3.2 Twinning and domain structures

This chapter forms the introduction to the treatment of twinning in Chapter 3.3 and of domain structures in Chapter 3.4 . It starts with a historical overview of twinning (beginning with a paper by Rome de l'Isle from 1783) and continues with the history of the various forms of domain structures: ferromagnetism, ferroelectricity and ferroelasticity, summarized as ferroic by Aizu in 1970. This historical survey is followed by a brief excursion into the rather new field of bicrystallography and grain boundaries. The major part of the chapter is concerned with an extended exposition of the mathematical tools needed in the subsequent parts, especially in Chapter 3.4 . One section introduces the basic concepts of set theory and explains the notion of unordered and ordered pairs, mappings of sets and the partition of a set into equivalence classes. The next section deals with basic group theory and is devoted mainly to group–subgroup relations and relevant notions, of which black-and-white and colour groups and coset decompositions of a group into left and double cosets are of central importance. In the final section, group theory is combined with set theory in the ‘action of a group on a set’ which represents an effective algebraic tool for the symmetry analysis of domain structures. The notions of stabilizer, orbit and stratum are explained and their significance in the analysis is illustrated by concrete examples. Keywords: bicrystallography; bicrystals; black and white symmetry groups; coincidence-site lattice; conjugate subgroups; cosets; daughter phase; dichromatic complexes; dichromatic groups; domain structures; domains; double cosets; equivalence classes; equivalence relation; ferroelectric domain structures; ferroic domains; mappings; normalizers; orbit; parent phases; partition; sets; stabilizers; twinning