Volume 230, Issue 12 - Special Issue: Mathematical Crystallography / Guest Editors: Massimo Nespolo and Mois Aroyo

As crystallography merges with materials science and engineering, mathematical crystallography is growing in new directions, including: Characterizing new materials with unusual properties; Imaging, including but not limited to diffraction; Exploring and exploiting superspaces; Mapping the aperiodic landscape, from chaos to classical periodicity and beyond; Re-modeling the structures of real crystals, both periodic and aperiodic; Modeling self-assembly and self-reorganization on the nanoscale. In short, it’s not (just) about space groups and tilings anymore.

A crystal structure may be described by a periodic graph, so the automatic generation of periodic graphs satisfying given criteria should prove useful in understanding crystal structure and developing a theory for crystal design. We present an algorithm for constructing periodic graphs, based on a discrete variant of “turtle geometry,” which we formalize using isometries represented as affine transformations. This algorithm is given information about the putative orbits (kinds) of vertices and edges and produces a finite graph which is then collapsed into a unit cell of a periodic graph. We verify that this algorithm will generate any periodic graph.

As the second paper of the series, the first of which was published in 1997 (M. Tanemura, T. Matsumoto, Density of the p2gg -4c1 packing of ellipses (I). Z. Kristallogr. 1997 , 212 , 637), the p2gg -4c1(b) packing of identical ellipses where four ellipses are included in the rectangular unit cell and where every ellipse has six contacting neighbors, is discussed. It is shown here that the p2gg -4c1(b) packing of ellipses does not exceed the maximum density ρ = π/12$\rho \; = \;\pi /\sqrt {12} $ through numerical computations and series expansions. It is also shown that the p2gg -2a2 packing of ellipses has the similar properties and that it is considered as a special case of p2gg -4c1(b) packing. The method of computing the density for every parameter values of aspect ratio and tilt angle is given.

Point groups consist of rotations, reflections, and roto-reflections and are foundational in crystallography. Symmorphic space groups are those that can be decomposed as a semi-direct product of pure translations and pure point subgroups. In contrast, Bieberbach groups consist of pure translations, screws, and glides. These “torsion-free” space groups are rarely mentioned as being a special class outside of the mathematics literature. Every space group can be thought of as lying along a spectrum with the symmorphic case at one extreme and Bieberbach space groups at the other. The remaining nonsymmorphic space groups lie somewhere in between. Many of these can be decomposed into semi-direct products of Bieberbach subgroups and point transformations. In particular, we show that those 3D Sohncke space groups most populated by macromolecular crystals obey such decompositions. We tabulate these decompositions for those Sohncke groups that admit such decompositions. This has implications to the study of packing arrangements in macromolecular crystals. We also observe that every Sohncke group can be written as a product of Bieberbach and symmorphic subgroups, and this has implications for new nomenclature for space groups.

The paper deals with | F | 2 , A and D Patterson functions. The information available in the various volumes of International Tables for Crystallography is detailed. A and D are defined and the reason for their use is explained. The symmetry of these functions is explained. Calculation by symmetry-adapted Patterson functions and the Fast-Fourier algorithm are described. There are two examples. There is a very useful table presenting the relevant information.

Even though a lattice and its sublattices have the same group of coincidence isometries, the coincidence index of a coincidence isometry with respect to a lattice Λ 1 and to a sublattice Λ 2 may differ. Here, we examine the coloring of Λ 1 induced by Λ 2 to identify how the coincidence indices with respect to Λ 1 and to Λ 2 are related. This leads to a generalization of the notion of color symmetries of lattices to what we call color coincidences of lattices. Examples involving the cubic and hypercubic lattices are given to illustrate these ideas.

The symmetry theory of halving objects into two identical fragments is developed and illustrated by examples. La coupe du roi is a fascinating way to slice an apple into chiral fragments with the same handedness. It may be applied to any object with point group symmetry 2 or 222 and their supergroups, e.g. to cones, cylinders or spheres. Variants of coupes du roi are presented and applied to a cube. An object is divided into halves by a number of cuts. The important cuts extend to the centre of the object, and these must form a single closed loop. The symmetry of the division into two fragments is given by the point group of the decorated object, i.e. the object including all the cuts. Point groups comprising only rotations result in chiral fragments with the same handedness. Such divisions may illustrate the reaction path for changing the sense of chirality of molecules via a dimeric achiral transition state rather than by deformation. Non-trivial divisions of objects into achiral fragments or into racemic chiral fragments are obtained with point groups, again of the decorated object, comprising roto-inversions X ̅, excepting X  = 4 n  + 2. All other point groups either do not lead to half-objects, or they result in a trivial single planar cut parallel to a mirror plane.

Neutron powder diffraction measurements on the 35 % La-substituted Bi 1− x La x Fe 0.5 Sc 0.5 O 3 composition revealed that the samples obtained under high-pressure (6 GPa) and high-temperature (1500 K) conditions crystalize into a distorted perovskite structure with the orthorhombic Pnma symmetry and the unit cell parameters: a o = 5.6745(2) Å, b o = 7.9834(3) Å and c o = 5.6310(2) Å. A long-range magnetic ordering takes place below 220 K and implies a G -type magnetic structure with the moments 4.10(4) μ B per Fe aligned predominately along the orthorhombic c -axis. The space group representation theory using the orthorhombic symmetry yields four bi-linear coupling schemes for the magnetic order parameters imposed by antisymmetric exchange interactions. The couplings are analysed based on symmetry adapted distortion modes defined in respect of the undistorted cubic perovskite structure. The approach allows a quantitative estimation of the coupling strength. It is shown that the experimentally found spin configuration combines the magnetic order parameters coupled by the atomic displacement modes with the largest amplitudes. The results indicate that the antisymmetric exchange is the dominant anisotropic term which fully controls the direction of the Fe 3+ spins in the distorted perovskite lattice.