Chapter 3.2 Twinning and domain structures

Physical properties of crystals

First Online Edition (2006)

Part 3. Symmetry aspects of phase transitions, twinning and domain structures

  1. V. Janovec1,
  2. Th. Hahn2,
  3. H. Klapper3

Published Online: 1 JAN 2006

DOI: 10.1107/97809553602060000643

International Tables for Crystallography

International Tables for Crystallography

How to Cite

Janovec, V., Hahn, T. and Klapper, H. 2006. Twinning and domain structures. International Tables for Crystallography. D:3:3.2:377–392.

Author Information

  1. 1

    Department of Physics, Technical University of Liberec, Hálkova 6, 461 17 Liberec 1, Czech Republic

  2. 2

    Institut für Kristallographie, Rheinisch–Westfälische Technische Hochschule, D‐52056 Aachen, Germany

  3. 3

    Mineralogisch‐Petrologisches Institut, Universität Bonn, D‐53113 Bonn, Germany

Publication History

  1. Published Online: 1 JAN 2006



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 Romé 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.


  • 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