Many compounds adopting the spinel AB(2)X(4) structure are technologically important owing to their tunable physical and chemical properties enabling diverse applications in energy storage, catalysis, magnetism, and functional ceramics. Most of them are traditionally assigned to the centrosymmetric space group Fd (3) over barm. However, the physical properties of some spinels are incompatible with centrosymmetry. This discrepancy is often accounted for by reducing the symmetry to the non-centrosymmetric space group F (4) over bar 3m, allowing thus small atomic displacements from their original position in Fd (3) over barm. In this work, we demonstrate that the loss of the inversion symmetry can occur without any atomic displacements, since the centrosymmetric Fd (3) over barm and non-centrosymmetric F4(1)32 space groups are equivalent for structure determination and refinement based on X-ray diffraction data. If consistent with experiment, only the use of an anharmonic model of atomic displacements can distinguish these space groups. This study aims to clarify certain misconceptions regarding the structural symmetry and physical properties of spinel-type compounds.

Many compounds belonging to the spinel AB2X4 structure play an important role due to their wide range of practical applications. Most of them are traditionally assigned to the centrosymmetric space group Fd-3m. However, the physical properties of some spinels are incompatible with centrosymmetry. This discrepancy is often accounted for by reducing the symmetry to the non-centrosymmetric space group F-43m, allowing thus small atomic displacements from their original position in Fd-3m. In this work, we demonstrate that the loss of the inversion symmetry can occur without any atomic displacements, since the centrosymmetric Fd-3m and non-centrosymmetric F4132 space groups are equivalent for structure determination and refinement based on X-ray diffraction data. If consistent with experiment, only the use of an anharmonic model of atomic displacements can distinguish these space groups. This study aims to clarify certain misconceptions regarding the structural symmetry and physical properties of spinel type compounds.

Obituary for Dieter Schwarzenbach.

Incommensurately modulated crystalline phases are part of a more general family called aperiodic crystals. Their symmetry is treated within the theoretical framework of superspace groups that is a generalization of the 3D space groups that are used for conventional crystalline structures. Aperiodic structures are conveniently embedded in superspace with dimensions varying from (3 + 1) to (3 + 3)D. They do occur in all types of material, organic and inorganic, metal and alloys, minerals and macromolecules under various pressure and temperature. Some examples of incommensurate structures are presented along with their chemical and physical properties.

Although W. L. Bragg's law can be easily derived for beginners in the field of crystallography, its interpretation however seems to cause some difficulties which lies essentially in the relation between the concept of lattice planes and the unit cell constants characterizing the lattice periodicity of the crystal structure. Our approach is certainly not new and is based on a more physical approach where every single point in the crystal participates in the diffraction process. From the early stages of developing a model of diffraction, we make abundant use the dual reference frames namely the direct and reciprocal reference frames. With this approach, W. L. Bragg's law can be reformulated directly in terms of the reciprocal unit cell constants avoiding thus the necessity to introduce a priori the notion of lattice planes. Following the derivation of the diffraction law, different steps and methods leading to the complete determination of a crystal structure are derived. We present also some simulation tools to explain in particular the crystal diffraction phenomenon based on the Ewald sphere and the solution of crystalline structures based on the dual space iteration techniques which are currently used.

A detailed synchrotron X-ray diffraction (XRD) study performed with a single crystal of BaVS3 (barium vanadium trisulfide) in the temperature range between 10 and 295 K is reported. Aside from the known tetragonal–orthorhombic (240 K) and orthorhombic–monoclinic (69 K) phase transitions, in the 130 < T ≤ 295 K range the overall structure can be viewed as a host–guest (H–G) composite. The BaS3 matrix is the host, while the V-chains form the guest. The two subsystems lock in at T LOCK = 130 ± 20 K. This temperature is marked by a symmetry change from orthorhombic to monoclinic. This results in the formation of twins, implying a structural phase transition identified here for the first time. From the refined structural data, it is possible to follow, starting already at 295 K downwards, the stepwise transformation of VS6 octahedra into VS5 tetragonal pyramids as the origin of the structure evolution. The new findings will yield a better understanding of the complex electronic phase diagram of BaVS3.

A commentary is given on the article by Carolyn Pratt Brock and Robin Taylor [Acta Cryst. (2020), B76, 630–642] in this issue.

Through the years, mineralogical studies have produced a tremendous amount of data on the atomic arrangement and mineral properties. Quite often, structural analysis has led to elucidate the role played by minor components, giving interesting insights into the physico-chemical conditions of mineral crystallization and allowing the description of unpredictable structures that represented a body of knowledge critical for assessing their technological potentialities. Using such a rich database, containing many basic acquisitions, further steps became appropriate and possible, into the directions of more advanced knowledge frontiers. Some of these frontiers assume the name of modularity, complexity, aperiodicity, and matter organization at not conventional levels, and will be discussed in this review.

This chapter discusses tilings as mathematical models for quasicrystals. In a first approximation quasicrystals may be described as being space filling with copies of two or more types of tiles. This description gives a connection with the mathematical notion of tilings, which have been well studied. A brief introduction of tilings is presented in this chapter along with the method of substitution to create aperiodic tilings. The symmetry of the tilings is also treated in this chapter, as are model sets and random tilings. Quasiperiodic crystals often have approximants, that is, periodic structures that are close to the aperiodic ones. The relations between quasiperiodic crystals and approximants also is described in this chapter.

This chapter first introduces the mathematical concept of aperiodic and quasiperiodic functions, which will form the theoretical basis of the superspace description of the new recently discovered forms of matter. They are divided in three groups, namely modulated phases, composites, and quasicrystals. It is shown how the atomic structures and their symmetry can be characterized and described by the new concept. The classification of superspace groups is introduced along with some examples. For quasicrystals, the notion of approximants is also introduced for a better understanding of their structures. Finally, alternatives for the descriptions of the new materials are presented along with scaling symmetries. Magnetic systems and time-reversal symmetry are also introduced.

Physical properties of aperiodic crystals present some theoretical challenges due to the lack of three-dimensional periodicity. For the description of the structure there is a periodic representation in higher-dimensional space. For physical properties, however, this scheme cannot be used because the mapping between interatomic forces and the high-dimensional representation is not straightforward. In this chapter methods are described to deal with these problems. First, the hydrodynamic theory of aperiodic crystals and then the phonons and phasons theory are developed and illustrated with some examples. The properties of electrons in aperiodic crystals are also presented. Finally, the experimental findings of phonon and phason modes for modulated and quasicrystals are presented. The chapter also discusses diffuse scattering, the Debye–Waller factor, and electrical conductivity.

Until the 1970s all materials studied consisted of periodic arrays of unit cells, or were amorphous. In the following decades a new class of solid state matter, called aperiodic crystals, has been found. It is a long-range ordered structure, but without lattice periodicity. It is found in a wide range of materials: organic and inorganic compounds, minerals (including a substantial portion of the earth’s crust), and metallic alloys, under various pressures and temperatures. Because of the lack of periodicity the usual techniques for the study of structure and physical properties no longer work, and new techniques have to be developed. This book deals with the characterization of the structure, the structure determination, and the study of the physical properties, especially the dynamical and electronic properties of aperiodic crystals. The treatment is based on a description in a space with more dimensions than three, the so-called superspace. This allows us to generalize the standard crystallography and to look differently at the dynamics. The three main classes of aperiodic crystals, modulated phases, incommensurate composites, and quasicrystals are treated from a unified point of view which stresses the similarities of the various systems. The book assumes as a prerequisite a knowledge of the fundamental techniques of crystallography and the theory of condensed matter, and covers the literature at the forefront of the field.

The host-guest structures of elements at high pressure discovered a decade ago still leave many open questions due to the lack of precise models based on full exploitation of the diffraction data. This concerns in particular Ba IV, which is stable in the range 12-45 GPa. With the example of phase Ba IVb, which is characterized here for the first time, a systematic analysis is presented of possible host-guest structure models based on high-quality single-crystal diffraction data obtained with synchrotron radiation at six different pressures between 16.5 and 19.6 GPa. It is shown that a new incommensurately modulated (IM) structure model better fits the experimental data. Unlike the composite models which are commonly reported for the Ba IV phases, the IM model reveals a density wave and its pressure-dependent evolution. The crucial role played by the selected model in the interpretation of structure evolution under pressure is discussed. The findings give a new experimental basis for a better understanding of the nature of host-guest structures.

The concept of superspace was introduced with the aim to extend the application of crystallographic symmetry to incommensurately modulated (IM) and composite (COMP) structures, both belonging to the class of aperiodic structures.Superspace symmetry is now universally accepted and routinely applied to describe them.Superspace can however do much more.First, its application is not only limited to the description of aperiodic structures in (3+n)-dimensions where n varies between 1 and 3.It can also be applied to the description of superstructures and it can lead also to the description of conventional structures.In other words, the concept of superspace can be used to describe families of compounds where both aperiodic and periodic cases occur.Moreover, it appears frequently that the superspace description of one of its aperiodic member allows to derive the possible three dimensional symmetries of the periodic members of the same family of compounds.Examples are scheelites [1], calaverite, hexagonal ferrites, palmierite and PbO2.In particular, it is possible to derive and/or predict polytypic modifications as in the temperature dependent phases of K5Yb(MoO4)4 [2] and pharmaceutical Cimetidine, C10H16N6S.Another interesting aspect of crystal chemistry concerns characteristic interatomic distances.IM structures can essentially simplify the statistical estimation of this property, which is traditionally based on analysis of many different compounds.For instance, it was found that the estimation of K-O, In-O and P-O distances based on about 3000 3D compounds is equal to the statistical distribution in a single K3In(PO4)2 IM structure [3].All coordination numbers (CN = 6, 8, 10, 12) found for K in about 1500 3D structures occur in this single IM structure.The example of Eu-containing molybdates with variable compositions and (3+n)D IM structures is an interesting illustration of the superspace approach to explain physical properties like . .luminescence.It was shown that the luminescent efficiency is correlated to specific associations of Eu atoms in the cationic substructure, which are aperiodically ordered.The superspace approach is particularly efficient for the investigation of high-pressure structures of chemical elements where a majority of them exhibits aperiodic structures above normal pressures.Our recent study of BaIV IM phase reveals an interesting case of atomic density waves in the range of 16.6 -19.5 GPa.Our presentation will illustrate a few examples of the use of superspace to discover new relations between crystal structures and possible applications in crystal chemistry.

Alkali–silica reaction (ASR) is one of the most important deterioration mechanisms in concrete leading to substantial damages of structures worldwide. Synchrotron-based micro-X-ray diffraction (micro-XRD) was employed to characterize the mineral phases formed in micro-cracks of concrete aggregates as a consequence of ASR. This high spatial resolution technique enables to directly gain structural information on ASR products formed in a 40-year old motorway bridge damaged due to ASR. Micro-X-ray-fluorescence was applied on thin sections to locate the reaction products formed in veins within concrete aggregates. Micro-XRD pattern were collected at selected points of interest along a vein by rotating the sample. Rietveld refinement determined the structure of the ASR product consisting of a new layered framework similar to mountainite and rhodesite. It is conceivable that understanding the structure of the ASR product may help developing new technical treatments inhibiting ASR.

When dealing with descriptions of structures, it is not seldom that we are faced with the problem of comparing identical crystal structures described with respect to different coordinate systems. For example, two descriptions of the same structure can differ by an origin shift or by a different choice of the basis. Different phases of the same compound often differ in their symmetry at various temperatures or pressures. Any detailed comparison of their structures requires the selection of a common basis and consequently the transformation of the original data to a different coordinate system. The purpose of this chapter is to provide the mathematical tools to accomplish these transformations. The method for transforming the crystallographic data following a change of origin or a change of the basis is given and illustrated with some examples. The transformation rules of the metric tensor characterizing both the direct and reciprocal space and of the space-group symmetry operations under coordinate transformations are further derived and discussed. More than 40 different types of coordinate-system transformations representing the most frequently encountered cases are listed and illustrated. Finally, synoptic tables of space (plane) groups show different types of symmetry operations belonging to the same coset with respect to the translation subgroup and a large selection of alternative settings of space (plane) groups and their Hermann–Mauguin symbols covering most practical cases.

This chapter provides an introduction to the various crystallographic items used for the presentation of the symmetry data in the space-group tables of this volume. It starts with a detailed introduction to the Hermann–Mauguin symbols for space, plane and crystallographic point groups, and to their Schoenflies symbols. A description is given of the symbols of the symmetry operations applied in the volume, and their listings in the general-position and the symmetry-operations blocks. This is followed by analysis of some specific features of the symmetry-element and general-position graphical representations of space groups. Seitz symbols for crystallographic symmetry operations are discussed and illustrated, along with the so-called additional symmetry operations of space groups, which result from the periodicity of the space groups. The classification of points in direct space into general and special Wyckoff positions, and the study of their site-symmetry groups and Wyckoff multiplicities are presented in detail. In addition, more advanced topics like Wyckoff sets, eigensymmetry groups and non-characteristic orbits are treated. The final sections offer a useful introduction to two-dimensional sections and projections of space groups and their symmetry properties.