Collisional stopping power of ions in warm dense matter

TL;DR

The paper addresses accurate modeling of ion stopping power in warm dense matter, where neither classical plasma theory nor standard solid-state methods alone are adequate. The authors develop a BUU-based kinetic framework augmented with a potential of mean force derived from an average-atom two-component plasma model to incorporate degeneracy and strong coupling, and compute quantum scattering cross sections for electron-ion collisions. Compared with TDDFT results for deuterium near solid density at a few electronvolts, the model achieves comparable accuracy at a fraction of the computational cost and captures the classical-to-degenerate transition as Θ crosses unity. The approach yields practical stopping-power tables for ICF hydrodynamics and reveals Barkas-type asymmetries, representing a significant step toward bridging plasma physics and condensed-matter descriptions in warm dense matter.

Abstract

A model for the collisional stopping of ions on free electrons in warm dense matter is developed and explored. It is based on plasma kinetic theory, but with modifications to address the warm dense matter regime. Specifically, it uses the Boltzmann-Uehling-Uhlenbeck kinetic equation to incorporate effects of Fermi degeneracy of electrons. The cross section is computed from quantum scattering of electrons and ions occuring via the potential of mean force derived from an average atom model, which incorporates effects of strong Coulomb correlations. Predictions from this model show comparable accuracy to results from time-dependent density functional theory calculations for deuterium near solid density and a temperature of several electronvolts, at a fraction of the computational cost. Further, the model captures the transition of a plasma from the classical limit to the degenerate limit, including qualitative behaviors of solid state theory.

Figures (6)

• Figure 1: The density temperature phase space around the warm dense matter regime denoted by the green shaded region. The phase space is split up by the lines $\Gamma=1$, $\Theta=1$, and $r_s =1$, into four regions determined by the dominate energy in the system. The two dashed gray lines mark where classical plasma physics theories fail at $\Gamma=0.1$ and $\Theta=10$ and where the Boltzmann-Uehling-Uhlenbeck based theory extends this to at $\Gamma=30$ and $\Theta=0.1$.
• Figure 2: Scattering potentials for hydrogen at 5eV and 1.67 g/cc. This translates to $\Gamma=4.65$ and $\Theta=0.14$, putting this example firmly in the warm dense matter regime. The blue line is the potential of mean force calculated from the average-atom two-component plasma model StarrettPRE2013StarrettHEDP2017. The red dashed line is a screened Coulomb potential with a screening length determined by a combination of the Debye length and the Thomas-Fermi length StantonPRE2016. The green dashed line shows the Coulomb potential. Notice how the potential of mean force contains screening absent from the Coulomb potential, and the correlation effects absent from the other potentials.
• Figure 3: Stopping power of a deuteron in warm dense deuterium at various densities and temperatures. The results of the present method are shown as solid purple lines. This is compared to two analytic theories: Brown-Preston-Singleton BrownPhysRep2005 (dashed blue lines) and Li-Petrasso LiPRL1993 (dashed-dotted red lines). We also compare to orbital-free (green triangles) and Kohn-Sham (orange circles) TDDFT calculations WhitePRB2018. Our method matches the density functional theory calculations in both the low speed and high speed limits. Neither analytic theory is consistent with both the high and low velocity limits of the density functional theory, as they are not well suited for the warm dense matter regime. Units are given in atomic units where $E_\textrm{H}=27.2$eV is the Hartree energy, $a_0$ is the Bohr radius, and $\alpha=e^2/(4\pi \epsilon_o \hbar c)$ is the fine structure constant.
• Figure 4: Predictions of proton stopping power in hydrogen at 1.67 g/cc and various temperatures. Panel (a) shows the results from 0.5eV to 20eV, which are characterized by a degenerate regime, $\Theta <1$. Panel (b) shows the results for 20eV to 800eV, which are characterized by a classical regime, $\Theta>1$. Notice that the stopping power curves for the degenerate cases in (a) do not change very much over the span of temperatures. This because the ballistic proton is stopping on electrons at the Fermi energy, which is temperature independent. Whereas in (b), the curve changes substantially with temperature as the proton is stopping on electrons at the thermal velocity, which increases with temperature. Units are given in atomic units where $E_H=27.2$eV is the Hartree energy, $a_0$ is the Bohr radius, and $\alpha$ is the fine structure constant.
• Figure 5: Electron ion collision frequency of a hydrogen plasma computed using Eq. (\ref{['eqn:coll']}). The two plots show the (a) temperature and (b) density dependence in both the degenerate and classical limits. Notice that the behavior of the two limits is qualitatively very different, suggesting that the characteristic energies involved in collisions has changed from the thermal speed of the plasma to the Fermi energy. The black vertical line on both plots denotes the point $\Theta =1$, and represents the location of the transition, as well as where warm dense matter conditions are present. Units are given in atomic units where $E_H=27.2$eV is the Hartree energy, $a_0$ is the Bohr radius, and $\alpha$ is the fine structure constant. Note that the collision frequency calculated is actually divided by the ion density, $n_i$, as a density of ions for a ballistic projectile has no physical meaning.