Domain states with the same stabilizer

Janovec, V.; Přívratská

doi:10.1107/97809553602060000645

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

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International Tables for Crystallography (2006). Vol. D. ch. 3.4, p. 457

Section 3.4.2.2.3. Domain states with the same stabilizer

V. Janovec^a ^* and J. Přívratská^b

^a Department of Physics, Technical University of Liberec, Hálkova 6, 461 17 Liberec 1, 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), 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 , 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 in Section 3.2.3.3.5 and equation (3.2.3.95 )], $[d_F=[N_G(F_1):F_1]=|N_G(F_1)|:|F_1|.\eqno(3.4.2.35) ]$ 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 )] $[n_F=[G:N_G(F_1)]=|G|:|N_G(F_1)|.\eqno(3.4.2.36) ]$ 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 , 3.2.3.29 and 3.2.3.34 in Section 3.2.3.3 ).

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 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).

References

International Tables for Crystallography (2006). Vol. D. ch. 3.4, p. 457