Real Space Structure Factor for Different Quasicrystals

Wolny, Janusz; Kozakowski, B.; Kuczera, Pawel; Strzałka, Radoslaw; Wnęk, Anna

doi:10.1002/ijch.201100144

Cited by 19 publications

(18 citation statements)

References 40 publications

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“…It was proven [15,21] that the AUC can be considered as equivalent to an oblique projection of atomic surface onto physical space along direction defined by k perp /k par , which is the ratio of perpendicular-to parallel-space component of any scattering vector from reciprocal space. In fact, variables (u, v) can be obtained by an oblique projection of the perpendicular-space components of atomic positions (the atomic surface).…”

Section: Statistical Approach and Superspace Methodsmentioning

confidence: 99%

“…TAU2-scaling applies for both cases. The shapes of AUCs are the following: 4-parameter distribution P(u 1 , u 2 , v 1 , v 2 ) for Penrose tiling is spanned by 4 two-dimensional pentagons (2 smaller and 2 bigger) [20][21][22]; 6-parameter distribution P(u 1 , u 2 , u 3 , v 1 , v 2 , v 3 ) for Ammann tiling takes form of 3D Keplerian solid-rhombic triacontahedron [15,23]. To construct AUCs for 2D and 3D quasicrystals we need to span pairs of reference lattices which are 2D or 3D, respectively.…”

Section: Auc For Quasicrystalsmentioning

confidence: 99%

“…For instance, if the number of atoms in sequence is given by multiples of 6 at scaling factor = 3, the probability distribution values take 9 non-zero positions of heights proportional to 1/6. For further details see [15,30]. Summarizing, for the Thue-Morse aperiodic sequence the reference lattice concept allows to designate intensities and scaling properties for the singular continuous part with scattering vector = /(2 + 1) with being integer.…”

Section: Auc For Thue-morse Sequencementioning

confidence: 99%

“…Details can be found in refs. [15,30]. The complete diffraction pattern of Thue-Morse chain is presented in Figure 4c.…”

Section: Structure Factor For Thue-morse Sequencementioning

confidence: 99%

“…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. The distribution gives us a perfect tool for structural considerations of periodic and any type of aperiodic systems with no interpretational difficulties, and also whole spectrum of analyzes available to perform [15].…”

Section: Introductionmentioning

confidence: 99%

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

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.

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Section: Statistical Approach and Superspace Methodsmentioning

confidence: 99%

Section: Auc For Quasicrystalsmentioning

confidence: 99%

Section: Auc For Thue-morse Sequencementioning

confidence: 99%

“…Details can be found in refs. [15,30]. The complete diffraction pattern of Thue-Morse chain is presented in Figure 4c.…”

Section: Structure Factor For Thue-morse Sequencementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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

2016

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25 Years of Quasiperiodic Crystallography in Physical Space using the Average Unit Cell Approach

Wolny,

Bugański,

Strzałka

et al. 2024

Israel Journal of Chemistry

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Since the discovery of quasicrystals 40 years ago, many new paradigms and methods have been introduced to crystallography. 25 years ago, a statistical method of structure and diffraction analysis of aperiodic materials was proposed and, over these years, developed to describe model and real systems. This short review paper briefly invokes the basic concepts of the method: a reference lattice and an average unit cell, but also gives an overview of its application to atomic structure and diffraction analysis of various systems. Results are briefly discussed for mathematical sequences (Fibonacci and Thue‐Morse), model quasilattices in 2D and 3D (Penrose and Ammann tiling), refinements of real decagonal and icosahedral quasicrystals, analysis of structure disorder in quasicrystals, description of modulated systems, including macromolecular biological systems, and beyond usual application in crystallography.

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The estimation of phason flips in 1D quasicrystal from the diffraction pattern

Bugański

Strzałka

Wolny

2015

Physica Status Solidi (b)

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The statistical approach based on the average unit‐cell concept and the envelop functions for the diffraction pattern were used to estimate the number of phason flips in the model 1D quasicrystal (Fibonacci chain). The characteristic function of a statistical distribution expanded to a power series with distribution moments as coefficients can be used to retrieve phases of diffraction peaks. In addition, the number of flips in the structure can be designated in two ways: with the value of the second moment's value and directly from the shape of a probability density function retrieved from the diffraction pattern. In this paper, all calculations are performed for a nondecorated Fibonacci chain.

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Real Space Structure Factor for Different Quasicrystals

Cited by 19 publications

References 40 publications

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

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

25 Years of Quasiperiodic Crystallography in Physical Space using the Average Unit Cell Approach

The estimation of phason flips in 1D quasicrystal from the diffraction pattern

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