Open-access and free articles in Acta Crystallographica Section A: Foundations and Advances
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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.en-gbCopyright (c) 2021 International Union of CrystallographyInternational Union of CrystallographyInternational Union of Crystallographyhttps://journals.iucr.orgurn:issn:0108-7673Acta 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/htmlOpen-access and free articles in Acta Crystallographica Section A Foundations and Advancestextyearly62002-01-01T00:00+00:00med@iucr.orgActa Crystallographica Section A Foundations and AdvancesCopyright (c) 2021 International Union of Crystallographyurn:issn:0108-7673Open-access and free articles in Acta Crystallographica Section A: Foundations and Advanceshttp://journals.iucr.org/logos/rss10a.gif
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Still imageSmall-angle X-ray scattering from GaN nanowires on Si(111): facet truncation rods, facet roughness and Porod's law
http://scripts.iucr.org/cgi-bin/paper?iv5011
Small-angle X-ray scattering from GaN nanowires grown on Si(111) is measured in the grazing-incidence geometry and modelled by means of a Monte Carlo simulation that takes into account the orientational distribution of the faceted nanowires and the roughness of their side facets. It is found that the scattering intensity at large wavevectors does not follow Porod's law I(q) ∝ q−4. The intensity depends on the orientation of the side facets with respect to the incident X-ray beam. It is maximum when the scattering vector is directed along a facet normal, reminiscent of surface truncation rod scattering. At large wavevectors q, the scattering intensity is reduced by surface roughness. A root-mean-square roughness of 0.9 nm, which is the height of just 3–4 atomic steps per micrometre-long facet, already gives rise to a strong intensity reduction.https://creativecommons.org/licenses/by/4.0/urn:issn:2053-2733Kaganer, V.M.Konovalov, O.V.Fernández-Garrido, S.2021-01-05doi:10.1107/S205327332001548XInternational Union of CrystallographyThe intensity of small-angle X-ray scattering from GaN nanowires on Si(111) depends on the orientation of the side facets with respect to the incident beam. This reminiscence of truncation rod scattering gives rise to a deviation from Porod's law. A roughness of just 3–4 atomic steps per micrometre-long side facet notably changes the intensity curves.enNANOWIRES; POROD'S LAW; FACET TRUNCATION RODS; SMALL-ANGLE X-RAY SCATTERING; SAXS; GRAZING-INCIDENCE SMALL-ANGLE X-RAY SCATTERING; GISAXSSmall-angle X-ray scattering from GaN nanowires grown on Si(111) is measured in the grazing-incidence geometry and modelled by means of a Monte Carlo simulation that takes into account the orientational distribution of the faceted nanowires and the roughness of their side facets. It is found that the scattering intensity at large wavevectors does not follow Porod's law I(q) ∝ q−4. The intensity depends on the orientation of the side facets with respect to the incident X-ray beam. It is maximum when the scattering vector is directed along a facet normal, reminiscent of surface truncation rod scattering. At large wavevectors q, the scattering intensity is reduced by surface roughness. A root-mean-square roughness of 0.9 nm, which is the height of just 3–4 atomic steps per micrometre-long facet, already gives rise to a strong intensity reduction.text/htmlSmall-angle X-ray scattering from GaN nanowires on Si(111): facet truncation rods, facet roughness and Porod's lawtext1772021-01-05Acta Crystallographica Section A: Foundations and Advanceshttps://creativecommons.org/licenses/by/4.0/2053-2733research papers42med@iucr.orgJanuary 2021532053-2733Macromolecular phasing using diffraction from multiple crystal forms
http://scripts.iucr.org/cgi-bin/paper?sc5137
A phasing algorithm for macromolecular crystallography is proposed that utilizes diffraction data from multiple crystal forms – crystals of the same molecule with different unit-cell packings (different unit-cell parameters or space-group symmetries). The approach is based on the method of iterated projections, starting with no initial phase information. The practicality of the method is demonstrated by simulation using known structures that exist in multiple crystal forms, assuming some information on the molecular envelope and positional relationships between the molecules in the different unit cells. With incorporation of new or existing methods for determination of these parameters, the approach has potential as a method for ab initio phasing.https://creativecommons.org/licenses/by/4.0/urn:issn:2053-2733Metz, M.Arnal, R.D.Brehm, W.Chapman, H.N.Morgan, A.J.Millane, R.P.2021-01-05doi:10.1107/S2053273320013650International Union of CrystallographyA phasing algorithm for protein crystallography using diffraction data from multiple crystal forms is proposed. The algorithm is evaluated by simulation, and practical aspects and potential for ab initio phasing are discussed.enMULTIPLE CRYSTAL FORMS; AB INITIO PHASING; ITERATIVE PROJECTION ALGORITHMS; X-RAY FREE-ELECTRON LASERS; XFELSA phasing algorithm for macromolecular crystallography is proposed that utilizes diffraction data from multiple crystal forms – crystals of the same molecule with different unit-cell packings (different unit-cell parameters or space-group symmetries). The approach is based on the method of iterated projections, starting with no initial phase information. The practicality of the method is demonstrated by simulation using known structures that exist in multiple crystal forms, assuming some information on the molecular envelope and positional relationships between the molecules in the different unit cells. With incorporation of new or existing methods for determination of these parameters, the approach has potential as a method for ab initio phasing.text/htmlMacromolecular phasing using diffraction from multiple crystal formstext1772021-01-05Acta Crystallographica Section A: Foundations and Advanceshttps://creativecommons.org/licenses/by/4.0/2053-2733research papers19med@iucr.orgJanuary 2021352053-2733A cloud platform for atomic pair distribution function analysis: PDFitc
http://scripts.iucr.org/cgi-bin/paper?ae5091
A cloud web platform for analysis and interpretation of atomic pair distribution function (PDF) data (PDFitc) is described. The platform is able to host applications for PDF analysis to help researchers study the local and nanoscale structure of nanostructured materials. The applications are designed to be powerful and easy to use and can, and will, be extended over time through community adoption and development. The currently available PDF analysis applications, structureMining, spacegroupMining and similarityMapping, are described. In the first and second the user uploads a single PDF and the application returns a list of best-fit candidate structures, and the most likely space group of the underlying structure, respectively. In the third, the user can upload a set of measured or calculated PDFs and the application returns a matrix of Pearson correlations, allowing assessment of the similarity between different data sets. structureMining is presented here as an example to show the easy-to-use workflow on PDFitc. In the future, as well as using the PDFitc applications for data analysis, it is hoped that the community will contribute their own codes and software to the platform.https://creativecommons.org/licenses/by/4.0/urn:issn:2053-2733Yang, L.Culbertson, E.A.Thomas, N.K.Vuong, H.T.Kjær, E.T.S.Jensen, K.M.Ø.Tucker, M.G.Billinge, S.J.L.2021-01-05doi:10.1107/S2053273320013066International Union of CrystallographyA new web platform is presented for the pair distribution function (PDF) community to use and share advanced PDF analysis software in the cloud.enPAIR DISTRIBUTION FUNCTION; PDF; DATA ANALYSIS; WEB APPLICATIONS; CLOUD COMPUTINGA cloud web platform for analysis and interpretation of atomic pair distribution function (PDF) data (PDFitc) is described. The platform is able to host applications for PDF analysis to help researchers study the local and nanoscale structure of nanostructured materials. The applications are designed to be powerful and easy to use and can, and will, be extended over time through community adoption and development. The currently available PDF analysis applications, structureMining, spacegroupMining and similarityMapping, are described. In the first and second the user uploads a single PDF and the application returns a list of best-fit candidate structures, and the most likely space group of the underlying structure, respectively. In the third, the user can upload a set of measured or calculated PDFs and the application returns a matrix of Pearson correlations, allowing assessment of the similarity between different data sets. structureMining is presented here as an example to show the easy-to-use workflow on PDFitc. In the future, as well as using the PDFitc applications for data analysis, it is hoped that the community will contribute their own codes and software to the platform.text/htmlA cloud platform for atomic pair distribution function analysis: PDFitctext1772021-01-05Acta Crystallographica Section A: Foundations and Advanceshttps://creativecommons.org/licenses/by/4.0/2053-2733research papers2med@iucr.orgJanuary 202162053-2733Algorithms for target transformations of lattice basis vectors
http://scripts.iucr.org/cgi-bin/paper?ae5090
Simple algorithms are proposed for the transformation of lattice basis vectors to a specific target. In the first case, one of the new basis vectors is aligned to a predefined lattice direction, while in the second case, two of the new basis vectors are brought to a lattice plane with predefined Miller indices. The multi-dimensional generalization of the algorithm is available in the supporting materials. The algorithms are useful for such crystallographic operations as simulation of zone planes (i.e. geometry of electron diffraction patterns) or transformation of a unit cell for surface or cleavage energy calculations. The most general multi-dimensional version of the algorithm may be useful for the analysis of quasiperiodic crystals or as an alternative method of calculating Bézout coefficients. The algorithms are demonstrated both graphically and numerically.https://creativecommons.org/licenses/by/4.0/urn:issn:2053-2733Gorfman, S.2020-10-29doi:10.1107/S2053273320012668International Union of CrystallographyPresented here are algorithms for the transformation of lattice basis vectors to a specific target. The algorithms are useful for crystallographic operations in direct and reciprocal spaces alike. The algorithms are demonstrated graphically and numerically.enCRYSTAL LATTICE; TRANSFORMATIONS; LATTICE PLANES; ZONESSimple algorithms are proposed for the transformation of lattice basis vectors to a specific target. In the first case, one of the new basis vectors is aligned to a predefined lattice direction, while in the second case, two of the new basis vectors are brought to a lattice plane with predefined Miller indices. The multi-dimensional generalization of the algorithm is available in the supporting materials. The algorithms are useful for such crystallographic operations as simulation of zone planes (i.e. geometry of electron diffraction patterns) or transformation of a unit cell for surface or cleavage energy calculations. The most general multi-dimensional version of the algorithm may be useful for the analysis of quasiperiodic crystals or as an alternative method of calculating Bézout coefficients. The algorithms are demonstrated both graphically and numerically.text/htmlAlgorithms for target transformations of lattice basis vectorstext6762020-10-29Acta Crystallographica Section A: Foundations and Advanceshttps://creativecommons.org/licenses/by/4.0/2053-2733research papers713med@iucr.orgNovember 20207182053-2733A flexible and standalone forward simulation model for laboratory X-ray diffraction contrast tomography
http://scripts.iucr.org/cgi-bin/paper?iv5008
Laboratory X-ray diffraction contrast tomography (LabDCT) has recently been developed as a powerful technique for non-destructive mapping of grain microstructures in bulk materials. As the grain reconstruction relies on segmentation of diffraction spots, it is essential to understand the physics of the diffraction process and resolve all the spot features in detail. To this aim, a flexible and standalone forward simulation model has been developed to compute the diffraction projections from polycrystalline samples with any crystal structure. The accuracy of the forward simulation model is demonstrated by good agreements in grain orientations, boundary positions and shapes between a virtual input structure and that reconstructed based on the forward simulated diffraction projections of the input structure. Further experimental verification is made by comparisons of diffraction spots between simulations and experiments for a partially recrystallized Al sample, where a satisfactory agreement is found for the spot positions, sizes and intensities. Finally, applications of this model to analyze specific spot features are presented.https://creativecommons.org/licenses/by/4.0/urn:issn:2053-2733Fang, H.Juul Jensen, D.Zhang, Y.2020-09-18doi:10.1107/S2053273320010852International Union of CrystallographyA flexible and standalone forward simulation model has been developed to compute the diffraction projections for laboratory diffraction contrast tomography (LabDCT). The outputs are expected to be of great value for all present users of LabDCT as well as interested new users.en3D GRAIN MAPPING; DIFFRACTION CONTRAST TOMOGRAPHY; X-RAY DIFFRACTION; FORWARD SIMULATION; GRAIN RECONSTRUCTIONLaboratory X-ray diffraction contrast tomography (LabDCT) has recently been developed as a powerful technique for non-destructive mapping of grain microstructures in bulk materials. As the grain reconstruction relies on segmentation of diffraction spots, it is essential to understand the physics of the diffraction process and resolve all the spot features in detail. To this aim, a flexible and standalone forward simulation model has been developed to compute the diffraction projections from polycrystalline samples with any crystal structure. The accuracy of the forward simulation model is demonstrated by good agreements in grain orientations, boundary positions and shapes between a virtual input structure and that reconstructed based on the forward simulated diffraction projections of the input structure. Further experimental verification is made by comparisons of diffraction spots between simulations and experiments for a partially recrystallized Al sample, where a satisfactory agreement is found for the spot positions, sizes and intensities. Finally, applications of this model to analyze specific spot features are presented.text/htmlA flexible and standalone forward simulation model for laboratory X-ray diffraction contrast tomographytext6762020-09-18Acta Crystallographica Section A: Foundations and Advanceshttps://creativecommons.org/licenses/by/4.0/2053-2733research papers652med@iucr.orgNovember 20206632053-2733Quaternions: what are they, and why do we need to know?
http://scripts.iucr.org/cgi-bin/paper?me6092
Copyright (c) 2020 International Union of Crystallographyurn:issn:2053-2733Horn, B.K.P.2020-08-06doi:10.1107/S2053273320010359International Union of CrystallographyThe significance of the work by A. J. Hanson [Acta Cryst. (2020), A76, 432–457] on finding the optimal alignment of pairs of spatial and/or orientation data sets is discussed.enQUATERNIONS; DATA ALIGNMENT; ROTATION; ORIENTATION; ORTHOGONAL PROCRUSTES PROBLEM; ORIENTATION DISTRIBUTION FUNCTION; ODFtext/htmlQuaternions: what are they, and why do we need to know?text765Copyright (c) 2020 International Union of CrystallographyActa Crystallographica Section A: Foundations and Advances2020-08-06556scientific commentaries2053-2733September 2020med@iucr.org5582053-2733Pure discrete spectrum and regular model sets in d-dimensional unimodular substitution tilings
http://scripts.iucr.org/cgi-bin/paper?ug5007
Primitive substitution tilings on {\bb R}^d whose expansion maps are unimodular are considered. It is assumed that all the eigenvalues of the expansion maps are algebraic conjugates with the same multiplicity. In this case, a cut-and-project scheme can be constructed with a Euclidean internal space. Under some additional condition, it is shown that if the substitution tiling has pure discrete spectrum, then the corresponding representative point sets are regular model sets in that cut-and-project scheme.https://creativecommons.org/licenses/by/4.0/urn:issn:2053-2733Lee, D.-I.Akiyama, S.Lee, J.-Y.2020-08-21doi:10.1107/S2053273320009717International Union of CrystallographyThe equivalence between pure discrete spectrum and regular model sets in d-dimensional unimodular substitution tilings is discussed.enPISOT FAMILY SUBSTITUTION TILINGS; PURE DISCRETE SPECTRUM; REGULAR MODEL SETS; MEYER SETS; RIGIDITYPrimitive substitution tilings on {\bb R}^d whose expansion maps are unimodular are considered. It is assumed that all the eigenvalues of the expansion maps are algebraic conjugates with the same multiplicity. In this case, a cut-and-project scheme can be constructed with a Euclidean internal space. Under some additional condition, it is shown that if the substitution tiling has pure discrete spectrum, then the corresponding representative point sets are regular model sets in that cut-and-project scheme.text/htmlPure discrete spectrum and regular model sets in d-dimensional unimodular substitution tilingstext5762020-08-21Acta Crystallographica Section A: Foundations and Advanceshttps://creativecommons.org/licenses/by/4.0/2053-2733research papers600med@iucr.orgSeptember 20206102053-2733Embedding-theory-based simulations using experimental electron densities for the environment
http://scripts.iucr.org/cgi-bin/paper?ug5015
The basic idea of frozen-density embedding theory (FDET) is the constrained minimization of the Hohenberg–Kohn density functional EHK[ρ] performed using the auxiliary functional E_{v_{AB}}^{\rm FDET}[\Psi _A, \rho _B], where ΨA is the embedded NA-electron wavefunction and ρB(r) is a non-negative function in real space integrating to a given number of electrons NB. This choice of independent variables in the total energy functional E_{v_{AB}}^{\rm FDET}[\Psi _A, \rho _B] makes it possible to treat the corresponding two components of the total density using different methods in multi-level simulations. The application of FDET using ρB(r) reconstructed from X-ray diffraction data for a molecular crystal is demonstrated for the first time. For eight hydrogen-bonded clusters involving a chromophore (represented as ΨA) and the glycylglycine molecule [represented as ρB(r)], FDET is used to derive excitation energies. It is shown that experimental densities are suitable for use as ρB(r) in FDET-based simulations.https://creativecommons.org/licenses/by/4.0/urn:issn:2053-2733Ricardi, N.Ernst, M.Macchi, P.Wesolowski, T.A.2020-07-20doi:10.1107/S2053273320008062International Union of CrystallographyFor the first time, the use of experimentally derived molecular electron densities as ρB(r) in calculations based on frozen-density embedding theory (FDET) of environment-induced shifts of electronic excitations for chromophores in clusters is demonstrated. ρB(r) was derived from X-ray restrained molecular wavefunctions of glycylglycine to obtain environment densities for simulating electronic excitations in clusters.enQUANTUM CRYSTALLOGRAPHY; DENSITY EMBEDDING; MULTI-SCALE SIMULATIONS; ELECTRONIC STRUCTURE; CHROMOPHORESThe basic idea of frozen-density embedding theory (FDET) is the constrained minimization of the Hohenberg–Kohn density functional EHK[ρ] performed using the auxiliary functional E_{v_{AB}}^{\rm FDET}[\Psi _A, \rho _B], where ΨA is the embedded NA-electron wavefunction and ρB(r) is a non-negative function in real space integrating to a given number of electrons NB. This choice of independent variables in the total energy functional E_{v_{AB}}^{\rm FDET}[\Psi _A, \rho _B] makes it possible to treat the corresponding two components of the total density using different methods in multi-level simulations. The application of FDET using ρB(r) reconstructed from X-ray diffraction data for a molecular crystal is demonstrated for the first time. For eight hydrogen-bonded clusters involving a chromophore (represented as ΨA) and the glycylglycine molecule [represented as ρB(r)], FDET is used to derive excitation energies. It is shown that experimental densities are suitable for use as ρB(r) in FDET-based simulations.text/htmlEmbedding-theory-based simulations using experimental electron densities for the environmenttext765https://creativecommons.org/licenses/by/4.0/Acta Crystallographica Section A: Foundations and Advances2020-07-20571research papers2053-2733September 2020med@iucr.org5792053-2733Inflation versus projection sets in aperiodic systems: the role of the window in averaging and diffraction
http://scripts.iucr.org/cgi-bin/paper?ae5086
Tilings based on the cut-and-project method are key model systems for the description of aperiodic solids. Typically, quantities of interest in crystallography involve averaging over large patches, and are well defined only in the infinite-volume limit. In particular, this is the case for autocorrelation and diffraction measures. For cut-and-project systems, the averaging can conveniently be transferred to internal space, which means dealing with the corresponding windows. In this topical review, this is illustrated by the example of averaged shelling numbers for the Fibonacci tiling, and the standard approach to the diffraction for this example is recapitulated. Further, recent developments are discussed for cut-and-project structures with an inflation symmetry, which are based on an internal counterpart of the renormalization cocycle. Finally, a brief review is given of the notion of hyperuniformity, which has recently gained popularity, and its application to aperiodic structures.https://creativecommons.org/licenses/by/4.0/urn:issn:2053-2733Baake, M.Grimm, U.2020-07-09doi:10.1107/S2053273320007421International Union of CrystallographyAveraged quantities such as mean shelling numbers, scaling behaviour or diffraction for cut-and-project sets can conveniently be computed in internal space, also for systems with fractally bounded windows.enQUASICRYSTALS; PROJECTION METHOD; INFLATION RULES; DIFFRACTION; HYPERUNIFORMITYTilings based on the cut-and-project method are key model systems for the description of aperiodic solids. Typically, quantities of interest in crystallography involve averaging over large patches, and are well defined only in the infinite-volume limit. In particular, this is the case for autocorrelation and diffraction measures. For cut-and-project systems, the averaging can conveniently be transferred to internal space, which means dealing with the corresponding windows. In this topical review, this is illustrated by the example of averaged shelling numbers for the Fibonacci tiling, and the standard approach to the diffraction for this example is recapitulated. Further, recent developments are discussed for cut-and-project structures with an inflation symmetry, which are based on an internal counterpart of the renormalization cocycle. Finally, a brief review is given of the notion of hyperuniformity, which has recently gained popularity, and its application to aperiodic structures.text/htmlInflation versus projection sets in aperiodic systems: the role of the window in averaging and diffractiontext765https://creativecommons.org/licenses/by/4.0/Acta Crystallographica Section A: Foundations and Advances2020-07-09559topical reviews2053-2733September 2020med@iucr.org5702053-2733On Cayley graphs of {\bb Z}^4
http://scripts.iucr.org/cgi-bin/paper?eo5107
The generating sets of {\bb Z}^4 have been enumerated which consist of integral four-dimensional vectors with components −1, 0, 1 and allow Cayley graphs without edge intersections in a straight-edge embedding in a four-dimensional Euclidean space. Owing to computational restrictions the valency of enumerated graphs has been fixed to 10. Up to isomorphism 58 graphs have been found and characterized by coordination sequences, shortest cycles and automorphism groups. To compute automorphism groups, a novel strategy is introduced that is based on determining vertex stabilizers from the automorphism group of a sufficiently large finite ball cut out from an infinite graph. Six exceptional, rather `dense' graphs have been identified which are locally isomorphic to a five-dimensional cubic lattice within a ball of radius 10. They could be built by either interconnecting interpenetrated three- or four-dimensional cubic lattices and therefore necessarily contain Hopf links between quadrangular cycles. As a consequence, a local combinatorial isomorphism does not extend to a local isotopy.https://creativecommons.org/licenses/by/4.0/urn:issn:2053-2733Baburin, I.A.2020-07-16doi:10.1107/S2053273320007159International Union of CrystallographyCayley graphs of {\bb Z}^4 with valency 10 have been enumerated which correspond to generating sets of integral vectors with components −1, 0, 1 and which are embedded in a four-dimensional Euclidean space without edge intersections.enCAYLEY GRAPHS; FREE ABELIAN GROUPS; COMPUTATIONAL GROUP THEORY; VERTEX-TRANSITIVE GRAPHS; ISOTOPYThe generating sets of {\bb Z}^4 have been enumerated which consist of integral four-dimensional vectors with components −1, 0, 1 and allow Cayley graphs without edge intersections in a straight-edge embedding in a four-dimensional Euclidean space. Owing to computational restrictions the valency of enumerated graphs has been fixed to 10. Up to isomorphism 58 graphs have been found and characterized by coordination sequences, shortest cycles and automorphism groups. To compute automorphism groups, a novel strategy is introduced that is based on determining vertex stabilizers from the automorphism group of a sufficiently large finite ball cut out from an infinite graph. Six exceptional, rather `dense' graphs have been identified which are locally isomorphic to a five-dimensional cubic lattice within a ball of radius 10. They could be built by either interconnecting interpenetrated three- or four-dimensional cubic lattices and therefore necessarily contain Hopf links between quadrangular cycles. As a consequence, a local combinatorial isomorphism does not extend to a local isotopy.text/htmlOn Cayley graphs of {\bb Z}^4text765https://creativecommons.org/licenses/by/4.0/Acta Crystallographica Section A: Foundations and Advances2020-07-16584research papers2053-2733September 2020med@iucr.org5882053-2733Multiplicity-weighted Euler's formula for symmetrically arranged space-filling polyhedra
http://scripts.iucr.org/cgi-bin/paper?sc5138
The famous Euler's rule for three-dimensional polyhedra, F − E + V = 2 (F, E and V are the numbers of faces, edges and vertices, respectively), when extended to many tested cases of space-filling polyhedra such as the asymmetric unit (ASU), takes the form Fn − En + Vn = 1, where Fn, En and Vn enumerate the corresponding elements, normalized by their multiplicity, i.e. by the number of times they are repeated by the space-group symmetry. This modified formula holds for the ASUs of all 230 space groups and 17 two-dimensional planar groups as specified in the International Tables for Crystallography, and for a number of tested Dirichlet domains, suggesting that it may have a general character. The modification of the formula stems from the fact that in a symmetrical space-filling arrangement the polyhedra (such as the ASU) have incomplete bounding elements (faces, edges, vertices), since they are shared (in various degrees) with the space-filling neighbors.https://creativecommons.org/licenses/by/4.0/urn:issn:2053-2733Dauter, Z.Jaskolski, M.2020-07-09doi:10.1107/S2053273320007093International Union of CrystallographyFor many tested cases of identical space-filling polyhedra, such as the space-group-specific asymmetric units or Dirichlet domains, the numbers of their faces (Fn), edges (En) and vertices (Vn), in each case normalized by division by the multiplicity of their (potentially special) symmetry position, fulfill a modified Euler's formula Fn − En + Vn = 1.enASYMMETRIC UNIT; UNIT CELL; EULER'S FORMULA; SPACE-FILLING POLYHEDRA; DIRICHLET DOMAINSThe famous Euler's rule for three-dimensional polyhedra, F − E + V = 2 (F, E and V are the numbers of faces, edges and vertices, respectively), when extended to many tested cases of space-filling polyhedra such as the asymmetric unit (ASU), takes the form Fn − En + Vn = 1, where Fn, En and Vn enumerate the corresponding elements, normalized by their multiplicity, i.e. by the number of times they are repeated by the space-group symmetry. This modified formula holds for the ASUs of all 230 space groups and 17 two-dimensional planar groups as specified in the International Tables for Crystallography, and for a number of tested Dirichlet domains, suggesting that it may have a general character. The modification of the formula stems from the fact that in a symmetrical space-filling arrangement the polyhedra (such as the ASU) have incomplete bounding elements (faces, edges, vertices), since they are shared (in various degrees) with the space-filling neighbors.text/htmlMultiplicity-weighted Euler's formula for symmetrically arranged space-filling polyhedratext765https://creativecommons.org/licenses/by/4.0/Acta Crystallographica Section A: Foundations and Advances2020-07-09580research papers2053-2733September 2020med@iucr.org5832053-2733Isotopy classification of three-dimensional embedded nets
http://scripts.iucr.org/cgi-bin/paper?me6077
Copyright (c) 2020 International Union of Crystallographyurn:issn:2053-2733Schulte, E.2020-04-29doi:10.1107/S2053273320005616International Union of CrystallographyThe article by Power et al. [Acta Cryst. (2020), A76, 275–301] on the isotopy classification of crystal nets is discussed.enEMBEDDED NETS; ISOTOPY CLASSIFICATION; TOPOLOGYtext/htmlIsotopy classification of three-dimensional embedded netstext763Copyright (c) 2020 International Union of CrystallographyActa Crystallographica Section A: Foundations and Advances2020-04-29273scientific commentaries2053-2733May 2020med@iucr.org2742053-2733Theoretical study of the properties of X-ray diffraction moiré fringes. III. Theoretical simulation of previous experimental moiré images
http://scripts.iucr.org/cgi-bin/paper?td5063
As a practical confirmation of a recently published X-ray moiré-fringe theory [Yoshimura (2015). Acta Cryst. A71, 368–381], computer simulations using this theory were conducted for previous experimental moiré images of a strained bicrystal specimen [Yoshimura (1996). Acta Cryst. A52, 312–325]. Simulated moiré images with a good or fairly good likeness are presented as a result of this simulation, in which the characteristic fringe-and-band and local strain patterns in the experimental images are reproduced well. Experimental moiré images taken when the inclination of the lattice planes was forcedly increased in one of the component crystals of the bicrystal specimen were also fairly well simulated in this computation, and their fringe patterns of inclined fringes are shown to be in accordance with the prediction by the theory. This moiré-fringe theory is thus considered to be widely applicable to the study of moiré images. Furthermore, the successful simulation of the previous experimental moiré images means that a satisfactory theoretical explanation was given for the experimental images, with respect to their characteristic global features. However, this study by the theoretical simulation shows explicitly that some significant peculiarities in the fringe profiles of the experimental images still remain unexplained by this moiré-fringe theory.https://creativecommons.org/licenses/by/4.0/urn:issn:2053-2733Yoshimura, J.2020-06-30doi:10.1107/S205327332000532XInternational Union of CrystallographyUsing a recently developed moiré-fringe theory of X-ray diffraction, computer simulations of previous experimental moiré images are presented, for an experimental verification of the moiré-fringe theory and for a theoretical explanation of the peculiar experimental moiré images.enX-RAY MOIRE FRINGES; STRAINED CRYSTALS; LOW-CONTRAST BAND PATTERN; PECULIAR EXPERIMENTAL FRINGE PROFILESAs a practical confirmation of a recently published X-ray moiré-fringe theory [Yoshimura (2015). Acta Cryst. A71, 368–381], computer simulations using this theory were conducted for previous experimental moiré images of a strained bicrystal specimen [Yoshimura (1996). Acta Cryst. A52, 312–325]. Simulated moiré images with a good or fairly good likeness are presented as a result of this simulation, in which the characteristic fringe-and-band and local strain patterns in the experimental images are reproduced well. Experimental moiré images taken when the inclination of the lattice planes was forcedly increased in one of the component crystals of the bicrystal specimen were also fairly well simulated in this computation, and their fringe patterns of inclined fringes are shown to be in accordance with the prediction by the theory. This moiré-fringe theory is thus considered to be widely applicable to the study of moiré images. Furthermore, the successful simulation of the previous experimental moiré images means that a satisfactory theoretical explanation was given for the experimental images, with respect to their characteristic global features. However, this study by the theoretical simulation shows explicitly that some significant peculiarities in the fringe profiles of the experimental images still remain unexplained by this moiré-fringe theory.text/htmlTheoretical study of the properties of X-ray diffraction moiré fringes. III. Theoretical simulation of previous experimental moiré imagestext764https://creativecommons.org/licenses/by/4.0/Acta Crystallographica Section A: Foundations and Advances2020-06-30503research papers2053-2733July 2020med@iucr.org5202053-2733The quaternion-based spatial-coordinate and orientation-frame alignment problems
http://scripts.iucr.org/cgi-bin/paper?ib5072
The general problem of finding a global rotation that transforms a given set of spatial coordinates and/or orientation frames (the `test' data) into the best possible alignment with a corresponding set (the `reference' data) is reviewed. For 3D point data, this `orthogonal Procrustes problem' is often phrased in terms of minimizing a root-mean-square deviation (RMSD) corresponding to a Euclidean distance measure relating the two sets of matched coordinates. This article focuses on quaternion eigensystem methods that have been exploited to solve this problem for at least five decades in several different bodies of scientific literature, where they were discovered independently. While numerical methods for the eigenvalue solutions dominate much of this literature, it has long been realized that the quaternion-based RMSD optimization problem can also be solved using exact algebraic expressions based on the form of the quartic equation solution published by Cardano in 1545; focusing on these exact solutions exposes the structure of the entire eigensystem for the traditional 3D spatial-alignment problem. The structure of the less-studied orientation-data context is then explored, investigating how quaternion methods can be extended to solve the corresponding 3D quaternion orientation-frame alignment (QFA) problem, noting the interesting equivalence of this problem to the rotation-averaging problem, which also has been the subject of independent literature threads. The article concludes with a brief discussion of the combined 3D translation–orientation data alignment problem. Appendices are devoted to a tutorial on quaternion frames, a related quaternion technique for extracting quaternions from rotation matrices and a review of quaternion rotation-averaging methods relevant to the orientation-frame alignment problem. The supporting information covers novel extensions of quaternion methods to the 4D Euclidean spatial-coordinate alignment and 4D orientation-frame alignment problems, some miscellaneous topics, and additional details of the quartic algebraic eigenvalue problem.https://creativecommons.org/licenses/by/4.0/urn:issn:2053-2733Hanson, A.J.2020-06-18doi:10.1107/S2053273320002648International Union of CrystallographyQuaternion methods for obtaining solutions to the problem of finding global rotations that optimally align pairs of corresponding lists of 3D spatial and/or orientation data are critically studied. The existence of multiple literatures and historical contexts is pointed out, and the algebraic solutions of the quaternion approach to the classic 3D spatial problem are emphasized. The treatment is extended to novel quaternion-based solutions to the alignment problems for 4D spatial and orientation data.enDATA ALIGNMENT; SPATIAL-COORDINATE ALIGNMENT; ORIENTATION-FRAME ALIGNMENT; QUATERNIONS; QUATERNION FRAMES; QUATERNION EIGENVALUE METHODSThe general problem of finding a global rotation that transforms a given set of spatial coordinates and/or orientation frames (the `test' data) into the best possible alignment with a corresponding set (the `reference' data) is reviewed. For 3D point data, this `orthogonal Procrustes problem' is often phrased in terms of minimizing a root-mean-square deviation (RMSD) corresponding to a Euclidean distance measure relating the two sets of matched coordinates. This article focuses on quaternion eigensystem methods that have been exploited to solve this problem for at least five decades in several different bodies of scientific literature, where they were discovered independently. While numerical methods for the eigenvalue solutions dominate much of this literature, it has long been realized that the quaternion-based RMSD optimization problem can also be solved using exact algebraic expressions based on the form of the quartic equation solution published by Cardano in 1545; focusing on these exact solutions exposes the structure of the entire eigensystem for the traditional 3D spatial-alignment problem. The structure of the less-studied orientation-data context is then explored, investigating how quaternion methods can be extended to solve the corresponding 3D quaternion orientation-frame alignment (QFA) problem, noting the interesting equivalence of this problem to the rotation-averaging problem, which also has been the subject of independent literature threads. The article concludes with a brief discussion of the combined 3D translation–orientation data alignment problem. Appendices are devoted to a tutorial on quaternion frames, a related quaternion technique for extracting quaternions from rotation matrices and a review of quaternion rotation-averaging methods relevant to the orientation-frame alignment problem. The supporting information covers novel extensions of quaternion methods to the 4D Euclidean spatial-coordinate alignment and 4D orientation-frame alignment problems, some miscellaneous topics, and additional details of the quartic algebraic eigenvalue problem.text/htmlThe quaternion-based spatial-coordinate and orientation-frame alignment problemstext764https://creativecommons.org/licenses/by/4.0/Acta Crystallographica Section A: Foundations and Advances2020-06-18432lead articles2053-2733July 2020med@iucr.org4572053-2733Multiple Bragg reflection by a thick mosaic crystal. II. Simplified transport equation solved on a grid
http://scripts.iucr.org/cgi-bin/paper?ae5082
The generalized Darwin–Hamilton equations [Wuttke (2014). Acta Cryst. A70, 429–440] describe multiple Bragg reflection from a thick, ideally imperfect crystal. These equations are simplified by making full use of energy conservation, and it is demonstrated that the conventional two-ray Darwin–Hamilton equations are obtained as a first-order approximation. Then an efficient numeric solution method is presented, based on a transfer matrix for discretized directional distribution functions and on spectral collocation in the depth coordinate. Example solutions illustrate the orientational spread of multiply reflected rays and the distortion of rocking curves, especially if the detector only covers a finite solid angle.https://creativecommons.org/licenses/by/4.0/urn:issn:2053-2733Bornemann, F.Li, Y.Y.Wuttke, J.2020-04-16doi:10.1107/S2053273320002065International Union of CrystallographyTo describe multiple Bragg reflection from a thick, ideally imperfect crystal, the transport equations are reformulated in three-dimensional phase space and solved by spectral collocation in the depth coordinate. Example solutions illustrate the orientational spread of multiply reflected rays and the distortion of rocking curves, especially for finite detectors.enMOSAIC CRYSTALS; MULTIPLE SCATTERING; DARWIN-HAMILTON EQUATIONS; SPECTRAL COLLOCATIONThe generalized Darwin–Hamilton equations [Wuttke (2014). Acta Cryst. A70, 429–440] describe multiple Bragg reflection from a thick, ideally imperfect crystal. These equations are simplified by making full use of energy conservation, and it is demonstrated that the conventional two-ray Darwin–Hamilton equations are obtained as a first-order approximation. Then an efficient numeric solution method is presented, based on a transfer matrix for discretized directional distribution functions and on spectral collocation in the depth coordinate. Example solutions illustrate the orientational spread of multiply reflected rays and the distortion of rocking curves, especially if the detector only covers a finite solid angle.text/htmlMultiple Bragg reflection by a thick mosaic crystal. II. Simplified transport equation solved on a gridtext763https://creativecommons.org/licenses/by/4.0/Acta Crystallographica Section A: Foundations and Advances2020-04-16376research papers2053-2733May 2020med@iucr.org3892053-2733Structure-mining: screening structure models by automated fitting to the atomic pair distribution function over large numbers of models
http://scripts.iucr.org/cgi-bin/paper?vk5039
A new approach is presented to obtain candidate structures from atomic pair distribution function (PDF) data in a highly automated way. It fetches, from web-based structural databases, all the structures meeting the experimenter's search criteria and performs structure refinements on them without human intervention. It supports both X-ray and neutron PDFs. Tests on various material systems show the effectiveness and robustness of the algorithm in finding the correct atomic crystal structure. It works on crystalline and nanocrystalline materials including complex oxide nanoparticles and nanowires, low-symmetry and locally distorted structures, and complicated doped and magnetic materials. This approach could greatly reduce the traditional structure searching work and enable the possibility of high-throughput real-time auto-analysis PDF experiments in the future.https://creativecommons.org/licenses/by/4.0/urn:issn:2053-2733Yang, L.Juhás, P.Terban, M.W.Tucker, M.G.Billinge, S.J.L.2020-04-28doi:10.1107/S2053273320002028International Union of CrystallographyStructure-mining finds and returns the best-fit structures from structural databases given a measured pair distribution function data set. Using databases and heuristics for automation, it has the potential to save experimenters a large amount of time as they explore candidate structures from the literature.enPAIR DISTRIBUTION FUNCTION; PDF; AUTOMATED FITTING; ATOMIC STRUCTURE; STRUCTURE DISCOVERYA new approach is presented to obtain candidate structures from atomic pair distribution function (PDF) data in a highly automated way. It fetches, from web-based structural databases, all the structures meeting the experimenter's search criteria and performs structure refinements on them without human intervention. It supports both X-ray and neutron PDFs. Tests on various material systems show the effectiveness and robustness of the algorithm in finding the correct atomic crystal structure. It works on crystalline and nanocrystalline materials including complex oxide nanoparticles and nanowires, low-symmetry and locally distorted structures, and complicated doped and magnetic materials. This approach could greatly reduce the traditional structure searching work and enable the possibility of high-throughput real-time auto-analysis PDF experiments in the future.text/htmlStructure-mining: screening structure models by automated fitting to the atomic pair distribution function over large numbers of modelstext763https://creativecommons.org/licenses/by/4.0/Acta Crystallographica Section A: Foundations and Advances2020-04-28395research papers2053-2733May 2020med@iucr.org4092053-2733SPIND-TC: an indexing method for two-color X-ray diffraction data
http://scripts.iucr.org/cgi-bin/paper?ib5084
Recent developments of two-color operation modes at X-ray free-electron laser facilities provide new research opportunities, such as X-ray pump/X-ray probe experiments and multiple-wavelength anomalous dispersion phasing methods. However, most existing indexing methods were developed for indexing diffraction data from monochromatic X-ray beams. Here, a new algorithm is presented for indexing two-color diffraction data, as an extension of the sparse-pattern indexing algorithm SPIND, which has been demonstrated to be capable of indexing diffraction patterns with as few as five peaks. The principle and implementation of the two-color indexing method, SPIND-TC, are reported in this paper. The algorithm was tested on both simulated and experimental data of protein crystals. The results show that the diffraction data can be accurately indexed in both cases. Source codes are publicly available at https://github.com/lixx11/SPIND-TC.https://creativecommons.org/licenses/by/4.0/urn:issn:2053-2733Li, X.Li, C.Liu, H.2020-04-02doi:10.1107/S2053273320001916International Union of CrystallographyAn auto-indexing method for two-color X-ray diffraction data is presented, which has been tested on both simulated and experimental protein diffraction data. The indexing yield is increased significantly compared with the previous approach using conventional indexers.enSERIAL CRYSTALLOGRAPHY; TWO-COLOR DIFFRACTION; INDEXING ALGORITHMRecent developments of two-color operation modes at X-ray free-electron laser facilities provide new research opportunities, such as X-ray pump/X-ray probe experiments and multiple-wavelength anomalous dispersion phasing methods. However, most existing indexing methods were developed for indexing diffraction data from monochromatic X-ray beams. Here, a new algorithm is presented for indexing two-color diffraction data, as an extension of the sparse-pattern indexing algorithm SPIND, which has been demonstrated to be capable of indexing diffraction patterns with as few as five peaks. The principle and implementation of the two-color indexing method, SPIND-TC, are reported in this paper. The algorithm was tested on both simulated and experimental data of protein crystals. The results show that the diffraction data can be accurately indexed in both cases. Source codes are publicly available at https://github.com/lixx11/SPIND-TC.text/htmlSPIND-TC: an indexing method for two-color X-ray diffraction datatext763https://creativecommons.org/licenses/by/4.0/Acta Crystallographica Section A: Foundations and Advances2020-04-02369research papers2053-2733May 2020med@iucr.org3752053-2733An efficient method for indexing grazing-incidence X-ray diffraction data of epitaxially grown thin films
http://scripts.iucr.org/cgi-bin/paper?wo5036
Crystal structure identification of thin organic films entails a number of technical and methodological challenges. In particular, if molecular crystals are epitaxially grown on single-crystalline substrates a complex scenario of multiple preferred orientations of the adsorbate, several symmetry-related in-plane alignments and the occurrence of unknown polymorphs is frequently observed. In theory, the parameters of the reduced unit cell and its orientation can simply be obtained from the matrix of three linearly independent reciprocal-space vectors. However, if the sample exhibits unit cells in various orientations and/or with different lattice parameters, it is necessary to assign all experimentally obtained reflections to their associated individual origin. In the present work, an effective algorithm is described to accomplish this task in order to determine the unit-cell parameters of complex systems comprising different orientations and polymorphs. This method is applied to a polycrystalline thin film of the conjugated organic material 6,13-pentacenequinone (PQ) epitaxially grown on an Ag(111) surface. All reciprocal vectors can be allocated to unit cells of the same lattice constants but grown in various orientations [sixfold rotational symmetry for the contact planes (102) and (102)]. The as-determined unit cell is identical to that reported in a previous study determined for a fibre-textured PQ film. Preliminary results further indicate that the algorithm is especially effective in analysing epitaxially grown crystallites not only for various orientations, but also if different polymorphs are present in the film.https://creativecommons.org/licenses/by/4.0/urn:issn:2053-2733Simbrunner, J.Schrode, B.Domke, J.Fritz, T.Salzmann, I.Resel, R.2020-04-02doi:10.1107/S2053273320001266International Union of CrystallographyA method is described for indexing grazing-incidence X-ray diffraction data of epitaxially grown thin films comprising various crystal orientations and/or polymorphs by measuring reciprocal-lattice vectors.enEPITAXY; INDEXING; MATHEMATICAL CRYSTALLOGRAPHYCrystal structure identification of thin organic films entails a number of technical and methodological challenges. In particular, if molecular crystals are epitaxially grown on single-crystalline substrates a complex scenario of multiple preferred orientations of the adsorbate, several symmetry-related in-plane alignments and the occurrence of unknown polymorphs is frequently observed. In theory, the parameters of the reduced unit cell and its orientation can simply be obtained from the matrix of three linearly independent reciprocal-space vectors. However, if the sample exhibits unit cells in various orientations and/or with different lattice parameters, it is necessary to assign all experimentally obtained reflections to their associated individual origin. In the present work, an effective algorithm is described to accomplish this task in order to determine the unit-cell parameters of complex systems comprising different orientations and polymorphs. This method is applied to a polycrystalline thin film of the conjugated organic material 6,13-pentacenequinone (PQ) epitaxially grown on an Ag(111) surface. All reciprocal vectors can be allocated to unit cells of the same lattice constants but grown in various orientations [sixfold rotational symmetry for the contact planes (102) and (102)]. The as-determined unit cell is identical to that reported in a previous study determined for a fibre-textured PQ film. Preliminary results further indicate that the algorithm is especially effective in analysing epitaxially grown crystallites not only for various orientations, but also if different polymorphs are present in the film.text/htmlAn efficient method for indexing grazing-incidence X-ray diffraction data of epitaxially grown thin filmstext763https://creativecommons.org/licenses/by/4.0/Acta Crystallographica Section A: Foundations and Advances2020-04-02345research papers2053-2733May 2020med@iucr.org3572053-2733Isotopy classes for 3-periodic net embeddings
http://scripts.iucr.org/cgi-bin/paper?ib5087
Entangled embedded periodic nets and crystal frameworks are defined, along with their dimension type, homogeneity type, adjacency depth and periodic isotopy type. Periodic isotopy classifications are obtained for various families of embedded nets with small quotient graphs. The 25 periodic isotopy classes of depth-1 embedded nets with a single-vertex quotient graph are enumerated. Additionally, a classification is given of embeddings of n-fold copies of pcu with all connected components in a parallel orientation and n vertices in a repeat unit, as well as demonstrations of their maximal symmetry periodic isotopes. The methodology of linear graph knots on the flat 3-torus [0,1)3 is introduced. These graph knots, with linear edges, are spatial embeddings of the labelled quotient graphs of an embedded net which are associated with its periodicity bases.https://creativecommons.org/licenses/by/4.0/urn:issn:2053-2733Power, S.C.Baburin, I.A.Proserpio, D.M.2020-03-05doi:10.1107/S2053273320000625International Union of CrystallographyEntangled embedded periodic nets and crystal frameworks are defined, along with their dimension type, homogeneity type, adjacency depth and periodic isotopy type.enPERIODIC NETS; EMBEDDED NETS; COORDINATION POLYMERS; ISOTOPY TYPES; CRYSTALLOGRAPHIC FRAMEWORKSEntangled embedded periodic nets and crystal frameworks are defined, along with their dimension type, homogeneity type, adjacency depth and periodic isotopy type. Periodic isotopy classifications are obtained for various families of embedded nets with small quotient graphs. The 25 periodic isotopy classes of depth-1 embedded nets with a single-vertex quotient graph are enumerated. Additionally, a classification is given of embeddings of n-fold copies of pcu with all connected components in a parallel orientation and n vertices in a repeat unit, as well as demonstrations of their maximal symmetry periodic isotopes. The methodology of linear graph knots on the flat 3-torus [0,1)3 is introduced. These graph knots, with linear edges, are spatial embeddings of the labelled quotient graphs of an embedded net which are associated with its periodicity bases.text/htmlIsotopy classes for 3-periodic net embeddingstext763https://creativecommons.org/licenses/by/4.0/Acta Crystallographica Section A: Foundations and Advances2020-03-05275lead articles2053-2733May 2020med@iucr.org3012053-2733