An implementation of Slater-type spherical scattering factors for X-ray and electron diffraction for elements in the range Z = 1–103 is presented within the software Olex2. Both high- and low-angle Fourier behaviour of atomic electron density and electrostatic potential can thus be addressed, in contrast to the limited flexibility of the four Gaussian plus constant descriptions which are currently the most widely used method for calculating atomic scattering factors during refinement. The implementation presented here accommodates the increasing complexity of the electronic structure of heavier elements by using complete atomic wavefunctions without any interpolation between precalculated tables or intermediate fitting functions. Atomic wavefunctions for singly charged ions are implemented and made accessible, and these show drastic changes in electron diffraction scattering factors compared with the neutral atom. A comparison between the two different spherical models of neutral atoms is presented as an example for four different kinds of X-ray and two electron diffraction structures, and comparisons of refinement results using the existing diffraction data are discussed. A systematic but slight improvement in R values and residual densities can be observed when using the new scattering factors, and this is discussed relative to effects on the atomic displacement parameters and atomic positions, which are prominent near the heavier elements in a structure.

Rietveld refinements are widely used for many purposes in the physical sciences. Conducting a Rietveld refinement typically requires expert input because correct results may require that parameters be added to the fit in the proper order. This order will depend on the nature of the data and the initial parameter values. A mechanism for computing the next parameter to add to the refinement is shown. The fitting function is evaluated with the current parameter value set and each parameter incremented and decremented by a small offset. This provides the partial derivatives with respect to each parameter, along with information to discriminate meaningful values from numerical computational errors. The implementation of this mechanism in the open-source GSAS-II program is discussed. This new method is discussed as an important step towards the development of automated Rietveld refinement technology.

INSIGHT is a Python-based software tool for processing and reducing 2D grazing-incidence wide- and small-angle X-ray scattering (GIWAXS/GISAXS) data. It offers the geometric transformation of the 2D GIWAXS/GISAXS detector image to reciprocal space, including vectorized and parallelized pixel-wise intensity correction calculations. An explicit focus on efficient data management and batch processing enables full control of large time-resolved synchrotron and laboratory data sets for a detailed analysis of kinetic GIWAXS/GISAXS studies of thin films. It processes data acquired with arbitrarily rotated detectors and performs vertical, horizontal, azimuthal and radial cuts in reciprocal space. It further allows crystallographic indexing and GIWAXS pattern simulation, and provides various plotting and export functionalities. Customized scripting offers a one-step solution to reduce, process, analyze and export findings of large in situ and operando data sets.

An anomalous ultra-small-angle X-ray scattering (AUSAXS) system has been constructed at BL28XU at SPring-8 for time-resolved AUSAXS experiments. The path length was extended to 9.1 m and a minimum of q = 0.0069 nm−1 was attained. Scattering profiles at 0.0069 to 0.3 nm−1 were successfully obtained at 17 different X-ray energies in 30 s using the BL28XU optical setup, which enables adjustment of the energy of the incident X-rays quickly without the beam position drifting. Time-resolved measurements were conducted to investigate changes in the structure of zinc compounds in poly(styrene-ran-butadiene) rubber during vulcanization. A change in energy dependence of the scattered intensity with time was found during vulcanization, suggesting the transformation of zinc in the reaction.

Data collection at X-ray free electron lasers has particular experimental challenges, such as continuous sample delivery or the use of novel ultrafast high-dynamic-range gain-switching X-ray detectors. This can result in a multitude of data artefacts, which can be detrimental to accurately determining structure-factor amplitudes for serial crystallography or single-particle imaging experiments. Here, a new data-classification tool is reported that offers a variety of machine-learning algorithms to sort data trained either on manual data sorting by the user or by profile fitting the intensity distribution on the detector based on the experiment. This is integrated into an easy-to-use graphical user interface, specifically designed to support the detectors, file formats and software available at most X-ray free electron laser facilities. The highly modular design makes the tool easily expandable to comply with other X-ray sources and detectors, and the supervised learning approach enables even the novice user to sort data containing unwanted artefacts or perform routine data-analysis tasks such as hit finding during an experiment, without needing to write code.

Hendrickson & Lattman [Acta Cryst. (1970), B26, 136–143] introduced a method for representing crystallographic phase probabilities defined on the unit circle. Their approach could model the bimodal phase probability distributions that can result from experimental phase determination procedures. It also provided simple and highly effective means to combine independent sources of phase information. The present work discusses the equivalence of the Hendrickson–Lattman distribution and the generalized von Mises distribution of order two, which has been studied in the statistical literature. Recognizing this connection allows the Hendrickson–Lattman distribution to be expressed in an alternative form which is easier to interpret, as it involves the location and concentration parameters of the component von Mises distributions. It also allows clarification of the conditions for bimodality and access to a simplified analytical method for evaluating the trigonometric moments of the distribution, the first of which is required for computing the best Fourier synthesis in the presence of phase, but not amplitude, uncertainty.

NdGa hydride and deuteride phases were prepared from high-quality NdGa samples and their structures characterized by powder and single-crystal X-ray diffraction and neutron powder diffraction. NdGa with the orthorhombic CrB-type structure absorbs hydrogen at hydrogen pressures ≤ 1 bar until reaching the composition NdGaH(D)1.1, which maintains a CrB-type structure. At elevated hydrogen pressure additional hydrogen is absorbed and the maximum composition recovered under standard temperature and pressure conditions is NdGaH(D)1.6 with the Cmcm LaGaH1.66-type structure. This structure is a threefold superstructure with respect to the CrB-type structure. The hydrogen atoms are ordered and distributed on three fully occupied Wyckoff positions corresponding to tetrahedral (4c, 8g) and trigonal–bipyramidal (8g) voids in the parent structure. The threefold superstructure is maintained in the H-deficient phases NaGaH(D)x until 1.6 ≥ x ≥ 1.2. At lower H concentrations, coinciding with the composition of the hydride obtained from hydrogenation at atmospheric pressure, the unit cell of the CrB-type structure is resumed. This phase can also display H deficiency, NdGaH(D)y (1.1 ≥ y ≥ 0.9), with H(D) exclusively situated in partially empty tetrahedral voids. The phase boundary between the threefold superstructure (LaGaH1.66 type) and the onefold structure (NdGaH1.1 type) is estimated on the basis of phase–composition isotherms and neutron powder diffraction to be x = 1.15.

The maximum range of perpendicular momentum transfer (qz) has been tripled for X-ray scattering from liquid surfaces when using a double-crystal deflector setup to tilt the incident X-ray beam. This is achieved by employing a higher-energy X-ray beam to access Miller indices of reflecting crystal atomic planes that are three times higher than usual. The deviation from the exact Bragg angle condition induced by misalignment between the X-ray beam axis and the main rotation axis of the double-crystal deflector is calculated, and a fast and straightforward procedure to align them is deduced. An experimental method of measuring scattering intensity along the qz direction on liquid surfaces up to qz = 7 Å−1 is presented, with liquid copper serving as a reference system for benchmarking purposes.

A multi-slit very small angle neutron scattering (MS-VSANS) instrument has been finally accepted at the China Spallation Neutron Source (CSNS). It is the first spallation neutron source based VSANS instrument. MS-VSANS has a good signal-to-noise ratio and can cover a wide scattering vector magnitude range from 0.00028 to 1.4 Å−1. In its primary flight path, a combined curved multichannel beam bender and sections of rotary exchange drums are installed to minimize the background downstream of the instrument. An exchangeable multi-slit beam focusing system is integrated into the primary flight path, enabling access to a minimum scattering vector magnitude of 0.00028 Å−1. MS-VSANS has three modes, namely conventional SANS, polarizing SANS and VSANS modes. In the SANS mode, three motorized high-efficiency 3He tube detectors inside the detector tank cover scattering angles from 0.12 to 35° simultaneously. In the polarizing SANS mode, a double-V cavity provides highly polarized neutrons and a high-efficiency 3He polarization analyser allows full polarization analysis. In the VSANS mode, an innovative high-resolution gas electron multiplier detector covers scattering angles from 0.016 to 0.447°. The absolute scattering intensities of a selection of standard samples are obtained using the direct-beam technique; the effectiveness of this method is verified by testing the standard samples and comparing the results with those from a benchmark instrument. The MS-VSANS instrument is designed to be flexible and versatile and all the design goals have been achieved.

Fluctuation X-ray scattering (FXS) offers a complementary approach for nano- and bioparticle imaging with an X-ray free-electron laser (XFEL), by extracting structural information from correlations in scattered XFEL pulses. Here a workflow is presented for single-particle structure determination using FXS. The workflow includes procedures for extracting the rotational invariants from FXS patterns, performing structure reconstructions via iterative phasing of the invariants, and aligning and averaging multiple reconstructions. The reconstruction pipeline is implemented in the open-source software xFrame and its functionality is demonstrated on several simulated structures.

Serial crystallography experiments at synchrotron and X-ray free-electron laser (XFEL) sources are producing crystallographic data sets of ever-increasing volume. While these experiments have large data sets and high-frame-rate detectors (around 3520 frames per second), only a small percentage of the data are useful for downstream analysis. Thus, an efficient and real-time data classification pipeline is essential to differentiate reliably between useful and non-useful images, typically known as `hit' and `miss', respectively, and keep only hit images on disk for further analysis such as peak finding and indexing. While feature-point extraction is a key component of modern approaches to image classification, existing approaches require computationally expensive patch preprocessing to handle perspective distortion. This paper proposes a pipeline to categorize the data, consisting of a real-time feature extraction algorithm called modified and parallelized FAST (MP-FAST), an image descriptor and a machine learning classifier. For parallelizing the primary operations of the proposed pipeline, central processing units, graphics processing units and field-programmable gate arrays are implemented and their performances compared. Finally, MP-FAST-based image classification is evaluated using a multi-layer perceptron on various data sets, including both synthetic and experimental data. This approach demonstrates superior performance compared with other feature extractors and classifiers.

In addressing the challenges faced by laboratories and universities with limited (or no) cryo-electron microscopy (cryo-EM) infrastructure, the ESRF, in collaboration with the Grenoble Institute for Structural Biology (IBS), has implemented the cryo-EM Solution-to-Structure (SOS) pipeline. This inclusive process, spanning grid preparation to high-resolution data collection, covers single-particle analysis and cryo-electron tomography (cryo-ET). Accessible through a rolling access route, proposals undergo scientific merit and technical feasibility evaluations. Stringent feasibility criteria demand robust evidence of sample homogeneity. Two distinct entry points are offered: users can either submit purified protein samples for comprehensive processing or initiate the pipeline with already vitrified cryo-EM grids. The SOS pipeline integrates negative stain imaging (exclusive to protein samples) as a first quality step, followed by cryo-EM grid preparation, grid screening and preliminary data collection for single-particle analysis, or only the first two steps for cryo-ET. In both cases, if the screening steps are successfully completed, high-resolution data collection will be carried out using a Titan Krios microscope equipped with a latest-generation direct electron counting detector coupled to an energy filter. The SOS pipeline thus emerges as a comprehensive and efficient solution, further democratizing access to cryo-EM research.

Analysis of small-angle scattering (SAS) data requires intensive modeling to infer and characterize the structures present in a sample. This iterative improvement of models is a time-consuming process. Presented here is Scattering Equation Builder (SEB), a C++ library that derives exact analytic expressions for the form factors of complex composite structures. The user writes a small program that specifies how the sub-units should be linked to form a composite structure and calls SEB to obtain an expression for the form factor. SEB supports e.g. Gaussian polymer chains and loops, thin rods and circles, solid spheres, spherical shells and cylinders, and many different options for how these can be linked together. The formalism behind SEB is presented and simple case studies are given, such as block copolymers with different types of linkage, as well as more complex examples, such as a random walk model of 100 linked sub-units, dendrimers, polymers and rods attached to the surfaces of geometric objects, and finally the scattering from a linear chain of five stars, where each star is built up of four diblock copolymers. These examples illustrate how SEB can be used to develop complex models and hence reduce the cost of analyzing SAS data.

The strong metal–support interaction (SMSI) is a phenomenon observed in supported metal catalyst systems in which reducible metal oxide supports can form overlayers over the surface of active metal nanoparticles (NPs) under a hydrogen (H2) environment at elevated temperatures. SMSI has been shown to affect catalyst performance in many reactions by changing the type and number of active sites on the catalyst surface. Laboratory methods for the analysis of SMSI at the nanoparticle-ensemble level are lacking and mostly based on indirect evidence, such as gas chemisorption. Here, we demonstrate the possibility to detect and characterize SMSIs in Co/TiOx model catalysts using the laboratory X-ray standing wave (XSW) technique for a large ensemble of NPs at the bulk scale. We designed a thermally stable MoNx/SiNx periodic multilayer to retain XSW generation after reduction with H2 gas at 600°C. The model catalyst system was synthesized here by deposition of a thin TiOx layer on top of the periodic multilayer, followed by Co NP deposition via spare ablation. A partial encapsulation of Co NPs by TiOx was identified by analyzing the change in Ti atomic distribution. This novel methodological approach can be extended to observe surface restructuring of model catalysts in situ at high temperature (up to 1000°C) and pressure (≤3 mbar), and can also be relevant for fundamental studies in the thermal stability of membranes, as well as metallurgy.

This article describes a correction procedure for the removal of indirect background contributions to measured small-angle X-ray scattering patterns. The high scattering power of a sample in the ultra-small-angle region may serve as a secondary source for a window placed in front of the detector. The resulting secondary scattering appears as a sample-dependent background in the measured pattern that cannot be directly subtracted. This is an intricate problem in measurements at ultra-low angles, which can significantly reduce the useful dynamic range of detection. Two different procedures are presented to retrieve the real scattering profile of the sample.

Due to the ambiguity related to the lack of phase information, determining the physical parameters of multilayer thin films from measured neutron and X-ray reflectivity curves is, on a fundamental level, an underdetermined inverse problem. This ambiguity poses limitations on standard neural networks, constraining the range and number of considered parameters in previous machine learning solutions. To overcome this challenge, a novel training procedure has been designed which incorporates dynamic prior boundaries for each physical parameter as additional inputs to the neural network. In this manner, the neural network can be trained simultaneously on all well-posed subintervals of a larger parameter space in which the inverse problem is underdetermined. During inference, users can flexibly input their own prior knowledge about the physical system to constrain the neural network prediction to distinct target subintervals in the parameter space. The effectiveness of the method is demonstrated in various scenarios, including multilayer structures with a box model parameterization and a physics-inspired special parameterization of the scattering length density profile for a multilayer structure. In contrast to previous methods, this approach scales favourably when increasing the complexity of the inverse problem, working properly even for a five-layer multilayer model and a periodic multilayer model with up to 17 open parameters.

X-ray crystallography is an established tool to probe the structure of macromolecules with atomic resolution. Compared with alternative techniques such as single-particle cryo-electron microscopy and micro-electron diffraction, X-ray crystallography is uniquely suited to room-temperature studies and for obtaining a detailed picture of macromolecules subjected to an external electric field (EEF). The impact of an EEF on proteins has been extensively explored through single-crystal X-ray crystallography, which works well with larger high-quality protein crystals. This article introduces a novel design for a 3D-printed in situ crystallization plate that serves a dual purpose: fostering crystal growth and allowing the concurrent examination of the effects of an EEF on crystals of varying sizes. The plate's compatibility with established X-ray crystallography techniques is evaluated.

This paper introduces a new 2D representation of the orientation distribution function for an arbitrary material texture. The approach is based on the isometric square torus mapping of the Clifford torus, which allows for points on the unit quaternion hypersphere (each corresponding to a 3D orientation) to be represented in a periodic 2D square map. The combination of three such orthogonal mappings into a single RGB (red–green–blue) image provides a compact periodic representation of any set of orientations. Square torus representations of five different orientation sampling methods are compared and analyzed in terms of the Riesz s energies that quantify the uniformity of the samplings. The effect of crystallographic symmetry on the square torus map is analyzed in terms of the Rodrigues fundamental zones for the rotational symmetry groups. The paper concludes with example representations of important texture components in cubic and hexagonal materials. The new RGB representation provides a convenient and compact way of generating training data for the automated analysis of material textures by means of neural networks.

Three-dimensional cryo electron microscopy reconstructions are obtained by extracting information from a large number of projections of the object. These projections correspond to different `views' or `orientations', i.e. directions in which these projections show the reconstructed object. Uneven distribution of these views and the presence of dominating preferred orientations may distort the reconstructed spatial images. This work describes the program VUE (views on uniform grids for cryo electron microscopy), designed to study such distributions. Its algorithms, based on uniform virtual grids on a sphere, allow an easy calculation and accurate quantitative analysis of the frequency distribution of the views. The key computational element is the Lambert azimuthal equal-area projection of a spherical uniform grid onto a disc. This projection keeps the surface area constant and represents the frequency distribution with no visual bias. Since it has multiple tunable parameters, the program is easily adaptable to individual needs, and to the features of a particular project or of the figure to be produced. It can help identify problems related to an uneven distribution of views. Optionally, it can modify the list of projections, distributing the views more uniformly. The program can also be used as a teaching tool.

Macromolecular crystallography contributes significantly to understanding diseases and, more importantly, how to treat them by providing atomic resolution 3D structures of proteins. This is achieved by collecting X-ray diffraction images of protein crystals from important biological pathways. Spotfinders are used to detect the presence of crystals with usable data, and the spots from such crystals are the primary data used to solve the relevant structures. Having fast and accurate spot finding is essential, but recent advances in synchrotron beamlines used to generate X-ray diffraction images have brought us to the limits of what the best existing spotfinders can do. This bottleneck must be removed so spotfinder software can keep pace with the X-ray beamline hardware improvements and be able to see the weak or diffuse spots required to solve the most challenging problems encountered when working with diffraction images. In this paper, we first present Bragg Spot Detection (BSD), a large benchmark Bragg spot image dataset that contains 304 images with more than 66 000 spots. We then discuss the open source extensible U-Net-based spotfinder Bragg Spot Finder (BSF), with image pre-processing, a U-Net segmentation backbone, and post-processing that includes artifact removal and watershed segmentation. Finally, we perform experiments on the BSD benchmark and obtain results that are (in terms of accuracy) comparable to or better than those obtained with two popular spotfinder software packages (Dozor and DIALS), demonstrating that this is an appropriate framework to support future extensions and improvements.

The pair angle distribution function (PADF) is a three- and four-atom correlation function that characterizes the local angular structure of disordered materials, particles or nanocrystalline materials. The PADF can be measured using X-ray or electron fluctuation diffraction data, which can be collected by scanning or flowing a structurally disordered sample through a focused beam. It is a natural generalization of established pair distribution methods, which do not provide angular information. The software package pypadf provides tools to calculate the PADF from fluctuation diffraction data. The package includes tools for calculating the intensity correlation function, which is a necessary step in the PADF calculation and also the basis for other fluctuation scattering analysis techniques.

The capillary wave model of a liquid surface predicts both the X-ray specular reflection and the diffuse scattering around it. A quantitative method is presented to obtain the X-ray reflectivity (XRR) from a liquid surface through the diffuse scattering data around the specular reflection measured using a grazing incidence X-ray off-specular scattering (GIXOS) geometry at a fixed horizontal offset angle with respect to the plane of incidence. With this approach the entire Qz-dependent reflectivity profile can be obtained at a single, fixed incident angle. This permits a much faster acquisition of the profile than with conventional reflectometry, where the incident angle must be scanned point by point to obtain a Qz-dependent profile. The XRR derived from the GIXOS-measured diffuse scattering, referred to in this paper as pseudo-reflectivity, provides a larger Qz range compared with the reflectivity measured by conventional reflectometry. Transforming the GIXOS-measured diffuse scattering profile to pseudo-XRR opens up the GIXOS method to widely available specular XRR analysis software tools. Here the GIXOS-derived pseudo-XRR is compared with the XRR measured by specular reflectometry from two simple vapor–liquid interfaces at different surface tension, and from a hexadecyltri­methyl­ammonium bromide monolayer on a water surface. For the simple liquids, excellent agreement (beyond 11 orders of magnitude in signal) is found between the two methods, supporting the approach of using GIXOS-measured diffuse scattering to derive reflectivities. Pseudo-XRR obtained at different horizontal offset angles with respect to the plane of incidence yields indistinguishable results, and this supports the robustness of the GIXOS-XRR approach. The pseudo-XRR method can be extended to soft thin films on a liquid surface, and criteria are established for the applicability of the approach.

Polymer-derived ceramics (PDCs) remain at the forefront of research for a variety of applications including ultra-high-temperature ceramics, energy storage and functional coatings. Despite their wide use, questions remain about the complex structural transition from polymer to ceramic and how local structure influences the final microstructure and resulting properties. This is further complicated when nanofillers are introduced to tailor structural and functional properties, as nanoparticle surfaces can interact with the matrix and influence the resulting structure. The inclusion of crystalline nanofiller produces a mixed crystalline–amorphous composite, which poses characterization challenges. With this study, we aim to address these challenges with a local-scale structural study that probes changes in a polysiloxane matrix with incorporated copper nanofiller. Composites were processed at three unique temperatures to capture mixing, pyrolysis and initial crystallization stages for the pre-ceramic polymer. We observed the evolution of the nanofiller with electron microscopy and applied synchrotron X-ray diffraction with differential pair distribution function (d-PDF) analysis to monitor changes in the matrix's local structure and interactions with the nanofiller. The application of the d-PDF to PDC materials is novel and informs future studies to understand interfacial interactions between nanofiller and matrix throughout PDC processing.

Controlling the shape and size dispersivity and crystallinity of nanoparticles (NPs) has been a challenge in identifying these parameters' role in the physical and chemical properties of NPs. The need for reliable quantitative tools for analyzing the dispersivity and crystallinity of NPs is a considerable problem in optimizing scalable synthesis routes capable of controlling NP properties. The most common tools are electron microscopy (EM) and X-ray scattering techniques. However, each technique has different susceptibility to these parameters, implying that more than one technique is necessary to characterize NP systems with maximum reliability. Wide-angle X-ray scattering (WAXS) is mandatory to access information on crystallinity. In contrast, EM or small-angle X-ray scattering (SAXS) is required to access information on whole NP sizes. EM provides average values on relatively small ensembles in contrast to the bulk values accessed by X-ray techniques. Besides the fact that the SAXS and WAXS techniques have different susceptibilities to size distributions, SAXS is easily affected by NP–NP interaction distances. Because of all the variables involved, there have yet to be proposed methodologies for cross-analyzing data from two techniques that can provide reliable quantitative results of dispersivity and crystallinity. In this work, a SAXS/WAXS-based methodology is proposed for simultaneously quantifying size distribution and degree of crystallinity of NPs. The most reliable easy-to-access size result for each technique is demonstrated by computer simulation. Strategies on how to compare these results and how to identify NP–NP interaction effects underneath the SAXS intensity curve are presented. Experimental results are shown for cubic-like CeO2 NPs. WAXS size results from two analytical procedures are compared, line-profile fitting of individual diffraction peaks in opposition to whole pattern fitting. The impact of shape dispersivity is also evaluated. Extension of the proposed methodology for cross-analyzing EM and WAXS data is possible.

Serial crystallography (SX) involves combining observations from a very large number of diffraction patterns coming from crystals in random orientations. To compile a complete data set, these patterns must be indexed (i.e. their orientation determined), integrated and merged. Introduced here is TORO (Torch-powered robust optimization) Indexer, a robust and adaptable indexing algorithm developed using the PyTorch framework. TORO is capable of operating on graphics processing units (GPUs), central processing units (CPUs) and other hardware accelerators supported by PyTorch, ensuring compatibility with a wide variety of computational setups. In tests, TORO outpaces existing solutions, indexing thousands of frames per second when running on GPUs, which positions it as an attractive candidate to produce real-time indexing and user feedback. The algorithm streamlines some of the ideas introduced by previous indexers like DIALS real-space grid search [Gildea, Waterman, Parkhurst, Axford, Sutton, Stuart, Sauter, Evans & Winter (2014). Acta Cryst. D70, 2652–2666] and XGandalf [Gevorkov, Yefanov, Barty, White, Mariani, Brehm, Tolstikova, Grigat & Chapman (2019). Acta Cryst. A75, 694–704] and refines them using faster and principled robust optimization techniques which result in a concise code base consisting of less than 500 lines. On the basis of evaluations across four proteins, TORO consistently matches, and in certain instances outperforms, established algorithms such as XGandalf and MOSFLM [Powell (1999). Acta Cryst. D55, 1690–1695], occasionally amplifying the quality of the consolidated data while achieving superior indexing speed. The inherent modularity of TORO and the versatility of PyTorch code bases facilitate its deployment into a wide array of architectures, software platforms and bespoke applications, highlighting its prospective significance in SX.

Understanding the symmetries described by subperiodic groups – frieze, rod and layer groups – has been instrumental in predicting various properties (band structures, optical absorption, Raman spectra, diffraction patterns, topological properties etc.) of `low-dimensional' crystals. This knowledge is crucial in the tailored design of materials for specific applications across electronics, photonics and materials engineering. However, there are materials that have the property of being periodic only in one direction and whose symmetry cannot be described by the subperiodic rod groups. Describing the symmetry of these materials necessitates the application of line group theory. This paper gives an overview of subperiodic groups while briefly introducing line groups in order to acquaint the crystallographic community with these symmetries and direct them to pertinent literature. Since line groups are generally not sub­periodic, they have thus far remained outside the realm of symmetries traditionally considered in crystallography, although there are numerous `one-dimensional' crystals (i.e. monoperiodic structures) possessing line group symmetry.

From the historical roots of metalworking to the forefront of modern nanotechnology, functional materials have played a pivotal role in transforming societies, and their influence is poised to persist into the future. Encompassing a wide array of solid-state materials, spanning semiconductors to polymers, molecular crystals to nanoparticles, functional materials find application in critical sectors such as electronics, computers, information, communication, bio­technology, aerospace, defense, environment, energy, medicine and consumer products. This feature article delves into diverse instances of functional materials, exploring their structures, their properties and the underlying mechanisms that contribute to their outstanding performance across fields like batteries, photovoltaics, magnetics and heterogeneous catalysts. The field of structural sciences serves as the cornerstone for unraveling the intricate relationship between structure, dynamics and function. Acting as a bridge, it connects the fundamental understanding of materials to their practical applications.

Recent developments in synchrotron radiation facilities have increased the amount of data generated during acquisitions considerably, requiring fast and efficient data processing techniques. Here, the application of dense neural networks (DNNs) to data treatment of X-ray diffraction computed tomography (XRD-CT) experiments is presented. Processing involves mapping the phases in a tomographic slice by predicting the phase fraction in each individual pixel. DNNs were trained on sets of calculated XRD patterns generated using a Python algorithm developed in-house. An initial Rietveld refinement of the tomographic slice sum pattern provides additional information (peak widths and integrated intensities for each phase) to improve the generation of simulated patterns and make them closer to real data. A grid search was used to optimize the network architecture and demonstrated that a single fully connected dense layer was sufficient to accurately determine phase proportions. This DNN was used on the XRD-CT acquisition of a mock-up and a historical sample of highly heterogeneous multi-layered decoration of a late medieval statue, called `applied brocade'. The phase maps predicted by the DNN were in good agreement with other methods, such as non-negative matrix factorization and serial Rietveld refinements performed with TOPAS, and outperformed them in terms of speed and efficiency. The method was evaluated by regenerating experimental patterns from predictions and using the R-weighted profile as the agreement factor. This assessment allowed us to confirm the accuracy of the results.

The spatial orientation of α lamellae in a metastable β-Ti matrix of Timetal LCB (Ti–6.8 Mo–4.5 Fe–1.5 Al in wt%) was examined and the orientation of the hexagonal close-packed α lattice in the α lamella was determined. For this purpose, a combination of methods of small-angle X-ray scattering, scanning electron microscopy and electron backscatter diffraction was used. The habit planes of α laths are close to {111}β, which corresponds to (1320)α in the hexagonal coordinate system of the α phase. The longest α lamella direction lies approximately along one of the 〈110〉β directions which are parallel to the specific habit plane. Taking into account the average lattice parameters of the β and α phases in aged conditions in Timetal LCB, it was possible to index all main axes and faces of an α lath not only in the cubic coordinate system of the parent β phase but also in the hexagonal system of the α phase.

The influence of various combinations of residual stress, composition and grain interaction gradients in polycrystalline materials with cubic symmetry on energy-dispersive X-ray stress analysis is theoretically investigated. For the evaluation of the simulated sin2ψ distributions, two different strategies are compared with regard to their suitability for separating the individual gradients. It is shown that the separation of depth gradients of the strain-free lattice parameter a0(z) from residual stress gradients σ(z) is only possible if the data analysis is carried out in section planes parallel to the surface. The impact of a surface layer z* that is characterized by a direction-dependent grain interaction model in contrast to the volume of the material is quantified by comparing a ferritic and an austenitic steel, which feature different elastic anisotropy. It is shown to be of minor influence on the resulting residual stress depth profiles if the data evaluation is restricted to reflections hkl with orientation factors Γhkl close to the model-independent orientation Γ*. Finally, a method is proposed that allows the thickness of the anisotropic surface layer z* to be estimated on the basis of an optimization procedure.

Presented and discussed here is the implementation of a software solution that provides prompt X-ray diffraction data analysis during fast dynamic compression experiments conducted within the dynamic diamond anvil cell technique. It includes efficient data collection, streaming of data and metadata to a high-performance cluster (HPC), fast azimuthal data integration on the cluster, and tools for controlling the data processing steps and visualizing the data using the DIOPTAS software package. This data processing pipeline is invaluable for a great number of studies. The potential of the pipeline is illustrated with two examples of data collected on ammonia–water mixtures and multiphase mineral assemblies under high pressure. The pipeline is designed to be generic in nature and could be readily adapted to provide rapid feedback for many other X-ray diffraction techniques, e.g. large-volume press studies, in situ stress/strain studies, phase transformation studies, chemical reactions studied with high-resolution diffraction etc.

Small-angle scattering (SAS) is a key experimental technique for analyzing nanoscale structures in various materials. In SAS data analysis, selecting an appropriate mathematical model for the scattering intensity is critical, as it generates a hypothesis of the structure of the experimental sample. Traditional model selection methods either rely on qualitative approaches or are prone to overfitting. This paper introduces an analytical method that applies Bayesian model selection to SAS measurement data, enabling a quantitative evaluation of the validity of mathematical models. The performance of the method is assessed through numerical experiments using artificial data for multicomponent spherical materials, demonstrating that this proposed analysis approach yields highly accurate and interpretable results. The ability of the method to analyze a range of mixing ratios and particle size ratios for mixed components is also discussed, along with its precision in model evaluation by the degree of fitting. The proposed method effectively facilitates quantitative analysis of nanoscale sample structures in SAS, which has traditionally been challenging, and is expected to contribute significantly to advancements in a wide range of fields.

Using the well known Rn ratio method, a protocol has been elaborated for determining the lattice direction for the 15 most common cubic zone axis spot patterns. The method makes use of the lengths of the three shortest reciprocal-lattice vectors in each pattern and the angles between them. No prior pattern calibration is required for the method to work, as the Rn ratio method is based entirely on geometric relationships. In the first step the pattern is assigned to one of three possible pattern types according to the angles that are measured between the three reciprocal-lattice vectors. The lattice direction [uvw] and possible Bravais type(s) and Laue indices of the corresponding reflections can then be determined by using lookup tables. In addition to determining the lattice direction, this simple geometric analysis allows one to distinguish between the P, I and F Bravais lattices for spot patterns aligned along [013], [112], [114] and [233]. Moreover, the F lattice can always be uniquely identified from the [011] and [123] patterns.

X-ray photon correlation spectroscopy (XPCS) is a powerful tool for the investigation of dynamics covering a broad range of timescales and length scales. The two-time correlation function (TTC) is commonly used to track non-equilibrium dynamical evolution in XPCS measurements, with subsequent extraction of one-time correlations. While the theoretical foundation for the quantitative analysis of TTCs is primarily established for equilibrium systems, where key parameters such as the diffusion coefficient remain constant, non-equilibrium systems pose a unique challenge. In such systems, different projections (`cuts') of the TTC may lead to divergent results if the underlying fundamental parameters themselves are subject to temporal variations. This article explores widely used approaches for TTC calculations and common methods for extracting relevant information from correlation functions, particularly in the light of comparing dynamics in equilibrium and non-equilibrium systems.

X-ray ptychography is a lensless imaging technique that visualizes the nano­structure of a thick specimen which cannot be observed with an electron microscope. It reconstructs a complex-valued refractive index of the specimen from observed diffraction patterns. This reconstruction problem is called phase retrieval (PR). For further improvement in the imaging capability, including expansion of the depth of field, various PR algorithms have been proposed. Since a high-quality PR method is built upon a base PR algorithm such as ePIE, developing a well performing base PR algorithm is important. This paper proposes an improved iterative algorithm named CRISP. It exploits subgradient projection which allows adaptive step size and can be expected to avoid yielding a poor image. The proposed algorithm was compared with ePIE, which is a simple and fast-convergence algorithm, and its modified algorithm, rPIE. The experiments confirmed that the proposed method improved the reconstruction performance for both simulation and real data.

For a reliable characterization of materials and systems featuring multiple structural levels, a broad length scale from a few ångström to hundreds of nanometres must be analyzed and an extended Q range must be covered in X-ray and neutron scattering experiments. For certain samples or effects, it is advantageous to perform such characterization with a single instrument. Neutrons offer the unique advantage of contrast variation and matching by D-labeling, which is of great value in the characterization of natural or synthetic polymers. Some time-of-flight small-angle neutron scattering (TOF-SANS) instruments at neutron spallation sources can cover an extended Q range by using a broad wavelength band and a multitude of detectors. The detectors are arranged to cover a wide range of scattering angles with a resolution that allows both large-scale morphology and crystalline structure to be resolved simultaneously. However, for such analyses, the SANS instruments at steady-state sources operating in conventional monochromatic pinhole mode rely on additional wide-angle neutron scattering (WANS) detectors. The resolution must be tuned via a system of choppers and a TOF data acquisition option to reliably measure the atomic to mesoscale structures. The KWS-2 SANS diffractometer at Jülich Centre for Neutron Science allows the exploration of a wide Q range using conventional pinhole and lens focusing modes and an adjustable resolution Δλ/λ between 2 and 20%. This is achieved through the use of a versatile mechanical velocity selector combined with a variable slit opening and rotation frequency chopper. The installation of WANS detectors planned on the instrument required a detailed analysis of the quality of the data measured over a wide angular range with variable resolution. This article presents an assessment of the WANS performance by comparison with a McStas [Willendrup, Farhi & Lefmann (2004). Physica B, 350, E735–E737] simulation of ideal experimental conditions at the instrument.

With the emergence of ultrafast X-ray sources, interest in following fast processes in small molecules and macromolecules has increased. Most of the current research into ultrafast structural dynamics of macromolecules uses X-ray free-electron lasers. In parallel, small-scale laboratory-based laser-driven ultrafast X-ray sources are emerging. Continuous development of these sources is underway, and as a result many exciting applications are being reported. However, because of their low flux, such sources are not commonly used to study the structural dynamics of macromolecules. This article examines the feasibility of time-resolved powder diffraction of macromolecular microcrystals using a laboratory-scale laser-driven ultrafast X-ray source.

Unlike most of the periodic table, many rare-earth elements display considerable resonant scattering for thermal neutrons. Although this property is accompanied by strong neutron absorption, modern high-intensity neutron sources make diffraction experiments possible with these elements. Computation of scattering intensities is accomplished by fitting the variation in resonant scattering lengths (b0, b' and b'') to a semi-empirical Breit–Wigner formalism, which can be evaluated over the range of neutron energies useful for diffraction, typically E = 10–600 meV; λ = 0.4–2.8 Å (with good extrapolation to longer wavelengths).

Structural modelling of operando pair distribution function (PDF) data of complex functional materials can be highly challenging. To aid the understanding of complex operando PDF data, this article demonstrates a toolbox for PDF analysis. The tools include denoising using principal component analysis together with the structureMining, similarityMapping and nmfMapping apps available through the online service `PDF in the cloud' (PDFitc, https://pdfitc.org/). The toolbox is used for both ex situ and operando PDF data for 3 nm TiO2-bronze nanocrystals, which function as the active electrode material in a Li-ion battery. The tools enable structural modelling of the ex situ and operando PDF data, revealing two pristine TiO2 phases (bronze and anatase) and two lithiated LixTiO2 phases (lithiated versions of bronze and anatase), and the phase evolution during galvanostatic cycling is characterized.

This article demonstrates the feasibility of obtaining accurate pair distribution functions of thin amorphous films down to 80 nm, using modern laboratory-based X-ray sources. The pair distribution functions are obtained using a single diffraction scan without the requirement of additional scans of the substrate or of the air. By using a crystalline substrate combined with an oblique scattering geometry, most of the Bragg scattering of the substrate is avoided, rendering the substrate Compton scattering the primary contribution. By utilizing a discriminating energy filter, available in the latest generation of modern detectors, it is demonstrated that the Compton intensity can further be reduced to negligible levels at higher wavevector values. Scattering from the sample holder and the air is minimized by the systematic selection of pixels in the detector image based on the projected detection footprint of the sample and the use of a 3D-printed sample holder. Finally, X-ray optical effects in the absorption factors and the ratios between the Compton intensity of the substrate and film are taken into account by using a theoretical tool that simulates the electric field inside the film and the substrate, which aids in planning both the sample design and the measurement protocol.

Ptychography is a powerful computational imaging technique with microscopic imaging capability and adaptability to various specimens. To obtain an imaging result, it requires a phase-retrieval algorithm whose performance directly determines the imaging quality. Recently, deep neural network (DNN)-based phase retrieval has been proposed to improve the imaging quality from the ordinary model-based iterative algorithms. However, the DNN-based methods have some limitations because of the sensitivity to changes in experimental conditions and the difficulty of collecting enough measured specimen images for training the DNN. To overcome these limitations, a ptychographic phase-retrieval algorithm that combines model-based and DNN-based approaches is proposed. This method exploits a DNN-based denoiser to assist an iterative algorithm like ePIE in finding better reconstruction images. This combination of DNN and iterative algorithms allows the measurement model to be explicitly incorporated into the DNN-based approach, improving its robustness to changes in experimental conditions. Furthermore, to circumvent the difficulty of collecting the training data, it is proposed that the DNN-based denoiser be trained without using actual measured specimen images but using a formula-driven supervised approach that systemically generates synthetic images. In experiments using simulation based on a hard X-ray ptychographic measurement system, the imaging capability of the proposed method was evaluated by comparing it with ePIE and rPIE. These results demonstrated that the proposed method was able to reconstruct higher-spatial-resolution images with half the number of iterations required by ePIE and rPIE, even for data with low illumination intensity. Also, the proposed method was shown to be robust to its hyperparameters. In addition, the proposed method was applied to ptychographic datasets of a Simens star chart and ink toner particles measured at SPring-8 BL24XU, which confirmed that it can successfully reconstruct images from measurement scans with a lower overlap ratio of the illumination regions than is required by ePIE and rPIE.

Energy-dispersive Laue diffraction (EDLD) is a powerful method to obtain position-resolved texture information in inhomogeneous biological samples without the need for sample rotation. This study employs EDLD texture scanning to investigate the impact of two salivary peptides, statherin (STN) and histatin-1 (HTN) 21 N-terminal peptides (STN21 and HTN21), on the crystallographic structure of dental enamel. These proteins are known to play crucial roles in dental caries progression. Three healthy incisors were randomly assigned to three groups: artificially demineralized, demineralized after HTN21 peptide pre-treatment and demineralized after STN21 peptide pre-treatment. To understand the micro-scale structure of the enamel, each specimen was scanned from the enamel surface to a depth of 250 µm using microbeam EDLD. Via the use of a white beam and a pixelated detector, where each pixel functions as a spectrometer, pole figures were obtained in a single exposure at each measurement point. The results revealed distinct orientations of hydroxyapatite crystallites and notable texture variation in the peptide-treated demineralized samples compared with the demineralized control. Specifically, the peptide-treated demineralized samples exhibited up to three orientation populations, in contrast to the demineralized control which displayed only a single orientation population. The texture index of the demineralized control (2.00 ± 0.21) was found to be lower than that of either the STN21 (2.32 ± 0.20) or the HTN21 (2.90 ± 0.46) treated samples. Hence, texture scanning with EDLD gives new insights into dental enamel crystallite orientation and links the present understanding of enamel demineralization to the underlying crystalline texture. For the first time, the feasibility of EDLD texture measurements for quantitative texture evaluation in demineralized dental enamel samples is demonstrated.

X-ray scattering has become a major tool in the structural characterization of nanoscale materials. Thanks to the widely available experimental and computational atomic models, coordinate-based X-ray scattering simulation has played a crucial role in data interpretation in the past two decades. However, simulation of real-space pair distance distribution functions (PDDFs) from small- and wide-angle X-ray scattering, SAXS/WAXS, has been relatively less exploited. This study presents a comparison of PDDF simulation methods, which are applied to molecular structures that range in size from β-cyclo­dextrin [1 kDa molecular weight (MW), 66 non-hydrogen atoms] to the satellite tobacco mosaic virus capsid (1.1 MDa MW, 81 960 non-hydrogen atoms). The results demonstrate the power of interpretation of experimental SAXS/WAXS from the real-space view, particularly by providing a more intuitive method for understanding of partial structure contributions. Furthermore, the computational efficiency of PDDF simulation algorithms makes them attractive as approaches for the analysis of large nanoscale materials and biological assemblies. The simulation methods demonstrated in this article have been implemented in stand-alone software, SolX 3.0, which is available to download from https://12idb.xray.aps.anl.gov/solx.html.

The Q resolution in Bonse–Hart double-crystal diffractometers is determined for a given Bragg angle by the value of the crystallographic structure factor. To date, the reflections Si 220 or Si 111 have been used exclusively in neutron scattering, which provide resolutions for triple-bounce crystals of about 2 × 10−5 Å−1 (FWHM). The Darwin width of the GaAs 200 reflection is about a factor of 10 smaller, offering the possibility of a Q resolution of 2 × 10−6 Å−1 provided crystals of sufficient quality are available. This article reports a feasibility study with single-bounce GaAs 200, yielding a Q resolution of 4.6 × 10−6 Å−1, six times superior in comparison with a Si 220 setup.

A procedure is presented to exactly obtain the apparent average crystallite size (ACS) of powder samples using standard in-house powder diffraction experiments without any restriction originating from the Scherrer equation. Additionally, the crystallite size distribution within the sample can be evaluated. To achieve this, powder diffractograms are background corrected and long-range radial distribution functions G(r) up to 300 nm are calculated from the diffraction data. The envelope function fenv of G(r) is approximated by a procedure determining the absolute maxima of G(r) in a certain interval (r range). Fitting of an ACS distribution envelope function to this approximation gives the ACS and its distribution. The method is tested on diffractograms of LaB6 standard reference materials measured with different wavelengths to demonstrate the validity of the approach and to clarify the influence of the wavelength used. The latter results in a general description of the maximum observable average crystallite size, which depends on the instrument and wavelength used. The crystallite site distribution is compared with particle size distributions based on transmission electron microscopy investigations, providing an approximation of the average number of crystallites per particle.

Small-angle neutron scattering is a widely used technique to study large-scale structures in bulk samples. The largest accessible length scale in conventional Bragg scattering is determined by the combination of the longest available neutron wavelength and smallest resolvable scattering angle. A method is presented that circumvents this limitation and is able to extract larger length scales from the low-q power-law scattering using a modification of the well known Porod law connecting the scattered intensity of randomly distributed objects to their specific surface area. It is shown that in the special case of a highly aligned domain structure the specific surface area extracted from the modified Porod law can be used to determine specific length scales of the domain structure. The analysis method is applied to study the micrometre-sized domain structure found in the intermediate mixed state of the superconductor niobium. The analysis approach allows the range of accessible length scales to be extended from 1 µm to up to 40 µm using a conventional small-angle neutron scattering setup.

Compared with batch and vapor diffusion methods, counter diffusion can generate larger and higher-quality protein crystals yielding improved diffraction data and higher-resolution structures. Typically, counter-diffusion experiments are conducted in elongated chambers, such as glass capillaries, and the crystals are either directly measured in the capillary or extracted and mounted at the X-ray beamline. Despite the advantages of counter-diffusion protein crystallization, there are few fixed-target devices that utilize counter diffusion for crystallization. In this article, different designs of user-friendly counter-diffusion chambers are presented which can be used to grow large protein crystals in a 2D polymer microfluidic fixed-target chip. Methods for rapid chip fabrication using commercially available thin-film materials such as Mylar, propyl­ene and Kapton are also detailed. Rules of thumb are provided to tune the nucleation and crystal growth to meet users' needs while minimizing sample consumption. These designs provide a reliable approach to forming large crystals and maintaining their hydration for weeks and even months. This allows ample time to grow, select and preserve the best crystal batches before X-ray beam time. Importantly, the fixed-target microfluidic chip has a low background scatter and can be directly used at beamlines without any crystal handling, enabling crystal quality to be preserved. The approach is demonstrated with serial diffraction of photoactive yellow protein, yielding 1.32 Å resolution at room temperature. Fabrication of this standard microfluidic chip with commercially available thin films greatly simplifies fabrication and provides enhanced stability under vacuum. These advances will further broaden microfluidic fixed-target utilization by crystallographers.

High-pressure neutron powder diffraction data from PbNCN were collected on the high-pressure diffraction beamline SNAP located at the Spallation Neutron Source (SNS) of Oak Ridge National Laboratory (Tennessee, USA). The diffraction data were analyzed using the novel method of multidimensional (two dimensions for now, potentially more in the future) Rietveld refinement and, for comparison, employing the conventional Rietveld method. To achieve two-dimensional analysis, a detailed description of the SNAP instrument characteristics was created, serving as an instrument parameter file, and then yielding both cell and spatial parameters as refined under pressure for the first time for solid-state cyanamides/carbodi­imides. The bulk modulus B0 = 25.1 (15) GPa and its derivative B'0 = 11.1 (8) were extracted for PbNCN following the Vinet equation of state. Surprisingly, an internal transition was observed beyond 2.0 (2) GPa, resulting from switching the bond multiplicities (and bending direction) of the NCN2− complex anion. The results were corroborated using electronic structure calculation from first principles, highlighting both local structural and chemical bonding details.

High-pressure crystallographic data can be measured using a diamond anvil cell (DAC), which allows the sample to be viewed only along a cell vector which runs perpendicular to the diamond anvils. Although centring a sample perpendicular to this direction is straightforward, methods for centring along this direction often rely on sample focusing, measurements of the direct beam or short data collections followed by refinement of the crystal offsets. These methods may be inaccurate, difficult to apply or slow. Described here is a method based on precise measurement of the offset in this direction using a confocal optical device, whereby the cell centre is located at the mid-point of two measurements of the distance between a light source and the external faces of the diamond anvils viewed along the forward and reverse directions of the cell vector. It is shown that the method enables a DAC to be centred to within a few micrometres reproducibly and quickly.

Understanding the structure–property relationship in electrocatalysts under working conditions is crucial for the rational design of novel and improved catalytic materials. This paper presents the Aarhus University reactor for electrochemical studies using X-rays (AUREX) operando electrocatalytic flow cell, designed as an easy-to-use versatile setup with a minimal background contribution and a uniform flow field to limit concentration polarization and handle gas formation. The cell has been employed to measure operando total scattering, diffraction and absorption spectroscopy as well as simultaneous combinations thereof on a commercial silver electrocatalyst for proof of concept. This combination of operando techniques allows for monitoring of the short-, medium- and long-range structure under working conditions, including an applied potential, liquid electrolyte and local reaction environment. The structural transformations of the Ag electrocatalyst are monitored with non-negative matrix factorization, linear combination analysis, the Pearson correlation coefficient matrix, and refinements in both real and reciprocal space. Upon application of an oxidative potential in an Ar-saturated aqueous 0.1 M KHCO3/K2CO3 electrolyte, the face-centered cubic (f.c.c.) Ag gradually transforms first to a trigonal Ag2CO3 phase, followed by the formation of a monoclinic Ag2CO3 phase. A reducing potential immediately reverts the structure to the Ag (f.c.c.) phase. Following the electrochemical-reaction-induced phase transitions is of fundamental interest and necessary for understanding and improving the stability of electrocatalysts, and the operando cell proves a versatile setup for probing this. In addition, it is demonstrated that, when studying electrochemical reactions, a high energy or short exposure time is needed to circumvent beam-induced effects.