Eliashberg function in amorphous metals

An expression for the Eliashberg function ${\sigma }^{2}F\left(\omega \right)$ is derived for amorphous metals beginning with a formulation in terms of the Van Hove dynamical structure factor. The result is equivalent to one derived from a different starting point by Poon and Geballe. At low energy, ${\sigma }^{2}F\left(\omega \right)$ is shown to vary linearly with $\omega$ and inversely with the electron mean free path $\Lambda$ in agreement with Bergmann's expression derived for a Gaussian-disordered crystalline metal. Modification of the theory for short mean free paths is discussed in terms of the Pippard-Ziman condition on the electron-phonon interaction. Invoking a prescription derived by Pippard for the reduction of the electron-phonon interaction in ultrasonic attentuation, one finds a quadratic dependence of ${\sigma }^{2}F\left(\omega \right)$ on $\omega$ at low energies in high-resistivity amorphous metals; an even sharper reduction in the electron-phonon interaction and hence in ${\sigma }^{2}F\left(\omega \right)$ has been found by Poon, who treated the problem in transition-metal systems in the context of the Barisic-Labbe-Friedel rigid-ion approximation.