The Lorentz Correction in Barium Titanate

Abstract

It is assumed, following Devonshire, that the ferroelectric behavior of BaTi${\mathrm{O}}_{3}$ arises because of the Lorentz correction, leading to a vanishing term in the denominator of the expression for dielectric constant. If the polarizability varies slowly with temperature, the temperature variation of dielectric constant follows. This temperature variation is assumed to come from that part of the polarization resulting from the displacement of the Ti ion, in a field whose potential energy has fourth-power as well as second-power terms in the displacement. The main object of this paper is to compute the Lorentz correction exactly, not assuming spherical symmetry, but taking account of the precise crystal structure. When this is done, it is found that the Ti ions, and those oxygen ions which are in the same line with them, the line being parallel to the electric field, exert very strong fields on each other, the resulting local field at the Ti ion being much greater than if computed on the assumption of spherical symmetry. This enhanced field makes it clear that even a relatively small ionic polarizability for the Ti ions will be enough to lead to ferroelectricity. The polarization of the Ti ions is however an essential feature of the theory; if they are not polarized, the Lorentz correction is profoundly modified, leading almost exactly to the value given by the approximate theory assuming spherical symmetry, and not resulting in ferroelectricity. Detailed formulas are given for comparison of the present theory with Devonshire's results, so that the present methods can be incorporated in his treatment of the effect of elastic strain energy on the stability of the various phases below the Curie point.