Improved Kelbg Potentials for $Z>1$ and Application to Carbon Plasmas

TL;DR

The study addresses the challenge of efficiently capturing quantum effects in warm dense plasmas for complex elements beyond hydrogen. It develops a generalized improved Kelbg potential for $Z>1$ by fitting to exact Slater-sum pair densities across $Z=1$ to 54 and expresses the fitting parameter with a Padé form in terms of $T$ and $Z$; this is used in classical MD simulations of carbon plasmas to compute internal energy and pressure. The MD results are compared to a quantum EOS model based on PIMC/DFT (L9061), showing generally good agreement in regimes where the K-shell is not fully bound, with discrepancies arising as bound-state occupancy increases, indicating the limits of quantum statistical potentials and the need for full quantum treatment in such regimes. The work demonstrates that improved Kelbg/QSP-based MD can provide a fast, scalable EOS tool for hot dense carbon plasmas and potentially for mixtures, with explicit caveats about three-body effects and bound-state physics.

Abstract

In this work, we present a general form for the electron-ion diffractive potential derived from the quantum pair density matrix and fit to the improved Kelbg potential for atomic numbers up to $Z = 54$. We apply classical molecular dynamics using the improved Kelbg potential for carbon with various forms of the Pauli potential to compute internal energies and pressures for hot, dense plasma conditions. Our results are compared to an equation of state model based on path integral Monte Carlo and density functional theory simulations to examine the extent to which the improved Kelbg potential reproduces the internal energy and pressure of carbon plasmas. The regions of validity for carbon agree generally with those derived previously for hydrogen once pressure ionization effects are incorporated. Based on our carbon results and previously published hydrogen studies, we discuss the general applicability and limitations of these potentials for equation of state studies in warm dense matter and high energy density plasmas.

Figures (4)

• Figure 1: Diffractive potentials for the electron--carbon ion interaction at $2.5\times 10^5$ K (a) and $1.0\times 10^6$ K (b). Potentials computed from the exact pair density matrix are shown as purple symbols. The red line denotes the Kelbg potential Kelbg, while the green and blue lines correspond to the improved Kelbg potential using the Padé approximant of this work and of Filinov et al.Filinov04, respectively.
• Figure 2: Fitted parameter $\gamma_{ei}$ as a function of temperature (a) and as a function of the parameter $x(T,Z)=\sqrt{8\pi k_B T/(Z^2\mathrm{Ha})}$ (b). Figure a) shows the fitted values from the exact Slater sum (crosses), the Padé approximant of Filinov et al.Filinov04 (blue dashed lines), and the present work (green lines). Curves correspond to atomic numbers $Z=1$, 4, 6, 13, 26, and 54, from left to right. Figure b) shows the fitted parameter as a function of $x(T,Z)$ for the same $Z$ values, illustrating the utility of $x(T,Z)$ for fitting $\gamma_{ei}$.
• Figure 3: Simulated ion-ion pair distribution functions for carbon at 3.5g/cc from ddcMD simulations at a variety of temperatures plotted as a function of distance relative to the ion sphere radius defined as $r_s=\left(\frac{3}{4\pi\rho_i}\right)^{1/3}$ where $\rho_i$ is the ion number density. We note that 1 eV is equivalent to $1.1604 \times 10^4$ K.
• Figure 4: Qualitative assessment of the validity of the fitted improved Kelbg potential for carbon as a function of density in grams per cubic centimeter and temperature in Kelvin. Colormap shows the relative K-shell occupation on a scale of 0 (completely unoccupied) to 1 (completely occupied) based on the computed $1s$ orbital energy and electron chemical potential from Purgatorio. The solid black contour line corresponds to 50% occupation of the K-shell, the dotted black line corresponds to the density-temperature track for the principal Hugoniot of diamond from L9061, the dashed black line corresponds to $T=T_F$, and the dashed blue line corresponds to the boundary set by the equality in Eq. \ref{['validline']} assuming $Z=6$. Filled squares denote densities and temperatures for which the ddcMD simulations with QSPs were fully converged and did not exhibit unphysical clustering. The crosses indicate densities and temperatures for which the ddcMD calculations did not converge and the open squares indicate densities and temperatures where the internal energy and pressure were converged but evidence of unphysical quasi-bound states between carbon ions was observed.