Exact asymptotic expansions for the thermodynamics of hydrogen gas in the Saha regime

Abstract

We consider the hydrogen quantum plasma in the Saha regime, where it almost reduces to a partially ionized atomic gas. We briefly review the construction of systematic expansions of thermodynamical functions beyond Saha theory, which describes an ideal mixture of ionized protons, ionized electrons and hydrogen atoms in their ground-state. Thanks to the existence of rigorous results, we first identify the simultaneous low-temperature and low-density limit in which Saha theory becomes asymptotically exact. Then, we argue that the screened cluster representation is well suited for calculating corrections, since that formalism accounts for all screening and recombination phenomena at work in a more tractable way than other many-body methods. We sketch the corresponding diagrammatical analysis, which leads to an exact asymptotic expansion for the equation of state. That scaled low-temperature expansion improves the analytical knowledge of the phase diagram. It also provides reliable numerical values over a rather wide range of temperatures and densities, as confirmed by comparisons to quantum Monte Carlo data.

Thermodynamic Properties of Gaseous Plasmas in the Zero‐Temperature Limit

“…Both discussed above ionization "stairs" in thermal and caloric EOS are natural temperature independent zero-order terms Y 0 (μ) in such global asymptotic expansion (1). Such asymptotic expansion by the functions of temperature l i (μ) ~ exp{-const i (μ)/Т)} was rigorously developed for thermal EOS of hydrogen P(T,r) in the case of partially ionized electron-ion-atomic hydrogen (SAHA-limit, see [7,8] and references therein). The main claim of present paper is existence of such asymptotic expansion for any substance and in whole range of chemical potential within the energy scale of the substance, including creation of any bound complexes, multi-electronic ions, atoms, molecules etc.…”

“…Examples of this limiting structure are exposed at figures 1 and 2 for thermal and caloric EOS of lithium and helium plasmas [4][5][6]. For rigorous theoretical proof of existing the limit, which is under discussion (Saha-limit) in the case of hydrogen see [7,8] and references therein.The same stepped structure is valid in the zero-temperature limit for any molecular gases, for example for hydrogen (Fig. 3) [4] [6].…”

Pressure of a Partially Ionized Hydrogen Gas: Numerical Results from Exact Low Temperature Expansions

“…[11,12], the SLT expansion (4) provides an useful analytical knowledge of thermodynamics in an extended domain of the phase diagram. That domain is restricted to sufficiently low temperatures, typically k B T ≪ |E H |, and includes of course the fully ionized region ρ ≪ ρ * where SLT expansion (4) does reduces to virial expansion (1).…”

The divergent atomic partition function or how to assign correct statistical weights to bound states

“…Deriving accurate equations, without any double counting of states, for the hydrogen gas in regimes where a substantial fraction of electrons and protons are recombined into hydrogen atoms (or even molecules) is a challenging theoretical task with important practical implications in astrophysics and other fields. Significant progress has been made recently on that topic [1,[6][7][8][9][10]. Ebeling et al [1] derive an equation of state (EOS) for a partially ionized hydrogen gas by performing partial resummations in the virial expansion in such a way that the resulting EOS has a structure compatible not only with the virial expansion, but also with the Saha equation for the ionization/recombination equilibrium of hydrogen atoms (implemented with the BPL partition function).…”

“…The statistical weight of any bound entity in the SCR involves a finite internal partition function that depends solely on the fundamental physical constants, and that can be viewed as a generalization of Ebeling's two-body virial functions Q and E to a higher number of interacting particles [6]. The SCR has been used to derive the Scaled Low-Temperature (SLT) equation of state [7][8][9] for a partially ionized hydrogen gas, which is asymptotically exact in a coupled low-density -lowtemperature limit. Though it is obtained in the vicinity of a T → 0 limit, the SLT EOS reduces, as it should, to the virial expansion when one lets ρ → 0 at fixed T , providing thus a rigorous extension of the validity domain of the usual virial expansion into the partially (or fully) recombined atomic phase.…”