Tables for
Volume D
Physical properties of crystals
Edited by A. Authier

International Tables for Crystallography (2013). Vol. D, ch. 3.4, p. 528

Section Domain pairs in the microscopic description

V. Janoveca* and J. Přívratskáb

aInstitute of Physics, Academy of Sciences of the Czech Republic, Na Slovance 2, CZ-18221 Prague 8, Czech Republic, and bDepartment of Mathematics and Didactics of Mathematics, Technical University of Liberec, Hálkova 6, 461 17 Liberec 1, Czech Republic
Correspondence e-mail: Domain pairs in the microscopic description

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In the microscopic description, two microscopic domain states [{\sf S}_{i}] and [{\sf S}_{k}] with space-group symmetries [{\cal F}_i] and [{\cal F}_k], respectively, can form an ordered domain pair ([{\sf S}_{i},{\sf S}_{k}]) and an unordered domain pair [\{{\sf S}_{i},{\sf S}_{k}\}] in a similar way to in the continuum description, but one additional aspect has to be considered. The definition of the symmetry group [{\cal F}_{ik}] of an ordered domain pair ([{\sf S}_{i},{\sf S}_{k} ]), [{\cal F}_{ik} = {\cal F}_i \cap {\cal F}_k, \eqno( ]is meaningful only if the group [{\cal F}_{ik}] 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[link] ) [{\cal T}_{ik}={\cal T}_i \cap {\cal T}_k, \eqno( ]where [{\cal T}_i] and [{\cal T}_k] are translation subgroups of [{\cal F}_i] and [{\cal F}_k], respectively. This condition is fulfilled if both domain states [{\sf S}_{i}] and [{\sf S}_{k} ] 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[link], equation ([link])].

Example Domain pairs in calomel.  Calomel undergoes a non-equitranslational phase transition from a tetragonal parent phase to an orthorhombic ferroelastic phase (see Example[link] in Section[link]). Four basic microscopic single-domain states are displayed in Fig.[link]. 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.[link], where the first microscopic domain state [{\sf S}_{i} ] participating in a domain pair is displayed in the upper cell (light grey) and the second domain state [{\sf S}_{j}], [j=2,3], in the lower white cell. The overlapping structure in the middle (dark grey) is a geometrical representation of the domain pair [\{{\sf S}_1,{\sf S}_j\} ].


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Domain pairs in calomel. Single-domain states in the parent clamping approximation are those from Fig. The first domain state of a domain pair is shown shaded in grey (`black'), the second domain state is colourless (`white'), and the domain pair of two interpenetrating domain states is shown shaded in dark grey. (a) Ferroelastic domain pair [({{\sf S}_1},{{\sf S}_3}) ] in the parent clamping approximation. This is a partially transposable domain pair. (b) Translational domain pair [({{\sf S}_1},{{\sf S}_2}) ]. This is a completely transposable domain pair.

The domain pair [\{{\sf S}_1,{\sf S}_3\}], depicted in Fig.[link](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 [{\sf S}_1 ] and [{\sf S}_3].

Domain pair [\{{\sf S}_1,{\sf S}_2\}], shown in Fig.[link](b), is a translational (antiphase) domain pair in which domain states [{\sf S}_1] and [{\sf S}_2] differ only in location but not in orientation. The unit cell (heavily outlined small square) of the domain pair [\{{\sf S}_1,{\sf S}_2\}] is identical with the unit cell of the tetragonal parent phase (cf. Fig.[link]).

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.[link]

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