Tables for
Volume D
Physical properties of crystals
Edited by A. Authier

International Tables for Crystallography (2013). Vol. D, ch. 3.4, pp. 506-508

Section Domain pairs and their symmetry, twin law

V. Janoveca* and J. Přívratskáb

aInstitute of Physics, Academy of Sciences of the Czech Republic, Na Slovance 2, CZ-18221 Prague 8, Czech Republic, and bDepartment of Mathematics and Didactics of Mathematics, Technical University of Liberec, Hálkova 6, 461 17 Liberec 1, Czech Republic
Correspondence e-mail: Domain pairs and their symmetry, twin law

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A pair of two domain states, in short a domain pair, consists of two domain states, say [{\bf S}_i] and [{\bf S}_k], that are considered irrespective of their possible coexistence (Janovec, 1972[link]). Geometrically, domain pairs can be visualized as two interpenetrating structures of [{\bf S}_i] and [{\bf S}_k]. Algebraically, two domain states [{\bf S}_i] and [{\bf S}_k] can be treated in two ways: as an ordered or an unordered pair (see Section[link] ).

An ordered domain pair, denoted ([{\bf S}_i,{\bf S}_k]), consists of the first domain state [{\bf S}_i] and the second domain state. Occasionally, it is convenient to consider a trivial ordered domain pair ([{\bf S}_i,{\bf S}_i]) composed of two identical domain states [{\bf S}_i].

An ordered domain pair is a construct that in bicrystallography is called a dichromatic complex (see Section 3.3.3[link] ; Pond & Vlachavas, 1983[link]; Sutton & Balluffi, 1995[link]; Wadhawan, 2000[link]).

An ordered domain pair ([{\bf S}_i,{\bf S}_k]) is defined by specifying [{\bf S}_i] and [{\bf S}_k] or by giving [{\bf S}_i] and a switching operation [g_{ik}] that transforms [{\bf S}_i ] into [{\bf S}_k], [{\bf S}_k = g_{ik}{\bf S}_i, \quad {\bf S}_i,{\bf S}_k \in G{\bf S}_1, \quad g_{ik}\in G.\eqno( ]For a given [{\bf S}_i] and [{\bf S}_k], the switching operation [g_{ik}] is not uniquely defined since each operation from the left coset [g_{ik}{F}_i] [where [F_i] is the stabilizer (symmetry group) of [{\bf S}_i]] transforms [{\bf S}_i] into [{\bf S}_k], [g_{ik}{\bf S}_i =] [ (g_{ik}{F}_{i}){\bf S}_i =] [{\bf S}_k].

An ordered domain pair [({\bf S}_k,{\bf S}_i)] with a reversed order of domain states is called a transposed domain pair and is denoted [({\bf S}_i,{\bf S}_k)^t\equiv({\bf S}_k,{\bf S}_i)]. A non-trivial ordered domain pair [({\bf S}_i,{\bf S}_k)] is different from the transposed ordered domain pair, [({\bf S}_k,{\bf S}_i) \neq ({\bf S}_i,{\bf S}_k) \ \ {\rm for} \ \ i\neq k. \eqno( ]

If [g_{ik}] is a switching operation of an ordered domain pair [({\bf S}_i,{\bf S}_k)], then the inverse operation [g_{ik}^{-1}] of [g_{ik}] is a switching operation of the transposed domain pair [({\bf S}_k,{\bf S}_i)]: [{\rm if} \,\, ({\bf S}_i,{\bf S}_k)=({\bf S}_i,g_{ik}{\bf S}_i) \,\, {\rm and} \,\, ({\bf S}_k,{\bf S}_i)=({\bf S}_k,g_{ki}{\bf S}_k), \,\, {\rm then} \,\, g_{ki}=g_{ik}^{-1}.\eqno( ]

An unordered domain pair, denoted by [\{{\bf S}_i,{\bf S}_k\} ], is defined as an unordered set consisting of two domain states [{\bf S}_i] and [{\bf S}_k]. In this case, the sequence of domains states in a domain pair is irrelevant, therefore[\{{\bf S}_i,{\bf S}_k\}= \{{\bf S}_k,{\bf S}_i\}. \eqno(]

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 [({\bf S}_i,{\bf S}_k)] can be transformed by an operation [g\in G] into another domain pair, [g({\bf S}_i,{\bf S}_k)\equiv (g{\bf S}_i,g{\bf S}_k)=({\bf S}_l,{\bf S}_m), \quad {\bf S}_i, {\bf S}_k, {\bf S}_l, {\bf S}_m\in G{\bf S}_1, \quad g\in G. \eqno( ]These two domain pairs will be called crystallographically equivalent (in G) domain pairs and will be denoted [({\bf S}_i,{\bf S}_k)] [{\buildrel {G}\over {\sim}}] [({\bf S}_l,{\bf S}_m)].

If the transformed domain pair is a transposed domain pair [({\bf S}_k,{\bf S}_i) ], then the operation g will be called a transposing operation, [g^{\star}({\bf S}_i,{\bf S}_k)=(g^{\star}{\bf S}_i,g^{\star}{\bf S}_k)=({\bf S}_k,{\bf S}_i), \quad {\bf S}_i, {\bf S}_k \in G{\bf S}_1, \quad g^{\star}\in G.\eqno( ]We see that a transposing operation [g^{\star}\in G] exchanges domain states [{\bf S}_i] and [{\bf S}_k]: [g^{\star}{\bf S}_i = {\bf S}_k, \quad g^{\star}{\bf S}_k = {\bf S}_i, \quad {\bf S}_i, {\bf S}_k \in G{\bf S}_1, \quad g^{\star}\in G.\eqno( ]Thus, comparing equations ([link]) and ([link]), we see that a transposing operation [g^{\star}] is a switching operation that transforms [{\bf S}_i] into [{\bf S}_k ], and, in addition, switches [{\bf S}_k] into [{\bf S}_i]. Then a product of two transposing operations is an operation that changes neither [{\bf S}_i] nor [{\bf S}_k].

What we call in this chapter a transposing operation is usually denoted as a twin operation (see Section 3.3.5[link] and e.g. Holser, 1958[link]a; Curien & Donnay, 1959[link]; Koch, 2004[link]). 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 non-ferro­elastic domains (see Section[link]) but may differ in ferroelastic domain twins, where only some transposing operations of a single-domain pair survive as twin operations of the corresponding ferroelastic twin with a nonzero disorientation angle (see Section[link]).

Transposing operations are marked in this chapter by a star, [^\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 GI[\star]KoBo-1 and in the tables in Kopský (2001[link]). A prime, ′, is often used to designate transposing (twin) operations (see Section 3.3.5[link] ; Curien & Le Corre, 1958[link]; Curien & Donnay, 1959[link]). We have reserved the prime for operations involving time inversion, as is customary in magnetism (see Chapter 1.5[link] ). This choice allows one to analyse domain structures in magnetic and magnetoelectric materials (see e.g. Přívratská & Janovec, 1997[link]).

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[link] ; Koch, 2004[link]; Cahn, 1954[link]):

  • (i) A twin law describes the geometrical relation between twin components of a twin. This relation is expressed by a twin operation that brings one of the twin components into parallel orientation with the other, and vice versa. A symmetry element corresponding to the twin operation is called the twin element. (Requirement `and vice versa' is included in the definition of Cahn but not in that of Koch; for the most common twin operations of the second order the `vice versa' condition is fulfilled automatically.)

  • (ii) The relation between twin components deserves the name `twin law' only if it occurs frequently, is reproducible and represents an inherent feature of the crystal.

An analogous definition of a domain twin law can be formulated for domain twins by replacing the term `twin components' by `domains', say [{\bf D}_i({\bf S}_j,Q_k)] and [{\bf D}_m({\bf S}_n,Q_p)], where [{\bf S}_j], [Q_k] and [{\bf S}_n], [Q_p] are, respectively, the domain state and the domain region of the domains [{\bf D}_i({\bf S}_j,Q_k) ] and [{\bf D}_m({\bf S}_n,Q_p)], respectively (see Section[link]). The term `transposing operation' corresponds to transposing operation [g_{12}^{\star}] of domain pair [({\bf S}_1,{\bf S}_2) =] [({\bf S}_j, g_{jn}^{\star}{\bf S}_n)] as we have defined it above if two domains with domain states [{\bf S}_1] and [{\bf S}_2 ] coexist along a domain wall of the domain twin.

Domain twin laws can be conveniently expressed by crystallographic groups. This specification is simpler for non-ferroelastic twins, where a twin law can be expressed by a dichromatic space group (see Section[link]), whereas for ferroelastic twins with a compatible domain wall dichromatic layer groups are adequate (see Section[link]).

Restriction (ii)[link], formulated by Georges Friedel (1926[link]) and explained in detail by Cahn (1954[link]), 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)[link] 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)[link]. Condition (ii)[link] 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 [({\bf S}_1, {\bf S}_2)] and [({\bf S}_1, {\bf S}_3) ], formed from domain states of our illustrative example (see Fig.[link]), are displayed in Fig.[link](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.


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Transposable domain pairs. Single-domain states are those from Fig.[link]. (a) Completely transposable non-ferroelastic domain pair. (b) Partially transposable ferroelastic domain pair.

In Fig.[link], the ordered domain pair [({\bf S}_1, {\bf S}_2) ] and the transposed domain pair [({\bf S}_2, {\bf S}_1)] are depicted in a similar way for another example with symmetry descent [G=] [6_z/m_z \supset 2_z/m_z] = [ F_1 ].


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Non-transposable domain pairs. (a) The parent phase with symmetry [G=6_z/m_z] is represented by a dotted hexagon and the three ferroelastic single-domain states with symmetry [F_1=F_2=F_3=2_z/m_z] are depicted as drastically squeezed hexagons. (b) Domain pair [({\bf S}_1,{\bf S}_2) ] and transposed domain pair [({\bf S}_2,{\bf S}_1)]. There exists no operation from the group [6_z/m_z] that would exchange domain states [{\bf S}_1] and [{\bf S}_2], i.e. that would transform one domain pair into a transposed domain pair.

Let us now examine the symmetry of domain pairs. The symmetry group [{F}_{ik}] of an ordered domain pair [({\bf S}_i,{\bf S}_k) =] [({\bf S}_i,g_{ik}{\bf S}_i)] consists of all operations that leave invariant both [{\bf S}_i] and [{\bf S}_k ], i.e. [{F}_{ik}] comprises all operations that are common to stabilizers (symmetry groups) [F_i] and [F_k] of domain states [{\bf S}_i] and [{\bf S}_k], respectively, [{F}_{ik} \equiv {F}{_i} \cap {F}_k = {F}_i \cap g_{ik}{F}_{i}g_{ik}^{-1}, \eqno( ]where the symbol [\cap] denotes the intersection of groups [F_i] and [F_k]. The group [{F}_{ik}] is in Section 3.3.4[link] denoted by [{\cal H}^*] and is called an intersection group.

From equation ([link]), it immediately follows that the symmetry [F_{ki}] of the transposed domain pair [({\bf S}_k,{\bf S}_i) ] is the same as the symmetry [F_{ik}] of the initial domain pair [({\bf S}_i,{\bf S}_k)]: [{F}_{ki} = {F}{_k} \cap {F}_i={F}{_i} \cap {F}_k=F_{ik}.\eqno( ]

Symmetry operations of an unordered domain pair [\{{\bf S}_i,{\bf S}_k\} ] include, besides operations of [F_{ik}] that do not change either [{\bf S}_i] or [{\bf S}_k], all transposing operations, since for an unordered domain pair a transposed domain pair is identical with the initial domain pair [see equation ([link])]. If [g^{\star}_{ik} ] is a transposing operation of [({\bf S}_i,{\bf S}_k)], then all operations from the left coset [g^{\star}_{ik}F_{ik}] are transposing operations of that domain pair as well. Thus the symmetry group [J_{ik}] of an unordered domain pair [\{{\bf S}_i,{\bf S}_k\} ] can be, in a general case, expressed in the following way:[J_{ik} = F_{ik} \cup g^{\star}_{ik}F_{ik}, \quad g^{\star}_{ik}\in G.\eqno( ]

Since, for an unordered domain, the order of domain states in a domain pair is not significant, the transposition of indices [i, k] in [J_{ik}] does not change this group, [J_{ik}=F_{ik} \cup g^{\star}_{ik}F_{ik}=F_{ki} \cup g^{\star}_{ki}F_{ki}=J_{ki}, \eqno( ]which also follows from equations ([link]) and ([link]).

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 [({\bf S}_i,{\bf S}_k)] is given by equation ([link]).

The star in the symbol [J_{ik}^{\star}] indicates that this group contains transposing operations, i.e. that the corresponding domain pair [({\bf S}_i,{\bf S}_k)] is a transposable domain pair.

A transposable domain pair [({\bf S}_i,{\bf S}_k)] and transposed domain pair [({\bf S}_k,{\bf S}_i)] belong to the same G-orbit: [G({\bf S}_i,{\bf S}_k)=G({\bf S}_k,{\bf S}_i).\eqno( ]

If [\{{\bf S}_i,{\bf S}_k\}] is a transposable pair and, moreover, [F_i =F_k =F_{ik}], then all operations of the left coset [g_{ik}^{\star}F_i] simultaneously switch [{\bf S}_i] into [{\bf S}_k] and [{\bf S}_k] into [{\bf S}_i]. We call such a pair a completely transposable domain pair. The symmetry group [J_{ik}] of a completely transposable pair [\{{\bf S}_i,{\bf S}_k\} ] is [J_{ik}^{\star} = F_i \cup g^{\star}_{ik}F_i, \quad g^{\star}_{ik}\in G, \quad F_i=F_k. \eqno( ]We shall use for symmetry groups of completely transposable domain pairs the symbol [J_{ik}^{\star}].

If [F_i \neq F_k], then [F_{ik}\subset F_i] and the number of transposing operations is smaller than the number of operations switching [{\bf S}_i] into [{\bf S}_k]. We therefore call such pairs partially transposable domain pairs. The symmetry group [J_{ik}] of a partially transposable domain pair [\{{\bf S}_i,{\bf S}_k\}] is given by equation ([link]).

The symmetry groups [J_{ik}] and [J_{ik}^{\star}], expressed by ([link]) or by ([link]), respectively, consists of two left cosets only. The first is equal to [F_{ik}] and the second one [g^{\star}_{ik}F_{ik}] comprises all the transposing operations marked by a star. An explicit symbol [J_{ik}(F_{ik}) ] of these groups contains both the group [J_{ik}] and [F_{ik} ], which is a subgroup of [J_{ik}] of index 2.

If one `colours' one domain state, e.g. [{\bf S}_i], `black' and the other, e.g. [{\bf S}_k], `white', then the operations without a star can be interpreted as `colour-preserving' operations and operations with a star as `colour-exchanging' operations. Then the group [J_{ik}(F_{ik})] can be treated as a `black-and-white' or dichromatic group (see Section[link] ). These groups are also called Shubnikov groups (Bradley & Cracknell, 1972[link]), two-colour or Heesch–Shubnikov groups (Opechowski, 1986[link]), or antisymmetry groups (Vainshtein, 1994[link]).

The advantage of this notation is that instead of an explicit symbol [J_{ik}(F_{ik})], the symbol of a dichromatic group specifies both the group [J_{ik}] and the subgroup [F_{ij}] or [F_1], and thus also the transposing operations that define, according to equation ([link]), the second domain state [{\bf S}_j] of the pair.

We have agreed to use a special symbol [J_{ik}^{\star}] only for completely transposable domain pairs. Then the star in this case indicates that the subgroup [F_{ik}] is equal to the symmetry group of the first domain state [{\bf S}_i] in the pair, [F_{ik}=F_i]. Since the group [F_i] 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 [J_{ik}].

Domain pairs for which an exchanging operation [g^{\star}_{ik}] cannot be found are called non-transposable (or polar) domain pairs. The symmetry [J_{ij}] of a non-transposable domain pair is reduced to the usual `monochromatic' symmetry group [F_{ik}] of the corresponding ordered domain pair [({\bf S}_i,{\bf S}_k)]. The G-orbits of mutually transposed polar domain pairs are disjoint (Janovec, 1972[link]): [G({\bf S}_i,{\bf S}_k) \cap G({\bf S}_k,{\bf S}_i)= \emptyset.\eqno( ]Transposed polar domain pairs, which are always non-equivalent, are called complementary domain pairs.

If, in particular, [F_{ik}=F_i=F_k], then the symmetry group of the unordered domain pair is [J_{ik}=F_{i}=F_k.\eqno( ]In this case, the unordered domain pair [\{{\bf S}_i,{\bf S}_k\}] is called a non-transposable simple domain pair.

If [F_i\neq F_k], then the number of operations of [F_{ik}] is smaller than that of [F_i] and the symmetry group [J_{ik}] is equal to the symmetry group [F_{ik}] of the ordered domain pair [({\bf S}_i,{\bf S}_k)], [J_{ik}=F_{ik}, \quad F_{ik}\subset F_i. \eqno( ]Such an unordered domain pair [\{{\bf S}_i,{\bf S}_k\}] is called a non-transposable 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 single-domain 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  Now we examine domain pairs in our illustrative example of a phase transition with symmetry descent G = [4_z/m_zm_xm_{xy}\supset 2_xm_ym_z =] [F_1] and with four single-domain states [{\bf S}_1,] [{\bf S}_2,] [{\bf S}_3] and [{\bf S}_3], which are displayed in Fig.[link]. The domain pair [\{{\bf S}_1,{\bf S}_2\}] depicted in Fig.[link](a) is a completely transposable domain pair since transposing operations exist, e.g. [g^{\star}_{12}=m^{\star}_x], and the symmetry group [F_{12}] of the ordered domain pair [({\bf S}_1,{\bf S}_2) ] is [F_{12}=F_1\cap F_2=F_1=F_2=2_xm_ym_z.\eqno( ]The symmetry group [J_{12}] of the unordered pair [\{{\bf S}_1,{\bf S}_2\} ] is a dichromatic group, [J_{12}^{\star}=2_xm_ym_z \cup m^{\star}_x\{2_xm_ym_z\} = m_x^{\star}m_ym_z.\eqno( ]

The domain pair [\{{\bf S}_1,{\bf S}_3\}] in Fig.[link](b) is a partially transposable domain pair, since there are operations exchanging domain states [{\bf S}_1] and [{\bf S}_3], e.g. [g^{\star}_{13}=m^{\star}_{x\bar{y}}], but the symmetry group [F_{13}] of the ordered domain pair [({\bf S}_1,{\bf S}_3)] is smaller than [F_1]: [F_{13}=F_1 \cap F_3 = 2_xm_ym_z \cap m_x2_ym_z = \{1,m_z\}\equiv \{m_z\},\eqno( ]where 1 is an identity operation and [\{1,m_z\}] denotes the group [m_z]. The symmetry group of the unordered domain pair [\{{\bf S}_1,{\bf S}_3\} ] is equal to a dichromatic group, [J_{13} = \{m_z\} \cup 2^{\star}_{xy}.\{m_z\} = 2_{xy}^{\star}m_{\bar{x}y}^{\star}m_z.\eqno( ]

The domain pair [({\bf S}_1,{\bf S}_2)] in Fig.[link](b) is a non-transposable simple domain pair, since there is no transposing operation of [G=6_z/m_z] that would exchange domain states [{\bf S}_1 ] and [{\bf S}_2], and [F_1=F_2=2_z/m_z]. The symmetry group [J_{12}] of the unordered domain pair [\{{\bf S}_1,{\bf S}_3\}] is a `monochromatic' group, [J_{12}=F_{12}=F_1=F_2=2_z/m_z. \eqno( ]The G-orbit [6_z/m_z({\bf S}_1,{\bf S}_2)] of the pair [({\bf S}_1,{\bf S}_2)] has no common domain pair with the G-orbit [6_z/m_z({\bf S}_2,{\bf S}_1)] of the transposed domain pair [({\bf S}_2,{\bf S}_1) ]. 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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