Abstract
In this paper, we show the fundamentals of statistical method of structure analysis. Basic concept of a method is the average unit cell, which is a probability distribution of atomic positions with respect to some reference lattices. The distribution carries complete structural information required for structure determination via diffraction experiment regardless of the inner symmetry of diffracting medium. The shape of envelope function that connects all diffraction maxima can be derived as the Fourier transform of a distribution function. Moreover, distributions are sensitive to any disorder introduced to ideal structure—phonons and phasons. The latter are particularly important in case of quasicrystals. The statistical method deals very well with phason flips and may be used to redefine phasonic Debye-Waller correction factor. The statistical approach can be also successfully applied to the peak’s profile interpretation. It will be shown that the average unit cell can be equally well applied to a description of Bragg peaks as well as other components of diffraction pattern, namely continuous and singular continuous components. Calculations performed within statistical method are equivalent to the ones from multidimensional analysis. The atomic surface, also called occupation domain, which is the basic concept behind multidimensional models, acquires physical interpretation if compared to average unit cell. The statistical method applied to diffraction analysis is now a complete theory, which deals equally well with periodic and non-periodic crystals, including quasicrystals. The method easily meets also any structural disorder.
Highlights
Quasicrystals are materials exhibiting aperiodic arrangement of atoms in the atomic structure.Aperiodicity violating translational symmetry allows for non-crystallographic symmetry elements in the diffraction diagrams of quasicrystals and other aperiodic materials [1,2]
One of the very few alternatives to superspace description is the statistical method where atomic positions in the physical space are replaced by their relative values with respect to some periodic reference lattices [14]
Any kind of structure can be expressed in terms of the statistical distribution of atomic positions with respect to the periodic reference lattice with lattice constant related to characteristic length-scale present in the structure
Summary
Quasicrystals are materials exhibiting aperiodic arrangement of atoms in the atomic structure. One of the very few alternatives to superspace description is the statistical method where atomic positions in the physical space are replaced by their relative values with respect to some periodic reference lattices [14]. Obtained distribution of such relative positions is limited to the so-called average unit cell and exists in the physical space—exactly as the atomic structure. ItIntroducing another which thethe golden value (so called τ-based quasicrystals for which diffractiondescription—it peaks’ positions lattice is has samemean reasons as introducing doubled dimensionality in superspace is scale τ).
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