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. 538
Section 3.4.4.7. Microscopic structure and symmetry of 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 
The thermodynamic theory of domain walls outlined above is efficient in providing quantitative results (wall thickness, energy) in any specific material. However, since this is a continuum theory, it is not able to treat local structural changes on a microscopic level and, moreover, owing to the small thickness of domain walls (several lattice constants), the reliability of its conclusions is to some extent uncertain.
Discrete theories either use simplified models [e.g. pseudospin ANNNI (axial next nearest neighbour Ising) model] that yield quantitative results on profiles, energies and interaction energies of walls but do not consider real crystal structures, or calculate numerically for a certain structure the atomic positions within a wall from interatomic potentials.
Symmetry analysis of domain walls provides useful qualitative conclusions about the microscopic structure of walls. Layer groups with discrete twodimensional translations impose, via the site symmetries, restrictions on possible displacements and/or ordering of atoms or molecules. From these conclusions, combined with a reasonable assumption that these shifts or ordering vary continuously within a wall, one gets topological constraints on the field of local displacements and/or ordering of atoms or molecules in the wall. The advantage of this treatment is its simplicity and general validity, since no approximations or simplified models are needed. The analysis can also be applied to domain walls of zero thickness, where thermodynamic theory fails. However, this method does not yield any quantitative results, such as values of displacements, wall thickness, energy etc.
The procedure is similar to that in the continuum description. The main relations equations (3.4.4.12)–(3.4.4.17) and the classification given in Table 3.4.4.3 hold for a microscopic description as well; one has only to replace point groups by space groups.
A significant difference is that the sectional layer groups and the wall symmetry depend on the location of the plane p in the crystal lattice. This position can by expressed by a vector , where d is the scanning vector (see IT E , 2010 and the example below) and s is a nonnegative number smaller than 1, . An extended symbol of a twin in the microscopic description, corresponding to the symbol (3.4.4.1) in the continuum description, is The main features of the analysis are demonstrated on the following example.
Example 3.4.4.5. Ferroelastic domain wall in calomel. We examine a ferroelastic compatible domain wall in a calomel crystal (Janovec & Zikmund, 1993; IT E, 2010, Chapter 5.2 ). In Section 3.4.2.5, Example 3.4.2.7, we found the microscopic domain states (see Fig. 3.4.2.5) and, in Section 3.4.3.7, the corresponding ordered domain pair and unordered domain pair (depicted in Fig. 3.4.3.10). These pairs have symmetry groups and , respectively. Both groups have an orthorhombic basis , , , with a shift of origin for both groups.
Compatible domain walls in this ferroelastic domain pair have orientations (100) and (010) in the tetragonal coordinate system (see Table 3.4.3.6). We shall examine the former case – the latter is crystallographically equivalent. Sectional layer groups of this plane in groups and have a twodimensional translation group (net) with basic vectors and , and the scanning vector expresses the repetition period of the layer structure (cf. Fig. 3.4.3.10a). From the diagram of symmetry elements of the group and , available in IT A (2005), one can deduce the sectional layer groups at any location . These sectional layer groups are listed explicitly in IT E (2010) in the scanning tables of the respective space groups.
The resulting sectional layer groups and are given in Table 3.4.4.7 in two notations, in which the letter p signifies a twodimensional net with the basic translations introduced above. Standard symbols are related to the basis . Subscripts in noncoordinate notation specify the orientation of symmetry elements in the reference Cartesian coordinate system of the tetragonal phase, the partial translation in the glide plane a and in the screw axis is equal to , i.e. the symbols a and are also related to the basis . At special locations and , sectional groups contain both sidepreserving and sidereversing operations, whereas for any other location these layer groups are trivial (face) layer groups consisting of sidepreserving operations only and are, therefore, also called floating groups in the direction d (IT E , 2010).
^{†}Shift of origin .

The wall (twin) symmetry can be easily deduced from sectional layer groups and : the floating group is just the sectional layer group at a general location, = = . Two other generators in the group symbol of are nontrivial twinning operations (underlined with a star) of . The classification in the last column of Table 3.4.4.7 is defined in Table 3.4.4.3.
Local symmetry exerts constraints on possible displacements of the atoms within a wall. The site symmetry of atoms in a wall of zero thickness, or at the central plane of a finitethickness domain wall, are defined by the layer group . The site symmetry of the offcentre atoms at are determined by floating group and the limiting structures at and by space groups and , respectively. A reasonable condition that the displacements of atoms change continuously if one passes through the wall from to allows one to deduce a qualitative picture of the displacements within a wall.
Symmetry groups of domain pairs, sectional layer groups and the twin symmetry have been derived in the parent clamping approximation (PCA) (see Section 3.4.2.5). As can be seen from Fig. 3.4.3.5, a relaxation process, accompanying a lifting of this approximation, consists of a simple shear (shear vector parallel to q) and an elongation (or contraction) in the domain wall along the shear direction (change of the vector into the vector ). These deformations influence neither the layer group nor its floating group . Hence the wall (twin) symmetry derived in the parent clamping approximation expresses also the symmetry of a ferroelastic domain wall (twin) with nonzero spontaneous shear unless the simple shear is accompanied by a reshuffling of atoms or molecules in both domains. This useful statement holds for any ferroelastic domain wall (twin).
A microscopic structure of the ferroelastic domain wall in two symmetrically prominent positions is depicted in Fig. 3.4.4.11. For better recognition, displacements of molecules are exaggerated and the changes of the displacement lengths are neglected. Since the symmetry of all groups involved contains a reflection , the atomic shifts are confined to planes (001). It can be seen in the figure that when one moves through the wall in the direction [110] or , the vector of the molecular shift experiences rotations through about the direction in opposite senses for the `black' and `white' molecules.
The `black' molecules in the central layer at location or [wall (a) on the lefthand side of Fig. 3.4.4.11] exhibit nearly antiparallel displacements perpendicular to the wall. Strictly perpendicular shifts would represent `averaged' displacements compatible with the layer symmetry , which is, however, broken by a simple shear that decreases the symmetry to , which does not require perpendicular displacements of `black' molecules.
The wall with central plane location or (Fig. 3.4.4.11b) has symmetry , which restricts displacements of `white' molecules of the central layer to the y direction only; the `averaged' displacements compatible with (origin shift ) would have equal lengths of shifts in the and directions, but the relaxed central layer with symmetry allows unequal shifts in the and directions.
Walls (a) and (b) with two different prominent locations have different layer symmetries and different structures of the central layer. These two walls have extremal energy, but symmetry cannot decide which one has the minimum energy. The two walls have the same polar pointgroup symmetry , which permits a spontaneous polarization along y.
Similar analysis of the displacement and ordering fields in domain walls has been performed for KSCN crystals (Janovec et al., 1989), sodium superoxide NaO_{2} (Zieliński, 1990) and for the simple cubic phase of fullerene C_{60} (SaintGrégoire et al., 1997).
References
International Tables for Crystallography (2005). Vol. A, SpaceGroup Symmetry, 5th ed., edited by Th. Hahn. Heidelberg: Springer.International Tables for Crystallography (2010). Vol. E, Subperiodic groups, 2nd. ed., edited by V. Kopský & D. B. Litvin. Chichester: Wiley.
Janovec, V., Schranz, W., Warhanek, H. & Zikmund, Z. (1989). Symmetry analysis of domain structure in KSCN crystals. Ferroelectrics, 98, 171–189.
Janovec, V. & Zikmund, Z. (1993). Microscopic structure of domain walls and antiphase boundaries in calomel crystals. Ferroelectrics, 140, 89–93.
SaintGrégoire, P., Janovec, V. & Kopský, V. (1997). A sample analysis of domain walls in simple cubic phase of C_{60}. Ferroelectrics, 191, 73–78.
Zieliński, P. (1990). Grouptheoretical description of domains and phase boundaries in crystalline solids. Surf. Sci. Rep. 11, 179–223.