Efficient and consistent methodology for the calculation of population densities, partition functions, and thermodynamic properties of ideal and weakly non-ideal multicomponent plasma mixtures in the P-T phase space

A set of nonideal Saha equations, for multicomponent plasma mixtures, can be derived from the conditions of the minimization of the free energy function in the ionization processes. Population densities derived from the solution of this set of equations subjected to electro-neutrality and conservation of nuclei are fully consistent with the thermodynamic properties derived from the same free energy function and, therefore, can be used to calculate these properties. Depending on the degree of complexity of the free energy function, nonideal Saha equations are obtained with different forms of nonideality corrections. In most practical situations, the problem of solving the resulting system of coupled nonlinear Saha equations, subjected to the above mentioned constraints, in the pressure-temperature (P-T) phase space, is found to be effectively a one-dimensional nonlinear problem i.e., solving a single transcendental equation. The methodology and algorithm presented herein are based on "the chemical picture," with all thermodynamic properties derived self-consistently from the free energy function, and on the mapping of the resulting set of governing equations into a one-variable zero-search. The algorithm is successful for most practical models for nonideality corrections to the free energy function, which are fully reflected in the derived nonideal Saha equations as lowering of ionization potentials, corrected equation of state, and truncated partition functions. The ease and efficiency of the introduced algorithm allows, with significant simplicity, the computations of population densities of all plasma species (neutral, 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 is analytically known to be safe, fast, efficient, and solves the problem to the machine accuracy. It shows no numerical instabilities, no convergence problems, and no accuracy limitations or lack-of-change problems, which have been repeatedly reported in the literature. A nontrivial problem is worked out and presented in detail showing the effectiveness of the present methodology. For completeness, a criterion for the validity of the assumption of local thermodynamic equilibrium is applied to the results of the sample problem showing regions of the pressure-temperature phase space over which the assumption is valid.