On ν+1-lattices

Abstract

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].

Symmetry breaking in monoatomic 2-lattices

“…See [7,8,23,24] for more details and a proof of this Proposition. A set N (K) with the properties above is called an 'Ericksen-Pitteri (E-P) neighbourhood' of the centre K. As a consequence of the above Proposition, we have that, near K, 'symmetry cannot increase':…”

“…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 ).…”

Cauchy–Born Rule and the Stability of Crystalline Solids: Static Problems

“…We will denote the lattice by L. Lattices are divided into simple and complex lattices (ν + 1−lattices in the terminology of [24]). A complex lattice can be viewed as the union of congruent simple lattices.…”

A polycrystal based numerical investigation on the temperature dependence of slip resistance and texture evolution in magnesium alloy AZ31B

“…Different to dislocation slip, however, deformation twinning will result in the rotation of the lattice in the region of the crystal that has gone through twinning (Kelly and Knowles, 2012;Pitteri, 1985Pitteri, , 1986). …”