International
Tables for Crystallography Volume B Reciprocal space Edited by U. Shmueli © International Union of Crystallography 2006 |
International Tables for Crystallography (2006). Vol. B. ch. 4.6, p. 501
Section 4.6.3.3.1.4. Intensity statistics
aLaboratory of Crystallography, Swiss Federal Institute of Technology, CH-8092 Zurich, Switzerland |
In the following, only the properties of the quasiperiodic component of the 3D structure, namely the Fourier module , are discussed. The intensities of the Fibonacci chain decorated with point atoms are only a function of the perpendicular-space component of the diffraction vector. and are illustrated in Figs. 4.6.3.5 and 4.6.3.6 as a function of and of . The distribution of 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 of the limiting sphere in reciprocal space. The number of weak reflections increases with the square of , that of strong reflections only linearly (strong reflections always have small 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.