Population densities and partition functions of partially ionized ideal and non-ideal multi-component plasma mixtures in $P$–$T$ phase space

Abstract

In most practical situations, for multi-component non-ideal complex plasma mixtures in pressure–temperature ($P$–$T$) phase space, the problem of solving the system of coupled nonlinear Saha equations subjected to the constraints of electro-neutrality and conservation of nuclei is found to be effectively a one-dimensional nonlinear problem, i.e. solving a single transcendental equation. Computation of ionization stages and partition functions in the $P$–$T$ phase space is particularly important for non-ideal plasmas generated in devices in which pressure is a reliably measurable parameter while it is difficult to measure the number density of heavy particles. The methodology and algorithm presented herein are based on deriving an equivalent single transcendental equation, for which the solution is eminently trivial. The algorithm takes into account different practical models for non-ideality corrections (lowering of ionization potentials, truncation of partition functions and a corrected equation of state). The ease and efficiency of the introduced algorithm allows, with significant simplicity, the computations of population densities of all plasma species (ionized and excited) up to maximum ionization states equal to the atomic numbers of the involved elements with minimal computational work. It also considers an extensive database of energy levels of the excited states. The algorithm presented herein is analytically known to be safe, fast and efficient. It shows no numerical instabilities, no convergence problems and no accuracy limitations or lack of change problems, which have been reported in the literature. A couple of non-trivial problems are worked out and presented herein showing the effectiveness of the present methodology. For completeness, a criterion for the validity of the assumption of local thermodynamic equilibrium (LTE) is applied to the results from the sample problems, showing the regions of the pressure–temperature phase space over which the assumption is valid.