Quantum Effects in the Amplification of Sound in the Presence of a Magnetic Field

A quantum-mechanical treatment is used to calculate the magnetic-field dependence of sound, which is amplified by interaction with conduction electrons in the presence of crossed dc electric and magnetic fields. It is shown that oscillatory behavior as a function of the strength of the applied magnetic field occurs under conditions of amplification. The oscillations which occur in the amplification have an amplitude equal to the nonoscillatory part of the amplification for nontransverse orientations of sound-wave vector q and magnetic field H, as long as there are electrons that have a component of drift velocity in the direction of q which is equal to the sound velocity $V_s$. The amplification occurs when the drift velocity in the crossed fields, ${\mathrm{V}}_H$, has a component in the direction of q which exceeds $V_s$.