Beyond space groups: the arithmetic symmetry of deformable multilattices

Abstract: It is well known that the problem of classifying the symmetry of simple lattices leads to consideration of the conjugacy properties of the holohedral crystallographic point groups (`holohedries'). Classical results for the three-dimensional case then state that: (i) the orthogonal classification of the holohedries subdivides the simple lattices into the familiar seven crystal systems (this gives the `geometric symmetry' of simple lattices); (ii) the stricter arithmetic classification of the holohedries subdivides the three-dimensional simple lattices into the well known fourteen Bravais lattice types (this gives the `arithmetic symmetry' of simple lattices, which is more refined than the geometric one). There exists an analogous problem of studying the symmetry of the more complex periodic structures in three dimensions (`multilattices', that is, finite unions of translates of a given simple lattice), which describe in more detail the atomic lattices of the crystalline materials found in nature. In this case, the groups of affine isometries that leave a multilattice invariant, called the `space groups', must be considered. Well known results subdivide the space groups into 219 affine conjugacy (or isomorphism) classes. This corresponds to classifying the `geometric symmetry' of tridimensional multilattices. In crystallography, there does not exist a classical counterpart for multilattices of the above-mentioned arithmetic symmetry of simple lattices. In this paper, a natural framework is proposed in which to study the `arithmetic symmetry of multilattices' and it is shown that the latter gives a finer classification than that based on the 219 classes of space groups, even if site symmetry is taken into account. This approach originates from the investigation of the changes of symmetry in deformable crystalline solids and proves useful for the modelling of phase transitions in crystals and related phenomena.