Lattice Dynamics of a Rigid-Ion Model for Gadolinium Molybdate

Abstract

Results of lattice-dynamical calculations are presented which support the view that the ferroelectric phase transition in gadolinium molybdate (GMO) arises from the softening and ultimate instability of a doubly degenerate zone-edge mode of the high-temperature paraelectric phase. We have used a rigid-ion model in which the short-range force constants are obtained from a detailed knowledge of the crystal structure together with the conditions imposed by the requirement that the crystal must be in static equilibrium under the combined influence of both Coulomb and short-range forces. Our results show that this type of approach is very useful when one is dealing with complex structures such as GMO, which has thirty-four ions per unit cell in the paraelectric phase. In view of the simplicity of our model we are able to obtain a surprisingly good correlation with experimental results. In particular, our calculated zone-center frequencies reproduce the basic features of the observed Raman spectrum. Dispersion curves are presented which show a pronounced softening of two phonon branches which become doubly degenerate at the $M$ point. This result is in agreement with the results obtained by inelastic neutron scattering. The displacements associated with the soft $M$-point modes correlate with the difference in the structures of the high- and low-temperature phases determined by x-ray diffraction. This provides further evidence that the ferroelectric domains in GMO are to be interpreted as "frozen-in" soft zone-boundary modes of the paraelectric phase.