Pair-correlation functions for partially ionized hydrogen

A method is presented for calculating the pair-correlation functions for a high-density quantum-mechanical plasma of protons and electrons. The Slater sum is approximated in the form of a classical Boltzmann factor with the Coulomb potentials replaced by effective potentials. The effective potentials contain the classical potential plus quantum corrections. The quantum effects may be separated into symmetry effects, diffraction effects, and coupling between the two. This method is exact at infinite temperatures and is tested here down to temperatures at which there is about 40% ionization. The pair-correlation functions are obtained from a Monte Carlo calculation. The fraction of electrons in the ground state of an atom is also calculated; an approximate formula for this fraction is $\left(\frac{1}{2}\right)[1-\mathrm{erf}(\frac{b\ln T}{T_0}\left)\right]$, where $b$ and $T_0$ are simple fitted functions of the density.