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

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

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:  janovec@fzu.cz

3.4.2.2.3. Domain states with the same stabilizer

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In our illustrative example (see Fig. 3.4.2.2[link]), we have seen that two domain states [{\bf S}_1] and [{\bf S}_2] have the same symmetry group (stabilizer) [2_xm_ym_z]. 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[link] , the role of an intermediate group [L_1] is played in this case by the normalizer [N_G(F_1)] of the symmetry group [F_1] of the first domain state [{\bf S}_1]. The number [d_F] of domain states with the same symmetry group is given by [see Example 3.2.3.34[link] in Section 3.2.3.3.5[link] and equation (3.2.3.95[link] )], [d_F=[N_G(F_1):F_1]=|N_G(F_1)|:|F_1|.\eqno(3.4.2.35) ]The number [n_F] of subgroups that are conjugate under G to [F_1] can be calculated from the formula [see equation (3.2.3.96[link] )][n_F=[G:N_G(F_1)]=|G|:|N_G(F_1)|.\eqno(3.4.2.36) ]The product of [n_F] and [d_F] is equal to the number n of ferroic domain states, [n=n_Fd_F.\eqno(3.4.2.37) ]

The normalizer [N_G(F_1)] enables one not only to determine which domain states have the symmetry [F_1] but also to calculate all subgroups that are conjugate under G to [F_1] (see Examples 3.2.3.22[link] , 3.2.3.29[link] and 3.2.3.34[link] in Section 3.2.3.3[link] ).

Normalizers [N_G(F_1)] and the number [d_F] of principal domain states with the same symmetry are given in Table 3.4.2.7[link] for all symmetry descents [G \supset F_1]. The number [n_F] of subgroups conjugate to [F_1] is given by [n_F=n:d_F ].

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[link]).








































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