Electron correlations in the antiproton energy-loss distribution in He

We present ab initio calculations of the electronic differential energy-transfer cross sections for antiprotons with energies between 3 keV and 1 MeV interacting with helium. By comparison with simulations employing the mean-field description based on the single-active electron approximation we are able to identify electron correlation effects in the stopping and straggling cross sections. Most remarkably, we find that straggling exceeds the celebrated Bohr straggling limit when correlated shake-up processes are included.

Figure 1

(a) The SI0 impact-parameter-resolved DET $d\sigma (b,\varepsilon )/d\varepsilon$ and (b) the energy-transfer square (${\varepsilon }^2$) rescaled DET as a function of electron ejection energy for the 1 MeV antiproton projectile at fixed the $b=1\mathrm{a}.\mathrm{u}.$ impact parameter. This impact-parameter value was chosen to coincide with the maximum of the impact-parameter-dependent DET. TDCC and SAE results with different angular momentum basis sizes ($L_{max}$) are compared for the SI0 channel. In (b) the binary collision energy loss ($2v_p^2$) is indicated with a vertical dotted line.

Figure 2

The truncation error for stopping and straggling as a function of the angular momentum basis size $L_{max}$ at $E=1\mathrm{MeV}(v_p=6.32\mathrm{a}.\mathrm{u}.)$ and $b=1\mathrm{a}.\mathrm{u}.$

Figure 3

The energy-transfer cross section $d\sigma \left(\varepsilon \right)/d\varepsilon$ as a function of $\varepsilon$ for different antiproton impact energies $E$. The inset shows the finite cross section at and the continuous transition across the first ionization threshold.

Figure 4

The mean energy loss $S(b;v_p)$ [Eq. (13)] as a function of the impact parameter for different antiproton energies: (a) 16 keV and (b) 1000 keV. The present TDCC results contain only the inelastic one-electron channels: single ionization (SI0) and single excitation (EX0) with the second electron remaining in the ground state and are compared to the predictions of the Bohr model for close and distant collisions [1, 12]. The oscillator frequencies as given in the figures are obtained by fitting the distant collision model to the TDCC results at large impact parameters.

Figure 5

Exit-channel decomposition of the energy loss $S(b;v_p)$ as a function of the impact parameter for different antiproton energies (16, 100, and 1000 keV) obtained within the framework of the present TDCC and SAE-IEM models. Single ionization (SI0), single excitation (EX0), correlated excitation-ionization (SI-EX), and double ionization (DI). In the SAE approximation the IEM is invoked to approximate multielectron transitions.

Figure 6

Exit-channel decomposition of the straggling probability $T(b;v_p)$ as a function of the impact parameter for different antiproton energies (16, 100, and 1000 keV) obtained in the framework of the present TDCC and SAE-IEM models. Single ionization (SI0), single excitation (EX0), correlated excitation-ionization (SI-EX), and double ionization (DI). Within the SAE approximation, the IEM is used to approximate multielectron transitions.

Figure 7

The stopping cross section $S\left(v_p\right)$ as a function of antiproton impact energy. The present TDCC results are compared to the experimental data for $\overline{p}$ [26, 27] (experimental uncertainty as the shaded blue area) and ${\mu }^{-}$ colliding with He [57] and to other theoretical calculations: CCC of Bailey et al. [5]; atomic-orbital close coupling of Schiwietz et al. [24]; semiclassical $B$-spline close-coupling calculations of Lühr and Saenz [25]; and the binary collision theory of Sigmund and Schinner [15]. The TDCC results also include the contribution from nuclear stopping.

Figure 8

Comparison between straggling cross sections for protons and antiprotons colliding with helium, normalized to the Bohr straggling number $T_B=4\pi Z_p^2{Z}_T$. Shown are the present TDCC and SAE results for antiprotons, the experimental data by Bonderup and Hvelplund [9] for protons, and the analytic predictions by Sigmund [14] for protons and antiprotons. The smaller frame is a zoom in on the high antiproton energy region showing the (non)convergence of the presented results towards the classical Bohr limit.

Figure 9

Straggling cross-section ratios as a function of the inverse projectile energy. The TDCC data for the ratios $R$ are fitted to the following function: $R\left(E^{-1}\right)=R_0+a/E+b/E^2+c/E^3$. The $R_0$ asymptotic values of the ratios are 9.4% [SI-EX/(SI0 + EX0)] and 9.7% [(SI-EX + DI)/(SI0 + EX0)].