The form of a physical property tensor of a quasi-one-dimensional material such as a nanotube or a polymer is determined from the material's axial point group. Tables of the form of rank 1, 2, 3 and 4 property tensors are presented for a wide variety of magnetic and non-magnetic tensor types invariant under each point group in all 31 infinite series of axial point groups. An application of these tables is given in the prediction of the net polarization and magnetic-field-induced polarization in a one-dimensional longitudinal conical magnetic structure in multiferroic hexaferrites.

]]>In this paper periodically domain-inverted (PDI) ferroelectric crystals are studied using high-resolution X-ray diffraction. Rocking curves and reciprocal-space maps of the principal symmetric Bragg reflections in LiNbO_{3} (LN) (Λ = 5 µm), KTiOPO_{4} (KTP) (Λ = 9 µm) and KTiOAsO_{4} (KTA) (Λ = 39 µm) are presented. For all the samples strong satellite reflections were observed as a consequence of the PDI structure. Analysis of the satellites showed that they were caused by a combination of *coherent* and *incoherent* scattering between the adjacent domains. Whilst the satellites contained phase information regarding the structure of the domain wall, this information could not be rigorously extracted without *a priori* knowledge of the twinning mechanism. Analysis of the profiles reveals strain distributions of Δ*d*/*d* = 1.6 × 10^{−4} and 2.0 × 10^{−4} perpendicular to domain walls in KTP and LN samples, respectively, and lateral correlation lengths of 63 µm (KTP), 194 µm (KTA) and 10 µm (LN). The decay of crystal truncation rods in LN and KTP was found to support the occurrence of surface corrugations.

The main result of this work is extension of the famous characterization of Bravais lattices according to their metrical, algebraic and geometric properties onto a wide class of primitive lattices (including Buerger-reduced, nearly Buerger-reduced and a substantial part of Delaunay-reduced) related to low-restricted *semi-reduced descriptions* (*s.r.d.'s*). While the `geometric' operations in Bravais lattices map the basis vectors into themselves, the `arithmetic' operators in s.r.d. transform the basis vectors into cell vectors (basis vectors, face or space diagonals) and are represented by matrices from the set of all 960 matrices with the determinant ±1 and elements {0, ±1} of the matrix powers. A lattice is in s.r.d. if the moduli of off-diagonal elements in both the metric tensors *M* and *M*^{−1} are smaller than corresponding diagonal elements sharing the same column or row. Such lattices are split into 379 s.r.d. types relative to the arithmetic holohedries. Metrical criteria for each type do not need to be explicitly given but may be modelled as linear derivatives , where denotes the set of 39 highest-symmetry metric tensors, and describe changes of appropriate interplanar distances. A sole filtering of according to an experimental s.r.d. metric and subsequent geometric interpretation of the filtered matrices lead to mathematically stable and rich information on the Bravais-lattice symmetry and deviations from the exact symmetry. The emphasis on the crystallographic features of lattices was obtained by shifting the focus (i) from analysis of a lattice metric to analysis of symmetry matrices [Himes & Mighell (1987). *Acta Cryst.* A**43**, 375–384], (ii) from the *isometric approach* and *invariant subspaces* to the *orthogonality concept* {some ideas in Le Page [*J. Appl. Cryst.* (1982), **15**, 255–259]} and *splitting indices* [Stróż (2011). *Acta Cryst.* A**67**, 421–429] and (iii) from fixed cell transformations to transformations derivable *via* geometric information (Himes & Mighell, 1987; Le Page, 1982). It is illustrated that corresponding arithmetic and geometric holohedries share space distribution of symmetry elements. Moreover, completeness of the s.r.d. types reveals their combinatorial structure and simplifies the crystallographic description of structural phase transitions, especially those observed with the use of powder diffraction. The research proves that there are excellent theoretical and practical reasons for looking at crystal lattice symmetry from an entirely new and surprising point of view – the combinatorial set of matrices, their semi-reduced lattice context and their geometric properties.

This paper describes a detailed derivation of a structural model for an icosahedral quasicrystal based on a primitive icosahedral tiling (three-dimensional Penrose tiling) within a statistical approach. The average unit cell concept, where all calculations are performed in three-dimensional physical space, is used as an alternative to higher-dimensional analysis. Comprehensive analytical derivation of the structure factor for a primitive icosahedral lattice with monoatomic decoration (atoms placed in the nodes of the lattice only) presents in detail the idea of the statistical approach to icosahedral quasicrystal structure modelling and confirms its full agreement with the higher-dimensional description. The arbitrary decoration scheme is also discussed. The complete structure-factor formula for arbitrarily decorated icosahedral tiling is derived and its correctness is proved. This paper shows in detail the concept of a statistical approach applied to the problem of icosahedral quasicrystal modelling.

]]>A pair of enantiomer crystals is used to demonstrate how X-ray phase measurements provide reliable information for absolute identification and improvement of atomic model structures. Reliable phase measurements are possible thanks to the existence of intervals of phase values that are clearly distinguishable beyond instrumental effects. Because of the high susceptibility of phase values to structural details, accurate model structures were necessary for succeeding with this demonstration. It shows a route for exploiting physical phase measurements in the crystallography of more complex crystals.

]]>The implications of the paper by Grimmer [Acta Cryst. (2015), A71, 143149] are discussed.

]]>Neither *International Tables for Crystallography* (ITC) nor available crystallography textbooks state explicitly which of the 14 Bravais types of lattices are special cases of others, although ITC contains the information necessary to derive the result in two ways, considering either the symmetry or metric properties of the lattices. The first approach is presented here for the first time, the second has been given by Michael Klemm in 1982. Metric relations between conventional bases of special and general lattice types are tabulated and applied to continuous equi-translation phase transitions.

The affine and Euclidean normalizers of the subperiodic groups, the frieze groups, the rod groups and the layer groups, are derived and listed. For the layer groups, the special metrics used for plane-group Euclidean normalizers have been considered.

]]>The generalized Penrose tiling is, in fact, an infinite set of decagonal tilings. It is constructed with the same rhombs (thick and thin) as the conventional Penrose tiling, but its long-range order depends on the so-called shift parameter (*s*∈〈0; 1)). The structure factor is derived for the arbitrarily decorated generalized Penrose tiling within the average unit cell approach. The final formula works in physical space only and is directly dependent on the *s* parameter. It allows one to straightforwardly change the long-range order of the refined structure just by changing the *s* parameter and keeping the tile decoration unchanged. This gives a great advantage over the higher-dimensional method, where every change of the tiling (change in the *s* parameter) requires the structure model to be built from scratch, *i.e.* the fine division of the atomic surfaces has to be redone.

A simple combination of riding motion and an additive term is sufficient to estimate the temperature-dependent isotropic displacement parameters of hydrogen atoms, for use in X-ray structure refinements. The approach is validated against neutron diffraction data, and gives reasonable estimates in a very large temperature range (10–300 K). The model can be readily implemented in common structure refinement programs without auxiliary software.

]]>A group-theoretical discussion on the hypercubic lattice described by the affine Coxeter–Weyl group *W _{a}*(

The main goal of molecular replacement in macromolecular crystallography is to find the appropriate rigid-body transformations that situate identical copies of model proteins in the crystallographic unit cell. The search for such transformations can be thought of as taking place in the coset space Γ\*G* where Γ is the Sohncke group of the macromolecular crystal and *G* is the continuous group of rigid-body motions in Euclidean space. This paper, the third in a series, is concerned with viewing nonsymmorphic Γ in a new way. These space groups, rather than symmorphic ones, are the most common ones for protein crystals. Moreover, their properties impact the structure of the space Γ\*G*. In particular, nonsymmorphic space groups contain both Bieberbach subgroups and symmorphic subgroups. A number of new theorems focusing on these subgroups are proven, and it is shown that these concepts are related to the preferences that proteins have for crystallizing in different space groups, as observed in the Protein Data Bank.

The occurrence frequency of the {110} twin in aragonite is explained by the existence of an important substructure (60% of the atoms) which crosses the composition surface with only minor perturbation (about 0.2 Å) and constitutes a common atomic network facilitating the formation of the twin. The existence of such a common substructure is shown by the *C*2/*c* pseudo-eigensymmetry of the crystallographic orbits, which contains restoration operations whose linear part coincides with the twin operation. Furthermore, the local analysis of the composition surface in the aragonite structure shows that the structure is built from slices which are fixed by the twin operation, confirming and reinforcing the crystallographic orbit analysis of the structural continuity across the composition surface.

Statistical tests are applied for the detection of systematic errors in data sets from least-squares refinements or other residual-based reconstruction processes. Samples of the residuals of the data are tested against the hypothesis that they belong to the same distribution. For this it is necessary that they show the same mean values and variances within the limits given by statistical fluctuations. When the samples differ significantly from each other, they are not from the same distribution within the limits set by the significance level. Therefore they cannot originate from a single Gaussian function in this case. It is shown that a significance cutoff results in exactly this case. Significance cutoffs are still frequently used in charge-density studies. The tests are applied to artificial data with and without systematic errors and to experimental data from the literature.

]]>Dehn–Sommerville relations for simple (simplicial) polytopes are applied to primitive parallelohedra. New restrictions on numbers of *k*-faces of non-principal primitive parallelohedra are explicitly formulated for five-, six- and seven-dimensional parallelohedra.

One of the main goals in the study of color symmetry is to classify colorings of symmetrical objects through their color groups. The term color group is taken to mean the subgroup of the symmetry group of the uncolored symmetrical object which induces a permutation of colors in the coloring. This work looks for methods of determining the color group of a colored symmetric object. It begins with an index *n* subgroup *H* of the symmetry group *G* of the uncolored object. It then considers *H*-invariant colorings of the object, so that the color group *H*^{*} will be a subgroup of *G* containing *H*. In other words, *H*≤*H*^{*}≤*G*. It proceeds to give necessary and sufficient conditions for the equality of *H*^{*} and *G*. If *H*^{*}≠*G* and *n* is prime, then *H*^{*} = *H*. On the other hand, if *H*^{*}≠*G* and *n* is not prime, methods are discussed to determine whether *H*^{*} is *G*, *H* or some intermediate subgroup between *H* and *G*.

The distributions of bond topological properties (BTPs) of the electron density upon thermal vibrations of the nuclei are computationally examined to estimate different statistical figures, especially uncertainties, of these properties. The statistical analysis is based on a large ensemble of BTPs of the electron densities for thermally perturbed nuclear geometries of the formamide molecule. Each bond critical point (BCP) is found to follow a normal distribution whose covariance correlates with the displacement amplitudes of the nuclei involved in the bond. The BTPs are found to be markedly affected not only by normal modes of the significant bond-stretching component but also by modes that involve mainly hydrogen-atom displacements. Their probability distribution function can be decently described by Gumbel-type functions of positive (negative) skewness for the bonds formed by non-hydrogen (hydrogen) atoms.

]]>Accurate structure refinement from electron-diffraction data is not possible without taking the dynamical-diffraction effects into account. A complete three-dimensional model of the structure can be obtained only from a sufficiently complete three-dimensional data set. In this work a method is presented for crystal structure refinement from the data obtained by electron diffraction tomography, possibly combined with precession electron diffraction. The principle of the method is identical to that used in X-ray crystallography: data are collected in a series of small tilt steps around a rotation axis, then intensities are integrated and the structure is optimized by least-squares refinement against the integrated intensities. In the dynamical theory of diffraction, the reflection intensities exhibit a complicated relationship to the orientation and thickness of the crystal as well as to structure factors of other reflections. This complication requires the introduction of several special parameters in the procedure. The method was implemented in the freely available crystallographic computing system *Jana2006*.

The widely used pseudoatom formalism [Stewart (1976). *Acta Cryst*. A**32**, 565–574; Hansen & Coppens (1978). *Acta Cryst.* A**34**, 909–921] in experimental X-ray charge-density studies makes use of real spherical harmonics when describing the angular component of aspherical deformations of the atomic electron density in molecules and crystals. The analytical form of the density-normalized Cartesian spherical harmonic functions for up to *l*≤ 7 and the corresponding normalization coefficients were reported previously by Paturle & Coppens [*Acta Cryst.* (1988), A**44**, 6–7]. It was shown that the analytical form for normalization coefficients is available primarily for *l*≤ 4 [Hansen & Coppens, 1978; Paturle & Coppens, 1988; Coppens (1992). *International Tables for Crystallography*, Vol. B, *Reciprocal space*, 1st ed., edited by U. Shmueli, ch. 1.2. Dordrecht: Kluwer Academic Publishers; Coppens (1997). *X-ray Charge Densities and Chemical Bonding*. New York: Oxford University Press]. Only in very special cases it is possible to derive an analytical representation of the normalization coefficients for 4 < *l*≤ 7 (Paturle & Coppens, 1988). In most cases for *l* > 4 the density normalization coefficients were calculated numerically to within seven significant figures. In this study we review the literature on the density-normalized spherical harmonics, clarify the existing notations, use the Paturle–Coppens (Paturle & Coppens, 1988) method in the Wolfram *Mathematica* software to derive the Cartesian spherical harmonics for *l*≤ 20 and determine the density normalization coefficients to 35 significant figures, and computer-generate a Fortran90 code. The article primarily targets researchers who work in the field of experimental X-ray electron density, but may be of some use to all who are interested in Cartesian spherical harmonics.