The full quantitative characterization of nanopowders using transmission electron microscopy scattering patterns is shown. This study demonstrates the feasibility of the application of so-called combined analysis, a global approach for phase identification, structure refinement, characterization of anisotropic crystallite sizes and shapes, texture analysis and texture variations with the probed scale, using electron diffraction patterns of TiO_{2} and Mn_{3}O_{4} nanocrystal aggregates and platinum films. Electron diffraction pattern misalignments, positioning, and slight changes from pattern to pattern are directly integrated and refined within this approach. The use of a newly developed full-pattern search–match methodology for phase identification of nanopowders and the incorporation of the two-wave dynamical correction for diffraction patterns are also reported and proved to be efficient.

Diffraction profiles for different models of dislocation arrangements are calculated directly by the Monte Carlo method and compared with the strain distributions for the same arrangements, which corresponds to the Stokes–Wilson approximation. It is shown that the strain distributions and the diffraction profiles are in close agreement as long as long-range order is absent. Analytical calculation of the strain distribution for uncorrelated defects is presented. For straight dislocations, the Stokes–Wilson and the Krivoglaz–Wilkens approximations give the same diffraction profiles, with the Gaussian central part and ∝*q*^{−3} power law at the tails.

Symmetric Bragg-case reflections from a thick, ideally imperfect, crystal slab are studied mostly by analytical means. The scattering transfer function of a thin mosaic layer is derived and brought into a form that allows for analytical approximations or easy quadrature. The Darwin–Hamilton equations are generalized, lifting the restriction of wavevectors to a two-dimensional scattering plane. A multireflection expansion shows that wavevector diffusion can be studied independently of the real-space coordinate. Combining analytical arguments and Monte Carlo simulations, multiple Bragg reflections are found to result in a minor correction of the reflected intensity, a moderate broadening of the reflected azimuth angle distribution, a considerable modification of the polar angle distribution, and a noticeable shift and distortion of rocking curves.

]]>The subgroup structure of the hyperoctahedral group in six dimensions is investigated. In particular, the subgroups isomorphic to the icosahedral group are studied. The orthogonal crystallographic representations of the icosahedral group are classified and their intersections and subgroups analysed, using results from graph theory and their spectra.

]]>A low-discrepancy cubic variant of β-Mn is presented exhibiting local octagonal symmetry upon projection along any of the three mutually perpendicular 〈100〉 axes. Ideal structural parameters are derived to be and for the *P*4_{1}32 enantiomorph. A comparison of the actual and ideal structure models of β-Mn is made in terms of the newly devised concept of geometrical discrepancy maps. Two-dimensional maps of both the geometrical star discrepancy *D*^{*} and the minimal interatomic distance *d*_{min} are calculated over the combined structural parameter range and of generalized β-Mn type structures, showing that the `octagonal' variant of β-Mn is almost optimal in terms of globally minimizing *D*^{*} while at the same time globally maximizing *d*_{min}. Geometrical discrepancy maps combine predictive and discriminatory powers to appear useful within a wide range of structural chemistry studies.

The temperature dependence of H-*U*_{iso} in *N*-acetyl-L-4-hydroxyproline monohydrate is investigated. Imposing a constant temperature-independent multiplier of 1.2 or 1.5 for the riding hydrogen model is found to be inaccurate, and severely underestimates H-*U*_{iso} below 100 K. Neutron diffraction data at temperatures of 9, 150, 200 and 250 K provide benchmark results for this study. X-ray diffraction data to high resolution, collected at temperatures of 9, 30, 50, 75, 100, 150, 200 and 250 K (synchrotron and home source), reproduce neutron results only when evaluated by aspherical-atom refinement models, since these take into account bonding and lone-pair electron density; both invariom and Hirshfeld-atom refinement models enable a more precise determination of the magnitude of H-atom displacements than independent-atom model refinements. Experimental efforts are complemented by computing displacement parameters following the TLS+ONIOM approach. A satisfactory agreement between all approaches is found.

The resurgence in mathematical crystallography motivated the formation of the IUCr Commission on Mathematical Crystallography, as well as this virtual special issue of *Acta Crystallographica Section A*. This foreword describes some of the current activities of the Commission and introduces the articles in the special issue.

The concept of an orbifold is particularly suited to classification and enumeration of crystalline groups in the euclidean (flat) plane and its elliptic and hyperbolic counterparts. Using Conway's orbifold naming scheme, this article explicates conventional point, frieze and plane groups, and describes the advantages of the orbifold approach, which relies on simple rules for calculating the orbifold topology. The article proposes a simple taxonomy of orbifolds into seven classes, distinguished by their underlying topological connectedness, boundedness and orientability. Simpler `crystallographic hyperbolic groups' are listed, namely groups that result from hyperbolic sponge-like sections through three-dimensional euclidean space related to all known genus-three triply periodic minimal surfaces (*i.e.* the *P*, *D*, *Gyroid*, *CLP* and *H* surfaces) as well as the genus-four *I-WP* surface.

It is a scientific and engineering challenge to characterize materials under nonequilibrium conditions. In recent years, X-ray photon correlation spectroscopy (XPCS), a synchrotron-based coherent X-ray scattering technique, has been found useful in determining the timescales associated with various nonequilibrium processes, with detailed descriptions of the underlying processes lacking. Here, both static ultra small angle X-ray scattering (USAXS) and dynamic USAXS-based XPCS were used to investigate a transient structural change (a nonequilibrium process) associated with an isothermal anneal in a glass polymer composite system. While the bulk USAXS technique lacked the required sensitivity to detect the change in the microstructures, the local structural reorganization was apparent in the XPCS study. The structural changes were modeled using a three-dimensional finite element analysis approach and wave-propagation theory was used to simulate the resulting reciprocal-space coherent scattering intensity. Qualitative agreement was found between the modeling and experimental results, which validates that stress relaxation in the viscous polymer matrix was responsible for the observed changes. This analysis demonstrates that multi-physics modeling of complex systems can be used to interpret XPCS measurements of nonequilibrium processes.

]]>Staurolite has been long considered an enigma because of its remarkable pseudosymmetry and the frequent twinning. Staurolite gives two twins whose occurrence frequency seems to contradict the condition of lattice restoration requested by the reticular theory of twinning, in that the more frequent one (Saint Andrews cross twin) has a twin index of 12, whereas the less frequent one (Greek cross twin) has a twin index of 6. The hybrid theory of twinning shows that the former is actually a hybrid twin with two concurrent sublattices and an effective twin index of 6.0. However, this is still not sufficient to explain the observed higher occurrence frequency of the Saint Andrews twin. The (pseudo)-eigensymmetry of the crystallographic orbits of staurolite has been analysed and it was found that the whole substructure built on anions is restored (with small deviations) by both twin laws, which explains why twinning is frequent in staurolite. On the other hand, 45% of the cation sites are quasi-restored in the Saint Andrews cross twin, against only 19% for the Greek cross twin: this difference finally explains the different occurrence frequencies of the two twins.

]]>Although the ambiguity of the crystal structures determined directly from diffraction intensities has been historically recognized, it is not well understood in quantitative terms. Bernstein's theorem has recently been used to obtain the number of one-dimensional crystal structures of equal point atoms, given a minimum set of diffraction intensities. By a similar approach, the number of two- and three-dimensional crystal structures that can be determined from a minimum intensity data set is estimated herein. The ambiguity of structure determination from the algebraic minimum of data increases at least exponentially fast with the increasing structure size. Substituting lower-resolution intensities by higher-resolution ones in the minimum data set has little or no effect on this ambiguity if the number of such substitutions is relatively small.

]]>This paper considers Platonic solids/polytopes in the real Euclidean space of dimension 3 ≤*n* < ∞. The Platonic solids/polytopes are described together with their faces of dimensions 0 ≤*d*≤*n*− 1. Dual pairs of Platonic polytopes are considered in parallel. The underlying finite Coxeter groups are those of simple Lie algebras of types *A _{n}*,

With the development of X-ray free-electron lasers (XFELs), it is possible to determine the three-dimensional structures of noncrystalline objects with coherent X-ray diffraction imaging. In this diffract-and-destroy mode, many snapshot diffraction patterns are obtained from the identical objects which are presented one by one in random orientations to the XFEL beam. Determination of the orientation of an individual object is essential for reconstruction of a three-dimensional structure. Here a new method, called the multiple-common-lines method, has been proposed to determine the orientations of high- and low-signal snapshot diffraction patterns. The mean errors of recovered orientations (α, β, γ) of high- and low-signal patterns are about 0.14, 0.06, 0.12 and 0.77, 0.31, 0.60°, respectively; both sets of errors can meet the requirements of the reconstruction of a three-dimensional structure.

]]>This paper presents crystallographic data of double antisymmetry space groups, including symmetry-element diagrams, general-position diagrams and positions, with multiplicities, site symmetries, coordinates, spin vectors, roto vectors and displacement vectors.

]]>The physical property coefficients that arise in a phase transition which are zero in the high-symmetry phase and nonzero in the low-symmetry phase are called *spontaneous coefficients*. For all 1601 Aizu species of phase transitions, matrices have been constructed which show the nonzero coefficients of a wide variety of magnetic and nonmagnetic physical properties including toroidal property coefficients in the high-symmetry phase and their corresponding spontaneous coefficients in the low-symmetry phase. It is also shown that these spontaneous coefficients provide for the distinction of and switching between nonferroelastic domain pairs.

The report of the Executive Committee for 2012 is presented.

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