Time-dependent orbital-free density functional theory for electronic stopping power: Comparison to the Mermin-Kohn-Sham theory at high temperatures

Electronic stopping power in warm dense matter can affect energy transport and heating in astrophysical processes and internal confinement fusion. For cold condensed matter systems, stopping power can be modeled from first-principles using real-time time-dependent density functional theory (DFT). However, high temperatures (10's to 100's of eV) may be computationally prohibitive for traditional Mermin-Kohn-Sham DFT. New experimental measurements in the warm dense regime motivates the development of first-principles approaches, which can reach these temperatures. We have developed a time-dependent orbital-free density functional theory, which includes a novel nonadiabatic and temperature-dependent kinetic energy density functional, for the simulation of stopping power at any temperature. The approach is nonlinear with respect to the projectile perturbation, includes all ions and electrons, and does not require a priori determination of screened interaction potentials. Our results compare favorably with Kohn-Sham for temperatures in the WDM regime, especially nearing 100 eV.

Figure 1

Dynamic density-density response functions for homogeneous electron gas as a function of wave vector. The electron density, $\rho =0.004$ gives a Fermi-wave vector, $k_F=0.491$. The frequency is $\omega =0.125+i\times 0.0001\sim k_F^2/2+i\delta$, Lindhard response function (black solid), Thomas-Fermi–von-Weizsäcker response (blue dotted), the current-dependent response (green dash-dot).

Figure 2

$\Im {\left[\frac{\partial {\tilde{\chi }}_L^{-1}(q,\omega ,T)}{\partial w}\right]}_{\omega \to 0}$ as a function of the scaled wave vector $q/k_F$. Four density-temperature examples are shown. Markers are numerical calculations of Eq. (20) and its derivative, lines are calculated by Eqs. (21) and (24). The inset shows the best fits of the scaling factor $c_T$ to the $\Im {\left[\frac{\partial {\tilde{\chi }}_L^{-1}(q,\omega ,T)}{\partial w}\right]}_{\omega \to 0}\text{v.s.}q$ lines compared to the asymptotic limit fits, Eqs. (22) and (23), as well as the final $c_T$, Eq. (24).

Figure 3

0.7 g/cc deuterium stopping power (ESP) for deuteron. Top: ${10}^4$ Kelvin; bottom: ${10}^5$ Kelvin. Time-dependent (TD) Kohn-Sham (black solid); current-dependent TD orbital-free (CD) (red fashed); TFW TD orbital-free (blue dotted).

Figure 4

0.7 g/cc deuterium stopping power (ESP) for deuteron. Top, left: Current-dependent TD orbital-free; top, middle, left: TD Kohn-Sham; top, middle, right: dielectric function (DF) method with projectile screening; top, right: DF method with bare projectile, ${\tilde{\rho }}_p\left(q\right)=0$. Bottom, left: Current-dependent TD orbital-free, with $C_T=1$; bottom, middle, left: Li Petrasso model; bottom, middle, right: Brown-Preston-Singleton model; bottom, right: comparison of all methods at 600000 Kelvin.

Figure 5

3.5 g/cc deuterium stopping power (ESP) for deuteron. Top, left: Current-dependent TD orbital-free; top, middle, left: TD Kohn-Sham; top, middle, right: dielectric function (DF) method with projectile screening; Top, right: DF method with bare projectile, ${\tilde{\rho }}_p\left(q\right)=0$; bottom, left: current-dependent TD orbital-free, with $C_T=1$; bottom, middle, left: Li Petrasso (LP) model; bottom, middle, right: Brown-Preston-Singleton (BPS) model; bottom, right: comparison of all methods at 1 mega-Kelvin.

Figure 6

Static density-density response functions for homogeneous electron gas as a function of normalized wave vector. Lindhard function (black solid), Thomas-Fermi–von-Weizsäcker (TFW) (blue dotted), random phase approximation (RPA) (red dashed), and the approximate RPA with TFW for the noninteracting response.

Figure 7

Work on the deuteron projectile as a function of distance traveled. $v_p=2.5a.u.$ and $\delta t$ = 0.29 $as$. Linear fit to this line leads gives the electronic stopping power (ESP). time-dependent (TD) Kohn-Sham (black solid), Current-dependent TD orbital-free (CD) (blue dotted). Inset: Long trajectory shows instability (a deviation of linearity) for the TFW TD orbital-free (red dashed). For TFW $\delta t=0.096as$.

Figure 8

3.5 g/cc, 100 kK deuterium stopping power for deuteron. Convergence of TD-OF-DFT result with respect to the dimensions of the periodic simulation cell. BPS result also shown for reference. Length shown in Bohr. See text for description.