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 imageA new method for lattice reduction using directional and hyperplanar shearing
http://scripts.iucr.org/cgi-bin/paper?lu5009
A geometric method of lattice reduction based on cycles of directional and hyperplanar shears is presented. The deviation from cubicity at each step of the reduction is evaluated by a parameter called `basis rhombicity' which is the sum of the absolute values of the elements of the metric tensor associated with the basis. The levels of reduction are quite similar to those obtained with the Lenstra–Lenstra–Lovász (LLL) algorithm, at least up to the moderate dimensions that have been tested (lower than 20). The method can be used to reduce unit cells attached to given hyperplanes.https://creativecommons.org/licenses/by/4.0/urn:issn:2053-2733Cayron, C.2022-01-01doi:10.1107/S2053273321011037International Union of CrystallographyA new algorithm for lattice reduction based on a series of directional and hyperplanar shears and driven by the decrease of the basis rhombicity is proposed. It can be used to reduce unit cells in dimension 3 and higher.enLATTICE REDUCTION; HYPERPLANE; LEFT INVERSE; ALGORITHMA geometric method of lattice reduction based on cycles of directional and hyperplanar shears is presented. The deviation from cubicity at each step of the reduction is evaluated by a parameter called `basis rhombicity' which is the sum of the absolute values of the elements of the metric tensor associated with the basis. The levels of reduction are quite similar to those obtained with the Lenstra–Lenstra–Lovász (LLL) algorithm, at least up to the moderate dimensions that have been tested (lower than 20). The method can be used to reduce unit cells attached to given hyperplanes.text/htmlA new method for lattice reduction using directional and hyperplanar shearingtext1782022-01-01Acta Crystallographica Section A: Foundations and Advanceshttps://creativecommons.org/licenses/by/4.0/2053-2733research papers1med@iucr.orgJanuary 202292053-2733Parameterization of magnetic vector potentials and fields for efficient multislice calculations of elastic electron scattering
http://scripts.iucr.org/cgi-bin/paper?ou5020
The multislice method, which simulates the propagation of the incident electron wavefunction through a crystal, is a well established method for analysing the multiple scattering effects that an electron beam may undergo. The inclusion of magnetic effects into this method proves crucial towards simulating enhanced magnetic interaction of vortex beams with magnetic materials, calculating magnetic Bragg spots or searching for magnon signatures, to name a few examples. Inclusion of magnetism poses novel challenges to the efficiency of the multislice method for larger systems, especially regarding the consistent computation of magnetic vector potentials A and magnetic fields B over large supercells. This work presents a tabulation of parameterized magnetic (PM) values for the first three rows of transition metal elements computed from atomic density functional theory (DFT) calculations, allowing for the efficient computation of approximate A and B across large crystals using only structural and magnetic moment size and direction information. Ferromagnetic b.c.c. (body-centred cubic) Fe and tetragonal FePt are chosen to showcase the performance of PM values versus directly obtaining A and B from the unit-cell spin density by DFT. The magnetic fields of b.c.c. Fe are well described by the PM approach while for FePt the PM approach is less accurate due to deformations in the spin density. Calculations of the magnetic signal, namely the change due to A and B of the intensity of diffraction patterns, show that the PM approach for both b.c.c. Fe and FePt is able to describe the effects of magnetism in these systems to a good degree of accuracy.https://creativecommons.org/licenses/by/4.0/urn:issn:2053-2733Lyon, K.Rusz, J.2021-10-29doi:10.1107/S2053273321008792International Union of CrystallographyA tabulation is presented of parameterized magnetic fields computed from atomic density functional theory calculations that allows for the efficient computation of approximate magnetic vector fields in materials using only structural and magnetic moment size and direction information. Multislice calculations of the change in the intensity of diffraction patterns due to magnetism for body-centred cubic Fe and FePt show that this approach is able to describe the effects of magnetism in these kinds of systems to a good degree of accuracy.enMAGNETISM; MULTISLICE; PARAMETERIZATION; SIMULATION; DFTThe multislice method, which simulates the propagation of the incident electron wavefunction through a crystal, is a well established method for analysing the multiple scattering effects that an electron beam may undergo. The inclusion of magnetic effects into this method proves crucial towards simulating enhanced magnetic interaction of vortex beams with magnetic materials, calculating magnetic Bragg spots or searching for magnon signatures, to name a few examples. Inclusion of magnetism poses novel challenges to the efficiency of the multislice method for larger systems, especially regarding the consistent computation of magnetic vector potentials A and magnetic fields B over large supercells. This work presents a tabulation of parameterized magnetic (PM) values for the first three rows of transition metal elements computed from atomic density functional theory (DFT) calculations, allowing for the efficient computation of approximate A and B across large crystals using only structural and magnetic moment size and direction information. Ferromagnetic b.c.c. (body-centred cubic) Fe and tetragonal FePt are chosen to showcase the performance of PM values versus directly obtaining A and B from the unit-cell spin density by DFT. The magnetic fields of b.c.c. Fe are well described by the PM approach while for FePt the PM approach is less accurate due to deformations in the spin density. Calculations of the magnetic signal, namely the change due to A and B of the intensity of diffraction patterns, show that the PM approach for both b.c.c. Fe and FePt is able to describe the effects of magnetism in these systems to a good degree of accuracy.text/htmlParameterization of magnetic vector potentials and fields for efficient multislice calculations of elastic electron scatteringtext6772021-10-29Acta Crystallographica Section A: Foundations and Advanceshttps://creativecommons.org/licenses/by/4.0/2053-2733research papers509med@iucr.orgNovember 20215182053-2733Layer groups: Brillouin-zone and crystallographic databases on the Bilbao Crystallographic Server
http://scripts.iucr.org/cgi-bin/paper?ug5030
The section of the Bilbao Crystallographic Server (https://www.cryst.ehu.es/) dedicated to subperiodic groups contains crystallographic and Brillouin-zone databases for the layer groups. The crystallographic databases include the generators/general positions (GENPOS), Wyckoff positions (WYCKPOS) and maximal subgroups (MAXSUB). The Brillouin-zone database (LKVEC) offers k-vector tables and Brillouin-zone figures of all 80 layer groups which form the background of the classification of their irreducible representations. The symmetry properties of the wavevectors are described applying the so-called reciprocal-space-group approach and this classification scheme is compared with that of Litvin & Wike [(1991), Character Tables and Compatibility Relations of the Eighty Layer Groups and Seventeen Plane Groups. New York: Plenum Press]. The specification of independent parameter ranges of k vectors in the representation domains of the Brillouin zones provides a solution to the problems of uniqueness and completeness of layer-group representations. The Brillouin-zone figures and k-vector tables are described in detail and illustrated by several examples.https://creativecommons.org/licenses/by/4.0/urn:issn:2053-2733de la Flor, G.Souvignier, B.Madariaga, G.Aroyo, M.I.2021-09-24doi:10.1107/S205327332100783XInternational Union of CrystallographyThe set of databases of the Bilbao Crystallographic Server (https://www.cryst.ehu.es/) including crystallographic data on generators, general positions, Wyckoff positions, maximal subgroups and Brillouin-zone figures and k-vector tables for all 80 layer groups are discussed in detail and illustrated.enBILBAO CRYSTALLOGRAPHIC SERVER; LAYER GROUPS; BRILLOUIN-ZONE DATABASEThe section of the Bilbao Crystallographic Server (https://www.cryst.ehu.es/) dedicated to subperiodic groups contains crystallographic and Brillouin-zone databases for the layer groups. The crystallographic databases include the generators/general positions (GENPOS), Wyckoff positions (WYCKPOS) and maximal subgroups (MAXSUB). The Brillouin-zone database (LKVEC) offers k-vector tables and Brillouin-zone figures of all 80 layer groups which form the background of the classification of their irreducible representations. The symmetry properties of the wavevectors are described applying the so-called reciprocal-space-group approach and this classification scheme is compared with that of Litvin & Wike [(1991), Character Tables and Compatibility Relations of the Eighty Layer Groups and Seventeen Plane Groups. New York: Plenum Press]. The specification of independent parameter ranges of k vectors in the representation domains of the Brillouin zones provides a solution to the problems of uniqueness and completeness of layer-group representations. The Brillouin-zone figures and k-vector tables are described in detail and illustrated by several examples.text/htmlLayer groups: Brillouin-zone and crystallographic databases on the Bilbao Crystallographic Servertext6772021-09-24Acta Crystallographica Section A: Foundations and Advanceshttps://creativecommons.org/licenses/by/4.0/2053-2733research papers559med@iucr.orgNovember 20215712053-2733The role of an objective function in the mathematical modelling of wide-angle X-ray diffraction curves of semi-crystalline polymers
http://scripts.iucr.org/cgi-bin/paper?ae5105
To decompose a wide-angle X-ray diffraction (WAXD) curve of a semi-crystalline polymer into crystalline peaks and amorphous halos, a theoretical best-fitted curve, i.e. a mathematical model, is constructed. In fitting the theoretical curve to the experimental one, various functions can be used to quantify and minimize the deviations between the curves. The analyses and calculations performed in this work have proved that the quality of the model, its parameters and consequently the information on the structure of the investigated polymer are considerably dependent on the shape of an objective function. It is shown that the best models are obtained employing the least-squares method in which the sum of squared absolute errors is minimized. On the other hand, the methods in which the objective functions are based on the relative errors do not give a good fit and should not be used. The comparison and evaluation were performed using WAXD curves of seven polymers: isotactic polypropylene, polyvinylidene fluoride, cellulose I, cellulose II, polyethylene, polyethylene terephthalate and polyamide 6. The methods were compared and evaluated using statistical tests and measures of the quality of fitting.https://creativecommons.org/licenses/by/4.0/urn:issn:2053-2733Rabiej, M.Rabiej, S.2021-09-24doi:10.1107/S2053273321007762International Union of CrystallographyThe choice of an objective function in constructing a mathematical model of an experimental wide-angle X-ray diffraction curve of semi-crystalline polymers is discussed.enPOLYMERS; DIFFRACTION CURVES; ABSOLUTE ERROR; RELATIVE ERROR; OBJECTIVE FUNCTION; MATHEMATICAL MODELLINGTo decompose a wide-angle X-ray diffraction (WAXD) curve of a semi-crystalline polymer into crystalline peaks and amorphous halos, a theoretical best-fitted curve, i.e. a mathematical model, is constructed. In fitting the theoretical curve to the experimental one, various functions can be used to quantify and minimize the deviations between the curves. The analyses and calculations performed in this work have proved that the quality of the model, its parameters and consequently the information on the structure of the investigated polymer are considerably dependent on the shape of an objective function. It is shown that the best models are obtained employing the least-squares method in which the sum of squared absolute errors is minimized. On the other hand, the methods in which the objective functions are based on the relative errors do not give a good fit and should not be used. The comparison and evaluation were performed using WAXD curves of seven polymers: isotactic polypropylene, polyvinylidene fluoride, cellulose I, cellulose II, polyethylene, polyethylene terephthalate and polyamide 6. The methods were compared and evaluated using statistical tests and measures of the quality of fitting.text/htmlThe role of an objective function in the mathematical modelling of wide-angle X-ray diffraction curves of semi-crystalline polymerstext6772021-09-24Acta Crystallographica Section A: Foundations and Advanceshttps://creativecommons.org/licenses/by/4.0/2053-2733research papers534med@iucr.orgNovember 20215472053-2733On incoherent diffractive imaging
http://scripts.iucr.org/cgi-bin/paper?iv5016
Incoherent diffractive imaging (IDI) promises structural analysis with atomic resolution based on intensity interferometry of pulsed X-ray fluorescence emission. However, its experimental realization is still pending and a comprehensive theory of contrast formation has not been established to date. Explicit expressions are derived for the equal-pulse two-point intensity correlations, as the principal measured quantity of IDI, with full control of the prefactors, based on a simple model of stochastic fluorescence emission. The model considers the photon detection statistics, the finite temporal coherence of the individual emissions, as well as the geometry of the scattering volume. The implications are interpreted in view of the most relevant quantities, including the fluorescence lifetime, the excitation pulse, as well as the extent of the scattering volume and pixel size. Importantly, the spatiotemporal overlap between any two emissions in the sample can be identified as a crucial factor limiting the contrast and its dependency on the sample size can be derived. The paper gives rigorous estimates for the optimum sample size, the maximum photon yield and the expected signal-to-noise ratio under optimal conditions. Based on these estimates, the feasibility of IDI experiments for plausible experimental parameters is discussed. It is shown in particular that the mean number of photons per detector pixel which can be achieved with X-ray fluorescence is severely limited and as a consequence imposes restrictive constraints on possible applications.https://creativecommons.org/licenses/by/4.0/urn:issn:2053-2733Lohse, L.M.Vassholz, M.Salditt, T.2021-08-27doi:10.1107/S2053273321007300International Union of CrystallographyStarting from a simple model of stochastic fluorescence emission, a theory is derived of contrast formation and signal-to-noise ratio for incoherent diffractive imaging; its feasibility for plausible experimental parameters is discussed.enFEMTOSECOND STUDIES; FREE-ELECTRON LASER; CORRELATED FLUCTUATIONS; DIFFRACT-THEN-DESTROY; SINGLE PARTICLES; XFELIncoherent diffractive imaging (IDI) promises structural analysis with atomic resolution based on intensity interferometry of pulsed X-ray fluorescence emission. However, its experimental realization is still pending and a comprehensive theory of contrast formation has not been established to date. Explicit expressions are derived for the equal-pulse two-point intensity correlations, as the principal measured quantity of IDI, with full control of the prefactors, based on a simple model of stochastic fluorescence emission. The model considers the photon detection statistics, the finite temporal coherence of the individual emissions, as well as the geometry of the scattering volume. The implications are interpreted in view of the most relevant quantities, including the fluorescence lifetime, the excitation pulse, as well as the extent of the scattering volume and pixel size. Importantly, the spatiotemporal overlap between any two emissions in the sample can be identified as a crucial factor limiting the contrast and its dependency on the sample size can be derived. The paper gives rigorous estimates for the optimum sample size, the maximum photon yield and the expected signal-to-noise ratio under optimal conditions. Based on these estimates, the feasibility of IDI experiments for plausible experimental parameters is discussed. It is shown in particular that the mean number of photons per detector pixel which can be achieved with X-ray fluorescence is severely limited and as a consequence imposes restrictive constraints on possible applications.text/htmlOn incoherent diffractive imagingtext5772021-08-27Acta Crystallographica Section A: Foundations and Advanceshttps://creativecommons.org/licenses/by/4.0/2053-2733research papers480med@iucr.orgSeptember 20214962053-2733Moiré, Euler and self-similarity – the lattice parameters of twisted hexagonal crystals
http://scripts.iucr.org/cgi-bin/paper?ug5017
A real-space approach for the calculation of the moiré lattice parameters for superstructures formed by a set of rotated hexagonal 2D crystals such as graphene or transition-metal dichalcogenides is presented. Apparent moiré lattices continuously form for all rotation angles, and their lattice parameter to a good approximation follows a hyperbolical angle dependence. Moiré crystals, i.e. moiré lattices decorated with a basis, require more crucial assessment of the commensurabilities and lead to discrete solutions and a non-continuous angle dependence of the moiré-crystal lattice parameter. In particular, this lattice parameter critically depends on the rotation angle, and continuous variation of the angle can lead to apparently erratic changes of the lattice parameter. The solutions form a highly complex pattern, which reflects number-theoretical relations between formation parameters of the moiré crystal. The analysis also provides insight into the special case of a 30° rotation of the constituting lattices, for which a dodecagonal quasicrystalline structure forms.https://creativecommons.org/licenses/by/4.0/urn:issn:2053-2733Feuerbacher, M.2021-08-19doi:10.1107/S2053273321007245International Union of CrystallographyThe moiré lattice parameters are calculated for superstructures formed by a set of rotated hexagonal 2D crystals such as graphene or transition-metal dichalcogenides, and the highly complex pattern of solutions is discussed.en2D MATERIALS; MOIRE PATTERN; TWISTED BILAYERS; TWISTRONICS; GRAPHENEA real-space approach for the calculation of the moiré lattice parameters for superstructures formed by a set of rotated hexagonal 2D crystals such as graphene or transition-metal dichalcogenides is presented. Apparent moiré lattices continuously form for all rotation angles, and their lattice parameter to a good approximation follows a hyperbolical angle dependence. Moiré crystals, i.e. moiré lattices decorated with a basis, require more crucial assessment of the commensurabilities and lead to discrete solutions and a non-continuous angle dependence of the moiré-crystal lattice parameter. In particular, this lattice parameter critically depends on the rotation angle, and continuous variation of the angle can lead to apparently erratic changes of the lattice parameter. The solutions form a highly complex pattern, which reflects number-theoretical relations between formation parameters of the moiré crystal. The analysis also provides insight into the special case of a 30° rotation of the constituting lattices, for which a dodecagonal quasicrystalline structure forms.text/htmlMoiré, Euler and self-similarity – the lattice parameters of twisted hexagonal crystalstext5772021-08-19Acta Crystallographica Section A: Foundations and Advanceshttps://creativecommons.org/licenses/by/4.0/2053-2733research papers460med@iucr.orgSeptember 20214712053-2733A fast algorithm to find reduced hyperplane unit cells and solve N-dimensional Bézout's identities
http://scripts.iucr.org/cgi-bin/paper?lu5010
Deformation twinning on a plane is a simple shear that transforms a unit cell attached to the plane into another unit cell equivalent by mirror symmetry or 180° rotation. Thus, crystallographic models of twinning require the determination of the short unit cells attached to the planes, or hyperplanes for dimensions higher than 3. Here, a method is presented to find them. Equivalently, it gives the solutions of the N-dimensional Bézout's identity associated with the Miller indices of the hyperplane.https://creativecommons.org/licenses/by/4.0/urn:issn:2053-2733Cayron, C.2021-08-13doi:10.1107/S2053273321006835International Union of CrystallographyThe paper describes a method to determine a short unit cell attached to any hyperplane given by its integer vector p. Equivalently, it gives all the solutions of the N-dimensional Bézout's identity associated with the coordinates of p.enN-DIMENSIONAL BEZOUT'S IDENTITY; HYPERPLANE UNIT CELL; INTEGER RELATION; TWINNINGDeformation twinning on a plane is a simple shear that transforms a unit cell attached to the plane into another unit cell equivalent by mirror symmetry or 180° rotation. Thus, crystallographic models of twinning require the determination of the short unit cells attached to the planes, or hyperplanes for dimensions higher than 3. Here, a method is presented to find them. Equivalently, it gives the solutions of the N-dimensional Bézout's identity associated with the Miller indices of the hyperplane.text/htmlA fast algorithm to find reduced hyperplane unit cells and solve N-dimensional Bézout's identitiestext775https://creativecommons.org/licenses/by/4.0/Acta Crystallographica Section A: Foundations and Advances2021-08-13453research papers2053-2733September 2021med@iucr.org4592053-2733A 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.https://creativecommons.org/licenses/by/4.0/urn: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 ALGORITHM; IPP PROCEDURE; |[RHO]|-BASED PHASING RESIDUAL; DIRECT METHODS; ORIGIN-FREE MODULUS SUM FUNCTION; STRUCTURE 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 structurestext4772021-06-21Acta Crystallographica Section A: Foundations and Advanceshttps://creativecommons.org/licenses/by/4.0/2053-2733research papers339med@iucr.orgJuly 20213472053-2733From crystal color symmetry to quantum spacetime
http://scripts.iucr.org/cgi-bin/paper?me6135
urn: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 SYMMETRY; QUANTUM SPACETIME; RENORMALIZED BLENDED SPACETIME; RELATIVISTIC SPACETIME CRYSTALStext/htmlFrom crystal color symmetry to quantum spacetimetext4772021-05-27Acta Crystallographica Section A: Foundations and Advances2053-2733scientific commentaries239med@iucr.orgJuly 20212412053-2733A 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.https://creativecommons.org/licenses/by/4.0/urn: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 CHARACTERISTIC; ORBIFOLDS; SPACE-FILLING POLYHEDRA; SPACE GROUPS; ASYMMETRIC 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 concepttext4772021-06-21Acta Crystallographica Section A: Foundations and Advanceshttps://creativecommons.org/licenses/by/4.0/2053-2733research papers317med@iucr.orgJuly 20213262053-2733Resolution 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.https://creativecommons.org/licenses/by/4.0/urn: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 LASERS; X-RAY SPECTROSCOPY; BENT CRYSTALS; DIAMOND CRYSTAL OPTICS; FEMTOSECOND 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 silicontext4772021-05-27Acta Crystallographica Section A: Foundations and Advanceshttps://creativecommons.org/licenses/by/4.0/2053-2733research papers268med@iucr.orgJuly 20212762053-2733Relativistic 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.https://creativecommons.org/licenses/by/4.0/urn: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.enSPACETIME; SPECIAL RELATIVITY; RENORMALIZED BLENDED SPACETIME; RELATIVISTIC 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 crystalstext4772021-05-27Acta Crystallographica Section A: Foundations and Advanceshttps://creativecommons.org/licenses/by/4.0/2053-2733research papers242med@iucr.orgJuly 20212562053-2733Combining X-rays, neutrons and electrons, and NMR, for precision and accuracy in structure–function studies
http://scripts.iucr.org/cgi-bin/paper?ae5100
The distinctive features of the physics-based probes used in understanding the structure of matter focusing on biological sciences, but not exclusively, are described in the modern context. This is set in a wider scope of holistic biology and the scepticism about `reductionism', what is called the `molecular level', and how to respond constructively. These topics will be set alongside the principles of accuracy and precision, and their boundaries. The combination of probes and their application together is the usual way of realizing accuracy. The distinction between precision and accuracy can be blurred by the predictive force of a precise structure, thereby lending confidence in its potential accuracy. These descriptions will be applied to the comparison of cryo and room-temperature protein crystal structures as well as the solid state of a crystal and the same molecules studied by small-angle X-ray scattering in solution and by electron microscopy on a sample grid. Examples will include: time-resolved X-ray Laue crystallography of an enzyme Michaelis complex formed directly in a crystal equivalent to in vivo; a new iodoplatin for radiation therapy predicted from studies of platin crystal structures; and the field of colouration of carotenoids, as an effective assay of function, i.e. their colouration, when unbound and bound to a protein. The complementarity of probes, as well as their combinatory use, is then at the foundation of real (biologically relevant), probe-artefacts-free, structure–function studies. The foundations of our methodologies are being transformed by colossal improvements in technologies of X-ray and neutron sources and their beamline instruments, as well as improved electron microscopes and NMR spectrometers. The success of protein structure prediction from gene sequence recently reported by CASP14 also opens new doors to change and extend the foundations of the structural sciences.https://creativecommons.org/licenses/by/4.0/urn:issn:2053-2733Helliwell, J.R.2021-05-04doi:10.1107/S205327332100317XInternational Union of CrystallographyThe distinctive features of the probes used in understanding the structure of matter focusing on biological sciences, but not exclusively, are described in the modern context to minimize the consequences of artefactual information in data interpretation. The precision and accuracy of both data and technique are revisited. A variety of structural results are described that reach beyond reductionism to the whole biological organism. All these aspects open new doors to change and extend the foundations of the structural sciences.enX-RAYS; NEUTRONS; ELECTRONS; NMR; STRUCTURE AND FUNCTIONThe distinctive features of the physics-based probes used in understanding the structure of matter focusing on biological sciences, but not exclusively, are described in the modern context. This is set in a wider scope of holistic biology and the scepticism about `reductionism', what is called the `molecular level', and how to respond constructively. These topics will be set alongside the principles of accuracy and precision, and their boundaries. The combination of probes and their application together is the usual way of realizing accuracy. The distinction between precision and accuracy can be blurred by the predictive force of a precise structure, thereby lending confidence in its potential accuracy. These descriptions will be applied to the comparison of cryo and room-temperature protein crystal structures as well as the solid state of a crystal and the same molecules studied by small-angle X-ray scattering in solution and by electron microscopy on a sample grid. Examples will include: time-resolved X-ray Laue crystallography of an enzyme Michaelis complex formed directly in a crystal equivalent to in vivo; a new iodoplatin for radiation therapy predicted from studies of platin crystal structures; and the field of colouration of carotenoids, as an effective assay of function, i.e. their colouration, when unbound and bound to a protein. The complementarity of probes, as well as their combinatory use, is then at the foundation of real (biologically relevant), probe-artefacts-free, structure–function studies. The foundations of our methodologies are being transformed by colossal improvements in technologies of X-ray and neutron sources and their beamline instruments, as well as improved electron microscopes and NMR spectrometers. The success of protein structure prediction from gene sequence recently reported by CASP14 also opens new doors to change and extend the foundations of the structural sciences.text/htmlCombining X-rays, neutrons and electrons, and NMR, for precision and accuracy in structure–function studiestext3772021-05-04Acta Crystallographica Section A: Foundations and Advanceshttps://creativecommons.org/licenses/by/4.0/2053-2733lead articles173med@iucr.orgMay 20211852053-2733Diffraction line profiles from polydisperse crystalline systems. Corrigenda
http://scripts.iucr.org/cgi-bin/paper?me6128
Equation (16) and some entries in Table 1 in the article by Scardi & Leoni [(2001), Acta Cryst. A57, 604–613] are corrected.Copyright (c) 2021 International Union of Crystallographyurn:issn:2053-2733Scardi, P.2021-03-17doi:10.1107/S2053273321002813International Union of CrystallographyErrors in the article by Scardi & Leoni [(2001), Acta Cryst. A57, 604–613] are corrected.enLINE PROFILE ANALYSIS; LPA; WHOLE POWDER PATTERN MODELLING; WPPM; CRYSTALLINE DOMAIN SIZE; CORRIGENDAEquation (16) and some entries in Table 1 in the article by Scardi & Leoni [(2001), Acta Cryst. A57, 604–613] are corrected.text/htmlDiffraction line profiles from polydisperse crystalline systems. Corrigendatext3772021-03-17Acta Crystallographica Section A: Foundations and AdvancesCopyright (c) 2021 International Union of Crystallography2053-2733addenda and errata232med@iucr.orgMay 20212322053-2733A new electron diffraction approach for structure refinement applied to Ca3Mn2O7
http://scripts.iucr.org/cgi-bin/paper?lu5005
The digital large-angle convergent-beam electron diffraction (D-LACBED) technique is applied to Ca3Mn2O7 for a range of temperatures. Bloch-wave simulations are used to examine the effects that changes in different parameters have on the intensity in D-LACBED patterns, and atomic coordinates, thermal atomic displacement parameters and apparent occupancy are refined to achieve a good fit between simulation and experiment. The sensitivity of the technique to subtle changes in structure is demonstrated. Refined structures are in good agreement with previous determinations of Ca3Mn2O7 and show the decay of anti-phase oxygen octahedral tilts perpendicular to the c axis of the A21am unit cell with increasing temperature, as well as the robustness of oxygen octahedral tilts about the c axis up to ∼400°C. The technique samples only the zero-order Laue zone and is therefore insensitive to atom displacements along the electron-beam direction. For this reason it is not possible to distinguish between in-phase and anti-phase oxygen octahedral tilting about the c axis using the [110] data collected in this study.https://creativecommons.org/licenses/by/4.0/urn:issn:2053-2733Beanland, R.Smith, K.Vaněk, P.Zhang, H.Hubert, A.Evans, K.Römer, R.A.Kamba, S.2021-03-17doi:10.1107/S2053273321001546International Union of CrystallographyThe `digital' large-angle convergent-beam electron diffraction (D-LACBED) method uses computer control of a transmission electron microscope to collect hundreds of diffraction patterns from a region a few nanometres in size, which are combined into a single data set. The sensitivity of the resulting patterns to crystal structure is explored using the Ruddlesden–Popper oxide Ca3Mn2O7 and it is found that refinement of atomic coordinates can be performed to sub-picometre precision.enDIGITAL DIFFRACTION; ELECTRON DIFFRACTION; CA3MN2O7; CBED; LACBEDThe digital large-angle convergent-beam electron diffraction (D-LACBED) technique is applied to Ca3Mn2O7 for a range of temperatures. Bloch-wave simulations are used to examine the effects that changes in different parameters have on the intensity in D-LACBED patterns, and atomic coordinates, thermal atomic displacement parameters and apparent occupancy are refined to achieve a good fit between simulation and experiment. The sensitivity of the technique to subtle changes in structure is demonstrated. Refined structures are in good agreement with previous determinations of Ca3Mn2O7 and show the decay of anti-phase oxygen octahedral tilts perpendicular to the c axis of the A21am unit cell with increasing temperature, as well as the robustness of oxygen octahedral tilts about the c axis up to ∼400°C. The technique samples only the zero-order Laue zone and is therefore insensitive to atom displacements along the electron-beam direction. For this reason it is not possible to distinguish between in-phase and anti-phase oxygen octahedral tilting about the c axis using the [110] data collected in this study.text/htmlA new electron diffraction approach for structure refinement applied to Ca3Mn2O7text3772021-03-17Acta Crystallographica Section A: Foundations and Advanceshttps://creativecommons.org/licenses/by/4.0/2053-2733research papers196med@iucr.orgMay 20212072053-2733A discussion on `Domain formation and phase transitions in the wurtzite-based heterovalent ternaries: a Landau theory analysis'
http://scripts.iucr.org/cgi-bin/paper?ug5021
Heterovalent ternary nitrides are considered one of the promising classes of materials for photovoltaics, combining attractive physical properties with low toxicity and element abundance. One of the front-runner systems under consideration is ZnSnN2. Although it is nominally a ternary compound, no clear crystallographic evidence for cation ordering has been observed so far. An attempt to elucidate this discrepancy [Quayle (2020). Acta Cryst. A76, 410–420] was the trigger for an intensive discussion between the authors, and an agreement was reached to elaborate on some points in order to set things in perspective. Rather than using a conventional comment–answer scheme, this is published in the form of a joint discussion to celebrate constructive criticism and collegiality.https://creativecommons.org/licenses/by/4.0/urn:issn:2053-2733Quayle, P.C.Breternitz, J.2021-03-23doi:10.1107/S2053273321001376International Union of CrystallographyA scientific exchange on an earlier paper [Quayle (2020). Acta Cryst. A76, 410–420] has led to the clarification of some of the points.enGROUP-SUBGROUP RELATIONSHIPS; NITRIDE MATERIALS; WURTZITE TYPEHeterovalent ternary nitrides are considered one of the promising classes of materials for photovoltaics, combining attractive physical properties with low toxicity and element abundance. One of the front-runner systems under consideration is ZnSnN2. Although it is nominally a ternary compound, no clear crystallographic evidence for cation ordering has been observed so far. An attempt to elucidate this discrepancy [Quayle (2020). Acta Cryst. A76, 410–420] was the trigger for an intensive discussion between the authors, and an agreement was reached to elaborate on some points in order to set things in perspective. Rather than using a conventional comment–answer scheme, this is published in the form of a joint discussion to celebrate constructive criticism and collegiality.text/htmlA discussion on `Domain formation and phase transitions in the wurtzite-based heterovalent ternaries: a Landau theory analysis'text3772021-03-23Acta Crystallographica Section A: Foundations and Advanceshttps://creativecommons.org/licenses/by/4.0/2053-2733research papers217med@iucr.orgMay 20212212053-2733Nothing trumps good data
http://scripts.iucr.org/cgi-bin/paper?me6117
urn:issn:2053-2733Pinkerton, A.A.2021-01-29doi:10.1107/S2053273321000759International Union of CrystallographyThe advantages of a powerful new tool for determining the electron density of small inorganic systems using high-quality powder diffraction data from the MYTHEN microstrip detector [Svane et al. (2021). Acta Cryst. A77, 85–95] are considered.enELECTRON DENSITY; DATA QUALITY; CHARGE DENSITY; POWDER DIFFRACTION; MYTHEN DETECTORtext/htmlNothing trumps good datatext2772021-01-29Acta Crystallographica Section A: Foundations and Advances2053-2733scientific commentaries83med@iucr.orgMarch 2021842053-2733Coordination sequences of crystals are of quasi-polynomial type
http://scripts.iucr.org/cgi-bin/paper?pl5008
The coordination sequence of a graph measures how many vertices the graph has at each distance from a fixed vertex and is a generalization of the coordination number. Here it is proved that the coordination sequence of the graph obtained from a crystal is of quasi-polynomial type, as had been postulated by Grosse-Kunstleve et al. [Acta Cryst. (1996), A52, 879–889].https://creativecommons.org/licenses/by/4.0/urn:issn:2053-2733Nakamura, Y.Sakamoto, R.Mase, T.Nakagawa, J.2021-02-18doi:10.1107/S2053273320016769International Union of CrystallographyIt is proved that the coordination sequence of the graph obtained from a crystal is of quasi-polynomial type, as had been postulated by Grosse-Kunstleve et al. [Acta Cryst. (1996), A52, 879–889] in their study of coordination sequences of zeolites.enCOORDINATION SEQUENCES; GRAPH THEORY; HILBERT POLYNOMIAL; MONOID THEORYThe coordination sequence of a graph measures how many vertices the graph has at each distance from a fixed vertex and is a generalization of the coordination number. Here it is proved that the coordination sequence of the graph obtained from a crystal is of quasi-polynomial type, as had been postulated by Grosse-Kunstleve et al. [Acta Cryst. (1996), A52, 879–889].text/htmlCoordination sequences of crystals are of quasi-polynomial typetext2772021-02-18Acta Crystallographica Section A: Foundations and Advanceshttps://creativecommons.org/licenses/by/4.0/2053-2733research papers138med@iucr.orgMarch 20211482053-2733