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. 492-493
Section 3.4.2.2.3. Domain states with the same stabilizer^{a}Institute of Physics, Academy of Sciences of the Czech Republic, Na Slovance 2, CZ-18221 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 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 point-group symmetry descents can be easily generalized to microscopic domain states and space-group symmetry descents (see Section 3.4.2.5).