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.

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(r) = ρ(r) + g(r) of the δ direct methods (δ-GEQ) is established which, when expressed in the form δ(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(χ) = ∫ [ρ(χ) - ρ(Φ)s] dV and R(Φ) = ∫ m[δ(χ) - ρ(Φ)s] dV in each iteration of the algorithm by maximizing the respective S(Φ) and S(Φ) sum functions. While R(χ) converges to zero, R(Φ) converges, as predicted by the theory, to a positive quantity. These two independent residuals combine δ 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 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(r) - g(r), which provides a simple new derivation of the already known δ 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(χ) = FT{c(|E| - 〈|E|〉)×exp(iα)} with χ = {α}, the set of phases of |ρ| and c = scaling constant; (iii) m = mask, being either 0 or 1; s is 1 or -1 depending on whether ρ(Φ) is positive or negative.

A 6D structure model for face-centred icosahedral quasicrystals consisting of so-called pseudo-Mackay and mini-Bergman-type atomic clusters is proposed based on the structure model of the AlPdCrFe 3/2 cubic approximant crystal (with space group Pa3, a = 40.5 Å) [Fujita et al. (2013). Acta Cryst. A69, 322-340]. The cluster centres form an icosahedral close sphere packing generated by the occupation domains similar to those in the model proposed by Katz & Gratias [J. Non-Cryst. Solids (1993), 153-154, 187-195], but their size is smaller by a factor τ [τ = (1 + (5))/2]. The clusters cover approximately 99.46% of the atomic structure, and the cluster arrangement exhibits 15 and 19 different local configurations, respectively, for the pseudo-Mackay and mini-Bergman-type clusters. The occupation domains that generate cluster shells are modelled and discussed in terms of structural disorder and local reorganization of the cluster arrangements (phason flip).

In powder diffraction, lattice symmetry relaxation causes a peak to split into several components which are not resolved if the degree of desymmetrization is small (pseudosymmetry). Here the equations which rule peak splitting are elaborated for the six minimal symmetry transitions, showing that the resulting split peaks are generally broader and asymmetric, and suffer an hkl-dependent displacement with respect to the high-symmetry parent peak. These results will be of help in Rietveld refinement of pseudosymmetric structures where an exact interpretation of peak deformation is required.

This article describes the diffraction pattern (2-periodic Fourier transform) from the vertices of a large patch of the recently discovered `Spectre' tiling - a strictly chiral aperiodic monotile. It was reported recently that the diffraction pattern of the related weakly chiral aperiodic `Hat' monotile was 2-periodic with chiral plane-group symmetry p6 [Kaplan et al. (2024). Acta Cryst. A80, 72-78]. The diffraction periodicity arises because the Hat tiling is a systematic aperiodic deletion of vertices from the 2-periodic hexagonal mta tiling. Despite the similarity of the Hat and Spectre tilings, the Spectre tiling is not aligned with a 2-periodic lattice, and its diffraction pattern is non-periodic with chiral point symmetry 6 about the origin.

The crystal structure uniformity is numerically estimated as the standard deviation of the crystal space quantizer 〈G〉. This criterion has been applied to explore the uniformity of ionic sublattices in 21465 crystal structures of inorganic ionic compounds. In most cases, at least one kind of sublattice (whole ionic lattice, cationic or anionic sublattice) was found to be highly uniform with a small 〈G〉 value. Non-uniform structures appeared to be either erroneous or essentially non-ionic. As a result, a set of uniformity criteria is proposed for the estimation of the stability of ionic crystal structures.

Algebraic expressions for averaging linear and nonlinear stiffness tensors from general anisotropy to different effective symmetries (11 Laue classes elastically representing all 32 crystal classes, and two non-crystalline symmetries: isotropic and cylindrical) have been derived by automatic symbolic computations of the arithmetic mean over the set of rotational transforms determining a given symmetry. This approach generalizes the Voigt average to nonlinear constants and desired approximate symmetries other than isotropic, which can be useful for a description of textured polycrystals and rocks preserving some symmetry aspects. Low-symmetry averages have been used to derive averages of higher symmetry to speed up computations. Relationships between the elastic constants of each symmetry have been deduced from their corresponding averages by resolving the rank-deficient system of linear equations. Isotropy has also been considered in terms of generalized Lamé constants. The results are published in the form of appendices in the supporting information for this article and have been deposited in the Mendeley database.

In electron microscopy the principle of reciprocity is often used to imply time reversal symmetry. While this is true for elastic scattering, its applicability to inelastic scattering is less well established. From the second law of thermodynamics, the entropy for a thermally isolated system must be constant for any reversible process. Using entropy and statistical fluctuation arguments, it is shown that, while reversibility is possible at the microscopic level, it becomes statistically less likely for higher energy transfers. The implications for reciprocal imaging modes, including energy loss and energy gain measurements, as well as Kainuma's reciprocal wave model are also discussed.

IUCr Journals subscription prices for 2025.

Small-angle X-ray scattering (SAXS) is a widely used method for nanoparticle characterization. A common approach to analysing nanoparticles in solution by SAXS involves fitting the curve using a parametric model that relates real-space parameters, such as nanoparticle size and electron density, to intensity values in reciprocal space. Selecting the optimal model is a crucial step in terms of analysis quality and can be time-consuming and complex. Several studies have proposed effective methods, based on machine learning, to automate the model selection step. Deploying these methods in software intended for both researchers and industry raises several issues. The diversity of SAXS instrumentation requires assessment of the robustness of these methods on data from various machine configurations, involving significant variations in the q-space ranges and highly variable signal-to-noise ratios (SNR) from one data set to another. In the case of laboratory instrumentation, data acquisition can be time-consuming and there is no universal criterion for defining an optimal acquisition time. This paper presents an approach that revisits the nanoparticle model selection method proposed by Monge et al. [Acta Cryst. (2024), A80, 202-212], evaluating and enhancing its robustness on data from device configurations not seen during training, by expanding the data set used for training. The influence of SNR on predictor robustness is then assessed, improved, and used to propose a stopping criterion for optimizing the trade-off between exposure time and data quality.

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.

The atomic pair distribution function (PDF) is a real-space representation of the structure of a material. Experimental PDFs are obtained using a Fourier transform from total scattering data which may or may not have Bragg diffraction peaks. The determination of Bragg peak resolution in scattering data from the fundamental physical parameters of the diffractometer used is well established, but after the Fourier transform from reciprocal to direct space, these contributions are harder to identify. Starting from an existing definition of the resolution function of large-area detectors for X-ray diffraction, this approach is expanded into direct space. The effect of instrumental parameters on PDF peak resolution is developed mathematically, then studied with modelling and comparison with experimental PDFs of LaB from measurements made in different-sized capillaries.

The superstructure spots that appear in diffraction patterns of tilted perovskites are well documented and easily calculated using crystallographic software. Here, by considering a distortion mode as a perturbation of the prototype perovskite structure, it is shown how the structure-factor equation yields Boolean conditions for the presence of superstructure reflections. This approach may have some advantages for the analysis of electron diffraction patterns of perovskites.

An edge-n-coloring of a uniform tiling {\cal T} is uniform if for any two vertices of {\cal T} there is a symmetry of {\cal T} that preserves the colors of the edges and maps one vertex onto the other. This paper gives a method based on group theory and color symmetry theory to arrive at uniform edge-n-colorings of uniform tilings. The method is applied to give a complete enumeration of uniform edge-n-colorings of the uniform tilings of the Euclidean plane, for which the results point to a total of 114 colorings, n = 1, 2, 3, 4, 5. Examples of uniform edge-n-colorings of tilings in the hyperbolic plane and two-dimensional sphere are also presented.

Neutron time-of-flight transmission spectra of mosaic crystals contain Bragg dips, i.e., minima at wavelengths corresponding to diffraction reflections. The positions of the dips are used for investigating crystal lattices. By rotating the sample around a fixed axis and recording a spectrum at each rotation step, the intensity of the transmitted beam is obtained as a function of the rotation angle and wavelength. The questions addressed in this article concern the determination of lattice parameters and orientations of centrosymmetric crystals from such data. It is shown that if the axis of sample rotation is inclined to the beam direction, the reflection positions unambiguously determine reciprocal-lattice vectors, which is not the case when the axis is perpendicular to the beam. Having a set of such vectors, one can compute the crystal orientation or lattice parameters using existing indexing software. The considerations are applicable to arbitrary Laue symmetry. The work contributes to the automation of the analysis of diffraction data obtained in the neutron imaging mode.

Obituary for Dieter Schwarzenbach.

Fundamental equations describing the X-ray and electron diffraction scattering in imperfect crystals have been derived in the form of the matrix Fredholm-Volterra integral equation of the second kind. A theoretical approach has been developed using the perfect-crystal Green function formalism. In contrast, another approach utilizes the wavefield eigenfunctions related to the diagonalized matrix propagators of the conventional Takagi-Taupin and Howie-Whelan equations. Using the Liouville-Neumann-type series formalism for building up the matrix Fredholm-Volterra integral equation solutions, the general resolvent function solutions of the X-ray and electron diffraction boundary-valued Cauchy problems have been obtained. Based on the resolvent-type solutions, the aim is to reveal the features of the diffraction scattering onto the crystal lattice defects, including the mechanisms of intra- and interbranch wave scattering in the strongly deformed regions in the vicinity of crystal lattice defect cores. Using the two-stage resolvent solution of the second order, this approach has been supported by straightforward calculation of the electron bright- and dark-field contrasts of an edge dislocation in a thick foil. The results obtained for the bright- and dark-field profiles of the edge dislocation are discussed and compared with analogous ones numerically calculated by Howie & Whelan [Proc. R. Soc. A (1962), 267, 206].

Analytical calculations of absorption corrections for X-ray powder diffraction experiments on non-ideal samples with surface roughness, porosity or absorption contrasts from multiple phases require complex mathematical models to represent their material distribution. In a computational approach to this problem, a practicable ray-tracing algorithm is formulated which is capable of simulating angle-dependent absorption corrections in reflection geometry for any given rasterized sample model. Single or multiphase systems with arbitrary surface roughness, porosity and spatial distribution of the phases in any combination can be modeled on a voxel grid by assigning respective values to each voxel. The absorption corrections are calculated by tracing the attenuation of X-rays along their individual paths via a modified shear-warp algorithm. The algorithm is presented in detail and the results of simulated absorption corrections on samples with various surface modulations are discussed in the context of published experimental results.

In ab initio indexing, for a given diffraction/scattering pattern, the unit-cell parameters and the Miller indices assigned to reflections in the pattern are determined simultaneously. `Ab initio' means a process performed without any good prior information on the crystal lattice. Newly developed ab initio indexing software is frequently reported in crystallography. However, it is not widely recognized that use of a Bravais lattice determination method, which is tolerant of experimental errors, can simplify indexing algorithms and increase their success rates. One of the goals of this article is to collect information on the lattice-basis reduction theory and its applications. The main result is a Bravais lattice determination algorithm for 2D lattices, along with a mathematical proof that it works even for parameters containing large observational errors. It uses two lattice-basis reduction methods that seem to be optimal for different symmetries, similarly to the algorithm for 3D lattices implemented in the CONOGRAPH software. In indexing, a method for error-stable unit-cell identification is also required to exclude duplicate solutions. Several methods are introduced to measure the difference in unit cells known in crystallography and mathematics.

This paper proposes a new order parameter model which satisfactorily explains complicated symmetry changes, the temperature-pressure (T-P) phase diagram and elastic anomalies observed experimentally with the improper ferroelastic phase transitions in multiferroic KMnF single crystal. First, it is shown that the order parameter model is transformed according to the four-dimensional reducible representation of the wavevector star channel group. Second, based on the order parameter model and the singularity theory, the sixth-order structurally stable Landau potential model is constructed. Finally, the theoretical T-P phase diagram is plotted and the elastic anomalies possible for each of the phase transitions are discussed.