Symmetry considerations

Tolédano, J.-C.

doi:10.1107/97809553602060000642

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. 3.1, pp. 343-344

Section 3.1.2.2.6. Symmetry considerations

J.-C. Tolédano^d ^*

3.1.2.2.6. Symmetry considerations

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3.1.2.2.6.1. Order-parameter symmetry

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Up to now, we have defined the order parameter as a set of degrees of freedom determining a second-degree contribution to the free energy, the coefficient of which has a specific temperature dependence proportional to [(T - T_c)] . Actually, the order parameter can also be defined on the basis of its specific symmetry characteristics.

Let us consider the manner by which the components ( [d_x] , [d_y] , [d_z] ) transform when we apply to the crystal each of the 16 symmetry operations of the group [G = 4/mmm] . Table 3.1.2.1 specifies the results of these transformations.

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Transformation of the components of $[\bf d]$ under the symmetry operations of group [G = 4/mmm]

G	E		$[\sigma_{xy}]$	$[\sigma_{xy'}]$



G	I	$[\sigma_z]$	$[U_{xy}]$	$[U_{xy'}]$

In the first place, we note that [d_z] is transformed either into itself or into ( [- d_z] ). If we consider this coordinate as the basis vector of a one-dimensional vector space, we can conclude that this vector space (i.e. the space formed by the set of vectors that are linear combinations of the basis) is invariant by all the transformations of the group G. Such a space, containing obviously no space of smaller dimension, is, according to the definitions given in Chapter 1.2 , a one-dimensional irreducible invariant space with respect to the group G.

Each of the components ( [d_x, d_y] ) is not transformed into a proportional component by all the elements of G. Certain of these elements transform [d_x] into $[\pm d_y]$ , and conversely. Hence [d_x] and [d_y] are not, separately, bases for one-dimensional irreducible invariant spaces. However, their set generates a two-dimensional vector space that has the property to be invariant and irreducible by all the transformations of G.

Note that the set of the three components ( [d_x] , [d_y] , [d_z] ) carries a three-dimensional vector space which, obviously, has the property to be invariant by all the transformations of G. However, this vector space contains the two invariant spaces carried respectively by [d_z] and by ( [d_x, d_y] ). Hence it is not irreducible.

In conclusion, from a symmetry standpoint, the order parameter of a phase transition is a set of degrees of freedom that carries an irreducible vector space (an irreducible representation) with respect to the action of the group G, the latter group being the symmetry group of the high-symmetry phase.

3.1.2.2.6.2. Degeneracy of the low-symmetry phase

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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