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

International Tables for Crystallography (2013). Vol. D, ch. 3.4, pp. 511-512

Section Classes of equivalent domain pairs and their classifications

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: Classes of equivalent domain pairs and their classifications

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Two domain pairs that are crystallographically equivalent, [({\bf S}_i,{\bf S}_k)\,{\buildrel{G}\over {\sim}}\,({\bf S}_l,{\bf S}_m) ] [see equation ([link])], 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 [({\bf S}_i,{\bf S}_k)], can be obtained by applying to [({\bf S}_i,{\bf S}_k)] all operations of G, i.e. by forming a G-orbit [G({\bf S}_i,{\bf S}_k)].

One can always find in this orbit a domain pair [({\bf S}_1,{\bf S}_j) ] that has in the first place the first domain state [{\bf S}_1]. We shall call such a pair a representative domain pair of the orbit. The initial orbit [G({\bf S}_i,{\bf S}_k)] and the orbit [G({\bf S}_1,{\bf S}_j) ] are identical: [G({\bf S}_i,{\bf S}_k)=G({\bf S}_1,{\bf S}_j). ]

The set [{\sf P}] of [n^2] ordered pairs (including trivial ones) that can be formed from n domain states can be divided into G-orbits (classes of equivalent domain pairs): [{\sf P} = G({\bf S}_1,{\bf S}_1) \cup G({\bf S}_1,g_2{\bf S}_1) \cup\ldots\cup ({\bf S}_1,g_j{\bf S}_1) \cup\ldots\cup G({\bf S}_1,g_q{\bf S}_1).\eqno( ]

Similarly, as there is a one-to-one correspondence between domain states and left cosets of the stabilizer (symmetry group) [F_1] of the first domain state [see equation ([link])], there is an analogous relation between G-orbits of domain pairs and so-called double cosets of [F_1].

A double coset [F_1g_jF_1] of [F_1] is a set of left cosets that can be expressed as [fg_jF_1], where [f\in F_1 ] runs over all operations of [F_1] (see Section[link] ). A group G can be decomposed into disjoint double cosets of [F_1\subset G]: [\displaylines{G=F_{1}eF_1 \cup F_{1}g_2F_{1} \cup\ldots\cup F_{1}g_jF_{1}\cup\ldots \cup F_{1}g_qF_{1},\cr\hfill j=1,2,\ldots, q,\hfill(} ]where [g_1=e,g_2,{\ldots}g_j,{\ldots}g_q] is the set of representatives of double cosets.

There is a one-to-one correspondence between double cosets of the decomposition ([link]) and G-orbits of domain pairs ([link]) (see Section[link] , Proposition[link] ): [G({\bf S}_1,{\bf S}_j)\leftrightarrow F_{1}g_jF_{1}, \ \ {\rm where} \ \ {\bf S}_j=g_j{\bf S}_1, \quad j=1,2,{\ldots},q.\eqno( ]

We see that the representatives [g_j] of the double cosets in decomposition ([link]) define domain pairs [({\bf S}_1,g_j{\bf S}_1) ] which represent all different G-orbits of domain pairs. Just as different left cosets [g_iF_1] specify all domain states, different double cosets determine all classes of equivalent domain pairs (G-orbits 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, [(F_1g_{j}F_1)^{-1}=F_1g_j^{-1}F_1=F_1g_jF_1,\eqno(]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, [(F_1g_{j}F_1)^{-1}=F_1g_j^{-1}F_1\cap F_1g_jF_1=\emptyset, \eqno( ]is a non-invertible (polar) double coset ([\emptyset] denotes an empty set) and the corresponding class of domain pairs comprises non-transposable (polar) domain pairs. A double coset [F_1g_jF_1] and its inverse [(F_1g_{j}F_1)^{-1}] are called complementary double cosets. Corresponding classes called complementary classes of equivalent domain pairs consist of transposed domain pairs that are non-equivalent.

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, [F_1g_jF_1=g_jF_1=(g_jF_1)^{-1}, \eqno( ]then the domain pairs in the G-orbit [G({\bf S}_1,g_j{\bf S}_1) ] are completely transposable. An invertible double coset comprising several left cosets is associated with a G-orbit consisting of partially transposable domain pairs. Non-invertible double cosets can be divided into simple non-transposable double cosets (complementary double cosets consist of one left coset each) and multiple non-transposable double cosets (complementary double cosets comprise more than one left coset each).

Thus there are four types of double cosets (see Table[link] in Section[link] ) to which there correspond the four basic types of domain pairs presented in Table[link].

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Four types of domain pairs

[F_{1j}][J_{1j}][K_{1j}] Double cosetDomain pair name symbol
[F_{1}=F_j ] [F_{1}\cup g_{1j}^{\star}F_1] [F_{1}\cup g_{1j}^{\star}F_1] [F_1g_{1j}F_1=g_{1j}F_1=(g_{1j}F_1)^{-1} ] [\underline{t}]ransposable [\underline{c} ]ompletely tc
[F_{1j}\subset F_1 ] [F_{1j}\cup g_{1j}^{\star}F_{1j} ] [F_{1}\cup g_{1j}^{\star}F_{1}\cup\,\ldots \, ] [F_1g_{1j}F_1=(F_1g_{1j}F_1)^{-1}] [\underline{t}]ransposable [\underline{p} ]artially tp
[F_{1}=F_j ] [F_{1} ] [F_{1}\cup g_{1j}F_{1}\cup g_{1j}^{-1}F_{1} ] [F_1g_{1j}F_1=g_{1j}F_1\cap (g_{1j}F_1)^{-1}=\emptyset ] [\underline{n}]on-transposable [\underline{s}]imple ns
[F_{1j}\subset F_1 ] [F_{1j} ] [F_{1}\cup g_{1j}F_{1}\cup (g_{1j}F_{1})^{-1}\cup\,\ldots ] [F_1g_{1j}F_1\cap (F_1g_{1j}F_1)^{-1}=\emptyset ] [\underline{n}]on-transposable [\underline{m}]ultiple nm

These results can be illustrated using the example of a phase transition with G = [4_z/m_zm_xm_{xy}\supset 2_xm_ym_z =] [F_1] with four domain states (see Fig.[link]). The corresponding four left cosets of [2_xm_ym_z] are given in Table[link]. Any operation from the first left coset (identical with [F_1]) 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 [({\bf S}_1,\bar 1{\bf S}_1)=({\bf S}_1,{\bf S}_2) ]. By applying operations of [G=4_z/m_zm_xm_{xy}] on [({\bf S}_1,{\bf S}_2) ], one gets the class of equivalent domain pairs (G-orbit): [({\bf S}_1,{\bf S}_2)] [{\buildrel {G}\over {\sim}}] [({\bf S}_2,{\bf S}_1) ] [{\buildrel {G} \over {\sim}}] [({\bf S}_3,{\bf S}_4)] [{\buildrel {G} \over {\sim}}] [({\bf S}_4,{\bf S}_3)]. 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, [({\bf S}_1,2_{xy}{\bf S}_1)=] [({\bf S}_1,{\bf S}_3) ] which is indeed a partially transposable domain pair [cf. ( and (]. The class of equivalent ordered domain pairs is [({\bf S}_1,{\bf S}_3) ] [{\buildrel {G}\over{\sim}}] [({\bf S}_3,{\bf S}_1)] [{\buildrel{G}\over {\sim}}] [({\bf S}_1,{\bf S}_4)] [{\buildrel{G}\over {\sim}} ] [({\bf S}_4,{\bf S}_1)] [{\buildrel{G}\over {\sim}}] [({\bf S}_3,{\bf S}_2)] [{\buildrel{G}\over {\sim}}] [({\bf S}_2,{\bf S}_3) ] [{\buildrel{G}\over {\sim}}] [({\bf S}_2,{\bf S}_4)] [{\buildrel{G}\over {\sim}}] [({\bf S}_4,{\bf S}_2)]. These are `90° domain pairs'.

An example of non-invertible double cosets is provided by the decomposition of the group [G=6_z/m_z] into left and double cosets of [F_1=2_z/m_z ] displayed in Table[link]. 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 non-transposable domain pairs are [({\bf S}_1,{\bf S}_2)] and [({\bf S}_2,] [{\bf S}_1)], and are depicted in Fig.[link].

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Decomposition of [G=6_z/m_z] into left cosets of [F_1=2_z/m_z ]

Left cosetPrincipal domain state
1 [2_z] [\bar 1] [m_z] [{\bf S}_1]
[3_z ] [6_z^5 ] [{\bar 3}_z ] [{\bar 6}_z^5 ] [{\bf S}_2]
[3_z^2 ] [6_z ] [{\bar 3}_z^5 ] [{\bar 6}_z ] [{\bf S}_3]

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 [G\supset F_1]. Left coset and double coset decompositions for all crystallographic point-group descents are available in the software GI[\star]KoBo-1, 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 [K_{1j}] can represent classes of equivalent domain pairs [G({\bf S}_1,{\bf S}_j)] with two exceptions:

  • (i) Two complementary classes of non-transposable domain pairs have the same twinning group. This follows from the fact that if a twinning group contains the double coset, then it must comprise also the inverse double coset.

  • (ii) Different classes of transposable domain pairs have different twining groups except in the following case (which corresponds to the orthorhombic ferroelectric phase in perov­skites): the group [F_1 =m_{x\bar y}2_{xy}m_z] generates with switching operations [g=2_{yz} ] and [g_3=m_{yz}] two different double cosets with the same twinning group [K_{12}=K_{13}=m\bar 3m] (one can verify this in the software GI[\star]KoBo-1, path: Subgroups\View\Twinning groups). Domain states are characterized in this ferroelectric phase by the direction of the spontaneous polarization. The angles between the spontaneous polarizations of the domain states in domain pairs [({\bf S}_1,2_{yz}{\bf S}_1)] and [({\bf S}_1,m_{yz}{\bf S}_1) ] are 120 and 60°, respectively; this shows that these representative domain pairs are not equivalent and belong to two different G-orbits of domain pairs. To distinguish these two cases, we add to the twinning group [m\bar 3m[m_{x\bar y}2_{xy}m_z]] either the switching operation [2_{yz}] or [m_{yz}], i.e. the two distinct orbits are labelled by the symbols [m\bar 3m(2_{xy})] and [m\bar 3m(m_{xy}) ], respectively.

Bearing in mind these two exceptions, one can, in the continuum description, represent G-orbits of domain pairs [G({\bf S}_1,{\bf S}_j) ] by twinning groups [K_{1j}(F_1)].

We have used this correspondence in synoptic Table[link] of symmetry descents at ferroic phase transitions. For each sym­metry descent [G\supset F_1], the twinning groups given in column [K_{1j}] specify possible G-orbits of domain pairs that can appear in the domain structure of the ferroic phase (Litvin & Janovec, 1999[link]). We divide all orbits of domain pairs (represented by corresponding twinning groups [K_{1j}]) that appear in Table[link] into classes of non-ferroelastic and ferroelastic domain pairs and present them with further details in the three synoptic Tables[link],[link] and[link] described in Sections[link] and[link].

As we have seen, a classification of domain pairs according to their internal symmetry (summarized in Table[link]) introduces a partition of all domain pairs that can be formed from domain states of the G-orbit [G{\bf S}_1] 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[link], columns two and three), introduces a division of domain pairs into corresponding equivalence classes.


Litvin, D. B. & Janovec, V. (1999). Classification of domain pairs and tensor distinction. Ferroelectrics, 222, 87–93.

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