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. 512515
Section 3.4.3.5. Nonferroelastic domain pairs: twin laws, domain distinction and switching fields, synoptic table^{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 
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).
References
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