Acta Crystallographica Section A: Foundations and Advances最新文献

Quartet invariants play a minor role in modern direct methods. In practice, only the quartets whose cosine is estimated to be negative are used, as they have no correlation with the triplet invariants. However, their role remains marginal: in fact, the quartet relations are of order 1/N while the triplet relations are of order 1/√N. The reliability of the quartets is therefore relatively low, in particular for the quartets estimated to be negative. Two papers have recently appeared (Papers I and II of this series) that describe procedures able to exploit the information contained in the Patterson map to estimate the triplet phases. The improvements in estimates are notable, apparently capable of resolving macromolecular structures even at non-atomic resolution. It therefore seems useful to develop a theory of quartet invariants that is able to exploit the Patterson information. This is the main purpose of this article. The method of joint probability distribution functions is used to obtain a von Mises-type distribution which associates a probability with each quartet phase. It is expected that the Patterson map, used as a priori information, can significantly increase the reliabilities of quartet invariants, particularly those whose cosine is estimated to be negative. The quartets may thus be able to play a more prominent role in future.

The IUCr Commission on Crystallographic Nomenclature (the CCN) has considered the question of whether fractional values of Z, the number of formula units in the unit cell, can be permitted. The consideration was initiated by a crystallographer who had determined a very unusual molecular structure, but the discussion was later broadened to cover all non-polymeric structures. Reliable structures having a fractional Z were located in the structural databases and in the literature, and were studied carefully. The CCN concluded that in some rare cases the Z value reported for a structure of a stoichiometric compound may be a fraction, but that that choice would have to be justified very carefully and convincingly. For solid solutions Z should be an integer.

We provide a systematic account using symmetry and coordinates to explore symmetric (vertex- and edge-transitive) 3-periodic weavings of piecewise linear threads. The dia-w graph, derived from the diamond structure, generates an infinite family of supersymmetric 3-periodic thread weaves - a unique property shared only with the 2-periodic sql-w family of fabric weaves. Additionally, we describe a selection of symmetric 3-periodic, 3-coordinated weavings related to the srs graph and a supersymmetric Borromean weave of hcb graphs.

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.

The phase-seeding method proposed by Carrozzini et al. [(2025), Acta Cryst. A81, 188-201] introduces a strategy for integrating artificial intelligence (AI) with established ab initio phasing techniques. Rather than presenting an AI-based phasing solution itself, the authors demonstrate how traditional crystallographic methods can be significantly enhanced if provided with a small subset of approximate phase values - a `phase seed' - that could, in principle, be generated by a machine learning model. By discretizing phase values into a few angular bins, the method transforms the continuous phase problem into a classification task, thereby reducing the computational burden on AI training. This hybrid approach shows promise for improving structure solution, particularly for large and complex non-centrosymmetric crystals, and opens a pathway for future AI-assisted crystallographic workflows.

As `2D' materials (i.e. materials just a few atoms thick) continue to gain prominence, understanding their symmetries is critical for unlocking their full potential. In this work, we present comprehensive tables that tabulate the rod group symmetries of all crystallographic lines in all 80 layer groups, which describe the symmetries of 2D materials. These tables are analogous to the scanning tables for space groups found in Volume E of the International Tables for Crystallography, but are specifically tailored for layer groups and their applications to 2D materials. This resource will aid in the analysis of line defects, such as domain walls, which play a crucial role in determining the properties and functionality of 2D materials.

A new theory for the probabilistic estimation of first-rank one-phase semi-invariants is presented. In this approach, atomic positions are treated as primitive random variables but are constrained by the a priori knowledge of interatomic vectors. This information is always available, thus allowing the new technique to be considered an ab initio probabilistic method conditioned by the knowledge of the Patterson map. The theoretical foundation for the estimation of triplet invariants was outlined in the first paper of this series [Giacovazzo (2019). Acta Cryst. A75, 142-157]. Subsequent experimental tests, shown in the second paper of this series [Burla et al. (2024). J. Appl. Cryst. 57, 1011-1022], have demonstrated the significant superiority of this new approach over existing methods. The improvements were so notable that it has been suggested this technique could be valuable for the ab initio solution of macromolecular structures. This work expands the probabilistic approach to include the estimation of first-rank one-phase semi-invariants, The hope is that they can contribute to the ab initio solution of macromolecular structures. Only in this way can one-phase semi-invariants go from being a historical curiosity to an effective tool for solving macromolecular structures.

The paper discusses the contradiction between the 7 band (on a cylinder with infinite radius) symmetry groups and the 5 uniaxial Curie limit symmetry groups. Logical difficulties in understanding the symmetry axis ∞ as a true crystallographic one are shown. The formula n → ∞ is proposed to be understood as if the order n of the axis becomes as large as desired, but retains the properties of a natural number (even, odd etc.). In this way, the true inversion axes of symmetry and one-to-one correspondence of bands and limit groups are restored. Such an analysis may be useful in teaching a university course in crystallography.

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 (Cu_1.95Se, Nb_1-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 Cu_1.95Se, 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 Nb_1-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.

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 2θ_B, 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.