High-density jellium-model calculation of force between half-planes of a nearly-free-electron metal at small separation

Abstract

The high-density limit of the jellium model is used to study the kinetic energy $T(z)$ of the electron gas as bulk jellium is separated into two half-planes at distance $z$. It is shown that, for $\mathrm{qz}\ensuremath{\ll}1$, where ${q}^{\ensuremath{-}1}$ is the Thomas-Fermi screening length, the Taylor expansion of $T(z)$ around $z=0$ contains, in particular, a quadratic term with a coefficient proportional to ${r}_{s}^{\frac{\ensuremath{-}11}{2}}$, where ${r}_{s}$ is the mean interelectronic separation. Using the virial theorem, this same ${r}_{s}$ dependence is shown to appear in the quadratic term in the expansion of the total energy $E(z)$. It is thereby argued that in the limit ${r}_{s}\ensuremath{\rightarrow}0$ the constant in the force $F(z)=Az$ for small $z$ in real metals, calculated from phonondispersion relations, must tend to a limit proportional to ${r}_{s}^{\frac{\ensuremath{-}11}{2}}$. Possible implications of this result for prediction of the surface energy of simple metals are briefly considered.