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.