The rotating substrate method of crystallite deposition is modeled, allowing computation of effective material coefficients of the layers resulting from the averaging. A worked numerical example particularized to 6mm ZnO is provided.

Low-concentration poly(ethylene oxide) exhibit the polymorph-dependent effects on both the surface and bulk crystal growth of carbamazepine polymorphs. These polymorph-dependent effects of PEO were mainly attributed to the polymer enrichment at the interface and different crystal surface-polymer interactions.

This study presents the first theoretical predictions of the phase transitions, elastic properties, and thermal behavior of AuMTe2 (M = Ga, In) chalcopyrite compounds. Using density functional theory and the quasi-harmonic Debye model, key mechanical and thermodynamic properties are analyzed, offering insights valuable for future experimental validation.

Single-crystal growth, differential thermal analysis (DTA), derivative thermogravimetry (DTG), differential scanning calorimetry (DSC), X-ray structural studies and polarized microscopy observations of bis(ethylenediammonium) tetrabromozincate(II) bromide [NH3(CH2)2NH3]2[ZnBr4]Br2 are presented. A reversible phase transition is described. At room temperature, the complex crystallizes in the monoclinic system. In some cases, the single crystals are twinned into two or more large domains of ferroelastic type with domain walls in the (100) crystallographic plane. DTA and DTG measurements show chemical stability of the crystal up to ∼538 K. In the DSC studies, a reversible isostructural phase transition was revealed at ∼526/522 K on heating/cooling run, respectively. Optical observation on the heating run reveals that at the phase transition the plane of twinning (domain wall) does not disappear and additionally the appearance of a new domain structure of ferroelastic type with domain walls in the planes (101), (101), (100) and (001) is observed. The domain structure pattern is preserved after cooling to the room-temperature phase and the symmetry of this phase is unchanged.

Sublimation methods utilizing the surface properties of substrates can address the challenge of controlling hollow morphologies in rod crystals. Spherulites were formed on the hydrophilic surface of the (0001) planes of α-quartz and sapphire substrates by sublimation of 1,2-bis(3,5-dimethyl-2-thienyl)perfluorocyclopentene (1a). Various types of hollow morphologies, distinguished by the size and shape of their cross sections and by the presence or absence of branching structures, were formed separately on α-quartz and sapphire substrates. Such precise control of the hollow morphologies was attributed to the wettability of each substrate, leading to the formation of spherulites of 1a. In addition, it was indicated that the formation process of the surface morphologies of spherulites was associated with the hollow morphologies of rod crystals of 1a.

YxOs4B4 (x = 1.161) crystallizes as a tetragonal incommensurate composite of columns of Y atoms extending along [001] in an Os4B4 framework. The structure was refined using the superspace approach. The basic structure of the Y subsystem can be idealized as having I4/mmm symmetry, with a crystallographically unique Y atom located on the 4/mmm position. The actual superspace symmetry is P42/nmc(00σ3)s0s0. The Y atoms feature only subtle positional modulation in the [001] direction. The Os4B4 subsystem [P42/ncm(00σ3)00ss superspace symmetry] is built of columns of edge-sharing Os4 tetrahedra extending along [001] and B2 dumbbells. The Os4 tetrahedra feature pronounced positional modulation with a distinct variation of the Os—Os bond lengths. Modulation of the B2 dumbbells is best described as a rotation about the [001] axis.

Nickel orthovanadate is a promising material with potential applications in energy storage and photocatalytic devices. The crystal structure of Ni3V2O8 at 299 (3) K and 1323 (8) K was studied using X-ray powder diffraction. The sample was a single-phase orthorhombic kagome-staircase-Ni3(VO4)2-type structure (space group Cmca) at both temperatures. The phase purity and morphology was studied using energy-dispersive X-ray spectroscopy and scanning electron microscopy. The refined unit-cell parameters at 299 (3) K are a = 5.93384 (4) Å, b = 11.38318 (7) Å and c = 8.23818 (5) Å, and at 1323 (8) K are a = 6.02077 (7) Å, b = 11.48838 (7) Å and c = 8.32611 (9) Å. The obtained results indicate thermal expansion anisotropy, with a largest expansivity along a. Variations in Ni—O and V—O bonds with temperature are observed. The variation in the Ni—O bond is about one order higher in magnitude than that of the V—O bond, signifying the high rigidity of V—O bonds. The unit-cell size variations with rising effective ionic volume of the divalent A ion in the A3B2O8 family [A = Ni, Mg, Zn, Co, Mn (experimental data) and also A = Cu, Cd (theoretical data), B = V or As] are analyzed. Based on experimental and theoretical data, trends within the family are observed and the unit-cell size for reported solid solution of nickel (87%) and copper (13%) mixture in (Ni1–xCux)3V2O8 are predicted. Predictions are also provided for some hypothetical A3B2O8 ternary compound and solid solutions.

The odd hydration number has so far been missing in the water-rich magnesium chloride hydrate series (MgCl2·nH2O). In this study, magnesium chloride heptahydrate, MgCl2·7H2O (or MgCl2·7D2O), which forms at high pressures above 2 GPa and high temperatures above 300 K, has been identified. Its structure has been determined by a combination of in-situ single-crystal X-ray diffraction at 2.5 GPa and 298 K and powder neutron diffraction at 3.1 GPa and 300 K. The single-crystal specimen was grown by mixing alcohols to prevent nucleation of undesired crystalline phases. The results show orientational disorder of water molecules, which was also examined using density functional theory calculations. The disorder involves the reconnection of hydrogen bonds, which differs from those in water ice phases and known disordered salt hydrates. Shrinkage by compression occurs mainly in one direction. In the plane perpendicular to this most compressible direction, oxygen and chlorine atoms are in a hexagonal-like arrangement.

A detailed structural analysis of the Zn-rich δ1-MnZn9.7 phase using single-crystal X-ray diffraction is presented. The δ1 phase has been synthesized by the high-temperature synthetic route. The structure crystallizes in space group P63/mmc (Pearson symbol hP556) with unit-cell parameters: a = b = 12.9051 (2) Å and c = 57.640 (1) Å. The 556 atoms are distributed over 52 Wyckoff positions in the hexagonal unit cell: seven ordered Mn sites, 37 ordered Zn sites and eight positionally disordered Zn sites. The structure predominantly consists of Frank–Kasper polyhedra (endohedral icosahedra Zn12 and icosioctahedron Zn16) and four distinct types of glue Zn atoms. The structure comprises a 127-atom supercluster (Mn13Zn114), a 38-atom extended Pearce cluster (Mn3Zn35), a 46-atom L-tetrahedron (Mn4Zn42), a Friauf polyhedron (Zn17), a disordered icosahedral cluster (MnZn12) and four glue Zn atoms. Positionally disordered Zn sites around an Mn site can be visualized as the superimposition of three differently oriented Zn12 icosahedra.

With ever-improving quantum-mechanical computational methods, the accuracy requirements for experimental crystal structures increase. The crystal structure of calcium atorvastatin trihydrate, which has 56 degrees of freedom when determined with a real-space algorithm, was determined from powder diffraction data by Hodge et al. [Powder Diffr. (2020), 35, 136–143]. The crystal structure was a good fit to the experimental data, indicating that the electron density had been captured essentially correctly, but two independent quantum-mechanical calculations disagreed with the experimental structure and with each other. Using the same experimental data, the crystal structure was redetermined from scratch and it was shown that it can be reproduced within a root-mean-square Cartesian displacement of 0.1 Å by two independent quantum-mechanical calculations. The consequences for the calculated energies and solubilities are described.

The crystal structure of the incommensurate modulated phase of potassium guaninate monohydrate has been solved on the basis of high-pressure single-crystal X-ray diffraction data. The modulated structure was described as a `mosaic' sequence of three different local configurations of two neighbouring guaninate rings. In contrast to known examples of incommensurate modulated organic compounds, the modulation functions of all atoms are discontinuous. This is the first example of the experimental detection of an incommensurate modulated crystal structure that can be modelled using the special `soliton mode' modulation function proposed by Aramburu et al. [(1995), J. Phys. Condens. Matter, 7, 6187–6196].

Through a combination of magnetic susceptibility, specific heat, and neutron powder diffraction measurements we have revealed a sequence of four magnetic phase transitions in the columnar quadruple perovskite Er2CuMnMn4O12. A key feature of the quadruple perovskite structural framework is the complex interplay of multiple magnetic sublattices via frustrated exchange topologies and competing magnetic anisotropies. It is shown that in Er2CuMnMn4O12, this phenomenology gives rise to multiple spin-reorientation transitions driven by the competition of easy-axis single ion anisotropy and the Dzyaloshinskii–Moriya interaction; both within the manganese B-site sublattice. At low temperature, one Er sublattice orders due to a finite f-d exchange field aligned parallel to its Ising axis, while the other Er sublattice remains non-magnetic until a final, symmetry-breaking phase transition into the ground state. This non-trivial low-temperature interplay of transition metal and rare-earth sublattices, as well as an observed k = (0, 0, ½) periodicity in both manganese spin canting and Er ordering, raises future challenges to develop a complete understanding of the R2CuMnMn4O12 family.

Accurate modeling of conformational energies is key to the crystal structure prediction of conformational polymorphs. Focusing on molecules XXXI and XXXII from the seventh blind test of crystal structure prediction, this study employs various electronic structure methods up to the level of domain-local pair natural orbital coupled cluster singles and doubles with perturbative triples [DLPNO-CCSD(T1)] to benchmark the conformational energies and to assess their impact on the crystal energy landscapes. Molecule XXXI proves to be a relatively straightforward case, with the conformational energies from generalized gradient approximation (GGA) functional B86bPBE-XDM changing only modestly when using more advanced density functionals such as PBE0-D4, ωB97M-V, and revDSD-PBEP86-D4, dispersion-corrected second-order Møller–Plesset perturbation theory (SCS-MP2D), or DLPNO-CCSD(T1). In contrast, the conformational energies of molecule XXXII prove difficult to determine reliably, and variations in the computed conformational energies appreciably impact the crystal energy landscape. Even high-level methods such as revDSD-PBEP86-D4 and SCS-MP2D exhibit significant disagreements with the DLPNO-CCSD(T1) benchmarks for molecule XXXII, highlighting the difficulty of predicting conformational energies for complex, drug-like molecules. The best-converged predicted crystal energy landscape obtained here for molecule XXXII disagrees significantly with what has been inferred about the solid-form landscape experimentally. The identified limitations of the calculations are probably insufficient to account for the discrepancies between theory and experiment on molecule XXXII, and further investigation of the experimental solid-form landscape would be valuable. Finally, assessment of several semi-empirical methods finds r2SCAN-3c to be the most promising, with conformational energy accuracy intermediate between the GGA and hybrid functionals and a low computational cost.

The seventh blind test of crystal structure prediction (CSP) methods substantially increased the level of complexity of the target compounds relative to the previous tests organized by the Cambridge Crystallographic Data Centre. In this work, the performance of density-functional methods is assessed using numerical atomic orbitals and the exchange-hole dipole moment dispersion correction (XDM) for the energy-ranking phase of the seventh blind test. Overall, excellent performance was seen for the two rigid molecules (XXVII, XXVIII) and for the organic salt (XXXIII). However, for the agrochemical (XXXI) and pharmaceutical (XXXII) targets, the experimental polymorphs were ranked fairly high in energy amongst the provided candidate structures and inclusion of thermal free-energy corrections from the lattice vibrations was found to be essential for compound XXXI. Based on these results, it is proposed that the importance of vibrational free-energy corrections increases with the number of rotatable bonds.

A novel approach to computationally enhance the sampling of molecular crystal structures is proposed and tested. This method is based on the use of extended variables coupled to a Monte Carlo based crystal polymorph generator. Inspired by the established technique of quasi-random sampling of polymorphs using the rigid molecule constraint, this approach represents molecular clusters as extended variables within a thermal reservoir. Polymorph unit-cell variables are generated using pseudo-random sampling. Within this framework, a harmonic coupling between the extended variables and polymorph configurations is established. The extended variables remain fixed during the inner loop dedicated to polymorph sampling, enforcing a stepwise propagation of the extended variables to maintain system exploration. The final processing step results in a polymorph energy landscape, where the raw structures sampled to create the extended variable trajectory are re-optimized without the thermal coupling term. The foundational principles of this approach are described and its effectiveness using both a Metropolis Monte Carlo type algorithm and modifications that incorporate replica exchange is demonstrated. A comparison is provided with pseudo-random sampling of polymorphs for the molecule coumarin. The choice to test a design of this algorithm as relevant for enhanced sampling of crystal structures was due to the obvious relation between molecular structure variables and corresponding crystal polymorphs as representative of the inherent vapor to crystal transitions that exist in nature. Additionally, it is shown that the trajectories of extended variables can be harnessed to extract fluctuation properties that can lead to valuable insights. A novel thermodynamic variable is introduced: the free energy difference between ensembles of Z' = 1 and Z' = 2 crystal polymorphs.

A seventh blind test of crystal structure prediction has been organized by the Cambridge Crystallographic Data Centre. The results are presented in two parts, with this second part focusing on methods for ranking crystal structures in order of stability. The exercise involved standardized sets of structures seeded from a range of structure generation methods. Participants from 22 groups applied several periodic DFT-D methods, machine learned potentials, force fields derived from empirical data or quantum chemical calculations, and various combinations of the above. In addition, one non-energy-based scoring function was used. Results showed that periodic DFT-D methods overall agreed with experimental data within expected error margins, while one machine learned model, applying system-specific AIMnet potentials, agreed with experiment in many cases demonstrating promise as an efficient alternative to DFT-based methods. For target XXXII, a consensus was reached across periodic DFT methods, with consistently high predicted energies of experimental forms relative to the global minimum (above 4 kJ mol−1 at both low and ambient temperatures) suggesting a more stable polymorph is likely not yet observed. The calculation of free energies at ambient temperatures offered improvement of predictions only in some cases (for targets XXVII and XXXI). Several avenues for future research have been suggested, highlighting the need for greater efficiency considering the vast amounts of resources utilized in many cases.

A seventh blind test of crystal structure prediction was organized by the Cambridge Crystallographic Data Centre featuring seven target systems of varying complexity: a silicon and iodine-containing molecule, a copper coordination complex, a near-rigid molecule, a cocrystal, a polymorphic small agrochemical, a highly flexible polymorphic drug candidate, and a polymorphic morpholine salt. In this first of two parts focusing on structure generation methods, many crystal structure prediction (CSP) methods performed well for the small but flexible agrochemical compound, successfully reproducing the experimentally observed crystal structures, while few groups were successful for the systems of higher complexity. A powder X-ray diffraction (PXRD) assisted exercise demonstrated the use of CSP in successfully determining a crystal structure from a low-quality PXRD pattern. The use of CSP in the prediction of likely cocrystal stoichiometry was also explored, demonstrating multiple possible approaches. Crystallographic disorder emerged as an important theme throughout the test as both a challenge for analysis and a major achievement where two groups blindly predicted the existence of disorder for the first time. Additionally, large-scale comparisons of the sets of predicted crystal structures also showed that some methods yield sets that largely contain the same crystal structures.

The synthesis of a chiral isothiourea, namely, (4aR,8aR)-3-phenyl-4a,5,6,7,8,8a-hexahydrobenzo[4,5]imidazo[2,1-b]thiazol-9-ium bromide, C15H17N2S+·Br−, with potential organocatalytic and anti-inflammatory activity is reported. The preparation of the heterocycle of interest was carried out in two high-yielding steps. The hydrobromide salt of the isothiourea of interest provided suitable crystals for X-ray diffraction analysis, the results of which are reported. Salient observations from this analysis are the near perpendicular arrangement of the phenyl ring and the mean plane of the heterocycle. This conformational characteristic may be relevant with regard the stereoselectivity induced by the chiral isothiourea in asymmetric reactions. Furthermore, evidence was found for the existence of an S...Br− halogen bond.

We report the structural characterization of a new quaternary telluride, Ba2Y0.87(1)Mn1.71(1)Te5, which was synthesized by the direct reaction of the elements inside a vacuum-sealed fused-silica tube. The quaternary phase is the first member of the Ba–M–Mn–Te system (M = Sc and Y). The composition and structure of the phase were elucidated using SEM–EDX (scanning electron microscopy–energy dispersive X-ray spectrometry) and single-crystal X-ray diffraction (SCXRD) studies. The title phase is nonstoichiometric and crystallizes in the monoclinic system (space group C2/m) having the refined unit-cell parameters a = 15.1466 (8), b = 4.5782 (3), c = 10.6060 (7) Å and β = 116.956 (2)°, with two formula units (Z = 2). The pseudo-two-dimensional crystal structure of Ba2Y0.87(1)Mn1.71(1)Te5 consists of distorted YTe6 octahedra and MnTe4 tetrahedra as the building blocks of the structure. The YTe6 octahedra are arranged to form infinite one-dimensional chains by sharing edges along the [010] direction. These chains are further connected to the MnTe4 tetrahedra along the c axis to create layered two-dimensional polyanionic [Y0.87(1)Mn1.71(1)Te5]4− units. The stuffing of Ba2+ cations in between the layers of [Y0.87(1)Mn1.71(1)Te5]4− anions brings the charge neutrality of the structure. Each Ba atom in the structure sits at the centre of a distorted monocapped trigonal prism-like polyhedron of seven Te atoms.

Hydraulic fracking exposes shale plays to acidic hydraulic fracking fluid (HFF), releasing toxic uranium (U) along with the desired oil and gas. With no existing methods to ensure U remains sequestered in the shale, this study sought to add organic ligands to HFF to explore potential U retention in shale plays. To test this possibility, incubations were set up in which uranyl acetate and one organic bipyridine ligand (either 2,2'-, 2,3'-, 2,4'-, or 4,4'-bipyridine) were added to pristine HFF as the crystallization medium. After several months and complete evaporation of all volatiles, bulk yellow crystalline material was obtained from the incubations, three of which yielded crystals suitable for single-crystal analysis, resulting in two novel structures and a high-quality structure of a previously described compound. The UO2VI acetate complexes bis(acetato-κ2O,O')(2,2'-bipyridine-κ2N,N')dioxidouranium(VI), [U(C2H3O2)2O2(C10H8N2)2] or [2,2'-bipyridine]UVIO2(CH3CO2)2, (I), and bis(acetato-κ2O,O')(2,4'-bipyridine-κN1')dioxidouranium(VI), [U(C2H3O2)2O2(C10H8N2)2] or [2,4'-bipyridine]2UVIO2(CH3CO2)2, (III), contain eight-coordinate UVI in a pseudo-hexagonal bipyramidal coordination geometry and are molecular, packing via weak C—H...O/N interactions, whereas catena-poly[bis(2,3'-bipyridinium) [di-μ-acetato-μ3-hydroxido-μ-hydroxido-di-μ3-oxido-hexaoxidotriuranium(VI)]–2,3'-bipyridine–water (1/1/1)], (C10H9N2)2[U3(C2H3O2)2O8(OH)2]·C10H8N2·H2O or {[2,3'-bipyridinium]2[2,3'-bipyridine][(UVIO2)3(O)2(OH)2(CH3CO2)2·H2O]}n, (II), forms an ionic one-dimensional polymer with seven-coordinate pentagonal bipyramidal UVI centers and hydrogen-bonding interactions within each chain. The formation of these crystals could indicate the potential for bipyridine to bind with U in shale during fracking, which will be explored in a future study via ICP-MS (inductively coupled plasma mass spectrometry) analyses of U concentration in HFF/bipyridine/shale incubations. The variation seen here between the molecular structures may indicate variance in the ability of bipyridine isomers to form complexes with U, which could impact their ability to retain U within shale in the context of fracking.

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.

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.

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.

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.

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.

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.

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.

A crystal structure with N atoms in its unit cell can be solved starting from a model with atoms 1 to j − 1 being located. To locate the next atom j, the method uses a modified definition of the traditional R1 factor where its dependencies on the locations of atoms j + 1 to N are removed. This modified R1 is called the single-atom R1 (sR1), because the locations of atoms 1 to j − 1 in sR1 are the known parameters, and only the location of atom j is unknown. Finding the correct position of atom j translates thus into the optimization of the sR1 function, with respect to its fractional coordinates, xj, yj, zj. Using experimental data, it has been verified that an sR1 has a hole near each missing atom. Further, it has been verified that an algorithm based on sR1, hereby called the sR1 method, can solve crystal structures (with up to 156 non-hydrogen atoms in the unit cell). The strategy to carry out this calculation has also been optimized. The main feature of the sR1 method is that, starting from a single arbitrarily positioned atom, the structure is gradually revealed. With the user's help to delete poorly determined parts of the structure, the sR1 method can build the model to a high final quality. Thus, sR1 is a viable and useful tool for solving crystal structures.

Recent advances in quantum crystallography have shown that, beyond conventional charge density refinement, a one-electron reduced density matrix (1-RDM) satisfying N-representability conditions can be reconstructed using jointly experimental X-ray structure factors and directional Compton profiles (DCP) through semidefinite programming. So far, such reconstruction methods for 1-RDM, not constrained to idempotency, have been tested only on a toy model system (CO2). In this work, a new method is assessed on crystalline urea [CO(NH2)2] using static (0 K) and dynamic (50 K) artificial experimental data. An improved model, including symmetry constraints and frozen core-electron contribution, is introduced to better handle the increasing system complexity. Reconstructed 1-RDMs, deformation densities and DCP anisotropy are analysed, and it is demonstrated that the changes in the model significantly improve the reconstruction quality, even when there is insufficient information and data corruption. The robustness of the model and the strategy are thus shown to be well adapted to address the reconstruction problem from actual experimental scattering data.

Monoclinic ferroelectric phases are prevalent in various functional materials, most notably mixed-ion perovskite oxides. These phases can manifest as regularly ordered long-range crystallographic structures or as macroscopic averages of the self-assembled tetragonal/rhombohedral nanodomains. The structural and physical properties of monoclinic ferroelectric phases play a pivotal role when exploring the interplay between ferroelectricity, ferroelasticity, giant piezoelectricity and multiferroicity in crystals, ceramics and epitaxial thin films. However, the complex nature of this subject presents challenges, particularly in deciphering the microstructures of monoclinic domains. In Paper I [Biran & Gorfman (2024). Acta Cryst. A80, 112–128] the geometrical principles governing the connection of domain microstructures formed by pairing MAB type monoclinic domains were elucidated. Specifically, a catalog was established of `permissible domain walls', where `permissible', as originally introduced by Fousek & Janovec [J. Appl. Phys. (1969), 40, 135–142], denotes a mismatch-free connection between two monoclinic domains along the corresponding domain wall. The present article continues the prior work by elaborating on the formalisms of permissible domain walls to describe domain microstructures formed by pairing the MC type monoclinic domains. Similarly to Paper I, 84 permissible domain walls are presented for MC type domains. Each permissible domain wall is characterized by Miller indices, the transformation matrix between the crystallographic basis vectors of the domains and, crucially, the expected separation of Bragg peaks diffracted from the matched pair of domains. All these parameters are provided in an analytical form for easy and intuitive interpretation of the results. Additionally, 2D illustrations are provided for selected instances of permissible domain walls. The findings can prove valuable for various domain-related calculations, investigations involving X-ray diffraction for domain analysis and the description of domain-related physical properties.

In Part I of this series, all topologically possible 1-periodic infinite graphs (chain graphs) representing chains of tetrahedra with up to 6–8 vertices (tetrahedra) per repeat unit were generated. This paper examines possible restraints on embedding these chain graphs into Euclidean space such that they are compatible with the metrics of chains of tetrahedra in observed crystal structures. Chain-silicate minerals with T = Si4+ (plus P5+, V5+, As5+, Al3+, Fe3+, B3+, Be2+, Zn2+ and Mg2+) have a grand nearest-neighbour 〈T–T〉 distance of 3.06±0.15 Å and a minimum T⋯T separation of 3.71 Å between non-nearest-neighbour tetrahedra, and in order for embedded chain graphs (called unit-distance graphs) to be possible atomic arrangements in crystals, they must conform to these metrics, a process termed equalization. It is shown that equalization of all acyclic chain graphs is possible in 2D and 3D, and that equalization of most cyclic chain graphs is possible in 3D but not necessarily in 2D. All unique ways in which non-isomorphic vertices may be moved are designated modes of geometric modification. If a mode (m) is applied to an equalized unit-distance graph such that a new geometrically distinct unit-distance graph is produced without changing the lengths of any edges, the mode is designated as valid (mv); if a new geometrically distinct unit-distance graph cannot be produced, the mode is invalid (mi). The parameters mv and mi are used to define ranges of rigidity of the unit-distance graphs, and are related to the edge-to-vertex ratio, e/n, of the parent chain graph. The program GraphT–T was developed to embed any chain graph into Euclidean space subject to the metric restraints on T–T and T⋯T. Embedding a selection of chain graphs with differing e/n ratios shows that the principal reason why many topologically possible chains cannot occur in crystal structures is due to violation of the requirement that T⋯T > 3.71 Å. Such a restraint becomes increasingly restrictive as e/n increases and indicates why chains with stoichiometry TO<2.5 do not occur in crystal structures.

Following the work of Day & Hawthorne [Acta Cryst. (2022), A78, 212–233] and Day et al. [Acta Cryst. (2024), A80, 258–281], the program GraphT–T has been developed to embed graphical representations of observed and hypothetical chains of (SiO4)4− tetrahedra into 2D and 3D Euclidean space. During embedding, the distance between linked vertices (T–T distances) and the distance between unlinked vertices (T⋯T separations) in the resultant unit-distance graph are restrained to the average observed distance between linked Si tetrahedra (3.06±0.15 Å) and the minimum separation between unlinked vertices is restrained to be equal to or greater than the minimum distance between unlinked Si tetrahedra (3.713 Å) in silicate minerals. The notional interactions between vertices are described by a 3D spring-force algorithm in which the attractive forces between linked vertices behave according to Hooke's law and the repulsive forces between unlinked vertices behave according to Coulomb's law. Embedding parameters (i.e. spring coefficient, k, and Coulomb's constant, K) are iteratively refined during embedding to determine if it is possible to embed a given graph to produce a unit-distance graph with T–T distances and T⋯T separations that are compatible with the observed T–T distances and T⋯T separations in crystal structures. The resultant unit-distance graphs are denoted as compatible and may form crystal structures if and only if all distances between linked vertices (T–T distances) agree with the average observed distance between linked Si tetrahedra (3.06±0.15 Å) and the minimum separation between unlinked vertices is equal to or greater than the minimum distance between unlinked Si tetrahedra (3.713 Å) in silicate minerals. If the unit-distance graph does not satisfy these conditions, it is considered incompatible and the corresponding chain of tetrahedra is unlikely to form crystal structures. Using GraphT–T, Day et al. [Acta Cryst. (2024), A80, 258–281] have shown that several topological properties of chain graphs influence the flexibility (and rigidity) of the corresponding chains of Si tetrahedra and may explain why particular compatible chain arrangements (and the minerals in which they occur) are more common than others and/or why incompatible chain arrangements do not occur in crystals despite being topologically possible.

Analytical calculations of absorption corrections for X-ray powder diffraction experiments on non-ideal samples with surface roughness, porosity or absorption contrasts from multiple phases require complex mathematical models to represent their material distribution. In a computational approach to this problem, a practicable ray-tracing algorithm is formulated which is capable of simulating angle-dependent absorption corrections in reflection geometry for any given rasterized sample model. Single or multiphase systems with arbitrary surface roughness, porosity and spatial distribution of the phases in any combination can be modeled on a voxel grid by assigning respective values to each voxel. The absorption corrections are calculated by tracing the attenuation of X-rays along their individual paths via a modified shear-warp algorithm. The algorithm is presented in detail and the results of simulated absorption corrections on samples with various surface modulations are discussed in the context of published experimental results.

The atomic pair distribution function (PDF) is a real-space representation of the structure of a material. Experimental PDFs are obtained using a Fourier transform from total scattering data which may or may not have Bragg diffraction peaks. The determination of Bragg peak resolution in scattering data from the fundamental physical parameters of the diffractometer used is well established, but after the Fourier transform from reciprocal to direct space, these contributions are harder to identify. Starting from an existing definition of the resolution function of large-area detectors for X-ray diffraction, this approach is expanded into direct space. The effect of instrumental parameters on PDF peak resolution is developed mathematically, then studied with modelling and comparison with experimental PDFs of LaB6 from measurements made in different-sized capillaries.

The superstructure spots that appear in diffraction patterns of tilted perovskites are well documented and easily calculated using crystallographic software. Here, by considering a distortion mode as a perturbation of the prototype perovskite structure, it is shown how the structure-factor equation yields Boolean conditions for the presence of superstructure reflections. This approach may have some advantages for the analysis of electron diffraction patterns of perovskites.

This article describes the diffraction pattern (2-periodic Fourier transform) from the vertices of a large patch of the recently discovered `Spectre' tiling – a strictly chiral aperiodic monotile. It was reported recently that the diffraction pattern of the related weakly chiral aperiodic `Hat' monotile was 2-periodic with chiral plane-group symmetry p6 [Kaplan et al. (2024). Acta Cryst. A80, 72–78]. The diffraction periodicity arises because the Hat tiling is a systematic aperiodic deletion of vertices from the 2-periodic hexagonal mta tiling. Despite the similarity of the Hat and Spectre tilings, the Spectre tiling is not aligned with a 2-periodic lattice, and its diffraction pattern is non-periodic with chiral point symmetry 6 about the origin.

In electron microscopy the principle of reciprocity is often used to imply time reversal symmetry. While this is true for elastic scattering, its applicability to inelastic scattering is less well established. From the second law of thermodynamics, the entropy for a thermally isolated system must be constant for any reversible process. Using entropy and statistical fluctuation arguments, it is shown that, while reversibility is possible at the microscopic level, it becomes statistically less likely for higher energy transfers. The implications for reciprocal imaging modes, including energy loss and energy gain measurements, as well as Kainuma's reciprocal wave model are also discussed.

The crystal structure uniformity is numerically estimated as the standard deviation of the crystal space quantizer 〈G3〉. This criterion has been applied to explore the uniformity of ionic sublattices in 21465 crystal structures of inorganic ionic compounds. In most cases, at least one kind of sublattice (whole ionic lattice, cationic or anionic sublattice) was found to be highly uniform with a small 〈G3〉 value. Non-uniform structures appeared to be either erroneous or essentially non-ionic. As a result, a set of uniformity criteria is proposed for the estimation of the stability of ionic crystal structures.

In powder diffraction, lattice symmetry relaxation causes a peak to split into several components which are not resolved if the degree of desymmetrization is small (pseudosymmetry). Here the equations which rule peak splitting are elaborated for the six minimal symmetry transitions, showing that the resulting split peaks are generally broader and asymmetric, and suffer an hkl-dependent displacement with respect to the high-symmetry parent peak. These results will be of help in Rietveld refinement of pseudosymmetric structures where an exact interpretation of peak deformation is required.

A 6D structure model for face-centred icosahedral quasicrystals consisting of so-called pseudo-Mackay and mini-Bergman-type atomic clusters is proposed based on the structure model of the Al69.1Pd22Cr2.1Fe6.8 3/2 cubic approximant crystal (with space group Pa3, a = 40.5 Å) [Fujita et al. (2013). Acta Cryst. A69, 322–340]. The cluster centres form an icosahedral close sphere packing generated by the occupation domains similar to those in the model proposed by Katz & Gratias [J. Non-Cryst. Solids (1993), 153–154, 187–195], but their size is smaller by a factor τ2 [τ = (1 + (5)1/2)/2]. The clusters cover approximately 99.46% of the atomic structure, and the cluster arrangement exhibits 15 and 19 different local configurations, respectively, for the pseudo-Mackay and mini-Bergman-type clusters. The occupation domains that generate cluster shells are modelled and discussed in terms of structural disorder and local reorganization of the cluster arrangements (phason flip).

This study introduces an alternative method to the Takagi–Taupin equations for investigating the dark-field X-ray microscopy (DFXM) of deformed crystals. In scenarios where dynamical diffraction cannot be disregarded, it is essential to assess the potential inaccuracies of data interpretation based on the kinematic diffraction theory. Unlike the Takagi–Taupin equations, this new method utilizes an exact dispersion relation, and a previously developed finite difference scheme with minor modifications is used for the numerical implementation. The numerical implementation has been validated by calculating the diffraction of a diamond crystal with three components, wherein dynamical diffraction is applicable to the first component and kinematic diffraction pertains to the remaining two. The numerical convergence is tested using diffraction intensities. In addition, the DFXM image of a diamond crystal containing a stacking fault is calculated using the new method and compared with the experimental result. The new method is also applied to calculate the DFXM image of a twisted diamond crystal, which clearly shows a result different from those obtained using the Takagi–Taupin equations.

Small-angle X-ray scattering (SAXS) is a widely used method for nanoparticle characterization. A common approach to analysing nanoparticles in solution by SAXS involves fitting the curve using a parametric model that relates real-space parameters, such as nanoparticle size and electron density, to intensity values in reciprocal space. Selecting the optimal model is a crucial step in terms of analysis quality and can be time-consuming and complex. Several studies have proposed effective methods, based on machine learning, to automate the model selection step. Deploying these methods in software intended for both researchers and industry raises several issues. The diversity of SAXS instrumentation requires assessment of the robustness of these methods on data from various machine configurations, involving significant variations in the q-space ranges and highly variable signal-to-noise ratios (SNR) from one data set to another. In the case of laboratory instrumentation, data acquisition can be time-consuming and there is no universal criterion for defining an optimal acquisition time. This paper presents an approach that revisits the nanoparticle model selection method proposed by Monge et al. [Acta Cryst. (2024), A80, 202–212], evaluating and enhancing its robustness on data from device configurations not seen during training, by expanding the data set used for training. The influence of SNR on predictor robustness is then assessed, improved, and used to propose a stopping criterion for optimizing the trade-off between exposure time and data quality.

Algebraic expressions for averaging linear and nonlinear stiffness tensors from general anisotropy to different effective symmetries (11 Laue classes elastically representing all 32 crystal classes, and two non-crystalline symmetries: isotropic and cylindrical) have been derived by automatic symbolic computations of the arithmetic mean over the set of rotational transforms determining a given symmetry. This approach generalizes the Voigt average to nonlinear constants and desired approximate symmetries other than isotropic, which can be useful for a description of textured polycrystals and rocks preserving some symmetry aspects. Low-symmetry averages have been used to derive averages of higher symmetry to speed up computations. Relationships between the elastic constants of each symmetry have been deduced from their corresponding averages by resolving the rank-deficient system of linear equations. Isotropy has also been considered in terms of generalized Lamé constants. The results are published in the form of appendices in the supporting information for this article and have been deposited in the Mendeley database.