International
Tables for Crystallography Volume A Spacegroup symmetry Edited by M. I. Aroyo © International Union of Crystallography 2016 
International Tables for Crystallography (2016). Vol. A, ch. 1.7, pp. 138139

The cubic phase of barium titanate BaTiO_{3}, of symmetry type , undergoes a phase transition to a tetragonal phase of symmetry type which can give rise to six distinct singledomain states (Janovec et al., 2004). This is represented in Fig. 1.7.3.2, where at the centre are four unit cells of the cubic phase with barium and titanium atoms represented by large and small filled circles, respectively, and oxygen atoms, which are located at the centre of each unitcell face, as open circles. A cubictotetragonal phase transition gives rise to atomic displacements represented by arrows, and to six singledomain states, four of which are depicted in the figure. The polar tetragonal symmetry of each of these tetragonal domain states is also shown.
In determining the symmetry of a domain wall, we first construct a domain twin (Janovec & Přívratská, 2014): we choose two singledomain states, for this example the two in Fig. 1.7.3.2 with symmetry P4_{z}m_{x}m_{xy}, and construct a domain pair consisting of the superposition of these two singledomain states, see Fig. 1.7.3.3. The domain twin we choose to construct is obtained by passing a plane of orientation (010) through this domain pair at the origin and deleting from one side of the plane the atoms of one of the singledomain states, and the atoms of the second singledomain state from the other side of the plane, see Fig. 1.7.3.4. The plane is referred to as the central plane of the domain wall, and the atoms in and near this plane as the domain wall.
The symmetry of the central plane of the domain wall is determined from the symmetry of the domain pair and the scanning tables: The symmetry of the domain pair is the group of operations that either leaves both singledomain states invariant or simultaneously switches the two domain states. P4_{z}m_{x}m_{xy} leaves both singledomain states invariant and the symmetry operation of spatial inversion switches the two singledomain states, see Fig. 1.7.3.3. Consequently, the domainpair symmetry is . The symmetry of the central plane is determined from the scanning table for the space group P4_{z}/m_{z}m_{x}m_{xy}, the orientation orbit (010), the orientation of the domain wall and the linear orbit 0d, since the central plane of the wall passes through the origin (Volume E, 2010). The symmetry of the central plane is the sectional layer group pm_{x}m_{z}m_{y}, where p denotes the lattice of translations in the x, 0, z plane.
Let n denote a unit vector perpendicular to the central plane of the domain wall; in this example n is in the [010] direction. The symmetry of the domain wall consists of:
The symmetry of the domain wall is then p2_{x}/m_{x}.
References
International Tables for Crystallography (2010). Vol. E, Subperiodic Groups, edited by V. Kopský & D. B. Litvin, 2nd ed. Chichester: John Wiley & Sons.Janovec, V., Grocký, M., Kopský, V. & Kluiber, Z. (2004). On atomic displacements in 90° ferroelectric domain walls of tetragonal BaTiO_{3} crystals. Ferroelectrics, 303, 65–68.
Janovec, V. & Přívratská, J. (2014). Domain structures. In International Tables for Crystallography, Vol. D, Physical Properties of Crystals, edited by A. Authier, 2nd ed., ch. 3.4. Chichester: Wiley.