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. 506508
Section 3.4.3.1. Domain pairs and their symmetry, twin law^{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 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.
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