Degeneracy of the low-symmetry phase

Tolédano, J.-C.

doi:10.1107/97809553602060000642

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

pdf | chapter contents | chapter index | related articles

International Tables for Crystallography (2006). Vol. D. ch. 3.1, pp. 343-344

Section 3.1.2.2.6.2. Degeneracy of the low-symmetry phase

J.-C. Tolédano^d ^*

3.1.2.2.6.2. Degeneracy of the low-symmetry phase

| top | pdf |

We had noted above that the structure is invariant by G in the stable state of the system above [T_c] . When $[{\bf d} \ne 0]$ , the structure becomes invariant by a smaller set of transformations. Let us enumerate these transformations for each possible stable state of the system below [T_c] .

When the order parameter coincides with [d_z] , we determined, below [T_c] , two stable states, $[d_z^0 = \pm [\alpha(T_c - T)/\beta]^{1/2}]$ . The crystalline structures determined by these displacements of the [M^+] ion parallel to the z axis are both invariant by the same set of eight symmetry transformations. These comprise the cyclic group of order 4 generated by the fourfold rotation around z, and by the reflections in planes containing this axis. This set is the group $[C_{4v} = 4mm]$ , a subgroup F of G. The transition is thus accompanied by a lowering of the symmetry of the system.

Also note that the two states $[\pm d_z^z]$ are transformed into each other by certain of the symmetry operations such as the mirror symmetry $[\sigma_z]$ `lost' below [T_c] . These two states correspond to the same value of the free energy [the minimum value determined in equation (3.1.2.3)]: they are equally stable. This can also be checked by applying to the system the mirror symmetry $[\sigma_z]$ . This transformation keeps unchanged the value of F since the free energy is invariant by all the transformations belonging to G (to which $[\sigma_z]$ belongs). The state [d_z] is, however, not preserved, and is transformed into ( [- d_z] ).

We have not determined explicitly the stable states of the system in the case of a two-dimensional order parameter ( [d_x, d_y] ). A simple discussion along the line developed for the one-dimensional order parameter [d_z] would show that the relevant form of the free energy is $[F= F_0 + {\alpha(T-T_c)\over 2}\left(d_x^2 + d_y^2\right) + \beta_1 \left(d_x^4 + d_y^4\right) +\beta_2 d_x^2 d_y^2 \eqno (3.1.2.12)]$ and that the possible stable states below [T_c] are:

(i) $[d_x^0 = \pm [\alpha (T_c - T)/\beta_1]^{1/2}]$ , ;
(ii) $[d_y^0 = \pm [\alpha (T_c - T)/\beta_2]^{1/2}]$ , ;
(iii) and (iv) $[d_x^0 = \pm d_y = \pm[\alpha (T_c - T)/(\beta_1 + \beta_2)]^{1/2}]$ .

Like the case of [d_z] , there is a lowering of the crystal symmetry below [T_c] . In the four cases, one finds that the respective symmetry groups of the structure are (i) $[F = C_{2v} = mm2_x]$ ; (ii) $[F' = C_{2v} = mm2_y]$ ; (iii) $[F = C_{2v} = mm2_{xy}]$ ; (iv) [F'=] $[C_{2v}=]$ $[mm2_{xy}]$ .

States (i) and (ii) correspond to each other through one of the `lost' transformations of G (the rotations by $[\pi / 2]$ ). They therefore possess the same free energy and stability. The second set of states (iii) and (iv) also constitute, for the same reason, a pair of states with the same value of the equilibrium free energy.

Note that the symmetry groups associated with equally stable states are conjugate relative to G, that is they satisfy the relationship $[F' = gFg^{-1}]$ , with g belonging to G.

References

International Tables for Crystallography (2006). Vol. D. ch. 3.1, pp. 343-344