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) 2023 International Union of CrystallographyInternational Union of CrystallographyInternational Union of CrystallographytextActa 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/htmlurn:issn:0108-7673https://journals.iucr.orgOpen-access and free articles in Acta Crystallographica Section A Foundations and Advances2002-01-01T00:00+00:006yearlymed@iucr.orgActa Crystallographica Section A Foundations and AdvancesCopyright (c) 2023 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 imageOptimal estimated standard uncertainties of reflection intensities for kinematical refinement from 3D electron diffraction data
http://scripts.iucr.org/cgi-bin/paper?pl5027
Estimating the error in the merged reflection intensities requires a full understanding of all the possible sources of error arising from the measurements. Most diffraction-spot integration methods focus mainly on errors arising from counting statistics for the estimation of uncertainties associated with the reflection intensities. This treatment may be incomplete and partly inadequate. In an attempt to fully understand and identify all the contributions to these errors, three methods are examined for the correction of estimated errors of reflection intensities in electron diffraction data. For a direct comparison, the three methods are applied to a set of organic and inorganic test cases. It is demonstrated that applying the corrections of a specific model that include terms dependent on the original uncertainty and the largest intensity of the symmetry-related reflections improves the overall structure quality of the given data set and improves the final Rall factor. This error model is implemented in the data reduction software PETS2.textenhttps://creativecommons.org/licenses/by/4.0/Several models for estimating the standard uncertainties of reflection intensities are analysed for refinement against 3D electron diffraction data. A new model is proposed which results in more accurate structure models.Estimating the error in the merged reflection intensities requires a full understanding of all the possible sources of error arising from the measurements. Most diffraction-spot integration methods focus mainly on errors arising from counting statistics for the estimation of uncertainties associated with the reflection intensities. This treatment may be incomplete and partly inadequate. In an attempt to fully understand and identify all the contributions to these errors, three methods are examined for the correction of estimated errors of reflection intensities in electron diffraction data. For a direct comparison, the three methods are applied to a set of organic and inorganic test cases. It is demonstrated that applying the corrections of a specific model that include terms dependent on the original uncertainty and the largest intensity of the symmetry-related reflections improves the overall structure quality of the given data set and improves the final Rall factor. This error model is implemented in the data reduction software PETS2.text/htmldoi:10.1107/S2053273323005053Khouchen, M.Klar, P.B.Chintakindi, H.Suresh, A.Palatinus, L.2023-08-14ERROR MODELLING; ERROR ANALYSIS; DATA REDUCTION; ELECTRON DIFFRACTIONurn:issn:2053-2733International Union of CrystallographyOptimal estimated standard uncertainties of reflection intensities for kinematical refinement from 3D electron diffraction data427September 20232053-273352023-08-1479https://creativecommons.org/licenses/by/4.0/Acta Crystallographica Section A: Foundations and Advances2053-2733research papers439med@iucr.orgPatch frequencies in rhombic Penrose tilings
http://scripts.iucr.org/cgi-bin/paper?nv5007
This exposition presents an efficient algorithm for an exact calculation of patch frequencies for rhombic Penrose tilings. A construction of Penrose tilings via dualization is recalled and, by extending the known method for obtaining vertex configurations, the desired algorithm is obtained. It is then used to determine the frequencies of several particularly large patches which appear in the literature. An analogous approach works for a particular class of tilings and this is also explained in detail for the Ammann–Beenker tiling.textenhttps://creativecommons.org/licenses/by/4.0/An algorithm is presented for an exact calculation of patch frequencies for a family of tilings which can be obtained via dualization.This exposition presents an efficient algorithm for an exact calculation of patch frequencies for rhombic Penrose tilings. A construction of Penrose tilings via dualization is recalled and, by extending the known method for obtaining vertex configurations, the desired algorithm is obtained. It is then used to determine the frequencies of several particularly large patches which appear in the literature. An analogous approach works for a particular class of tilings and this is also explained in detail for the Ammann–Beenker tiling.text/htmldoi:10.1107/S2053273323004990Mazáč, J.2023-07-24PATCH FREQUENCY; TILING; DUALIZATION METHODurn:issn:2053-2733International Union of CrystallographyPatch frequencies in rhombic Penrose tilingsActa Crystallographica Section A: Foundations and Advances2023-07-24https://creativecommons.org/licenses/by/4.0/5792053-2733September 2023399med@iucr.org411research papers2053-2733Algorithms for magnetic symmetry operation search and identification of magnetic space group from magnetic crystal structure
http://scripts.iucr.org/cgi-bin/paper?ib5114
A crystal symmetry search is crucial for computational crystallography and materials science. Although algorithms and implementations for the crystal symmetry search have been developed, their extension to magnetic space groups (MSGs) remains limited. In this paper, algorithms for determining magnetic symmetry operations of magnetic crystal structures, identifying magnetic space-group types of given MSGs, searching for transformations to a Belov–Neronova–Smirnova (BNS) setting, and symmetrizing the magnetic crystal structures using the MSGs are presented. The determination of magnetic symmetry operations is numerically stable and is implemented with minimal modifications from the existing crystal symmetry search. Magnetic space-group types and transformations to the BNS setting are identified by a two-step approach combining space-group-type identification and the use of affine normalizers. Point coordinates and magnetic moments of the magnetic crystal structures are symmetrized by projection operators for the MSGs. An implementation is distributed with a permissive free software license in spglib v2.0.2: https://github.com/spglib/spglib.textenhttps://creativecommons.org/licenses/by/4.0/This paper presents algorithms for determining magnetic symmetry operations of magnetic crystal structures, identifying magnetic space-group types from a given magnetic space group (MSG), searching for transformations to a Belov–Neronova–Smirnova setting, and symmetrizing the magnetic crystal structures on the basis of the determined MSGs.A crystal symmetry search is crucial for computational crystallography and materials science. Although algorithms and implementations for the crystal symmetry search have been developed, their extension to magnetic space groups (MSGs) remains limited. In this paper, algorithms for determining magnetic symmetry operations of magnetic crystal structures, identifying magnetic space-group types of given MSGs, searching for transformations to a Belov–Neronova–Smirnova (BNS) setting, and symmetrizing the magnetic crystal structures using the MSGs are presented. The determination of magnetic symmetry operations is numerically stable and is implemented with minimal modifications from the existing crystal symmetry search. Magnetic space-group types and transformations to the BNS setting are identified by a two-step approach combining space-group-type identification and the use of affine normalizers. Point coordinates and magnetic moments of the magnetic crystal structures are symmetrized by projection operators for the MSGs. An implementation is distributed with a permissive free software license in spglib v2.0.2: https://github.com/spglib/spglib.text/htmldoi:10.1107/S2053273323005016Shinohara, K.Togo, A.Tanaka, I.2023-09-06MAGNETIC SPACE GROUP; MAGNETIC SPACE-GROUP TYPE; MAGNETIC STRUCTURE; CRYSTAL STRUCTURE ANALYSIS; AFFINE NORMALIZERurn:issn:2053-2733International Union of CrystallographyAlgorithms for magnetic symmetry operation search and identification of magnetic space group from magnetic crystal structure2053-2733med@iucr.org398research papers2053-2733September 2023390Acta Crystallographica Section A: Foundations and Advances2023-09-06https://creativecommons.org/licenses/by/4.0/579Machine learning for classifying narrow-beam electron diffraction data
http://scripts.iucr.org/cgi-bin/paper?lu5027
As an alternative approach to X-ray crystallography and single-particle cryo-electron microscopy, single-molecule electron diffraction has a better signal-to-noise ratio and the potential to increase the resolution of protein models. This technology requires collection of numerous diffraction patterns, which can lead to congestion of data collection pipelines. However, only a minority of the diffraction data are useful for structure determination because the chances of hitting a protein of interest with a narrow electron beam may be small. This necessitates novel concepts for quick and accurate data selection. For this purpose, a set of machine learning algorithms for diffraction data classification has been implemented and tested. The proposed pre-processing and analysis workflow efficiently distinguished between amorphous ice and carbon support, providing proof of the principle of machine learning based identification of positions of interest. While limited in its current context, this approach exploits inherent characteristics of narrow electron beam diffraction patterns and can be extended for protein data classification and feature extraction.textenhttps://creativecommons.org/licenses/by/4.0/Neural networks were trained for robust classification of narrow electron beam diffraction patterns and may significantly decrease the need for storage space.As an alternative approach to X-ray crystallography and single-particle cryo-electron microscopy, single-molecule electron diffraction has a better signal-to-noise ratio and the potential to increase the resolution of protein models. This technology requires collection of numerous diffraction patterns, which can lead to congestion of data collection pipelines. However, only a minority of the diffraction data are useful for structure determination because the chances of hitting a protein of interest with a narrow electron beam may be small. This necessitates novel concepts for quick and accurate data selection. For this purpose, a set of machine learning algorithms for diffraction data classification has been implemented and tested. The proposed pre-processing and analysis workflow efficiently distinguished between amorphous ice and carbon support, providing proof of the principle of machine learning based identification of positions of interest. While limited in its current context, this approach exploits inherent characteristics of narrow electron beam diffraction patterns and can be extended for protein data classification and feature extraction.text/htmldoi:10.1107/S2053273323004680Matinyan, S.Demir, B.Filipcik, P.Abrahams, J.P.van Genderen, E.2023-06-20DIFFRACTION; SINGLE-MOLECULE ELECTRON DIFFRACTION; TEM; TRANSMISSION ELECTRON MICROSCOPY; MACHINE LEARNING; NEURAL NETWORKSurn:issn:2053-2733International Union of CrystallographyMachine learning for classifying narrow-beam electron diffraction datahttps://creativecommons.org/licenses/by/4.0/2023-06-20360368Acta Crystallographica Section A: Foundations and Advances4792053-2733July 2023med@iucr.orgresearch papers2053-2733New benchmarks in the modelling of X-ray atomic form factors
http://scripts.iucr.org/cgi-bin/paper?ae5130
Analytical representations of X-ray atomic form factor data have been determined. The original data, f0(s;Z), are reproduced to a high degree of accuracy. The mean absolute errors calculated for all s = sin θ/λ and Z values in question are primarily determined by the precision of the published data. The inverse Mott–Bethe formula is the underlying basis with the electron scattering factor expressed by an expansion in Gaussian basis functions. The number of Gaussians depends upon the element and the data and is in the range 6–20. The refinement procedure, conducted to obtain the parameters of the models, is carried out for seven different form factor tables published in the span Cromer & Mann [(1968), Acta Cryst. A24, 321–324] to Olukayode et al. [(2023), Acta Cryst. A79, 59–79]. The s ranges are finite, the most common span being [0.0, 6.0] Å−1. Only one function for each element is needed to model the full range. This presentation to a large extent makes use of a detailed graphical account of the results.textenhttps://creativecommons.org/licenses/by/4.0/Improved analytical representations of X-ray atomic form factors are put forward based on the inverse Mott–Bethe formula. Applying these representations, the mean absolute errors calculated for the complete set of form factors given in Table 6.1.1.1 in International Tables for Crystallography, Vol. C, 3rd ed., are reduced by a factor of ∼50 from previous published analyses. Various form factor compilations are examined to record the applicability of the approach outlined.Analytical representations of X-ray atomic form factor data have been determined. The original data, f0(s;Z), are reproduced to a high degree of accuracy. The mean absolute errors calculated for all s = sin θ/λ and Z values in question are primarily determined by the precision of the published data. The inverse Mott–Bethe formula is the underlying basis with the electron scattering factor expressed by an expansion in Gaussian basis functions. The number of Gaussians depends upon the element and the data and is in the range 6–20. The refinement procedure, conducted to obtain the parameters of the models, is carried out for seven different form factor tables published in the span Cromer & Mann [(1968), Acta Cryst. A24, 321–324] to Olukayode et al. [(2023), Acta Cryst. A79, 59–79]. The s ranges are finite, the most common span being [0.0, 6.0] Å−1. Only one function for each element is needed to model the full range. This presentation to a large extent makes use of a detailed graphical account of the results.text/htmldoi:10.1107/S2053273323003996Thorkildsen, G.2023-06-02ATOMIC FORM FACTORS; ANALYTICAL REPRESENTATIONS; INVERSE MOTT-BETHE FORMULAurn:issn:2053-2733International Union of CrystallographyNew benchmarks in the modelling of X-ray atomic form factors2053-2733research papersmed@iucr.orgJuly 20232053-2733479Acta Crystallographica Section A: Foundations and Advances3303182023-06-02https://creativecommons.org/licenses/by/4.0/Crystallography of homophase twisted bilayers: coincidence, union lattices and space groups
http://scripts.iucr.org/cgi-bin/paper?nv5002
This paper presents the basic tools used to describe the global symmetry of so-called bilayer structures obtained when two differently oriented crystalline monoatomic layers of the same structure are superimposed and displaced with respect to each other. The 2D nature of the layers leads to the use of complex numbers that allows for simple explicit analytical expressions of the symmetry properties involved in standard bicrystallography [Gratias & Portier (1982). J. Phys. Colloq. 43, C6-15–C6-24; Pond & Vlachavas (1983). Proc. R. Soc. Lond. Ser. A, 386, 95–143]. The focus here is on the twist rotations such that the superimposition of the two layers generates a coincidence lattice. The set of such coincidence rotations plotted as a function of the lengths of their coincidence lattice unit-cell nodes exhibits remarkable arithmetic properties. The second part of the paper is devoted to determination of the space groups of the bilayers as a function of the rigid-body translation associated with the coincidence rotation. These general results are exemplified with a detailed study of graphene bilayers, showing that the possible symmetries of graphene bilayers with a coincidence lattice, whatever the rotation and the rigid-body translation, are distributed in only six distinct types of space groups. The appendix discusses some generalized cases of heterophase bilayers with coincidence lattices due to specific lattice constant ratios, and mechanical deformation by elongation and shear of a layer on top of an undeformed one.textenhttps://creativecommons.org/licenses/by/4.0/A general scheme is proposed to classify and determine the crystallographic properties of twisted bilayers of any homophase 2D structures using complex numbers.This paper presents the basic tools used to describe the global symmetry of so-called bilayer structures obtained when two differently oriented crystalline monoatomic layers of the same structure are superimposed and displaced with respect to each other. The 2D nature of the layers leads to the use of complex numbers that allows for simple explicit analytical expressions of the symmetry properties involved in standard bicrystallography [Gratias & Portier (1982). J. Phys. Colloq. 43, C6-15–C6-24; Pond & Vlachavas (1983). Proc. R. Soc. Lond. Ser. A, 386, 95–143]. The focus here is on the twist rotations such that the superimposition of the two layers generates a coincidence lattice. The set of such coincidence rotations plotted as a function of the lengths of their coincidence lattice unit-cell nodes exhibits remarkable arithmetic properties. The second part of the paper is devoted to determination of the space groups of the bilayers as a function of the rigid-body translation associated with the coincidence rotation. These general results are exemplified with a detailed study of graphene bilayers, showing that the possible symmetries of graphene bilayers with a coincidence lattice, whatever the rotation and the rigid-body translation, are distributed in only six distinct types of space groups. The appendix discusses some generalized cases of heterophase bilayers with coincidence lattices due to specific lattice constant ratios, and mechanical deformation by elongation and shear of a layer on top of an undeformed one.text/htmldoi:10.1107/S2053273323003662Gratias, D.Quiquandon, M.2023-06-02BICRYSTALLOGRAPHY WITH COMPLEX NUMBERS; BILAYERS; COINCIDENCE LATTICES; SPACE GROUPSurn:issn:2053-2733International Union of CrystallographyCrystallography of homophase twisted bilayers: coincidence, union lattices and space groups479Acta Crystallographica Section A: Foundations and AdvancesJuly 20232053-2733research papersmed@iucr.org2053-27332023-06-02https://creativecommons.org/licenses/by/4.0/301317Efficient structure-factor modeling for crystals with multiple components
http://scripts.iucr.org/cgi-bin/paper?pl5025
Diffraction intensities from a crystallographic experiment include contributions from the entire unit cell of the crystal: the macromolecule, the solvent around it and eventually other compounds. These contributions cannot typically be well described by an atomic model alone, i.e. using point scatterers. Indeed, entities such as disordered (bulk) solvent, semi-ordered solvent (e.g. lipid belts in membrane proteins, ligands, ion channels) and disordered polymer loops require other types of modeling than a collection of individual atoms. This results in the model structure factors containing multiple contributions. Most macromolecular applications assume two-component structure factors: one component arising from the atomic model and the second one describing the bulk solvent. A more accurate and detailed modeling of the disordered regions of the crystal will naturally require more than two components in the structure factors, which presents algorithmic and computational challenges. Here an efficient solution of this problem is proposed. All algorithms described in this work have been implemented in the computational crystallography toolbox (CCTBX) and are also available within Phenix software. These algorithms are rather general and do not use any assumptions about molecule type or size nor about those of its components.textenhttps://creativecommons.org/licenses/by/4.0/A multi-component description of the unit-cell content is introduced. Efficient algorithms to define the contribution of these components to structure factors are described and implemented in CCTBX and Phenix.Diffraction intensities from a crystallographic experiment include contributions from the entire unit cell of the crystal: the macromolecule, the solvent around it and eventually other compounds. These contributions cannot typically be well described by an atomic model alone, i.e. using point scatterers. Indeed, entities such as disordered (bulk) solvent, semi-ordered solvent (e.g. lipid belts in membrane proteins, ligands, ion channels) and disordered polymer loops require other types of modeling than a collection of individual atoms. This results in the model structure factors containing multiple contributions. Most macromolecular applications assume two-component structure factors: one component arising from the atomic model and the second one describing the bulk solvent. A more accurate and detailed modeling of the disordered regions of the crystal will naturally require more than two components in the structure factors, which presents algorithmic and computational challenges. Here an efficient solution of this problem is proposed. All algorithms described in this work have been implemented in the computational crystallography toolbox (CCTBX) and are also available within Phenix software. These algorithms are rather general and do not use any assumptions about molecule type or size nor about those of its components.text/htmldoi:10.1107/S205327332300356XAfonine, P.V.Adams, P.D.Urzhumtsev, A.G.2023-06-20STRUCTURE FACTORS; MULTIPLE COMPONENTS; SCATTERING FUNCTIONS; BULK SOLVENT; REFINEMENT; DENSITY MAPSurn:issn:2053-2733International Union of CrystallographyEfficient structure-factor modeling for crystals with multiple components2053-2733research papersmed@iucr.orgJuly 20232053-2733794Acta Crystallographica Section A: Foundations and Advances3523452023-06-20https://creativecommons.org/licenses/by/4.0/A note on the wedge reversion antisymmetry operation and 51 types of physical quantities in arbitrary dimensions
http://scripts.iucr.org/cgi-bin/paper?ib5117
The paper by Gopalan [(2020). Acta Cryst. A76, 318–327] presented an enumeration of the 41 physical quantity types in non-relativistic physics, in arbitrary dimensions, based on the formalism of Clifford algebra. Gopalan considered three antisymmetries: spatial inversion, 1, time reversal, 1′, and wedge reversion, 1†. A consideration of the set of all seven antisymmetries (1, 1′, 1†, 1′†, 1†, 1′, 1′†) leads to an extension of the results obtained by Gopalan. It is shown that there are 51 types of physical quantities with distinct symmetry properties in total.textenhttps://creativecommons.org/licenses/by/4.0/It is shown that there are 51 types of physical quantities in arbitrary dimensions with distinct transformations by wedge reversion symmetry. In the paper by Gopalan [(2020). Acta Cryst. A76, 318–327] only 41 types were enumerated.The paper by Gopalan [(2020). Acta Cryst. A76, 318–327] presented an enumeration of the 41 physical quantity types in non-relativistic physics, in arbitrary dimensions, based on the formalism of Clifford algebra. Gopalan considered three antisymmetries: spatial inversion, 1, time reversal, 1′, and wedge reversion, 1†. A consideration of the set of all seven antisymmetries (1, 1′, 1†, 1′†, 1†, 1′, 1′†) leads to an extension of the results obtained by Gopalan. It is shown that there are 51 types of physical quantities with distinct symmetry properties in total.text/htmldoi:10.1107/S2053273323003303Fabrykiewicz, P.2023-06-05MULTIVECTORS; WEDGE REVERSION; ANTISYMMETRY; CLIFFORD ALGEBRAurn:issn:2053-2733International Union of CrystallographyA note on the wedge reversion antisymmetry operation and 51 types of physical quantities in arbitrary dimensions381https://creativecommons.org/licenses/by/4.0/2023-06-05384July 20232053-2733Acta Crystallographica Section A: Foundations and Advances4792053-2733med@iucr.orgshort communicationsApproximating lattice similarity
http://scripts.iucr.org/cgi-bin/paper?uv5018
A method is proposed for choosing unit cells for a group of crystals so that they all appear as nearly similar as possible to a selected cell. Related unit cells with varying cell parameters or indexed with different lattice centering can be accommodated.textenhttps://creativecommons.org/licenses/by/4.0/A method is proposed for transforming unit cells for a group of crystals so that they all appear as similar as possible to a selected cell.A method is proposed for choosing unit cells for a group of crystals so that they all appear as nearly similar as possible to a selected cell. Related unit cells with varying cell parameters or indexed with different lattice centering can be accommodated.text/htmldoi:10.1107/S2053273323003200Andrews, L.C.Bernstein, H.J.Sauter, N.K.2023-07-24LATTICE MATCHING; DELAUNAY; DELONE; NIGGLI; SELLINGurn:issn:2053-2733International Union of CrystallographyApproximating lattice similarityActa Crystallographica Section A: Foundations and Advances52023-07-2479https://creativecommons.org/licenses/by/4.0/September 20232053-2733480484med@iucr.orgresearch papers2053-2733Fourier-synthesis approach for static charge-density reconstruction from theoretical structure factors of CaB6
http://scripts.iucr.org/cgi-bin/paper?pl5022
In a pilot study, electron-density (ED) and ED Laplacian distributions were reconstructed for the challenging case of CaB6 (Pearson symbol cP7) with conceptually fractional B—B bonds from quantum-chemically calculated structure-factor sets with resolutions 0.5 Å–1 ≤ [sin(θ)/λ]max ≤ 5.0 Å–1 by means of Fourier-synthesis techniques. Convergence of norm deviations of the distributions obtained with respect to the reference ones was obtained in the valence region of the unit cell. The QTAIM (quantum theory of atoms in molecules) atomic charges, and the ED and ED Laplacian values at the characteristic critical points of the Fourier-synthesized distributions have been analysed for each resolution and found to display a convergent behaviour with increasing resolution. The presented method(exponent) (ME) type of Fourier-synthesis approach can qualitatively reconstruct all characteristic chemical bonding features of the ED from valence-electron structure-factor sets with resolutions of about 1.2 Å–1 and beyond, and from all-electron structure-factor sets with resolutions of about 2.0 Å–1 and beyond. Application of the ME type of Fourier-synthesis approach for reconstruction of ED and ED Laplacian distributions at experimental resolution is proposed to complement the usual extrapolation to infinite resolution in Hansen–Coppens multipole model derived static ED distributions.textenhttps://creativecommons.org/licenses/by/4.0/A novel type of Fourier-synthesis approach is reported for determining electron-density distributions and their Laplacians from static structure factors of CaB6. The approach relies on mathematical weighting functions to yield a data set, reproducing all characteristic chemical bonding features of the original quantum-chemically calculated distributions.In a pilot study, electron-density (ED) and ED Laplacian distributions were reconstructed for the challenging case of CaB6 (Pearson symbol cP7) with conceptually fractional B—B bonds from quantum-chemically calculated structure-factor sets with resolutions 0.5 Å–1 ≤ [sin(θ)/λ]max ≤ 5.0 Å–1 by means of Fourier-synthesis techniques. Convergence of norm deviations of the distributions obtained with respect to the reference ones was obtained in the valence region of the unit cell. The QTAIM (quantum theory of atoms in molecules) atomic charges, and the ED and ED Laplacian values at the characteristic critical points of the Fourier-synthesized distributions have been analysed for each resolution and found to display a convergent behaviour with increasing resolution. The presented method(exponent) (ME) type of Fourier-synthesis approach can qualitatively reconstruct all characteristic chemical bonding features of the ED from valence-electron structure-factor sets with resolutions of about 1.2 Å–1 and beyond, and from all-electron structure-factor sets with resolutions of about 2.0 Å–1 and beyond. Application of the ME type of Fourier-synthesis approach for reconstruction of ED and ED Laplacian distributions at experimental resolution is proposed to complement the usual extrapolation to infinite resolution in Hansen–Coppens multipole model derived static ED distributions.text/htmldoi:10.1107/S2053273323002644Bergner, C.Grin, Y.Wagner, F.R.2023-05-05ELECTRON DENSITY; FOURIER TRANSFORMATION; FOURIER SYNTHESIS; HEXABORIDES; LAPLACIANurn:issn:2053-2733International Union of CrystallographyFourier-synthesis approach for static charge-density reconstruction from theoretical structure factors of CaB6research papersmed@iucr.org2053-2733793Acta Crystallographica Section A: Foundations and Advances2053-2733May 2023272https://creativecommons.org/licenses/by/4.0/2023-05-05246On the combinatorics of crystal structures. II. Number of Wyckoff sequences of a given subdivision complexity
http://scripts.iucr.org/cgi-bin/paper?uv5014
Wyckoff sequences are a way of encoding combinatorial information about crystal structures of a given symmetry. In particular, they offer an easy access to the calculation of a crystal structure's combinatorial, coordinational and configurational complexity, taking into account the individual multiplicities (combinatorial degrees of freedom) and arities (coordinational degrees of freedom) associated with each Wyckoff position. However, distinct Wyckoff sequences can yield the same total numbers of combinatorial and coordinational degrees of freedom. In this case, they share the same value for their Shannon entropy based subdivision complexity. The enumeration of Wyckoff sequences with this property is a combinatorial problem solved in this work, first in the general case of fixed subdivision complexity but non-specified Wyckoff sequence length, and second for the restricted case of Wyckoff sequences of both fixed subdivision complexity and fixed Wyckoff sequence length. The combinatorial results are accompanied by calculations of the combinatorial, coordinational, configurational and subdivision complexities, performed on Wyckoff sequences representing actual crystal structures.textenhttps://creativecommons.org/licenses/by/4.0/The number of Wyckoff sequences of a given subdivision complexity is calculated by means of a generating polynomial approach and a dynamic programming approach. The result depends on the choice of space-group symmetry (which is obligatory) and Wyckoff sequence length (which is optional). It also takes into account specified values for the total number of combinatorial and coordinational degrees of freedom, thereby representing crystal structures of invariant subdivision complexity.Wyckoff sequences are a way of encoding combinatorial information about crystal structures of a given symmetry. In particular, they offer an easy access to the calculation of a crystal structure's combinatorial, coordinational and configurational complexity, taking into account the individual multiplicities (combinatorial degrees of freedom) and arities (coordinational degrees of freedom) associated with each Wyckoff position. However, distinct Wyckoff sequences can yield the same total numbers of combinatorial and coordinational degrees of freedom. In this case, they share the same value for their Shannon entropy based subdivision complexity. The enumeration of Wyckoff sequences with this property is a combinatorial problem solved in this work, first in the general case of fixed subdivision complexity but non-specified Wyckoff sequence length, and second for the restricted case of Wyckoff sequences of both fixed subdivision complexity and fixed Wyckoff sequence length. The combinatorial results are accompanied by calculations of the combinatorial, coordinational, configurational and subdivision complexities, performed on Wyckoff sequences representing actual crystal structures.text/htmldoi:10.1107/S2053273323002437Hornfeck, W.Červený, K.2023-05-11WYCKOFF SEQUENCES; COMBINATORICS; SHANNON ENTROPY; STRUCTURAL COMPLEXITYurn:issn:2053-2733International Union of CrystallographyOn the combinatorics of crystal structures. II. Number of Wyckoff sequences of a given subdivision complexityresearch papersmed@iucr.org2053-2733379Acta Crystallographica Section A: Foundations and AdvancesMay 20232053-2733294https://creativecommons.org/licenses/by/4.0/2023-05-11280Boris Gruber's contributions to mathematical crystallography
http://scripts.iucr.org/cgi-bin/paper?uv5015
Boris Gruber made fundamental contributions to the study of crystal lattices, leading to a finer classification of lattice types than those of Paul Niggli and Boris Delaunay before him.textenBoris Gruber's fundamental contributions to the classification of crystal lattices are reviewed.Boris Gruber made fundamental contributions to the study of crystal lattices, leading to a finer classification of lattice types than those of Paul Niggli and Boris Delaunay before him.text/htmldoi:10.1107/S2053273323001961Grimmer, H.2023-05-05BORIS GRUBER; CRYSTAL LATTICES; LATTICE CHARACTERS; BUERGER CELLS; NIGGLI CELL; BRAVAIS TYPES; GENERAurn:issn:2053-2733International Union of CrystallographyBoris Gruber's contributions to mathematical crystallographyscientific commentariesmed@iucr.org2053-2733793Acta Crystallographica Section A: Foundations and Advances2053-2733May 20233002023-05-05295Crystal search – feasibility study of a real-time deep learning process for crystallization well images
http://scripts.iucr.org/cgi-bin/paper?ik5007
To avoid the time-consuming and often monotonous task of manual inspection of crystallization plates, a Python-based program to automatically detect crystals in crystallization wells employing deep learning techniques was developed. The program uses manually scored crystallization trials deposited in a database of an in-house crystallization robot as a training set. Since the success rate of such a system is able to catch up with manual inspection by trained persons, it will become an important tool for crystallographers working on biological samples. Four network architectures were compared and the SqueezeNet architecture performed best. In detecting crystals AlexNet accomplished a better result, but with a lower threshold the mean value for crystal detection was improved for SqueezeNet. Two assumptions were made about the imaging rate. With these two extremes it was found that an image processing rate of at least two times, but up to 58 times in the worst case, would be needed to reach the maximum imaging rate according to the deep learning network architecture employed for real-time classification. To avoid high workloads for the control computer of the CrystalMation system, the computing is distributed over several workstations, participating voluntarily, by the grid programming system from the Berkeley Open Infrastructure for Network Computing (BOINC). The outcome of the program is redistributed into the database as automatic real-time scores (ARTscore). These are immediately visible as colored frames around each crystallization well image of the inspection program. In addition, regions of droplets with the highest scoring probability found by the system are also available as images.textenhttps://creativecommons.org/licenses/by/4.0/Four deep learning architectures were applied and SqueezeNet scored best. It was combined with the grid programming system BOINC to realize automatic real-time scoring of crystallization well images. Scores are written to a database and displayed to facilitate image inspection for users.To avoid the time-consuming and often monotonous task of manual inspection of crystallization plates, a Python-based program to automatically detect crystals in crystallization wells employing deep learning techniques was developed. The program uses manually scored crystallization trials deposited in a database of an in-house crystallization robot as a training set. Since the success rate of such a system is able to catch up with manual inspection by trained persons, it will become an important tool for crystallographers working on biological samples. Four network architectures were compared and the SqueezeNet architecture performed best. In detecting crystals AlexNet accomplished a better result, but with a lower threshold the mean value for crystal detection was improved for SqueezeNet. Two assumptions were made about the imaging rate. With these two extremes it was found that an image processing rate of at least two times, but up to 58 times in the worst case, would be needed to reach the maximum imaging rate according to the deep learning network architecture employed for real-time classification. To avoid high workloads for the control computer of the CrystalMation system, the computing is distributed over several workstations, participating voluntarily, by the grid programming system from the Berkeley Open Infrastructure for Network Computing (BOINC). The outcome of the program is redistributed into the database as automatic real-time scores (ARTscore). These are immediately visible as colored frames around each crystallization well image of the inspection program. In addition, regions of droplets with the highest scoring probability found by the system are also available as images.text/htmldoi:10.1107/S2053273323001948Thielmann, Y.Luft, T.Zint, N.Koepke, J.2023-06-02BIOCRYSTALLIZATION; HIGH-THROUGHPUT SCREENING; DEEP LEARNING; NEURAL NETWORK; U-NET; ALEXNET; VGGNET; RESNET; SQUEEZENET; BOINCurn:issn:2053-2733International Union of CrystallographyCrystal search – feasibility study of a real-time deep learning process for crystallization well images3312023-06-02https://creativecommons.org/licenses/by/4.0/338July 20232053-2733794Acta Crystallographica Section A: Foundations and Advances2053-2733research papersmed@iucr.orgCrystal diffraction prediction and partiality estimation using Gaussian basis functions
http://scripts.iucr.org/cgi-bin/paper?ik5005
The recent diversification of macromolecular crystallographic experiments including the use of pink beams, convergent electron diffraction and serial snapshot crystallography has shown the limitations of using the Laue equations for diffraction prediction. This article gives a computationally efficient way of calculating approximate crystal diffraction patterns given varying distributions of the incoming beam, crystal shapes and other potentially hidden parameters. This approach models each pixel of a diffraction pattern and improves data processing of integrated peak intensities by enabling the correction of partially recorded reflections. The fundamental idea is to express the distributions as weighted sums of Gaussian functions. The approach is demonstrated on serial femtosecond crystallography data sets, showing a significant decrease in the required number of patterns to refine a structure to a given error.textenhttps://creativecommons.org/licenses/by/4.0/Reflection position, size and shape prediction and partiality estimation of crystal diffraction by integrating using a Gaussian basis are described.The recent diversification of macromolecular crystallographic experiments including the use of pink beams, convergent electron diffraction and serial snapshot crystallography has shown the limitations of using the Laue equations for diffraction prediction. This article gives a computationally efficient way of calculating approximate crystal diffraction patterns given varying distributions of the incoming beam, crystal shapes and other potentially hidden parameters. This approach models each pixel of a diffraction pattern and improves data processing of integrated peak intensities by enabling the correction of partially recorded reflections. The fundamental idea is to express the distributions as weighted sums of Gaussian functions. The approach is demonstrated on serial femtosecond crystallography data sets, showing a significant decrease in the required number of patterns to refine a structure to a given error.text/htmldoi:10.1107/S2053273323000682Brehm, W.White, T.Chapman, H.N.2023-02-17PARTIALITY ESTIMATION; DIFFRACTION PREDICTION; MERGING; SERIAL SNAPSHOT CRYSTALLOGRAPHYurn:issn:2053-2733International Union of CrystallographyCrystal diffraction prediction and partiality estimation using Gaussian basis functionsmed@iucr.orgresearch papers2053-2733Acta Crystallographica Section A: Foundations and Advances2792053-2733March 20231622023-02-17https://creativecommons.org/licenses/by/4.0/145Dynamic tilting in perovskites
http://scripts.iucr.org/cgi-bin/paper?lu5021
A new computational analysis of tilt behaviour in perovskites is presented. This includes the development of a computational program – PALAMEDES – to extract tilt angles and the tilt phase from molecular dynamics simulations. The results are used to generate simulated selected-area electron and neutron diffraction patterns which are compared with experimental patterns for CaTiO3. The simulations not only reproduced all symmetrically allowed superlattice reflections associated with tilt but also showed local correlations that give rise to symmetrically forbidden reflections and the kinematic origin of diffuse scattering.textenhttps://creativecommons.org/licenses/by/4.0/A new computational program to analyse and extract tilt data from molecular dynamics simulations of perovskites is presented and results compared with experimental data.A new computational analysis of tilt behaviour in perovskites is presented. This includes the development of a computational program – PALAMEDES – to extract tilt angles and the tilt phase from molecular dynamics simulations. The results are used to generate simulated selected-area electron and neutron diffraction patterns which are compared with experimental patterns for CaTiO3. The simulations not only reproduced all symmetrically allowed superlattice reflections associated with tilt but also showed local correlations that give rise to symmetrically forbidden reflections and the kinematic origin of diffuse scattering.text/htmldoi:10.1107/S2053273322011949Handley, C.M.Ward, R.E.Freeman, C.L.Reaney, I.M.Sinclair, D.C.Harding, J.H.2023-01-23PEROVSKITES; TILT; DIFFRACTION; MOLECULAR DYNAMICS; SUPERLATTICEurn:issn:2053-2733International Union of CrystallographyDynamic tilting in perovskites163https://creativecommons.org/licenses/by/4.0/2023-01-231702053-2733March 2023Acta Crystallographica Section A: Foundations and Advances7922053-2733med@iucr.orgresearch papersDynamical diffraction of high-energy electrons by light-atom structures: a multiple forward scattering interpretation
http://scripts.iucr.org/cgi-bin/paper?lu5020
Because of the strong electron–atom interaction, the kinematic theory of diffraction cannot be used to describe the scattering of electrons by an assembly of atoms due to the strong dynamical diffraction that needs to be taken into account. In this paper, the scattering of high-energy electrons by a regular array of light atoms is solved exactly by applying the T-matrix formalism to the corresponding Schrödinger's equation in spherical coordinates. The independent atom model is used, where each atom is represented by a sphere with an effective constant potential. The validity of the forward scattering approximation and the phase grating approximation, assumed by the popular multislice method, is discussed, and an alternative interpretation of multiple scattering is proposed and compared with existing interpretations.textenhttps://creativecommons.org/licenses/by/4.0/The T-matrix is used to compute the scattering of fast electrons by a regular array of effective spherical potential wells. An assessment of the forward scattering approximation and a real-space multiple scattering interpretation are provided.Because of the strong electron–atom interaction, the kinematic theory of diffraction cannot be used to describe the scattering of electrons by an assembly of atoms due to the strong dynamical diffraction that needs to be taken into account. In this paper, the scattering of high-energy electrons by a regular array of light atoms is solved exactly by applying the T-matrix formalism to the corresponding Schrödinger's equation in spherical coordinates. The independent atom model is used, where each atom is represented by a sphere with an effective constant potential. The validity of the forward scattering approximation and the phase grating approximation, assumed by the popular multislice method, is discussed, and an alternative interpretation of multiple scattering is proposed and compared with existing interpretations.text/htmldoi:10.1107/S2053273322011779Drevon, T.R.Waterman, D.G.Krissinel, E.2023-02-09HIGH-ENERGY ELECTRON DIFFRACTION; T-MATRIX; MULTIPLE SCATTERING; INDEPENDENT ATOM APPROXIMATIONurn:issn:2053-2733International Union of CrystallographyDynamical diffraction of high-energy electrons by light-atom structures: a multiple forward scattering interpretationmed@iucr.orgresearch papers2053-2733Acta Crystallographica Section A: Foundations and Advances792March 20232053-27331912023-02-09https://creativecommons.org/licenses/by/4.0/180A phase retrieval algorithm for triply periodic minimal surface like structures
http://scripts.iucr.org/cgi-bin/paper?ik5006
A method to solve the crystallographic phase problem of materials with triply periodic minimal surface like structures, such as lyotropic liquid crystal bicontinuous cubic phases, is reported. In triply periodic minimal surface like structures, the difference between the maximum and minimum electron densities tends to be the smallest for the true phase combination among the possible combinations [Oka (2022). Acta Cryst. A78, 430–436]. Using this feature, a new iterative phase retrieval algorithm for structure determination was developed. The algorithm modifies electron densities outside the upper and lower thresholds in the iterative Fourier transformation process with fixed amplitudes for the structure factors, and efficiently searches for the structure with the smallest difference between the maximum and minimum electron densities. The proper structure was determined by this algorithm for all tested data for lyotropic liquid crystal bicontinuous cubic phases and mesoporous silicas. Although some cases required constraints such as the volume fraction for structure determination, more than half could be determined without any constraints, including space groups.textenhttps://creativecommons.org/licenses/by/4.0/A method to solve the crystallographic phase problem of materials with triply periodic minimal surface like structures, such as lyotropic liquid crystal bicontinuous cubic phases, is reported.A method to solve the crystallographic phase problem of materials with triply periodic minimal surface like structures, such as lyotropic liquid crystal bicontinuous cubic phases, is reported. In triply periodic minimal surface like structures, the difference between the maximum and minimum electron densities tends to be the smallest for the true phase combination among the possible combinations [Oka (2022). Acta Cryst. A78, 430–436]. Using this feature, a new iterative phase retrieval algorithm for structure determination was developed. The algorithm modifies electron densities outside the upper and lower thresholds in the iterative Fourier transformation process with fixed amplitudes for the structure factors, and efficiently searches for the structure with the smallest difference between the maximum and minimum electron densities. The proper structure was determined by this algorithm for all tested data for lyotropic liquid crystal bicontinuous cubic phases and mesoporous silicas. Although some cases required constraints such as the volume fraction for structure determination, more than half could be determined without any constraints, including space groups.text/htmldoi:10.1107/S2053273322010786Oka, T.2023-01-01CRYSTALLOGRAPHIC PHASE RETRIEVAL; LYOTROPIC LIQUID CRYSTALS; MESOPOROUS SILICA; TRIPLY PERIODIC MINIMAL SURFACESurn:issn:2053-2733International Union of CrystallographyA phase retrieval algorithm for triply periodic minimal surface like structures5851https://creativecommons.org/licenses/by/4.0/2023-01-012053-2733research papersmed@iucr.orgJanuary 20232053-2733791Acta Crystallographica Section A: Foundations and AdvancesElectron density and thermal motion of diamond at elevated temperatures
http://scripts.iucr.org/cgi-bin/paper?pl5020
The electron density and thermal motion of diamond are determined at nine temperatures between 100 K and 1000 K via synchrotron powder X-ray diffraction (PXRD) data collected on a high-accuracy detector system. Decoupling of the thermal motion from the thermally smeared electron density is performed via an iterative Wilson–Hansen–Coppens–Rietveld procedure using theoretical static structure factors from density functional theory (DFT) calculations. The thermal motion is found to be harmonic and isotropic in the explored temperature range, and excellent agreement is observed between experimental atomic displacement parameters (ADPs) and those obtained via theoretical harmonic phonon calculations (HPC), even at 1000 K. The Debye temperature of diamond is determined experimentally to be ΘD = 1883 (35) K. A topological analysis of the electron density explores the temperature dependency of the electron density at the bond critical point. The properties are found to be constant throughout the temperature range. The robustness of the electron density confirms the validity of the crystallographic convolution approximation for diamond in the explored temperature range.textenhttps://creativecommons.org/licenses/by/4.0/The electron densities and atomic displacement parameters of diamond are determined from 100 K to 1000 K using synchrotron powder X-ray diffraction.The electron density and thermal motion of diamond are determined at nine temperatures between 100 K and 1000 K via synchrotron powder X-ray diffraction (PXRD) data collected on a high-accuracy detector system. Decoupling of the thermal motion from the thermally smeared electron density is performed via an iterative Wilson–Hansen–Coppens–Rietveld procedure using theoretical static structure factors from density functional theory (DFT) calculations. The thermal motion is found to be harmonic and isotropic in the explored temperature range, and excellent agreement is observed between experimental atomic displacement parameters (ADPs) and those obtained via theoretical harmonic phonon calculations (HPC), even at 1000 K. The Debye temperature of diamond is determined experimentally to be ΘD = 1883 (35) K. A topological analysis of the electron density explores the temperature dependency of the electron density at the bond critical point. The properties are found to be constant throughout the temperature range. The robustness of the electron density confirms the validity of the crystallographic convolution approximation for diamond in the explored temperature range.text/htmldoi:10.1107/S2053273322010154Beyer, J.Grønbech, T.B.E.Zhang, J.Kato, K.Brummerstedt Iversen, B.2023-01-01X-RAY ELECTRON DENSITY; SYNCHROTRON POWDER X-RAY DIFFRACTION; DIAMOND; CONVOLUTION APPROXIMATIONurn:issn:2053-2733International Union of CrystallographyElectron density and thermal motion of diamond at elevated temperaturesmed@iucr.orgresearch papers2053-2733Acta Crystallographica Section A: Foundations and Advances791January 20232053-2733502023-01-01https://creativecommons.org/licenses/by/4.0/41Geographic style maps for two-dimensional lattices
http://scripts.iucr.org/cgi-bin/paper?uv5012
This paper develops geographic style maps containing two-dimensional lattices in all known periodic crystals parameterized by recent complete invariants. Motivated by rigid crystal structures, lattices are considered up to rigid motion and uniform scaling. The resulting space of two-dimensional lattices is a square with identified edges or a punctured sphere. The new continuous maps show all Bravais classes as low-dimensional subspaces, visualize hundreds of thousands of lattices of real crystal structures from the Cambridge Structural Database, and motivate the development of continuous and invariant-based crystallography.textenhttps://creativecommons.org/licenses/by/4.0/Continuous invariant-based maps visualize for the first time all two-dimensional lattices extracted from hundreds of thousands of known crystal structures in the Cambridge Structural Database.This paper develops geographic style maps containing two-dimensional lattices in all known periodic crystals parameterized by recent complete invariants. Motivated by rigid crystal structures, lattices are considered up to rigid motion and uniform scaling. The resulting space of two-dimensional lattices is a square with identified edges or a punctured sphere. The new continuous maps show all Bravais classes as low-dimensional subspaces, visualize hundreds of thousands of lattices of real crystal structures from the Cambridge Structural Database, and motivate the development of continuous and invariant-based crystallography.text/htmldoi:10.1107/S2053273322010075Bright, M.Cooper, A.I.Kurlin, V.2023-01-01TWO-DIMENSIONAL LATTICES; REDUCED BASIS; OBTUSE SUPERBASE; ISOMETRY; COMPLETE INVARIANTS; METRIC TENSOR; CONTINUITYurn:issn:2053-2733International Union of CrystallographyGeographic style maps for two-dimensional latticeshttps://creativecommons.org/licenses/by/4.0/2023-01-01113791Acta Crystallographica Section A: Foundations and AdvancesJanuary 20232053-2733research papersmed@iucr.org2053-2733