Order-parameter symmetry

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, p. 343

Section 3.1.2.2.6.1. Order-parameter symmetry

J.-C. Tolédano^d ^*

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.

Table 3.1.2.1 | top | pdf |
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.

References

International Tables for Crystallography (2006). Vol. D. ch. 3.1, p. 343