Order parameter symmetries and free-energy expansions for purely ferroelastic transitions

Abstract

Purely ferroelastic transitions, i.e., transitions associated with a spontaneous deformation but no spontaneous polarization, appear from recent experimental investigations as a very common type of structural phase transitions. A theoretical analysis is presented for this class of transitions on the basis of Landau's theory of continuous phase transitions. The possible symmetries of their order parameter are systematically worked out both for the cell-preserving transitions and for those which modify the number of atoms in the crystal unit cell. Accordingly, we examine the transitions induced by the space-groups irreducible representations complying with the Landau and Lifshitz symmetry criteria. For each representation attention is given, in particular, to the construction of the finite group constituted by the set of distinct matrices of the representation, since this set (the image of the representation) determines the form of the Landau free energy and the number and characteristics of the low-symmetry phases. The images corresponding to the four-, six-, and eight-dimensional order parameters, which were not known previously, have been systematically determined for the 230 crystallographic space groups. We find 22 four-dimensional images associated with the active irreducible representations, ten six-dimensional, and five eight-dimensional ones. The corresponding Landau free energies have been expanded as a function of the order-parameter components, up to the lowest-degree terms which influence the stability of the low-symmetry phases just below a line of continuous transitions. It is pointed out that, in some cases, polynomials of degree as high as 12 must be considered. Whenever the order parameter and the spontaneous deformation have different symmetry properties, (improper ferroelastics), we have examined the additional contributions to the Landau free energy which represent the coupling between those two quantities. Their form depends on the particular space-group representations and not only on the images of these representations. We have worked out the lowest-degree coupling term for all the considered transitions. Its degree in the order parameter (the faintness index), which governs the qualitative temperature dependence of the macroscopic deformation and elastic constants is found equal to 2 in most cases, though values of 3 and 4 are also found possible. For each of the predicted types of ferroelastic transitions the space-symmetry change and unit-cell expansion have been determined. These crystallographic results are presented in table form and discussed. It is pointed out, in particular, that purely ferroelastic transitions constitute the major part of the ferroic phase transitions which are inferred to be possible in the framework of the Landau theory. Finally, the former theoretical results are compared to the available experimental data.