Open-access and free articles in Acta Crystallographica Section A: Foundations and Advances
Acta Crystallographica Section A: Foundations and Advances covers theoretical and fundamental aspects of the structure of matter. The journal is the prime forum for research in diffraction physics and the theory of crystallographic structure determination by diffraction methods using X-rays, neutrons and electrons. The structures include periodic and aperiodic crystals, and non-periodic disordered materials, and the corresponding Bragg, satellite and diffuse scattering, thermal motion and symmetry aspects. Spatial resolutions range from the subatomic domain in charge-density studies to nanodimensional imperfections such as dislocations and twin walls. The chemistry encompasses metals, alloys, and inorganic, organic and biological materials. Structure prediction and properties such as the theory of phase transformations are also covered.
Copyright (c) 2025 International Union of Crystallography
Open-access and free articles in Acta Crystallographica Section A Foundations and Advances
Acta Crystallographica Section A Foundations and Advances
Periodic Platonic solids: the royal family of periodic nets
Among all the infinitely many 3D periodic networks you could draw in space, which ones are the most symmetric, in a strict, mathematically defined sense? This is the lattice analog of a very old question: if Platonic solids are the most symmetric finite polyhedra, what plays the same role for repeating structures in 3D?
Acta Crystallographica Section A: Foundations and Advances
On artificial crystal structure generation for solving the phase problem with deep learning
We discuss and present approaches for generating artificial crystal structures for training neural networks to solve the phase problem. Structure generation is considered as a two-step process involving sampling unit-cell parameters and filling the unit cell with atoms. The former step includes generating lattice basis vectors from randomly sampled unit-cell volume. Apart from randomly placing atoms, we use database data to guide fast and scalable generation of molecule-like fragments. The recently developed neural network PhAI is then used as a benchmark and retrained with various sets of training data to assess how the corresponding models perform on experimental crystal structure data. We found a significant improvement in PhAI retrained on a new kind of artificial data to generalize the phase problem solution for larger unit-cell structures.
PHASE PROBLEM; DEEP LEARNING; ARTIFICIAL CRYSTAL STRUCTURES
Computing the bridge length: the key ingredient in a continuous isometry classification of periodic point sets
The fundamental model of any periodic crystal is a periodic set of points at all atomic centres. Since crystal structures are determined in a rigid form, their strongest equivalence is rigid motion (composition of translations and rotations) or isometry (also including reflections). The recent classification of periodic point sets under rigid motion used a complete invariant isoset whose size essentially depends on the bridge length, defined as the minimum `jump' that suffices to connect any points in the given set. We propose a practical algorithm to compute the bridge length of any periodic point set given by a motif of points in a periodically translated unit cell. The algorithm has been tested on a large crystal dataset and is required for an efficient continuous classification of all periodic crystals. The exact computation of the bridge length is a key step to realizing the inverse design of materials from new invariant values.
PERIODIC POINT SETS; LABELLED QUOTIENT GRAPHS; ISOMETRY INVARIANT
Artificial intelligence in action: building simulation and analysis tools for powder diffraction
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.
X-RAY DIFFRACTION; ARTIFICIAL INTELLIGENCE; LLMS; LARGE LANGUAGE MODELS; MACHINE LEARNING; SOFTWARE DEVELOPMENT
Atomic pair distribution functions from textured polycrystalline samples: fundamentals
Equations for the reduced structure function and atomic pair distribution function (PDF) of a textured polycrystalline sample are formulated in terms of the orientational distribution function (ODF) and the structure function from a single crystallite. This ODF is sensitive to orientational distributions of inter­atomic vectors and to differentiate it from the crystallographic case we call it the bond orientational distribution function (BODF). This BODF may be obtained from experimental data when the structure of the reference crystallite is known. It can be applied to nanocrystalline and amorphous samples going beyond the information present in a conventional crystallographic texture study.
PAIR DISTRIBUTION FUNCTION; PDF; CRYSTALLOGRAPHIC TEXTURE; PREFERRED ORIENTATION; TOTAL SCATTERING; NANOSTRUCTURE DETERMINATION; ORIENTATION DISTRIBUTION FUNCTION; ODF; BOND ORIENTATION DISTRIBUTION FUNCTION; BODF
Machine learning for the classification of serial electron diffraction patterns: synthetic data
Serial electron crystallography faces a fundamental challenge due to the flat Ewald sphere resulting from the short electron wavelength, leading to limited 3D information in individual patterns. Recently, an algorithm for unit-cell determination from zonal electron diffraction patterns (GM algorithm) [Miehe (1997). Ber. Dtsch. Miner. Ges. Beih. z. Eur. J. Miner. 9, 250; Gorelik et al. (2025). Acta Cryst. A81, 124–136] was introduced in the context of serial electron crystallography. This algorithm requires the extraction of 2D zonal patterns from the complete serial dataset. Here, we present a machine learning approach for pattern sorting and apply it initially to simulated electron diffraction patterns.
SERIAL ELECTRON CRYSTALLOGRAPHY; MACHINE LEARNING
Direct derivation of anisotropic atomic displacement parameters from molecular dynamics simulations in extended solids with substitutional disorder using a neural network potential
Atomic displacement parameters (ADPs) are crystallographic information describing the statistical distribution of atoms around an atom site. Anisotropic ADPs by atom were directly derived from classical molecular dynamics (MD) simulations using a universal machine-learned potential. The (co)valences of atom positions were taken over recordings at different time steps in a single MD simulation. The procedure is demonstrated on extended solids, namely rocksalt structure MgO and three thermoelectric materials, Ag8SnSe6, Na2In2Sn4 and BaCu1.14In0.86P2. Unlike the very frequently used lattice dynamics approach, the MD approach can obtain ADPs in crystals with substitutional disorder and explicitly at finite temperature, but not under conditions where atoms migrate in the crystal. The calculated ADP approaches 0 when the temperature approaches 0, and the ADP is proportional to the temperature when the atom is in a harmonic potential and the sole contribution to the actual non-zero ADP is from the zero-point motion. The zero-point motion contribution can be estimated from the proportionality constant assuming this Einstein model. ADPs from MD simulations could act as a tool complementing experimental efforts to understand the crystal structure including the distribution of atoms around atom sites.
ANISOTROPIC ATOMIC DISPLACEMENT PARAMETERS; ADPS; NEURAL NETWORK POTENTIALS; MOLECULAR DYNAMICS SIMULATIONS; THERMOELECTRIC MATERIALS
Temporal ghost imaging for pump–probe X-ray solution scattering
Time-resolved small- and wide-angle X-ray scattering is a valuable tool for investigating biomolecular dynamics on a wide variety of timescales, without cryo-freezing or crystallization. However, some systems, such as the initial excitation of photo-active proteins, evolve dynamically on timescales that may be faster than the duration of the pump and probe beams. Data from a single pump–probe pulse pair therefore contain information from a mixture of time points. In this work, a simple algorithm is developed to recover the dynamics of solution scattering profiles. It leverages information about the pump and probe pulse beams' temporal profiles by using the same mathematical framework as ghost imaging [Pittman et al. (1995). Phys. Rev. A 52, R3429–R3432; Bennink et al. (2002). Phys. Rev. Lett. 89, 113601; Gatti et al. (2004). Phys. Rev. Lett. 93, 093602]. Results from several simulated data sets are presented.
SAXS; SMALL-ANGLE X-RAY SCATTERING; GHOST IMAGING; BAYESIAN; XFELS; X-RAY FREE ELECTRON LASERS; PUMP-PROBE
Crystal tensor properties of magnetic materials with and without spin–orbit coupling. Application of spin point groups as approximate symmetries
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.
SPIN POINT GROUPS; MAGNETIC POINT GROUPS; SYMMETRY-ADAPTED TENSOR FORMS; SPIN-ORBIT COUPLING
Benchmarking 3D-ΔPDF analysis using in-house X-ray sources
New detector technology has in recent years improved the data quality available from in-house X-ray diffractometers. A recent study compared high-resolution low-temperature X-ray diffraction data obtained from modern in-house diffractometers with synchrotron data in relation to extracting subtle electron-density details using the multipole model [Vosegaard et al. (2023). Acta Cryst. B79, 380–391]. It was concluded that for organic molecular crystals excellent agreement can be obtained, and only subtle electron-density details are better resolved at the synchrotron sources. This study aims to benchmark the quality of weak diffuse scattering data and three-dimensional difference pair-distribution function (3D-ΔPDF) analysis for in-house X-ray sources against more accurate and better resolved synchrotron data using three examples (Cu1.95Se, Nb1−xCoSb and InTe). Since the 3D-ΔPDF method is still relatively new in crystallographic research, we also provide a general description of the pipeline of analysis. The three selected systems highlight important differences in correlated disorder and the corresponding analysis. In all three cases, the synchrotron data have better signal-to-noise ratios and extend to higher scattering vectors. Using the in-house 3D-ΔPDF on Cu1.95Se, the same ordered 2D superstructure can be determined as for the synchrotron data, although additional arguments based on order within a 2D supercell or on ionic radii must be used to obtain an adequate model. For Nb1−xCoSb, the preference for vacancies to avoid each other and the size effect associated with structural relaxation of the lattice near vacancies can also be observed and assigned in the in-house 3D-ΔPDF. For InTe, the weak diffuse scattering, radial broadening and higher temperature than the original study mean that, although most of the important features are visible in the in-house data, some features are obscured, and the full correlated disorder model cannot be constructed. Overall, it is found that many of the conclusions derived from synchrotron data can also be extracted from in-house data, but in some cases additional postulates are needed, and in general subtle details may be too noisy to be properly interpreted in the in-house data.
3D-[DELTA]PDF; DIFFUSE SCATTERING; CORRELATED DISORDER; PAIR-DISTRIBUTION FUNCTION
A note on the relation of anisotropic peak broadening with lattice symmetry in powder diffraction
A bridge is established between the Gregorkiewitz & Boschetti [Acta Cryst. (2024), A80, 439–445] and Stephens [J. Appl. Cryst. (1999), 32, 281–289] formalisms of anisotropic peak broadening in powder diffraction. The paper by Gregorkiewitz & Boschetti presented formulas describing position shifts of low-symmetry peaks due to different lattice relaxation schemes. Anisotropic peak broadening caused by lattice relaxation can be parameterized by the variance of slightly dispersed peaks' positions. The calculated variances are compared with formulas from the widely used phenomenological model of anisotropic peak broadening by Stephens. Specific relations between anisotropic peak broadening parameters can be a hint of a possible unresolved peak splitting due to lattice symmetry lowering.
RIETVELD REFINEMENT; PSEUDOSYMMETRY; MICROSTRUCTURES
The phase-seeding method for solving non-centrosymmetric crystal structures: a challenge for artificial intelligence
The overall crystallographic process involves acquiring experimental data and using crystallographic software to find the structure solution. Unfortunately, while diffracted intensities can be measured, the corresponding phases – needed to determine atomic positions – remain experimentally inaccessible (phase problem). Direct methods and the Patterson approach have been successful in solving crystal structures but face limitations with large structures or low-resolution data. Current artificial intelligence (AI) based approaches, such as those recently developed by Larsen et al. [Science (2024), 385, 522–528], have been applied with success to solve centrosymmetric structures, where the phase is binary (0 or π). The current work proposes a new phasing method designed for AI integration, applicable also to non-centrosymmetric structures, where the phase is a continuous variable. The approach involves discretizing the initial phase values for non-centrosymmetric structures into a few distinct values (e.g. values corresponding to the four quadrants). This reduces the complex phase problem from a continuous regression task to a multi-class classification problem, where only a few phase seed values need to be determined. This discretization allows the use of a smaller training dataset for deep learning models, reducing computational complexity. Our feasibility study results show that this method can effectively solve small, medium and large structures, with the minimum percentage of phase seeds (three or four points in the interval [0, 2π]), and 10% to 30% of seed symmetry-independent reflections. This phase-seeding method has the potential to extend AI-based approaches to solve crystal structures ab initio, regardless of complexity or symmetry, by combining AI classification algorithms with classical phasing procedures.
CRYSTAL STRUCTURE SOLUTION; CRYSTALLOGRAPHIC METHODS; PHASE SEEDING; ARTIFICIAL INTELLIGENCE
Uncertainties of recalculated bond lengths, angles and polyhedral volumes as implemented in the Crystal Palace program for parametric crystal structure analysis
Crystal Palace is a new Windows program for Parametric Analysis of Least-squares and Atomic Coordination with Estimated standard uncertainties (e.s.u.'s). The primary purpose of the program is to organize the refined structures from parametric structural studies (as a function of pressure or temperature or a series of compositions) for analysis of the structural trends, and the production of tables for publication without the risks associated with manual editing. The program reads structural information from one or more crystallographic information format (cif) files. It organizes the data by finding the structurally equivalent atoms in each structure and therefore can correctly organize structural information even if atom names or site occupancies are different, or the atom lists in the cif files are ordered differently. A major shortcoming of cif files as currently used is that they do not contain the full variance–covariance matrix from the structure refinement, but only the uncertainties of the individual positional parameters. Without the covariance of positional parameters, the e.s.u.'s of bond lengths and angles cannot be determined. Crystal Palace uses symmetry to estimate the major contributions to the covariance of atomic coordinates and thus realistic uncertainties of bond lengths, angles and polyhedral volumes. Crystal Palace also calculates various polyhedral distortion parameters and rigid-body corrections to bond lengths.
PARAMETRIC DATA ANALYSIS; BOND LENGTHS; POLYHEDRAL VOLUMES; COVARIANCE; SOFTWARE; CRYSTAL PALACE
Topological coordination numbers and coordination reciprocity from electron-density distributions
Triangulated surface data sets of quantum theory of atoms in molecules (QTAIM) interatomic surfaces have been employed to calculate solid angles subtended at the nuclear positions by each diatomic contact surface. On this basis, topological effective coordination numbers were evaluated. This corresponds to a generalization of the established Voronoi–Dirichlet partitioning (VDP) based procedure. The topological coordination number (tCN) approach developed includes coordination reciprocity requirements necessary to extract coordination-consistent sub-coordination scenarios for identification of chemically meaningful coordination numbers. The ranking between different sub-coordination scenarios is accomplished by weighting functions derived from purely geometrical properties of square and semicircle areas. Exemplary cases analyzed using theoretical electron-density distributions span the range from the face centered cubic, body centered cubic, hexagonal close packed and diamond types of element structures, to rocksalt, CsCl and zincblende types of structures, to compounds of the TiNiSi structure type. An important difference compared with VDP-based coordination numbers arises from the natural inclusion of the effect of different atomic sizes in the tCN approach. Even in highly symmetrical element structures, differences between VDP and tCN results are obtained as an effect of atomic electron-density decay utilizing still available degrees of freedom in the crystal structure. Especially in the TiNiSi type of examples, the advantage of numerically ranking between different sub-coordination scenarios of similar importance emerges. Instead of being obliged to choose only one of them, a more precise characterization contains a listing of different scenarios with their relative weights and associated effective coordination numbers. This seems to be generally the more appropriate way to analyze atomic coordination, especially in more complex structures such as intermetallic phases, opening up its possible use as input for AI applications on structure–property relationships.
COORDINATION NUMBERS; QTAIM; QUANTUM THEORY OF ATOMS IN MOLECULES; ELECTRON DENSITY; VORONOI-DIRICHLET
Homophase bilayers: more than just the sum of their monolayers
Twisted homophase bilayers, stacks of two rotated monolayers such as graphene, exhibit remarkable physical properties absent in their constituent monolayers. The structure of bilayer systems is dominated by a moiré effect and critically depends on the twist angle. Quiquandon & Gratias [Acta Cryst. (2025), A81, 94–106] develop a crystallographic framework for rigorous description of the structure of bilayers, including systems without a coincidence lattice. They offer a set of tools that can describe the structure of any arbitrary bilayer system and enable the connection with its physical properties.
CRYSTALLOGRAPHY; GRAPHENE; BILAYERS; MOIRE PATTERNS; 0-LATTICE; QUASIPERIODICITY
Honeycombs – their variety, topology and symmetry
The double-layer honeycomb with hexagonal cells, three rhombic faces between the two layers and p3m1 layer space-group symmetry, used universally by honeybees, is often considered to be the most efficient (from the point of view of wax economy) and the only honeycomb manufactured by bees. However, another variant of a symmetric and periodic double-layer hexagonal honeycomb with two hexagons and two rhombi between the two layers and slightly better wax economy was discovered theoretically in 1964 by Fejes Tóth and found in nature some years later. The present work shows that there is yet another possibility, with the interface formed by one hexagon and two quadrangles, in addition to the trivial case with flat hexagonal cell bottoms and very poor wax economy. Moreover, we demonstrate that the geometry of the Fejes Tóth honeycomb can be optimized for even better wax economy. All the theoretical honeycomb types are derived using the principle of Dirichlet-domain construction and shown to have more and less symmetric variants. Wax economy is calculated for each case, confirming that indeed the modified Fejes Tóth honeycomb is the most efficient, while the trivial flat-bottom case is the least.
HONEYCOMBS; WAX ECONOMY; LAYER SYMMETRY; DIRICHLET DOMAINS
Refuting `a new theory for X-ray diffraction' – a reciprocal-space approach
Some ten years ago, Fewster proposed `a new theory for X-ray diffraction' in order to explain the completeness of powder diffraction patterns from samples with very few crystals, claiming to find extra intensity at Bragg scattering angles 2θB, even when a grain was not oriented in the Bragg condition, and claiming this to be a new approach to X-ray scattering [Fewster (2014). Acta Cryst. A70, 257–282]. Fraser & Wark [Acta Cryst. (2018), A74, 447–456] gave a detailed account of the errors and issues in the approach by Fewster, but the situation appears to be still undecided. To address this issue, we use a different perspective, based on conventional scattering theory and using a simpler description in reciprocal space, rather than the angular space used by Fewster and by Fraser & Wark. This allows us to focus on the crucial conceptual errors in the proposed theory. We show that Fewster is in fact not proposing a new theory, but finds effects that disagree with conventional theory because of errors in the path length calculation. We also discuss extensively the effect of residual intensity in reciprocal space, away from the Bragg peaks, and caused by the termination of crystals. We show that the residual intensity has no significant effect on the intensity of typical powder diffraction patterns. We hope that, with this account, we can put the discussion about the new theory to rest, along with the theory itself.
X-RAY DIFFRACTION; SCATTERING THEORY; CRYSTAL TERMINATION
Deconvoluting thermomechanical effects in X-ray diffraction data using machine learning
X-ray diffraction is ideal for probing the sub-surface state during complex or rapid thermomechanical loading of crystalline materials. However, challenges arise as the size of diffraction volumes increases due to spatial broadening and because of the inability to deconvolute the effects of different lattice deformation mechanisms. Here, we present a novel approach that uses combinations of physics-based modeling and machine learning to deconvolve thermal and mechanical elastic strains for diffraction data analysis. The method builds on a previous effort to extract thermal strain distribution information from diffraction data. The new approach is applied to extract the evolution of the thermo­mechanical state during laser melting of an Inconel 625 wall specimen which produces significant residual stress upon cooling. A combination of heat transfer and fluid flow, elasto-plasticity and X-ray diffraction simulations is used to generate training data for machine-learning (Gaussian process regression, GPR) models that map diffracted intensity distributions to underlying thermomechanical strain fields. First-principles density functional theory is used to determine accurate temperature-dependent thermal expansion and elastic stiffness used for elasto-plasticity modeling. The trained GPR models are found to be capable of deconvoluting the effects of thermal and mechanical strains, in addition to providing information about underlying strain distributions, even from complex diffraction patterns with irregularly shaped peaks.
STRAIN; SYNCHROTRON X-RAY DIFFRACTION; FIRST-PRINCIPLES CALCULATIONS; STRESS; GAUSSIAN PROCESS REGRESSION; SUPERALLOYS; MACHINE LEARNING; PHYSICS-BASED MODELING
Unit-cell parameters determination from a set of independent electron diffraction zonal patterns
Due to the short de Broglie wavelength of electrons compared with X-rays, the curvature of their Ewald sphere is low, and individual electron diffraction patterns are nearly flat in reciprocal space. As a result, a reliable unit-cell determination from a set of randomly oriented electron diffraction patterns, an essential step in serial electron diffraction, becomes a non-trivial task. Here we describe an algorithm for unit-cell determination from a set of independent electron diffraction patterns, as implemented in the program PIEP (Program for Interpreting Electron diffraction Patterns), written in the early 1990s. We evaluate the performance of the algorithm by unit-cell determination of two known structures – copper perchlorophthalocyanine (CuPcCl16) and lysozyme, challenging the algorithm by high-index zone patterns and long crystallographic axes. Finally, we apply the procedure to a new, structurally uncharacterized five amino acid peptide.
ELECTRON CRYSTALLOGRAPHY; ELECTRON DIFFRACTION; 3D ED; MICROED; SERIAL ELECTRON DIFFRACTION; UNIT-CELL PARAMETERS DETERMINATION
A physical optics formulation of Bloch waves and its application to 4D STEM, 3D ED and inelastic scattering simulations
Bloch waves are often used in dynamical diffraction calculations, such as simulating electron diffraction intensities for crystal structure refinement. However, this approach relies on matrix diagonalization and is therefore computationally expensive for large unit cell crystals. Here Bloch wave theory is re-formulated using the physical optics concepts underpinning the multislice method. In particular, the multislice phase grating and propagator functions are expressed in matrix form using elements of the Bloch wave structure matrix. The specimen is divided into thin slices, and the evolution of the electron wavefunction through the specimen calculated using the Bloch phase grating and propagator matrices. By decoupling specimen scattering from free space propagation of the electron beam, many computationally demanding simulations, such as 4D STEM imaging modes, 3D ED precession and rotation electron diffraction, phonon and plasmon inelastic scattering, are considerably simplified. The computational cost scales as {\cal O}({N^2} ) per slice, compared with {\cal O}({N^3} ) for a standard Bloch wave calculation, where N is the number of diffracted beams. For perfect crystals the performance can at times be better than multislice, since only the important Bragg reflections in the otherwise sparse diffraction plane are calculated. The physical optics formulation of Bloch waves is therefore an important step towards more routine dynamical diffraction simulation of large data sets.
BLOCH WAVES; MULTISLICE; 4D STEM SIMULATION; PRECESSION ELECTRON DIFFRACTION; PHONON SCATTERING; PLASMON SCATTERING; INELASTIC SCATTERING