Intensity statistics

Steurer, W.; Haibach, T.

doi:10.1107/97809553602060000568

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

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International Tables for Crystallography (2006). Vol. B. ch. 4.6, p. 507 | 1 | 2 |

Section 4.6.3.3.2.4. Intensity statistics

W. Steurer^a ^* and T. Haibach^a

^aLaboratory of Crystallography, Swiss Federal Institute of Technology, CH-8092 Zurich, Switzerland
Correspondence e-mail: w.steurer@kristall.erdw.ethz.ch

4.6.3.3.2.4. Intensity statistics

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This section deals with the reciprocal-space characteristics of the 2D quasiperiodic component of the 3D structure, namely the Fourier module $[M_{1}^{*}]$ . The radial structure-factor distributions of the Penrose tiling decorated with point scatterers are plotted in Figs. 4.6.3.21 and 4.6.3.22 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 4D 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 to the power of four, that of strong reflections only quadratically (strong reflections always have small $[{\bf H}^{\perp}]$ components). The radial distribution of the structure-factor amplitudes as a function of perpendicular space clearly shows three branches, corresponding to the reflection classes $[{\textstyle\sum_{i = 1}^{4}} h_{i} = m\hbox{ mod }5]$ with [|m| = 0] , [|m| = 1] and [|m| = 2] (Fig. 4.6.3.23).

Figure 4.6.3.21| top | pdf |

Radial distribution function of the structure factors $[F({\bf H})]$ of the Penrose tiling (edge length of the Penrose unit rhombs a_r = 4.04 Å) decorated with point atoms as a function of $[{\bf H}^{\parallel}]$ . All structure factors within $[10^{-4} |F({\bf 0})|^{2} \;\lt\; |F({\bf H})|^{2} \;\lt\; |F({\bf 0})|^{2}]$ and $[0 \leq |{\bf H}^{\parallel}| \leq 2.5\;\hbox{\AA}^{-1}]$ have been used and normalized to [F(0000) = 1] .

Figure 4.6.3.22| top | pdf |

Radial distribution function of the structure factors $[F({\bf H})]$ of the Penrose tiling (edge length of the Penrose unit rhombs a_r = 4.04 Å) decorated with point atoms as a function of $[{\bf H}^{\perp}]$ . All structure factors within $[10^{-4} |F({\bf 0})|^{2} \;\lt\; |F({\bf H})|^{2} \;\lt\; |F({\bf 0})|^{2}]$ and $[0 \leq |{\bf H}^{\parallel}| \leq 2.5\;\hbox{\AA}^{-1}]$ have been used and normalized to [F(0000) = 1] .

Figure 4.6.3.23| top | pdf |

Radial distribution function of the structure-factor magnitudes $[|F({\bf H})|]$ of the Penrose tiling (edge length of the Penrose unit rhombs a_r = 4.04 Å) decorated with point atoms as a function of $[{\bf H}^{\perp}]$ . All structure factors within $[10^{-4} |F({\bf 0})|^{2} \;\lt\; |F({\bf H})|^{2} \;\lt\; |F({\bf 0})|^{2}]$ and $[0 \leq |{\bf H}^{\parallel}| \leq 2.5\;\hbox{\AA}^{-1}]$ have been used and normalized to [F(0000) = 1] . The branches with (a) $[\left|{\textstyle\sum_{i = 1}^{4}} h_{i}\right| = 0\hbox{ mod }5]$ , (b) $[\left|{\textstyle\sum_{i = 1}^{4}} h_{i}\right| = 1\hbox{ mod }5]$ and (c) $[\left|{\textstyle\sum_{i = 1}^{4}} h_{i}\right| = 2\hbox{ mod }5]$ are shown.

The weighted reciprocal space of the 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 (Figs. 4.6.3.19, 4.6.3.20 and 4.6.3.22).

References

International Tables for Crystallography (2006). Vol. B. ch. 4.6, p. 507