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. 515528
Section 3.4.3.6. Ferroelastic domain pairs^{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 
A ferroelastic domain pair consists of two domain states that have different spontaneous strain. A domain pair is a ferroelastic domain pair if the crystal family of its twinning group differs from the crystal family of the symmetry group of domain state ,
Before treating compatible domain walls and disorientations, we explain the basic concept of spontaneous strain.
A strain describes a change of crystal shape (in a macroscopic description) or a change of the unit cell (in a microscopic description) under the influence of mechanical stress, temperature or electric field. If the relative changes are small, they can be described by a secondrank symmetric tensor called the Lagrangian strain. The values of the strain components (or in matrix notation ) can be calculated from the `undeformed' unitcell parameters before deformation and `deformed' unitcell parameters after deformation (see Schlenker et al., 1978; Salje, 1990; Carpenter et al., 1998).
A spontaneous strain describes the change of an `undeformed' unit cell of the highsymmetry phase into a `deformed' unit cell of the lowsymmetry phase. To exclude changes connected with thermal expansion, one demands that the parameters of the undeformed unit cell are those that the highsymmetry phase would have at the temperature at which parameters of the lowsymmetry phase are measured. To determine these parameters directly is not possible, since the parameters of the highsymmetry phase can be measured only in the highsymmetry phase. One uses, therefore, different procedures in order to estimate values for the highsymmetry parameters under the external conditions to which the measured values of the lowsymmetry phase refer (see e.g. Salje, 1990; Carpenter et al., 1998). Three main strategies are illustrated using the example of leucite (see Fig. 3.4.3.4):
Spontaneous strain has been examined in detail in many ferroic crystals by Carpenter et al. (1998).
Spontaneous strain can be divided into two parts: one that is different in all ferroelastic domain states and the other that is the same in all ferroelastic domain states. This division can be achieved by introducing a modified strain tensor (Aizu, 1970b), also called a relative spontaneous strain (Wadhawan, 2000): where is the matrix of relative (modified) spontaneous strain in the ferroelastic domain state , is the matrix of an `absolute' spontaneous strain in the same ferroelastic domain state and is the matrix of an average spontaneous strain that is equal to the sum of the matrices of absolute spontaneous strains over all ferroelastic domain states,
The relative spontaneous strain is a symmetrybreaking strain that transforms according to a nonidentity representation of the parent group G, whereas the average spontaneous strain is a nonsymmetry breaking strain that transforms as the identity representation of G.
Example 3.4.3.6. We illustrate these concepts with the example of symmetry descent with two ferroelastic domain states and (see Fig. 3.4.2.2). The absolute spontaneous strain in the first ferroelastic domain state is where and are the lattice parameters of the orthorhombic and tetragonal phases, respectively.
The spontaneous strain in domain state is obtained by applying to any switching operation that transforms into (see Table 3.4.2.1),
The average spontaneous strain is, according to equation (3.4.3.45), This deformation is invariant under any operation of G.
The relative spontaneous strains in ferroelastic domain states and are, according to equation (3.4.3.44),
Symmetrybreaking nonzero components of the relative spontaneous strain are identical, up to the factor , with the secondary tensor parameters and of the transition with the stabilizer . The nonsymmetrybreaking component does not appear in the relative spontaneous strain.
The form of relative spontaneous strains for all ferroelastic domain states of all full ferroelastic phases are listed in Aizu (1970b).
We start with the example of a phase transition with the symmetry descent , which generates two ferroelastic singledomain states and (see Fig. 3.4.2.2). An `elementary cell' of the parent phase is represented in Fig. 3.4.3.5(a) by a square and the corresponding domain state is denoted by .
In the ferroic phase, the square can change either under spontaneous strain into a spontaneously deformed rectangular cell representing a domain state , or under a spontaneous strain into rectangular representing domain state . We shall use the letter as a symbol of the parent phase and as symbols of two ferroelastic singledomain states.
Let us now choose in the parent phase a vector . This vector changes into in ferroelastic domain state and into in ferroelastic domain state . We see that the resulting vectors and have different direction but equal length: . This consideration holds for any vector in the plane p, which can therefore be called an equally deformed plane (EDP). One can find that the perpendicular plane is also an equally deformed plane, but there is no other plane with this property.
The intersection of the two perpendicular equally deformed planes p and is a line called an axis of the ferroelastic domain pair (in Fig. 3.4.3.5 it is a line at H perpendicular to the paper). This axis is the only line in which any vector chosen in the parent phase exhibits equal deformation and has its direction unchanged in both singledomain states and of a ferroelastic domain pair.
This consideration can be expressed analytically as follows (Fousek & Janovec, 1969; Sapriel, 1975). We choose in the parent phase a plane p and a unit vector in this plane. The changes of lengths of this vector in the two ferroelastic domain states and are and , respectively, where and are spontaneous strains in and , respectively (see e.g. Nye, 1985). (We are using the Einstein summation convention: when a letter suffix occurs twice in the same term, summation with respect to that suffix is to be understood.) If these changes are equal, i.e. if for any vector in the plane p, then this plane will be an equally deformed plane. If we introduce a differential spontaneous strainthe condition (3.4.3.51) can be rewritten as This equation describes a cone with the apex at the origin. The cone degenerates into two planes if the determinant of the differential spontaneous strain tensor equals zero, If this condition is satisfied, two solutions of (3.4.3.53) exist: These are equations of two planes p and passing through the origin. Their normal vectors are and . It can be shown that from the equation which holds for the trace of the matrix , it follows that these two planes are perpendicular:
The intersection of these equally deformed planes (3.4.3.53) is the axis h of the ferroelastic domain pair .
Let us illustrate the application of these results to the domain pair depicted in Fig. 3.4.3.1(b) and discussed above. From equations (3.4.3.41) and (3.4.3.47), or (3.4.3.49) and (3.4.3.50) we find the only nonzero components of the difference strain tensor areCondition (3.4.3.54) is fulfilled and equation (3.4.3.53) is There are two solutions of this equation: These two equally deformed planes p and have the normal vectors and . The axis h of this domain pair is directed along [001].
Equally deformed planes in our example have the same orientations as have the mirror planes and lost at the transition . From Fig. 3.4.3.5(a) it is clear why: reflection , which is a transposing operation of the domain pair (), ensures that the vectors and arising from have equal length. A similar conclusion holds for a 180° rotation and a plane perpendicular to the corresponding twofold axis. Thus we come to two useful rules:
Any reflection through a plane that is a transposing operation of a ferroelastic domain pair ensures the existence of two planes of equal deformation: one is parallel to the corresponding mirror plane and the other one is perpendicular to this mirror plane.
Any 180° rotation that is a transposing operation of a ferroelastic domain pair ensures the existence of two equally deformed planes: one is perpendicular to the corresponding twofold axis and the other one is parallel to this axis.
A reflection in a plane or a 180° rotation generates at least one equally deformed plane with a fixed prominent crystallographic orientation independent of the magnitude of the spontaneous strain; the other perpendicular equally deformed plane may have a noncrystallographic orientation which depends on the spontaneous strain and changes with temperature. If between switching operations there are two reflections with corresponding perpendicular mirror planes, or two 180° rotations with corresponding perpendicular twofold axes, or a reflection and a 180° rotation with a corresponding twofold axis parallel to the mirror, then both perpendicular equally deformed planes have fixed crystallographic orientations. If there are no switching operations of the second order, then both perpendicular equally deformed planes may have noncrystallographic orientations, or equally deformed planes may not exist at all.
Equally deformed planes in ferroelastic–ferroelectric phases have been tabulated by Fousek (1971). Sapriel (1975) lists equations (3.4.3.55) of equally deformed planes for all ferroelastic phases. Table 3.4.3.6 contains the orientation of equally deformed planes (with further information about the walls) for representative domain pairs of all orbits of ferroelastic domain pairs. Table 3.4.3.7 lists representative domain pairs of all ferroelastic orbits for which no compatible walls exist.
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.
As we have seen, for each ferroelastic domain pair for which condition (3.4.3.54) for the existence of coherent domain walls is fulfilled, there exist two perpendicular equally deformed planes. On each of these planes two ferroelastic twins can be formed; these two twins are in a simple relation (one is a reversed twin of the other), have the same symmetry, and can therefore be represented by one of these twins. Then we can say that from one ferroelastic domain pair two different twins can be formed. Each of these twins represents a different `twin law' that has arisen from the initial domain pair. All four ferroelastic twins can be described in terms of mechanical twinning with the same value of the shear angle .
Table 3.4.3.6 presents representative domain pairs of all classes of ferroelastic domain pairs for which compatible domain walls exist. The first five columns concern the domain pair. In subsequent columns, each row splits into two rows describing the orientation of two associated perpendicular equally deformed planes and the symmetry properties of the four domain twins that can be formed from the given domain pair. We explain the meaning of each column in detail.

The first three columns specify domain pairs.
Example 3.4.3.7. The rhombohedral phase of perovskite crystals. Examples include PZNPT and PMNPT solid solutions (see e.g. Erhart & Cao, 2001) and BaTiO_{3} below 183 K. The phase transition has symmetry descent .
In Table 3.4.2.7 we find that there are eight domain states and eight ferroelectric domain states. In this fully ferroelectric phase, domain states can be specified by unit vectors representing the direction of spontaneous polarization. We choose with corresponding symmetry group .
From eight domain states one can form domain pairs. These pairs can be divided into classes of equivalent pairs which are specified by different twinning groups. In column of Table 3.4.2.7 we find three twinning groups:
These conclusions are useful in deciphering the `domainengineered structures' of these crystals (Yin & Cao, 2000).
Ferroelastic domain pairs for which condition (3.4.3.54) for the existence of coherent domain walls is violated are listed in Table 3.4.3.7. All these pairs are nontransposable pairs. It is expected that domain walls between ferroelastic domain states would be stressed and would contain dislocations. Dudnik & Shuvalov (1989) have shown that in thin samples, where elastic stresses are reduced, `almost coherent' ferroelastic domain walls may exist.

Example 3.4.3.8. Ferroelastic crystal of langbeinite. Langbeinite K_{2}Mg_{2}(SO_{4})_{3} undergoes a phase transition with symmetry descent that appears in Table 3.4.3.7. The ferroelastic phase has three ferroelastic domain states. Dudnik & Shuvalov (1989) found, in accord with their theoretical predictions, nearly linear `almost coherent' domain walls accompanied by elastic stresses in crystals thinner than 0.5 mm. In thicker crystals, elastic stresses became so large that crystals were cracking and no domain walls were observed.
Similar effects were reported by the same authors for the partial ferroelastic phase of CH_{3}NH_{3}Al(SO_{4})_{2}·12H_{2}O (MASD) with symmetry descent , where ferroelastic domain walls were detected only in thin samples.
References
Aizu, K. (1970b). Determination of the state parameters and formulation of spontaneous strain for ferroelastics. J. Phys. Soc. Jpn, 28, 706–716.Cahn, R. W. (1954). Twinned crystals. Adv. Phys. 3, 363–445.
Carpenter, M. A., Salje, E. K. H. & GraemeBarber, A. (1998). Spontaneous strain as a determinant of thermodynamic properties for phase transitions in minerals. Eur. J. Mineral. 10, 621–691.
Christian, J. W. (1975). The Theory of Transformations in Metals and Alloys. Oxford: Pergamon Press.
Dudnik, E. F. & Shuvalov, L. A. (1989). Domain structure and phase boundaries in ferroelastics. Ferroelectrics, 98, 207–234.
Erhart, J. & Cao, W. (2001). Effective symmetry and physical properties of twinned perovskite ferroelectric single crystals. J. Mater. Res. 16, 570–578.
Fousek, J. (1971). Permissible domain walls in ferroelectric species. Czech. J. Phys. B, 21, 955–968.
Fousek, J. & Janovec, V. (1969). The orientation of domain walls in twinned ferroelectric crystals. J. Appl. Phys. 40, 135–142.
KlassenNeklyudova, M. V. (1964). Mechanical Twinning of Crystals. New York: Consultants Bureau.
Litvin, D. B. & Janovec, V. (1999). Classification of domain pairs and tensor distinction. Ferroelectrics, 222, 87–93.
Nye, J. F. (1985). Physical Properties of Crystals. Oxford: Clarendon Press.
Rosová, A. (1999). Real domain structure origination in (110) mechanical twinning in YBa_{2}Cu_{3}O_{7−y}. In Studies of High Temperature Superconductors, Vol. 28, edited by A. Narlikar, pp. 125–148. New York: Nova Science Publishers.
Rosová, A., Boulesteix, C. & Vávra, I. (1993). Role of microtwins in twin lamella intersections and interconnections in YBa_{2}Cu_{3}O_{7−y}. Physica C, 214, 247–256.
Salje, E. K. H. (1990). Phase Transitions in Ferroelastic and Coelastic Crystals, 1st ed. Cambridge University Press.
Sapriel, J. (1975). Domainwall orientations in ferroelastics. Phys. Rev. B, 12, 5128–5140.
Schlenker, J. L., Gibbs, G. V. & Boisen, M. B. (1978). Straintensor components expressed in terms of lattice parameters. Acta Cryst. A34, 52–54.
Schmid, H. (1991). Polarized light microscopy of the ferroelastic domains of YBa_{2}Cu_{3}O_{7−x}. Phase Transit. 30, 205–214.
Schmid, H. (1993). Polarized light microscopy (PLM) of ferroelectric and ferroelastic domains in transmitted and reflected light. In Ferroelectric Ceramics, edited by N. Setter & E. L. Colla, pp. 107–126. Basel: Birkhäuser.
Schmid, H., Burkhardt, E., Walker, E., Brixel, W., Clin, M., Rivera, J.P., Jorda, J.L., François, M. & Yvon, K. (1988). Polarized light and Xray precession study of the ferroelastic domains of YBa_{2}Cu_{3}O_{7−δ}. Z. Phys. B Condens. Matter, 72, 305–322.
Shmyt'ko, I. M., Shekhtman, V. Sh., Ossipyan, Yu. A. & Afonikova, N. S. (1987). Twin structure and structure of twin boundaries in 1–2–3O_{7−δ} crystals. Ferroelectrics, 97, 151–170.
Shuvalov, L. A., Dudnik, E. F. & Wagin, S. V. (1985). Domain structure geometry of real ferroelastics. Ferroelectrics, 65, 143–152.
Wadhawan, V. K. (2000). Introduction to Ferroic Materials. The Netherlands: Gordon and Breach.
Yin, J. & Cao, W. (2000). Domain configurations in domain engineered 0.995Pb(Zn_{1/3}Nb_{2/3})O_{3}–0.045PbTiO_{3} single crystals. J. Appl. Phys. 87, 7438–7441.