This article presents a review of the landscape of Open Science development and its relevance to crystallography. As open as possible as closed as necessary is the model for Open Science as commended by the UK and China. Mainland Europe requires compulsory release after 3 years since measurement of raw data. The USA policy situation has fluctuated considerably in recent years. Altogether, this article addresses the International Union of Crystallography's (IUCr) input into these activities. It considers its participation in international scientific and societal organizations, as well as its compliance with the policy recommendations. It considers the practical landscape in which crystallographers work, including their funding agencies, governments, central facilities and universities. This article sets the wider scene before describing how crystallography benchmarks against those under development. The IUCr's Teaching Pamphlets and the Online Dictionary of Crystallography are educational resources fully open to Global South and Global North readers and authors. Crystallography provides a mature, practice-based model of responsible openness that predates and can inform contemporary Open Science policy.
Nanocrystals exhibit size-dependent structural behaviour because surface and interface effects become increasingly important as the characteristic domain size decreases. In this work, we investigate the size dependence of three diffraction-derived structural parameters - lattice parameter, isotropic Debye-Waller coefficient and microstrain - by combining molecular dynamics simulations of spherical Pd, Fe and Ti nanocrystals with powder diffraction analysis and comparison with representative literature data. The atomistic results show that small nanocrystals are characterized by a mean compressive state together with pronounced surface-stress inhomogeneity, while enhanced atomic displacements are concentrated in the outer coordination shells. On this basis, three simple nanoscale hypotheses are proposed. The lattice parameter is described as an ideal capillarity-driven reference trend with leading 1/D (where D is the particle diameter) behaviour, although comparison with experimental data confirms that this quantity is not universal and may be strongly modified by surface chemistry, defects, non-stoichiometry and morphology. By contrast, the Debye-Waller coefficient follows a more general 1/D decrease, consistent with a surface-shell picture of enhanced vibrational and static disorder. Microstrain arises from surface-stress gradients and, in the small-particle limit, exhibits a natural 1/D2 dependence associated with surface-stress heterogeneity; over broader size ranges, a mixed 1/D + 1/D2 form provides a more effective description. These results support a heuristic surface-driven interpretation of nanoscale structural disorder and clarify the different degrees of generality of the three scaling laws.
Relativistic X-ray scattering factors are reported for a broad and systematically defined set of chemically relevant atomic anions, including mono- and multivalent species, for elements with atomic numbers Z = 1-85 [Greenwood & Earnshaw (1997). Chemistry of the Elements]. Dirac-Hartree-Fock wavefunctions were generated using the DFRATOM code [Matsuoka & Watanabe (2001). Comput. Phys. Commun. 139, 218-234], with soft radial confinement applied where necessary to stabilize diffuse anionic states. The influence of this confinement on the resulting X-ray scattering factors is explicitly and systematically analyzed to ensure that the reported data reflect physically meaningful electronic structure trends rather than artifacts of the stabilization procedure. The quality of the resulting wavefunctions was assessed using previously reported high-quality benchmark energies [Visscher & Dyall (1997). At. Data. Nucl. Tables 67, 207-224], radial metrics, and direct comparison of X-ray scattering factors with established reference data for neutral atoms and several monovalent anions [Rez et al. (1994). Acta Cryst. A50, 481-497; Su & Coppens (1997). Acta Cryst. A53, 749-762; Macchi & Coppens (2001). Acta Cryst. A57, 656-662]. To ensure physically meaningful results, an X-ray scattering based perturbation metric was introduced to quantify the impact of confinement on the calculated X-ray scattering factors, together with additional high-sin θ/λ consistency checks. The resulting X-ray scattering factors were computed and tabulated in a format consistent with that used in the International Tables for Crystallography [Maslen et al. (2006). Vol. C, Section 6.1.1, pp. 554-589], facilitating direct crystallographic use. Analysis of representative main-group, transition-metal and heavy p-block elements demonstrates that electron attachment leads to a systematic enhancement of the X-ray scattering factors at small momentum transfer, governed primarily by the expansion of the valence electron density, while the high-sin θ/λ region remains dominated by largely invariant core electron contributions. The data presented here provide the first broad, internally consistent reference set of fully relativistic X-ray scattering factors spanning a wide range of multivalent atomic anions of chemical and crystallographic interest, and represent a natural extension of the studies by Olukayode et al. [Acta Cryst. (2023), A79, 59-79; Acta Cryst. (2023), A79, 229-245].
Quasicrystals, materials with long-range order but no periodicity, were first discovered in nature within the Khatyrka meteorite, a CV3 carbonaceous chondrite. Their occurrence demonstrated that hypervelocity impacts can generate quasiperiodic phases under transient conditions far from equilibrium, which survived for billions of years. Icosahedral and decagonal quasicrystals from Khatyrka formed at pressures exceeding 5 GPa and temperatures above 1200°C, as shown by their microstructures and association with shock-melted silicates and high-pressure polymorphs. Laboratory shock-recovery experiments reproduced these phases, confirming their synthesis during microsecond-scale shock pulses and their persistence after release. The presence of metallic aluminium, rarely stabilized in natural systems, indicates that extreme redox conditions are transiently established during impacts, enabling unusual alloy chemistries. Although rare in the meteoritic record, quasicrystals may be more widespread, their scarcity reflecting preservation biases and limited analytical focus on metallic phases. Advanced nanoscale diffraction and tomography methods, coupled with systematic surveys, are essential to uncover their distribution. Beyond meteorites, terrestrial craters, lunar breccias, Martian meteorites and asteroid samples are promising targets. Quasicrystals thus represent durable witnesses of impact processes, expanding the mineralogical tools for tracing high-energy events that shaped the early solar system.
The inversion of large-scale diffraction datasets from modern synchrotron sources presents a fundamental challenge in computational crystallography. This paper presents a unified algorithmic framework for the analysis of both near-field (morphological) and far-field (orientational and strain) high-energy diffraction microscopy (HEDM) data. We detail the mathematical formalisms and physical models that form the foundation of this methodology. Key aspects include a generalized model for detector distortion correction, robust algorithms for peak identification in noisy and overlapping patterns, a computationally efficient indexing formalism based on Friedel pair symmetry, and a decoupled iterative refinement scheme that exploits the differing sensitivities of position, orientation and lattice parameters to diffraction observables. We also describe the synergistic integration of near-field and far-field data streams, a critical feature of a truly comprehensive approach. The framework is validated in Part II of this series [Sharma et al. (2026). Acta Cryst. A82, 305-320] using both experimental Ti-7 Al datasets and synthetic reconstructions with known ground truth, achieving orientation accuracy of ∼0.05° and position accuracy of ∼10 µm on experimental data, and a 190× improvement in lattice parameter precision over conventional simultaneous parameter refinement on synthetic data. This integrated framework provides a powerful and extensible solution for turning raw diffraction images into actionable microstructural and micromechanical information.
Polytypes can occur in many inorganic and organic compounds alike. In the most basic scenario, different polytypes of a compound are based on translationally equivalent layers and two equivalent arrangements of adjacent layers. This means there are only two distinct stacking vectors, t1 and t2. For this particular situation, we have enumerated the number of principally different polytypes for a given number m of layers in the stacking sequence, under the assumption of the validity of four constraints: (i) periodicity, (ii) exchangeability, (iii) reversibility and (iv) simplicity. Different results are obtained when the reversibility constraint is overturned, a factor that depends on the polar or non-polar character of the underlying layers. The results of this theoretical study have been applied to derive the existing polytypes up to m = 6 for two different polysomes of the family of so-called silico-ferrites of calcium and aluminium (SFCA and SFCA-I, respectively). These phases are of considerable interest to the steel industry, as they represent the primary constituents of the matrix in iron-ore sinters. Finally, the impact of polytypism on the powder diffraction patterns of the SFCAs was investigated and the influence of stacking disorder on the diffractograms was analysed. The resulting collection of simulated powder diffraction patterns may serve as a reference database for the identification of polytype effects in industrial sinters.
The Lorentz factor is a fundamental correction factor in quantitative analysis of X-ray diffraction experiments, enabling measured integrated peak intensities to be related to calculated structure factors. In this review, the physical origin of the Lorentz factor, its derivation and practical implementations are presented based on a unified approach, treating the Lorentz factor as the Jacobian relating the experimental measurement coordinates to the reciprocal-space volume element. This approach clarifies the trigonometric and wavelength-dependent contributions to the Lorentz factor and explains the origin of the commonly used `angular-velocity' formulation for rotational measurements. Equations for the Lorentz factor are derived systematically for a broad range of modern X-ray diffraction geometries, including single-crystal and powder diffraction, grazing-incidence diffraction methods, as well as small-angle X-ray scattering. Hereby it is demonstrated that the Lorentz factor sensitively depends not only on the experimental geometry, but also on the type of sample under investigation. In addition, we discuss how the Lorentz correction may be avoided by direct reciprocal-space integration, and the practical limitations of such approaches are pointed out. By providing a unified and comprehensive overview of the Lorentz factor, this review aims to support reliable and consistent intensity evaluation across various modern X-ray diffraction methods.
A report of the Twenty-Sixth General Assembly and International Congress of Crystallography is given.
The root lattice An and its dual weight lattice An* acquire remarkable quasicrystallographic significance through their projections onto the Coxeter plane, where (n + 1)-fold symmetry emerges naturally. The primary objective of this work is to present a general technique for projection of the Voronoi tessellation of the weight lattice An* and apply it for the lattice A4*. The projection of the Voronoi tessellation of the weight lattice A4* produces a totally different tiling scheme than the tiling obtained from the Voronoi cell projection of the lattice A4 which leads to the Penrose tiling with thick and thin rhombuses. The Voronoi cell of the weight lattice A4* is a permutohedron of order 5, comprising 2D faces of regular hexagons and squares in 4D but which project into two types of hexagons and two types of rhombuses with edges of two lengths in proportion to the golden ratio. A detailed structure of the permutohedron with its 3D and 2D facets has been worked out. A centrally symmetric tessellation of the Coxeter plane exhibiting fivefold symmetry has been constructed using four distinct tile types.
The increasing complexity of in situ high-energy diffraction microscopy (HEDM) experiments demands a quantitative understanding of the data analysis pipeline to ensure reproducible science. However, the influence of key analysis parameters on the accuracy and precision of microstructural reconstructions is often not well quantified, creating a barrier to progress. This paper addresses this critical gap by presenting a rigorous, systematic validation of the HEDM data reduction methodology as implemented in the MIDAS software suite. Using a new, dedicated Ti-7 Al dataset, we investigate both far-field (FF) and near-field (NF) HEDM. Our results reveal critical sensitivities, demonstrating that grain position accuracy in FF-HEDM is highly dependent on the diversity of sampled diffraction vectors, while orientation precision in NF-HEDM improves dramatically with increased detector separation. We demonstrate the methodology's robustness against common experimental challenges, such as severe diffraction peak overlap, which is effectively filtered by requiring crystallographic consistency. Based on these quantitative findings, we establish a framework of best practices for HEDM data acquisition and analysis to guide the community towards more accurate and reliable results.
A technical correction and update of the widely used recipes by Becker & Coppens [Acta Cryst. (1974), A30, 129-147], for the estimation of primary and secondary extinction factors in perfect spherical crystallites or grains, is presented. In the original work, these extinction factors were evaluated numerically from a complicated integral, and simplified analytical approximations to these evaluations constituted the provided recipes. However, these original recipes are plagued by issues of numerical precision in general, and even suffer from complete numerical breakdown in the case of strong extinction effects, especially in backwards scattering. Using modern computing capabilities, the numerical evaluations of the integrals are revisited, and improved recipes are provided with consistent precision guarantees for all Bragg angles and levels of extinction. The new recipes are provided both in a `standard' version, believed to be of suitable precision for any actual analysis of diffraction or transmission data, and a `luxury' reference version with even higher precision. The performance of the new recipes is compared with that of the original work, and in order to facilitate easy adoption by the community, reference implementations are provided for C, C++ and Python languages.
We propose a novel geometric characterization method for flat-faceted crystals bounded by a single simple crystallographic form - the rhombic dodecahedron. The approach utilizes distances between six pairs of parallel facets and the lengths of three false edges (out of six possible) resulting from anisotropic crystal growth. Our study has established the existence of exactly 34 combinatorially distinct, full-faceted real rhombic dodecahedra. For each type, we determined the range of crystal shape variations using the semi-axial lengths of the minimal-volume circumscribed ellipsoid. The results demonstrate that all real rhombic dodecahedra, except for eight special forms, can exhibit the complete spectrum of possible shape variations. These eight exceptional forms cannot develop tabular morphologies with bidirectional isometry. The obtained results provide valuable tools for precise analysis of rhombic dodecahedral crystal morphology and for establishing morphogenetic correlations between the degree of crystal distortions and the dissymmetry of the crystal-forming environment. This information is relevant for assessing the typomorphic features of natural diamonds, garnets, sodalites and magnetite crystals, as well as their synthetic analogs and other chemical compounds.
Tangles are defined as embeddings of connected graphs that contain knots and/or links as substructures. We enumerate and describe piecewise-linear embeddings of cubic and icosahedral tangles and linked polyhedra with transitivity [1 2]; that is, with one kind of vertex (all related by symmetry) and two kinds of edge. There are eight families of catenated polyhedra and ten families of tangles. For each family, we only describe the members with the largest girth, where girth is defined as the ratio of the shortest distance between edges with no shared vertices to the length of the longest edge.
The multipole model provides a significantly improved description of electron density compared with the spherical atom approximation and is widely applied in the crystallographic refinement of X-ray diffraction data. The multipolar atom types from theory and statistical clustering (MATTS) data bank collects multipole model parameters for atom types. These parameters are derived from quantum chemical calculations performed on experimental geometries of model molecules and rely on the concept of transferability between chemically similar atoms. An essential component of each atom type is the definition of the local coordinate system (LCS) and the symmetry of the electron density, which so far have been selected individually using expert judgment during data bank construction. In this work, we focus on the electron densities of atoms from model molecules and atom types from the MATTS data bank. We introduce a systematic procedure in which symmetry constraints are removed during multipole model refinement of model molecules and multiple types of LCS are tested using chemically meaningful directions. We examine how different LCS choices influence the pseudosymmetry, the apparent symmetry of electron density, identified empirically based on the refined multipole model parameters and their statistical significance. Our results show that refinement without symmetry constraints improves the representation of pseudosymmetry and, in some cases, leads to changes in both the values of the multipole model parameters and the assigned symmetry of the electron density. We propose an optimal LCS for each topological kind of atom type and provide clear criteria for symmetry assignment. This work contributes to the future development of the MATTS data bank with improved descriptions of LCSs and transferability of electron-density symmetry.
Exact renormalization techniques are important and powerful, particularly for inflation-generated systems. We review recent results in this direction. We recall the necessary notions for inflation systems and show the renormalization principle, which allows us to obtain exact values of highly erratic functions, such as window covariograms. We apply these techniques to compute the diffraction pattern of the new monotile tilings with arbitrary precision. We also recall a recent invariant for a system with pure-point spectrum, the orbit separation dimension, and its relation to renormalization. Lastly, we recall results beyond the pure-point spectrum setting and show how renormalization and Lyapunov exponents can be used to exclude the presence of absolutely continuous parts of the spectra.
Tsai-type 1/1 quasicrystalline approximants in the Au-Si-RE (RE = Ho, Tb) system are body-centered cubic phases decorated by clusters having their inner part occupied by either a disordered tetrahedron (IT) or a single rare-earth atom (CC). The system investigated in the present study is the Au-Si-Ho(CC) compound having full occupation of the cluster centers by single Ho atoms. We present a detailed study of the structure and reactivity of its (100) surface using scanning tunneling microscopy and X-ray photoelectron spectroscopy. After annealing Au-Si-Ho(CC) between 725 K and 880 K, the surface exhibits a (2 × 2) surface reconstruction. The surface appears to terminate at specific bulk planes intersecting Tsai-type cluster centers. Adsorption of C60 molecules on this surface leads to a film with a hexagonal structure corresponding to a (111) plane of the C60 bulk structure. Compared with a previous report on the (100) surface of the Au-Si-Ho 1/1 approximant having a mixed IT and CC decoration, it appears that cluster center decoration does not affect the surface plane selection rule in Tsai-type approximants. However, the inner decoration affects the atomic structure within the selected surface termination and the structure of C60 thin films formed on these surfaces.
The notion of defects in crystalline phases of matter has been extremely powerful for understanding crystal growth, deformation and melting. Many of these discontinuities in the periodic order of crystals are well described by the Burgers vector, derived from the particle displacements, which encapsulates the direction and magnitude of slip relative to the undeformed state. Since the reference structure of the crystal is known a priori, the Burgers vector can be determined experimentally using both imaging and diffraction methods to measure the final lattice distortion, and thus infer the particle displacements. Glasses have structures that lack the periodicity of crystals, and thus a well defined reference state. Yet, measurable structural parameters can still be obtained from diffraction from a glass. Here we examine the usefulness of these parameters to probe deformation in glasses. We find that coordinated transformations in the centrosymmetry of local particle arrangements are a strong marker of plastic events. For a glass, determining the local distortions corresponding to these plastic events requires measurements before and after deformation. We investigate two geometric indicators that can be derived from these distortions, namely the continuous Burgers vector and the quadrupolar strain. We find that the Burgers vector again emerges as a robust and sensitive metric for understanding local structural transformations due to mechanical deformation, even in disordered glasses.
In this paper we classify the two-orbit polyhedra in classes 201 and 21 that have a line preserved by their symmetry groups. Other classes of two-orbit polyhedra that have been previously classified include the chiral (or type 2) polyhedra, and the polyhedra in classes 20 and 22. Since all of those polyhedra are affinely irreducible, none of their symmetry groups preserve a line as we see in the examples studied here. The skeleta of the polyhedra described in this paper bear some similarities to carbon nanotubes; however there the vertices of their skeleta all have degree three, whereas the polyhedra studied here all have degree four.
Spin space groups, formed by operations where the rotation of the spins is independent of the accompanying operation acting on the crystal structure, are appropriate groups to describe the symmetry of magnetic structures with null spin-orbit coupling. Their corresponding spin point groups are the symmetry groups to be considered for deriving the symmetry constraints on the form of the crystal tensor properties of such idealized structures. These groups can also be taken as approximate symmetries (with some restrictions) of real magnetic structures, where spin-orbit coupling and magnetic anisotropy are however present. Here we formalize the invariance transformation properties that must satisfy the most important crystal tensors under a spin point group. This is done using modified Jahn symbols, which generalize those applicable to ordinary magnetic point groups [Gallego et al. (2019). Acta Cryst. A75, 438-447]. The analysis includes not only equilibrium tensors, but also transport, optical and non-linear optical susceptibility tensors. The constraints imposed by spin collinearity and coplanarity within the spin group formalism on a series of representative tensors are discussed and compiled. As illustrative examples, the defined tensor invariance equations have been applied to some known magnetic structures, showing the differences in the symmetry-adapted form of some relevant tensors, when considered under the constraints of its spin point group or its magnetic point group. This comparison, with the spin point group implying additional constraints in the tensor form, can allow one to distinguish those magnetic-related properties that can be solely attributed to spin-orbit coupling from those that are expected even when spin-orbit coupling is negligible.
This paper explores the application of generative pre-trained transformer (GPT)-based large language models (LLMs) in the development of simulation and analysis tools for X-ray powder diffraction. We demonstrate how these models enable users with minimal programming experience to generate functional and efficient code through natural language prompts. The discussion highlights both the capabilities and limitations of LLM-assisted coding, offering insights into the practical integration of artificial intelligence for simulating and analysing simple X-ray powder diffraction patterns.