Boris Gruber's contributions to mathematical crystallography.

Boris Gruber made fundamental contributions to the study of crystal lattices, leading to a finer classification of lattice types than those of Paul Niggli and Boris Delaunay before him.

Thermoelectric transport properties in magnetically ordered crystals. Further corrigenda and addenda.

Further corrigenda and addenda for the article by Grimmer [Acta Cryst. (2017), A73, 333-345] are reported. New figures in the supporting information show how the restrictions on the forms of galvanomagnetic and thermomagnetic tensors are related to those on corresponding thermoelectromagnetic tensors.

Thermoelectric transport properties in magnetically ordered crystals. Corrigendum and addenda.

A correction and additions concerning the limiting point groups are made to the article by Grimmer [Acta Cryst. (2017), A73, 333-345].

Thermoelectric transport properties in magnetically ordered crystals.

The forms of the tensors describing thermoelectric transport properties in magnetically ordered crystals are given for frequently used orientations of the 122 space-time point groups up to second order in an applied magnetic field. It is shown which forms are interchanged for the point groups of the hexagonal crystal family by two different conventions for the connection between the Hermann-Mauguin symbol and the orientation of the Cartesian coordinate system. The forms are given in Nye notation, which conspicuously shows how the forms for different point groups are related.

Partial order among the 14 Bravais types of lattices: basics and applications.

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.

Density functional calculations of polysynthetic Brazil twinning in α-quartz.

Polysynthetic Brazil twinning in α-quartz, which occurs commonly in amethyst, is interpreted in the literature as having its composition planes parallel to one of the faces of the major rhombohedron r. It is shown that, instead, the composition planes are parallel to one of the faces of the minor rhombohedron z. The proposed translation 0.

Opechowski-Guccione-like symbols labelling magnetic space groups independent of tabulated (0, 0, 0)+ sets.

For the magnetic space-group types with a black and white lattice two sets of symbols have been proposed: the BNS symbols [Belov et al. (1957). Sov.

Comments on tables of magnetic space groups.

Litvin [Acta Cryst. (2008), A64, 419-424 and supplementary material] extends much of the information contained in Volume A of International Tables for Crystallography for the 230 space-group types to the 1651 types of Shubnikov space groups, using Opechowski-Guccione (OG) notation for the space groups with a black-white lattice. It is pointed out that OG notation has crucial disadvantages compared to Belov-Neronova-Smirnova (BNS) notation.

Elastic properties of two-dimensional quasicrystals.

Quasicrystals (QC) with two-dimensional quasiperiodic and one-dimensional periodic structure are considered. Their symmetry can be described by embedding the three-dimensional physical space V(E) in a five-dimensional superspace V, which is the direct sum of V(E) and a two-dimensional internal space V(I). A displacement v in V can be written as v = u + w, where u in V(E) and w in V(I).

The piezoelectric effect of second order in stress or strain: its form for crystals and quasicrystals of any symmetry.

The restrictions on the coefficients describing physical effects depend on the orientation of the symmetry elements of the crystal or quasicrystal with respect to the Cartesian coordinate system employed. They are given for the piezoelectric effect of second order in stress or strain for all the orientations that can be expressed by the sequence of elements in the Hermann-Mauguin symbol of the point group. In the literature, the restrictions are usually given only for a particular orientation, which sometimes is not specified.

A simple method of determining the form of the phonon part of tensors describing quasicrystal properties.

It is shown that the restrictions on the form of property tensors of rank <5 that follow from the Neumann principle for the point groups describing quasicrystals can easily be deduced from the restrictions for the point groups describing ordinary crystals. For octagonal and dodecagonal point groups, this is true even for property tensors of rank<8 and <12, respectively. The results derived in a number of papers for various physical properties of quasicrystals with certain point-group symmetries are generalized to all quasicrystal point groups, and it is shown that the results become more lucid if the classification of quasicrystal point groups with a principal axis into pentagonal, decagonal, octagonal and dodecagonal ones is done appropriately.

Spectral decomposition of the linear elastic tensor for trigonal symmetry; classification of symmetry restrictions for arbitrary point groups.

The linear compliance tensor for trigonal symmetry has four different eigenvalues, two of which have multiplicity 1, the others multiplicity 2. They and the corresponding eigenvectors have been calculated in terms of the seven parameters of the corresponding Voigt matrix. Necessary and sufficient conditions have been derived for these components to guarantee positive eigenvalues and thus a positive strain energy.

Quartz aggregates revisited.

Quartz aggregates that formed by coalescence of quartz crystals in the magma or in hydrothermal solution are considered. If the individuals have rhombohedral faces in contact, there will be two special cases: parallel intergrowths and intergrowths that agree in orientation and contact plane with Esterel twins grown from a twinned nucleus. For all other known cases, i.

Twinning by reticular pseudo-merohedry in trigonal, tetragonal and hexagonal crystals.

Twin laws for trigonal, tetragonal and hexagonal crystals describing twins with principal axes inclined by an angle Phi > 0 are analysed. Twins by reticular merohedry (i.e.

Determination of all misorientations of tetragonal lattices with low multiplicity; connection with Mallard's rule of twinning.

Two congruent lattices are considered, which are misoriented in such a way that they have a fraction 1/Sigma of symmetry translations in common. Whereas for cubic lattices body or face centring does not affect the 'multiplicity' or 'twin index' Sigma, this is not generally true for tetragonal lattices. Consider a fixed misorientation and let Sigma(P) and Sigma(I) be the multiplicities for tP and tI lattices with the same axial ratio c/a.