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 Crystallography2021-06-29International 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 4, 2021textweekly62002-01-01T00:00+00:004772021-06-29Copyright (c) 2021 International Union of CrystallographyActa Crystallographica Section A: Foundations and Advances239urn:issn:2053-2733med@iucr.orgJune 20212021-06-29Acta Crystallographica Section Ahttp://journals.iucr.org/logos/rss10a.gif
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Still imageFrom crystal color symmetry to quantum spacetime
http://scripts.iucr.org/cgi-bin/paper?me6135
Copyright (c) 2021 International Union of Crystallographyurn:issn:2053-2733Bojowald, M.Saxena, A.2021-05-27doi:10.1107/S2053273321005234International Union of CrystallographyThis perspective article elucidates both the importance and the implications of relativistic spacetime crystals as well as the renormalized blended coordinates transformation. It alludes to possible applications in materials science, condensed matter physics and quantum gravity.ENcolor symmetryquantum spacetimerenormalized blended spacetimerelativistic spacetime crystalstext/htmlFrom crystal color symmetry to quantum spacetimetext4772021-05-27Copyright (c) 2021 International Union of CrystallographyActa Crystallographica Section Ascientific commentaries239241Relativistic spacetime crystals
http://scripts.iucr.org/cgi-bin/paper?ib5098
Periodic space crystals are well established and widely used in physical sciences. Time crystals have been increasingly explored more recently, where time is disconnected from space. Periodic relativistic spacetime crystals on the other hand need to account for the mixing of space and time in special relativity through Lorentz transformation, and have been listed only in 2D. This work shows that there exists a transformation between the conventional Minkowski spacetime (MS) and what is referred to here as renormalized blended spacetime (RBS); they are shown to be equivalent descriptions of relativistic physics in flat spacetime. There are two elements to this reformulation of MS, namely, blending and renormalization. When observers in two inertial frames adopt each other's clocks as their own, while retaining their original space coordinates, the observers become blended. This process reformulates the Lorentz boosts into Euclidean rotations while retaining the original spacetime hyperbola describing worldlines of constant spacetime length from the origin. By renormalizing the blended coordinates with an appropriate factor that is a function of the relative velocities between the various frames, the hyperbola is transformed into a Euclidean circle. With these two steps, one obtains the RBS coordinates complete with new light lines, but now with a Euclidean construction. One can now enumerate the RBS point and space groups in various dimensions with their mapping to the well known space crystal groups. The RBS point group for flat isotropic RBS spacetime is identified to be that of cylinders in various dimensions: mm2 which is that of a rectangle in 2D, (∞/m)m which is that of a cylinder in 3D, and that of a hypercylinder in 4D. An antisymmetry operation is introduced that can swap between space-like and time-like directions, leading to color spacetime groups. The formalism reveals RBS symmetries that are not readily apparent in the conventional MS formulation. Mathematica script is provided for plotting the MS and RBS geometries discussed in the work.Copyright (c) 2021 International Union of Crystallographyurn:issn:2053-2733Gopalan, V.2021-05-27doi:10.1107/S2053273321003259International Union of CrystallographyBy appropriate reformulation of relativistic spacetime geometry, a direct mapping to Euclidean space crystals is shown. Using this mapping, hidden symmetries in relativistic spacetime crystals are uncovered.ENspacetimespecial relativityrenormalized blended spacetimerelativistic spacetime crystalsPeriodic space crystals are well established and widely used in physical sciences. Time crystals have been increasingly explored more recently, where time is disconnected from space. Periodic relativistic spacetime crystals on the other hand need to account for the mixing of space and time in special relativity through Lorentz transformation, and have been listed only in 2D. This work shows that there exists a transformation between the conventional Minkowski spacetime (MS) and what is referred to here as renormalized blended spacetime (RBS); they are shown to be equivalent descriptions of relativistic physics in flat spacetime. There are two elements to this reformulation of MS, namely, blending and renormalization. When observers in two inertial frames adopt each other's clocks as their own, while retaining their original space coordinates, the observers become blended. This process reformulates the Lorentz boosts into Euclidean rotations while retaining the original spacetime hyperbola describing worldlines of constant spacetime length from the origin. By renormalizing the blended coordinates with an appropriate factor that is a function of the relative velocities between the various frames, the hyperbola is transformed into a Euclidean circle. With these two steps, one obtains the RBS coordinates complete with new light lines, but now with a Euclidean construction. One can now enumerate the RBS point and space groups in various dimensions with their mapping to the well known space crystal groups. The RBS point group for flat isotropic RBS spacetime is identified to be that of cylinders in various dimensions: mm2 which is that of a rectangle in 2D, (∞/m)m which is that of a cylinder in 3D, and that of a hypercylinder in 4D. An antisymmetry operation is introduced that can swap between space-like and time-like directions, leading to color spacetime groups. The formalism reveals RBS symmetries that are not readily apparent in the conventional MS formulation. Mathematica script is provided for plotting the MS and RBS geometries discussed in the work.text/htmlRelativistic spacetime crystalstext4772021-05-27Copyright (c) 2021 International Union of CrystallographyActa Crystallographica Section Aresearch papers242256Multiwavelength anomalous X-ray diffraction for combined imaging of atomic displacement and strain
http://scripts.iucr.org/cgi-bin/paper?iv5014
The X-ray Bragg coherent diffractive imaging (CDI) technique assumes that the structure factor holds constant over the measured crystal. This approximation breaks down for materials exhibiting variations in the unit-cell configuration, such as piezo- and ferroelectrics. In that case, the strain field cannot be reliably determined from the reconstruction because the lattice deformation and the structure factor contribute concomitantly. Proposed here is a solution to this problem achieved by combining Bragg CDI and the multiwavelength anomalous diffraction approach that measures a Friedel pair of reflections at two different photon energies near an absorption edge. Comparing the obtained reconstructions with a parametric model that includes calculating the scattering amplitude as a function of wavelength and the unit-cell configuration, the contributions of the lattice deformation and the structure factor are separated. Simulations of the ferroelectric material BaTiO3 demonstrate the possibility of simultaneous probing of the strain and displacement of the Ti atoms. The proposed method opens up an opportunity to apply coherent X-ray diffraction for nanoscale-resolved 3D mapping of polarization domains in micro- and nanocrystals.Copyright (c) 2021 International Union of Crystallographyurn:issn:2053-2733Shabalin, A.G.Shpyrko, O.G.2021-06-21doi:10.1107/S2053273321004976International Union of CrystallographyA new method is presented combining multiwavelength anomalous diffraction and Bragg coherent diffractive imaging. Combining the advantages of both approaches, it opens the possibility of simultaneously probing the strain and the unit-cell configuration.ENcoherent X-ray diffractive imagingmultiwavelength anomalous diffractionimaging polarization domainsatomic displacementlattice deformationThe X-ray Bragg coherent diffractive imaging (CDI) technique assumes that the structure factor holds constant over the measured crystal. This approximation breaks down for materials exhibiting variations in the unit-cell configuration, such as piezo- and ferroelectrics. In that case, the strain field cannot be reliably determined from the reconstruction because the lattice deformation and the structure factor contribute concomitantly. Proposed here is a solution to this problem achieved by combining Bragg CDI and the multiwavelength anomalous diffraction approach that measures a Friedel pair of reflections at two different photon energies near an absorption edge. Comparing the obtained reconstructions with a parametric model that includes calculating the scattering amplitude as a function of wavelength and the unit-cell configuration, the contributions of the lattice deformation and the structure factor are separated. Simulations of the ferroelectric material BaTiO3 demonstrate the possibility of simultaneous probing of the strain and displacement of the Ti atoms. The proposed method opens up an opportunity to apply coherent X-ray diffraction for nanoscale-resolved 3D mapping of polarization domains in micro- and nanocrystals.text/htmlMultiwavelength anomalous X-ray diffraction for combined imaging of atomic displacement and straintext4772021-06-21Copyright (c) 2021 International Union of CrystallographyActa Crystallographica Section Ashort communications257261Polarization effects of X-ray monochromators modeled using dynamical scattering theory
http://scripts.iucr.org/cgi-bin/paper?ae5102
The difference in the diffracted intensity of the σ- and π-polarized components of an X-ray beam in powder diffraction has generally been treated according to equations based on dipole scattering, also known as kinematic X-ray scattering. Although this treatment is correct for powders and post-sample analyzers known to be of high mosaicity, it does not apply to systems configured with nearly perfect crystal incident-beam monochromators. Equations are presented for the polarization effect, based on dynamical diffraction theory applied to the monochromator crystal. The intensity of the π component relative to the σ component then becomes approximately proportional to |cos 2θm| rather than to cos22θm, where θm is the Bragg diffraction angle of the monochromator crystal. This changes the predicted intensities of X-ray powder diffraction patterns produced on instruments with incident-beam monochromators, especially in the regions far from 2θ = 90° in the powder pattern. Experimental data, based on well known standard reference materials, are presented, confirming that the dynamical polarization correction is required when a Ge 111 incident-beam monochromator is used. The dynamical correction is absent as an option in the Rietveld analysis codes with which the authors are familiar.Copyright (c) 2021 International Union of Crystallographyurn:issn:2053-2733Mendenhall, M.H.Black, D.Windover, D.Cline, J.P.2021-05-27doi:10.1107/S2053273321003879International Union of CrystallographyTreatment of the effects of a single-crystal incident-beam monochromator in a powder diffraction experiment must be carried out using dynamical diffraction to obtain the correct intensities for reflections, due to the modification of the beam polarization by the monochromator. Theory and data are presented to verify the effect.ENdynamical diffractionpowder diffractionincident-beam monochromatorspolarizationThe difference in the diffracted intensity of the σ- and π-polarized components of an X-ray beam in powder diffraction has generally been treated according to equations based on dipole scattering, also known as kinematic X-ray scattering. Although this treatment is correct for powders and post-sample analyzers known to be of high mosaicity, it does not apply to systems configured with nearly perfect crystal incident-beam monochromators. Equations are presented for the polarization effect, based on dynamical diffraction theory applied to the monochromator crystal. The intensity of the π component relative to the σ component then becomes approximately proportional to |cos 2θm| rather than to cos22θm, where θm is the Bragg diffraction angle of the monochromator crystal. This changes the predicted intensities of X-ray powder diffraction patterns produced on instruments with incident-beam monochromators, especially in the regions far from 2θ = 90° in the powder pattern. Experimental data, based on well known standard reference materials, are presented, confirming that the dynamical polarization correction is required when a Ge 111 incident-beam monochromator is used. The dynamical correction is absent as an option in the Rietveld analysis codes with which the authors are familiar.text/htmlPolarization effects of X-ray monochromators modeled using dynamical scattering theorytext4772021-05-27Copyright (c) 2021 International Union of CrystallographyActa Crystallographica Section Aresearch papers262267Resolution of a bent-crystal spectrometer for X-ray free-electron laser pulses: diamond versus silicon
http://scripts.iucr.org/cgi-bin/paper?wo5037
The resolution function of a spectrometer based on a strongly bent single crystal (bending radius of 10 cm or less) is evaluated. It is shown that the resolution is controlled by two parameters: (i) the ratio of the lattice spacing of the chosen reflection to the crystal thickness and (ii) a single parameter comprising crystal thickness, its bending radius, distance to a detector, and anisotropic elastic constants of the chosen crystal. The results allow the optimization of the parameters of bent-crystal spectrometers for the hard X-ray free-electron laser sources.Copyright (c) 2021 International Union of Crystallographyurn:issn:2053-2733Kaganer, V.M.Petrov, I.Samoylova, L.2021-05-27doi:10.1107/S2053273321003697International Union of CrystallographyThe resolution function of a bent-crystal spectrometer for pulses of an X-ray free-electron laser is evaluated. Under appropriate conditions, the energy resolution reaches the ratio of the lattice spacing to the crystal thickness.ENX-ray free-electron lasersX-ray spectroscopybent crystalsdiamond crystal opticsfemtosecond X-ray diffractionThe resolution function of a spectrometer based on a strongly bent single crystal (bending radius of 10 cm or less) is evaluated. It is shown that the resolution is controlled by two parameters: (i) the ratio of the lattice spacing of the chosen reflection to the crystal thickness and (ii) a single parameter comprising crystal thickness, its bending radius, distance to a detector, and anisotropic elastic constants of the chosen crystal. The results allow the optimization of the parameters of bent-crystal spectrometers for the hard X-ray free-electron laser sources.text/htmlResolution of a bent-crystal spectrometer for X-ray free-electron laser pulses: diamond versus silicontext4772021-05-27Copyright (c) 2021 International Union of CrystallographyActa Crystallographica Section Aresearch papers268276A reference-based multi-lattice indexing method integrating prior information correction and iterative refinement in protein crystallography
http://scripts.iucr.org/cgi-bin/paper?ae5097
A new multi-lattice indexing method based on the principle of whole-pattern matching given cell dimensions and space-group symmetry is presented for macromolecular crystallography. The proposed method, termed the multi-crystal data processing suite (MCDPS), features a local correction for prior information accompanied by iterative refinement of experimental parameters, both of which are numerically and experimentally demonstrated to be critical for accurately identifying multiple crystal lattices. Further analysis of data reduction and structure determination with conventional single-crystal programs reveals that the processed multi-lattice data sets are comparable in quality to typical single-crystal ones in terms of crystallographic metrics. Importantly, it is confirmed that careful exclusion of overlapping reflections prior to scaling is necessary to guarantee an accurate data reduction result. The potential for multi-lattice indexing in solving the general macroscopic twinning problem is also explored.Copyright (c) 2021 International Union of Crystallographyurn:issn:2053-2733Zhou, Q.Gao, Z.-Q.Dong, Z.Jiang, Y.-M.She, Z.Geng, Z.Dong, Y.-H.2021-05-27doi:10.1107/S2053273321003521International Union of CrystallographyA new multi-lattice indexing method based on the principle of whole-pattern matching given cell dimensions and space-group symmetry is presented for macromolecular crystallography. The proposed method features a local correction for prior information accompanied by iterative refinement of experimental parameters, both of which are numerically and experimentally demonstrated to be critical for accurately identifying multiple crystal lattices.ENmulti-crystal diffractionindexing algorithmmulti-crystal data processing suiteMCDPSprotein crystallographydata reductionA new multi-lattice indexing method based on the principle of whole-pattern matching given cell dimensions and space-group symmetry is presented for macromolecular crystallography. The proposed method, termed the multi-crystal data processing suite (MCDPS), features a local correction for prior information accompanied by iterative refinement of experimental parameters, both of which are numerically and experimentally demonstrated to be critical for accurately identifying multiple crystal lattices. Further analysis of data reduction and structure determination with conventional single-crystal programs reveals that the processed multi-lattice data sets are comparable in quality to typical single-crystal ones in terms of crystallographic metrics. Importantly, it is confirmed that careful exclusion of overlapping reflections prior to scaling is necessary to guarantee an accurate data reduction result. The potential for multi-lattice indexing in solving the general macroscopic twinning problem is also explored.text/htmlA reference-based multi-lattice indexing method integrating prior information correction and iterative refinement in protein crystallographytext4772021-05-27Copyright (c) 2021 International Union of CrystallographyActa Crystallographica Section Aresearch papers277288Improvement of precision in refinements of structure factors using convergent-beam electron diffraction patterns taken at Bragg-excited conditions
http://scripts.iucr.org/cgi-bin/paper?ou5018
A local structure analysis method based on convergent-beam electron diffraction (CBED) has been used for refining isotropic atomic displacement parameters and five low-order structure factors with sin θ/λ ≤ 0.28 Å−1 of potassium tantalate (KTaO3). Comparison between structure factors determined from CBED patterns taken at the zone-axis (ZA) and Bragg-excited conditions is made in order to discuss their precision and sensitivities. Bragg-excited CBED patterns showed higher precision in the refinement of structure factors than ZA patterns. Consistency between higher precision and sensitivity of the Bragg-excited CBED patterns has been found only for structure factors of the outer zeroth-order Laue-zone reflections with larger reciprocal-lattice vectors. Correlation coefficients among the refined structure factors in the refinement of Bragg-excited patterns are smaller than those of the ZA ones. Such smaller correlation coefficients lead to higher precision in the refinement of structure factors.Copyright (c) 2021 International Union of Crystallographyurn:issn:2053-2733Aryal, B.Morikawa, D.Tsuda, K.Terauchi, M.2021-06-10doi:10.1107/S2053273321004137International Union of CrystallographyA new strategy for improving precision and sensitivity in the refinement of crystal structure factors is proposed. This technique will help to precisely determine the chemical bonding states of crystalline materials, which are closely related to their physical properties.ENCBEDstructure factorselectron densityelectrostatic potentialprecisionA local structure analysis method based on convergent-beam electron diffraction (CBED) has been used for refining isotropic atomic displacement parameters and five low-order structure factors with sin θ/λ ≤ 0.28 Å−1 of potassium tantalate (KTaO3). Comparison between structure factors determined from CBED patterns taken at the zone-axis (ZA) and Bragg-excited conditions is made in order to discuss their precision and sensitivities. Bragg-excited CBED patterns showed higher precision in the refinement of structure factors than ZA patterns. Consistency between higher precision and sensitivity of the Bragg-excited CBED patterns has been found only for structure factors of the outer zeroth-order Laue-zone reflections with larger reciprocal-lattice vectors. Correlation coefficients among the refined structure factors in the refinement of Bragg-excited patterns are smaller than those of the ZA ones. Such smaller correlation coefficients lead to higher precision in the refinement of structure factors.text/htmlImprovement of precision in refinements of structure factors using convergent-beam electron diffraction patterns taken at Bragg-excited conditionstext4772021-06-10Copyright (c) 2021 International Union of CrystallographyActa Crystallographica Section Aresearch papers289295On symmetry breaking of dual polyhedra of non-crystallographic group H3
http://scripts.iucr.org/cgi-bin/paper?ug5018
The study of the polyhedra described in this paper is relevant to the icosahedral symmetry in the assembly of various spherical molecules, biomolecules and viruses. A symmetry-breaking mechanism is applied to the family of polytopes {\cal V}_{H_{3}}(\lambda) constructed for each type of dominant point λ. Here a polytope {\cal V}_{H_{3}}(\lambda) is considered as a dual of a {\cal D}_{H_{3}}(\lambda) polytope obtained from the action of the Coxeter group H3 on a single point \lambda\in{\bb R}^{3}. The H3 symmetry is reduced to the symmetry of its two-dimensional subgroups H2, A1 × A1 and A2 that are used to examine the geometric structure of {\cal V}_{H_{3}}(\lambda) polytopes. The latter is presented as a stack of parallel circular/polygonal orbits known as the `pancake' structure of a polytope. Inserting more orbits into an orbit decomposition results in the extension of the {\cal V}_{H_{3}}(\lambda) structure into various nanotubes. Moreover, since a {\cal V}_{H_{3}}(\lambda) polytope may contain the orbits obtained by the action of H3 on the seed points (a, 0, 0), (0, b, 0) and (0, 0, c) within its structure, the stellations of flat-faced {\cal V}_{H_{3}}(\lambda) polytopes are constructed whenever the radii of such orbits are appropriately scaled. Finally, since the fullerene C20 has the dodecahedral structure of {\cal V}_{H_{3}}(a,0,0), the construction of the smallest fullerenes C24, C26, C28, C30 together with the nanotubes C20+6N, C20+10N is presented.Copyright (c) 2021 International Union of Crystallographyurn:issn:2053-2733Myronova, M.2021-06-16doi:10.1107/S2053273321002254International Union of CrystallographyThe ubiquity of the icosahedral symmetry among various spherical molecules, fullerenes and viruses motivates the current study. The structures of dual polytopes with the H3 symmetry obtained for seven types of dominant points are described in detail. The existence of the corresponding nanotubes is explained using a symmetry-breaking mechanism. The structure of the fullerene C20 is expanded into the nanotubes C20+6N and C20+10N, using the reduction of the icosahedral symmetry to the subgroups A2 and H2, respectively.ENCoxeter groupsdual polytopesorbit decompositionsfullerenesnanotubesstellated polyhedraThe study of the polyhedra described in this paper is relevant to the icosahedral symmetry in the assembly of various spherical molecules, biomolecules and viruses. A symmetry-breaking mechanism is applied to the family of polytopes {\cal V}_{H_{3}}(\lambda) constructed for each type of dominant point λ. Here a polytope {\cal V}_{H_{3}}(\lambda) is considered as a dual of a {\cal D}_{H_{3}}(\lambda) polytope obtained from the action of the Coxeter group H3 on a single point \lambda\in{\bb R}^{3}. The H3 symmetry is reduced to the symmetry of its two-dimensional subgroups H2, A1 × A1 and A2 that are used to examine the geometric structure of {\cal V}_{H_{3}}(\lambda) polytopes. The latter is presented as a stack of parallel circular/polygonal orbits known as the `pancake' structure of a polytope. Inserting more orbits into an orbit decomposition results in the extension of the {\cal V}_{H_{3}}(\lambda) structure into various nanotubes. Moreover, since a {\cal V}_{H_{3}}(\lambda) polytope may contain the orbits obtained by the action of H3 on the seed points (a, 0, 0), (0, b, 0) and (0, 0, c) within its structure, the stellations of flat-faced {\cal V}_{H_{3}}(\lambda) polytopes are constructed whenever the radii of such orbits are appropriately scaled. Finally, since the fullerene C20 has the dodecahedral structure of {\cal V}_{H_{3}}(a,0,0), the construction of the smallest fullerenes C24, C26, C28, C30 together with the nanotubes C20+6N, C20+10N is presented.text/htmlOn symmetry breaking of dual polyhedra of non-crystallographic group H3text4772021-06-16Copyright (c) 2021 International Union of CrystallographyActa Crystallographica Section Aresearch papers296316A topological proof of the modified Euler characteristic based on the orbifold concept
http://scripts.iucr.org/cgi-bin/paper?ug5026
The notion of the Euler characteristic of a polyhedron or tessellation has been the subject of in-depth investigations by many authors. Two previous papers worked to explain the phenomenon of the vanishing (or zeroing) of the modified Euler characteristic of a polyhedron that underlies a periodic tessellation of a space under a crystallographic space group. The present paper formally expresses this phenomenon as a theorem about the vanishing of the Euler characteristic of certain topological spaces called topological orbifolds. In this new approach, it is explained that the theorem in question follows from the fundamental properties of the orbifold Euler characteristic. As a side effect of these considerations, a theorem due to Coxeter about the vanishing Euler characteristic of a honeycomb tessellation is re-proved in a context which frees the calculations from the assumptions made by Coxeter in his proof. The abstract mathematical concepts are visualized with down-to-earth examples motivated by concrete situations illustrating wallpaper and 3D crystallographic space groups. In a way analogous to the application of the classic Euler equation to completely bounded solids, the formula proven in this paper is applicable to such important crystallographic objects as asymmetric units and Dirichlet domains.Copyright (c) 2021 International Union of Crystallographyurn:issn:2053-2733Naskręcki, B.Dauter, Z.Jaskolski, M.2021-06-21doi:10.1107/S2053273321004320International Union of CrystallographyThe vanishing of the modified Euler characteristic for symmetrically arranged space-filling polytopes is given a general proof based on the topological concept of orbifolds. The modified Euler characteristic is applicable to such important crystallographic objects as asymmetric units and Dirichlet domains.ENEuler characteristicorbifoldsspace-filling polyhedraspace groupsasymmetric unitsThe notion of the Euler characteristic of a polyhedron or tessellation has been the subject of in-depth investigations by many authors. Two previous papers worked to explain the phenomenon of the vanishing (or zeroing) of the modified Euler characteristic of a polyhedron that underlies a periodic tessellation of a space under a crystallographic space group. The present paper formally expresses this phenomenon as a theorem about the vanishing of the Euler characteristic of certain topological spaces called topological orbifolds. In this new approach, it is explained that the theorem in question follows from the fundamental properties of the orbifold Euler characteristic. As a side effect of these considerations, a theorem due to Coxeter about the vanishing Euler characteristic of a honeycomb tessellation is re-proved in a context which frees the calculations from the assumptions made by Coxeter in his proof. The abstract mathematical concepts are visualized with down-to-earth examples motivated by concrete situations illustrating wallpaper and 3D crystallographic space groups. In a way analogous to the application of the classic Euler equation to completely bounded solids, the formula proven in this paper is applicable to such important crystallographic objects as asymmetric units and Dirichlet domains.text/htmlA topological proof of the modified Euler characteristic based on the orbifold concepttext4772021-06-21Copyright (c) 2021 International Union of CrystallographyActa Crystallographica Section Aresearch papers317326Magnetic modes compatible with the symmetry of crystals
http://scripts.iucr.org/cgi-bin/paper?ib5097
A classification of magnetic point groups is presented which gives an answer to the question: which magnetic groups can describe a given magnetic mode? There are 32 categories of magnetic point groups which describe 64 unique magnetic modes: 16 with a ferromagnetic component and 48 without. This classification focused on magnetic modes is helpful for finding the magnetic space group which can describe the magnetic symmetry of the material.Copyright (c) 2021 International Union of Crystallographyurn:issn:2053-2733Fabrykiewicz, P.Przeniosło, R.Sosnowska, I.2021-06-21doi:10.1107/S2053273321004551International Union of CrystallographyThe magnetic point groups are classified into 32 categories which lead to a total of 64 unique magnetic modes with the same set of directions of M. This classification gives hints on the neutron powder diffraction analysis of collinear magnetic ordering.ENmagnetic orderingmagnetic space groupssymmetrysite symmetrypoint groupspin reorientationA classification of magnetic point groups is presented which gives an answer to the question: which magnetic groups can describe a given magnetic mode? There are 32 categories of magnetic point groups which describe 64 unique magnetic modes: 16 with a ferromagnetic component and 48 without. This classification focused on magnetic modes is helpful for finding the magnetic space group which can describe the magnetic symmetry of the material.text/htmlMagnetic modes compatible with the symmetry of crystalstext4772021-06-21Copyright (c) 2021 International Union of CrystallographyActa Crystallographica Section Aresearch papers327338A new density-modification procedure extending the application of the recent |ρ|-based phasing algorithm to larger crystal structures
http://scripts.iucr.org/cgi-bin/paper?ik5001
The incorporation of the new peakness-enhancing fast Fourier transform compatible ipp procedure (ipp = inner-pixel preservation) into the recently published SM algorithm based on |ρ| [Rius (2020). Acta Cryst A76, 489–493] improves its phasing efficiency for larger crystal structures with atomic resolution data. Its effectiveness is clearly demonstrated via a collection of test crystal structures (taken from the Protein Data Bank) either starting from random phase values or by using the randomly shifted modulus function (a Patterson-type synthesis) as initial ρ estimate. It has been found that in the presence of medium scatterers (e.g. S or Cl atoms) crystal structures with 1500 × c atoms in the unit cell (c = number of centerings) can be routinely solved. In the presence of strong scatterers like Fe, Cu or Zn atoms this number increases to around 5000 × c atoms. The implementation of this strengthened SM algorithm is simple, since it only includes a few easy-to-adjust parameters.Copyright (c) 2021 International Union of Crystallographyurn:issn:2053-2733Rius, J.Torrelles, X.2021-06-21doi:10.1107/S2053273321004915International Union of CrystallographyThe insertion of a peakness-enhancing fast Fourier transform compatible module in the novel SM,|ρ| phasing algorithm improves its efficiency for larger crystal structures as shown with a collection of representative X-ray diffraction data sets taken from the Protein Data Bank.ENSM phasing algorithmipp procedure|ρ|-based phasing residualdirect methodsorigin-free modulus sum functionstructure solutionThe incorporation of the new peakness-enhancing fast Fourier transform compatible ipp procedure (ipp = inner-pixel preservation) into the recently published SM algorithm based on |ρ| [Rius (2020). Acta Cryst A76, 489–493] improves its phasing efficiency for larger crystal structures with atomic resolution data. Its effectiveness is clearly demonstrated via a collection of test crystal structures (taken from the Protein Data Bank) either starting from random phase values or by using the randomly shifted modulus function (a Patterson-type synthesis) as initial ρ estimate. It has been found that in the presence of medium scatterers (e.g. S or Cl atoms) crystal structures with 1500 × c atoms in the unit cell (c = number of centerings) can be routinely solved. In the presence of strong scatterers like Fe, Cu or Zn atoms this number increases to around 5000 × c atoms. The implementation of this strengthened SM algorithm is simple, since it only includes a few easy-to-adjust parameters.text/htmlA new density-modification procedure extending the application of the recent |ρ|-based phasing algorithm to larger crystal structurestext4772021-06-21Copyright (c) 2021 International Union of CrystallographyActa Crystallographica Section Aresearch papers339347Sidney C. Abrahams (1924–2021)
http://scripts.iucr.org/cgi-bin/paper?es5034
Copyright (c) 2021 International Union of Crystallographyurn:issn:2053-2733Brock, C.P.Glazer, A.M.2021-06-28doi:10.1107/S2053273321004927International Union of CrystallographyObituary for Sidney C. AbrahamsENobituarylithium niobatelithium tantalateferroelectricitytungsten bronzestext/htmlSidney C. Abrahams (1924–2021)text4772021-06-28Copyright (c) 2021 International Union of CrystallographyActa Crystallographica Section Aobituaries348350An Invitation to Applied Category Theory. Seven Sketches in Compositionality. By Brendan Fong and David I. Spivak. Cambridge University Press, 2019. Paperback, pp. 348. Price GBP 37.99. ISBN 9781108711821.
http://scripts.iucr.org/cgi-bin/paper?xo0167
Copyright (c) 2021 International Union of Crystallographyurn:issn:2053-2733Stöger, B.2021-06-21doi:10.1107/S2053273321003685International Union of CrystallographyENbook reviewcategory theorytext/htmlAn Invitation to Applied Category Theory. Seven Sketches in Compositionality. By Brendan Fong and David I. Spivak. Cambridge University Press, 2019. Paperback, pp. 348. Price GBP 37.99. ISBN 9781108711821.text4772021-06-21Copyright (c) 2021 International Union of CrystallographyActa Crystallographica Section Abook reviews351352Space Group Visualizer. By Eckhard Hitzer and Christian Perwass. Independently published, 2021. Paperback, pp. 162. Price EUR 22.25, USD 24.99. ISBN 979-8719838618.
http://scripts.iucr.org/cgi-bin/paper?xo0181
Copyright (c) 2021 International Union of Crystallographyurn:issn:2053-2733Müller, U.2021-06-21doi:10.1107/S2053273321004174International Union of CrystallographyENspace-group theorycrystallographic softwaretext/htmlSpace Group Visualizer. By Eckhard Hitzer and Christian Perwass. Independently published, 2021. Paperback, pp. 162. Price EUR 22.25, USD 24.99. ISBN 979-8719838618.text4772021-06-21Copyright (c) 2021 International Union of CrystallographyActa Crystallographica Section Abook reviews353354