Acta Crystallographica Section A: Foundations and Advances最新文献

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

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

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

The kisrhombille tiling is the dual tessellation of one of the semi-regular tessellations. It consists of right-angled triangle tiles with 12 different orientations. An adequate coordinate system for the tiles of the grid has been defined that allows a formal description of the grid. In this paper, two tiles are considered to be neighbors if they share at least one point in their boundary. Paths are sequences of tiles such that any two consecutive tiles are neighbors. The digital distance is defined as the minimum number of steps in a path between the tiles, and the distance formula is proven through constructing minimum paths. In fact, the distance between triangles is almost twice the hexagonal distance of their embedding hexagons.

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

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

Extended homometry is a phenomenon in which distinct structures have the same X-ray diffraction (XRD) intensities, which may lead to incorrect results of structural analysis based on XRD methods. It is proposed and proved herein that half of a crystallographic orbit has the same powder X-ray diffraction intensity as its complementary set; three more theorems are deduced. These results are conducive to understanding the formation of extended homometric structures. Also analyzed are some reported or potential homometric or weakly homometric structures in the Inorganic Crystal Structure Database to confirm the theorems. This work presents a quick approach to analyze and construct extended homometric structures based on crystallographic orbits.

In electron diffraction, thermal atomic motion produces incoherent scattering over a relatively wide angular range, which appears as a diffuse background that is usually subtracted from measurements of Bragg spot intensities in structure solution methods. The transfer of electron flux from Bragg spots to diffuse scatter is modelled using complex scattering factors f + if' in the Bloch wave methodology. In a two-beam Einstein model the imaginary `absorptive' scattering factor f' can be obtained by the evaluation of an integral containing f over all possible scattering angles. While more sophisticated models of diffuse scatter are widely used in the electron microscopy community, it is argued in this paper that this simple model is appropriate for current structure solution and refinement methods. The two-beam model is a straightforward numerical calculation, but even this simplistic approach can become time consuming for simulations of materials with large numbers of atoms in the unit cell and/or many incident beam orientations. Here, a parameterized form of f' is provided for 103 elements as neutral, spherical atoms that reduces calculation time considerably.

A novel automated high-throughput screening approach, ClusterFinder, is reported for finding candidate structures for atomic pair distribution function (PDF) structural refinements. Finding starting models for PDF refinements is notoriously difficult when the PDF originates from nanoclusters or small nanoparticles. The reported ClusterFinder algorithm can screen 10⁴ to 10⁵ candidate structures from structural databases such as the Inorganic Crystal Structure Database (ICSD) in minutes, using the crystal structures as templates in which it looks for atomic clusters that result in a PDF similar to the target measured PDF. The algorithm returns a rank-ordered list of clusters for further assessment by the user. The algorithm has performed well for simulated and measured PDFs of metal-oxido clusters such as Keggin clusters. This is therefore a powerful approach to finding structural cluster candidates in a modelling campaign for PDFs of nanoparticles and nanoclusters.

The MIF (multiple implication function) group symmetry was assigned to all 230 space groups. Knowledge of MIF symmetry allows the calculation of an asymmetric unit. A more accurate procedure for calculating MIFs has been developed. Extensive tables of MIF symmetry and asymmetric units were computer generated. The development of implication theory for crystal structure determination seems to have reached completion.