Classical Theory of the Pair Distribution Function of Plasmas

The diagrammatic method used by Abe to obtain the Helmholtz free energy of the classical electron gas is used to derive the pair distribution function beyond the Debye-Hückel result. The calculation can be done systematically either in configuration space or wave-number space and is carried out exactly to $\mathrm{O}\left({\Lambda }^2\right)$ where $\Lambda =\frac{1}{4}\pi \rho \lambda _D^{}{}_{}{}^3$ is the classical plasma parameter. It is shown that $g\left(r\right)\mathrm{}$ has the form $g\left(r\right)=e^{-\beta u_s}[1+g_2^{}{}_{}{}^b\mathrm{}(r)+g_2^{}{}_{}{}^c\mathrm{}(r\left)\right]$, where $u_s=\left(\frac{e^2}{r}\right)\exp \left(\frac{-r}{{\lambda }_D}\right)$ is the Debye screened potential, and $g_2^{}{}_{}{}^b$ and $g_2^{}{}_{}{}^c$ are functions of $\frac{r}{{\lambda }_D}$ which are evaluated analytically to $\mathrm{O}\left({\Lambda }^2\right)$. For the region $\beta e^2<r<\mathrm{}{\lambda }_D$ the correction in brackets multiplying the Boltzmann factor $\exp (-\beta u_s)$ is less than 1. A surprising result is that for $r\gg {\lambda }_D$ the linear Debye-Hückel theory does not correctly describe the disappearance of particle correlations, and instead the correction function $g_2^{}{}_{}{}^c$ dominates. As $r\to \infty$ it is found that $g\left(r\right)-1\mathrm{}\sim \frac{1}{8}\left(\ln 3\right){\Lambda }^2\exp \left(\frac{-r}{{\lambda }_D}\right)$.