Kinetic model for electron-ion transport in warm dense matter

Shane Rightley and Scott D. Baalrud

Phys. Rev. E 103, 063206 – Published 14 June 2021

Abstract

We present a model for electron-ion transport in warm dense matter that incorporates Coulomb coupling effects into the quantum Boltzmann equation of Uehling and Uhlenbeck through the use of a statistical potential of mean force. Although the model presented here can be derived rigorously in the classical limit [S. D. Baalrud and J. Daligault, Phys. Plasmas 26, 082106 (2019)], its quantum generalization is complicated by the uncertainty principle. Here we apply an existing model for the potential of mean force based on the quantum Ornstein-Zernike equation coupled with an average-atom model [C. E. Starrett, High Energy Density Phys. 25, 8 (2017)]. This potential contains correlations due to both Coulomb coupling and exchange, and the collision kernel of the kinetic theory enforces Pauli blocking while allowing for electron diffraction and large-angle collisions. We use the Uehling-Uhlenbeck equation to predict the momentum and temperature relaxation times and electrical conductivity of solid density aluminum plasma based on electron-ion collisions. We present results for density and temperature conditions that span the transition from classical weakly-coupled plasma to degenerate moderately-coupled plasma. Our findings agree well with recent quantum molecular dynamics simulations.

Received 9 February 2021
Revised 3 May 2021
Accepted 3 May 2021

DOI:https://doi.org/10.1103/PhysRevE.103.063206

Physics Subject Headings (PhySH)

High-energy-density plasmas Nonequilibrium statistical mechanics Plasma transport Quantum kinetic theory Quantum statistical mechanics

Quantum plasmas Strongly-coupled plasmas Warm-dense matter

Plasma kinetic theory Semiclassical methods

Plasma PhysicsStatistical Physics & Thermodynamics

Authors & Affiliations

Shane Rightley ^*

Department of Physics and Astronomy, University of Iowa, Iowa City, Iowa 52242, USA

Scott D. Baalrud ^†

Department of Nuclear Engineering and Radiological Sciences, University of Michigan, Ann Arbor, Michigan 48109, USA

^*shane-rightley@uiowa.edu
^†baalrud@umich.edu

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Vol. 103, Iss. 6 — June 2021

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Images

Figure 1
Parameter regimes of fully ionized hydrogen plasma. The solid black line is the boundary between weak and strong electron coupling $Γ_{e} = 1$ and turns over due to the electron degeneracy; the dotted lined is the separation between weak and strong ion coupling $Γ_{i} = 1$ ; and the dashed line is the separation between classical and degenerate electrons $Θ = 1$ . The darker blue oval denotes the sector of WDM. Region 1 (yellow) is classical weakly coupled plasma; region 2 (light blue) is characterized by classical strong coupling; region 3 (pink) by quantum weak coupling; and region 4 (green) by both quantum electrons and strongly coupled ions. We expect the theory presented here to apply to each region 1–4. The red line demarcates the region of validity of plasma-type transport theories; beyond this is the regime of condensed matter.
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Figure 2
Electron-ion potential of mean force as obtained via the model described in Ref. [36] for solid-density ( $2.7 g {cm}^{- 3}$ ) warm dense aluminum at 1, 25, and 500 eV (thick curves). In the high-temperature limit the potential approaches the screened Coulomb potential of a classical plasma (thin curves); as the temperature decreases it is altered by both degeneracy and correlations leading to different scale lengths of the potential in addition to the nonmonotonic behavior.
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Figure 3
Statistical weighting function $G$ from the integrand for temperature and momentum relaxation, spanning the transition from classical to degenerate conditions. The relevant collision velocities become narrowly centered around the Fermi velocity at strong degeneracy. This represents the availability of regions of phase space for scattering particles to leave/enter, with the Pauli principle greatly restricting the available states at collision energies below the Fermi energy when the plasma is degenerate.
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Figure 4
Electron-ion collisional relaxation times ( $τ = ν_{e i,}^{- 1}$ ) as a function of temperature in solid density ( $2.7 {g cm}^{- 3}$ ) aluminum. This demonstrates the influence of different physical effects, as the LFP equation does not contain effects due to strong coupling or strong scattering, and the Landau-Spitzer theory additionally does not account for electron degeneracy in the scattering physics.
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Figure 5
Electron-ion contribution to the electrical conductivity of solid density aluminum ( $2.7 {g cm}^{- 3}$ ) as derived through the current work (solid line) the Starrett and Shaffer model evaluated at first order (dashed line) and at high order (dot-dashed line) in the Chapman-Enskog expansion, the Lee-More model (thin dashes), the LS conductivity (dotted line), along with QMD results of Witte et al. [28] using the Perdew–Burke-Ernzerhof and Heyd–Scuseria–Ernzerhof exchange-correlation functionals.
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