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) 2024 International Union of CrystallographyInternational Union of CrystallographyInternational Union of Crystallographyurn:issn:0108-7673https://journals.iucr.orgtexttext/htmlActa 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.Open-access and free articles in Acta Crystallographica Section A Foundations and Advancesyearly62002-01-01T00:00+00:00urn:issn:0108-7673med@iucr.orgActa Crystallographica Section A Foundations and AdvancesCopyright (c) 2024 International Union of CrystallographyOpen-access and free articles in Acta Crystallographica Section A: Foundations and Advanceshttp://journals.iucr.org/logos/rss10a.gif
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Still imageClusterFinder: a fast tool to find cluster structures from pair distribution function data
http://scripts.iucr.org/cgi-bin/paper?tw5008
A novel automated high-throughput screening approach, ClusterFinder, is reported for finding candidate structures for atomic pair distribution function (PDF) structural refinements. Finding starting models for PDF refinements is notoriously difficult when the PDF originates from nanoclusters or small nanoparticles. The reported ClusterFinder algorithm can screen 104 to 105 candidate structures from structural databases such as the Inorganic Crystal Structure Database (ICSD) in minutes, using the crystal structures as templates in which it looks for atomic clusters that result in a PDF similar to the target measured PDF. The algorithm returns a rank-ordered list of clusters for further assessment by the user. The algorithm has performed well for simulated and measured PDFs of metal–oxido clusters such as Keggin clusters. This is therefore a powerful approach to finding structural cluster candidates in a modelling campaign for PDFs of nanoparticles and nanoclusters.https://creativecommons.org/licenses/by/4.0/PAIR DISTRIBUTION FUNCTION ANALYSIS; NANOCLUSTERS; NANOMATERIALS; SCREENINGAn automated high-throughput screening approach is presented for identifying starting structure models for pair distribution function analysis of nanoclusters.doi:10.1107/S2053273324001116urn:issn:2053-2733texttext/htmlA novel automated high-throughput screening approach, ClusterFinder, is reported for finding candidate structures for atomic pair distribution function (PDF) structural refinements. Finding starting models for PDF refinements is notoriously difficult when the PDF originates from nanoclusters or small nanoparticles. The reported ClusterFinder algorithm can screen 104 to 105 candidate structures from structural databases such as the Inorganic Crystal Structure Database (ICSD) in minutes, using the crystal structures as templates in which it looks for atomic clusters that result in a PDF similar to the target measured PDF. The algorithm returns a rank-ordered list of clusters for further assessment by the user. The algorithm has performed well for simulated and measured PDFs of metal–oxido clusters such as Keggin clusters. This is therefore a powerful approach to finding structural cluster candidates in a modelling campaign for PDFs of nanoparticles and nanoclusters.ClusterFinder: a fast tool to find cluster structures from pair distribution function dataInternational Union of CrystallographyAnker, A.S.Friis-Jensen, U.Johansen, F.L.Billinge, S.J.L.Jensen, K.M.Ø.en2024-02-2922080March 20242053-2733Acta Crystallographica Section A: Foundations and Advanceshttps://creativecommons.org/licenses/by/4.0/research papers22053-2733med@iucr.org2132024-02-29Automated selection of nanoparticle models for small-angle X-ray scattering data analysis using machine learning
http://scripts.iucr.org/cgi-bin/paper?iv5034
Small-angle X-ray scattering (SAXS) is widely used to analyze the shape and size of nanoparticles in solution. A multitude of models, describing the SAXS intensity resulting from nanoparticles of various shapes, have been developed by the scientific community and are used for data analysis. Choosing the optimal model is a crucial step in data analysis, which can be difficult and time-consuming, especially for non-expert users. An algorithm is proposed, based on machine learning, representation learning and SAXS-specific preprocessing methods, which instantly selects the nanoparticle model best suited to describe SAXS data. The different algorithms compared are trained and evaluated on a simulated database. This database includes 75 000 scattering spectra from nine nanoparticle models, and realistically simulates two distinct device configurations. It will be made freely available to serve as a basis of comparison for future work. Deploying a universal solution for automatic nanoparticle model selection is a challenge made more difficult by the diversity of SAXS instruments and their flexible settings. The poor transferability of classification rules learned on one device configuration to another is highlighted. It is shown that training on several device configurations enables the algorithm to be generalized, without degrading performance compared with configuration-specific training. Finally, the classification algorithm is evaluated on a real data set obtained by performing SAXS experiments on nanoparticles for each of the instrumental configurations, which have been characterized by transmission electron microscopy. This data set, although very limited, allows estimation of the transferability of the classification rules learned on simulated data to real data.https://creativecommons.org/licenses/by/4.0/MACHINE LEARNING; NANOPARTICLES; SAXS; SMALL-ANGLE X-RAY SCATTERING; DATA ANALYSIS; MODEL SELECTIONMany models have been developed for analyzing SAXS data; however choosing the optimal model is difficult and time-consuming, especially for non-expert users. This paper proposes an algorithm, based on machine learning, representation learning and SAXS-specific preprocessing methods, which instantly selects the nanoparticle model best suited to describe SAXS data.doi:10.1107/S2053273324000950urn:issn:2053-2733texttext/htmlSmall-angle X-ray scattering (SAXS) is widely used to analyze the shape and size of nanoparticles in solution. A multitude of models, describing the SAXS intensity resulting from nanoparticles of various shapes, have been developed by the scientific community and are used for data analysis. Choosing the optimal model is a crucial step in data analysis, which can be difficult and time-consuming, especially for non-expert users. An algorithm is proposed, based on machine learning, representation learning and SAXS-specific preprocessing methods, which instantly selects the nanoparticle model best suited to describe SAXS data. The different algorithms compared are trained and evaluated on a simulated database. This database includes 75 000 scattering spectra from nine nanoparticle models, and realistically simulates two distinct device configurations. It will be made freely available to serve as a basis of comparison for future work. Deploying a universal solution for automatic nanoparticle model selection is a challenge made more difficult by the diversity of SAXS instruments and their flexible settings. The poor transferability of classification rules learned on one device configuration to another is highlighted. It is shown that training on several device configurations enables the algorithm to be generalized, without degrading performance compared with configuration-specific training. Finally, the classification algorithm is evaluated on a real data set obtained by performing SAXS experiments on nanoparticles for each of the instrumental configurations, which have been characterized by transmission electron microscopy. This data set, although very limited, allows estimation of the transferability of the classification rules learned on simulated data to real data.Automated selection of nanoparticle models for small-angle X-ray scattering data analysis using machine learningInternational Union of CrystallographyMonge, N.Deschamps, A.Amini, M.-R.en2024-02-29212https://creativecommons.org/licenses/by/4.0/research papersActa Crystallographica Section A: Foundations and Advances2053-2733March 20248022024-02-29med@iucr.org2022053-2733Universal parameters of bulk-solvent masks
http://scripts.iucr.org/cgi-bin/paper?pl5035
The bulk solvent is a major component of biomacromolecular crystals that contributes significantly to the observed diffraction intensities. Accurate modelling of the bulk solvent has been recognized as important for many crystallographic calculations. Owing to its simplicity and modelling power, the flat (mask-based) bulk-solvent model is used by most modern crystallographic software packages to account for disordered solvent. In this model, the bulk-solvent contribution is defined by a binary mask and a scale (scattering) function. The mask is calculated on a regular grid using the atomic model coordinates and their chemical types. The grid step and two radii, solvent and shrinkage, are the three parameters that govern the mask calculation. They are highly correlated and their choice is a compromise between the computer time needed to calculate the mask and the accuracy of the mask. It is demonstrated here that this choice can be optimized using a unique value of 0.6 Å for the grid step irrespective of the data resolution, and the radii values adjusted correspondingly. The improved values were tested on a large sample of Protein Data Bank entries derived from X-ray diffraction data and are now used in the computational crystallography toolbox (CCTBX) and in Phenix as the default choice.https://creativecommons.org/licenses/by/4.0/BULK-SOLVENT MODELLING; FLAT MODELS; SOLVENT MASKS; GRID STEPS; MASK PARAMETERSThe optimal choice of bulk-solvent mask parameters (grid step, and solvent and shrinkage radii) has been revised.doi:10.1107/S2053273324000299urn:issn:2053-2733texttext/htmlThe bulk solvent is a major component of biomacromolecular crystals that contributes significantly to the observed diffraction intensities. Accurate modelling of the bulk solvent has been recognized as important for many crystallographic calculations. Owing to its simplicity and modelling power, the flat (mask-based) bulk-solvent model is used by most modern crystallographic software packages to account for disordered solvent. In this model, the bulk-solvent contribution is defined by a binary mask and a scale (scattering) function. The mask is calculated on a regular grid using the atomic model coordinates and their chemical types. The grid step and two radii, solvent and shrinkage, are the three parameters that govern the mask calculation. They are highly correlated and their choice is a compromise between the computer time needed to calculate the mask and the accuracy of the mask. It is demonstrated here that this choice can be optimized using a unique value of 0.6 Å for the grid step irrespective of the data resolution, and the radii values adjusted correspondingly. The improved values were tested on a large sample of Protein Data Bank entries derived from X-ray diffraction data and are now used in the computational crystallography toolbox (CCTBX) and in Phenix as the default choice.Universal parameters of bulk-solvent masksInternational Union of CrystallographyUrzhumtsev, A.Adams, P.Afonine, P.en2024-02-092024-02-09med@iucr.org1942053-273322053-273380March 2024research papershttps://creativecommons.org/licenses/by/4.0/Acta Crystallographica Section A: Foundations and Advances201Parameterized absorptive electron scattering factors
http://scripts.iucr.org/cgi-bin/paper?pl5034
In electron diffraction, thermal atomic motion produces incoherent scattering over a relatively wide angular range, which appears as a diffuse background that is usually subtracted from measurements of Bragg spot intensities in structure solution methods. The transfer of electron flux from Bragg spots to diffuse scatter is modelled using complex scattering factors f + if′ in the Bloch wave methodology. In a two-beam Einstein model the imaginary `absorptive' scattering factor f′ can be obtained by the evaluation of an integral containing f over all possible scattering angles. While more sophisticated models of diffuse scatter are widely used in the electron microscopy community, it is argued in this paper that this simple model is appropriate for current structure solution and refinement methods. The two-beam model is a straightforward numerical calculation, but even this simplistic approach can become time consuming for simulations of materials with large numbers of atoms in the unit cell and/or many incident beam orientations. Here, a parameterized form of f′ is provided for 103 elements as neutral, spherical atoms that reduces calculation time considerably.https://creativecommons.org/licenses/by/4.0/ELECTRON DIFFRACTION; ABSORPTION; 3D-ED; THREE-DIMENSIONAL ELECTRON DIFFRACTION; THERMAL DIFFUSE SCATTERINGThis paper provides a rapid parameterized calculation of absorptive scattering factors for 103 elements as neutral, spherical atoms, which reduces calculation time considerably.doi:10.1107/S2053273323010963urn:issn:2053-2733texttext/htmlIn electron diffraction, thermal atomic motion produces incoherent scattering over a relatively wide angular range, which appears as a diffuse background that is usually subtracted from measurements of Bragg spot intensities in structure solution methods. The transfer of electron flux from Bragg spots to diffuse scatter is modelled using complex scattering factors f + if′ in the Bloch wave methodology. In a two-beam Einstein model the imaginary `absorptive' scattering factor f′ can be obtained by the evaluation of an integral containing f over all possible scattering angles. While more sophisticated models of diffuse scatter are widely used in the electron microscopy community, it is argued in this paper that this simple model is appropriate for current structure solution and refinement methods. The two-beam model is a straightforward numerical calculation, but even this simplistic approach can become time consuming for simulations of materials with large numbers of atoms in the unit cell and/or many incident beam orientations. Here, a parameterized form of f′ is provided for 103 elements as neutral, spherical atoms that reduces calculation time considerably.Parameterized absorptive electron scattering factorsInternational Union of CrystallographyThomas, M.Cleverley, A.Beanland, R.en2024-01-252053-27331462024-01-25med@iucr.org2Acta Crystallographica Section A: Foundations and Advanceshttps://creativecommons.org/licenses/by/4.0/research papers80March 20242053-2733150Modelling dynamical 3D electron diffraction intensities. II. The role of inelastic scattering
http://scripts.iucr.org/cgi-bin/paper?tw5007
The strong interaction of high-energy electrons with a crystal results in both dynamical elastic scattering and inelastic events, particularly phonon and plasmon excitation, which have relatively large cross sections. For accurate crystal structure refinement it is therefore important to uncover the impact of inelastic scattering on the Bragg beam intensities. Here a combined Bloch wave–Monte Carlo method is used to simulate phonon and plasmon scattering in crystals. The simulated thermal and plasmon diffuse scattering are consistent with experimental results. The simulations also confirm the empirical observation of a weaker unscattered beam intensity with increasing energy loss in the low-loss regime, while the Bragg-diffracted beam intensities do not change significantly. The beam intensities include the diffuse scattered background and have been normalized to adjust for the inelastic scattering cross section. It is speculated that the random azimuthal scattering angle during inelastic events transfers part of the unscattered beam intensity to the inner Bragg reflections. Inelastic scattering should not significantly influence crystal structure refinement, provided there are no artefacts from any background subtraction, since the relative intensity of the diffracted beams (which includes the diffuse scattering) remains approximately constant in the low energy loss regime.https://creativecommons.org/licenses/by/4.0/INELASTIC DIFFUSE SCATTERING; PHONONS; PLASMONS; BLOCH WAVESAn experimental and computational investigation is presented of the role of inelastic scattering on electron diffraction intensities.doi:10.1107/S2053273323010690urn:issn:2053-2733texttext/htmlThe strong interaction of high-energy electrons with a crystal results in both dynamical elastic scattering and inelastic events, particularly phonon and plasmon excitation, which have relatively large cross sections. For accurate crystal structure refinement it is therefore important to uncover the impact of inelastic scattering on the Bragg beam intensities. Here a combined Bloch wave–Monte Carlo method is used to simulate phonon and plasmon scattering in crystals. The simulated thermal and plasmon diffuse scattering are consistent with experimental results. The simulations also confirm the empirical observation of a weaker unscattered beam intensity with increasing energy loss in the low-loss regime, while the Bragg-diffracted beam intensities do not change significantly. The beam intensities include the diffuse scattered background and have been normalized to adjust for the inelastic scattering cross section. It is speculated that the random azimuthal scattering angle during inelastic events transfers part of the unscattered beam intensity to the inner Bragg reflections. Inelastic scattering should not significantly influence crystal structure refinement, provided there are no artefacts from any background subtraction, since the relative intensity of the diffracted beams (which includes the diffuse scattering) remains approximately constant in the low energy loss regime.Modelling dynamical 3D electron diffraction intensities. II. The role of inelastic scatteringInternational Union of CrystallographyMendis, B.en2024-01-25Acta Crystallographica Section A: Foundations and Advancesresearch papershttps://creativecommons.org/licenses/by/4.0/80March 20242053-27331882053-2733med@iucr.org2024-01-251782Modelling dynamical 3D electron diffraction intensities. I. A scattering cluster algorithm
http://scripts.iucr.org/cgi-bin/paper?tw5006
Three-dimensional electron diffraction (3D-ED) is a powerful technique for crystallographic characterization of nanometre-sized crystals that are too small for X-ray diffraction. For accurate crystal structure refinement, however, it is important that the Bragg diffracted intensities are treated dynamically. Bloch wave simulations are often used in 3D-ED, but can be computationally expensive for large unit cell crystals due to the large number of diffracted beams. Proposed here is an alternative method, the `scattering cluster algorithm' (SCA), that replaces the eigen-decomposition operation in Bloch waves with a simpler matrix multiplication. The underlying principle of SCA is that the intensity of a given Bragg reflection is largely determined by intensity transfer (i.e. `scattering') from a cluster of neighbouring diffracted beams. However, the penalty for using matrix multiplication is that the sample must be divided into a series of thin slices and the diffracted beams calculated iteratively, similar to the multislice approach. Therefore, SCA is more suitable for thin specimens. The accuracy and speed of SCA are demonstrated on tri-isopropyl silane (TIPS) pentacene and rubrene, two exemplar organic materials with large unit cells.https://creativecommons.org/licenses/by/4.0/DYNAMICAL ELECTRON DIFFRACTION; BLOCH WAVES; STRUCTURE MATRIX; MULTISLICEA new method is developed for calculating dynamical electron diffraction intensities.doi:10.1107/S2053273323010689urn:issn:2053-2733texttext/htmlThree-dimensional electron diffraction (3D-ED) is a powerful technique for crystallographic characterization of nanometre-sized crystals that are too small for X-ray diffraction. For accurate crystal structure refinement, however, it is important that the Bragg diffracted intensities are treated dynamically. Bloch wave simulations are often used in 3D-ED, but can be computationally expensive for large unit cell crystals due to the large number of diffracted beams. Proposed here is an alternative method, the `scattering cluster algorithm' (SCA), that replaces the eigen-decomposition operation in Bloch waves with a simpler matrix multiplication. The underlying principle of SCA is that the intensity of a given Bragg reflection is largely determined by intensity transfer (i.e. `scattering') from a cluster of neighbouring diffracted beams. However, the penalty for using matrix multiplication is that the sample must be divided into a series of thin slices and the diffracted beams calculated iteratively, similar to the multislice approach. Therefore, SCA is more suitable for thin specimens. The accuracy and speed of SCA are demonstrated on tri-isopropyl silane (TIPS) pentacene and rubrene, two exemplar organic materials with large unit cells.Modelling dynamical 3D electron diffraction intensities. I. A scattering cluster algorithmInternational Union of CrystallographyMendis, B.en2024-01-252053-27331672024-01-25med@iucr.org280March 20242053-2733Acta Crystallographica Section A: Foundations and Advanceshttps://creativecommons.org/licenses/by/4.0/research papers177Analytical models representing X-ray form factors of ions
http://scripts.iucr.org/cgi-bin/paper?ae5139
Parameters in analytical models for X-ray form factors of ions f0(s), based on the inverse Mott–Bethe formula involving a variable number of Gaussians, are determined for a wide range of published data sets {s, f0(s)}. The models reproduce the calculated form-factor values close to what is expected from a uniform statistical distribution with limits determined by their precision. For different ions associated with the same atom, the number of Gaussians in the models decreases with increasing net positive charge.https://creativecommons.org/licenses/by/4.0/X-RAY FORM FACTORS; INVERSE MOTT-BETHE FORMULA; ANALYTICAL REPRESENTATIONS; IONSAnalytical representations of X-ray form factors for ions are examined based on the inverse Mott–Bethe formula. Twelve sources of form-factor data spanning the period 1961–2023 and representing various ranges and grids in sinθ/λ and different precisions are analysed. Generally, 96.1% of all form factors are exactly reproduced by the analytical models.doi:10.1107/S2053273323010550urn:issn:2053-2733texttext/htmlParameters in analytical models for X-ray form factors of ions f0(s), based on the inverse Mott–Bethe formula involving a variable number of Gaussians, are determined for a wide range of published data sets {s, f0(s)}. The models reproduce the calculated form-factor values close to what is expected from a uniform statistical distribution with limits determined by their precision. For different ions associated with the same atom, the number of Gaussians in the models decreases with increasing net positive charge.Analytical models representing X-ray form factors of ionsInternational Union of CrystallographyThorkildsen, G.en2024-01-01med@iucr.org1292024-01-012053-27331https://creativecommons.org/licenses/by/4.0/research papersActa Crystallographica Section A: Foundations and Advances2053-2733January 202480136Deep learning applications in protein crystallography
http://scripts.iucr.org/cgi-bin/paper?ae5136
Deep learning techniques can recognize complex patterns in noisy, multidimensional data. In recent years, researchers have started to explore the potential of deep learning in the field of structural biology, including protein crystallography. This field has some significant challenges, in particular producing high-quality and well ordered protein crystals. Additionally, collecting diffraction data with high completeness and quality, and determining and refining protein structures can be problematic. Protein crystallographic data are often high-dimensional, noisy and incomplete. Deep learning algorithms can extract relevant features from these data and learn to recognize patterns, which can improve the success rate of crystallization and the quality of crystal structures. This paper reviews progress in this field.https://creativecommons.org/licenses/by/4.0/PROTEIN CRYSTALLOGRAPHY; DEEP LEARNING; ARTIFICIAL INTELLIGENCE; MACHINE LEARNINGDeep learning applications are increasingly dominating many areas of science. This paper reviews their relevance for and impact on protein crystallography.doi:10.1107/S2053273323009300urn:issn:2053-2733texttext/htmlDeep learning techniques can recognize complex patterns in noisy, multidimensional data. In recent years, researchers have started to explore the potential of deep learning in the field of structural biology, including protein crystallography. This field has some significant challenges, in particular producing high-quality and well ordered protein crystals. Additionally, collecting diffraction data with high completeness and quality, and determining and refining protein structures can be problematic. Protein crystallographic data are often high-dimensional, noisy and incomplete. Deep learning algorithms can extract relevant features from these data and learn to recognize patterns, which can improve the success rate of crystallization and the quality of crystal structures. This paper reviews progress in this field.Deep learning applications in protein crystallographyInternational Union of CrystallographyMatinyan, S.Filipcik, P.Abrahams, J.P.en2024-01-01112024-01-01med@iucr.org2053-2733172053-2733January 202480https://creativecommons.org/licenses/by/4.0/lead articlesActa Crystallographica Section A: Foundations and AdvancesPermissible domain walls in monoclinic MAB ferroelectric phases
http://scripts.iucr.org/cgi-bin/paper?lu5030
The concept of monoclinic ferroelectric phases has been extensively used over recent decades for the understanding of crystallographic structures of ferroelectric materials. Monoclinic phases have been actively invoked to describe the phase boundaries such as the so-called morphotropic phase boundary in functional perovskite oxides. These phases are believed to play a major role in the enhancement of such functional properties as dielectricity and electromechanical coupling through rotation of spontaneous polarization and/or modification of the rich domain microstructures. Unfortunately, such microstructures remain poorly understood due to the complexity of the subject. The goal of this work is to formulate the geometrical laws behind the monoclinic domain microstructures. Specifically, the result of previous work [Gorfman et al. (2022). Acta Cryst. A78, 158–171] is implemented to catalog and outline some properties of permissible domain walls that connect `strain' domains with monoclinic (MA/MB type) symmetry, occurring in ferroelectric perovskite oxides. The term `permissible' [Fousek & Janovec (1969). J. Appl. Phys. 40, 135–142] pertains to the domain walls connecting a pair of `strain' domains without a lattice mismatch. It was found that 12 monoclinic domains may form pairs connected along 84 types of permissible domain walls. These contain 48 domain walls with fixed Miller indices (known as W-walls) and 36 domain walls whose Miller indices may change when free lattice parameters change as well (known as S-walls). Simple and intuitive analytical expressions are provided that describe the orientation of these domain walls, the matrices of transformation between crystallographic basis vectors and, most importantly, the separation between Bragg peaks, diffracted from each of the 84 pairs of domains, connected along a permissible domain wall. It is shown that the orientation of a domain wall may be described by the specific combination of the monoclinic distortion parameters r = [2/(γ − α)][(c/a) − 1], f = (π − 2γ)/(π − 2α) and p = [2/(π − α − γ)] [(c/a) − 1]. The results of this work will enhance understanding and facilitate investigation (e.g. using single-crystal X-ray diffraction) of complex monoclinic domain microstructures in both crystals and thin films.https://creativecommons.org/licenses/by/4.0/FERROELASTIC DOMAINS; MONOCLINIC SYMMETRY; X-RAY DIFFRACTIONAll the possibilities for permissible (mismatch-free) walls between monoclinic domains of pseudocubic ferroic perovskites are analyzed. Analytical expressions are derived for the orientation of such walls, the orientation relationship between the lattice vectors and the separation between Bragg peaks diffracted from matched domains.doi:10.1107/S205327332300921Xurn:issn:2053-2733texttext/htmlThe concept of monoclinic ferroelectric phases has been extensively used over recent decades for the understanding of crystallographic structures of ferroelectric materials. Monoclinic phases have been actively invoked to describe the phase boundaries such as the so-called morphotropic phase boundary in functional perovskite oxides. These phases are believed to play a major role in the enhancement of such functional properties as dielectricity and electromechanical coupling through rotation of spontaneous polarization and/or modification of the rich domain microstructures. Unfortunately, such microstructures remain poorly understood due to the complexity of the subject. The goal of this work is to formulate the geometrical laws behind the monoclinic domain microstructures. Specifically, the result of previous work [Gorfman et al. (2022). Acta Cryst. A78, 158–171] is implemented to catalog and outline some properties of permissible domain walls that connect `strain' domains with monoclinic (MA/MB type) symmetry, occurring in ferroelectric perovskite oxides. The term `permissible' [Fousek & Janovec (1969). J. Appl. Phys. 40, 135–142] pertains to the domain walls connecting a pair of `strain' domains without a lattice mismatch. It was found that 12 monoclinic domains may form pairs connected along 84 types of permissible domain walls. These contain 48 domain walls with fixed Miller indices (known as W-walls) and 36 domain walls whose Miller indices may change when free lattice parameters change as well (known as S-walls). Simple and intuitive analytical expressions are provided that describe the orientation of these domain walls, the matrices of transformation between crystallographic basis vectors and, most importantly, the separation between Bragg peaks, diffracted from each of the 84 pairs of domains, connected along a permissible domain wall. It is shown that the orientation of a domain wall may be described by the specific combination of the monoclinic distortion parameters r = [2/(γ − α)][(c/a) − 1], f = (π − 2γ)/(π − 2α) and p = [2/(π − α − γ)] [(c/a) − 1]. The results of this work will enhance understanding and facilitate investigation (e.g. using single-crystal X-ray diffraction) of complex monoclinic domain microstructures in both crystals and thin films.Permissible domain walls in monoclinic MAB ferroelectric phasesInternational Union of CrystallographyBiran, I.Gorfman, S.en2024-01-01Acta Crystallographica Section A: Foundations and Advancesresearch papershttps://creativecommons.org/licenses/by/4.0/80January 20242053-27331282053-27332024-01-01112med@iucr.org1Small-angle scattering tensor tomography algorithm for robust reconstruction of complex textures
http://scripts.iucr.org/cgi-bin/paper?ae5133
The development of small-angle scattering tensor tomography has enabled the study of anisotropic nanostructures in a volume-resolved manner. It is of great value to have reconstruction methods that can handle many different nanostructural symmetries. For such a method to be employed by researchers from a wide range of backgrounds, it is crucial that its reliance on prior knowledge about the system is minimized, and that it is robust under various conditions. Here, a method is presented that employs band-limited spherical functions to enable the reconstruction of reciprocal-space maps of a wide variety of nanostructures. This method has been thoroughly tested and compared with existing methods in its ability to retrieve known reciprocal-space maps, as well as its robustness to changes in initial conditions, using both simulations and experimental data. It has also been evaluated for its computational performance. The anchoring of this method in a framework of integral geometry and linear algebra highlights its possibilities and limitations.https://creativecommons.org/licenses/by/4.0/TENSOR TOMOGRAPHY; SAXS; SMALL-ANGLE X-RAY SCATTERING; SCATTERING; RECIPROCAL-SPACE MAPS; OPTIMIZATIONThe development of small-angle scattering tensor tomography has enabled the study of anisotropic nanostructures in a volume-resolved manner. The paper presents a method tested against both simulations and experimental data, and compared with existing methods, demonstrating its ability to handle several different classes of nanostructures.doi:10.1107/S205327332300863Xurn:issn:2053-2733texttext/htmlThe development of small-angle scattering tensor tomography has enabled the study of anisotropic nanostructures in a volume-resolved manner. It is of great value to have reconstruction methods that can handle many different nanostructural symmetries. For such a method to be employed by researchers from a wide range of backgrounds, it is crucial that its reliance on prior knowledge about the system is minimized, and that it is robust under various conditions. Here, a method is presented that employs band-limited spherical functions to enable the reconstruction of reciprocal-space maps of a wide variety of nanostructures. This method has been thoroughly tested and compared with existing methods in its ability to retrieve known reciprocal-space maps, as well as its robustness to changes in initial conditions, using both simulations and experimental data. It has also been evaluated for its computational performance. The anchoring of this method in a framework of integral geometry and linear algebra highlights its possibilities and limitations.Small-angle scattering tensor tomography algorithm for robust reconstruction of complex texturesInternational Union of CrystallographyNielsen, L.C.Erhart, P.Guizar-Sicairos, M.Liebi, M.en2023-10-1962053-27332023-10-19med@iucr.org515526Acta Crystallographica Section A: Foundations and Advanceshttps://creativecommons.org/licenses/by/4.0/research papersNovember 2023792053-2733Molecular replacement for small-molecule crystal structure determination from X-ray and electron diffraction data with reduced resolution
http://scripts.iucr.org/cgi-bin/paper?lu5029
The resolution of 3D electron diffraction (ED) data of small-molecule crystals is often relatively poor, due to either electron-beam radiation damage during data collection or poor crystallinity of the material. Direct methods, used as standard for crystal structure determination, are not applicable when the data resolution falls below the commonly accepted limit of 1.2 Å. Therefore an evaluation was carried out of the performance of molecular replacement (MR) procedures, regularly used for protein structure determination, for structure analysis of small-molecule crystal structures from 3D ED data. In the course of this study, two crystal structures of Bi-3812, a highly potent inhibitor of the oncogenic transcription factor BCL6, were determined: the structure of α-Bi-3812 was determined from single-crystal X-ray data, the structure of β-Bi-3812 from 3D ED data, using direct methods in both cases. These data were subsequently used for MR with different data types, varying the data resolution limit (1, 1.5 and 2 Å) and by using search models consisting of connected or disconnected fragments of BI-3812. MR was successful with 3D ED data at 2 Å resolution using a search model that represented 74% of the complete molecule.https://creativecommons.org/licenses/by/4.0/ELECTRON CRYSTALLOGRAPHY; SMALL MOLECULES; MOLECULAR REPLACEMENTMolecular replacement as implemented in Phaser was applied to structure analysis of small-molecule single-crystal X-ray and electron diffraction data.doi:10.1107/S2053273323008458urn:issn:2053-2733texttext/htmlThe resolution of 3D electron diffraction (ED) data of small-molecule crystals is often relatively poor, due to either electron-beam radiation damage during data collection or poor crystallinity of the material. Direct methods, used as standard for crystal structure determination, are not applicable when the data resolution falls below the commonly accepted limit of 1.2 Å. Therefore an evaluation was carried out of the performance of molecular replacement (MR) procedures, regularly used for protein structure determination, for structure analysis of small-molecule crystal structures from 3D ED data. In the course of this study, two crystal structures of Bi-3812, a highly potent inhibitor of the oncogenic transcription factor BCL6, were determined: the structure of α-Bi-3812 was determined from single-crystal X-ray data, the structure of β-Bi-3812 from 3D ED data, using direct methods in both cases. These data were subsequently used for MR with different data types, varying the data resolution limit (1, 1.5 and 2 Å) and by using search models consisting of connected or disconnected fragments of BI-3812. MR was successful with 3D ED data at 2 Å resolution using a search model that represented 74% of the complete molecule.Molecular replacement for small-molecule crystal structure determination from X-ray and electron diffraction data with reduced resolutionInternational Union of CrystallographyGorelik, T.E.Lukat, P.Kleeberg, C.Blankenfeldt, W.Mueller, R.en2023-10-1962053-2733med@iucr.org5042023-10-1951479November 20232053-2733Acta Crystallographica Section A: Foundations and Advanceshttps://creativecommons.org/licenses/by/4.0/research papersSite-occupancy factors in the Debye scattering equation. A theoretical discussion on significance and correctness
http://scripts.iucr.org/cgi-bin/paper?lu5032
The Debye scattering equation (DSE) [Debye (1915). Ann. Phys. 351, 809–823] is widely used for analyzing total scattering data of nanocrystalline materials in reciprocal space. In its modified form (MDSE) [Cervellino et al. (2010). J. Appl. Cryst. 43, 1543–1547], it includes contributions from uncorrelated thermal agitation terms and, for defective crystalline nanoparticles (NPs), average site-occupancy factors (s.o.f.'s). The s.o.f.'s were introduced heuristically and no theoretical demonstration was provided. This paper presents in detail such a demonstration, corrects a glitch present in the original MDSE, and discusses the s.o.f.'s physical significance. Three new MDSE expressions are given that refer to distinct defective NP ensembles characterized by: (i) vacant sites with uncorrelated constant site-occupancy probability; (ii) vacant sites with a fixed number of randomly distributed atoms; (iii) self-excluding (disordered) positional sites. For all these cases, beneficial aspects and shortcomings of introducing s.o.f.'s as free refinable parameters are demonstrated. The theoretical analysis is supported by numerical simulations performed by comparing the corrected MDSE profiles and the ones based on atomistic modeling of a large number of NPs, satisfying the structural conditions described in (i)–(iii).https://creativecommons.org/licenses/by/4.0/DEBYE SCATTERING EQUATION; SITE-OCCUPANCY FACTORS; DEFECTIVE NANOCRYSTALSThe modified Debye scattering equation, often used for characterizing defective nanoparticle ensembles, has been theoretically analyzed with a special focus on the significance and correctness of the site-occupancy factors, oi's.doi:10.1107/S2053273323008446urn:issn:2053-2733texttext/htmlThe Debye scattering equation (DSE) [Debye (1915). Ann. Phys. 351, 809–823] is widely used for analyzing total scattering data of nanocrystalline materials in reciprocal space. In its modified form (MDSE) [Cervellino et al. (2010). J. Appl. Cryst. 43, 1543–1547], it includes contributions from uncorrelated thermal agitation terms and, for defective crystalline nanoparticles (NPs), average site-occupancy factors (s.o.f.'s). The s.o.f.'s were introduced heuristically and no theoretical demonstration was provided. This paper presents in detail such a demonstration, corrects a glitch present in the original MDSE, and discusses the s.o.f.'s physical significance. Three new MDSE expressions are given that refer to distinct defective NP ensembles characterized by: (i) vacant sites with uncorrelated constant site-occupancy probability; (ii) vacant sites with a fixed number of randomly distributed atoms; (iii) self-excluding (disordered) positional sites. For all these cases, beneficial aspects and shortcomings of introducing s.o.f.'s as free refinable parameters are demonstrated. The theoretical analysis is supported by numerical simulations performed by comparing the corrected MDSE profiles and the ones based on atomistic modeling of a large number of NPs, satisfying the structural conditions described in (i)–(iii).Site-occupancy factors in the Debye scattering equation. A theoretical discussion on significance and correctnessInternational Union of CrystallographyFerri, F.Bossuto, M.C.Anzini, P.Cervellino, A.Guagliardi, A.Bertolotti, F.Masciocchi, N.en2023-11-022053-273379November 2023https://creativecommons.org/licenses/by/4.0/research papersActa Crystallographica Section A: Foundations and Advances5962023-11-02587med@iucr.org2053-27336Chiral spiral cyclic twins. II. A two-parameter family of cyclic twins composed of discrete circle involute spirals
http://scripts.iucr.org/cgi-bin/paper?ae5132
A mathematical toy model of chiral spiral cyclic twins is presented, describing a family of deterministically generated aperiodic point sets. Its individual members depend solely on a chosen pair of integer parameters, a modulus m and a multiplier μ. By means of their specific parameterization they comprise local features of both periodic and aperiodic crystals. In particular, chiral spiral cyclic twins are composed of discrete variants of continuous curves known as circle involutes, each discrete spiral being generated from an integer inclination sequence. The geometry of circle involutes does not only provide for a constant orthogonal separation distance between adjacent spiral branches but also yields an approximate delineation of the intrinsically periodic twin domains as well as a single aperiodic core domain interconnecting them. Apart from its mathematical description and analysis, e.g. concerning its circle packing densities, the toy model is studied in association with the crystallography and crystal chemistry of α-uranium and CrB-type crystal structures.https://creativecommons.org/licenses/by/4.0/CHIRAL PROPERTIES; SPIRAL STRUCTURES; CYCLIC TWINSA mathematical toy model of chiral spiral cyclic twins is presented, describing a family of deterministically generated aperiodic point sets. Its individual members depend solely on a chosen pair of integer parameters, a modulus m and a multiplier μ, comprising local features of both periodic and aperiodic crystals. Chiral spiral cyclic twins are composed of discrete variants of continuous curves known as circle involutes.doi:10.1107/S2053273323008276urn:issn:2053-2733texttext/htmlA mathematical toy model of chiral spiral cyclic twins is presented, describing a family of deterministically generated aperiodic point sets. Its individual members depend solely on a chosen pair of integer parameters, a modulus m and a multiplier μ. By means of their specific parameterization they comprise local features of both periodic and aperiodic crystals. In particular, chiral spiral cyclic twins are composed of discrete variants of continuous curves known as circle involutes, each discrete spiral being generated from an integer inclination sequence. The geometry of circle involutes does not only provide for a constant orthogonal separation distance between adjacent spiral branches but also yields an approximate delineation of the intrinsically periodic twin domains as well as a single aperiodic core domain interconnecting them. Apart from its mathematical description and analysis, e.g. concerning its circle packing densities, the toy model is studied in association with the crystallography and crystal chemistry of α-uranium and CrB-type crystal structures.Chiral spiral cyclic twins. II. A two-parameter family of cyclic twins composed of discrete circle involute spiralsInternational Union of CrystallographyHornfeck, W.en2023-10-3179November 20232053-2733Acta Crystallographica Section A: Foundations and Advanceshttps://creativecommons.org/licenses/by/4.0/research papers5862053-2733med@iucr.org5702023-10-316An efficient system matrix factorization method for scanning diffraction based strain tensor tomography
http://scripts.iucr.org/cgi-bin/paper?vk5050
Diffraction-based tomographic strain tensor reconstruction problems in which a strain tensor field is determined from measurements made in different crystallographic directions are considered in the context of sparse matrix algebra. Previous work has shown that the estimation of the crystal elastic strain field can be cast as a linear regression problem featuring a computationally involved assembly of a system matrix forward operator. This operator models the perturbation in diffraction signal as a function of spatial strain tensor state. The structure of this system matrix is analysed and a block-partitioned factorization is derived that reveals the forward operator as a sum of weighted scalar projection operators. Moreover, the factorization method is generalized for another diffraction model in which strain and orientation are coupled and can be reconstructed jointly. The proposed block-partitioned factorization method provides a bridge to classical absorption tomography and allows exploitation of standard tomographic ray-tracing libraries for implementation of the forward operator and its adjoint. Consequently, RAM-efficient, GPU-accelerated, on-the-fly strain/orientation tensor reconstruction is made possible, paving the way for higher spatial resolution studies of intragranular deformation.https://creativecommons.org/licenses/by/4.0/X-RAY DIFFRACTION; STRAIN TENSOR; TOMOGRAPHY; DIFFRACTION IMAGINGMatrix analysis is used to provide a computationally efficient mathematical framework for diffraction-based strain tomography.doi:10.1107/S2053273323008136urn:issn:2053-2733texttext/htmlDiffraction-based tomographic strain tensor reconstruction problems in which a strain tensor field is determined from measurements made in different crystallographic directions are considered in the context of sparse matrix algebra. Previous work has shown that the estimation of the crystal elastic strain field can be cast as a linear regression problem featuring a computationally involved assembly of a system matrix forward operator. This operator models the perturbation in diffraction signal as a function of spatial strain tensor state. The structure of this system matrix is analysed and a block-partitioned factorization is derived that reveals the forward operator as a sum of weighted scalar projection operators. Moreover, the factorization method is generalized for another diffraction model in which strain and orientation are coupled and can be reconstructed jointly. The proposed block-partitioned factorization method provides a bridge to classical absorption tomography and allows exploitation of standard tomographic ray-tracing libraries for implementation of the forward operator and its adjoint. Consequently, RAM-efficient, GPU-accelerated, on-the-fly strain/orientation tensor reconstruction is made possible, paving the way for higher spatial resolution studies of intragranular deformation.An efficient system matrix factorization method for scanning diffraction based strain tensor tomographyInternational Union of CrystallographyHenningsson, A.Hall, S.A.en2023-09-29549Acta Crystallographica Section A: Foundations and Advanceshttps://creativecommons.org/licenses/by/4.0/research papersNovember 2023792053-273362053-27332023-09-29542med@iucr.orgTERSE/PROLIX (TRPX) – a new algorithm for fast and lossless compression and decompression of diffraction and cryo-EM data
http://scripts.iucr.org/cgi-bin/paper?lu5031
High-throughput data collection in crystallography poses significant challenges in handling massive amounts of data. Here, TERSE/PROLIX (or TRPX for short) is presented, a novel lossless compression algorithm specifically designed for diffraction data. The algorithm is compared with established lossless compression algorithms implemented in gzip, bzip2, CBF (crystallographic binary file), Zstandard(zstd), LZ4 and HDF5 with gzip, LZF and bitshuffle+LZ4 filters, in terms of compression efficiency and speed, using continuous-rotation electron diffraction data of an inorganic compound and raw cryo-EM data. The results show that TRPX significantly outperforms all these algorithms in terms of speed and compression rate. It was 60 times faster than bzip2 (which achieved a similar compression rate), and more than 3 times faster than LZ4, which was the runner-up in terms of speed, but had a much worse compression rate. TRPX files are byte-order independent and upon compilation the algorithm occupies very little memory. It can therefore be readily implemented in hardware. By providing a tailored solution for diffraction and raw cryo-EM data, TRPX facilitates more efficient data analysis and interpretation while mitigating storage and transmission concerns. The C++20 compression/decompression code, custom TIFF library and an ImageJ/Fiji Java plugin for reading TRPX files are open-sourced on GitHub under the permissive MIT license.https://creativecommons.org/licenses/by/4.0/COMPRESSION; TERSE/PROLIX; TRPX; LOSSLESS; DIFFRACTION DATA; CRYO-EM DATA; LOSSLESS DATA COMPRESSIONThis article presents a fast and lossless algorithm for compressing diffraction data, achieving up to 85% reduction in file size while processing up to 2000 512 × 512 frames s−1. This breakthrough in compression technology is a significant step towards more efficient analysis and storage of large diffraction data sets.doi:10.1107/S205327332300760Xurn:issn:2053-2733texttext/htmlHigh-throughput data collection in crystallography poses significant challenges in handling massive amounts of data. Here, TERSE/PROLIX (or TRPX for short) is presented, a novel lossless compression algorithm specifically designed for diffraction data. The algorithm is compared with established lossless compression algorithms implemented in gzip, bzip2, CBF (crystallographic binary file), Zstandard(zstd), LZ4 and HDF5 with gzip, LZF and bitshuffle+LZ4 filters, in terms of compression efficiency and speed, using continuous-rotation electron diffraction data of an inorganic compound and raw cryo-EM data. The results show that TRPX significantly outperforms all these algorithms in terms of speed and compression rate. It was 60 times faster than bzip2 (which achieved a similar compression rate), and more than 3 times faster than LZ4, which was the runner-up in terms of speed, but had a much worse compression rate. TRPX files are byte-order independent and upon compilation the algorithm occupies very little memory. It can therefore be readily implemented in hardware. By providing a tailored solution for diffraction and raw cryo-EM data, TRPX facilitates more efficient data analysis and interpretation while mitigating storage and transmission concerns. The C++20 compression/decompression code, custom TIFF library and an ImageJ/Fiji Java plugin for reading TRPX files are open-sourced on GitHub under the permissive MIT license.TERSE/PROLIX (TRPX) – a new algorithm for fast and lossless compression and decompression of diffraction and cryo-EM dataInternational Union of CrystallographyMatinyan, S.Abrahams, J.P.en2023-09-25541research papershttps://creativecommons.org/licenses/by/4.0/Acta Crystallographica Section A: Foundations and Advances2053-2733November 2023796536med@iucr.org2023-09-252053-2733Optimal 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.https://creativecommons.org/licenses/by/4.0/ERROR MODELLING; ERROR ANALYSIS; DATA REDUCTION; ELECTRON DIFFRACTIONSeveral 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.doi:10.1107/S2053273323005053urn:issn:2053-2733texttext/htmlEstimating 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.Optimal estimated standard uncertainties of reflection intensities for kinematical refinement from 3D electron diffraction dataInternational Union of CrystallographyKhouchen, M.Klar, P.B.Chintakindi, H.Suresh, A.Palatinus, L.en2023-08-142023-08-14med@iucr.org4272053-273352053-273379September 2023research papershttps://creativecommons.org/licenses/by/4.0/Acta Crystallographica Section A: Foundations and Advances439Patch 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.https://creativecommons.org/licenses/by/4.0/PATCH FREQUENCY; TILING; DUALIZATION METHODAn algorithm is presented for an exact calculation of patch frequencies for a family of tilings which can be obtained via dualization.doi:10.1107/S2053273323004990urn:issn:2053-2733texttext/htmlThis 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.Patch frequencies in rhombic Penrose tilingsInternational Union of CrystallographyMazáč, J.en2023-07-245med@iucr.org2023-07-243992053-2733411https://creativecommons.org/licenses/by/4.0/research papersActa Crystallographica Section A: Foundations and Advances2053-2733September 202379Algorithms 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.https://creativecommons.org/licenses/by/4.0/MAGNETIC SPACE GROUP; MAGNETIC SPACE-GROUP TYPE; MAGNETIC STRUCTURE; CRYSTAL STRUCTURE ANALYSIS; AFFINE NORMALIZERThis 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.doi:10.1107/S2053273323005016urn:issn:2053-2733texttext/htmlA 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.Algorithms for magnetic symmetry operation search and identification of magnetic space group from magnetic crystal structureInternational Union of CrystallographyShinohara, K.Togo, A.Tanaka, I.en2023-09-062053-27333902023-09-06med@iucr.org5September 2023792053-2733Acta Crystallographica Section A: Foundations and Advanceshttps://creativecommons.org/licenses/by/4.0/research papers398Machine 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.https://creativecommons.org/licenses/by/4.0/DIFFRACTION; SINGLE-MOLECULE ELECTRON DIFFRACTION; TEM; TRANSMISSION ELECTRON MICROSCOPY; MACHINE LEARNING; NEURAL NETWORKSNeural networks were trained for robust classification of narrow electron beam diffraction patterns and may significantly decrease the need for storage space.doi:10.1107/S2053273323004680urn:issn:2053-2733texttext/htmlAs 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.Machine learning for classifying narrow-beam electron diffraction dataInternational Union of CrystallographyMatinyan, S.Demir, B.Filipcik, P.Abrahams, J.P.van Genderen, E.en2023-06-203602023-06-20med@iucr.org2053-27334research papershttps://creativecommons.org/licenses/by/4.0/Acta Crystallographica Section A: Foundations and Advances2053-2733July 202379368New 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.https://creativecommons.org/licenses/by/4.0/ATOMIC FORM FACTORS; ANALYTICAL REPRESENTATIONS; INVERSE MOTT-BETHE FORMULAImproved 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.doi:10.1107/S2053273323003996urn:issn:2053-2733texttext/htmlAnalytical 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.New benchmarks in the modelling of X-ray atomic form factorsInternational Union of CrystallographyThorkildsen, G.en2023-06-0242053-27332023-06-02med@iucr.org318330July 2023792053-2733Acta Crystallographica Section A: Foundations and Advanceshttps://creativecommons.org/licenses/by/4.0/research papersCrystallography 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.https://creativecommons.org/licenses/by/4.0/BICRYSTALLOGRAPHY WITH COMPLEX NUMBERS; BILAYERS; COINCIDENCE LATTICES; SPACE GROUPSA general scheme is proposed to classify and determine the crystallographic properties of twisted bilayers of any homophase 2D structures using complex numbers.doi:10.1107/S2053273323003662urn:issn:2053-2733texttext/htmlThis 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.Crystallography of homophase twisted bilayers: coincidence, union lattices and space groupsInternational Union of CrystallographyGratias, D.Quiquandon, M.en2023-06-02317July 2023792053-2733Acta Crystallographica Section A: Foundations and Advancesresearch papershttps://creativecommons.org/licenses/by/4.0/42053-27332023-06-02med@iucr.org301Efficient 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.https://creativecommons.org/licenses/by/4.0/STRUCTURE FACTORS; MULTIPLE COMPONENTS; SCATTERING FUNCTIONS; BULK SOLVENT; REFINEMENT; DENSITY MAPSA 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.doi:10.1107/S205327332300356Xurn:issn:2053-2733texttext/htmlDiffraction 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.Efficient structure-factor modeling for crystals with multiple componentsInternational Union of CrystallographyAfonine, P.V.Adams, P.D.Urzhumtsev, A.G.en2023-06-20352research papershttps://creativecommons.org/licenses/by/4.0/Acta Crystallographica Section A: Foundations and Advances2053-2733July 20237943452023-06-20med@iucr.org2053-2733A 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.https://creativecommons.org/licenses/by/4.0/MULTIVECTORS; WEDGE REVERSION; ANTISYMMETRY; CLIFFORD ALGEBRAIt 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.doi:10.1107/S2053273323003303urn:issn:2053-2733texttext/htmlThe 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.A note on the wedge reversion antisymmetry operation and 51 types of physical quantities in arbitrary dimensionsInternational Union of CrystallographyFabrykiewicz, P.en2023-06-05Acta Crystallographica Section A: Foundations and Advanceshttps://creativecommons.org/licenses/by/4.0/short communicationsJuly 2023792053-27333842053-2733381med@iucr.org2023-06-054Approximating 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.https://creativecommons.org/licenses/by/4.0/LATTICE MATCHING; DELAUNAY; DELONE; NIGGLI; SELLINGA 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.doi:10.1107/S2053273323003200urn:issn:2053-2733texttext/htmlA 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.Approximating lattice similarityInternational Union of CrystallographyAndrews, L.C.Bernstein, H.J.Sauter, N.K.en2023-07-2452023-07-24480med@iucr.org2053-27334842053-2733September 202379research papershttps://creativecommons.org/licenses/by/4.0/Acta Crystallographica Section A: Foundations and AdvancesFourier-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.https://creativecommons.org/licenses/by/4.0/ELECTRON DENSITY; FOURIER TRANSFORMATION; FOURIER SYNTHESIS; HEXABORIDES; LAPLACIANA 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.doi:10.1107/S2053273323002644urn:issn:2053-2733texttext/htmlIn 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.Fourier-synthesis approach for static charge-density reconstruction from theoretical structure factors of CaB6International Union of CrystallographyBergner, C.Grin, Y.Wagner, F.R.en2023-05-0532053-27332023-05-05246med@iucr.org272Acta Crystallographica Section A: Foundations and Advancesresearch papershttps://creativecommons.org/licenses/by/4.0/May 2023792053-2733On 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.https://creativecommons.org/licenses/by/4.0/WYCKOFF SEQUENCES; COMBINATORICS; SHANNON ENTROPY; STRUCTURAL COMPLEXITYThe 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.doi:10.1107/S2053273323002437urn:issn:2053-2733texttext/htmlWyckoff 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.On the combinatorics of crystal structures. II. Number of Wyckoff sequences of a given subdivision complexityInternational Union of CrystallographyHornfeck, W.Červený, K.en2023-05-11294Acta Crystallographica Section A: Foundations and Advancesresearch papershttps://creativecommons.org/licenses/by/4.0/May 2023792053-273332053-27332023-05-11med@iucr.org280