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. 505528
Section 3.4.3. Domain pairs: domain twin laws, distinction of domain states and switching^{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 
Different domains observed by a single apparatus can exhibit different properties even though their crystal structures are either the same or enantiomorphic and differ only in spatial orientation. Domains are usually distinguished by their bulk properties, i.e. according to their domain states. Then the problem of domain distinction is reduced to the distinction of domain states. To solve this task, we have to describe in a convenient way the distinction of any two of all possible domain states. For this purpose, we use the concept of domain pair.
Domain pairs allow one to express the geometrical relationship between two domain states (the `twin law'), determine the distinction of two domain states and define switching fields that may induce a change of one state into the other. Domain pairs also present the first step in examining domain twins and domain walls.
In this section, we define domain pairs, ascribe to them symmetry groups and socalled twinning groups, and give a classification of domain pairs. Then we divide domain pairs into equivalence classes (Gorbits of domain pairs) – which comprise domain pairs with the same inherent properties but with different orientations and/or locations in space – and examine the relation between Gorbits and twinning groups.
A qualitative difference between the coexistence of two domain states provides a basic division into nonferroelastic and ferroelastic domain pairs. The synoptic Table 3.4.3.4 lists representatives of all Gorbits of nonferroelastic domain pairs, contains information about the distinction of nonferroelastic domain states by means of diffraction techniques and specifies whether or not important property tensors can distinguish between domain states of a nonferroelastic domain pair. These data also determine the external fields needed to switch the first domain state into the second domain state of a domain pair. Synoptic Table 3.4.3.6 contains representative ferroelastic domain pairs of Gorbits of domain pairs for which there exist compatible (permissible) domain walls and gives for each representative pair the orientation of the two compatible domain walls, the expression for the disorientation angle (obliquity) and other data. Table 3.4.3.7 lists representatives of all classes of ferroelastic domain pairs for which no compatible domain walls exist. Since Table 3.4.2.7 contains for each symmetry descent all twinning groups that specify different Gorbits of domain pairs which can appear in the ferroic phase, one can get from this table and from Tables 3.4.3.4, 3.4.3.6 and 3.4.3.7 the significant features of the domain structure of any ferroic phase.
A pair of two domain states, in short a domain pair, consists of two domain states, say and , that are considered irrespective of their possible coexistence (Janovec, 1972). Geometrically, domain pairs can be visualized as two interpenetrating structures of and . Algebraically, two domain states and can be treated in two ways: as an ordered or an unordered pair (see Section 3.2.3.1.2 ).
An ordered domain pair, denoted (), consists of the first domain state and the second domain state. Occasionally, it is convenient to consider a trivial ordered domain pair () composed of two identical domain states .
An ordered domain pair is a construct that in bicrystallography is called a dichromatic complex (see Section 3.3.3 ; Pond & Vlachavas, 1983; Sutton & Balluffi, 1995; Wadhawan, 2000).
An ordered domain pair () is defined by specifying and or by giving and a switching operation that transforms into , For a given and , the switching operation is not uniquely defined since each operation from the left coset [where is the stabilizer (symmetry group) of ] transforms into , .
An ordered domain pair with a reversed order of domain states is called a transposed domain pair and is denoted . A nontrivial ordered domain pair is different from the transposed ordered domain pair,
If is a switching operation of an ordered domain pair , then the inverse operation of is a switching operation of the transposed domain pair :
An unordered domain pair, denoted by , is defined as an unordered set consisting of two domain states and . In this case, the sequence of domains states in a domain pair is irrelevant, therefore
In what follows, we shall omit the specification `ordered' or `unordered' if it is evident from the context, or if it is not significant.
A domain pair can be transformed by an operation into another domain pair, These two domain pairs will be called crystallographically equivalent (in G) domain pairs and will be denoted .
If the transformed domain pair is a transposed domain pair , then the operation g will be called a transposing operation, We see that a transposing operation exchanges domain states and : Thus, comparing equations (3.4.3.1) and (3.4.3.7), we see that a transposing operation is a switching operation that transforms into , and, in addition, switches into . Then a product of two transposing operations is an operation that changes neither nor .
What we call in this chapter a transposing operation is usually denoted as a twin operation (see Section 3.3.5 and e.g. Holser, 1958a; Curien & Donnay, 1959; Koch, 2004). We are reserving the term `twin operation' for operations that exchange domain states of a simple domain twin in which two ferroelastic domain states coexist along a domain wall. Then, as we shall see, the transposing operations are identical with the twin operations in nonferroelastic domains (see Section 3.4.3.5) but may differ in ferroelastic domain twins, where only some transposing operations of a singledomain pair survive as twin operations of the corresponding ferroelastic twin with a nonzero disorientation angle (see Section 3.4.3.6.3).
Transposing operations are marked in this chapter by a star, (with five points), which should be distinguished from an asterisk, (with six points), used to denote operations or symmetry elements in reciprocal space. The same designation is used in the software GIKoBo1 and in the tables in Kopský (2001). A prime, ′, is often used to designate transposing (twin) operations (see Section 3.3.5 ; Curien & Le Corre, 1958; Curien & Donnay, 1959). We have reserved the prime for operations involving time inversion, as is customary in magnetism (see Chapter 1.5 ). This choice allows one to analyse domain structures in magnetic and magnetoelectric materials (see e.g. Přívratská & Janovec, 1997).
In connection with this, we invoke the notion of a twin law. Since this term is not yet common in the context of domain structures, we briefly explain its meaning.
In crystallography, a twin is characterized by a twin law defined in the following way (see Section 3.3.2 ; Koch, 2004; Cahn, 1954):
An analogous definition of a domain twin law can be formulated for domain twins by replacing the term `twin components' by `domains', say and , where , and , are, respectively, the domain state and the domain region of the domains and , respectively (see Section 3.4.2.1). The term `transposing operation' corresponds to transposing operation of domain pair as we have defined it above if two domains with domain states and coexist along a domain wall of the domain twin.
Domain twin laws can be conveniently expressed by crystallographic groups. This specification is simpler for nonferroelastic twins, where a twin law can be expressed by a dichromatic space group (see Section 3.4.3.5), whereas for ferroelastic twins with a compatible domain wall dichromatic layer groups are adequate (see Section 3.4.3.6.3).
Restriction (ii), formulated by Georges Friedel (1926) and explained in detail by Cahn (1954), expresses a necessity to exclude from considerations crystal aggregates (intergrowths) with approximate or accidental `nearly exact' crystal components resembling twins (Friedel's macles d'imagination) and thus to restrict the definition to `true twins' that fulfil condition (i) exactly and are characteristic for a given material. If we confine our considerations to domain structures that are formed from a homogeneous parent phase, this requirement is fulfilled for all aggregates consisting of two or more domains. Then the definition of a `domain twin law' is expressed only by condition (i). Condition (ii) is important for growth twins.
We should note that the definition of a twin law given above involves only domain states and does not explicitly contain specification of the contact region between twin components or neighbouring domains. The concept of domain state is, therefore, relevant for discussing the twin laws. Moreover, there is no requirement on the coexistence of interpenetrating structures in a domain pair. One can even, therefore, consider cases where no real coexistence of both structures is possible. Nevertheless, we note that the characterization of twin laws used in mineralogy often includes specification of the contact region (e.g. twin plane or diffuse region in penetrating twins).
Ordered domain pairs and , formed from domain states of our illustrative example (see Fig. 3.4.2.2), are displayed in Fig. 3.4.3.1(a) and (b), respectively, as two superposed rectangles with arrows representing spontaneous polarization. In ordered domain pairs, the first and the second domain state are distinguished by shading [the first domain state is grey (`black') and the second clear (`white')] and/or by using dashed and dotted lines for the first and second domain state, respectively.

Transposable domain pairs. Singledomain states are those from Fig. 3.4.2.2. (a) Completely transposable nonferroelastic domain pair. (b) Partially transposable ferroelastic domain pair. 
In Fig. 3.4.3.2, the ordered domain pair and the transposed domain pair are depicted in a similar way for another example with symmetry descent = .
Let us now examine the symmetry of domain pairs. The symmetry group of an ordered domain pair consists of all operations that leave invariant both and , i.e. comprises all operations that are common to stabilizers (symmetry groups) and of domain states and , respectively, where the symbol denotes the intersection of groups and . The group is in Section 3.3.4 denoted by and is called an intersection group.
From equation (3.4.3.8), it immediately follows that the symmetry of the transposed domain pair is the same as the symmetry of the initial domain pair :
Symmetry operations of an unordered domain pair include, besides operations of that do not change either or , all transposing operations, since for an unordered domain pair a transposed domain pair is identical with the initial domain pair [see equation (3.4.3.4)]. If is a transposing operation of , then all operations from the left coset are transposing operations of that domain pair as well. Thus the symmetry group of an unordered domain pair can be, in a general case, expressed in the following way:
Since, for an unordered domain, the order of domain states in a domain pair is not significant, the transposition of indices in does not change this group, which also follows from equations (3.4.3.3) and (3.4.3.9).
A basic classification of domain pairs follows from their symmetry. Domain pairs for which at least one transposing operation exists are called transposable (or ambivalent) domain pairs. The symmetry group of a transposable unordered domain pair is given by equation (3.4.3.10).
The star in the symbol indicates that this group contains transposing operations, i.e. that the corresponding domain pair is a transposable domain pair.
A transposable domain pair and transposed domain pair belong to the same Gorbit:
If is a transposable pair and, moreover, , then all operations of the left coset simultaneously switch into and into . We call such a pair a completely transposable domain pair. The symmetry group of a completely transposable pair is We shall use for symmetry groups of completely transposable domain pairs the symbol .
If , then and the number of transposing operations is smaller than the number of operations switching into . We therefore call such pairs partially transposable domain pairs. The symmetry group of a partially transposable domain pair is given by equation (3.4.3.10).
The symmetry groups and , expressed by (3.4.3.10) or by (3.4.3.13), respectively, consists of two left cosets only. The first is equal to and the second one comprises all the transposing operations marked by a star. An explicit symbol of these groups contains both the group and , which is a subgroup of of index 2.
If one `colours' one domain state, e.g. , `black' and the other, e.g. , `white', then the operations without a star can be interpreted as `colourpreserving' operations and operations with a star as `colourexchanging' operations. Then the group can be treated as a `blackandwhite' or dichromatic group (see Section 3.2.3.2.7 ). These groups are also called Shubnikov groups (Bradley & Cracknell, 1972), twocolour or Heesch–Shubnikov groups (Opechowski, 1986), or antisymmetry groups (Vainshtein, 1994).
The advantage of this notation is that instead of an explicit symbol , the symbol of a dichromatic group specifies both the group and the subgroup or , and thus also the transposing operations that define, according to equation (3.4.3.7), the second domain state of the pair.
We have agreed to use a special symbol only for completely transposable domain pairs. Then the star in this case indicates that the subgroup is equal to the symmetry group of the first domain state in the pair, . Since the group is usually well known from the context (in our main tables it is given in the first column), we no longer need to add it to the symbol of .
Domain pairs for which an exchanging operation cannot be found are called nontransposable (or polar) domain pairs. The symmetry of a nontransposable domain pair is reduced to the usual `monochromatic' symmetry group of the corresponding ordered domain pair . The Gorbits of mutually transposed polar domain pairs are disjoint (Janovec, 1972): Transposed polar domain pairs, which are always nonequivalent, are called complementary domain pairs.
If, in particular, , then the symmetry group of the unordered domain pair is In this case, the unordered domain pair is called a nontransposable simple domain pair.
If , then the number of operations of is smaller than that of and the symmetry group is equal to the symmetry group of the ordered domain pair , Such an unordered domain pair is called a nontransposable multiple domain pair. The reason for this designation will be given later in this section.
We stress that domain states forming a domain pair are not restricted to singledomain states. Any two domain states with a defined orientation in the coordinate system of the parent phase can form a domain pair for which all definitions given above are applicable.
Example 3.4.3.1. Now we examine domain pairs in our illustrative example of a phase transition with symmetry descent G = and with four singledomain states and , which are displayed in Fig. 3.4.2.2. The domain pair depicted in Fig. 3.4.3.1(a) is a completely transposable domain pair since transposing operations exist, e.g. , and the symmetry group of the ordered domain pair is The symmetry group of the unordered pair is a dichromatic group,
The domain pair in Fig. 3.4.3.1(b) is a partially transposable domain pair, since there are operations exchanging domain states and , e.g. , but the symmetry group of the ordered domain pair is smaller than : where 1 is an identity operation and denotes the group . The symmetry group of the unordered domain pair is equal to a dichromatic group,
The domain pair in Fig. 3.4.3.2(b) is a nontransposable simple domain pair, since there is no transposing operation of that would exchange domain states and , and . The symmetry group of the unordered domain pair is a `monochromatic' group, The Gorbit of the pair has no common domain pair with the Gorbit of the transposed domain pair . These two `complementary' orbits contain mutually transposed domain pairs.
Symmetry groups of domain pairs provide a basic classification of domain pairs into the four types introduced above. This classification applies to microscopic domain pairs as well.
We have seen that for transposable domain pairs the symmetry group of a domain pair specifies transposing operations that transform into . This does not apply to nontransposable domain pairs, where the symmetry group does not contain any switching operation. Another group exists, called the twinning group, which is associated with a domain pair and which does not have this drawback. The twinning group determines the distinction of two domain states, specifies the external fields needed to switch one domain state into another one and enables one to treat domain pairs independently of the transition . This facilitates the tabulation of the properties of nonequivalent domain pairs that appear in all possible ferroic phases.
The twinning group of a domain pair is defined as the minimal subgroup of G that contains both and a switching operation of the domain pair , (Fuksa & Janovec, 1995; Fuksa, 1997), where no group exists such that
The twinning group is identical to the embracing (fundamental) group used in bicrystallography (see Section 3.2.2 ). In Section 3.3.4 it is called a composite symmetry of a twin.
Since is a group, it must contain all products of with operations of , i.e. the whole left coset . For completely transposable domain pairs, the union of and forms a group that is identical with the symmetry group of the unordered domain pair :
In a general case, the twinning group , being a supergroup of , can always be expressed as a decomposition of the left cosets of ,
We can associate with the twinning group a set of c domain states, the orbit of , which can be generated by applying to the representatives of the left cosets in decomposition (3.4.3.25), This orbit is called the generic orbit of domain pair .
Since the generic orbit (3.4.3.26) contains both domain states of the domain pair , one can find different and equal nonzero tensor components in two domain states and by a similar procedure to that used in Section 3.4.2.3 for ascribing principal and secondary tensor parameters to principal and secondary domain states. All we have to do is just replace the group G of the parent phase by the twinning group . There are, therefore, three kinds of nonzero tensor components in and :
Cartesian tensor components corresponding to the tensor parameters can be calculated by means of conversion equations [for details see the manual of the software GIKoBo1, path: Subgroups\View\Domains and Kopský (2001)].
Let us now illustrate the above recipe for finding tensor distinctions by two simple examples.
Example 3.4.3.2. The domain pair in Fig. 3.4.3.1(a) is a completely transposable pair, therefore, according to equations (3.4.3.24) and (3.4.3.18),
In Table 3.1.3.1 , we find that the first principal tensor parameter of the transition G = is the xcomponent of the spontaneous polarization, . Since the switching operation is for example the inversion , the tensor parameter in the second domain state is .
Other principal tensor parameters can be found in the software GIKoBo1 or in Kopský (2001), p. 185. They are: (the physical meaning of the components is explained in Table 3.4.3.5). In the second domain state , these components have the opposite sign. No other tensor components exist that would be different in and , since there is no intermediate group in between and .
Nonzero components that are the same in both domain states are nonzero components of property tensors in the group and are listed in Section 1.1.4.7 or in the software GIKoBo1 or in Kopský (2001).
The numbers of independent tensor components that are different and those that are the same in two domain states are readily available for all nonferroelastic domain pairs and important property tensors in Table 3.4.3.4.
Example 3.4.3.3. The twinning group of the partially transposable domain pair in Fig. 3.4.3.1(b) with has the twinning group Domain states and differ in the principal tensor parameter of the transition , which is twodimensional and which we found in Example 3.4.2.4: . Then in the domain state it is . Other principal tensors are: (the physical meaning of the components is explained in Table 3.4.3.5). In the domain state they keep their absolute value but appear as the second nonzero components, as with spontaneous polarization.
There is an intermediate group between and , since does not contain . The onedimensional secondary tensor parameters for the symmetry descent was also found in Example 3.4.2.4: . All these parameters have the opposite sign in .
The tensor distinction of two domain states and in a domain pair provides a useful classification of domain pairs given in the second and the third columns of Table 3.4.3.1. This classification can be extended to ferroic phases which are named according to domain pairs that exist in this phase. Thus, for example, if a ferroic phase contains ferroelectric (ferroelastic) domain pair(s), then this phase is a ferroelectric (ferroelastic) phase. Finer division into full and partial ferroelectric (ferroelastic) phases specifies whether all or only some of the possible domain pairs in this phase are ferroelectric (ferroelastic) ones. Another approach to this classification uses the notions of principal and secondary tensor parameters, and was explained in Section 3.4.2.2.

A discussion of and many examples of secondary ferroic phases are available in papers by Newnham & Cross (1974a,b) and Newnham & Skinner (1976), and tertiary ferroic phases are discussed by Amin & Newnham (1980).
We shall now show that the tensor distinction of domain states is closely related to the switching of domain states by external fields.
We saw in Section 3.4.2.1 that all domain states of the orbit have the same chance of appearing. This implies that they have the same free energy, i.e. they are degenerate. The same conclusion follows from thermodynamic theory, where domain states appear as equivalent solutions of equilibrium values of the order parameter, i.e. all domain states exhibit the same free energy (see Section 3.1.2 ). These statements hold under a tacit assumption of absent external electric and mechanical fields. If these fields are nonzero, the degeneracy of domain states can be partially or completely lifted.
The free energy per unit volume of a ferroic domain state , , with spontaneous polarization with components , , and with spontaneous strain components , is (Aizu, 1972)where the Einstein summation convention (summation with respect to suffixes that occur twice in the same term) is used with and . The symbols in equation (3.4.3.32) have the following meaning: and are components of the external electric field and of the mechanical stress, respectively, are components of the piezoelectric tensor, are components of the electric susceptibility, are compliance components, and are components of electrostriction (components with Greek indices are expressed in matrix notation) [see Section 3.4.5 (Glossary), Chapter 1.1 or Nye (1985); Sirotin & Shaskolskaya (1982)].
We shall examine two domain states and , i.e. a domain pair , in electric and mechanical fields. The difference of their free energies is given by
For a domain pair and given external fields, there are three possibilities:
A typical dependence of applied stress and corresponding strain in ferroelastic materials has a form of a elastic hysteresis loop (see Fig. 3.4.1.3). Similar dielectric hysteresis loops are observed in ferroelectric materials; examples can be found in books on ferroelectric crystals (e.g. Jona & Shirane, 1962).
A classification of switching (state shifts in Aizu's terminology) based on equation (3.4.3.33) was put forward by Aizu (1972, 1973) and is summarized in the second and fourth columns of Table 3.4.3.1. The order of the state shifts specifies the switching fields that are necessary for switching one domain state of a domain pair into the second state of the pair.
Another distinction related to switching distinguishes between actual and potential ferroelectric (ferroelastic) phases, depending on whether or not it is possible to switch the spontaneous polarization (spontaneous strain) by applying an electric field (mechanical stress) lower than the electrical (mechanical) breakdown limit under reasonable experimental conditions (Wadhawan, 2000). We consider in our classification always the potential ferroelectric (ferroelastic) phase.
A closer look at equation (3.4.3.33) reveals a correspondence between the difference coefficients in front of products of field components and the tensor distinction of domain states and in the domain pair : If a morphic Cartesian tensor component of a polar tensor is different in these two domain states, then the corresponding difference coefficient is nonzero and defines components of fields that can switch one of these domain states into the other. A similar statement holds for the symmetric tensors of rank two (e.g. the spontaneous strain tensor).
Tensor distinction for all representative nonferroelastic domain pairs is available in the synoptic Table 3.4.3.4. These data also carry information about the switching fields.
Two domain pairs that are crystallographically equivalent, [see equation (3.4.3.5)], have different orientations in space but their inherent properties are the same. It is, therefore, useful to divide all domain pairs of a ferroic phase into classes of equivalent domain pairs. All domain pairs that are equivalent (in G) with a given domain pair, say , can be obtained by applying to all operations of G, i.e. by forming a Gorbit .
One can always find in this orbit a domain pair that has in the first place the first domain state . We shall call such a pair a representative domain pair of the orbit. The initial orbit and the orbit are identical:
The set of ordered pairs (including trivial ones) that can be formed from n domain states can be divided into Gorbits (classes of equivalent domain pairs):
Similarly, as there is a onetoone correspondence between domain states and left cosets of the stabilizer (symmetry group) of the first domain state [see equation (3.4.2.9)], there is an analogous relation between Gorbits of domain pairs and socalled double cosets of .
A double coset of is a set of left cosets that can be expressed as , where runs over all operations of (see Section 3.2.3.2.8 ). A group G can be decomposed into disjoint double cosets of : where is the set of representatives of double cosets.
There is a onetoone correspondence between double cosets of the decomposition (3.4.3.35) and Gorbits of domain pairs (3.4.3.34) (see Section 3.2.3.3.6 , Proposition 3.2.3.35 ):
We see that the representatives of the double cosets in decomposition (3.4.3.35) define domain pairs which represent all different Gorbits of domain pairs. Just as different left cosets specify all domain states, different double cosets determine all classes of equivalent domain pairs (Gorbits of domain pairs).
The properties of double cosets are reflected in the properties of corresponding domain pairs and provide a natural classification of domain pairs. A specific property of a double coset is that it is either identical or disjoint with its inverse. A double coset that is identical with its inverse, is called an invertible (ambivalent) double coset. The corresponding class of domain pairs consists of transposable (ambivalent) domain pairs.
A double coset that is disjoint with its inverse, is a noninvertible (polar) double coset ( denotes an empty set) and the corresponding class of domain pairs comprises nontransposable (polar) domain pairs. A double coset and its inverse are called complementary double cosets. Corresponding classes called complementary classes of equivalent domain pairs consist of transposed domain pairs that are nonequivalent.
Another attribute of a double coset is the number of left cosets which it comprises. If an invertible double coset consists of one left coset, then the domain pairs in the Gorbit are completely transposable. An invertible double coset comprising several left cosets is associated with a Gorbit consisting of partially transposable domain pairs. Noninvertible double cosets can be divided into simple nontransposable double cosets (complementary double cosets consist of one left coset each) and multiple nontransposable double cosets (complementary double cosets comprise more than one left coset each).
Thus there are four types of double cosets (see Table 3.2.3.1 in Section 3.2.3.2 ) to which there correspond the four basic types of domain pairs presented in Table 3.4.3.2.

These results can be illustrated using the example of a phase transition with G = with four domain states (see Fig. 3.4.2.2). The corresponding four left cosets of are given in Table 3.4.2.1. Any operation from the first left coset (identical with ) transforms the second left coset into itself, i.e. this left coset is a double coset. Since it consists of an operation of order two, it is a simple invertible double coset. The corresponding representative domain pair is . By applying operations of on , one gets the class of equivalent domain pairs (Gorbit): . These domain pairs can be labelled as `180° pairs' according to the angle between the spontaneous polarization in the two domain states.
When one applies operations from the first left coset on the third left coset, one gets the fourth left coset, therefore a double coset consists of these two left cosets. An inverse of any operation of this double coset belongs to this double coset, hence it is a multiple invertible double coset. Corresponding domain pairs are partially transposable ones. A representative pair is, for example, which is indeed a partially transposable domain pair [cf. (3.4.3.19) and (3.4.3.20)]. The class of equivalent ordered domain pairs is . These are `90° domain pairs'.
An example of noninvertible double cosets is provided by the decomposition of the group into left and double cosets of displayed in Table 3.4.3.3. The inverse of the second left coset (second line) is equal to the third left coset (third line) and vice versa. Each of these two left cosets thus corresponds to a double coset and these double cosets are complementary double cosets. Corresponding representative simple nontransposable domain pairs are and , and are depicted in Fig. 3.4.3.2.

We conclude that double cosets determine classes of equivalent domain pairs that can appear in the ferroic phase resulting from a phase transition with a symmetry descent . Left coset and double coset decompositions for all crystallographic pointgroup descents are available in the software GIKoBo1, path: Subgroups\View\Twinning groups.
A double coset can be specified by any operation belonging to it. This representation is not very convenient, since it does not reflect the properties of corresponding domain pairs and there are many operations that can be chosen as representatives of a double coset. It turns out that in a continuum description the twinning group can represent classes of equivalent domain pairs with two exceptions:
Bearing in mind these two exceptions, one can, in the continuum description, represent Gorbits of domain pairs by twinning groups .
We have used this correspondence in synoptic Table 3.4.2.7 of symmetry descents at ferroic phase transitions. For each symmetry descent , the twinning groups given in column specify possible Gorbits of domain pairs that can appear in the domain structure of the ferroic phase (Litvin & Janovec, 1999). We divide all orbits of domain pairs (represented by corresponding twinning groups ) that appear in Table 3.4.2.7 into classes of nonferroelastic and ferroelastic domain pairs and present them with further details in the three synoptic Tables 3.4.3.4, 3.4.3.6 and 3.4.3.7 described in Sections 3.4.3.5 and 3.4.3.6.
As we have seen, a classification of domain pairs according to their internal symmetry (summarized in Table 3.4.3.2) introduces a partition of all domain pairs that can be formed from domain states of the Gorbit into equivalence classes of pairs with the same internal symmetry. Similarly, any inherent physical property of domain pairs induces a partition of all domain pairs into corresponding equivalence classes. Thus, for example, the classification of domain pairs, based on tensor distinction or switching of domain states (see Table 3.4.3.1, columns two and three), introduces a division of domain pairs into corresponding equivalence classes.
3.4.3.5. Nonferroelastic domain pairs: twin laws, domain distinction and switching fields, synoptic table
Two domain states and form a nonferroelastic domain pair if the spontaneous strain in both domain states is the same, . This is so if the twinning group of the pair and the symmetry group of domain state belong to the same crystal family (see Table 3.4.2.2):
It can be shown that all nonferroelastic domain pairs are completely transposable domain pairs (Janovec et al., 1993), i.e. and the twinning group is equal to the symmetry group of the unordered domain pair [see equation (3.4.3.24)]: (Complete transposability is only a necessary, but not a sufficient, condition of a nonferroelastic domain pair, since there are also ferroelastic domain pairs that are completely transposable – see Table 3.4.3.6.)
The relation between domain states in a nonferroelastic domain twin, in which two domain states coexist, is the same as that of a corresponding nonferroelastic domain pair consisting of singledomain states. Transposing operations are, therefore, also twinning operations.
Synoptic Table 3.4.3.4 lists representative domain pairs of all orbits of nonferroelastic domain pairs. Each pair is specified by the first domain state with symmetry group and by transposing operations that transform into , . Twin laws in dichromatic notation are presented and basic data for tensor distinction and switching of nonferroelastic domains are given.
The first three columns specify domain pairs.

The second part of the table concerns the distinction and switching of domain states of the nonferroelastic domain pair .
Table 3.4.3.5 lists important property tensors up to fourth rank. Property tensor components that appear in the column headings of Table 3.4.3.4 are given in the first column, where bold face is used for the polar tensors significant for specifying the switching fields appearing in schematic form in the last column. In the third and fourth columns, those propery tensors appear for which hold all the results presented in Table 3.4.3.4 for the symbols given in the first column of Table 3.4.3.5.

We turn attention to Section 3.4.5 (Glossary), which describes the difference between the notation of tensor components in matrix notation given in Chapter 1.1 and those used in the software GIKoBo1 and in Kopský (2001).
The numbers a in front of the vertical bar  in Table 3.4.3.4 provide global information about the tensor distinction of two domain states and enables one to classify domain pairs. Thus, for example, the first number a in column gives the number of nonzero components of the spontaneous polarization that differ in sign in both domain states; if , this domain pair can be classified as a ferroelectric domain pair.
Similarly, the first number a in column determines the number of independent components of the tensor of optical activity that have opposite sign in domain states and ; if , the two domain states in the pair can be distinguished by optical activity. Such a domain pair can be called a gyrotropic domain pair. As in Table 3.4.3.1 for the ferroelectric (ferroelastic) domain pairs, we can define a gyrotropic phase as a ferroic phase with gyrotropic domain pairs. The corresponding phase transition to a gyrotropic phase is called a gyrotropic phase transition (Koňák et al., 1978; Wadhawan, 2000). If it is possible to switch gyrotropic domain states by an external field, the phase is called a ferrogyrotropic phase (Wadhawan, 2000). Further division into full and partial subclasses is possible.
One can also define piezoelectric (electrooptic) domain pairs, electrostrictive (elastooptic) domain pairs and corresponding phases and transitions.
As we have already stated, domain states in a domain pair differ in principal tensor parameters of the transition . These principal tensor parameters are Cartesian tensor components or their linear combinations that transform according to an irreducible representation specifying the primary order parameter of the transition (see Section 3.1.3 ). Owing to a special form of expressed by equation (3.4.3.42), this representation is a real onedimensional irreducible representation of . Such a representation associates +1 with operations of and −1 with operations from the left coset . This means that the principal tensor parameters are onedimensional and have the same absolute value but opposite sign in and . Principal tensor parameters for symmetry descents and associated 's of all nonferroelastic domain pairs can be found for property tensors of lower rank in Table 3.1.3.1 and for all tensors appearing in Table 3.4.3.5 in the software GIKoBo1 and in Kopský (2001).
These specific properties of nonferroelastic domain pairs allow one to formulate simple rules for tensor distinction that do not use principal tensor parameters and that are applicable for property tensors of lower rank.
Example 3.4.3.4. Tensor distinction of domains and switching in lead germanate. Lead germanate (Pb_{5}Ge_{3}O_{11}) undergoes a phase transition with symmetry descent for which we find in Table 3.4.2.7, column , just one twinning group , i.e. . This means that there is only one Gorbit of domain pairs. Since Fam3 = Fam [see Table 3.4.2.2 and equation (3.4.3.40)] this orbit comprises nonferroelastic domain pairs. In Table 3.4.3.4, we find for and that the two domain states differ in some components of all property tensors listed in this table. The first polar tensor is the spontaneous polarization (the pair is ferroelectric) with one component that has opposite sign in the two domain states. In Table 3.1.3.1 , we find for and that this component is . From Table 3.4.3.1, it follows that the state shift is electrically first order with switching field .
The first optical tensor, which could enable the visualization of the domain states, is the optical activity with two independent components which have opposite sign in the two domain states. In the software GIKoBo1, path: Subgroups\View\Domains or in Kopský (2001) we find these components: . Shur et al. (1989) have visualized in this way the domain structure of lead germanate with excellent black and white contrast (see Fig. 3.4.3.3). Other examples are given in Shuvalov & Ivanov (1964) and especially in Koňák et al. (1978).
Table 3.4.3.4 can be used readily for twinning by merohedry [see Chapter 3.3 and e.g. Cahn (1954); Koch (2004)], where it enables an easy determination of the tensor distinction of twin components and the specification of external fields for possible switching and detwinning.
Example 3.4.3.5. Tensor distinction and switching of Dauphiné twins in quartz. Quartz undergoes a phase transition from to . Using the same procedure as in the previous example, we come to following conclusions: There are only two domain states , and the twinning group, expressing the twin law, is equal to the highsymmetry group . In Table 3.4.3.4, we find that these two states differ in one independent component of the piezoelectric tensor and in one elastic compliance component. Comparison of the matrices for and (see Sections 1.1.4.10.3 and 1.1.4.10.4 ) yields the following morphic tensor components in the first domain state : and . According to the rule given above, the values of morphic components in the second domain state are and [see Section 3.4.5 (Glossary)]. These results show that there is an elastic state shift of second order and an electromechanical state shift of second order. Nonzero components in are the same in both domain states. Similarly, one can find five independent components of the tensor that are nonzero in and equal in both domain states. For the piezooptic tensor , one can proceed in a similar way. Aizu (1973) has used the ferrobielastic character of the domain pairs for visualizing domains and realizing switching in quartz. Other methods for switching and visualizing domains in quartz are known (see e.g. Bertagnolli et al., 1978, 1979).
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.
In the microscopic description, two microscopic domain states and with spacegroup symmetries and , respectively, can form an ordered domain pair () and an unordered domain pair in a similar way to in the continuum description, but one additional aspect has to be considered. The definition of the symmetry group of an ordered domain pair (), is meaningful only if the group is a space group with a threedimensional translational subgroup (threedimensional twin lattice in the classical description of twinning, see Section 3.3.8 ) where and are translation subgroups of and , respectively. This condition is fulfilled if both domain states and have the same spontaneous strains, i.e. in nonferroelastic domain pairs, but in ferroelastic domain pairs one has to suppress spontaneous deformations by applying the parent clamping approximation [see Section 3.4.2.2, equation (3.4.2.49)].
Example 3.4.3.9. Domain pairs in calomel. Calomel undergoes a nonequitranslational phase transition from a tetragonal parent phase to an orthorhombic ferroelastic phase (see Example 3.4.2.7 in Section 3.4.2.5). Four basic microscopic singledomain states are displayed in Fig. 3.4.2.5. From these states, one can form 12 nontrivial ordered singledomain pairs that can be partitioned (by means of double coset decomposition) into two orbits of domain pairs.
Representative domain pairs of these orbits are depicted in Fig. 3.4.3.10, where the first microscopic domain state participating in a domain pair is displayed in the upper cell (light grey) and the second domain state , , in the lower white cell. The overlapping structure in the middle (dark grey) is a geometrical representation of the domain pair .
The domain pair , depicted in Fig. 3.4.3.10(a), is a ferroelastic domain pair in the parent clamping approximation. Then two overlapping structures of the domain pair have a common threedimensional lattice with a common unit cell (the dotted square), which is the same as the unit cells of domain states and .
Domain pair , shown in Fig. 3.4.3.10(b), is a translational (antiphase) domain pair in which domain states and differ only in location but not in orientation. The unit cell (heavily outlined small square) of the domain pair is identical with the unit cell of the tetragonal parent phase (cf. Fig. 3.4.2.5).
The two arrows attached to the circles in the domain pairs represent exaggerated displacements within the wall.
Domain pairs represent an intermediate step in analyzing microscopic structures of domain walls, as we shall see in Section 3.4.4.
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