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 [M_{1}^{*}], are discussed. The intensities [I({\bf H})] of the Fibonacci chain decorated with point atoms are only a function of the perpendicular-space component of the diffraction vector. [|F({\bf H})|] and [F({\bf H})] are illustrated in Figs. 4.6.3.5[link] and 4.6.3.6[link] as a function of [{\bf H}^{\parallel}] and of [{\bf H}^{\perp}]. The distribution of [|F({\bf H})|] 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 [H_{\max}] of the limiting sphere in reciprocal space. The number of weak reflections increases with the square of [H_{\max}], that of strong reflections only linearly (strong reflections always have small [{\bf H}^{\perp}] 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[link] and 4.6.3.9[link]). 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]

Figure 4.6.3.8 | top | pdf |

The structure factors [F({\bf H})] (below) and their magnitudes [|F({\bf H})|] (above) of the squared 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 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 4.6.3.9]

Figure 4.6.3.9 | top | pdf |

The structure factors [F({\bf H})] (below) and their magnitudes [|F({\bf H})|] (above) of the squared 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 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].








































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