Studies on electronic properties of atoms in plasmas with quantum bound and free electrons

Particularly challenging for both experiment and theory are plasmas at the solid and higher densities. In present opacity experiments, the plasmas have densities of the order of 0.01–0.001 g/cm³ and temperatures of a few tens eV. Plasmas at higher densities have been already routinely produced in several inertial fusion energy devices. However, direct access to atomic properties in such regimes is practically impossible since comparisons between theory and experiment are based on large scale numerical hydrodynamic simulations in which atomic data are in a complex manner related to hydrodynamics. This situation can improve with the large lasers of the next generation and to some extent with large Z-pinches (Sandia) and Free-Electron Lasers. One expects that these facilities will be able to provide more directly atomic data at densities much higher than presently available. At higher densities plasmas are strongly coupled and are difficult to model since the partition of electrons into bound and free becomes less clear. The presence of neighbouring ions can strongly impact the bound electron structures, leading for instance to pressure ionization of bound levels. Realistic numerical quantum simulations with many ionic centres seem still beyond the reach of present-day methods and computers. We believe that models of screened ions in equilibrium plasmas with bound and free electrons treated quantum-mechanically can be still used at high densities in opacity and equation of state calculations, provided the two following conditions are fulfilled : i) all electrons are treated within the same formalism, and ii) the equilibrium is obtained from a variational model. However, up to now models of atoms at finite temperature have not been fully variational. We have developed such a fully variational model for the first time [1, 2] and are currently implementing this new theory into a code. In our future studies, starting from a thermodynamically consistent model of the plasma equilibrium, we can consider a realistic theory of the dynamical response of atoms or ions immersed in plasmas. As it follows from our work [3], the linear response is an approach in which one can account for a mixture between atomic transitions (electron-hole) and collective excitations in plasmas. It could be a good model to describe dissipative phenomena in Warm Dense Matter at solid and higher densities.

1.1 Variational theory of atoms and superconfigurations in quantum plasmas

Collaborations: Institute of Theoretical Physics, Warsaw Poland, DAM/DIF

We have solved [1, 2] the classical problem of an atom immersed in quantum plasma considered in the Inferno model by Liberman (Liberman D. A., PRB, 20, 4981 (1979)). Our approach is for the first time a fully variational theory obeying the virial theorem - all variables are variational except the parameters defining the equilibrium, i.e., the temperature , the ion density and the atomic number . It is the quantum version of the classical Thomas-Fermi (TF) atom-in-plasma model proposed nearly 60 years ago (Feynman R. P. et al, PR 75, 1561 (1949)). Our theory has been derived first in the Density Functional formalism (DFT) [1] and then in a formalism beyond the DFT [3]. In our approach, we recover the Feynman et al. result in the TF case and show why the previous approaches to the quantum problem have not been variational. The derivation beyond DFT has been necessary in order to show that in the problem there is a unique ionization model (IM) :

Z-Z^*=∫d³r(n(r)-n₀) , where n₀=Z^*n_i, Z^* the mean ionization charge⁄atom and n(r) the electron density in the atom (n(r) tends to n₀ far from the atom). The same IM has simplicity been used in the TF case by Feynman et al. The variational principle leads to the following equation for n₀: ∫d³rθ(r-R)V_el(r)=0, where V_el(r) is the self-consistent-field electrostatic potential, R the WS radius and θ denotes the Heaviside function. In the case of superconfigurations-in-plasmas (SC), results are similar except that averages over all superconfigurations appear. In the TF case, this condition for n₀ gives the neutrality of the WS sphere and one gets the classical TF ion-in-cell Average Atom (AA) of Feynman et al. The situation is different in the quantum AA and in the SC cases in which the WS sphere is not neutral and the SCF potential V_el(r) outside the WS sphere is not zero. Due to the fully variational character of our approach, the expression for the thermodynamic pressure in all cases does not require any numerical differentiation and is consistent with the virial theorem. Numerical implementation of this theory in a relativistic code is in progress [4].

1.2 Linear response of atoms in plasmas within the quantum-mechanical formalism for bound and free electrons

In the independent electron (IE) model the electron-hole transitions in plasma can be divided into bound-bound, bound free and free-free modes. When the IE model is insufficient and one should take into account interactions, a mixing of the electron-hole modes appears and additional collective modes can appear. In the WDM regime, one may expects such effects to be of importance for the dissipative phenomena. In order to address this question, we have derived a system of equations giving the linear response of a quantum Average Atom (AA) in plasma to a frequency-dependent perturbing dipole potential [3]. Both bound and free states are treated quantum-mechanically, a necessary condition to take account of the mode mixing. In our linear response approach, the energy extinction cross-section per AA is calculated from the imaginary part of the induced dipole. We have derived a sum rule in which the induced dipole is localized using two well defined equilibrium AA quantities. The new sum rule can be viewed as a generalization of the known relation between different forms of dipole matrix element. The sum rule makes also possible the practical calculation of the induced dipole. We have further shown how the homogeneous plasma contribution to the induced potential leads to a renormalization resulting in the appearance of the cold plasma dielectric function.

[1] Variational approach to the Average-Atom-in-jellium and superconfigurations-in-jellium models with all electrons treated quantum-mechanically,
Blenski T., Cichocki B.
High Energy Density Physics 3, 34 (2007)

[2] Variational theory of Average-Atom and superconfigurations in quantum plasmas
Blenski T., Cichocki B.
Physical Review E 75, 056402 (2007)

[3] On the linear dynamic response of average atom in plasma
Blenski T.
Journal of Quantitative Spectroscopy and Radiative Transfer 99, 84, (2006)

[4] Variational Average-Atom in Quantum Plasmas
(VAAQP) - first numerical results.
R Piron, T Blenski and B Cichocki, http://arxiv.org/abs/0810.3156

#1117 - Màj : 26/09/2018