On the electrons really contributing to dc conductivity of warm dense matter

Atomic properties of warm dense matter is an active field of research. Understanding transport properties of these states is essential for providing coefficients needed by magneto-radiative hydrodynamics codes for many studies, including hydrodynamic instabilities, energy balances or heating in fusion plasmas, difficult to investigate by experimental means. In this paper, we present an average-atom approach for the calculation of direct-current electric conductivity within Ziman's theory. The mean ion charge $Z^*$, commonly called ionization, is an important input of the Ziman formula, but is not clearly defined within average-atom models. Our study spans a wide range of thermodynamical conditions, i.e., for the densities, from a few $10^{-2}$ to about 4 times the solid's density, and, for the temperatures, typically from 0.1 eV to 700 eV, favorable to large differences in the mean ion charge $Z^*$ according to its definition. We compare and discuss different ways of defining $Z^*$ while trying to figure out which electrons really contribute to electric conduction. We compare our results with experimental data and published theoretical values, in particular from the second transport code comparison workshop, which was held in July 2023 at Lawrence Livermore National Laboratory. These comparisons lead us to propose indicators for the relevance of including different charges predicted by our average-atom model in the definition of $Z^*$.