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

International Tables for Crystallography (2010). Vol. B, ch. 4.6, pp. 603-604   | 1 | 2 |

## Section 4.6.3.3.1.1. 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

#### 4.6.3.3.1.1. Indexing

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The reciprocal space of the Fibonacci chain is densely filled with Bragg reflections (Figs. 4.6.2.9 and 4.6.3.5 ). According to the nD embedding method, the shorter the parallel-space distance between two Bragg reflections, the larger the corresponding perpendicular-space distance becomes. Since the structure factor decreases rapidly as a function of (Fig. 4.6.3.6) , `neighbouring' reflections of strong Bragg peaks are extremely weak and, consequently, the reciprocal space appears to be filled with discrete Bragg peaks even for low-resolution experiments. Figure 4.6.3.5 | top | pdf |The structure factors (below) and their magnitudes (above) of a Fibonacci chain decorated with equal point atoms are shown as a function of the parallel-space component of the diffraction vector. The short distance in the Fibonacci chain is , all structure factors within have been calculated and normalized to . Figure 4.6.3.6 | top | pdf |The structure factors (below) and their magnitudes (above) of a Fibonacci chain decorated with equal point atoms are shown as a function of the perpendicular-space component of the diffraction vector. The short distance in the Fibonacci chain is , all structure factors within have been calculated and normalized to .

This property allows an unambiguous identification of a correct set of reciprocal-basis vectors. However, infinitely many sets allowing a correct indexing of the diffraction pattern with integer indices exist. Nevertheless, an optimum basis (low indices are assigned to strong reflections) can be derived: the intensity distribution, not the metrics, characterizes the best choice of indexing. Once the minimum distance S in the structure is identified from chemical considerations, the reciprocal basis should be chosen as described in Section 4.6.2.4 . It has to be kept in mind, however, that the identification of the metrics is not sufficient to distinguish in the 1D aperiodic case between an incommensurately modulated structure, a quasiperiodic structure or special kinds of structures with fractally shaped atomic surfaces.

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

 (1) Find pairs of strong reflections whose physical-space diffraction vectors are related to each other by the factor τ. (2) Index these reflections by assigning an appropriate value to . This value should be derived from the shortest interatomic distance S expected in the structure. (3) The reciprocal basis is correct if all observable Bragg reflections can be indexed with integer numbers.