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

International Tables for Crystallography (2010). Vol. B, ch. 4.6, pp. 605-606   | 1 | 2 |

## Section 4.6.3.3.1.4. Intensity statistics

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

#### 4.6.3.3.1.4. Intensity statistics

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In the following, only the properties of the quasiperiodic component of the 3D structure, namely the Fourier module , are discussed. The intensities of the Fibonacci chain decorated with point atoms are only a function of the perpendicular-space component of the diffraction vector. and are illustrated in Figs. 4.6.3.5 and 4.6.3.6 as a function of and of . The distribution of as a function of their frequencies clearly resembles a centric distribution, as can be expected from the centrosymmetric 2D subunit cell. The shape of the distribution function depends on the radius of the limiting sphere in reciprocal space. The number of weak reflections increases with the square of , that of strong reflections only linearly (strong reflections always have small components).

The weighted reciprocal space of the Fibonacci sequence contains an infinite number of Bragg reflections within a limited region of the physical space. Contrary to the diffraction pattern of a periodic structure consisting of point atoms on the lattice nodes, the Bragg reflections show intensities depending on the perpendicular-space components of their diffraction vectors.

The reciprocal space of a sequence generated from hyperatoms with fractally shaped atomic surfaces (squared Fibonacci sequence) is very similar to that of the Fibonacci sequence (Figs. 4.6.3.8 and 4.6.3.9). However, there are significantly more weak reflections in the diffraction pattern of the `fractal' sequence, caused by the geometric form factor.

 Figure 4.6.3.8 | top | pdf |The structure factors (below) and their magnitudes (above) of the squared Fibonacci chain decorated with equal point atoms are shown as a function of the parallel-space component of the diffraction vector. The short distance is , all structure factors within have been calculated and normalized to .
 Figure 4.6.3.9 | top | pdf |The structure factors (below) and their magnitudes (above) of the squared Fibonacci chain decorated with equal point atoms are shown as a function of the perpendicular-space component of the diffraction vector. The short distance is , all structure factors within have been calculated and normalized to .