Pressure of a partially ionized hydrogen gas : numerical results from exact low temperature expansions
Résumé
We consider a partially ionized hydrogen gas at low densities, where it reduces almost to an ideal mixture made with hydrogen atoms in their ground-state, ionized protons and ionized electrons. By performing systematic low-temperature expansions within the physical picture, in which the system is described as a quantum electron-proton plasma interacting via the Coulomb potential, exact formulae for the first five leading corrections to the ideal Saha equation of state have been derived [A. Alastuey, V. Ballenegger et al., J. Stat. Phys. 130, 1119 (2008)]. Those corrections account for all effects of interactions and thermal excitations up to order $\exp(E_H/kT)$ included, where $E_H \simeq -13.6$ eV is the ground state energy of the hydrogen atom. Among the five leading corrections, three are easy to evaluate, while the remaining ones involve suitably truncated internal partition functions of $H_2$ molecules and $H^-$ and $H_2^+$ ions, for which no analytical formulae are available in closed form. We estimate those partitions functions at finite temperature \textsl{via} a simple phenomenology based on known values of rotational and vibrational energies. This allows us to compute numerically the leading deviations to the Saha pressure along several isotherms and isochores. Our values are compared with those of the OPAL tables (for pure hydrogen) calculated within the ACTEX method.
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