The frequencies and the anharmonicities of the normal modes of oscillation of alkali halide crystals I. Lattice oscillations

The frequencies and the anharmonicities of the lattice oscillations of alkali halide crystals, i.e. the oscillations of the interpenetrating lattices of the alkali and the halide ions respectively, with respect to each other, are calculated on the basis of the Born model. If r be the small relative displacement of the two lattices, and Imn the direction cosines of r with reference to the cubic axes of the crystal, it is found that the potential energy can be expressed in the form U= U 0 + ar 3 + br 4 + cr 4 ( l 4 + m 4 + n 4 )+..., in which the constants U 0 , a , b and c are readily evaluated. The coefficient of r 2 determines the frequency, and of r 4 the anharmonicity, of the lattice oscillation. This oscillation is characterized by the development of a homogeneous electric polarization in the medium. It is found that the polarization field acting on an ion tending to displace the ion has just the Lorentz value, whereas the field tending to polarize the ion is almost nothing. The anharmonicity of the lattice oscillation, unlike its frequency, is found to vary with the direction of the oscillation, from a large positive value along [111]: to a small negative value along [100]. Its effect on the frequency of the octave, and on the specific heat at constant volume, are discussed.