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.

Symmetry breaking in monoatomic 2-lattices

“…5 We recall (see [25,26]) that the finite group Λ(εσ) carries more information than the mere point group of M(εσ): indeed, unlike with the latter, given the lattic group Λ(εσ) it is possible to reconstruct uniquely the (isomorphism class of the) space group of M(εσ) -see Proposition 5 in [25]. Furthermore, the arithmetic classification of multilattices is in general finer than their space-group classification.…”

“…We notice that certain nonessential descriptors ε σ of monoatomic 2-lattices actually give a 1-lattice in A 3 (see for instance [25,26]). Explicitly, in the configuration space D 3,1 the nonessential descriptors are those for which p = 1 2 β a e a + t, where t ∈ L(e a ) and the numbers β a are either (1, 1, 1) or a permutation of (1, 1, 0) or (1, 0, 0).…”

“…As discussed in [25,26] (see also Proposition 1 in [15]), the indeterminacy in the choice of the essential descriptors of a given 2-lattice (up to overall translations) leads to consider the following discrete group of 'global symmetry' (see [24]): 3…”

“…A general theory for the investigation of the arithmetic symmetry of crystalline lattices has been proposed in [25,26], based on earlier work [6,22,24] on crystal symmetry and the invariance of their constitutive equations. This approach, summarised in § §2-3, has been applied in [14,15] to obtain a systematic classification of the distinct arithmetic crystalline types in the two simplest cases, that is, for the two-and three-dimensional lattices with two identical atoms in their unit cells (called respectively the 'monoatomic 2-nets' and 'monoatomic 2-lattices' 1 ).…”

Frameworks with crystallographic symmetry

“…While not implicating an edge structure, the study of multi-lattices envisaged in [13,14] is related to the kinematics of phase transitions in crystalline materials. When approached from the point of view of periodic sphere packings, as in [6,15], multi-lattices do acquire an edge structure from contacts between spheres.…”