Interaction of Ultrasound with Electrons in Metals

The ultrasonic wave leads to a spatial separation of the ions from the free electrons at the Fermi surface. This causes an electrical field which eventually leads to a redistribution of the electrons and hence to an absorption coefficient $\alpha$ [Tucker/Rampton  (1972)]:

$$\alpha=\displaystyle\frac{nm}{2\rho v_{l}\tau}\left[{\displaystyle\frac{\displaystyle\frac{1}{3}k^{2}l_{e}^{2}\tan^{{-1}}(kl_{e})}{kl_{e}-\tan^{{-1}}(kl_{e})}-1}\right].$$

(1)

Here, n is the number of conduction electrons per volume, m is the electronic mass, l${}_{{e}}$ is the mean free path for electrons, v${}_{{F}}$ is the Fermi velocity, k the wave vector of the ultrasonic wave, and v ${}_{{l}}$ the longitudinal sound velocity.

In the limit of $kl_{e}\leq 1$ this leads to:

$$\alpha=\displaystyle\frac{2}{15}\displaystyle\frac{nmv_{F}^{2}\tau}{\rho v_{l}^{3}}(1-9/35(k_{e}l)^{2}+\ldots).$$

(2)

In the limit kl${}_{{e}}\gg 1$:

$$\alpha=\displaystyle\frac{\pi...