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

International Tables for Crystallography (2010). Vol. B, ch. 4.6, p. 613   | 1 | 2 |

Section 4.6.3.3.2.5. Relationships between structure factors at symmetry-related points of the Fourier image

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

4.6.3.3.2.5. Relationships between structure factors at symmetry-related points of the Fourier image

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Scaling the Penrose tiling by a factor [\tau^{-n}] by employing the matrix [S^{-n}] scales at the same time its reciprocal space by a factor [\tau^{n}]: [\eqalignno{ S{\bf H} &= \pmatrix{0 &1 &0 &\bar{1} &0\cr 0 &1 &1 &\bar{1} &0\cr \bar{1} &1 &1 &0 &0\cr \bar{1} &0 &1 &0 &0\cr 0 &0 &0 &0 &1\cr}_{D} \pmatrix{h_{1}\cr h_{2}\cr h_{3}\cr h_{4}\cr h_{5}\cr} = \pmatrix{h_{2} - h_{4}\cr h_{2} + h_{3} - h_{4}\cr - h_{1} + h_{2} + h_{3}\cr - h_{1} + h_{3}\cr h_{5}\cr}. &\cr}]Since this operation increases the lengths of the diffraction vectors by the factor τ in parallel space and decreases them by the factor [1/\tau] in perpendicular space, the following distribution of structure-factor magnitudes (for point atoms at rest) is obtained: [\displaylines{ \hfill \left|F(S^{n}{\bf H})\right| > \left|F(S^{n - 1}{\bf H})\right| > \ldots > \left|F(S^{1}{\bf H})\right| > \left|F({\bf H})\right|, \hfill \cr \hfill \left|F(\tau^{n}{\bf H}^{\|})\right| > \left|F(\tau^{n - 1}{\bf H}^{\|})\right| > \ldots > \left|F(\tau {\bf H}^{\|})\right| > \left|F({\bf H})\right|. \hfill \cr}]The scaling operations [S^{n}], [n \in {\bb Z}], the rotoscaling operations [(\Gamma (\alpha)S^{2})^{n}] (Fig. 4.6.3.14)[link] and the tenfold rotation [(\Gamma (\alpha))^{n}], where [(\Gamma (\alpha)S^{2})^{n} = \pmatrix{1 &1 &\bar{1} &\bar{1} &0\cr 1 &2 &0 &\bar{2} &0\cr 0 &2 &1 &\bar{1} &0\cr \bar{1} &1 &1 &0 &0\cr 0 &0 &0 &0 &1\cr}^{n}_{D},]connect all structure factors with diffraction vectors pointing to the nodes of an infinite series of pentagrams. The structure factors with positive signs are predominantly on the vertices of the pentagram while the ones with negative signs are arranged on circles around the vertices (Figs. 4.6.3.24[link][link][link] to 4.6.3.27[link]).

[Figure 4.6.3.24]

Figure 4.6.3.24 | top | pdf |

Pentagrammal relationships between scaling symmetry-related positive structure factors [F({\bf H})] of the Penrose tiling (edge length ar = 4.04 Å) in parallel space. The magnitudes of the structure factors are indicated by the diameters of the filled circles.

[Figure 4.6.3.25]

Figure 4.6.3.25 | top | pdf |

Pentagrammal relationships between scaling symmetry-related negative structure factors [F({\bf H})] of the Penrose tiling (edge length ar = 4.04 Å) in parallel space. The magnitudes of the structure factors are indicated by the diameters of the filled circles.

[Figure 4.6.3.26]

Figure 4.6.3.26 | top | pdf |

Pentagrammal relationships between scaling symmetry-related structure factors [F({\bf H})] of the Penrose tiling (edge length ar = 4.04 Å) in parallel space. Enlarged sections of Figs. 4.6.3.24[link] (above) and 4.6.3.25[link] (below) are shown.

[Figure 4.6.3.27]

Figure 4.6.3.27 | top | pdf |

Pentagrammal relationships between scaling symmetry-related structure factors [F({\bf H})] of the Penrose tiling (edge length ar = 4.04 Å) in perpendicular space. Enlarged sections of positive (above) and negative structure factors (below) are shown.








































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