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. 136139

The present volume in the series International Tables for Crystallography (Volume A: SpaceGroup Symmetry) treats one, two and threedimensional space groups. Volume E in the series, Subperiodic Groups (2010), treats two and threedimensional subperiodic groups: frieze groups (groups in twodimensional space with translations in a onedimensional subspace), rod groups (groups in threedimensional space with translations in a onedimensional subspace) and layer groups (groups in threedimensional space with translations in a twodimensional subspace). In the same way in which threedimensional space groups are used to classify the atomic structure of threedimensional crystals, the subperiodic groups are used to classify the atomic structure of other crystalline structures, such as liquid crystals, domain interfaces, twins and thin films.
In Volume A, the relationship between the space group of a crystal and the pointgroup symmetry of individual points in the crystal is given by site symmetries, the pointgroup subgroups of the space group that leave the points invariant. In Volume E, an analogous relationship is given between the space group of a crystal and the subperiodicgroup symmetry of planes that transect the crystal. Volume E contains scanning tables (with supplementary tables in Kopský & Litvin, 2004) from which the layergroup subgroups of the space group (called sectional layer groups) that leave the transecting planes invariant can be determined. The first attempts to derive sectional layer groups were made by Wondratschek (1971) and by using software written by Guigas (1971). Davies & Dirl (1993a,b) developed software for finding subgroups of space groups which was modified to find sectional layer groups. The use and determination of sectional layer groups have also been discussed by Janovec et al. (1988), Kopský & Litvin (1989) and Fuksa et al. (1993).
In Fig. 1.7.3.1, part of the scanning table for the space group (164) is given. From this one can determine the layergroup subgroups of that are symmetries of planes of orientation (hkil) = (0001). Vectors a′ and b′ are basic vectors of the translational subgroup of the layergroup symmetry of planes of this orientation. The vector d defines the scanning direction and is used to define the position of the plane within the crystal. The linear orbit is the set of all parallel planes obtained by applying all elements of the space group to any one plane. The sectional layer group is the layer subgroup of the space group that leaves the plane invariant.
Sectional layer groups were introduced by Holser (1958a,b) in connection with the consideration of domain walls and twin boundaries as symmetry groups of planes bisecting a crystal. The mutual orientation of the two domains separated by a domain wall or twin boundary is not arbitrary, but has crystallographic restrictions. The grouptheoretical basis for an analysis of domain pairs is given by Janovec (1972), and the structure of domain walls and twin boundaries is considered by Janovec (1981) and Zikmund (1984) [see also Janovec & Přívratská (2014)].
Layer symmetries have been used in bicrystallography. The term bicrystal was introduced by Pond & Bollmann (1979) in the study of grain boundaries [see also Pond & Vlachavas (1983) and Vlachavas (1985)]. A bicrystal is in general an edifice where two crystals, usually of the same structure but of different, possibly arbitrary, orientations, meet at a common boundary. The sectional layer groups describe the symmetries of such a boundary [see Volume E (2010), Section 5.2.5.2 ].
An example of the application of the scanning tables to determine the layergroup symmetry of planes in a crystal is given in Section 1.7.3.1. In Section 1.7.3.2 the derivation of the layergroup symmetry of a domain wall is described.
Fig. 1.7.3.1 shows the crystal structure of cadmium iodide, CdI_{2}. The space group of this crystal is of type (164). The anions form a hexagonal close packing of spheres and the cations occupy half of the octahedral holes, filling one of the alternate layers. In closepacking notation, the CdI_{2} structure isFrom the scanning tables, we obtain for planes with the (0001) orientation and at heights z = 0 or z = ½ a sectional layergroup symmetry type (layer group No. 72, or L72 for short), and for planes of this orientation at any other height a sectional layergroup symmetry type p3m1 (L69).
The plane contains cadmium ions. This plane is a constituent of the orbit of planes of orientation (0001) passing through the points with coordinates 0, 0, u, where u is an integer. All these planes contain cadmium ions in the same arrangement (C layer filled with Cd).
The plane at height z = ½ is a constituent of the orbit of planes of orientation (0001) passing through the points with coordinates 0, 0, u + ½. All these planes contain only voids and lie midway between A and B layers of iodine ions with the B layer below and the A layer above the plane.
The planes at levels z = ¼ and z = ¾ contain B and A layers of iodine ions, respectively. These planes and all planes related to them by translations t(0, 0, u) belong to the same orbit because the operations exchange the A and B layers.
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
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