Acta Crystallographica Section A: Foundations and Advances最新文献

Group-theoretical and linear-algebraic methods and tools have recently been developed that aim to exhaustively identify the small-angle rotational rigid-unit modes (RUMs) of a given framework material. But in their current form, they fail to detect RUMs that require a compensating lattice strain which grows linearly with the amplitude of the rigid-unit rotations. Here, we present a systematic approach to including linear strain compensation within the linear-algebraic RUM-search method, so that any geometrically possible small-angle RUM can be detected.

This work analyzes the rules governing the growth of the numbers of vertices, edges and faces in all possible periodic tessellations of the 2D Euclidean space, and encodes those rules in several types of polynomial growth functions. These encodings map the geometric, combinatorial and topological properties of the tessellations into sets of integer coefficients. Several general statements about these encodings are given with rigorous mathematical proof. The variation of the growth functions is represented graphically and analyzed in orphic diagrams, so named because of their similarity to orphic art. Several examples of 3D space groups are included, to emphasize the complexity of the growth functions in higher dimensions. A freely available Python library is presented to facilitate the discovery of the growth functions and the generation of orphic diagrams.

Ladder graphs admit a maximum-symmetry embedding in which edges coincide. In folding ladders, there are no zero-length edges. We give examples of high-symmetry 3-periodic ladders, particularly emphasizing the structures of 3-periodic vertex- and edge-transitive folding ladders. For these, the coincident-edge configuration is one of maximum volume for fixed edge length and has the same coordinates as (is isomeghethic to) a higher-symmetry 3-periodic graph.

The general equation δ_M(r) = ρ(r) + g(r) of the δ direct methods (δ-GEQ) is established which, when expressed in the form δ_M(r) - ρ(r) = g(r), is used in the SMAR phasing algorithm [Rius (2020). Acta Cryst A76, 489-493]. It is shown that SMAR is based on the alternating minimization of the two residuals R_ρ(χ) = ∫_V [ρ(χ) - ρ(Φ)s_ρ]² dV and R_δ(Φ) = ∫_V m_ρ[δ_M(χ) - ρ(Φ)s_ρ]² dV in each iteration of the algorithm by maximizing the respective S_ρ(Φ) and S_δ(Φ) sum functions. While R_ρ(χ) converges to zero, R_δ(Φ) converges, as predicted by the theory, to a positive quantity. These two independent residuals combine δ_M and ρ each with |ρ| while keeping the same unknowns, leading to overdetermination for diffraction data extending to atomic resolution. At the beginning of a SMAR phase refinement, the zero part of the m_ρ mask [resulting from the zero conversion of the slightly negative ρ(Φ) values] occupies ∼50% of the unit-cell volume and increases by ∼5% when convergence is reached. The effects on the residuals of the two SMAR phase refinement modes, i.e. only using density functions (slow mode) supplemented by atomic constraints (fast mode), are discussed in detail. Due to its architecture, the SMAR algorithm is particularly well suited for Deep Learning. Another way of using δ-GEQ is by solving it in the form ρ(r) = δ_M(r) - g(r), which provides a simple new derivation of the already known δ_M tangent formula, the core of the δ recycling phasing algorithm [Rius (2012). Acta Cryst. A68, 399-400]. The nomenclature used here is: (i) Φ is the set of φ structure factor phases of ρ to be refined; (ii) δ_M(χ) = FT^-1{c(|E| - 〈|E|〉)×exp(iα)} with χ = {α}, the set of phases of |ρ| and c = scaling constant; (iii) m_ρ = mask, being either 0 or 1; s_ρ is 1 or -1 depending on whether ρ(Φ) is positive or negative.

In this paper, we report on the full classification of generic iso-edge subdivisions of six-dimensional translational lattices. We obtain a complete list of 55083357 affine types of iso-edge subdivisions. We report on the use of the method of canonical forms that allows us to apply hashing techniques used in modern databases.

The complete classification of (primitive, generic) parallelohedra in a given dimension is a challenging computational task. Nearly 50 years have passed since the classification for the last dimension, n = 5, was completed. One application of such a classification is in solving the lattice sphere covering problem for the corresponding dimension. The paper by Dutour Sikirić & van Woerden [Acta Cryst. (2025), A81, https://doi.org/10.1107/S2053273324010143] marks a milestone in the classification effort for dimension n = 6. It provides a complete classification of all primitive iso-edge domains; here primitive parallelohedra are identified based on their facet vectors.

FFLUX is a multipolar machine-learned force field that uses Gaussian process regression models trained on data from quantum chemical topology calculations. It offers an efficient way of predicting both lattice and free energies of polymorphs, allowing their stability to be assessed at finite temperatures. Here the Ih, II and XV phases of ice are studied, building on previous work on formamide crystals and liquid water. A Gaussian process regression model of the water monomer was trained, achieving sub-kJ mol^-1 accuracy. The model was then employed in simulations with a Lennard-Jones potential to represent intermolecular repulsion and dispersion. Lattice constants of the FFLUX-optimized crystal structures were comparable with those calculated by PBE+D3, with FFLUX calculations estimated to be 10³-10⁵ times faster. Lattice dynamics calculations were performed on each phase, with ices Ih and XV found to be dynamically stable through phonon dispersion curves. However, ice II was incorrectly identified as unstable due to the non-bonded potential used, with a new phase (labelled here as II' and to our knowledge not found experimentally) identified as more stable. This new phase was also found to be dynamically stable using density functional theory but, unlike in FFLUX calculations, II remained the more stable phase. Finally, Gibbs free energies were accessed through the quasi-harmonic approximation for the first time using FFLUX, allowing thermodynamic stability to be assessed at different temperatures and pressures through the construction of a phase diagram.

In elastic crystals, a hyperelastic description is conventionally assumed, and the strain energy potential is idealized as a Taylor-series expansion in strain about an unstrained reference state. Coefficients of quadratic terms are second-order or linear elastic constants. Coefficients of higher-order terms are elastic constants of third order, fourth order, and so on. Recently published work by Telyatnik [Acta Cryst. (2024), A80, 394-404] extends prior knowledge of symmetry properties for anisotropic elastic constants of single crystals, as well as transversely isotropic and isotropic solids, to terms up to sixth order. Effective elastic constants for polycrystalline aggregates, with possible anisotropy, were reported by Telyatnik, in the same article, to the same order. A terse summary of nonlinear crystal elasticity and independent elastic constants of orders two and three are given in this commentary for context. Methods and results of Telyatnik, anticipated to be of great utility to crystal elasticity research, are then highlighted.

In the kinematical approximation, new analytical solutions are obtained that describe the diffraction of a restricted X-ray beam from a thin crystal. Calculation of the angular distribution of reflected X-ray beams within the framework of the developed approach significantly reduces the computational cost compared with numerical methods. For a thin silicon crystal, X-ray reciprocal-space mapping was simulated using analytical solutions, as well as calculated using numerical methods based on 2D recurrence relations and the Takagi-Taupin equations.

This study introduces an alternative method to the Takagi-Taupin equations for investigating the dark-field X-ray microscopy (DFXM) of deformed crystals. In scenarios where dynamical diffraction cannot be disregarded, it is essential to assess the potential inaccuracies of data interpretation based on the kinematic diffraction theory. Unlike the Takagi-Taupin equations, this new method utilizes an exact dispersion relation, and a previously developed finite difference scheme with minor modifications is used for the numerical implementation. The numerical implementation has been validated by calculating the diffraction of a diamond crystal with three components, wherein dynamical diffraction is applicable to the first component and kinematic diffraction pertains to the remaining two. The numerical convergence is tested using diffraction intensities. In addition, the DFXM image of a diamond crystal containing a stacking fault is calculated using the new method and compared with the experimental result. The new method is also applied to calculate the DFXM image of a twisted diamond crystal, which clearly shows a result different from those obtained using the Takagi-Taupin equations.