Variational atomic model of plasma accounting for ion radial correlations and electronic structure of ions.

We propose a model of ion-electron plasma (or nucleus-electron plasma) that accounts for the electronic structure around nuclei (i.e., ion structure) as well as for ion-ion correlations. The model equations are obtained through the minimization of an approximate free-energy functional, and it is shown that the model fulfills the virial theorem. The main hypotheses of this model are (1) nuclei are treated as classical indistinguishable particles, (2) electronic density is seen as a superposition of a uniform background and spherically symmetric distributions around each nucleus (system of ions in a plasma), (3) free energy is approached using a cluster expansion (nonoverlapping ions), and (4) resulting ion fluid is modeled through an approximate integral equation. In the present paper, the model is described only in its average-atom version.

Simpler free-energy functional of the Debye-Hückel model of fluids and the nonuniqueness of free-energy functionals in the theory of fluids.

In previous publications [Piron and Blenski, Phys. Rev. E 94, 062128 (2016)2470-004510.1103/PhysRevE.94.062128; Blenski and Piron, High Energy Density Phys. 24, 28 (2017)1574-181810.1016/j.hedp.2017.05.005], the authors have proposed Debye-Hückel-approximate free-energy functionals of the pair distribution functions for one-component fluids and two-component plasmas. These functionals yield the corresponding Debye-Hückel integral equations when they are minimized with respect to the pair distribution functions, lead to correct thermodynamic relations, and fulfill the virial theorem. In the present paper, we update our results by providing simpler functionals that have the same properties. We relate these functionals to the approaches of Lado [Phys. Rev. A 8, 2548 (1973)0556-279110.1103/PhysRevA.8.2548] and of Olivares and McQuarrie [J. Chem. Phys. 65, 3604 (1976)JCPSA60021-960610.1063/1.433545]. We also discuss briefly the nonuniqueness issue that is raised by these results.

Free-energy functional of the Debye-Hückel model of simple fluids.

The Debye-Hückel approximation to the free energy of a simple fluid is written as a functional of the pair correlation function. This functional can be seen as the Debye-Hückel equivalent to the functional derived in the hypernetted chain framework by Morita and Hiroike, as well as by Lado. It allows one to obtain the Debye-Hückel integral equation through a minimization with respect to the pair correlation function, leads to the correct form of the internal energy, and fulfills the virial theorem.

X-ray grating spectrometer for opacity measurements in the 50 eV to 250 eV spectral range at the LULI 2000 laser facility.

An x-ray grating spectrometer was built in order to measure opacities in the 50 eV to 250 eV spectral range with an average spectral resolution ∼ 50. It has been used at the LULI-2000 laser facility at École Polytechnique (France) to measure the Δn = 0, n = 3 transitions of several elements with neighboring atomic number: Cr, Fe, Ni, and Cu in the same experimental conditions. Hence a spectrometer with a wide spectral range is required. This spectrometer features one line of sight looking through a heated sample at backlighter emission. It is outfitted with one toroidal condensing mirror and several flat mirrors cutting off higher energy photons. The spectral dispersion is obtained with a flatfield grating. Detection consists of a streak camera sensitive to soft x-ray radiation. Some experimental results showing the performance of this spectrometer are presented.

Opacity of iron, nickel, and copper plasmas in the x-ray wavelength range: theoretical interpretation of 2p-3d absorption spectra.

This paper deals with theoretical studies on the 2p-3d absorption in iron, nickel, and copper plasmas related to LULI2000 (Laboratoire pour l'Utilisation des Lasers Intenses, 2000J facility) measurements in which target temperatures were of the order of 20 eV and plasma densities were in the range 0.004-0.01 g/cm(3). The radiatively heated targets were close to local thermodynamic equilibrium (LTE). The structure of 2p-3d transitions has been studied with the help of the statistical superconfiguration opacity code SCO and with the fine-structure atomic physics codes HULLAC and FAC. A new mixed version of the sco code allowing one to treat part of the configurations by detailed calculation based on the Cowan's code RCG has been also used in these comparisons. Special attention was paid to comparisons between theory and experiment concerning the term features which cannot be reproduced by SCO. The differences in the spin-orbit splitting and the statistical (thermal) broadening of the 2p-3d transitions have been investigated as a function of the atomic number Z. It appears that at the conditions of the experiment the role of the term and configuration broadening was different in the three analyzed elements, this broadening being sensitive to the atomic number. Some effects of the temperature gradients and possible non-LTE effects have been studied with the help of the radiative-collisional code SCRIC. The sensitivity of the 2p-3d structures with respect to temperature and density in medium-Z plasmas may be helpful for diagnostics of LTE plasmas especially in future experiments on the Δn=0 absorption in medium-Z plasmas for astrophysical applications.

Variational-average-atom-in-quantum-plasmas (VAAQP) code and virial theorem: equation-of-state and shock-Hugoniot calculations for warm dense Al, Fe, Cu, and Pb.

The numerical code VAAQP (variational average atom in quantum plasmas), which is based on a fully variational model of equilibrium dense plasmas, is applied to equation-of-state calculations for aluminum, iron, copper, and lead in the warm-dense-matter regime. VAAQP does not impose the neutrality of the Wigner-Seitz ion sphere; it provides the average-atom structure and the mean ionization self-consistently from the solution of the variational equations. The formula used for the electronic pressure is simple and does not require any numerical differentiation. In this paper, the virial theorem is derived in both nonrelativistic and relativistic versions of the model. This theorem allows one to express the electron pressure as a combination of the electron kinetic and interaction energies. It is shown that the model fulfills automatically the virial theorem in the case of local-density approximations to the exchange-correlation free-energy. Applications of the model to the equation-of-state and Hugoniot shock adiabat of aluminum, iron, copper, and lead in the warm-dense-matter regime are presented. Comparisons with other approaches, including the inferno model, and with available experimental data are given. This work allows one to understand the thermodynamic consistency issues in the existing average-atom models. Starting from the case of aluminum, a comparative study of the thermodynamic consistency of the models is proposed. A preliminary study of the validity domain of the inferno model is also included.

Variational theory of average-atom and superconfigurations in quantum plasmas.

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 differentiation and is consistent with the virial theorem.

L-band x-ray absorption of radiatively heated nickel.

Absorption of L-M and L-N transitions of nickel has been measured using point projection spectroscopy. The x-ray radiation from laser-irradiated gold cavities was used to heat volumetrically nickel foils "tamped with carbon" up to 20 eV. Experimental spectra have been analyzed with calculations based on the spin-orbit split arrays statistical approach and performed for each ionic species Ni5+ to Ni11+. Using a least-squares fit, this method provides an ion distribution broader than at local thermodynamic equilibrium, which is explained by spatial and temporal temperature gradients. A major improvement in the simulation of the absolute value of transmission is obtained with a resolved transition array statistical calculation that reproduces the experimental spectrum with the nominal areal mass density by taking into account the saturation of narrow lines.