Distribution of activation energies for impurity hopping in amorphous metals

Abstract

The distribution of activation energies $\Delta$ for classical over-the-barrier hopping is computed for a model amorphous metal. The spread in $\Delta$ is determined by the variation in equilibrium-site and saddle-point sizes for the assumed model of dense random packing (DRP) of soft spheres. The size distribution is related to the radial distribution function in a manner which reproduces recent numerical results for the interstitials in DRP models. Size (distance) variation in general is related to energy variation by the form of the potential energy $V\left(r\right)\mathrm{}$. We show, however, that the distribution of equilibrium-site energies can be related directly to the impurity-induced lattice expansion and bulk modulus without detailed knowledge of $V\left(r\right)\mathrm{}$. The form of $V\left(r\right)\mathrm{}$ is necessary for the saddle-point distribution, and we estimate this using simple analytic expressions which fit the observed lattice expansion and impurity (hydrogen) vibrational frequency. The effects of a hard core plus lattice relaxation at the saddle point are also considered. Specific account is taken of the correlation between saddle-point and equilibrium-site configurations in the computation of the distribution of $\Delta$. Results are compared with recent data on hydrogen internal friction in amorphous ${\mathrm{Pd}}_{80}$ ${\mathrm{Si}}_{20}$ and good agreement is found between our first-principles distribution and that used to fit the data.