High-Energy-Density Physics

Abstract

a b s t r a c tThe quantum-mechanical average-atom model is reviewed and applied to determine scattering phase shifts, mean-free paths, and relaxation times in warm-dense plasmas. Static conductivities s are based on an average-atom version of the Ziman formula. Applying linear response to the average-atom model leads to an average-atom version of the Kubo-Greenwood formula for the frequency-dependent conductivity s(u). The free-free contribution to s(u) is found to diverge as 1/u 2 at low frequencies; however, considering effects of multiple scattering leads to a modified version of s(u) that is finite and reduces to the Ziman formula at u ¼ 0. The resulting average-atom version of the Kubo-Greenwood formula satisfies the conductivity sum rule. The dielectric function e(u) and the complex index of refraction n(u) þ ik(u) are inferred from s(u) using dispersion relations. Applications to anomalous dispersion in laser-produced plasmas are discussed.Ó 2009 Elsevier B.V. All rights reserved. Average-atom and static conductivityLet us briefly reprise the average-atom model, which is a quantum-mechanical version of the temperature-dependent Thomas-Fermi theory of a plasma introduced sixty years ago by Feynman etal. [1]. In the average-atom model, the plasma is divided into neutral spherical cells, each containing a single nucleus (charge Z) and Z electrons. The radius of each cell is the Wigner-Seitz (WS) radius, determined from the material density r m (gm/cc), the atomic weight A (gm/mol), and Avogadro's number A ¼ 6:023 Â 10 23 , by R WS ¼ (3U/4p) 1/3 , where U ¼ A=Ar m is the cell volume. Individual electrons (bound and continuum) inside a neutral cell are assumed to move in a self-consistent potential. Outside the cell boundaries the potential vanishes. The continuum electrons penetrate the cell boundary and move into the region between atoms where the electron density approaches a constant value r 0 determined by the temperature T and chemical potential m. In the quantum-mechanical version of the average-atom model, the density oscillates about r 0 outside the cell with a small and ever decreasing amplitude [2]. To insure electrical neutrality, it is necessary to assume a uniform positive background charge r þ that precisely cancels r 0 . The average atom floats in this positive ''jellium'' sea. The average ionic charge of sea is Z * ¼ r þ U. The picture that evolves is an average atom of nuclear charge Z with Z bound and continuum electrons moving self-consistently inside a sphere of radius R WS ; outside is a neutral plasma consisting of electrons (density r 0 ) balanced by positive sea of ions (charge Z * ). The electron density is determined using the quantum-mechanical self-consistent field method. The average-atom model introduced here is a nonrelativistic version of Liberman's Inferno model [3] and is very similar to the model described previously by Blenski and Ishikawa [4].In the quantum-mechanical model, each electron is assumed to satisfy the central-field Schrö dinger equationwhere a ¼ (n, l) for bo...