Time-dependent density-functional calculation of the stopping power for protons and antiprotons in metals

Time-dependent density-functional theory is used to calculate the energy loss of antiprotons and protons traversing metal clusters of variable size. We find that the effective energy loss per unit path length inside the cluster shows no significant cluster size effects over the wide range of projectile velocities studied. This allows us to compare the calculated stopping power with the experimental values for a solid metal target. Excellent agreement between the theoretical results and recent experimental data is found for velocities below the inner-shell excitation threshold. We thus present a nonperturbative quantum-mechanical approach to obtain the energy loss of charges in solids.

Figure 1

(Color online) Contour plots for the antiproton-induced change $\Delta n(\mathbf{r},t)$ in the electron density of the spherical cluster. The cluster parameters are $r_s=2$ and $N_e=254$. Results are shown in the $(x,z)$ plane. The center of the cluster is at (0,0). The projectile trajectory is shown as a horizontal line in each plot. The projectile velocity is $v=1.5$. Plots (a) to (d) correspond to antiproton positions $z=-12.7$, 5.8, 12.6, and 42.5. Color codes are shown at the right side of the figure.

Figure 2

Force on an antiproton moving with $v=1.5$ as a function of its position with respect to the center of the cluster. The cluster parameters are $r_s=2$, $N_e=254$, and $R_{\mathrm{cl}}=12.7$.

Figure 3

(Color online) Stopping power $S$ of antiprotons moving inside clusters with $r_s=2.07$ and of different sizes (see legend), as a function of velocity. Lines are drawn to guide the eye. The inset compares the $N_e=556$ results with those of LRT and DFT.

Figure 4

(Color online) TDDFT calculation of the stopping power $S$ of Al $(r_s=2.07)$ for protons (black empty squares) and antiprotons (red empty circles) as a function of projectile velocity. Experimental results for antiprotons (green filled circles [18, 19, 20]) and protons (blue filled and empty triangles [20]), as well as tabulated stopping powers for protons (blue crossed [27] and filled [28, 29] squares) are also shown.