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Crystals 2016, 6(9), 104; doi:10.3390/cryst6090104

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

Faculty of Physics and Applied Computer Science, AGH University of Science and Technology, al. Mickiewicza 30, 30-059 Krakow, Poland
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Academic Editor: Enrique Maciá Barber
Received: 18 July 2016 / Revised: 12 August 2016 / Accepted: 22 August 2016 / Published: 26 August 2016
(This article belongs to the Special Issue Structure and Properties of Quasicrystals 2016)
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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. View Full-Text
Keywords: statistical approach; average unit cell; quasicrystals; aperiodic crystals; diffraction pattern statistical approach; average unit cell; quasicrystals; aperiodic crystals; diffraction pattern
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This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. (CC BY 4.0).

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Strzalka, R.; Buganski, I.; Wolny, J. Statistical Approach to Diffraction of Periodic and Non-Periodic Crystals—Review. Crystals 2016, 6, 104.

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