International
Tables for Crystallography Volume D Physical properties of crystals Edited by A. Authier © International Union of Crystallography 2013 |
International Tables for Crystallography (2013). Vol. D, ch. 3.4, p. 528
Section 3.4.3.7. Domain pairs in the microscopic description^{a}Institute of Physics, Academy of Sciences of the Czech Republic, Na Slovance 2, CZ-18221 Prague 8, Czech Republic, and ^{b}Department of Mathematics and Didactics of Mathematics, Technical University of Liberec, Hálkova 6, 461 17 Liberec 1, Czech Republic |
In the microscopic description, two microscopic domain states and with space-group symmetries and , respectively, can form an ordered domain pair () and an unordered domain pair in a similar way to in the continuum description, but one additional aspect has to be considered. The definition of the symmetry group of an ordered domain pair (), is meaningful only if the group is a space group with a three-dimensional translational subgroup (three-dimensional twin lattice in the classical description of twinning, see Section 3.3.8 ) where and are translation subgroups of and , respectively. This condition is fulfilled if both domain states and have the same spontaneous strains, i.e. in non-ferroelastic domain pairs, but in ferroelastic domain pairs one has to suppress spontaneous deformations by applying the parent clamping approximation [see Section 3.4.2.2, equation (3.4.2.49)].
Example 3.4.3.9. Domain pairs in calomel. Calomel undergoes a non-equitranslational phase transition from a tetragonal parent phase to an orthorhombic ferroelastic phase (see Example 3.4.2.7 in Section 3.4.2.5). Four basic microscopic single-domain states are displayed in Fig. 3.4.2.5. From these states, one can form 12 non-trivial ordered single-domain pairs that can be partitioned (by means of double coset decomposition) into two orbits of domain pairs.
Representative domain pairs of these orbits are depicted in Fig. 3.4.3.10, where the first microscopic domain state participating in a domain pair is displayed in the upper cell (light grey) and the second domain state , , in the lower white cell. The overlapping structure in the middle (dark grey) is a geometrical representation of the domain pair .
The domain pair , depicted in Fig. 3.4.3.10(a), is a ferroelastic domain pair in the parent clamping approximation. Then two overlapping structures of the domain pair have a common three-dimensional lattice with a common unit cell (the dotted square), which is the same as the unit cells of domain states and .
Domain pair , shown in Fig. 3.4.3.10(b), is a translational (antiphase) domain pair in which domain states and differ only in location but not in orientation. The unit cell (heavily outlined small square) of the domain pair is identical with the unit cell of the tetragonal parent phase (cf. Fig. 3.4.2.5).
The two arrows attached to the circles in the domain pairs represent exaggerated displacements within the wall.
Domain pairs represent an intermediate step in analyzing microscopic structures of domain walls, as we shall see in Section 3.4.4.