Pyrite contact twins.

Two examples of contact twins in pyrite from Peru are described. The first one, from Pasto Bueno ore deposit, shows the pyritohedron {120} as principal form, accompanied by the {111} octahedron and {100} cube as secondary forms, giving a lenticular aspect. (111) is the composition plane, and the twin operation is any one of the three binary axes ⟨110⟩ within this plane. The second one (unknown ore deposit) presents two forms, the octahedron {111} and the pyritohedron {120}; the two crystals in the twin are elongated along [101] and [011], respectively, producing a V profile. It is a reflection twin where the twin plane (110) coincides with the composition plane. These twins are by merohedry. Another contact twin is known in the literature, reported by Gaubert [Bull. Soc. Fr. Minéral. Cristallogr. (1928), 51, 211-212] who described it as a spinel twin, i.e. a reflection twin with twin and composition plane (111); here it is shown that it is actually a rotation twin in which the twin operation is a 180° rotation about any of the three equivalent directions ⟨211⟩, contained in the (111) composition plane. The occurrence of these twins as well as the doubtfulness of the spinel twin in pyrite shows a direct relationship with the structural interpretation based on the pseudo-symmetry of the crystallographic orbits.

Diffraction enhancement of symmetry and modular structures.

Diffraction enhancement of symmetry (DES) is the phenomenon according to which the symmetry of the diffraction pattern of a crystal can be higher than the point symmetry of the structure that has produced it. The most well known example is that of Friedel's law, which is however violated in the presence of resonant scattering. This phenomenon is addressed in monoarchetypal modular structures and it is shown that a sufficient condition for DES is that both the module and the family of stacking vectors are invariant under an isometry that is not a symmetry operation for the structure. In general, DES is still observed in the presence of significant resonant scattering. The example of SiC polytypes, where the phenomenon has been confirmed experimentally, is studied in detail but the conclusions depend only on the symmetry of the layer and on the stacking vectors: the same applies therefore to all compounds sharing these features, like ZnS.

N-Iodosaccharin-pyridine co-crystal system under pressure: experimental evidence of reversible twinning.

This work presents a single-crystal X-ray diffraction study of an organic co-crystal composed of N-iodosaccharin and pyridine (NISac·py) under hydrostatic pressure ranging from 0.00 (5) GPa to 4.5 (2) GPa. NISac·py crystallizes in the monoclinic system (space group B21/e). The unconventional setting of the space group is adopted (the conventional setting is P21/c, No. 14) to emphasise the strongly pseudo-orthorhombic symmetry of the lattice, with a ß angle very close to 90°. The crystal structure contains one molecule each of N-iodosaccharin (NISac) and pyridine (py) in the asymmetric unit (Z' = 1), linked via an Nsac...I...N'py halogen-bonding motif. A gradual modification of this motif is observed under pressure as a result of changes in the crystalline environment. Mechanical twinning is observed under compression and the sample splits into two domains, spanning an unequal volume that is mapped by a twofold rotation about the [100] direction of the B21/e unit cell. The twinning is particularly significant at high pressure, being reversible when the pressure is released. The structure of the twinned sample reveals the continuity of a substantial substructure across the composition plane. The presence of this common substructure in the two orientations of the twinned individuals can be interpreted as a structural reason for the formation of the twin and is the first observed example in a molecular crystal. These results indicate that the anisotropy of intermolecular interactions in the crystal structure results in an anisotropic strain generated upon the action of hydrostatic compression. Periodic density functional theory calculations were carried out by considering an isotropic external pressure, the results showing good agreement with the experimental findings. The bulk modulus of the crystal was obtained from the equations of state, being 7 (1) GPa for experimental data and 6.8 (5) GPa for theoretical data.

Crystal structure, XANES and charge distribution investigation of krennerite and sylvanite: analysis of Au-Te and Te-Te bonds in Au1-xAgxTe2 group minerals.

The structure refinement and XANES study of two gold-silver-tellurides [Au1+xAgxTe2, krennerite (x = 0.11-0.13) and sylvanite (x = 0.29-0.31)] are presented and the structures are compared with the prototype structure of calaverite (x = 0.08-0.10). Whereas the latter is well known for being incommensurately modulated at ambient conditions, neither krennerite nor sylvanite present any modulation. This is attributed to the presence of relatively strong Te-Te bonds (bond distances < 2.9 Å) in the two minerals, which are absent in calaverite (bond distances > 3.2 Å). In both tellurides, trivalent gold occurs in slightly distorted square planar coordination, whereas monovalent gold, partly substituted by monovalent silver, presents a 2+2+2 coordination, corresponding to distorted rhombic bipyramids. The differentiation between bonding and non-bonding contacts is obtained by computation of the Effective Coordination Number (ECoN). The CHARge DIstribution (CHARDI) analysis is satisfactory for both tellurides but suggests that the Te-Te bond in the [Te3]2- anion is not entirely homopolar. Both tellurides can therefore be described as Madelung-type compounds, despite the presence of Te-Te in both structures.

Zero-obliquity twin lattice quasi-symmetry threefold twinning in 1-{(R)-1-[(3-oxo-2-isoindolinoyl)methyl]-2-propenyl}-5-methyl-2,3-indolinedione.

Threefold twinning in 1-{(R)-1-[(3-oxo-2-isoindolinoyl)methyl]-2-propenyl}-5-methyl-2,3-indolinedione, C21H16N2O4, has been reported recently [Trost et al. (2020). Org. Lett. 22, 2584-2589] but the twin characterization was not published. This twinning presents several interesting features. The crystal structure is monoclinic, but its lattice is metrically strongly pseudo-orthorhombic and underpins a strongly pseudo-hexagonal sublattice. Several possible twin laws are compatible with these metric specializations, among which the one found experimentally corresponds to a trichromatic point group. Twinning is by reticular pseudo-merohedry with twin index 2 and zero obliquity but a non-zero twin misfit. The twin lattice coincides with the pseudo-hexagonal sublattice of the individual domain, which justifies the adoption of the unconventional setting B21 of the space group.

Producing highly complicated materials. Nature does it better.

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.

Groupoid description of modular structures.

Modular structures are crystal structures built by subperiodic (zero-, mono- or diperiodic) substructures, called modules. The whole set of partial operations relating substructures in a modular structure build up a groupoid; modular structures composed of identical substructures are described by connected groupoids, or groupoids in the sense of Brandt. A general approach is presented to describe modular structures by Brandt's groupoids and how to obtain the corresponding space groups, in which only the partial operations that have an extension to the whole crystal space appear.

The chromatic symmetry of twins and allotwins.

The symmetry of twins is described by chromatic point groups obtained from the intersection group {\cal H}^* of the oriented point groups of the individuals {\cal H}_i extended by the operations mapping different individuals. This article presents a revised list of twin point groups through the analysis of their groupoid structure, followed by the generalization to the case of allotwins. Allotwins of polytypes with the same type of point group can be described by a chromatic point group like twins. If the individuals are all differently oriented, the chromatic point group is obtained in the same way as in the case of twins; if they are mapped by symmetry operation of the individuals, the chromatic point group is neutral. If the same holds true for some but not all individuals, then the allotwin can be seen as composed of twinned regions described by a twin point group, that are then allotwinned and described by a colour identification group; the allotwin is then described by a chromatic group obtained as an extension of the former by the latter, and requires the use of extended symbols reminiscent of the extended Hermann-Mauguin symbols of space groups. In the case of allotwins of polytypes with different types of point groups, as well as incomplete (allo)twins, a chromatic point group does not reveal the full symmetry: the groupoid has to be specified instead.

Crystal structure and XANES investigation of petzite, Ag3AuTe2.

Petzite, Ag3AuTe2, crystallizes in the space group I4132, which is a Sohncke type of space group where chiral crystal structures can occur. The structure refinement of petzite reported long ago [Frueh (1959). Am. Mineral. 44, 693-701] did not provide any information about the absolute structure. A new single-crystal X-ray diffraction refinement has now been performed on a sample from Lake View Mine, Golden Mile, Kalgoorlie, Australia, which has resulted in a reliable absolute structure [a Flack parameter of 0.05 (3)], although this corresponds to the opposite enantiomorph reported previously. The minimum Te-Te distance is 3.767 (3) Å, slightly shorter than the van der Waals bonding distance, which suggests a weak interaction between the two chalcogens. XANES spectra near the Au and Te LIII edges suggest that the chemical-bonding character of Au in petzite is more metallic than in other gold minerals.

Plesiotwins versus diperiodic twins.

Plesiotwins and diperiodic twins have in common the fact of being characterized by a low degree of lattice restoration. Plesiotwins differ from twins by the fact that the relative orientation of the individuals is obtained by a non-crystallographic rotation about the normal to the composition plane, whereas for twins this rotation is crystallographic, apart from possible small deviations coming from metric pseudosymmetries. In the case of plesiotwins, the low degree of lattice restoration comes from a large coincidence site lattice (CSL) in the composition plane. Diperiodic twins, instead, have a small CSL in the composition plane but the second plane of the same family contributing to the overall lattice restoration is too far away from the composition plane to be considered significant. It is shown that plesiotwins can occur as reflection twins if the composition plane is not parallel to the twin plane, and as rotation twins in the case of parallel hemitropy. Diperiodic twins can in principle occur in any category, but either the metric conditions to obtain a diperiodic twin are actually in contrast with the metric pseudosymmetry required for twinning or the result is actually a hybrid twin. This justifies why no confirmed examples of diperiodic twins are known to date.

Comment on the article 'Protein crystal lattices are dynamic assemblies: the role of conformational entropy in the protein condensed phase'.

A comment is given on Dimova & Devedjiev [IUCrJ (2018), 5, 130-140].

Extrinsic faulting in 3C close-packed crystal structures: computational mechanics analysis.

Extrinsic faulting has been discussed previously within the so-called difference method and random walk calculation. In this contribution it is revisited under the framework of computational mechanics, which allows expressions to be derived for the statistical complexity, entropy density and excess entropy as a function of faulting probability. The approach allows one to compare the disordering process of an extrinsic fault with other faulting types. The ℇ-machine description of the faulting mechanics is presented. Several useful analytical expressions such as probability of consecutive symbols in the Hägg coding are presented, as well as hexagonality. The analytical expression for the pairwise correlation function of the layers is derived and compared with results previously reported. The effect of faulting on the interference function is discussed in relation to the diffraction pattern.

The crystallographic chameleon: when space groups change skin.

Volume A of International Tables for Crystallography is the reference for space-group information. However, the content is not exhaustive because for many space groups a variety of settings may be chosen but not all of them are described in detail or even fully listed. The use of alternative settings may seem an unnecessary complication when the purpose is just to describe a crystal structure; however, these are of the utmost importance for a number of tasks, such as the investigation of structure relations between polymorphs or derivative structures, the study of pseudo-symmetry and its potential consequences, and the analysis of the common substructure of twins. The aim of the article is twofold: (i) to present a guide to expressing the symmetry operations, the Hermann-Mauguin symbols and the Wyckoff positions of a space group in an alternative setting, and (ii) to point to alternative settings of space groups of possible practical applications and not listed in Volume A of International Tables for Crystallography.

Charge distribution as a tool to investigate structural details. IV. A new route to heteroligand polyhedra.

A new route to apply the charge distribution (CHARDI) method to structures based on heteroligand coordination polyhedra is presented. The previous algorithm used scale factors computed in an iterative way based on the assumption (which turned out to be not always correct) that a real over-under bonding effect affects mainly the anionic charges of each single anion, without grossly modifying the total charge of each type of anion. The new, more general approach is not based on any a priori assumption but treats separately the homoligand sub-polyhedra and attributes to each type of atom a fraction of the charge of the atom coordinated to it, computed in a self-consistent iterative way. The distinction between the bonding and non-bonding contact is also redefined in terms of the mean fictive ionic radii (MEFIR), without the need of an empirical parameter, used in the previous algorithm. CHARDI equations are generalized in terms of the new approach and a series of examples is presented.

Twinning of aragonite - the crystallographic orbit and sectional layer group approach.

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 C2/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.

Charge distribution as a tool to investigate structural details. III. Extension to description in terms of anion-centred polyhedra.

The charge distribution (CHARDI) method is a self-consistent generalization of Pauling's concept of bond strength which does not make use of empirical parameters but exploits the experimental geometry of the coordination polyhedra building a crystal structure. In the two previous articles of this series [Nespolo et al. (1999). Acta Cryst. B55, 902-916; Nespolo et al. (2001). Acta Cryst. B57, 652-664], we have presented the features and advantages of this approach and its extension to distorted and heterovalent polyhedra and to hydrogen bonds. In this third article we generalize CHARDI to structures based on anion-centred polyhedra, which have drawn attention in recent years, and we show that computations based on both descriptions can be useful to obtain a deeper insight into the structural details, in particular for mixed-valence compounds where CHARDI is able to give precise indications on the statistical distribution of atoms with different oxidation number. A graph-theoretical description of the structures rationalizes and gives further support to the conclusions obtained via the CHARDI approach.

Analysis of the structural continuity in twinned crystals in terms of pseudo-eigensymmetry of crystallographic orbits.

The reticular theory of twinning gives the necessary conditions on the lattice level for the formation of twins. The latter are based on the continuation, more or less approximate, of a substructure through the composition surface. The analysis of this structural continuity can be performed in terms of the eigensymmetry of the crystallographic orbits corresponding to occupied Wyckoff positions in the structure. If [Formula: see text] is the space group of the individual and [Formula: see text] a space group which fixes the twin lattice obtained as an intersection of the space groups of the individuals in their respective orientations, then a structural continuity is obtained if (1) the eigensymmetry of an orbit under [Formula: see text] contains the twin operation; (2) the eigensymmetry of a union of orbits under [Formula: see text] contains the twin operation; (3) the eigensymmetry of a split orbit under [Formula: see text] contains the twin operation; or (4) the eigensymmetry of a union of split orbits under [Formula: see text] contains the twin operation. The case of the twins in melilite is analysed: the (approximate) restoration of some of the orbits explains the formation of these twins.

Effects of merohedric twinning on the diffraction pattern.

In merohedric twinning, the lattices of the individuals are perfectly overlapped and the presence of twinning is not easily detected from the diffraction pattern, especially in the case of inversion twinning (class I). In general, the investigator has to consider three possible structural models: a crystal with space-group type H and point group P, either untwinned (H model) or twinned through an operation t in vector space (t-H model), and an untwinned crystal with space group G whose point group P' is obtained as an extension of P through the twin operation t (G model). In 71 cases, consideration of the reflection conditions may directly rule out the G model; in seven other cases the reflection conditions suggest a space group which does not correspond to the extension of H by the twin operation and the structure solution or at least the refinement will fail. When the twin operation belongs to a different crystal family (class IIB twinning: the crystal has a specialized metric), the presence of twinning can often be recognized by the peculiar effect it has on the reflection conditions.

The staurolite enigma solved.

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.