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

International Tables for Crystallography (2010). Vol. B, ch. 4.6, p. 619   | 1 | 2 |

Section 4.6.3.3.3.4. 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

4.6.3.3.3.4. Intensity statistics

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The radial structure-factor distributions of the 3D Penrose tiling decorated with point scatterers are plotted in Fig. 4.6.3.35[link] as a function of parallel and perpendicular space. The distribution of [|F({\bf H})|] as a function of their frequencies clearly resembles a centric distribution, as can be expected from the centrosymmetric unit cell. The shape of the distribution function depends on the radius of the limiting sphere in reciprocal space. The number of weak reflections increases as the power 6, that of strong reflections only as the power 3 (strong reflections always have small [{\bf H}^{\perp}] components).

[Figure 4.6.3.35]

Figure 4.6.3.35 | top | pdf |

Radial distribution function of the structure factors [F({\bf H})] of the 3D Penrose tiling (edge lengths of the Penrose unit rhombohedra ar = 5.0 Å) decorated with point atoms as a function of [|{\bf H}^{\parallel}|] (above) and [|{\bf H}^{\perp}|] (below). All reflections are shown within [10^{-6} |F({\bf 0})|^{2} \;\lt\; |F({\bf H})|^{2} \;\lt\; |F({\bf 0})|^{2}] and [-6 \leq h_{i} \leq 6, i = 1,\ldots, 6].

The weighted reciprocal space of the 3D Penrose tiling 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.








































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