Focusing Collisions in a Linear Chain of Atoms

Abstract

Focusing of the Silsbee type is studied in a row of identical atoms with the uniform spacing $S$, which interact during collisions through a repulsive potential $V\left(r\right)\mathrm{}$. Two energy-dependent parameters are introduced: the distance of closest approach in a head-on collision $c$, and $\gamma =-{\left(\frac{d\ln V}{d\ln r}\right)}_{r=c}$. In terms of these, the focusing parameter $\Lambda =\frac{{\theta }_2}{{\theta }_1}$ is approximately $\left(\frac{S{I}_0}{c}\right)-1\mathrm{}$, where $I_0$ is an integral involving $V\left(r\right)\mathrm{}$ which depends mainly on $\gamma$. For simple exponential potentials and large $\gamma$, $I_0=1+\frac{\left(\ln 4\right)}{\gamma }-\frac{1.368}{{\gamma }^2}+\frac{3.41}{{\gamma }^3}\cdots$. Appreciably less focusing is predicted than follows from earlier approximations which correspond to putting $I_0=1\mathrm{}$. For $〈110〉$ chains in copper, focusing is found at energies below 23 ev. This compares well with the 30 ev obtained by Gibson et al., for their three-dimensional model of copper.