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

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