A Tensor Classification of Twinning in Crystals

Abstract

A classification scheme for twinning in crystals is proposed. It is based on a tensor distinction of properties across the twin interface. The classification employs concepts from the theory of transformation twinning. The adequacy of such a scheme for other main types of twins, namely growth twins and nonferroelastic mechanical twins, is examined, and found to be satisfactory. All twins can be divided into four fundamentally distinct categories: ferroelastic or S-twins, nonferroelastic-ferroic or N-twins, Bollmann or B-twins and translation or T-twins. A comparison is made of the attributes of these four types of twins. A compact and informative twin symbol is introduced, which carries information about the tensor properties in which the twins differ. Information about any underlying total or partial coincidence sublattice is incorporated by attaching a subscript t or p to the main twin symbol. Several examples are discussed. This paper also introduces a rigorous definition of prototype symmetry in terms of the 'nondisruption condition'.

Multiple twinning in cubic crystals: geometric/algebraic study and its application for the identification of the Σ3ngrain boundaries

“…The colour group introduced by Shubnikov & Koptsik (1974) (see also Senechal, 1983) is now integrated in the modern theories of twinning. Crystallographers working in mineralogy (Wadhawan, 1997; and those working in physics of ferroelectric domains (Janovec, 1976) have made a synthesis of their respective approaches. This synthesis uses mathematical tools based on modern group theory such as orbits, stabilisers, coset partitioning etc.…”

Distinction of magnetic non-ferroelastic domains

“…In this paper, we turn our attention to a special property of morphic tensor components in non-ferroelastic domain pairs which simplifies the analysis of tensor distinction of non-ferroelastic domains. Wadhawan (1997Wadhawan ( , 2000 has tabulated examples of nonferroelastic domain pairs distinguished by physical property tensors, e.g. chromium oxide distinguished by components of the magnetoelectric tensor (Newnham & Cross, 1974a,b) and Pb 5 Ge 3 O 11 (Toledano & Toledano, 1976) distinguished by components of spontaneous polarization, compliance and optical gyration tensors.…”

Groupoid of orientational variants

“…Thus, K is in general not a group. A rigorous approach to twinning by applying some concepts of reconstructive transitions (in terms of the non-disruption condition) and by using the intersection group presented in equation (2) is given by Wadhawan (1997).…”