Acta Crystallographica Section A
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Acta Crystallographica Section A: Foundations and Advances covers theoretical and fundamental aspects of the structure of matter. The journal is the prime forum for research in diffraction physics and the theory of crystallographic structure determination by diffraction methods using X-rays, neutrons and electrons. The structures include periodic and aperiodic crystals, and non-periodic disordered materials, and the corresponding Bragg, satellite and diffuse scattering, thermal motion and symmetry aspects. Spatial resolutions range from the subatomic domain in charge-density studies to nanodimensional imperfections such as dislocations and twin walls. The chemistry encompasses metals, alloys, and inorganic, organic and biological materials. Structure prediction and properties such as the theory of phase transformations are also covered.enCopyright (c) 2023 International Union of Crystallography2023-09-01International Union of CrystallographyInternational Union of Crystallographyhttp://journals.iucr.orgurn:issn:2053-2733Acta Crystallographica Section A: Foundations and Advances covers theoretical and fundamental aspects of the structure of matter. The journal is the prime forum for research in diffraction physics and the theory of crystallographic structure determination by diffraction methods using X-rays, neutrons and electrons. The structures include periodic and aperiodic crystals, and non-periodic disordered materials, and the corresponding Bragg, satellite and diffuse scattering, thermal motion and symmetry aspects. Spatial resolutions range from the subatomic domain in charge-density studies to nanodimensional imperfections such as dislocations and twin walls. The chemistry encompasses metals, alloys, and inorganic, organic and biological materials. Structure prediction and properties such as the theory of phase transformations are also covered.text/htmlActa Crystallographica Section A: Foundations and Advances, Volume 79, Part 5, 2023textweekly62002-01-01T00:00+00:005792023-09-01Copyright (c) 2023 International Union of CrystallographyActa Crystallographica Section A: Foundations and Advances390urn:issn:2053-2733med@iucr.orgSeptember 20232023-09-01Acta Crystallographica Section Ahttp://journals.iucr.org/logos/rss10a.gif
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Still imageAlgorithms for magnetic symmetry operation search and identification of magnetic space group from magnetic crystal structure
http://scripts.iucr.org/cgi-bin/paper?ib5114
A crystal symmetry search is crucial for computational crystallography and materials science. Although algorithms and implementations for the crystal symmetry search have been developed, their extension to magnetic space groups (MSGs) remains limited. In this paper, algorithms for determining magnetic symmetry operations of magnetic crystal structures, identifying magnetic space-group types of given MSGs, searching for transformations to a Belov–Neronova–Smirnova (BNS) setting, and symmetrizing the magnetic crystal structures using the MSGs are presented. The determination of magnetic symmetry operations is numerically stable and is implemented with minimal modifications from the existing crystal symmetry search. Magnetic space-group types and transformations to the BNS setting are identified by a two-step approach combining space-group-type identification and the use of affine normalizers. Point coordinates and magnetic moments of the magnetic crystal structures are symmetrized by projection operators for the MSGs. An implementation is distributed with a permissive free software license in spglib v2.0.2: https://github.com/spglib/spglib.Copyright (c) 2023 International Union of Crystallographyurn:issn:2053-2733Shinohara, K.Togo, A.Tanaka, I.2023-09-06doi:10.1107/S2053273323005016International Union of CrystallographyThis paper presents algorithms for determining magnetic symmetry operations of magnetic crystal structures, identifying magnetic space-group types from a given magnetic space group (MSG), searching for transformations to a Belov–Neronova–Smirnova setting, and symmetrizing the magnetic crystal structures on the basis of the determined MSGs.ENmagnetic space groupmagnetic space-group typemagnetic structurecrystal structure analysisaffine normalizerA crystal symmetry search is crucial for computational crystallography and materials science. Although algorithms and implementations for the crystal symmetry search have been developed, their extension to magnetic space groups (MSGs) remains limited. In this paper, algorithms for determining magnetic symmetry operations of magnetic crystal structures, identifying magnetic space-group types of given MSGs, searching for transformations to a Belov–Neronova–Smirnova (BNS) setting, and symmetrizing the magnetic crystal structures using the MSGs are presented. The determination of magnetic symmetry operations is numerically stable and is implemented with minimal modifications from the existing crystal symmetry search. Magnetic space-group types and transformations to the BNS setting are identified by a two-step approach combining space-group-type identification and the use of affine normalizers. Point coordinates and magnetic moments of the magnetic crystal structures are symmetrized by projection operators for the MSGs. An implementation is distributed with a permissive free software license in spglib v2.0.2: https://github.com/spglib/spglib.text/htmlAlgorithms for magnetic symmetry operation search and identification of magnetic space group from magnetic crystal structuretext5792023-09-06Copyright (c) 2023 International Union of CrystallographyActa Crystallographica Section Aresearch papers390398Patch frequencies in rhombic Penrose tilings
http://scripts.iucr.org/cgi-bin/paper?nv5007
This exposition presents an efficient algorithm for an exact calculation of patch frequencies for rhombic Penrose tilings. A construction of Penrose tilings via dualization is recalled and, by extending the known method for obtaining vertex configurations, the desired algorithm is obtained. It is then used to determine the frequencies of several particularly large patches which appear in the literature. An analogous approach works for a particular class of tilings and this is also explained in detail for the Ammann–Beenker tiling.Copyright (c) 2023 International Union of Crystallographyurn:issn:2053-2733Mazáč, J.2023-07-24doi:10.1107/S2053273323004990International Union of CrystallographyAn algorithm is presented for an exact calculation of patch frequencies for a family of tilings which can be obtained via dualization.ENpatch frequencytilingdualization methodThis exposition presents an efficient algorithm for an exact calculation of patch frequencies for rhombic Penrose tilings. A construction of Penrose tilings via dualization is recalled and, by extending the known method for obtaining vertex configurations, the desired algorithm is obtained. It is then used to determine the frequencies of several particularly large patches which appear in the literature. An analogous approach works for a particular class of tilings and this is also explained in detail for the Ammann–Beenker tiling.text/htmlPatch frequencies in rhombic Penrose tilingstext5792023-07-24Copyright (c) 2023 International Union of CrystallographyActa Crystallographica Section Aresearch papers399411Background optimization of powder electron diffraction for implementation of the e-PDF technique and study of the local structure of iron oxide nanocrystals
http://scripts.iucr.org/cgi-bin/paper?tw5001
The local structural characterization of iron oxide nanoparticles is explored using a total scattering analysis method known as pair distribution function (PDF) (also known as reduced density function) analysis. The PDF profiles are derived from background-corrected powder electron diffraction patterns (the e-PDF technique). Due to the strong Coulombic interaction between the electron beam and the sample, electron diffraction generally leads to multiple scattering, causing redistribution of intensities towards higher scattering angles and an increased background in the diffraction profile. In addition to this, the electron–specimen interaction gives rise to an undesirable inelastic scattering signal that contributes primarily to the background. The present work demonstrates the efficacy of a pre-treatment of the underlying complex background function, which is a combination of both incoherent multiple and inelastic scatterings that cannot be identical for different electron beam energies. Therefore, two different background subtraction approaches are proposed for the electron diffraction patterns acquired at 80 kV and 300 kV beam energies. From the least-square refinement (small-box modelling), both approaches are found to be very promising, leading to a successful implementation of the e-PDF technique to study the local structure of the considered nanomaterial.Copyright (c) 2023 International Union of Crystallographyurn:issn:2053-2733Mogili, N.V.V.Verissimo, N.C.Abeykoon, A.M.M.Bozin, E.S.Bettini, J.Leite, E.R.Souza Junior, J.B.2023-07-25doi:10.1107/S2053273323005107International Union of CrystallographyIn the case of small nanocrystals (about 4 nm), with a change in electron beam energy the influence of multiple/inelastic scattering on the background signal of powder electron diffraction patterns is evident. Adopting different background removal approaches at lower (80 kV) and higher (300 kV) electron beam energies, e-PDF (electron pair distribution function) G(r) profiles are extracted. From small-box modelling of the structural parameters related to the iron oxide nanoparticles considered, the applicability of the subtraction procedures is discussed.ENelectron powder diffractionbackground subtractionelectron pair distribution function (e-PDF)electron reduced density function (e-RDF)nanocrystalline maghemiteThe local structural characterization of iron oxide nanoparticles is explored using a total scattering analysis method known as pair distribution function (PDF) (also known as reduced density function) analysis. The PDF profiles are derived from background-corrected powder electron diffraction patterns (the e-PDF technique). Due to the strong Coulombic interaction between the electron beam and the sample, electron diffraction generally leads to multiple scattering, causing redistribution of intensities towards higher scattering angles and an increased background in the diffraction profile. In addition to this, the electron–specimen interaction gives rise to an undesirable inelastic scattering signal that contributes primarily to the background. The present work demonstrates the efficacy of a pre-treatment of the underlying complex background function, which is a combination of both incoherent multiple and inelastic scatterings that cannot be identical for different electron beam energies. Therefore, two different background subtraction approaches are proposed for the electron diffraction patterns acquired at 80 kV and 300 kV beam energies. From the least-square refinement (small-box modelling), both approaches are found to be very promising, leading to a successful implementation of the e-PDF technique to study the local structure of the considered nanomaterial.text/htmlBackground optimization of powder electron diffraction for implementation of the e-PDF technique and study of the local structure of iron oxide nanocrystalstext5792023-07-25Copyright (c) 2023 International Union of CrystallographyActa Crystallographica Section Aresearch papers412426Optimal estimated standard uncertainties of reflection intensities for kinematical refinement from 3D electron diffraction data
http://scripts.iucr.org/cgi-bin/paper?pl5027
Estimating the error in the merged reflection intensities requires a full understanding of all the possible sources of error arising from the measurements. Most diffraction-spot integration methods focus mainly on errors arising from counting statistics for the estimation of uncertainties associated with the reflection intensities. This treatment may be incomplete and partly inadequate. In an attempt to fully understand and identify all the contributions to these errors, three methods are examined for the correction of estimated errors of reflection intensities in electron diffraction data. For a direct comparison, the three methods are applied to a set of organic and inorganic test cases. It is demonstrated that applying the corrections of a specific model that include terms dependent on the original uncertainty and the largest intensity of the symmetry-related reflections improves the overall structure quality of the given data set and improves the final Rall factor. This error model is implemented in the data reduction software PETS2.Copyright (c) 2023 International Union of Crystallographyurn:issn:2053-2733Khouchen, M.Klar, P.B.Chintakindi, H.Suresh, A.Palatinus, L.2023-08-14doi:10.1107/S2053273323005053International Union of CrystallographySeveral models for estimating the standard uncertainties of reflection intensities are analysed for refinement against 3D electron diffraction data. A new model is proposed which results in more accurate structure models.ENerror modellingerror analysisdata reductionelectron diffractionEstimating the error in the merged reflection intensities requires a full understanding of all the possible sources of error arising from the measurements. Most diffraction-spot integration methods focus mainly on errors arising from counting statistics for the estimation of uncertainties associated with the reflection intensities. This treatment may be incomplete and partly inadequate. In an attempt to fully understand and identify all the contributions to these errors, three methods are examined for the correction of estimated errors of reflection intensities in electron diffraction data. For a direct comparison, the three methods are applied to a set of organic and inorganic test cases. It is demonstrated that applying the corrections of a specific model that include terms dependent on the original uncertainty and the largest intensity of the symmetry-related reflections improves the overall structure quality of the given data set and improves the final Rall factor. This error model is implemented in the data reduction software PETS2.text/htmlOptimal estimated standard uncertainties of reflection intensities for kinematical refinement from 3D electron diffraction datatext5792023-08-14Copyright (c) 2023 International Union of CrystallographyActa Crystallographica Section Aresearch papers427439226816422681652268166226816722681682268169226817022681712268172226817322681742268175226817622681772268178Perfect precise colorings of plane semiregular tilings
http://scripts.iucr.org/cgi-bin/paper?nv5008
A coloring of a planar semiregular tiling {\cal T} is an assignment of a unique color to each tile of {\cal T}. If G is the symmetry group of {\cal T}, the coloring is said to be perfect if every element of G induces a permutation on the finite set of colors. If {\cal T} is k-valent, then a coloring of {\cal T} with k colors is said to be precise if no two tiles of {\cal T} sharing the same vertex have the same color. In this work, perfect precise colorings are obtained for some families of k-valent semiregular tilings in the plane, where k ≤ 6.Copyright (c) 2023 International Union of Crystallographyurn:issn:2053-2733Loquias, M.J.C.Santos, R.B.2023-08-17doi:10.1107/S2053273323006630International Union of CrystallographyThis contribution outlines how to obtain perfect precise colorings of tilings, and illustrates this by obtaining perfect precise colorings of some families of k-valent semiregular planar tilings with k colors.ENsemiregular tilingshyperbolic tilingsperfect coloringsprecise coloringstriangle groupsA coloring of a planar semiregular tiling {\cal T} is an assignment of a unique color to each tile of {\cal T}. If G is the symmetry group of {\cal T}, the coloring is said to be perfect if every element of G induces a permutation on the finite set of colors. If {\cal T} is k-valent, then a coloring of {\cal T} with k colors is said to be precise if no two tiles of {\cal T} sharing the same vertex have the same color. In this work, perfect precise colorings are obtained for some families of k-valent semiregular tilings in the plane, where k ≤ 6.text/htmlPerfect precise colorings of plane semiregular tilingstext5792023-08-17Copyright (c) 2023 International Union of CrystallographyActa Crystallographica Section Aresearch papers440451Distances in the face-centered cubic crystalline structure applying operational research
http://scripts.iucr.org/cgi-bin/paper?nv5006
The f.c.c. (face-centered cubic) grid is the structure of many crystals and minerals. It consists of four cubic lattices. It is supposed that there are two types of steps between two grid points. It is possible to step to one of the nearest neighbors of the same cubic lattice (type 1) or to step to one of the nearest neighbors of another cubic lattice (type 2). Steps belonging to the same type have the same length (weight). However, the two types have different lengths and thus may have different weights. This paper discusses the minimal path between any two points of the f.c.c. grid. The minimal paths are explicitly given, i.e. to obtain a minimal path one is required to perform only O(1) computations. The mathematical problem can be the model of different spreading phenomena in crystals having the f.c.c. structure.Copyright (c) 2023 International Union of Crystallographyurn:issn:2053-2733Stomfai, G.Kovacs, G.Nagy, B.Turgay, N.D.Vizvari, B.2023-08-25doi:10.1107/S2053273323004837International Union of CrystallographyThe face-centered cubic lattice describes the crystalline structure of various materials. Based on two weights assigned to the two types of neighbors among the nodes (representing, e.g., ions), formulae of chamfer distances are computed with the help of the simplex method.ENf.c.c. latticechamfer distancesshortest pathsimplex methodGomory methodThe f.c.c. (face-centered cubic) grid is the structure of many crystals and minerals. It consists of four cubic lattices. It is supposed that there are two types of steps between two grid points. It is possible to step to one of the nearest neighbors of the same cubic lattice (type 1) or to step to one of the nearest neighbors of another cubic lattice (type 2). Steps belonging to the same type have the same length (weight). However, the two types have different lengths and thus may have different weights. This paper discusses the minimal path between any two points of the f.c.c. grid. The minimal paths are explicitly given, i.e. to obtain a minimal path one is required to perform only O(1) computations. The mathematical problem can be the model of different spreading phenomena in crystals having the f.c.c. structure.text/htmlDistances in the face-centered cubic crystalline structure applying operational researchtext5792023-08-25Copyright (c) 2023 International Union of CrystallographyActa Crystallographica Section Aresearch papers452462Combinatorial aspects of the Löwenstein avoidance rule. Part III: the relational system of configurations
http://scripts.iucr.org/cgi-bin/paper?ae5129
This paper introduces a new method of determining the independence ratio of periodic nets, based on the observation that, in any maximum independent set of the whole net, be it periodic or not, the vertices of every unit cell should constitute an independent set, called here a configuration. For 1-periodic graphs, a configuration digraph represents possible sequences of configurations of the unit cell along the periodic line. It is shown that maximum independent sets of the periodic graph are based on directed cycles with the largest ratio. In the case of 2-periodic nets, it is necessary to draw a different configuration digraph for each crystallographic direction defining a linkage between neighbouring cells, a concept known as a binary relational system. The two possible systems are analysed in this paper: \overrightarrow{\bf{sql}} is associated to nets displaying linkages between unit cells along the directions 10 and 01, and \overrightarrow{\bf{hxl}} is associated to nets also displaying linkages between cells along the direction 11. For both kinds of nets, a maximum independent set is obtained as a homomorphic image from \overrightarrow{\bf{sql}} or \overrightarrow{\bf{hxl}} to the respective configuration system. The method is illustrated with some of the 2-periodic nets listed on the Reticular Chemistry Structure Resource site; it is shown that it provides a rigorous solution to the case of the net sdh that was not satisfactorily solved in Part II [Moreira de Oliveira, de Abreu Mendes & Eon (2022). Acta Cryst. A78, 115–127]. The method is extended to relational systems based on non-translational symmetry operations. The successive steps are then summarized and a simple application to the 3-periodic net qtz is discussed; analysis of zeolites and aluminosilicates may proceed along the same lines. It is shown that the new method enables the analysis of disordered distributions in periodic nets.Copyright (c) 2023 International Union of Crystallographyurn:issn:2053-2733Moreira de Oliveira Jr, M.Eon, J.-G.2023-08-25doi:10.1107/S2053273323006174International Union of CrystallographyMaximum independent sets in periodic nets are constructed as homomorphic images of Cayley colour graphs. The method applies to zeolites and aluminosilicates.ENmaximum independent setsperiodic netsquotient graphsrelational systemsThis paper introduces a new method of determining the independence ratio of periodic nets, based on the observation that, in any maximum independent set of the whole net, be it periodic or not, the vertices of every unit cell should constitute an independent set, called here a configuration. For 1-periodic graphs, a configuration digraph represents possible sequences of configurations of the unit cell along the periodic line. It is shown that maximum independent sets of the periodic graph are based on directed cycles with the largest ratio. In the case of 2-periodic nets, it is necessary to draw a different configuration digraph for each crystallographic direction defining a linkage between neighbouring cells, a concept known as a binary relational system. The two possible systems are analysed in this paper: \overrightarrow{\bf{sql}} is associated to nets displaying linkages between unit cells along the directions 10 and 01, and \overrightarrow{\bf{hxl}} is associated to nets also displaying linkages between cells along the direction 11. For both kinds of nets, a maximum independent set is obtained as a homomorphic image from \overrightarrow{\bf{sql}} or \overrightarrow{\bf{hxl}} to the respective configuration system. The method is illustrated with some of the 2-periodic nets listed on the Reticular Chemistry Structure Resource site; it is shown that it provides a rigorous solution to the case of the net sdh that was not satisfactorily solved in Part II [Moreira de Oliveira, de Abreu Mendes & Eon (2022). Acta Cryst. A78, 115–127]. The method is extended to relational systems based on non-translational symmetry operations. The successive steps are then summarized and a simple application to the 3-periodic net qtz is discussed; analysis of zeolites and aluminosilicates may proceed along the same lines. It is shown that the new method enables the analysis of disordered distributions in periodic nets.text/htmlCombinatorial aspects of the Löwenstein avoidance rule. Part III: the relational system of configurationstext5792023-08-25Copyright (c) 2023 International Union of CrystallographyActa Crystallographica Section Aresearch papers463479Approximating lattice similarity
http://scripts.iucr.org/cgi-bin/paper?uv5018
A method is proposed for choosing unit cells for a group of crystals so that they all appear as nearly similar as possible to a selected cell. Related unit cells with varying cell parameters or indexed with different lattice centering can be accommodated.Copyright (c) 2023 International Union of Crystallographyurn:issn:2053-2733Andrews, L.C.Bernstein, H.J.Sauter, N.K.2023-07-24doi:10.1107/S2053273323003200International Union of CrystallographyA method is proposed for transforming unit cells for a group of crystals so that they all appear as similar as possible to a selected cell.ENlattice matchingDelaunayDeloneNiggliSellingA method is proposed for choosing unit cells for a group of crystals so that they all appear as nearly similar as possible to a selected cell. Related unit cells with varying cell parameters or indexed with different lattice centering can be accommodated.text/htmlApproximating lattice similaritytext5792023-07-24Copyright (c) 2023 International Union of CrystallographyActa Crystallographica Section Aresearch papers480484Measuring lattices
http://scripts.iucr.org/cgi-bin/paper?uv5016
Unit cells are used to represent crystallographic lattices. Calculations measuring the differences between unit cells are used to provide metrics for measuring meaningful distances between three-dimensional crystallographic lattices. This is a surprisingly complex and computationally demanding problem. A review is presented of the current best practice using Delaunay-reduced unit cells in the six-dimensional real space of Selling scalar cells S6 and the equivalent three-dimensional complex space C3. The process is a simplified version of the process needed when working with the more complex six-dimensional real space of Niggli-reduced unit cells G6. Obtaining a distance begins with identification of the fundamental region in the space, continues with conversion to primitive cells and reduction, analysis of distances to the boundaries of the fundamental unit, and is completed by a comparison of direct paths with boundary-interrupted paths, looking for a path of minimal length.Copyright (c) 2023 International Union of Crystallographyurn:issn:2053-2733Andrews, L.C.Bernstein, H.J.2023-08-10doi:10.1107/S2053273323004692International Union of CrystallographyUnit cells are used to represent crystallographic lattices. Calculations measuring the differences between unit cells are used to provide metrics for measuring meaningful distances between three-dimensional crystallographic lattices. This is a surprisingly complex and computationally demanding problem. A review is presented of the current best practice using Delaunay-reduced unit cells in the six-dimensional real space of Selling scalar cells S6 and the equivalent three-dimensional complex space C3.ENDelaunayDeloneSellinglatticesunit cellUnit cells are used to represent crystallographic lattices. Calculations measuring the differences between unit cells are used to provide metrics for measuring meaningful distances between three-dimensional crystallographic lattices. This is a surprisingly complex and computationally demanding problem. A review is presented of the current best practice using Delaunay-reduced unit cells in the six-dimensional real space of Selling scalar cells S6 and the equivalent three-dimensional complex space C3. The process is a simplified version of the process needed when working with the more complex six-dimensional real space of Niggli-reduced unit cells G6. Obtaining a distance begins with identification of the fundamental region in the space, continues with conversion to primitive cells and reduction, analysis of distances to the boundaries of the fundamental unit, and is completed by a comparison of direct paths with boundary-interrupted paths, looking for a path of minimal length.text/htmlMeasuring latticestext5792023-08-10Copyright (c) 2023 International Union of CrystallographyActa Crystallographica Section Aresearch papers485498Delone lattice studies in C3, the space of three complex variables
http://scripts.iucr.org/cgi-bin/paper?uv5019
The Delone (Selling) scalars, which are used in unit-cell reduction and in lattice-type determination, are studied in C3, the space of three complex variables. The three complex coordinate planes are composed of the six Delone scalars. The transformations at boundaries of the Selling-reduced orthant are described as matrices of operators. A graphical representation as the projections onto the three coordinates is described. Note, in his later publications, Boris Delaunay used the Russian version of his surname, Delone.Copyright (c) 2023 International Union of Crystallographyurn:issn:2053-2733Andrews, L.C.Bernstein, H.J.2023-08-10doi:10.1107/S2053273323006198International Union of CrystallographyThe space C3 is explained in more detail than in the original description. Boundary transformations of the fundamental unit are described in detail. A graphical presentation of the basic coordinates is described and illustrated.ENlatticesDeloneDelaunaycell reductionSellingC3 spaceThe Delone (Selling) scalars, which are used in unit-cell reduction and in lattice-type determination, are studied in C3, the space of three complex variables. The three complex coordinate planes are composed of the six Delone scalars. The transformations at boundaries of the Selling-reduced orthant are described as matrices of operators. A graphical representation as the projections onto the three coordinates is described. Note, in his later publications, Boris Delaunay used the Russian version of his surname, Delone.text/htmlDelone lattice studies in C3, the space of three complex variablestext5792023-08-10Copyright (c) 2023 International Union of CrystallographyActa Crystallographica Section Aresearch papers499503