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.

]]>This paper completes the series of three independent articles [Bodner *et al.* (2013). *Acta Cryst*. A**69**, 583–591, (2014), *PLOS ONE*, 10.1371/journal.pone.0084079] describing the breaking of icosahedral symmetry to subgroups generated by reflections in three-dimensional Euclidean space as a mechanism of generating higher fullerenes from C_{60}. The icosahedral symmetry of C_{60} can be seen as the junction of 17 orbits of a symmetric subgroup of order 4 of the icosahedral group of order 120. This subgroup is noted by *A*_{1}×*A*_{1}, because it is isomorphic to the Weyl group of the semi-simple Lie algebra *A*_{1}×*A*_{1}. Thirteen of the *A*_{1}×*A*_{1} orbits are rectangles and four are line segments. The orbits form a stack of parallel layers centered on the axis of C_{60} passing through the centers of two opposite edges between two hexagons on the surface of C_{60}. These two edges are the only two line segment layers to appear on the surface shell. Among the 24 convex polytopes with shell formed by hexagons and 12 pentagons, having 84 vertices [Fowler & Manolopoulos (1992). *Nature (London)*, **355**, 428–430; Fowler & Manolopoulos (2007). *An Atlas of Fullerenes*. Dover Publications Inc.; Zhang *et al*. (1993). *J. Chem. Phys.***98**, 3095–3102], there are only two that can be identified with breaking of the *H*_{3} symmetry to *A*_{1}×*A*_{1}. The remaining ones are just convex shells formed by regular hexagons and 12 pentagons without the involvement of the icosahedral symmetry.

Group-theoretical and thermodynamic methods of the Landau theory of phase transitions are used to investigate the hyper-kagome atomic order in structures of ordered spinels and a spinel-like Na_{4}Ir_{3}O_{8} crystal. The formation of an atom hyper-kagome sublattice in Na_{4}Ir_{3}O_{8} is described theoretically on the basis of the archetype (hypothetical parent structure/phase) concept. The archetype structure of Na_{4}Ir_{3}O_{8} has a spinel-like structure (space group ) and composition [Na_{1/2}Ir_{3/2}]^{16d}[Na_{3/2}]^{16c}O^{32e}_{4}. The critical order parameter which induces hypothetical phase transition has been stated. It is shown that the derived structure of Na_{4}Ir_{3}O_{8} is formed as a result of the displacements of Na, Ir and O atoms, and ordering of Na, Ir and O atoms, ordering *d _{xy}*,

No deterministic approach to obtaining a crystal structure from a set of diffraction intensities exists, despite significant progress in traditional probabilistic direct methods. One of the biggest hurdles in determining a crystal structure algebraically is solving a system of many polynomial equations of high power on intensities in terms of atomic coordinates. In this study, homotopy continuation is used for exhaustive investigation of such systems and an optimized homotopy continuation method is developed with random restarts to determine small (*N* < 5) crystal structures from a minimum set of error-free intensities.

An experimental determination of the magnetic pair distribution function (mPDF) defined in an earlier paper [Frandsen *et al.* (2014). *Acta Cryst.* A**70**, 3–11] is presented for the first time. The mPDF was determined from neutron powder diffraction data from a reactor and a neutron time-of-flight total scattering source on a powder sample of the antiferromagnetic oxide MnO. A description of the data treatment that allowed the measured mPDF to be extracted and then modelled is provided and utilized to investigate the low-temperature structure of MnO. Atomic and magnetic co-refinements support the scenario of a locally monoclinic ground-state atomic structure, despite the average structure being rhombohedral, with the mPDF analysis successfully recovering the known antiferromagnetic spin configuration. The total scattering data suggest a preference for the spin axis to lie along the pseudocubic [10] direction. Finally, *r*-dependent PDF refinements indicate that the local monoclinic structure tends toward the average rhombohedral *R**m* symmetry over a length scale of approximately 100 Å.

How many different intermetallic compounds are known so far, and in how many different structure types do they crystallize? What are their chemical compositions, the most abundant ones and the rarest ones? These are some of the questions we are trying to find answers for in our statistical analysis of the structures of the 20 829 intermetallic phases included in the database *Pearson's Crystal Data*, with the goal of gaining insight into some of their ordering principles. In the present paper, we focus on the subset of 13 026 ternary intermetallics, which crystallize in 1391 different structure types; remarkably, 667 of them have just one representative. What makes these 667 structures so unique that they are not adopted by any other of the known intermetallic compounds? Notably, ternary compounds are known in only 5109 of the 85 320 theoretically possible ternary intermetallic systems so far. In order to get an overview of their chemical compositions we use structure maps with Mendeleev numbers as ordering parameters.

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.

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