Ionisation balance and equation of state of hot, dense carbon plasmas at pressures up to a few 100 Gbar

¹ School of Physics, Zhejiang University of Technology, Hangzhou, Zhejiang 310023, PR China
² Institute of Atomic and Molecular Physics, Jilin University, Changchun, Jilin 130012, PR China
³ College of Science, National University of Defense Technology, Changsha, Hunan 410073, PR China

^★ Corresponding authors: jlzeng@zjut.edu.cn; yuanjianmin@jlu.edu.cn

Received: 18 March 2025
Accepted: 1 May 2025

Abstract

Accurately determining the ionisation balance and the equation of state (EOS) of dense plasmas is crucial for investigating the structure, evolution, and interior of stars, as well as other high-density astrophysical objects. However, it is still challenging for current experiments and theories to achieve this objective for dense plasmas at pressures up to a few 100 Gbar. Here, we modify the chemical picture of the Saha equation of plasmas and extend it into a regime up to a density of a few hundred g cm⁻³ based on the minimisation of the free energy of the system. The non-ideal characteristics (NIC) of the free energy, particularly the part contributed by Coulomb interactions between electrons and ions, as well as among the electrons and ions themselves, are properly accounted for by employing a local-density, temperature-dependent ion-sphere model. In such a way, the NIC effects on the ionisation potential depression, the internal partition functions of the ions, and the partition functions of the free electrons are properly considered in the modified Saha equation. Hence, the ionisation balance and EOS are self-consistently determined in this theoretical formalism. We demonstrate the capability of this modified Saha equation for solving hot, dense carbon plasmas, and find that the accurate prediction of the critical density above which the K-shell electrons become delocalised by pressure is crucial for accurately obtaining the charge state distribution and the EOS. Comparisons are made for the average degree of ionisation and EOS of carbon plasmas with available data reported in the literature. Our predicted EOS gives a result in reasonable agreement with other theoretical results below the critical density of pressure-driven K-shell delocalisation. Nevertheless, above this critical density, our prediction is systematically higher by ~25% compared to most other theories. At mass densities around 5 g cm⁻³, our predicted opacity of dense hydrocarbon plasmas agrees with a recent experiment studying the EOS along the principal shock Hugoniot, which can be considered a validation of our theory at this density regime.