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

International Tables for Crystallography (2010). Vol. B, ch. 4.6, p. 599   | 1 | 2 |

Section 4.6.3.1.1. Indexing

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

4.6.3.1.1. Indexing

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The 3D reciprocal space [M^{*}] of a [(3 + d)\hbox{D}] IMS consists of two separable contributions, [M^{*} = \left\{{\bf H} = {\textstyle\sum\limits_{i = 1}^{3}} h_{i} {\bf a}_{i}^{*} + {\textstyle\sum\limits_{j = 1}^{d}} m_{j} {\bf q}_{j}\right\},]the set of main reflections [(m_{j} = 0)] and the set of satellite reflections [(m_{j} \neq 0)] (Fig. 4.6.3.1)[link]. In most cases, the modulation is only a weak perturbation of the crystal structure. The main reflections are related to the average structure, the satellites to the difference between the average and actual structure. Consequently, the satellite reflections are generally much weaker than the main reflections and can be easily identified. Once the set of main reflections has been separated, a conventional basis [{\bf a}_{i}^{*}, i = 1, \ldots, 3], for [\Lambda^{*}] is chosen.

The only ambiguity is in the assignment of rationally independent satellite vectors [{\bf q}_{i}]. They should be chosen inside the reciprocal-space unit cell (Brillouin zone) of [\Lambda^{*}] in such a way as to give a minimal number d of additional dimensions. If satellite vectors reach the Brillouin-zone boundary, centred [(3 + d)\hbox{D}] Bravais lattices are obtained. The star of satellite vectors has to be invariant under the point-symmetry group of the diffraction pattern. There should be no contradiction to a reasonable physical modulation model concerning period or propagation direction of the modulation wave. More detailed information on how to find the optimum basis and the correct setting is given by Janssen et al. (2004)[link] and Janner et al. (1983a[link],b[link]).

References

Janner, A., Janssen, T. & de Wolff, P. M. (1983a). Bravais classes for incommensurate crystal phases. Acta Cryst. A39, 658–666.
Janner, A., Janssen, T. & de Wolff, P. M. (1983b). Determination of the Bravais class for a number of incommensurate crystals. Acta Cryst. A39, 671–678.
Janssen, T., Janner, A., Looijenga-Vos, A. & de Wolff, P. M. (2004). Incommensurate and commensurate modulated crystal structures. In International Tables for Crystallography, Vol. C, edited by E. Prince, ch. 9.8. Dordrecht: Kluwer Academic Publishers.








































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