The reference lattice concept and its application to the analysis of diffraction patterns

Wolny, Janusz

doi:10.1080/01418619808223760

Cited by 60 publications

(34 citation statements)

References 17 publications

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“…Statistical definition of AUC is the distribution of atoms' position with respect to the knots of a periodic reference lattice with a constant [4]. If the coordinate of j-th point relative to the origin is x j = r ||j then with respect to the reference lattice it can be written as x j = j +u j , where j is an integer giving the number of the knot's position closest to x j and u j is a positive displacement of x j from the j knot.…”

Section: Average Unit Cellmentioning

confidence: 99%

“…Similarly to the modulated structures, quasicrystals can be described either in higherdimensional space (HDS) [1][2][3] or, alternatively, in only physical space [4]. In the multidimensional analysis, modulated structures become periodic and the atoms get some elongated inner structure called atomic surface (AS), which can be also interpreted as a distribution of all possible structural configurations.…”

Section: Introductionmentioning

confidence: 99%

“…It is possible though to enforce the periodicity by overlapping the structure with the reference lattice [4]. This operation would lead to the distribution of atoms' positions with respect to a periodic lattice called an average unit cell (AUC).…”

Section: Introductionmentioning

confidence: 99%

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Structure of quasicrystals described by statistical methods

Kozakowski

Wolny

Kuczera

2008

Philosophical Magazine

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Section: Average Unit Cellmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Structure of quasicrystals described by statistical methods

Kozakowski

Wolny

Kuczera

2008

Philosophical Magazine

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“…They are related by an oblique projection (Kozakowski & Wolny, 2006;Steurer & Haibach, 1999) of an atomic surface onto physical space. The projected atomic surface gains a new interpretation -it becomes a probability distribution of atomic displacements measured with respect to the reference frame introduced by Wolny (Wolny, 1998). This probability distribution is often called: Average Unit Cell (AUC).…”

Section: Introductionmentioning

confidence: 99%

Physical space structure refinement of decagonal quasicrystal in rhombic Penrose tiling model

Wolny¹,

Kozakowski²,

Kuczera³

et al. 2008

Zeitschrift Für Kristallographie

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Decagonal quasicrystal / Structure refinement / Penrose tiling / Average unit cell Abstract. For the first time, the refinement of a decagonal Al 72 Ni 20 Co 8 quasicrystal was conducted in physical space. We used a structure factor obtained in statistical approach. As a reference lattice we took a rhombic Penrose tiling to which we applied overlapping rules of clusters. The resulting structure gives R-factor ¼ 8.0% and R w -factor ¼ 6.1%. The density of the refined structure is very close to the experimental one.

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“…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.…”

Section: Introductionmentioning

confidence: 99%

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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The reference lattice concept and its application to the analysis of diffraction patterns

Cited by 60 publications

References 17 publications

Structure of quasicrystals described by statistical methods

Structure of quasicrystals described by statistical methods

Physical space structure refinement of decagonal quasicrystal in rhombic Penrose tiling model

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

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