Ab Initio Prediction of a Negative Barkas Coefficient for Slow Protons and Antiprotons in LiF

We report the ab initio prediction of a negative Barkas coefficient in lithium fluoride (LiF) insulator at low velocity ($v<0.25\mathrm{a}.\mathrm{u}.$, $E_{\mathrm{kin}}\sim 2\mathrm{keV}$). The electronic stopping power of protons in LiF has been extensively studied both experimentally and theoretically because of a controversial threshold effect. While our time-dependent density-functional theory simulations confirm the presence of a velocity threshold below which the proton stopping power vanishes, our calculations demonstrate that the antiprotons do not experience such a threshold. The combination of those two contrasting behaviors gives rise to an unprecedented negative Barkas effect: the stopping power of antiprotons is larger than that of protons. We identify that the slow antiproton at close encounter destabilizes a $p$ orbital of the ${\mathrm{F}}^{-}$ anion pointing toward the antiproton. This particular orbital becomes highly polarizable and hence contributes much to the stopping power.

Figure 1

Theoretical model for the LiF crystal. Upper panel: LiF cluster of 126 atoms with longitudinal and transversal views, extracted from the perfect rocksalt structure in a cylindrical shape along the $⟨111⟩$ axis. Cyan, pink, and white balls represent, respectively, Li, F, and the projectile. In the transversal view, the four impact parameters used in this study are shown in red, green, orange, and blue. Middle panel: Evolution of the total energy of the electrons $E(z)$ as a function of the penetration depth $z$. The zero was set to the value at $t=0$. Lower panel: Total electronic energy difference $E(z)-E(z-a)$, where $a$ is the periodicity length. In the middle and lower panels, the total energies are exhibited for $v=0.4\mathrm{a}.\mathrm{u}.$ for four impact parameters $p$ and two projectiles: proton (solid lines) and antiproton (dashed lines).

Figure 2

LiF random stopping power for protons and antiprotons in the low-velocity regime (below 30 keV). LR TDDFT that is insensitive to the projectile charge sign is represented with a short-dashed line. The solid lines with symbols are RT TDDFT $S_e$. The calculated total random stopping power, including both the electronic and the nuclear parts $S_e+S_n$, is drawn with the long-dashed lines. Several experiments are reported jointly [15, 17, 18] (triangles). The inset represents the RT TDDFT-calculated ESP for antiprotons for each of the four impact parameters, together with the earlier work of Pruneda et al. [20].

Figure 3

LiF RESP calculated when continuously varying the projectile charge $Z_1$ from $-1$ (antiproton) to 1 (proton). Solid lines and symbols are the RT TDDFT calculations, whereas the dashed lines are the LR TDDFT results.

Figure 4

${\mathrm{F}}^{-}$ anion stopping cross section (SCS) $S_e/\rho$ at $v=0.2\mathrm{a}.\mathrm{u}.$ (bars, left axis) and first excitation energy (diamond, right axis). The fluorine anion is considered in three different situations: alone, with a proton ${\mathrm{H}}^{+}$ at distance $p=0.45\AA$, or with an antiproton ${\overline{\mathrm{H}}}^{-}$ at the same distance. The $p_u$ orbital in the presence of an antiproton in the direction $u$ is drawn in an inset. The stopping cross section that originates from the $p_u$ orbital is separated from the rest.