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) 2021 International Union of Crystallography2021-01-01International 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 77, Part 1, 2021textweekly62002-01-01T00:00+00:001772021-01-01Copyright (c) 2021 International Union of CrystallographyActa Crystallographica Section A: Foundations and Advances1urn:issn:2053-2733med@iucr.orgJanuary 20212021-01-01Acta Crystallographica Section Ahttp://journals.iucr.org/logos/rss10a.gif
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Still imageModern crystallography and its foundations
http://scripts.iucr.org/cgi-bin/paper?me6113
Copyright (c) 2021 International Union of Crystallographyurn:issn:2053-2733Altomare, A.Billinge, S.J.L.2021-01-05doi:10.1107/S2053273320016678International Union of CrystallographyAuthors and readers are reminded of the history and scope of Acta Crystallographica Section A, and simple steps to help maintain its prominence and impact as the premier journal for foundational work in crystallography are outlined.ENEditorialActa Crystallographica Section Amodern crystallographytext/htmlModern crystallography and its foundationstext1772021-01-05Copyright (c) 2021 International Union of CrystallographyActa Crystallographica Section Aeditorial11A 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.Copyright (c) 2021 International Union of Crystallographyurn: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 functionPDFdata analysisweb applicationscloud 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: PDFitctext1772021-01-05Copyright (c) 2021 International Union of CrystallographyActa Crystallographica Section Aresearch papers26Spiral tetrahedral packing in the β-Mn crystal as symmetry realization of the 8D E8 lattice
http://scripts.iucr.org/cgi-bin/paper?ug5002
Experimental values of atomic positions in the β-Mn crystal permit one to distinguish among them a fragment of the helix containing 15 interpenetrating distorted icosahedra, 90 vertices and 225 tetrahedra. This fragment corresponds to the closed helix of 15 icosahedra in the 4D {3, 3, 5} polytope. The primitive cubic lattice of these icosahedral helices envelopes not only all atoms of β-Mn, but also all tetrahedra belonging to the tiling of the β-Mn structure. The 2D projection of all atomic positions in the β-Mn unit cells shows that they are situated (by neglecting small differences) on three circumferences containing 2D projections of 90 vertices of the {3, 3, 5} polytope on the same plane. Non-crystallographic symmetry of the β-Mn crystal is defined by mapping the closed icosahedral helix of the {3, 3, 5} polytope into 3D Euclidean space E3. This interpretation must be correlated also with the known previous determination of non-crystallographic symmetry of the β-Mn crystal by mapping into the 3D E3 space system of icosahedra from the 6D cubic B6 lattice. The recently proposed determination of non-crystallographic symmetry of the β-Mn crystal actually uses the symmetries of the 8D E8 lattice, in which both the 4D {3, 3, 5} polytope and cubic 6D B6 lattice can be inserted.Copyright (c) 2021 International Union of Crystallographyurn:issn:2053-2733Talis, A.Everstov, A.Kraposhin, V.2021-01-05doi:10.1107/S2053273320012978International Union of CrystallographyThe 2D projection of all atomic positions in the β-Mn unit cells shows that they are situated on three circumferences containing 2D projections of 90 vertices of the {3, 3, 5} polytope on the same plane. The exhaustive description of the non-crystallographic symmetry of the β-Mn crystal has been achieved by using the 8D E8 lattice in which both the 4D {3, 3, 5} polytope and cubic 6D B6 lattice can be inserted.ENβ-Mn crystaltetrahedral tilingnon-crystallographic symmetry4D {3, 3, 5} polytope8D E8 latticeExperimental values of atomic positions in the β-Mn crystal permit one to distinguish among them a fragment of the helix containing 15 interpenetrating distorted icosahedra, 90 vertices and 225 tetrahedra. This fragment corresponds to the closed helix of 15 icosahedra in the 4D {3, 3, 5} polytope. The primitive cubic lattice of these icosahedral helices envelopes not only all atoms of β-Mn, but also all tetrahedra belonging to the tiling of the β-Mn structure. The 2D projection of all atomic positions in the β-Mn unit cells shows that they are situated (by neglecting small differences) on three circumferences containing 2D projections of 90 vertices of the {3, 3, 5} polytope on the same plane. Non-crystallographic symmetry of the β-Mn crystal is defined by mapping the closed icosahedral helix of the {3, 3, 5} polytope into 3D Euclidean space E3. This interpretation must be correlated also with the known previous determination of non-crystallographic symmetry of the β-Mn crystal by mapping into the 3D E3 space system of icosahedra from the 6D cubic B6 lattice. The recently proposed determination of non-crystallographic symmetry of the β-Mn crystal actually uses the symmetries of the 8D E8 lattice, in which both the 4D {3, 3, 5} polytope and cubic 6D B6 lattice can be inserted.text/htmlSpiral tetrahedral packing in the β-Mn crystal as symmetry realization of the 8D E8 latticetext1772021-01-05Copyright (c) 2021 International Union of CrystallographyActa Crystallographica Section Aresearch papers718Macromolecular 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.Copyright (c) 2021 International Union of Crystallographyurn: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 formsab initio phasingiterative projection algorithmsX-ray free-electron lasersXFELsA 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 formstext1772021-01-05Copyright (c) 2021 International Union of CrystallographyActa Crystallographica Section Aresearch papers1935Gummelt versus Lück decagon covering and beyond. Implications for decagonal quasicrystals
http://scripts.iucr.org/cgi-bin/paper?ug5023
Specific structural repeat units can be used as quasi-unit cells of decagonal quasicrystals. So far, the most famous and almost exclusively employed one has been the Gummelt decagon. However, in an increasing number of cases Lück decagons have been found to be more appropriate without going into depth. The diversities and commonalities of these two basic decagonal clusters and of some more general ones are discussed. The importance of the type of underlying tiling for the correct classification of a quasi-unit cell is demonstrated.Copyright (c) 2021 International Union of Crystallographyurn:issn:2053-2733Steurer, W.2021-01-05doi:10.1107/S2053273320015181International Union of CrystallographyThe diversities and commonalities of the two basic decagonal clusters, the Gummelt and the Lück decagons, and of their coverings are discussed.ENQuasi-unit cellsGummelt decagonsLück decagonsSpecific structural repeat units can be used as quasi-unit cells of decagonal quasicrystals. So far, the most famous and almost exclusively employed one has been the Gummelt decagon. However, in an increasing number of cases Lück decagons have been found to be more appropriate without going into depth. The diversities and commonalities of these two basic decagonal clusters and of some more general ones are discussed. The importance of the type of underlying tiling for the correct classification of a quasi-unit cell is demonstrated.text/htmlGummelt versus Lück decagon covering and beyond. Implications for decagonal quasicrystalstext1772021-01-05Copyright (c) 2021 International Union of CrystallographyActa Crystallographica Section Aresearch papers3641Small-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.Copyright (c) 2021 International Union of Crystallographyurn: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.ENnanowiresPorod's lawfacet truncation rodssmall-angle X-ray scatteringSAXSgrazing-incidence small-angle X-ray scatteringGISAXSSmall-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 lawtext1772021-01-05Copyright (c) 2021 International Union of CrystallographyActa Crystallographica Section Aresearch papers4253HgH2 meets relativistic quantum crystallography. How to teach relativity to a non-relativistic wavefunction
http://scripts.iucr.org/cgi-bin/paper?pl5005
The capability of X-ray constrained wavefunction (XCW) fitting to introduce relativistic effects into a non-relativistic wavefunction is tested. It is quantified how much of the reference relativistic effects can be absorbed in the non-relativistic XCW calculation when fitted against relativistic structure factors of a model HgH2 molecule. Scaling of the structure-factor sets to improve the agreement statistics is found to introduce a significant systematic error into the XCW fitting of relativistic effects.Copyright (c) 2021 International Union of Crystallographyurn:issn:2053-2733Podhorský, M.Bučinský, L.Jayatilaka, D.Grabowsky, S.2021-01-05doi:10.1107/S2053273320014837International Union of CrystallographyRelativistic structure factors are used to teach a non-relativistic wavefunction relativity using the X-ray constrained wavefunction method, whereby resolution and scaling effects are critically assessed.ENquantum crystallographyX-ray constrained wavefunctionrelativistic effectsscalingcharge densityThe capability of X-ray constrained wavefunction (XCW) fitting to introduce relativistic effects into a non-relativistic wavefunction is tested. It is quantified how much of the reference relativistic effects can be absorbed in the non-relativistic XCW calculation when fitted against relativistic structure factors of a model HgH2 molecule. Scaling of the structure-factor sets to improve the agreement statistics is found to introduce a significant systematic error into the XCW fitting of relativistic effects.text/htmlHgH2 meets relativistic quantum crystallography. How to teach relativity to a non-relativistic wavefunctiontext1772021-01-05Copyright (c) 2021 International Union of CrystallographyActa Crystallographica Section Aresearch papers5466The maximum surface area polyhedron with five vertices inscribed in the sphere {\bb S}^{2}
http://scripts.iucr.org/cgi-bin/paper?pl5007
This article focuses on the problem of analytically determining the optimal placement of five points on the unit sphere {\bb S}^{2} so that the surface area of the convex hull of the points is maximized. It is shown that the optimal polyhedron has a trigonal bipyramidal structure with two vertices placed at the north and south poles and the other three vertices forming an equilateral triangle inscribed in the equator. This result confirms a conjecture of Akkiraju, who conducted a numerical search for the maximizer. As an application to crystallography, the surface area discrepancy is considered as a measure of distortion between an observed coordination polyhedron and an ideal one. The main result yields a formula for the surface area discrepancy of any coordination polyhedron with five vertices.Copyright (c) 2021 International Union of Crystallographyurn:issn:2053-2733Donahue, J.Hoehner, S.Li, B.2021-01-05doi:10.1107/S2053273320015089International Union of CrystallographyIt is shown that a polyhedron with the trigonal bipyramidal structure is the unique surface area maximizer among all polyhedra with five vertices inscribed in a sphere.ENpolyhedrasurface areatriangular bipyramidsoptimizationinequalitiesThis article focuses on the problem of analytically determining the optimal placement of five points on the unit sphere {\bb S}^{2} so that the surface area of the convex hull of the points is maximized. It is shown that the optimal polyhedron has a trigonal bipyramidal structure with two vertices placed at the north and south poles and the other three vertices forming an equilateral triangle inscribed in the equator. This result confirms a conjecture of Akkiraju, who conducted a numerical search for the maximizer. As an application to crystallography, the surface area discrepancy is considered as a measure of distortion between an observed coordination polyhedron and an ideal one. The main result yields a formula for the surface area discrepancy of any coordination polyhedron with five vertices.text/htmlThe maximum surface area polyhedron with five vertices inscribed in the sphere {\bb S}^{2}text1772021-01-05Copyright (c) 2021 International Union of CrystallographyActa Crystallographica Section Aresearch papers6774Algebraic approximations of a polyhedron correlation function stemming from its chord-length distribution
http://scripts.iucr.org/cgi-bin/paper?ib5095
An algebraic approximation, of order K, of a polyhedron correlation function (CF) can be obtained from γ′′(r), its chord-length distribution (CLD), considering first, within the subinterval [Di−1, Di] of the full range of distances, a polynomial in the two variables (r − Di−1)1/2 and (Di − r)1/2 such that its expansions around r = Di−1 and r = Di simultaneously coincide with the left and right expansions of γ′′(r) around Di−1 and Di up to the terms O(r − Di−1)K/2 and O(Di − r)K/2, respectively. Then, for each i, one integrates twice the polynomial and determines the integration constants matching the resulting integrals at the common end-points. The 3D Fourier transform of the resulting algebraic CF approximation correctly reproduces, at large q's, the asymptotic behaviour of the exact form factor up to the term O[q−(K/2+4)]. For illustration, the procedure is applied to the cube, the tetrahedron and the octahedron.Copyright (c) 2021 International Union of Crystallographyurn:issn:2053-2733Ciccariello, S.2021-01-05doi:10.1107/S2053273320014229International Union of CrystallographyA procedure is reported for obtaining an algebraic approximation of the correlation function of a polyhedron starting from its known chord-length distribution.ENsmall-angle scatteringpolyhedrachord-length distributioncorrelation functionsasymptotic behaviourAn algebraic approximation, of order K, of a polyhedron correlation function (CF) can be obtained from γ′′(r), its chord-length distribution (CLD), considering first, within the subinterval [Di−1, Di] of the full range of distances, a polynomial in the two variables (r − Di−1)1/2 and (Di − r)1/2 such that its expansions around r = Di−1 and r = Di simultaneously coincide with the left and right expansions of γ′′(r) around Di−1 and Di up to the terms O(r − Di−1)K/2 and O(Di − r)K/2, respectively. Then, for each i, one integrates twice the polynomial and determines the integration constants matching the resulting integrals at the common end-points. The 3D Fourier transform of the resulting algebraic CF approximation correctly reproduces, at large q's, the asymptotic behaviour of the exact form factor up to the term O[q−(K/2+4)]. For illustration, the procedure is applied to the cube, the tetrahedron and the octahedron.text/htmlAlgebraic approximations of a polyhedron correlation function stemming from its chord-length distributiontext1772021-01-05Copyright (c) 2021 International Union of CrystallographyActa Crystallographica Section Ashort communications7580Mathematics for Physicists. Introductory Concepts and Methods. By Alexander Altland and Jan von Delft. Cambridge University Press, 2019. Hardback, Pp. 720. Price GBP 39.99. ISBN 9781108471220.
http://scripts.iucr.org/cgi-bin/paper?xo0143
Copyright (c) 2021 International Union of Crystallographyurn:issn:2053-2733Glazer, M.2021-01-05doi:10.1107/S205327332001503XInternational Union of CrystallographyENbook reviewmathematics for physiciststext/htmlMathematics for Physicists. Introductory Concepts and Methods. By Alexander Altland and Jan von Delft. Cambridge University Press, 2019. Hardback, Pp. 720. Price GBP 39.99. ISBN 9781108471220.text1772021-01-05Copyright (c) 2021 International Union of CrystallographyActa Crystallographica Section Abook reviews8182