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. 490493
Section 3.4.2.2. Degenerate (secondary) domain states, partition of principal 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 
In this section we demonstrate that any morphic (spontaneous) property appears in the lowsymmetry phase in several equivalent variants and find what determines their number and basic properties.
As we saw in Fig. 3.4.2.2, the spontaneous polarization – a principal tensor parameter of the phase transition – can appear in four different directions that define four principal domain states. Another morphic property is a spontaneous strain describing the change of unitcell shape; it is depicted in Fig. 3.4.2.2 as a transformation of a square into a rectangle. This change can be expressed by a difference between two strain components , which is a morphic tensor parameter since it is zero in the parent phase and nonzero in the ferroic phase. The quantity is a secondary order parameter of the transition (for secondary order parameters see Section 3.1.3.2 ).
From Fig. 3.4.2.2, we see that two domain states and have the same spontaneous strain, whereas and exhibit another spontaneous strain . Thus we can infer that a property `to have the same value of spontaneous strain' divides the four principal domain states , , and into two classes: and with the same spontaneous strain and and with the same spontaneous strain . Spontaneous strain appears in two `variants': and .
We can define a ferroelastic domain state as a state of the crystal with a certain value of spontaneous strain , irrespective of the value of the principal order parameter. Values and thus specify two ferroelastic domain states and , respectively. The spontaneous strain in this example is a secondary order parameter and the ferroelastic domain states can therefore be called degenerate (secondary) domain states.
An algebraic version of the above consideration can be deduced from Table 3.4.2.1, where to each principal domain state (given in the second column) there corresponds a left coset of (presented in the first column). Thus to the partition of principal domain states into two subsets there corresponds, according to relation (3.4.2.9), a partition of left cosets where we use the fact that the union of the first two left cosets of is equal to the group . This group is the stabilizer of the first ferroelastic domain state , . Two left cosets of correspond to two ferroelastic domain states, and , respectively. Therefore, the number of ferroelastic domain states is equal to the number of left cosets of in , i.e. to the index of in , = : = : , and the number of principal domain states in one ferroelastic domain state is equal to the index of in , i.e. : : .
A generalization of these considerations, performed in Section 3.2.3.3.5 (see especially Proposition 3.2.3.30 and Examples 3.2.3.10 and 3.2.3.33 ), yields the following main results.
Assume that is a secondary order parameter of a transition with symmetry descent . Then the stabilizer of this parameter is an intermediate group, Lattices of subgroups in Figs. 3.1.3.1 and 3.1.3.2 are helpful in checking this condition.
The set of n principal domain states (the orbit ) splits into subsets
Each of these subsets consists of principal domain states,The number is called a degeneracy of secondary domain states.
The product of numbers and is equal to the number n of principal domain states [see equation (3.2.3.26 )]:
Principal domain states from each subset have the same value of the secondary order parameter , and any two principal domain states from different subsets have different values of . A state of the crystal with a given value of the secondary order parameter will be called a secondary domain state . Equivalent terms are degenerate or compound domain state.
In a limiting case , the parameter is identical with the principal tensor parameter and there is no degeneracy, .
Secondary domain states are in a onetoone correspondence with left cosets of in the decompositiontherefore
Principal domain states of the first secondary domain state can be determined from the first principal domain state : where is the representative of the kth left coset of of the decomposition
The partition of principal domain states according to a secondary order parameter offers a convenient labelling of principal domain states by two indices , where the first index j denotes the sequential number of the secondary domain state and the second index k gives the sequential number of the principal domain state within the jth secondary domain state [see equation (3.2.3.79 )]:where and are representatives of the decompositions (3.4.2.20) and (3.4.2.23), respectively.
The secondary order parameter can be identified with a principal order parameter of a phase transition with symmetry descent (see Section 3.4.2.3). The concept of secondary domain states enables one to define domain states that are characterized by a certain spontaneous property. We present the three most significant cases of such ferroic domain states.
The distinction ferroelastic–nonferroelastic is a basic division in domain structures. Ferroelastic transitions are ferroic transitions involving a spontaneous distortion of the crystal lattice that entails a change of shape of the crystallographic or conventional unit cell (Wadhawan, 2000). Such a transformation is accompanied by a change in the number of independent nonzero components of a symmetric secondrank tensor that describes spontaneous strain.
In discussing ferroelastic and nonferroelastic domain structures, the concepts of crystal family and holohedry of a point group are useful (IT A , 2005). Crystallographic point groups (and space groups as well) can be divided into seven crystal systems and six crystal families (see Table 3.4.2.2). A symmetry descent within a crystal family does not entail a qualitative change of the spontaneous strain – the number of independent nonzero tensor components of the strain tensor u remains unchanged.

We shall call the largest group of the crystal family to which the group M belongs the family group of M (symbol FamM). Then a simple criterion for a ferroic phase transition with symmetry descent to be a nonferroelastic phase transition is
A necessary and sufficient condition for a ferroelastic phase transition is
A ferroelastic domain state is defined as a state with a homogeneous spontaneous strain . [We drop the suffix `s' or `(s)' if the serial number of the domain state is given as the superscript . The definition of spontaneous strain is given in Section 3.4.3.6.1.] Different ferroelastic domain states differ in spontaneous strain. The symmetry of a ferroelastic domain state R_{i} is specified by the stabilizer of the spontaneous strain of the principal domain state [see (3.4.2.16)]. This stabilizer, which we shall denote by , can be expressed as an intersection of the parent group G and the family group of (see Table 3.4.2.2):This equation indicates that the ferroelastic domain state R_{i} has a prominent singledomain orientation. Further on, the term `ferroelastic domain state' will mean a `ferroelastic domain state in singledomain orientation'.
The number of ferroelastic domain states is given byIn our example, . In Table 3.4.2.7, last column, the number of ferroelastic domain states is given for all possible ferroic phase transitions.
The number of principal domain states compatible with one ferroelastic domain state (degeneracy of ferroelastic domain states) is given byIn our example, , i.e. two nonferroelastic principal domain states are compatible with each of the two ferroelastic domain states (cf. Fig. 3.4.2.2).
The product of and is equal to the number n of all principal domain states [see equation (3.4.2.19)],The number of principal domain states in one ferroelastic domain state can be calculated for all ferroic phase transitions from the ratio of numbers n and that are given in Table 3.4.2.7.
According to Aizu (1969), we can recognize three possible cases (see also Table 3.4.2.3):

Example 3.4.2.1. Domain states in leucite. Leucite (KAlSi_{2}O_{6}) (see e.g. Hatch et al., 1990) undergoes at about 938 K a ferroelastic phase transition from cubic symmetry to tetragonal symmetry . This phase can appear in singledomain states, which we denote , , . The symmetry group of the first domain state is . This group equals the stabilizer of the spontaneous strain of since Fam( (see Table 3.4.2.2), hence this phase is a full ferroelastic one.
At about 903 K, another phase transition reduces the symmetry to . Let us suppose that this transition has taken place in a domain state with symmetry ; then the roomtemperature ferroic phase has symmetry . The phase transition is a nonferroelastic one [] with nonferroelastic domain states, which we denote and . Similar considerations performed with initial domain states R_{2} and R_{3} generate another two couples of principal domain states , and , , respectively. Thus the roomtemperature phase is a partially ferroelastic phase with three degenerate ferroelastic domain states, each of which can contain two principal domain states. Both ferroelastic domains and nonferroelastic domains within each ferroelastic domain have been observed [see Fig. 3.3.10.13 in Chapter 3.3 , Palmer et al. (1988) and Putnis (1992)].
Ferroelectric domain states are defined as states with a homogeneous spontaneous polarization; different ferroelectric domain states differ in the direction of the spontaneous polarization. Ferroelectric domain states are specified by the stabilizer of the spontaneous polarization in the first principal domain state [see equation (3.4.2.16)]:The stabilizer is one of ten polar groups: 1, 2, 3, 4, 6, m, , , , . Since must be a polar group too, it is simple to find the stabilizer fulfilling relation (3.4.2.31).
The number of ferroelectric domain states is given byIf the polar group does not exist, we put . The number of ferroelectric domain states is given for all ferroic phase transitions in the eighth column of Table 3.4.2.7.
The number of principal domain states compatible with one ferroelectric domain state (degeneracy of ferroelectric domain states) is given by
The product of and is equal to the number n of all principal domain states [see equation (3.4.2.19)],The degeneracy of ferroelectric domain states can be calculated for all ferroic phase transitions from the ratio of the numbers n and that are given in Table 3.4.2.7.
Aizu (1969, 1970a) recognizes three possible cases (see also Table 3.4.2.3):
The classification of full, partial and nonferroelectrics and ferroelastics is summarized in Table 3.4.2.3.
Results for all symmetry descents follow readily from the numbers n, , in Table 3.4.2.7 and are given for all symmetry descents in Aizu (1970a). One can conclude that partial ferroelectrics are rather rare.
Example 3.4.2.3. Domain structure in tetragonal perovskites. Some perovskites (e.g. barium titanate, BaTiO_{3}) undergo a phase transition from the cubic parent phase with to a tetragonal ferroelectric phase with symmetry . The stabilizer Fam . There are 3 ferroelastic domain states each compatible with 2 principal ferroelectric domain states that are related e.g. by inversion , i.e. spontaneous polarization is antiparallel in two principal domain states within one ferroelastic domain state.
A similar situation, i.e. two nonferroelastic domain states with antiparallel spontaneous polarization compatible with one ferroelastic domain state, occurs in perovskites in the trigonal ferroic phase with symmetry and in the orthorhombic ferroic phase with symmetry .
Many other examples are discussed by Newnham (1974, 1975), Newnham & Cross (1974a,b), and Newnham & Skinner (1976).
In our illustrative example (see Fig. 3.4.2.2), we have seen that two domain states and have the same symmetry group (stabilizer) . In general, the condition `to have the same stabilizer (symmetry group)' divides the set of n principal domain states into equivalence classes. As shown in Section 3.2.3.3 , the role of an intermediate group is played in this case by the normalizer of the symmetry group of the first domain state . The number of domain states with the same symmetry group is given by [see Example 3.2.3.34 in Section 3.2.3.3.5 and equation (3.2.3.95 )], The number of subgroups that are conjugate under G to can be calculated from the formula [see equation (3.2.3.96 )]The product of and is equal to the number n of ferroic domain states,
The normalizer enables one not only to determine which domain states have the symmetry but also to calculate all subgroups that are conjugate under G to (see Examples 3.2.3.22 , 3.2.3.29 and 3.2.3.34 in Section 3.2.3.3 ).
Normalizers and the number of principal domain states with the same symmetry are given in Table 3.4.2.7 for all symmetry descents . The number of subgroups conjugate to is given by .
All these results obtained for pointgroup symmetry descents can be easily generalized to microscopic domain states and spacegroup symmetry descents (see Section 3.4.2.5).
References
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