The aim of this paper is to extend the results of [7, §2] to crystals having a more complex structure than monatomic crystals. These have been described in [7, (1.10)] by means of a simple Bravais lattice. The Introduction of [7] serves as a motivation for this paper too. Moreover, we presume the reader to be familiar with the notation and the content of [7], which we shall use here without further comment.Ericksen [2] and Parry [6] extended to diatomic crystals some of the considerations I presented in [7, §l(y)]. Moreover, rules similar to those discussed by Parry [6] are used in experimental work on crystals and may be found also in the physical literature on electron bands and wave propagation in crystals, where "Brillouin zones" or "Wigner-Seitz zones" are selected. An extensive treatment of zones in the framework of quantum theory of electron bands and wave propagation in crystals may be found in the works of Brillouin [1] and Seitz [10]. In the case of diatomic crystals in 2 dimensions, Parry [6] constructs energy density functions that have the general invariance required by the geometric symmetry of such crystals, and he gives rules for constructing the first Wigner-Seitz zone.Let us observe that the qualification "diatomic" above is misleading. We also had to state in [7, §1] the meaning of "monatomic", which is an equally ambiguous adjective. These do not refer to the fact that atoms of only two kinds or of only one kind, respectively, are present in the crystal, but rather specify the periodic structure of the crystal itself. There are crystals that are composed of only one kind of atoms, but cannot be fully described by the simple Bravais lattice associated with their translational invariance. For instance, this is the case of hexagonal close-packed metals like magnesium. These crystals are, strictly speaking, monatomic, but their periodic structure is much like the one of a diatomic crystal. As far as I know, there is no definition in the literature that avoids this confusion, so I need to state one of my own. Let 7/[] [ ++] denote the set of integers [of positive integers].
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.
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