Self-consistent treatment of the frequency spectrum of a model paraelectric

Abstract

The temperature dependence of the frequency spectrum of a model NaCl-structure paraelectric exhibiting soft-mode behavior at the zone center is treated self-consistently. A variational formulation is employed, yielding a set of nonlinear integral equations which are solved iteratively. The difficulties associated with perturbative treatments are avoided by starting from a renormalized phonon basis which represents a stable state of equilibrium for the crystal. In particular, the soft-mode branches, which may be imaginary in the harmonic approximation, are renormalized and treated self-consistently. The model consists of rigid anions and cations of unequal masses interacting via long-range Coulomb forces plus a short-range interaction extending to second neighbors. Anharmonicity is introduced through a parametrized quartic interaction. Calculations are carried out for a wide range of model parameters, including the situation where the phase space subtended by the imaginary harmonic frequencies may constitute a large fraction of the Brillouin zone. The importance of including the effect of the soft-mode branches is discussed in detail. It is pointed out that the self-consistent equations do not admit a zero solution for the soft-zone-center optic mode. An explanation is given of why the Curie-law dependence of the soft-zone-center frequency squared may extend to lower temperatures than one might expect on simple qualitative grounds. Attention is given to the coupling of the soft TO branch to the TA branch of the same symmetry, and the anomalous temperature dependence of the acoustic branch at finite wave vector is displayed. Finally, the temperature dependence of the intersublattice correlation function is calculated, and it is pointed out that the soft-mode branches contribute significantly to the temperature dependence of this function. The implications of the latter for a transition to an ordered phase are discussed. © 1971 The American Physical Society.