Nonlinear effects in light-ion stopping powers within real-time time-dependent density functional theory

TL;DR

The paper addresses the breakdown of linear-response scaling in electronic stopping powers for light ions in warm dense matter by leveraging real-time TDDFT to benchmark proton, alpha, and fractional-charge projectiles in aluminum and carbon. It shows nonlinear corrections of order about $10\%$ near the Bragg peak, challenging the common $S_Z(v) \propto Z^2$ prescription, and assesses several effective-charge models (e.g., Bohr, modified Bohr, Gus'kov) against TDDFT data. The study finds that some cases exhibit $Z_{\mathrm{eff}} > Z$, indicating nonlinearities beyond partial neutralization, and highlights material- and condition-dependent behavior that restricts universal modeling. Fractional-charge TDDFT tests help isolate linear-response contributions, underscoring the need to incorporate Barkas-type and higher-order effects into efficient stopping-power models for fusion and warm dense matter applications. These insights guide the development of more accurate, computationally efficient models that remain valid across relevant temperature, density, and projectile-velocity regimes.

Abstract

Electronic stopping power models describing fuel heating processes in inertial fusion energy concepts typically assume linear-response behavior through quadratic scaling with the projectile charge. We report the results of real-time time-dependent density functional theory (TDDFT) calculations indicating that even for low-Z ions, nonlinear processes modify stopping powers in warm dense matter by about 10% near and below the Bragg peak. By describing partial neutralization of slow ions, analytic effective charge models capture some qualitative aspects of the TDDFT results but do not always offer quantitative accuracy. Cases where the effective charge inferred from TDDFT exceeds the bare ion charge suggest that more complex nonlinear effects also contribute. These findings will inform future improvements to more efficient stopping power models.

Figures (6)

• Figure 1: Free-electron contributions to stopping powers of protons (small circles) and alpha particles (large circles) in solid-density aluminum at an electronic temperature of 1eV (blue) and 10eV (red) computed with TDDFT. Empty squares indicate the proton data scaled by a factor of 4.
• Figure 2: Stopping powers of protons (small circles) and alpha particles (large circles) in warm dense carbon at a) 1, 2eV and b) 10, 2eV computed with TDDFT. Results from all-electron (AE) calculations, all-electron calculations where carbon 1s states were frozen (AE$-$1s), and PAW calculations are shown in purple, teal, and orange, respectively. Empty squares show the proton data scaled by a factor of 4.
• Figure 3: Ratio of effective charges $Z_\alpha/Z_\mathrm{p}$ in warm dense aluminum as inferred from TDDFT stopping powers according to Eq. \ref{['eq:Zratio']} at temperatures of 1eV (blue squares) and 10eV (red diamonds). Shading indicates estimated errors from approximated electron-ion potentials and time step convergence. Also shown are analytic predictions from the Bohr stripping criterion bohr:1948 (Eq. \ref{['eq:bohr']}), its modified version (Eq. \ref{['eq:bohrmod']}), and the condition-dependent generalization by Gus'kov et al.gus2009method (Eq. \ref{['eq:guskov']}). Dotted gray indicates the high-velocity limit of $Z_\alpha/Z_\mathrm{p}\rightarrow 2$.
• Figure 4: Ratio of effective charges $Z_\alpha/Z_\mathrm{p}$ in warm dense carbon at a) 1, 2eV and b) 10, 2eV as inferred from TDDFT stopping power calculations that include all electrons (purple squares), freeze carbon 1s states (teal exes), or use PAW to treat only valence electrons (orange diamonds). Shades of gray indicate analytic predictions from the Bohr stripping criterion bohr:1948 (Eq. \ref{['eq:bohr']}), its modified version (Eq. \ref{['eq:bohrmod']}), the condition-dependent generalization by Gus'kov et al.gus2009method (Eq. \ref{['eq:guskov']}), the Gus'kov model with input from DFT (Eq. \ref{['eq:vrelmod']}), and a fitted version of the Gus'kov model (Eq. \ref{['eq:guskovmod']}). The high-velocity limit of $Z_\alpha/Z_\mathrm{p}\rightarrow 2$ is shown in dotted gray.
• Figure 5: Effective ionization of test charges stopping in carbon at 10gcm and 2eV. Brown triangles and green circles represent AE TDDFT results at two different velocities, while corresponding dashed curves indicate behavior predicted by the Gus'kov effective charge model gus2009method. The horizontal line indicates $Z_\mathrm{eff}=Z$.