Theory of Ultrasonic Absorption in Metals: the Collision-Drag Effect

A basic assumption of the semiclassical treatments of ultrasonic absorption in metals is that of "collision drag." This assumption states that, in the presence of an impressed ultrasonic wave, the velocity distribution toward which the conduction electrons relax is a Fermi distribution centered, not at the origin of velocity space, but at a point, ${\mathrm{v}}_i$, equal to the local, impressed lattice-displacement velocity. In the present paper, the explanation of this collision-drag effect in terms of basic electron-lattice theory is investigated for the case of collisions with thermal vibrations. The effect is found to originate from those higher-order terms in the electron-lattice interaction whose matrix elements are bilinear in the amplitudes of impressed and thermal lattice displacements. In the conventional perturbation-theory treatment, these matrix elements give rise to transitions in which both a thermal phonon and an impressed phonon are simultaneously either absorbed or emitted. However, in such a treatment, no collision-drag effects appear. In order to obtain them, it is necessary to alter the standard perturbation treatment so as to provide for space-time localization of collisions to within an interval small compared to the wavelength and period of the impressed sound wave. It is then found that the impressed ultrasonic wave produces a modification in the energy-conservation law of electron-lattice collisions, ${\epsilon }_{{\mathrm{k}}^{\prime }}={\epsilon }_{\mathrm{k}}\pm \hslash {\omega }_{\lambda }$, in which the effective electron energy, ${\epsilon }_{\mathrm{k}}$, is the Bloch energy, augmented by a term proportional to the impressed displacement velocity [Eq. (2.30)]. When this modification is introduced into the collision term of the Boltzmann transport equation, the equilibrium distribution (defined as that for which the collision term vanishes) is found to be a Fermi distribution centered about a point in k space equal to $\frac{m{\mathrm{v}}_i}{\hslash }$; for the free-electron model, this result is in accord with the collision-drag assumption as stated above. An additional result of the treatment is that the crystal-momentum conservation law of electron-lattice collisions is altered by the inclusion of terms linearly proportional to the impressed strain; this modification, however, turns out to have no effect in ultrasonic absorption. The final section of the paper is devoted to an investigation of the effects on energy transfer arising from the bilinear matrix elements; these effects are shown to be describable, also, in terms of the collision-drag picture.