International
Tables for Crystallography Volume D Physical properties of crystals Edited by A. Authier © International Union of Crystallography 2006 
International Tables for Crystallography (2006). Vol. D, ch. 3.4, pp. 491502
Section 3.4.4. Domain twins and domain walls^{a}Department of Physics, Technical University of Liberec, Hálkova 6, 461 17 Liberec 1, Czech Republic, and ^{b}Department of Mathematics and Didactics of Mathematics, Technical University of Liberec, Hálkova 6, 461 17 Liberec 1, Czech Republic 
In this section, we examine crystallographic properties of planar compatible domain walls and simple domain twins. The symmetry of these objects is described by layer groups. Since this concept is not yet common in crystallography, we briefly explain its meaning in Section 3.4.4.2. The exposition is performed in the continuum description, but most of the results apply with slight generalizations to the microscopic treatment that is illustrated with an example in Section 3.4.4.7.
We shall consider a simple domain twin that consists of two domains and which meet along a planar domain wall of zero thickness. Let us denote by p a plane of the domain wall, in brief wall plane of . This plane is specified by Miller indices , or by a normal to the plane which also defines the sidedness (plus and minus side) of the plane p. By orientation of the plane p we shall understand a specification which can, but may not, include the sidedness of p. If both the orientation and the sidedness are given, then the plane p divides the space into two halfspaces. Using the bra–ket symbols, mentioned in Section 3.4.3.6, we shall denote by the halfspace on the negative side of p and by the halfspace on the positive side of p.
A simple twin consists of two (theoretically semiinfinite) domains and with domain states and , respectively, that join along a planar domain wall the orientation of which is specified by the wall plane p with normal n. A symbol specifies the domain twin unequivocally: domain , with domain region filled with domain state , is on the negative side of p and domain is on the positive side of p (see Fig. 3.4.4.1a).

Symbols of a simple twin. (a) Two different symbols with antiparallel normal . (b) Symbols of the reversed twin. 
If we were to choose the normal of opposite direction, i.e. , the same twin would have the symbol (see Fig. 3.4.4.1a). Since these two symbols signify the same twin, we have the identity Thus, if we invert the normal n and simultaneously exchange domain states and in the twin symbol, we obtain an identical twin (see Fig. 3.4.4.1a). This identity expresses the fact that the specification of the twin by the symbol introduced above does not depend on the chosen direction of the wall normal .
The full symbol of the twin can be replaced by a shorter symbol if we accept a simple convention that the first lower index signifies the domain state that occupies the half space on the negative side of . Then the identity (3.4.4.1) in short symbols is
If the orientation and sidedness of the plane p of a wall is known from the context or if it is not relevant, the specification of n in the symbol of the domain twin and domain wall can be omitted.
A twin , or , can be formed by sectioning the ordered domain pair by a plane p with normal and removing the domain state on the negative side and domain state on the positive side of the normal . This is the same procedure that is used in bicrystallography when an ideal bicrystal is derived from a dichromatic complex (see Section 3.2.2 ).
A twin with reversed order of domain states is called a reversed twin. The symbol of the twin reversed to the initial twin is or A reversed twin is depicted in Fig. 3.4.4.1(b).
A planar domain wall is the interface between the domains and of the associated simple twin. Even a domain wall of zero thickness is specified not only by its orientation in space but also by the domain states that adhere to the minus and plus sides of the wall plane p. The symbol for the wall is, therefore, analogous to that of the twin, only in the explicit symbol the brackets ( ) are replaced by square brackets [ ] and T in the short symbol is replaced by W: or by a shorter equivalent symbol
An adequate concept for characterizing symmetry properties of simple domain twins and planar domain walls is that of layer groups. A layer group describes the symmetry of objects that exist in a threedimensional space and have twodimensional translation symmetry. Typical examples are twodimensional planes in threedimensional space [twosided planes and sectional layer groups (Holser, 1958a,b), domain walls and interfaces of zero thickness], layers of finite thickness (e.g. domain walls and interfaces of finite thickness) and two semiinfinite crystals joined along a planar and coherent (compatible) interface [e.g. simple domain twins with a compatible (coherent) domain wall, bicrystals].
A crystallographic layer group comprises symmetry operations (isometries) that leave invariant a chosen crystallographic plane p in a crystalline object. There are two types of such operations:
A layer group consists of two parts: where is a subgroup of that comprises all sidepreserving operations of ; this group is isomorphic to a plane group and is called a trivial layer group or a face group. An underlined character denotes a sidereversing operation and the left coset contains all sidereversing operations of . Since is a halving subgroup, the layer group can be treated as a dichromatic (blackandwhite) group in which sidepreserving operations are colourpreserving operations and sidereversing operations are colourexchanging operations.
There are 80 layer groups with discrete twodimensional translation subgroups [for a detailed treatment see IT E (2002), or e.g. Vainshtein (1994), Shubnikov & Kopcik (1974), Holser (1958a)]. Equivalent names for these layer groups are net groups (Opechowski, 1986), plane groups in three dimensions (Grell et al., 1989), groups in a twosided plane (Holser, 1958a,b) and others.
To these layer groups there correspond 31 point groups that describe the symmetries of crystallographic objects with twodimensional continuous translations. Holser (1958b) calls these groups point groups in a twosided plane, Kopský (1993) coins the term pointlike layer groups. We shall use the term `layer groups' both for layer groups with discrete translations, used in a microscopic description, and for crystallographic `pointlike layer groups' with continuous translations in the continuum approach. The geometrical meaning of these groups is similar and most of the statements and formulae hold for both types of layer groups.
Crystallographic layer groups with a continuous translation group [point groups of twosided plane (Holser, 1958b)] are listed in Table 3.4.4.1. The international notation corresponds to international symbols of layer groups with discrete translations; this notation is based on the Hermann–Mauguin (international) symbols of threedimensional space groups, where the c direction is the direction of missing translations and the character `1' represents a symmetry direction in the plane with no associated symmetry element (see IT E , 2002).

In the noncoordinate notation (Janovec, 1981), sidereversing operations are underlined. Thus e.g. denotes a rotation around a twofold axis in the plane p and a reflection through this plane, whereas 2 is a sidepreserving rotation around an axis perpendicular to the plane and m is a sidepreserving reflection through a plane perpendicular to the plane p. With exception of and , the symbol of an operation specifies the orientation of the plane p. This notation allows one to signify layer groups with different orientations in one reference coordinate system. Another noncoordinate notation has been introduced by Shubnikov & Kopcik (1974).
If a crystal with pointgroup symmetry G is bisected by a crystallographic plane p, then all operations of G that leave the plane p invariant form a sectional layer group = of the plane p in G. Operations of the group can be divided into two sets [see equation (3.4.4.7)]: where the trivial layer group expresses the symmetry of the crystal face with normal . These face symmetries are listed in IT A (2005), Part 10 , for all crystallographic point groups G and all orientations of the plane expressed by Miller indices . The underlined operation is a sidereversing operation that inverts the normal . The left coset contains all sidereversing operations of .
The number of planes symmetrically equivalent (in G) with the plane p is equal to the index of in G:
Example 3.4.4.1. As an example, we find the sectional layer group of the plane in the group (see Fig. 3.4.2.2).
In this example and the plane crystallographically equivalent with the plane is the plane (100) with sectional symmetry .
We shall examine the symmetry of a twin with a planar zerothickness domain wall with orientation and location defined by a plane p (Janovec, 1981; Zikmund, 1984; Zieliński, 1990). The symmetry properties of a planar domain wall are the same as those of the corresponding simple domain twin. Further, we shall consider twins but all statements also apply to the corresponding domain walls.
Operations that express symmetry properties of the twin must leave the orientation and location of the plane p invariant. We shall perform our considerations in the continuum description and shall assume that the plane p passes through the origin of the coordinate system. Then pointgroup symmetry operations leave the origin invariant and do not change the position of p.
If we apply an operation to the twin , we get a crystallographically equivalent twin with other domain states and another orientation of the domain wall,
It can be shown that the transformation of a domain pair by an operation defined by this relation fulfils the conditions of an action of the group G on a set of all domain pairs formed from the orbit (see Section 3.2.3.3 ). We can, therefore, use all concepts (stabilizer, orbit, class of equivalence etc.) introduced for domain states and also for domain pairs.
Operations g that describe symmetry properties of the twin must not change the orientation of the wall plane p but can reverse the sides of p, and must either leave invariant both domain states and or exchange these two states. There are four types of such operations and their action is summarized in Table 3.4.4.2. It is instructive to follow this action in Fig. 3.4.4.2 using an example of the twin with domain states and from our illustrative example (see Fig. 3.4.2.2).

We note that the star and the underlining do not represent any operation; they are just suitable auxiliary labels that can be omitted without changing the result of the operation.
To find all trivial symmetry operations of the twin , we recall that all symmetry operations that leave both and invariant constitute the symmetry group of the ordered domain pair , , where and are the symmetry groups of and , respectively. The sectional layer group of the plane p in group is (if we omit p) The trivial (sidepreserving) subgroup assembles all trivial symmetry operations of the twin . The left coset , where is a sidereversing operation, contains all sidereversing operations of this twin. In our example and (see Fig. 3.4.4.2).
Similarly, the left coset contains all stateexchanging operations, and all nontrivial symmetry operations of the twin . In the illustrative example, and .
The trivial group and its three cosets constitute the sectional layer group of the plane p in the symmetry group of the unordered domain pair , where is an operation of the left coset that leaves the normal invariant and .
Group can be interpreted as a symmetry group of a twin pair consisting of a domain twin and a superposed reversed twin with a common wall plane p. This construct is analogous to a domain pair (dichromatic complex in bicrystallography) in which two homogeneous domain states and are superposed (see Section 3.4.3.1). In the same way as the group of domain pair is divided into two cosets with different results of the action on this domain pair, the symmetry group of the twin pair can be decomposed into four cosets (3.4.4.13), each of which acts on a domain twin in a different way, as specified in Table 3.4.4.2.
We can associate with operations from each coset in (3.4.4.13) a label. If we denote operations from without a label by e, underlining by a and star by b, then the multiplication of labels is expressed by the relations The four different labels can be formally viewed as four colours, the permutation of which is defined by relations (3.4.3.14); then the group can be treated as a fourcolour layer group.
Since the symbol of a point group consists of generators from which any operation of the group can be derived by multiplication, one can derive from the international symbol of a sectional layer group, in which generators are supplied with adequate labels, the coset decomposition (3.4.4.13).
Thus for the domain pair in Fig. 3.4.4.2 with [see equation (3.4.3.18)] and we get the sectional layer group . Operations of this group (besides generators) are .
All operations that transform a twin into itself constitute the symmetry group (or in short of the twin . This is a layer group consisting of two parts: where is a face group comprising all trivial symmetry operations of the twin and the left coset contains all nontrivial operations of the twin that reverse the sides of the wall plane p and simultaneously exchange the states and .
One can easily verify that the symmetry of the twin is equal to the symmetry of the reversed twin ,
Similarly, for sectional layer groups,
Therefore, the symmetry of a twin and of sectional layer groups , is specified by the orientation of the plane p [expressed e.g. by Miller indices ] and not by the sidedness of p. However, the two layer groups and , and and express the symmetry of two different objects, which can in special cases (nontransposable pairs and irreversible twins) be symmetrically nonequivalent.
The symmetry also expresses the symmetry of the wall . This symmetry imposes constraints on the form of tensors describing the properties of walls. In this way, the appearance of spontaneous polarization in domain walls has been examined (Přívratská & Janovec, 1999; Přívratská et al., 2000).
According to their symmetry, twins and walls can be divided into two types: For a symmetric twin (domain wall), there exists a nontrivial symmetry operation and its symmetry is expressed by equation (3.4.4.15). A symmetric twin can be formed only from transposable domain pairs.
For an asymmetric twin (domain wall), there is no nontrivial symmetry operation and its symmetry group is, therefore, confined to trivial group , The difference between symmetric and asymmetric walls can be visualized in domain walls of finite thickness treated in Section 3.4.4.6.
The symmetry of a symmetric twin (wall), expressed by relation (3.4.4.15), is a layer group but not a sectional layer group of any point group. It can, however, be derived from the sectional layer group of the corresponding ordered domain pair [see equation (3.4.4.12)] and the sectional layer group of the unordered domain pair [see equation (3.4.4.13)],
This is particularly useful in the microscopic description, since sectional layer groups of crystallographic planes in threedimensional space groups are tabulated in IT E (2002), where one also finds an example of the derivation of the twin symmetry in the microscopic description.
The treatment of twin (wall) symmetry based on the concept of domain pairs and sectional layer groups of these pairs (Janovec, 1981; Zikmund, 1984) is analogous to the procedure used in treating interfaces in bicrystals (see Section 3.2.2 ; Pond & Bollmann, 1979; Pond & Vlachavas, 1983; Kalonji, 1985; Sutton & Balluffi, 1995). There is the following correspondence between terms: domain pair dichromatic complex; domain wall interface; domain twin with zerothickness domain wall ideal bicrystal; domain twin with finitethickness domain wall real (relaxed) bicrystal. Terms used in bicrystallography cover more general situations than domain structures (e.g. grain boundaries of crystals with noncrystallographic relations, phase interfaces). On the other hand, the existence of a highsymmetry phase, which is missing in bicrystallography, enables a more detailed discussion of crystallographically equivalent variants (orbits) of various objects in domain structures.
The symmetry group is the stabilizer of a domain twin (wall) in a certain group, and as such determines a class (orbit) of domain twins (walls) that are crystallographically equivalent with this twin (wall). The number of crystallographically equivalent twins is equal to the number of left cosets (index) of in the corresponding group. Thus the number of equivalent domain twins (walls) with the same orientation defined by a plane p of the wall is where is a sectional layer group of the plane p in the parent group G, is the index of in and absolute value denotes the number of operations in a group.
The set of all domain walls (twins) crystallographically equivalent in G with a given wall forms a Gorbit of walls, . The number of walls in this Gorbit is where is the number of planes equivalent with plane p expressed by equation (3.4.4.9) and is the number of equivalent domain walls with the plane p [see equation (3.4.4.20)]. Walls in one orbit have the same scalar properties (e.g. energy) and their structure and tensor properties are related by operations that relate walls from the same orbit.
Another aspect that characterizes twins and domain walls is the relation between a twin and the reversed twin. A twin (wall) which is crystallographically equivalent with the reversed twin (wall) will be called a reversible twin (wall). If a twin and the reversed twin are not crystallographically equivalent, the twin will be called an irreversible twin (wall). If a domain wall is reversible, then the properties of the reversed wall are fully specified by the properties of the initial wall, for example, these two walls have the same energy and their structures and properties are mutually related by a crystallographic operation. For irreversible walls, no relation exists between a wall and the reversed wall. Common examples of irreversible walls are electrically charged ferroelectric walls (walls carrying a nonzero polarization charge) and domain walls or discommensurations in phases with incommensurate structures.
A necessary and sufficient condition for reversibility is the existence of sidereversing and/or stateexchanging operations in the sectional layer group of the unordered domain pair [see equation (3.4.4.13)]. This group also contains the symmetry group of the twin [see equation (3.4.4.15)] and thus provides a full symmetry characteristic of twins and walls,
Sequences of walls and reversed walls appear in simple lamellar domain structures which are formed by domains with two alternating domain states, say and , and parallel walls and reversed walls (see Fig. 3.4.2.1).
The distinction `symmetric–asymmetric' and `reversible–irreversible' provides a natural classification of domain walls and simple twins. Five prototypes of domain twins and domain walls, listed in Table 3.4.4.3, correspond to five subgroups of the sectional layer group : the sectional layer group itself, the layer group of the twin the sectional layer group the trivial layer group and the trivial layer group .

An example of a symmetric reversible (SR) twin (and wall) is the twin in Fig. 3.4.4.2 with a nontrivial twinning operation and with reversing operations and . The twin and reversed twin in Fig. 3.4.3.8 are symmetric and irreversible (SI) twins with a twinning operation ; no reversing operations exist (walls are charged and charged walls are always irreversible, since a charge is invariant with respect to any transformation of the space). The twin and reversed twin in the same figure are asymmetric statereversible twins with statereversing operation and with no nontrivial twinning operation.
The same classification also applies to domain twins and walls in a microscopic description.
As in the preceding section, we shall now present separately the symmetries of nonferroelastic simple domain twins [`twinning without a change of crystal shape (or form)'] and of ferroelastic simple domain twins [`twinning with a change of crystal shape (or form)'; KlassenNeklyudova (1964), Indenbom (1982)].
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 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 symmetrically equivalent 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).
As explained in Section 3.4.3.6, from a domain pair of ferroelastic singledomain states with two perpendicular equally deformed planes p and one can form four different ferroelastic twins (see Fig. 3.4.3.8). Two mutually reversed twins and have the same twin symmetry and the same symmetry of the twin pair . The ferroelastic twin laws can be expressed by the layer group or, in a less complete way (without specification of reversibility), by the twin symmetry . The same holds for two mutually reversed twins and with a twin plane perpendicular to p.
Table 3.4.4.6 summarizes possible symmetries of ferroelastic domain twins and corresponding ferroelastic twin laws . Letters V and W signify straindependent and strainindependent (with a fixed orientation) domain walls, respectively. The classification of domain walls and twins is defined in Table 3.4.4.3. The last column contains twinning groups of ordered domain pairs from which these twins can be formed. The symbol of is followed by a symbol of the group given in square brackets. The twinning group specifies, up to two cases, a class of equivalent domain pairs [orbit ] (see Section 3.4.3.4). More details on particular cases (orientation of domain walls, disorientation angle, twin axis) can be found in synoptic Table 3.4.3.6. From this table follow two general conclusions:

This follows from simple reasoning: all twin symmetries in Table 3.4.4.6 have been derived in the parent clamping approximation and are expressed by the orthorhombic group or by some of its subgroups. As shown in Section 3.4.3.6.2, the maximal symmetry of a mechanically twinned crystal is also . An additional simple shear accompanying the lifting of the parent clamping approximation cannot, therefore, decrease the symmetry derived in the parent clamping approximation. In a similar way, one can prove the statement for the group of the twin pairs and .
A domain wall of zero thickness is a geometrical construct that enabled us to form a twin from a domain pair and to find a layer group that specifies the maximal symmetry of that twin. However, real domain walls have a finite, though small, thickness. Spatial changes of the structure within a wall may, or may not, lower the wall symmetry and can be conveniently described by a phenomenological theory.
We shall consider the simplest case of a one nonzero component of the order parameter (see Section 3.1.2 ). Two nonzero equilibrium homogeneous values of and of this parameter correspond to two domain states and . Spatial changes of the order parameter in a domain twin with a zerothickness domain wall are described by a steplike function for and for , where is the distance from the wall of zero thickness placed at .
A domain wall of finite thickness is described by a function with limiting values and : If the wall is symmetric, then the profile in one halfspace, say , determines the profile in the other halfspace . For continuous fulfilling conditions (3.4.4.23) this leads to the condition i.e. must be an odd function. This requirement is fulfilled if there exists a nontrivial symmetry operation of a domain wall (twin): a side reversal combined with an exchange of domain states results in an identical wall profile.
A particular form of the wall profile can be deduced from Landau theory. In the simplest case, the dependence of the domain wall would minimize the free energy where are phenomenological coefficients and T and T_{c} are the temperature and the temperature of the phase transition, respectively. The first three terms correspond to the homogeneous part of the Landau free energy (see Section 3.2.1 ) and the last term expresses the energy of the spatially changing order parameter. This variational task with boundary conditions (3.4.4.23) has the following solution (see e.g. Salje, 1990, 2000b; Ishibashi, 1990; Strukov & Levanyuk, 1998) where the value w specifies one half of the effective thickness of the domain wall and is given by This dependence, expressed in relative dimensionless variables and , is displayed in Fig. 3.4.4.6.

Profile of the onecomponent order parameter in a symmetric wall (S). The effective thickness of the wall is . 
The wall profile expressed by solution (3.4.4.26) is an odd function of , and fulfils thus the condition (3.4.4.24) of a symmetric wall.
The wall thickness can be estimated from electron microscopy observations, or more precisely by a diffuse Xray scattering technique (Locherer et al., 1998). The effective thickness [see equation (3.4.4.26)] in units of crystallographic repetition length A normal to the wall ranges from to , i.e. is about 10–100 nm (Salje, 2000b). The temperature dependence of the domain wall thickness expressed by equation (3.4.4.27) has been experimentally verified, e.g. on LaAlO_{3} (Chrosch & Salje, 1999).
The energy of the domain wall per unit area equals the difference between the energy of the twin and the energy of the singledomain crystal. For a one nonzero component order parameter with the profile (3.4.4.26), the wall energy is given by (Strukov & Levanyuk, 1998) where is the effective thickness of the wall [see equation (3.4.4.27)] and the coefficients are defined in equation (3.4.4.25).
The order of magnitude of the wall energy of ferroelastic and nonferroelastic domain walls is typically several millijoule per square metre (Salje, 2000b).
Example 3.4.4.3. In our example of a ferroelectric phase transition , one can identify with the component of spontaneous polarization and with the axis y. One can verify in Fig. 3.4.4.6 that the symmetry of the twin with a zerothickness domain wall is retained in the domain wall with symmetric profile (3.4.4.26): both nontrivial symmetry operations and transform the profile into an identical function.
This example illustrates another feature of a symmetric wall: All nontrivial symmetry operations of the wall are located at the central plane of the finitethickness wall. The sectional group of this plane thus expresses the symmetry of the central layer and also the global symmetry of a symmetric wall (twin). The local symmetry of the offcentre planes is equal to the face group of the the layer group (in our example ).
The relation between a wall profile of a symmetric reversible (SR) wall and the profile of the reversed wall is illustrated in Fig. 3.4.4.7, where the the dotted curve is the wall profile of the reversed wall. The profile of the reversed wall is completely determined by the the profile of the initial wall, since both profiles are related by equations The first part of the equation corresponds to a stateexchanging operation (cf. point in Fig. 3.4.4.7) and the second one to a sidereversing operation (point in the same figure). In a symmetric reversible wall, both types of reversing operations exist (see Table 3.4.4.3).

Profiles of the onecomponent order parameter in a symmetric wall (solid curve) and in the reversed wall (dotted curve). The wall is symmetric and reversible (SR). 
In a symmetric irreversible (SI) wall both initial and reversed wall profiles fulfil symmetry condition (3.4.4.24) but equations (3.4.4.30) relating both profiles do not exist. The profiles and may differ in shape and surface wall energy. Charged domain walls are always irreversible.
A possible profile of an asymmetric domain wall is depicted in Fig. 3.4.4.8 (full curve). There is no relation between the negative part and positive part of the wall profile . Owing to the absence of nontrivial twin operations, there is no central plane with higher symmetry. The local symmetry (sectional layer group) at any location within the wall is equal to the face group . This is also the global symmetry of the entire wall, .

Profiles of the onecomponent order parameter in an asymmetric wall (solid curve) and in the reversed asymmetric wall (dotted curve). The wall is asymmetric and statereversible (). 
The dotted curve in Fig. 3.4.4.8 represents the reversedwall profile of an asymmetric statereversible (AR) wall that is related to the initial wall by stateexchanging operations (see Table 3.4.4.5),
An example of an asymmetric sidereversible (A) wall is shown in Fig. 3.4.4.9. In this case, an asymmetric wall (full curve) and reversed wall (dotted curve) are related by sidereversing operations :

Profiles of the onecomponent order parameter in an asymmetric wall (solid curve) and in the reversed asymmetric wall (dotted curve). The wall is asymmetric and sidereversible (A). 
In an asymmetric irreversible (AI) wall, both profiles and are asymmetric and there is no relation between these two profiles.
The symmetry of a finitethickness wall with a profile is equal to or lower than the symmetry of the corresponding zerothickness domain wall, . A symmetry descent can be treated as a phase transition in the domain wall (see e.g. Bul'bich & Gufan, 1989a,b; Sonin & Tagancev, 1989). There are equivalent structural variants of the finitethickness domain wall with the same orientation and the same energy but with different structures of the wall,
Domainwall variants – twodimensional analogues of domain states – can coexist and meet along line defects – onedimensional analogues of a domain wall (Tagancev & Sonin, 1989).
Symmetry descent in domain walls of finite thickness may occur if the order parameter has more than one nonzero component. We can demonstrate this on ferroic phases with an order parameter with two components and . The profiles and can be found, as for a onecomponent order parameter, from the corresponding Landau free energy (see e.g. Cao & Barsch, 1990; Houchmandzadeh et al., 1991; Ishibashi, 1992, 1993; Rychetský & Schranz, 1993, 1994; Schranz, 1995; Huang et al., 1997; Strukov & Levanyuk, 1998; Hatt & Hatch, 1999; Hatch & Cao, 1999).
Let us denote by the symmetry of the profile and by the symmetry of the profile . Then the symmetry of the entire wall is a common part of the symmetries and ,
Example 3.4.4.4. In our illustrative phase transition , the order parameter has two components that can be associated with the x and y components and of the spontaneous polarization (see Table 3.1.3.1 and Fig. 3.4.2.2). We have seen that the domain wall of zero thickness has the symmetry . If one lets relax and keeps (a socalled linear structure), then (see Fig. 3.4.4.2 with ). If the last condition is lifted, a possible profile of a relaxed is depicted by the full curve in Fig. 3.4.4.10. If both components and are nonzero within the wall, one speaks about a rotational structure of domain wall. In this relaxed domain wall the spontaneous polarization rotates in the plane (001), resembling thus a Néel wall in magnetic materials. The even profile has the symmetry . Hence, according to (3.4.4.34), the symmetry of a relaxed wall with a rotational structure is . This is an asymmetric statereversible (AR) wall with two chiral variants [see equation (3.4.4.33)] that are related by and ; the profile of the second variant is depicted in Fig. 3.4.4.10 by a dashed curve.
Similarly, one gets for a zerothickness domain wall perpendicular to z the symmetry . For a relaxed domain wall with profiles and , displayed in Figs. 3.4.4.6 and 3.4.4.10 with , one gets , and . The relaxed domain wall with rotational structure has lower symmetry than the zerothickness wall or the wall with linear structure, but remains a symmetric and reversible (SR) domain wall in which spontaneous polarization rotates in a plane (001), resembling thus a Bloch wall in magnetic materials. Two chiral righthanded and lefthanded variants are related by operations and . This example illustrates that the structure of domain walls may differ with the wall orientation.
We note that the stability of a domain wall with a rotational structure and with a linear structure depends on the values of the coefficients in the Landau free energy, on temperature and on external fields. In favourable cases, a phase transition from a symmetric linear structure to a less symmetric rotational structure can occur. Such phase transitions in domain walls have been studied theoretically by Bul'bich & Gufan (1989a,b) and by Sonin & Tagancev (1989).
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 , 2002 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, 2002, 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 (2002) 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 , 2002).
^{†}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).
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