Variational approach to the average-atom-in-jellium and superconfigurations-in-jellium models with all electrons treated quantum-mechanically

Models of average-atom- and superconfigurations-in-plasmas with full quantum description of all electrons are important in opacity and EOS calculations, especially in case of high density plasmas in which pressure ionization phenomena can occur. However, variational formulation of models becomes difficult when quantum free electrons are considered.In our approach we use a cluster expansion of the free energy expression from which we retain the sum of the zero and of the first order terms. In the zero order the free energy is that of the homogeneous jellium of an unknown free electron density. In the first order we consider a one self-consistent-field (SCF) average atoms or ions submerged in this jellium. The SCF potential is assumed to be localized. We consider the first order atoms or ion in the whole space without imposing the neutrality of the Wigner–Seitz cell. The important part of the model is a relation between the average ion charge and the localized part of electron density that is called “ionization model”.The minimization procedure of the free energy is performed with respect to all variables except these characterizing the plasma equilibrium, i.e. except the temperature, the ion density and the atomic number. It is interesting that when the electron density is taken in the Thomas–Fermi (TF) approximation together with a specific choice of the ionization model, our approach correctly leads to the classical TF ion-in-cell average atom. It also appears that the same ionization model has to be used in the quantum case and that other ionization models lead to unphysical solutions. This observation is important with respect to the choice of correct ionization model in the case of superconfigurations-in-jellium.Due to the fully variational character of our approach the resulting expression for the thermodynamic pressure in all considered cases is simple and does not require any numerical differentiation.