Indexing

Steurer, W.; Haibach, T.

doi:10.1107/97809553602060000568

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

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International Tables for Crystallography (2006). Vol. B. ch. 4.6, p. 505 | 1 | 2 |

Section 4.6.3.3.2.1. Indexing

W. Steurer^a ^* and T. Haibach^a

^aLaboratory of Crystallography, Swiss Federal Institute of Technology, CH-8092 Zurich, Switzerland
Correspondence e-mail: w.steurer@kristall.erdw.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).
(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.Google Scholar

International Tables for Crystallography (2006). Vol. B. ch. 4.6, p. 505