefficients similar to those for IMSs.

The weights are the Jacobians of the transformations from [{\bf t}_{\nu}] to t, and [{\bf H}_{\nu}] are the reflection indices with respect to the subsystem Fourier modules [M_{\nu}^{*}] (van Smaalen, 1995[link], and references therein). The relative values of [|J_{\nu}|], where [J_{\nu} = \det \left[\left(V_{d\nu} - \sigma_{\nu} \cdot Z_{d\nu}\right)^{-1}\right],] are related to the volume ratios of the contributing subsystems. The subsystem structure factors correspond to those for IMSs (see Section 4.6.3.1[link]). Besides this formula, based on the publications of Yamamoto (1982)[link] and van Smaalen (1995)[link], different structure-factor equations have been discussed (Kato, 1990[link]; Petricek, Maly, Coppens et al., 1991[link]).

References

Kato, K. (1990). Strukturverfeinerung des Kompositkristalls im mehr­dimensionalen Raum. Acta Cryst. B46, 39–44.
Petricek, V., Maly, K., Coppens, P., Bu, X., Cisarova, I. & Frost-Jensen, A. (1991). The description and analysis of composite crystals. Acta Cryst. A47, 210–216.
Smaalen, S. van (1995). Incommensurate crystal structures. Crystallogr. Rev. 4, 79–202.
Yamamoto, A. (1982). Structure factor of modulated crystal structures. Acta Cryst. A38, 87–92.








































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