Statistical Approach to Diffraction of Periodic and Non-Periodic Crystals—Review

Abstract

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.

Atomic Structure of Decagonal Al-Cu-Rh Quasicrystal–Revisited: New Correction for Phonons

“…To our knowledge, there are at least two possible reasons for this phenomenon, which are namely (i) incorrect phasonic correction function and/or (ii) multiple scattering effect occurring in the diffraction experiment. Discussion on the influence of possible scenario (i) is explained in detail in references [5,6,40]. The standard exponential phasonic D-W factor with k perp vector in the exponent (see formula (3)) appears to artificially lower the intensities of reflections with large-k perp component, which are small by definition for quasicrystals.…”

“…The diffraction pattern considered within the statistical approach is strongly related to the probability distribution function of atomic positions in the AUC [3][4][5][6]. For periodic crystals, the distributions are discrete and can be described as a sum of Dirac-delta functions.…”

“…For quasicrystals, the distributions are continuous. For instance, in the case of the decagonal systems modelled by the Penrose tiling, which are scalable systems, the distributions are non-zero only along characteristic lines (see: τ 2 -scaling in quasicrystals in [3,5]). Defects cause the perturbation of these distributions.…”

“…That study showed the first refinement results of d-Al-Cu-{Rh,Ir} systems, whereas the d-Al-Cu-Co was previously refined using in-house X-ray diffraction data [2]. The refinement was conducted based on the Average Unit Cell (AUC) concept [3][4][5], which is a method that involves the modeling of the aperiodic systems (called the statistical method in the next parts of the paper), which is the main alternative to the multidimensional one. The same method is used in this paper.…”

Quo Vadis Quasicrystals?

“…On the other hand, the recourse to a higher dimensional space-based crystallography allows one to recover the very possibility of describing the order present in aperiodic crystals in terms of a periodic distribution of hyperatoms located in a hyperspace which embeds the three-dimensional physical space. In this context, the contribution by Radoslaw Strzalka, Irineusz Buganski, and Janusz Wolny introduces an alternative approach which is based on the average unit cell concept: a probability distribution of atomic positions with respect to a suitable reference lattice in physical space [18]. This distribution carries the complete structural information required for structure determination via diffraction experiments regardless of the inner symmetry of the diffracting medium.…”