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. 508510
Section 3.4.3.2. Twinning group, distinction of two domain states^{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 
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.
References
Amin, A. & Newnham, R. E. (1980). Tertiary ferroics. Phys. Status Solidi A, 61, 215–219.Fuksa, J. (1997). The role of the twinning group of a domain pair in tensor distinction of domain states. Ferroelectrics, 204, 135–155.
Fuksa, J. & Janovec, V. (1995). Permutation classification of domain pairs. Ferroelectrics, 172, 343–350.
Kopský, V. (2001). Tensor parameters of ferroic phase transitions I. Theory and tables. Phase Transit. 73, 1–422.
Newnham, R. E. & Cross, L. E. (1974a). Symmetry of secondary ferroics I. Mater. Res. Bull. 9, 927–934.
Newnham, R. E. & Cross, L. E. (1974b). Symmetry of secondary ferroics II. Mater. Res. Bull. 9, 1021–1032.
Newnham, R. E. & Skinner, D. P. Jr (1976). Polycrystalline secondary ferroics. Mater. Res. Bull. 11, 1273–1284.