Model of electron transport in dense plasmas spanning temperature regimes

We present a new model of electron transport in warm and hot dense plasmas which combines the quantum Landau-Fokker-Planck equation with the concept of mean-force scattering. We obtain electrical and thermal conductivities across several orders of magnitude in temperature, from warm dense matter conditions to hot, nondegenerate plasma conditions, including the challenging crossover regime between the two. The small-angle approximation characteristic of Fokker-Planck collision theories is mitigated to good effect by the construction of accurate effective Coulomb logarithms based on mean-force scattering, which allows the theory to remain accurate even at low temperatures, as compared with high-fidelity quantum simulation results. Electron-electron collisions are treated on equal footing as electron-ion collisions. Their accurate treatment is found to be essential for hydrogen, and is expected to be important to other low-$Z$ elements. We find that electron-electron scattering remains influential to the value of the thermal conductivity down to temperatures somewhat below the Fermi energy. The accuracy of the theory seems to falter only for the behavior of the thermal conductivity at very low temperatures due to a subtle interplay between the Pauli exclusion principle and the small-angle approximation as they pertain to electron-electron scattering. Even there, the model is in fair agreement with ab initio simulations.

Figure 1

Coulomb logarithms for (a) compressed hydrogen at 40 g ${\mathrm{cm}}^{-3}$ and (b) solid-density aluminum at 2.7 g ${\mathrm{cm}}^{-3}$. Thick solid curves are the mean-force model. The thin solid curve is the electron-electron mean-force model without low-temperature correction, Eq. (30) versus Eq. (33). The green dotted curve (LM) is the Lee-More Coulomb logarithm, Eq. (11), with their additional mean free path correction, Eq. (35). The red dashed curve is the Coulomb logarithm of Potekhin et al., Eq. (34).

Figure 2

Electrical conductivity (a) and thermal conductivity (b) of hot dense hydrogen plasma. The thin and thick qLFP curves are with and without low-temperature correction, as in Fig. 1.

Figure 3

Electrical conductivity (a) and thermal conductivity (b) of solid-density aluminum plasma ($\rho =2.7\mathrm{g}{\mathrm{cm}}^{-3}$). The thin and thick blue curves are qLFP curves with and without low-temperature correction, as in Fig. 1. The dotted green curves are the Lee-More model [45]. The dashed red curves are interpolations of Rinker's tables [59, 60]. The circles and triangles are QMD data by Witte et al. [67] using PBE and HSE exchange-correlation functionals, respectively. The black outlined regions are estimated ranges of thermal conductivity measured by McKelvey et al. [68].

Figure 4

Mean ionization state of aluminum at a density $\rho =2.7{\mathrm{g}\mathrm{cm}}^{-3}$ as predicted by the average-atom model of Ref. [48].

Figure 5

Low-temperature behavior of the thermal conductivity of solid-density aluminum. Curves and symbols have the same meaning as in Fig. 3.

Figure 6

Electron-electron correction factor to the electrical conductivity, $R_{\sigma }$, and thermal conductivity, $R_{\lambda }$, for compressed hydrogen at $\rho =1{\mathrm{g}\mathrm{cm}}^{-3}$. Solid blue and green lines are the present qLFP model. The dashed orange line is the practical formula by Reinholz et al. [10]. The dotted black lines are the Spitzer-Härm result [7].

Figure 7

Distribution of deflection angles, $D_{ij}(E,\theta )$, for elastic electron-ion (upper) and electron-electron (lower) scattering in solid-density aluminum at temperatures of 1, 10, 100, and $1000\mathrm{eV}$ (left to right). The color scale in each panel shows only the relative magnitude of the angle distribution, with dark and light shades being large and small values of the distribution, respectively. The dashed red line marks the energy corresponding to the root-mean-square relative velocity at each temperature.