Radial Distribution Functions and Bound Electronic Energy Levels in Hydrogen Plasmas

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Abstract
The radial distribution functions (RDF) for a hydrogen plasma are calculated, on the basis of the coupled proton-electron model, by solving a quantal hyper-netted chain equation (QHNC) for a mixture at densities 1018, 1020, 6 × 1021 and 6 × 1020 electrons/cm3 and at several temperatures of the order of 104K. The short-range divergence in the electron-proton RDF gep(r) due to the Coulomb interaction is automatically avoided by treating Schrödinger’s equation which is naturally requested to solve the QHNC equation. At the same time, the bound electronic energy levels are evaluated from such a self-consistent field around a proton in the plasma in the same region of densities and temperatures, that determines gep(r). There appears a peak in the proton-proton RDF which exhibits a tendency to form a hydrogen molecule in the plasma as its temperature decreases; for densities 1018, 1020, 6 × 1020 and 6 × 1021 electrons/cm3, this molecular formation occurs at about 3.3 × 104, 4.8 × 104, 5.4 × 104 and 5.9 ×104K, respectively. The approach of the plasma to this temperature may be observed by larger variations of the energy levels associated with temperature decrease. It is shown that a quantal version of the Debye-Hückel equation (QDH) can be derived from the QHNC equation within certain restriction; the QDH equation is also applied to this system and its results are compared with those of the QHNC equation. It is concluded that the QHNC equation is necessary for the description of a high-density plasma at low temperature with a larger plasma parameter Γ≫1 or near the molecular-formation temperature, while the QDH equation offers good results both for the RDF’s and for the bound energy levels, provided that the density and the temperature of the plasma belong to the region Γ≫1.