The frequencies and the anharmonicities of the normal modes of oscillation of alkali halide crystals - II. Low-frequency acoustic modes

Abstract

In the present part are deduced, on the basis of Born's model, expressions for the frequency and the ratio of the amplitudes of the alkali and the halide ions in an alkali halide crystal, for any general normal mode of oscillation of the crystal. The results are applied in detail to the special case of low-frequency acoustic modes. Since the amplitudes of the two ions are not in general the same, there is a resultant electric polarization of the medium accompanying the oscillations, and consequently a polarization field. The force acting on an ion due to this field is found to be comparable with the force of interaction with the other ions, not only in the optical branch, in which the displacements of adjacent positive and negative ions are in opposite directions and in which, therefore, the polarization is large, but also in general in the acoustic branch, in which their displacements are in the same direction. A detailed calculation, however, shows that for low frequency acoustic modes, though the ratio of the amplitudes of the two ions is affected by the polarization field, the frequency remains completely unaffected by it. The expressions deduced for the frequencies of the acoustic modes give us also the velocities of propagation of the corresponding acoustic waves, and since the latter are already known in terms of the elastic constants of the crystal, we obtain, incidentally, simple expressions for these constants. The elastic constants so calculated are found to agree with observation. Unlike the principal oscillation of the crystal dealt with in part I, these low-frequency acoustic modes have negligible anharmonicity.