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. 518521
Section 3.4.3.6.3. Disoriented domain states, ferroelastic domain twins and their twin laws^{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 
To examine another possible way of forming a ferroelastic domain twin, we return once again to Fig. 3.4.3.5(a) and split the space along the plane p into a halfspace on the negative side of the plane p (defined by a negative end of normal ) and another halfspace on the positive side of p. In the parent phase, the whole space is filled with domain state and we can, therefore, treat the crystal in region as a domain and the crystal in region as a domain (we remember that a domain is specified by its domain region, e.g. , and by a domain state, e.g. , in this region; see Section 3.4.2.1).
Now we cool the crystal down and exert the spontaneous strain on domain . The resulting domain contains domain state in the domain region with the planar boundary along (the overbar `−' signifies a rotation of the boundary in the positive sense). Similarly, domain changes after performing spontaneous strain into domain with domain state and the planar boundary along . This results in a disruption in the sector and in an overlap of and in the sector .
The overlap can be removed and the continuity recovered by rotating the domain through angle and the domain through about the domainpair axis A (see Fig. 3.4.3.5a and b). This rotation changes the domain into domain and domain into domain , where and are domain states rotated away from the singledomain state orientation through and , respectively. Domains and meet without additional strains or stresses along the plane p and form a simple ferroelastic twin with a compatible domain wall along p. This wall is stressfree and fulfils the conditions of mechanical compatibility.
Domain states and with new orientations are called disoriented (misoriented) domain states or suborientational states (Shuvalov et al., 1985; Dudnik & Shuvalov, 1989) and the angles and are the disorientation angles of and , respectively.
We have described the formation of a ferroelastic domain twin by rotating singledomain states into new orientations in which a stressfree compatible contact of two ferroelastic domains is achieved. The advantage of this theoretical construct is that it provides a visual interpretation of disorientations and that it works with ferroelastic singledomain states which can be easily derived and transformed.
There is an alternative approach in which a domain state in one domain is produced from the domain state in the other domain by a shear deformation. The same procedure is used in mechanical twinning [for mechanical twinning, see Section 3.3.8.4 and e.g. Cahn (1954); KlassenNeklyudova (1964); Christian (1975)].
We illustrate this approach again using our example. From Fig. 3.4.3.5(b) it follows that domain state in the second domain can be obtained by performing a simple shear on the domain state of the first domain. In this simple shear, a point is displaced in a direction parallel to the equally deformed plane p (in mechanical twinning called a twin plane) and to a plane perpendicular to the axis of the domain pair (plane of shear). The displacement is proportional to the distance d of the point from the domain wall. The amount of shear is measured either by the absolute value of this displacement at a unit distance, , or by an angle called a shear angle (sometimes is defined as the shear angle). There is no change of volume connected with a simple shear.
The angle is also called an obliquity of a twin (Cahn, 1954) and is used as a convenient measure of pseudosymmetry of the ferroelastic phase.
The highresolution electron microscopy image in Fig. 3.4.3.6 reveals the relatively large shear angle (obliquity) of a ferroelastic twin in the monoclinic phase of tungsten trioxide (WO_{3}). The plane (101) corresponds to the plane p of a ferroelastic wall in Fig. 3.4.3.5(b). The planes are crystallographic planes in the lower and upper ferroelastic domains, which correspond in Fig. 3.4.3.5(b) to domain and domain , respectively. The planes in these domains correspond to the diagonals of the elementary cells of and in Fig. 3.4.3.5(b) and are nearly perpendicular to the wall. The angle between these planes equals , where is the shear angle (obliquity) of the ferroelastic twin.

Highresolution electron microscopy image of a ferroelastic twin in the orthorhombic phase of WO_{3}. Courtesy of H. Lemmens, EMAT, University of Antwerp. 
Disorientations of domain states in a ferroelastic twin bring about a deviation of the optical indicatrix from a strictly perpendicular position. Owing to this effect, ferroelastic domains exhibit different colours in polarized light and can be easily visualized. This is illustrated for a domain structure of YBa_{2}Cu_{3}O_{7−δ} in Fig. 3.4.3.7. The symmetry descent G = gives rise to two ferroelastic domain states and . The twinning group of the nontrivial domain pair is The colour of a domain state observed in a polarizedlight microscope depends on the orientation of the index ellipsoid (indicatrix) with respect to a fixed polarizer and analyser. This index ellipsoid transforms in the same way as the tensor of spontaneous strain, i.e. it has different orientations in ferroelastic domain states. Therefore, different ferroelastic domain states exhibit different colours: in Fig. 3.4.3.7, the blue and pink areas (with different orientations of the ellipse representing the spontaneous strain in the plane of of figure) correspond to two different ferroelastic domain states. A rotation of the crystal that does not change the orientation of ellipses (e.g. a 180° rotation about an axis parallel to the fourfold rotation axis) does not change the colours (ferroelastic domain states). If one neglects disorientations of ferroelastic domain states (see Section 3.4.3.6) – which are too small to be detected by polarizedlight microscopy – then none of the operations of the group change the singledomain ferroelastic domain states , , hence there is no change in the colours of domain regions of the crystal. On the other hand, all operations with a star symbol (operations lost at the transition) exchange domain states and , i.e. also exchange the two colours in the domain regions. The corresponding permutation is a transposition of two colours and this attribute is represented by a star attached to the symbol of the operation. This exchange of colours is nicely demonstrated in Fig. 3.4.3.7 where a −90° rotation is accompanied by an exchange of the pink and blue colours in the domain regions (Schmid, 1991, 1993).

Ferroelastic twins in a very thin YBa_{2}Cu_{3}O_{7−δ} crystal observed in a polarizedlight microscope. Courtesy of H. Schmid, Université de Geneve. 
It can be shown (Shuvalov et al., 1985; Dudnik & Shuvalov, 1989) that for small spontaneous strains the amount of shear s and the angle can be calculated from the second invariant of the differential tensor : where
In our example, where there are only two nonzero components of the differential spontaneous strain tensor [see equation (3.4.3.58)], the second invariant = = and the angle is In this case, the angle can also be expressed as , where a and b are lattice parameters of the orthorhombic phase (Schmid et al., 1988).
The shear angle ranges in ferroelastic crystals from minutes to degrees (see e.g. Schmid et al., 1988; Dudnik & Shuvalov, 1989).
Each equally deformed plane gives rise to two compatible domain walls of the same orientation but with opposite sequence of domain states on each side of the plane. We shall use for a simple domain twin with a planar wall a symbol in which n denotes the normal to the wall. The bra–ket symbol and represents the halfspace domain regions on the negative and positive sides of , respectively, for which we have used letters and , respectively. Then and represent domains and , respectively. The symbol properly specifies a domain twin with a zerothickness domain wall.
A domain wall can be considered as a domain twin with domain regions restricted to nonhomogeneous parts near the plane p. For a domain wall in domain twin we shall use the symbol , which expresses the fact that a domain wall of zero thickness needs the same specification as the domain twin.
If we exchange domain states in the twin , we get a reversed twin (wall) with the symbol . These two ferroelastic twins are depicted in the lower right and upper left parts of Fig. 3.4.3.8, where – for ferroelastic–nonferroelectric twins – we neglect spontaneous polarization of ferroelastic domain states. The reversed twin has the opposite shear direction.

Exploded view of four ferroelastic twins with disoriented ferroelastic domain states and formed from a singledomain pair (in the centre). 
Twin and reversed twin can be, but may not be, crystallographically equivalent. Thus e.g. ferroelastic–nonferroelectric twins and in Fig. 3.4.3.8 are equivalent, e.g. via , whereas ferroelastic–ferroelectric twins and are not equivalent, since there is no operation in the group that would transform into .
As we shall show in the next section, the symmetry group of a twin and the symmetry group of a reverse twin are equal,
A sequence of repeating twins and reversed twins forms a lamellar ferroelastic domain structure that is very common in ferroelastic phases (see e.g. Figs. 3.4.1.1 and 3.4.1.4).
Similar considerations can be applied to the second equally deformed plane that is perpendicular to p. The two twins and corresponding compatible domain walls for the equally deformed plane have the symbols and , and are also depicted in Fig. 3.4.3.8. The corresponding lamellar domain structure is
Thus from one ferroelastic singledomain pair depicted in the centre of Fig. 3.4.3.8 four different ferroelastic domain twins can be formed. It can be shown that these four twins have the same shear angle and the same amount of shear s. They differ only in the direction of the shear.
Four disoriented domain states and that appear in the four domain twins considered above are related by lost operations (e.g. diagonal, vertical and horizontal reflections), i.e. they are crystallographically equivalent. This result can readily be obtained if we consider the stabilizer of a disoriented domain state , which is . Then the number of disoriented ferroelastic domain states is given by All these domain states appear in ferroelastic polydomain structures that contain coexisting lamellar structures (3.4.3.67) and (3.4.3.68).
Disoriented domain states in ferroelastic domain structures can be recognized by diffraction techniques (e.g. using an Xray precession camera). The presence of these four disoriented domain states results in splitting of the diffraction spots of the highsymmetry tetragonal phase into four or two spots in the orthorhombic ferroelastic phase. This splitting is schematically depicted in Fig. 3.4.3.9. For more details see e.g. Shmyt'ko et al. (1987), Rosová et al. (1993), and Rosová (1999).
Finally, we turn to twin laws of ferroelastic domain twins with compatible domain walls. In a ferroelastic twin, say , there are just two possible twinning operations that interchange two ferroelastic domain states and of the twin: reflection through the plane of the domain wall ( in our example) and 180° rotation with a rotation axis in the intersection of the domain wall and the plane of shear (). These are the only transposing operations of the domain pair that are preserved by the shear; all other transposing operations of the domain pair are lost. (This is a difference from nonferroelastic twins, where all transposing operations of the pair become twinning operations of a nonferroelastic twin.)
Consider the twin in Fig. 3.4.3.8. By nontrivial twinning operations we understand transposing operations of the domain pair , whereas trivial twinning operations leave invariant and . As we shall see in the next section, the union of trivial and nontrivial twinning operations forms a group . This group, called the symmetry group of the twin , comprises all symmetry operations of this twin and we shall use it for designating the twin law of the ferroelastic twin, just as the group of the domain pair specifies the twin law of a nonferroelastic twin. This group is a layer group (see Section 3.4.4.2) that keeps the plane p invariant, but for characterizing the twin law, which specifies the relation of domain states of two domains in the twin, one can treat as an ordinary (dichromatic) point group . Thus the twin law of the domain twin is designated by the group where (3.4.3.70) expresses the fact that a twin and the reversed twin have the same symmetry, see equation (3.4.3.66). We see that this group coincides with the symmetry group of the singledomain pair (see Fig. 3.4.3.1b).
The twin law of two twins and with the same equally deformed plane is expressed by the group which is different from the of the twin .
Representative domain pairs of all orbits of ferroelastic domain pairs (Litvin & Janovec, 1999) are listed in two tables. Table 3.4.3.6 contains representative domain pairs for which compatible domain walls exist and Table 3.4.3.7 lists ferroelastic domain pairs where compatible coexistence of domain states is not possible. Table 3.4.3.6 contains, beside other data, for each ferroelastic domain pair the orientation of two equally deformed planes and the corresponding symmetries of the corresponding four twins which express two twin laws.
References
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