Quantum Theory of Surface Energy and Tension

The surface energy $E_S$ of a system of interacting particles is $E_S=\int {-\infty }^{\infty }dz\left\{\epsilon \mathrm{}\right(z)-{\epsilon }_B[\frac{\rho \mathrm{}\left(z\right)\mathrm{}}{{\rho }_B}\left]\right\}$, where $\epsilon \mathrm{}\left(z\right)\mathrm{}$ and $\rho \mathrm{}\left(z\right)\mathrm{}$ are the energy and particle densities along the axis normal to the surface. ${\epsilon }_B$ and ${\rho }_B$ are the corresponding quantities in the bulk. Alternatively one may formulate an exact expression for the surface tension defined as $\gamma ={\left(\frac{\partial F}{\partial S}\right)}_V$, with the result that $\gamma =\int {-\infty }^{\infty }\mathrm{dz}\left\{2\left[t_z\mathrm{}\right(z)-t_{\perp }\mathrm{}(z\left)\right]+\frac{1}{4}\int d\mathrm{r}(r^2-3\mathrm{}{r_z}^2)\frac{1}{r}\frac{\mathrm{dv}}{\mathrm{dr}}{\rho }^{\mathrm{}\left(2\right)\mathrm{}}(z, z+r_z)\right\}.$ $t_i\mathrm{}\left(z\right)\mathrm{}$ are the components of the average kinetic energy density and ${\rho }^{\mathrm{}\left(2\right)\mathrm{}}(r_1, r_2)$ is the pair distribution function. In the ground state $\gamma =E_S$. We have applied these formulas to liquid ${\mathrm{He}}^4$ and ${\mathrm{He}}^3$ assuming that ${\rho }^{\mathrm{}\left(2\right)\mathrm{}}=\rho \mathrm{}\left(z\right)\mathrm{}\times \rho (z+r_z)g_B\mathrm{}\left(r\right)\mathrm{}$. The bulk radial distribution $g_B\mathrm{}\left(r\right)\mathrm{}$ is obtained from x-ray data. Use of the thermodynamic data ${\epsilon }_B$ and $p=0\mathrm{}$ gives $t_B$ and checks $g_B\mathrm{}\left(r\right)\mathrm{}$. Further we assume a free-volume form of the kinetic energy density $t\left(z\right)=t_B{\left[\frac{\rho \mathrm{}\left(z\right)\mathrm{}}{{\rho }_B}\right]}^{\frac{5}{3}}$ and an exponential density fall-off $\rho \mathrm{}\left(z\right)={\rho }_B{e}^{-\frac{z}{a}}$. Calculated values of $E_S$ show a minimum at $a=2.0\mathrm{}\mathrm{A}$. $E_S$ and $\gamma$ differ by 20% at this value, with $\gamma$ being 0.38 erg/${\mathrm{cm}}^2$ for ${\mathrm{He}}^4$. The experimental value of $\gamma$ is 0.35 erg/${\mathrm{cm}}^2$.