Acta Crystallographica Section A: Foundations and Advances最新文献

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.

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 BaCu_1.14In_0.86P₂. 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.

Understanding structure-property relationships is essential for advancing technologies based on thin films. X-ray pair distribution function (PDF) analysis can access relevant atomic structure details spanning local-, mid- and long-range structure. While X-ray PDF has been adapted for thin films on amorphous substrates, measurements on single-crystal substrates are necessary to accurately determine structure origins for some thin film materials, especially those for which the substrate changes the accessible structure and properties. However, when measuring films on single-crystal substrates, high-intensity anisotropic Bragg spots saturate 2D detector images, overshadowing the thin films' isotropic scattering signal. This renders previous data processing methods for films on amorphous substrates unsuitable for films on single-crystal substrates. To address this measurement need, we developed IsoDAT2D, an innovative data processing approach using unsupervised machine learning algorithms. The program combines dimensionality reduction and clustering algorithms to separate thin film and single-crystal substrate X-ray scattering signals. We use SimDAT2D, a program we developed to generate simulated thin film data, to validate IsoDAT2D. We also use IsoDAT2D to isolate X-ray total scattering signal from a thin film on a single-crystal substrate. The resulting PDF data are compared with similar data processed using previous methods, especially substrate subtraction for single-crystal and amorphous substrates. PDF data from IsoDAT2D-identified X-ray total scattering data are significantly better than from single-crystal substrate subtraction, but not as reliable as PDF data from amorphous substrate subtraction. With IsoDAT2D, there are new opportunities to expand PDF to a wider variety of thin films, including those on single-crystal substrates, with which new structure-property relationships can be elucidated to enable fundamental understanding and technological advances.

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.

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.

In a 4D polytope {3, 3, 5}, a 40-vertex toroidal helix is selected that unites the vertices of two orbits of the axis 20/9 with the angle of rotation 9 × 360°/20 = 162°. Symmetrization of this helix allows one to select in the 3D spherical space a helix {40/11} with the angle of rotation of 99°. Its mapping into the 3D Euclidean space E³ determines the helix {40/11}, which coincides with the helix of atoms C_α in the α-helix. A tube polytope with the symmetry group ±[O×D₂₀] contains a toroidal helix {40/11}, constructed of 40 prismatic cells. The symmetry of the polytope, as well as the partition it induces on the lateral face of the prismatic cell, allow one to find additional vertices that do not belong to the polytope. Putting the vertices of the helix {40/11} in correspondence with the atoms C_α and the additional vertices with the atoms O, C', N, H, determines the peptide plane of the α-helix; its multiplication by the axis 40/11 leads to a polytope model of the α-helix. A radial contraction of the polytope model, with subsequent mapping into E³, leads to its densely packed structural realization - the α-helix that is universal in proteins. A polytope with the group of symmetry ±[O×D₂₀] arises in the family of tube polytopes with the starting group ±1/2[O×C_2n] at n = 5. Along with the axis 40/11 of a single α-helix, the screw axes of this family of polytopes determine the axes 7/2, 11/3, 15/4, 18/5 realized as the axes of the α-helices included in superhelices.

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.

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.

In memory of George Sheldrick.

The report of the Executive Committee for 2023 is presented.