The transformation matrices (distortion, orientation, correspondence), their continuous forms and their variants

Three transformation matrices (distortion, orientation and correspondence) define the crystallography of displacive phase transformations. This article explains how to calculate them and their variants, and why they should be distinguished.

Automatic calculation of symmetry-adapted tensors in magnetic and non-magnetic materials: a new tool of the Bilbao Crystallographic Server

Two new tools hosted on the Bilbao Crystallographic Server are presented. The programs permit the automatic calculation of symmetry-adapted forms of tensor properties for magnetic and non-magnetic groups. The cases of equilibrium, transport, optical and nonlinear optical susceptibility tensors are studied separately.

Fast analytical evaluation of intermolecular electrostatic interaction energies using the pseudoatom representation of the electron density. II. The Fourier transform method

Numerical implementations of the presented Fourier transform method and the previously reported Löwdin α-function approach for analytical determination of the two-center Coulomb integrals that appear in calculations of the electrostatic interaction energies between pseudoatom-based charge distributions are carefully examined in terms of precision and speed. The refined Fortran-based computer code allows a fast evaluation of electrostatic interaction energies with a precision of 5 × 10⁻⁵ kJ mol⁻¹ or better using either of the two techniques.

Solving the disordered structure of β-Cu_2−xSe using the three-dimensional difference pair distribution function

Using three-dimensional difference pair distribution function analysis of single-crystal diffuse X-ray scattering, the disordered structure copper selenide (β-Cu_2−xSe) at room temperature is solved. The structure is ordered in two dimensions but disordered in the third.

Experimentally obtained and computer-simulated X-ray asymmetric eight-beam pinhole topographs for a silicon crystal

Experimentally obtained eight-beam pinhole topographs for a silicon crystal were compared with computer simulations based on the n-beam Takagi–Taupin equation and Ewald–Laue theory.

Experimentally obtained and computer-simulated X-ray non-coplanar 18-beam pinhole topographs for a silicon crystal

Experimentally obtained non-coplanar 18-beam pinhole topographs were compared with computer simulations based on the Ewald–Laue theory.

Identification of the impurity phase in high-purity CeB₆ by convergent-beam electron diffraction

The impurity phase in high-purity CeB₆ is unequivocally identified by first determining its space group, using an iterative convergent-beam electron diffraction approach, and following this with an atomic structure confirmation by quantitative convergent-beam electron diffraction and high-angle annular dark-field scanning transmission electron microscopy.

A hidden Markov model for describing turbostratic disorder applied to carbon blacks and graphene

A mathematical framework for analysing aperiodic crystals encompassing a wide range of turbostratic disorders, including disorder of the first and second type, is presented. The framework uses the theory of hidden Markov models and is applied to carbon blacks and graphene.

Dependence of X-ray asymmetrical Bragg case plane-wave rocking curves on the deviation from exact Bragg orientation in and perpendicular to the diffraction plane

The rocking-curve dependence on the deviation of an incident X-ray plane wave from the exact Bragg orientation in and perpendicular to the diffraction plane for the asymmetrical Bragg case is investigated.

The characteristic radiation of copper Kα_1,2,3,4

The characterization of Cu Kα_1,2,3,4 radiation is presented, including the 2p satellite. The details are robust enough to be transferable to other experiments for calibration and reference.

The polytopes of the H₃ group with 60 vertices and their orbit decompositions

A description of polytopes with 60 vertices generated by the finite reflection group H₃ is given and a decomposition of their vertices into orbits of lower-symmetry groups is provided.

The chromatic symmetry of twins and allotwins

The chromatic symmetry of twins is extended to the case of allotwins through a groupoid analysis.

On two special classes of parallelohedra in E⁶

The structures of quasicrystals can be considered as sections of lattices of translations in higher dimensions, which has greatly stimulated the investigation of lattices in arbitrary dimensions. The structure of the cone of positive definite quadratic forms in E^d×d is investigated for d = 6. A partition of the cone in Φ and Σ domains is performed and its local symmetries are determined.

Gröbner–Shirshov bases for non-crystallographic Coxeter groups

A Gröbner–Shirshov basis and the corresponding standard monomials for the non-crystallographic Coxeter group H₄ are constructed.

A space for lattice representation and clustering

Algorithms for defining the difference between two lattices are described. They are based on the work of Selling and Delone (Delaunay).