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) 2020 International Union of Crystallography2020-08-06International 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 76, Part 5, 2020textweekly62002-01-01T00:00+00:005762020-08-06Copyright (c) 2020 International Union of CrystallographyActa Crystallographica Section A: Foundations and Advances556urn:issn:2053-2733med@iucr.orgAugust 20202020-08-06Acta Crystallographica Section Ahttp://journals.iucr.org/logos/rss10a.gif
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Still imageQuaternions: 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.ENquaternionsdata alignmentrotationorientationorthogonal Procrustes problemorientation distribution functionODFtext/htmlQuaternions: what are they, and why do we need to know?text5762020-08-06Copyright (c) 2020 International Union of CrystallographyActa Crystallographica Section Ascientific commentaries00Inflation 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.Copyright (c) 2020 International Union of Crystallographyurn: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.ENquasicrystalsprojection methodinflation rulesdiffractionhyperuniformityTilings 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 diffractiontext5762020-07-09Copyright (c) 2020 International Union of CrystallographyActa Crystallographica Section Atopical reviews00Embedding-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.Copyright (c) 2020 International Union of Crystallographyurn: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 crystallographydensity embeddingmulti-scale simulationselectronic structurechromophoresThe 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 environmenttext5762020-07-20Copyright (c) 2020 International Union of CrystallographyActa Crystallographica Section Aresearch papers00Multiplicity-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.Copyright (c) 2020 International Union of Crystallographyurn: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 unitunit cellEuler's formulaspace-filling polyhedraDirichlet 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 polyhedratext5762020-07-09Copyright (c) 2020 International Union of CrystallographyActa Crystallographica Section Aresearch papers00On 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.Copyright (c) 2020 International Union of Crystallographyurn: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 graphsfree abelian groupscomputational group theoryvertex-transitive graphsisotopyThe 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}^4text5762020-07-16Copyright (c) 2020 International Union of CrystallographyActa Crystallographica Section Aresearch papers00X-ray scattering study of water confined in bioactive glasses: experimental and simulated pair distribution function
http://scripts.iucr.org/cgi-bin/paper?vk5044
Temperature-dependent total X-ray scattering measurements for water confined in bioactive glass samples with 5.9 nm pore diameter have been performed. Based on these experimental data, simulations were carried out using the Empirical Potential Structure Refinement (EPSR) code, in order to study the structural organization of the confined water in detail. The results indicate a non-homogeneous structure for water inside the pore, with three different structural organizations of water, depending on the distance from the pore surface: (i) a first layer (4 Å) of interfacial pore water that forms a strong chemical bond with the substrate, (ii) intermediate pore water forming a second layer (4–11 Å) on top of the interfacial pore water, (iii) bulk-like pore water in the centre of the pores. Analysis of the simulated site–site partial pair distribution function shows that the water–silica (Ow–Si) pair correlations occur at ∼3.75 Å. The tetrahedral network of bulk water with oxygen–oxygen (Ow–Ow) hydrogen-bonded pair correlations at ∼2.8, ∼4.1 and ∼4.5 Å is strongly distorted for the interfacial pore water while the second neighbour pair correlations are observed at ∼4.0 and ∼4.9 Å. For the interfacial pore water, an additional Ow–Ow pair correlation appears at ∼3.3 Å, which is likely caused by distortions due to the interactions of the water molecules with the silica at the pore surface.Copyright (c) 2020 International Union of Crystallographyurn:issn:2053-2733Khoder, H.Schaniel, D.Pillet, S.Bendeif, E.-E.2020-07-20doi:10.1107/S2053273320007834International Union of CrystallographyWater confined in bioactive glasses is studied by total X-ray scattering. Three structural configurations can be distinguished, from bulk-like water in the pore centre to a strongly distorted layer on the pore surface.ENconfined waterbioactive glassesstructural analysispair distribution functionTemperature-dependent total X-ray scattering measurements for water confined in bioactive glass samples with 5.9 nm pore diameter have been performed. Based on these experimental data, simulations were carried out using the Empirical Potential Structure Refinement (EPSR) code, in order to study the structural organization of the confined water in detail. The results indicate a non-homogeneous structure for water inside the pore, with three different structural organizations of water, depending on the distance from the pore surface: (i) a first layer (4 Å) of interfacial pore water that forms a strong chemical bond with the substrate, (ii) intermediate pore water forming a second layer (4–11 Å) on top of the interfacial pore water, (iii) bulk-like pore water in the centre of the pores. Analysis of the simulated site–site partial pair distribution function shows that the water–silica (Ow–Si) pair correlations occur at ∼3.75 Å. The tetrahedral network of bulk water with oxygen–oxygen (Ow–Ow) hydrogen-bonded pair correlations at ∼2.8, ∼4.1 and ∼4.5 Å is strongly distorted for the interfacial pore water while the second neighbour pair correlations are observed at ∼4.0 and ∼4.9 Å. For the interfacial pore water, an additional Ow–Ow pair correlation appears at ∼3.3 Å, which is likely caused by distortions due to the interactions of the water molecules with the silica at the pore surface.text/htmlX-ray scattering study of water confined in bioactive glasses: experimental and simulated pair distribution functiontext5762020-07-20Copyright (c) 2020 International Union of CrystallographyActa Crystallographica Section Aresearch papers00