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, pp. 533535
Section 3.4.4.4. Nonferroelastic domain twins and domain walls^{a}Institute of Physics, Academy of Sciences of the Czech Republic, Na Slovance 2, CZ18221 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 
Compatibility conditions impose no restriction on the orientation of nonferroelastic domain walls. Any of the nonferroelastic domain pairs listed in Table 3.4.3.4 can be sectioned on any crystallographic plane p and the sectional group specifies the symmetry properties of the corresponding twin and domain wall. The analysis can be confined to one representative orientation of each class of equivalent planes, but a listing of all possible cases is too voluminous for the present article. We give, therefore, in Table 3.4.4.4 only possible symmetries and of nonferroelastic domain twins and walls, together with their classification, without specifying the orientation of the wall plane p.

Nonferroelastic domain walls are usually curved with a slight preference for certain orientations (see Figs. 3.4.1.5 and 3.4.3.3). Such shapes indicate a weak anisotropy of the wall energy , i.e. small changes of with the orientation of the wall. The situation is different in ferroelectric domain structures, where charged domain walls have higher energies than uncharged ones.
A small energetic anisotropy of nonferroelastic domain walls is utilized in producing tailored domain structures (Newnham et al., 1975). A required domain pattern in a nonferroelastic ferroelectric crystal can be obtained by evaporating electrodes of a desired shape (e.g. stripes) onto a singledomain plate cut perpendicular to the spontaneous polarization . Subsequent poling by an electric field switches only regions below the electrodes and thus produces the desired antiparallel domain structure.
Periodically poled ferroelectric domain structures fabricated by this technique are used for example in quasiphasematching optical multipliers (see e.g. Shur et al., 1999, 2001; Rosenman et al., 1998). An example of such an engineered domain structure is presented in Fig. 3.4.4.3.
Anisotropic domain walls can also appear if the Landau free energy contains a socalled Lifshitz invariant (see Section 3.1.3.3 ), which lowers the energy of walls with certain orientations and can be responsible for the appearance of an incommensurate phase (see e.g. Dolino, 1985; Tolédano & Tolédano, 1987; Tolédano & Dmitriev, 1996; Strukov & Levanyuk, 1998). The irreversible character of domain walls in a commensurate phase of crystals also containing (at least theoretically) an incommensurate phase has been confirmed in the frame of phenomenological theory by Ishibashi (1992). The incommensurate structure in quartz that demonstrates such an anisotropy is discussed at the end of the next example.
Example 3.4.4.2. Domain walls in the α phase of quartz. Quartz (SiO_{2}) undergoes a structural phase transition from the parent phase (symmetry group ) to the ferroic phase (symmetry ). The phase can appear in two domain states and , which have the same symmetry . The symmetry of the unordered domain pair is given by .
Table 3.4.4.5 summarizes the results of the symmetry analysis of domain walls (twins). Each row of the table contains data for one representative domain wall from one orbit . The first column of the table specifies the normal of the wall plane p, further columns list the layer groups , and that describe the symmetry properties and classification of the wall (defined in Table 3.4.4.3), and is the number of symmetryequivalent domain walls [cf. equation (3.4.4.21)].

The last two columns give possible components of the spontaneous polarization P of the wall and the reversed wall . Except for walls with normals [001] and [100], all walls are polar, i.e. they can be spontaneously polarized. The reversal of the polarization in reversible domain walls requires the reversal of domain states. In irreversible domain walls, the reversal of into is accompanied by a change of the polarization P into P′, which may have a different absolute value and direction different to that of P.
The structure of two domain states and two mutually reversed domain walls obtained by molecular dynamics calculations are depicted in Fig. 3.4.4.4 (Calleja et al., 2001). This shows a projection on the ab plane of the structure represented by SiO_{4} tetrahedra, in which each tetrahedron shares four corners. The threefold symmetry axes in the centres of distorted hexagonal channels and three twofold symmetry axes (one with vertical orientation) perpendicular to the threefold axes can be easily seen. The two vertical dotted lines are the wall planes p of two mutually reversed walls and . In Table 3.4.4.5 we find that these walls have the symmetry , and in Fig. 3.4.4.4 we can verify that the operation is a `sidereversing' operation of the wall (and the whole twin as wall), operation is a `stateexchanging operation' and the operation is a nontrivial `sideandstate reversing' operation of the wall. The walls and are, therefore, symmetric and reversible walls.
During a small temperature interval above the appearance of the phase at 846 K, there exists an incommensurate phase that can be treated as a regular domain structure, consisting of triangular columnar domains with domain walls (discommensurations) of negative wall energy (see e.g. Dolino, 1985). Both theoretical considerations and electron microscopy observations (see e.g. Van Landuyt et al., 1985) show that the wall normal has the direction. From Table 3.4.4.5 it follows that there are six equivalent walls that are symmetric but irreversible, therefore any two equivalent walls differ in orientation.
This prediction is confirmed by electron microscopy in Fig. 3.4.4.5, where black and white triangles correspond to domains with domain states and , and the transition regions between black and white areas to domain walls (discommensurations). To a domain wall of a certain orientation no reversible wall appears with the same orientation but with a reversed order of black and white. Domain walls in homogeneous triangular parts of the structure are related by 120 and 240° rotations and carry, therefore, parallel spontaneous polarizations; wall orientations in two differently oriented blocks (the middle of the righthand part and the rest on the lefthand side) are related by 180° rotations about the axis in the plane of the photograph and are, therefore, polarized in antiparallel directions (for more details see SaintGrégoire & Janovec, 1989; Snoeck et al., 1994). After cooling down to room temperature, the wall energy becomes positive and the regular domain texture changes into a coarse domain structure in which these six symmetryrelated wall orientations still prevail (Van Landuyt et al., 1985).
References
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