On symmetries of crystals with defects related to a class of solvable groups (S₁)

Abstract

We find the geometrical symmetries of discrete structures which generalize the perfect lattices of crystallography to account for the existence of continuous distributions of defects.

Group Elastic Symmetries Common to Continuum and Discrete Defective Crystals

“…In this paper we focus on S 1 , one of the two nonisomorphic classes of connected, simply connected, non-compact, three-dimensional solvable groups with a discrete subgroup D. We define this group below and give the form of the corresponding dislocation density tensor. Further details regarding derivation of facts about S 1 that we state here can be found in Nicks and Parry [19].…”

“…In section 6, we improve upon a result given in [19] by invoking the 'substitution test' of Johnson [13] to give necessary and sufficient conditions that any symmetry of D (i.e. any change of generators of D) extend to an automorphism of S 1 .…”

“…In fact we concentrate on a particular class of (solvable) Lie groups, cf. Nicks and Parry [19], and show that some symmetries of the discrete subgroup can indeed be extended to automorphisms of G, and that others cannot be so extended i.e some symmetries of the discrete structure are elastic, some are inelastic.…”

“…Next, we consider the symmetries of the discrete subgroup D. Given that we restrict attention to a particular canonical Lie group G = (R 3 , ψ), and given the above interpretation of the arguments of w, the task in fact requires that we find all sets of generators of D. We give an algorithm which, given any set of generators, puts the set in the form (5.5), and quote necessary and sufficient conditions, from Nicks and Parry [19], that group elements of the form (5.5) generate D. Then, any set of generators is obtained from a set of the form (5.5) (satisfying the necessary and sufficient conditions given in [19]) by a sequence of elementary operations -the Nielsen transformations described in Johnson [13].…”

“…Recall that, given a group G with composition function ψ, the structure constants C ijk of the corresponding Lie algebra g are given by 20) where ψ = ψ i (x, y)e i , and the Lie bracket operation [·, ·] : 21) with respect to some basis {e 1 , e 2 , e 3 } of R 3 . The vector space R 3 and the operation [·, ·] make up the Lie algebra g corresponding to the Lie group G = (R 3 , ψ).…”