Theory of ultrasonic attenuation in metals and magneto-acoustic oscillations

Abstract

The attenuation of an ultrasonic wave by direct interaction with the conduction electrons in a metal is analyzed without making any special assumptions about the shape of the Fermi surface. The problem is reduced to a calculation of the currents set up in a stationary lattice by forces on the electrons, some real (due to electric fields) and some fictitious to describe the disturbances due to the passage of the wave. The fictitious forces have their origin in the relative motion of different parts of the metal and in the change in shape of the equilibrium form of the Fermi surface as a result of lattice deformation. It is assumed, as is valid for ultrasonic frequencies below about 10 9 c/s, that the electric fields serve to annul any electronic current relative to the lattice. The expressions obtained are qualitatively similar to the earlier results for a free-electron model, in particular the attenuation tends to a con­stant limit as the electronic free path becomes infinite. Only for a pure longitudinal wave is the limit simple in form and there it is determined by the Gaussian curvature and the mean square value of the deformation parameter round that zone of the Fermi surface on which the electrons move parallel to the wave front. The analysis is extended to cover the situation which arises in a transverse magnetic field, in the limit when the free path is long compared with the electronic orbit perimeters. The results are very complicated, but qualitatively similar to the free-electron results in the types of oscillatory behaviour predicted and in the limiting values of the attenuation as the magnetic field increases without limit; if there are open orbits present, however, the limiting behaviour is changed, the attenuation of longitudinal waves for instance tending to infinity as H 2 . Although the oscillatory behaviour may usually be governed by the extreme exten­sions of the Fermi surface in a direction normal to H and to the direction of propagation, it is suggested that this is not likely to hold if the deformation parameter is very variable over the Fermi surface, as is probable in the noble metals, and that the oscillations may then indicate the positions of parts of high deformability.