Variational theory of average-atom and superconfigurations in quantum plasmas

Abstract

Models of screened ions in equilibrium plasmas with all quantum electrons are important in opacity and equation of state calculations. Although such models have to be derived from variational principles, up to now existing models have not been fully variational. In this paper a fully variational theory respecting virial theorem is proposed-all variables are variational except the parameters defining the equilibrium, i.e., the temperature T, the ion density ni and the atomic number Z. The theory is applied to the quasiclassical Thomas-Fermi (TF) atom, the quantum average atom (QAA), and the superconfigurations (SC) in plasmas. Both the self-consistent-field (SCF) equations for the electronic structure and the condition for the mean ionization Z* are found from minimization of a thermodynamic potential. This potential is constructed using the cluster expansion of the plasma free energy from which the zero and the first-order terms are retained. In the zero order the free energy per ion is that of the quantum homogeneous plasma of an unknown free-electron density n0 = Z* ni occupying the volume 1/ni. In the first order, ions submerged in this plasma are considered and local neutrality is assumed. These ions are considered in the infinite space without imposing the neutrality of the Wigner-Seitz (WS) cell. As in the Inferno model, a central cavity of a radius R is introduced, however, the value of R is unknown a priori. The charge density due to noncentral ions is zero inside the cavity and equals en0 outside. The first-order contribution to free energy per ion is the difference between the free energy of the system "central ion+infinite plasma" and the free energy of the system "infinite plasma." An important part of the approach is an "ionization model" (IM), which is a relation between the mean ionization charge Z* and the first-order structure variables. Both the IM and the local neutrality are respected in the minimization procedure. The correct IM in the TF case is found to be Z-Z*= integral d3 r[n(r)-n0], where n(r) is the first-order electron density. It is shown that in the QAA case the same IM has to be used and that other IMs lead to unphysical solutions. With this IM R becomes in both cases (TF and QAA) equal to the WS radius and the variational calculation leads to SCF equations in an infinite plasma while n0 (or equivalently Z*) is to be found from the condition integral d3r theta(r-R)Vel(r)=0, where theta denotes Heaviside function and Vel(r) is the SCF electrostatic potential. In the SC case results are similar except that averages over all superconfigurations appear. In the TF case the condition for n0 gives the neutrality of the WS sphere and one gets the classical TF ion-in-cell average atom. The situation is different in the QAA and in the SC cases in which the cavity is not neutral and the SCF potential Vel(r) is not zero outside the cavity. Due to the fully variational character of our approach the expression for the thermodynamic pressure in all cases does not require any numerical differentiat...

Partial ionization in dense plasmas: Comparisons among average-atom density functional models

“…Then a "physical picture" [8,10,13,14] that views the plasma as a collection of nuclei and electrons supporting delocalized and localized states is needed. Activity expansions for effective composite particles [13] and models based on self-consistentfield calculations (mean-field approximations) for confined atoms [15] have been used to treat partially ionized atoms of differing charge states.…”

Variational Average-Atom in Quantum Plasmas (VAAQP) – A check of thermodynamic consistency

“…10, 9. As it is well known (see for instance [10]), this is equivalent to the explicit analytical differentiation of the grand potential Ω with respect to the ion density. Each solution of a calculation with the variational option of VAAQP corresponds to the minimum free energy respecting Eqs.…”

“…In the case of the present model the WS sphere is in principle not neutral. Let us mention also that in the case of the quasi-classical Thomas-Fermi expressions for the free energy and total number of electrons per ion, the approach from [9,10] leads correctly to the TF ion-in-cell model of Feynman et al [1]. This paper presents the main features of our VAAQP code which is based on the theory presented in Refs.…”

“…First satisfactory answer to the question how to formulate a fully variational approach to the quantum atom in jellium has been proposed in [9,10,11]. The difference with respect to the previous approaches [4,5,8] is the use of the cluster expansion for the two quantities : the plasma free energy per ion and the total number of electrons per ion.…”

“…This paper presents the main features of our VAAQP code which is based on the theory presented in Refs. [9,10] and reports some preliminary results from this code. A first exploratory study (see [11]) has already shown the feasibility of calculations with VAAQP in the Warm Dense Matter (WDM) regime.…”