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

TL;DR

This paper develops exact asymptotic expansions for the thermodynamics of hydrogen gas in the Saha regime, improving analytical understanding and providing reliable numerical results validated by quantum Monte Carlo data.

Contribution

It introduces a rigorous asymptotic expansion for hydrogen plasma thermodynamics using the screened cluster representation, surpassing previous Saha theory limitations.

Findings

Asymptotic expansion matches quantum Monte Carlo data

Identifies the low-temperature, low-density limit where Saha theory is exact

Provides a more accurate phase diagram for hydrogen plasma

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.