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. 501 | 1 | 2 |

Section 4.6.3.3.1.4. 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

4.6.3.3.1.4. 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 and 4.6.3.6 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 sub-unit 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 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 $[F({\bf H})]$ (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| 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] .

References

International Tables for Crystallography (2006). Vol. B. ch. 4.6, p. 501