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[link] and 4.6.3.5[link]). According to the nD embedding method, the shorter the parallel-space distance [\Delta {\bf H}^{\parallel} = {\bf H}_{2}^{\parallel} - {\bf H}_{1}^{\parallel}] between two Bragg reflections, the larger the corresponding perpendicular-space distance [\Delta {\bf H}^{\perp} = {\bf H}_{2}^{\perp} - {\bf H}_{1}^{\perp}] becomes. Since the structure factor [F({\bf H})] decreases rapidly as a function of [{\bf H}^{\perp}] (Fig. 4.6.3.6)[link], `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]

Figure 4.6.3.5 | top | pdf |

The structure factors [F({\bf H})] (below) and their magnitudes [|F({\bf H})|] (above) of a Fibonacci chain decorated with equal point atoms are shown as a function of the parallel-space component [|{\bf H}^{\parallel}|] of the diffraction vector. The short distance in the Fibonacci chain is [\hbox{S} = 2.5\;\hbox{\AA}], all structure factors within [0 \leq |{\bf H}| \leq 2.5\;\hbox{\AA}^{-1}] have been calculated and normalized to [F(00) = 1].

[Figure 4.6.3.6]

Figure 4.6.3.6 | top | pdf |

The structure factors [F({\bf H})] (below) and their magnitudes [|F({\bf H})|] (above) of a Fibonacci chain decorated with equal point atoms are shown as a function of the perpendicular-space component [|{\bf H}^{\perp}|] of the diffraction vector. The short distance in the Fibonacci chain is [\hbox{S} = 2.5\;\hbox{\AA}], all structure factors within [0 \leq |{\bf H}| \leq 2.5\;\hbox{\AA}^{-1}] have been calculated and normalized to [F(00) = 1].

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[link]. 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 [a^{*}]. 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.








































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