Effect of a magnetic field on the vibrations of an ionic lattice

If an ionic solid is regarded as a vibrating lattice of point charges, a static magnetic field will exert a Lorentz force on each ion. In this paper the standard lattice dynamical theory is modified to allow for such forces, and a perturbation approach is developed. The frequencies of lattice vibrations and the velocity of propagation of acoustic waves are found to be perturbed, with the perturbation parameter being the ratio of the ion-cyclotron frequency in the magnetic field to the characteristic frequency of lattice vibrations. It is found that in non-centro-symmetric crystals the perturbation will be first order in this parameter (and hence in the magnetic field), while for centro-symmetric crystals, with certain exceptions, they will be second order. The exceptions are for degenerate modes, such as 〈100〉 and 〈110〉 propagation in cubic crystals, when a first-order effect which is quadratic in the wave number is predicted. The possibility of experimental measurements of the first-order effects is pointed out. A simple two-dimensional model is examined for more detailed illustration.