Electron correlations in single-electron capture into any state of fast projectiles from heliumlike atomic systems

State-selective and total single-electron capture cross sections in fast collisions of a bare projectile with a heliumlike target are examined in the four-body formalism. A special emphasis is given to a proper inclusion of dynamic electron-electron correlation effects. For this purpose, the post form of the four-body boundary-corrected first Born approximation (CB1-4B) is utilized. With regard to our related previous study, where the prior version has been considered, in the present work an extensive analytical study of the post-transition amplitude for electron capture into the arbitrary final states $n^f{l}^f{m}^f$ of the projectile is carried out. The post-transition amplitude for single charge exchange encompassing symmetric and asymmetric collisions is derived in terms of five-dimensional integrals over real variables. The dielectronic interaction $V_{12}=1/r_{12}\equiv 1/|{\vec{r}}_1-{\vec{r}}_2|$ explicitly appears in the perturbation potential $V_f$ of the post-transition probability amplitude $T_{if}^{+},$ such that the CB1-4B method can provide information about the relative significance of the dynamic interelectron correlation in the collisions under study. An illustrative computation is performed involving state-selective and total single capture cross sections for the $p-\mathrm{He}$ collisions at intermediate and high impact energies. The so-called post-prior discrepancy, which plagues almost all the existing distorted wave approximations, is presently shown to be practically nonexistent in the CB1-4B method. The validity of our findings is critically assessed in comparisons with the available experimental data for both state-selective and total cross sections summed over all the discrete energy levels of the hydrogenlike atom formed with the projectile. Overall, excellent performance of the CB1-4B method is recorded, thus robustly establishing this formalism as the leading first-order description of high-energy single charge exchange, which is a collision of paramount theoretical and practical importance across interdisciplinary fields ranging from astrophysics to medicine.

Figure 1

State-selective cross sections $Q_{2s}^{+}$ and $Q_{2p}^{+}$ (in units of ${\mathrm{cm}}^2$) as a function of the laboratory impact energy $E$ (keV) for electron capture by protons from $\mathrm{He}\left(1s^2\right)$ in process (59). The solid and dashed curves are the present post cross sections of the CB1-4B method with and without $\Delta V_{12}$ in $V_f,$ respectively, obtained by using the helium ground-state wave functions of Löwdin [50]. Experimental data: $▾$ $(Q_{2s},Q_{2p})$ Cline et al. [51], $\square$ $(Q_{2s},Q_{2p})$ Hughes et al. [52], $▵$ $\left(Q_{2p}\right)$ Hippler et al. [53], and $\blacksquare$ $\left(Q_{2p}\right)$ Hippler et al. [54]. Both the theoretical and experimental cross sections for capture into the $2p$ states of H are divided by 10.

Figure 2

State-selective cross sections $Q_{3s}^{+},Q_{3p}^{+}$, and $Q_{3d}^{+}$ (in units of ${\mathrm{cm}}^2$) as a function of the laboratory impact energy $E$ (keV) for electron capture by protons from $\mathrm{He}\left(1s^2\right)$ in process (59). The solid and dashed curves are the present post cross sections of the CB1-4B method with and without potential $\Delta V_{12}$ in $V_f,$ respectively, obtained by using the helium ground-state wave functions of Löwdin [50]. Experimental data: $▾$ $\left(Q_{3s}\right)$ Conrads et al. [56], $\circ$ $\left(Q_{3s}\right)$ Cline et al. [51], $▿$ $(Q_{3s},Q_{3p},Q_{3d})$ Ford and Thomas [55], $\blacksquare$ $(Q_{3s},Q_{3p},Q_{3d})$ Brower and Pipkin [57], $▵$ $(Q_{3p},Q_{3d})$ Cline et al. [58], and $•$ Edwards and Thomas [60]. Both the theoretical and experimental cross sections for capture into the $3p$ and $3d$ states of H are divided by 10 and 1000, respectively.

Figure 3

State-selective cross sections $Q_{4s}^{+}$ (in units of ${\mathrm{cm}}^2$) as a function of the laboratory impact energy $E$ (keV) for electron capture by protons from $\mathrm{He}\left(1s^2\right)$ in process (59). The solid and dashed curves are the present post cross sections of the CB1-4B method with and without $\Delta V_{12}$ in $V_f,$ respectively, obtained by using the helium ground-state wave functions of Löwdin [50]. Experimental data: $▿$ $\left(Q_{4s}\right)$ Doughty et al. [59], $▾$ $\left(Q_{4s}\right)$ Brower and Pipkin [57], and $\circ$ $\left(Q_{4s}\right)$ Hughes et al. [61].

Figure 4

Total cross sections $Q^{+}\left(\Sigma \right)$ (in units of ${\mathrm{cm}}^2$) as a function of the laboratory impact energy $E$ (keV) for single capture from the ground state of helium by protons to all states $\left(\Sigma \right)$ of atomic hydrogen $\mathrm{H}\left(\Sigma \right)$ in process (60). The solid and the dashed curves are the present cross sections $Q^{+}\left(\Sigma \right)$ from (62) in the post CB1-4B method with and without $\Delta V_{12}$ in $V_f$, respectively, obtained by using the helium ground-state wave functions of Löwdin [50]. Experimental data: $▽$ Shah et al. [62], $△$ Schryber [63], $\circ$ Shah and Gilbody [64], $\square$ Horsdal-Pedersen et al. [65], $♦$ Berkner et al. [66], $▴$ Williams [67], $▾$ Martin et al. [68], and $•$ Welsh et al. [69].

Figure 5

Total cross sections $Q^{\pm }\left(\Sigma \right)$ (in units of ${\mathrm{cm}}^2$) as a function of the laboratory impact energy $E\left(\mathrm{keV}\right)$ for single capture from the ground state of helium by protons in process (60). The pairs of the top and bottom curves are the present post (solid curves) and prior (dashed curves) cross sections $Q^{+}\left(\Sigma \right)$ and $Q^{-}\left(\Sigma \right)$ of the CB1-4B method, obtained by using the helium ground-state wave functions of Löwdin [50]. The prior cross sections in the top and bottom pairs are both with the full perturbation potential $V_i.$ The post cross sections on the top and bottom pairs are with and without $\Delta V_{12}$ in $V_f,$ respectively. Both sets of the results for the bottom pair are divided by 100.

Figure 6

State-selective cross sections $Q_{2s}^{\pm }$ and $Q_{2p}^{\pm }$ (in units of ${\mathrm{cm}}^2$) as a function of the laboratory impact energy $E$ (keV) for electron capture by protons from $\mathrm{He}\left(1s^2\right)$ in process (59). The solid and dashed curves are the present post $\{Q_{2s}^{+},Q_{2p}^{+}\}$ and prior $\{Q_{2s}^{-},Q_{2p}^{-}\}$ cross sections in the CB1-4B method with the complete perturbation potentials $V_f$ and $V_i,$ respectively, obtained by using the helium ground-state wave functions of Löwdin [50].

Figure 7

State-selective cross sections $Q_{3s}^{\pm },Q_{3p}^{\pm }$ and $Q_{3d}^{\pm }$ (in units of ${\mathrm{cm}}^2$) as a function of the laboratory impact energy $E$ (keV) for electron capture by protons from $\mathrm{He}\left(1s^2\right)$ in process (59). The solid and dashed curves are the present post $\{Q_{3s}^{+},Q_{3p}^{+},Q_{3d}^{+}\}$ and prior $\{Q_{3s}^{-},Q_{3p}^{-},Q_{3d}^{-}\}$ cross sections of the CB1-4B method with the complete perturbation potentials $V_f$ and $V_i,$ respectively, obtained by using the helium ground-state wave functions of Löwdin [50].

Figure 8

State-selective cross sections $Q_{1s}^{\pm }$ (in units of ${\mathrm{cm}}^2$) as a function of the laboratory impact energy $E$ (keV) for electron capture by protons from $\mathrm{He}\left(1s^2\right)$ in process (59). The pairs of the top and bottom curves are the present post and prior cross sections of the CB1-4B method with the complete perturbation potential $V_f$ and $V_i$, respectively. The solid and dashed curves correspond to the helium ground-state wave functions of Löwdin [50] and Silverman et al. [70], respectively. Both sets of the results for the bottom pair are divided by 100.