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

International Tables for Crystallography (2010). Vol. B, ch. 4.6, p. 610   | 1 | 2 |

Section 4.6.3.3.2.1. 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

4.6.3.3.2.1. Indexing

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The indexing of the submodule [M_{1}^{*}] of the diffraction pattern of a decagonal phase is not unique. Since [M_{1}^{*}] corresponds to a [{\bb Z}] module of rank 4 with decagonal point symmetry, it is invariant under scaling by [\tau^{n}, n \in {\bb Z}]: [S^{n}M^{*} = \tau^{n} M^{*}]. Nevertheless, an optimum basis (low indices are assigned to strong reflections) can be derived: not the metrics, as for regular periodic crystals, but the intensity distribution characterizes the best choice of indexing.

A correct set of reciprocal-basis vectors can be identified experimentally in the following way:

  • (1) Find directions of systematic absences or pseudo-absences determining the possible orientations of the reciprocal-basis vectors (see Rabson et al., 1991[link]).

  • (2) Find pairs of strong reflections whose physical-space diffraction vectors are related to each other by the factor τ.

  • (3) Index these reflections by assigning an appropriate value to [a^{*}]. This value should be derived from the shortest interatomic distance S and the edge length of the unit tiles expected in the structure.

  • (4) The reciprocal basis is correct if all observable Bragg reflections can be indexed with integer numbers.

References

Rabson, D. A., Mermin, N. D., Rokhsar, D. S. & Wright, D. C. (1991). The space groups of axial crystals and quasicrystals. Rev. Mod. Phys. 63, 699–733.








































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