Boundary-corrected four-body continuum-intermediate-state method for charge exchange between hydrogenlike projectiles and atoms

Single-electron capture from one-electron and multielectron atoms colliding with hydrogenlike projectiles at intermediate and high incident energies is examined by using the post version of the boundary-corrected four-body continuum-intermediate-state (BCIS-4B) method. This method satisfies the correct boundary conditions in the entrance and exit channels. In the entrance configuration, the BCIS-4B method takes into account the ionization channel through the electronic continuum intermediate states described by the full Coulomb wave function centered on the screened nuclear charge of the projectile. The presented analytical calculation yields the transition amplitude in terms of an efficiently computed two-dimensional numerical quadrature over real variables. Total cross sections are computed for electron capture in the ${\mathrm{He}}^{+}\text{−}\mathrm{H}$, ${\mathrm{He}}^{+}\text{−}\mathrm{He}$, and ${\mathrm{Li}}^{2+}\text{−}\mathrm{He}$ collisions at intermediate and high impact energies. Also, differential cross sections are obtained for the ${\mathrm{He}}^{+}\text{−}\mathrm{He}$ collisions. The present results from the BCIS-4B method are found to be in very good agreement with the available experimental data on differential and total cross sections.

Figure 1

Theoretical total cross sections $Q$ (${\mathrm{cm}}^2$) for single charge exchange as a function of the laboratory incident energy $E$ (keV/amu). The curves show the results obtained by means of the CB1-4B method [55] for the process ${\text{H}}^{+}+\text{He}\left(1s^2\right)\to \text{H}\left(1s\right)+{\text{He}}^{+}\left(1s\right)$ (lower curve) and the process ${\text{He}}^{2+}+\text{He}\left(1s^2\right)\to {\text{He}}^{+}\left(1s\right)+{\text{He}}^{+}\left(1s\right)$ (upper curve). The open circles represent the theoretical results of the CBDW-4B method [56, 57] for the mentioned processes. All the computations were carried out using the same one-parameter wave function of Hylleraas for the helium target, $\text{He}\left(1s^2\right)$.

Figure 2

Theoretical total cross sections $Q$ (${\mathrm{cm}}^2$) for single charge exchange in the ${}^4{\mathrm{He}}^{+}\left(1s\right)+\mathrm{H}\left(1s\right)$ collisions as a function of the laboratory incident energy. The shown cross sections include only the term $-Z_T/x_1$ in the perturbation potential $V_f$ from (6). The open circles represent the results from the QBCIS-4B method, whereas the solid line shows results from the BCIS-4B method. All the computations were performed using the two-parameter wave function of Silverman et al. [59] for the helium atom in its ground state, $\text{He}\left(1s^2\right)$. The BCIS-4B method is the eikonal version of the QBCIS-4B method, which is the fully quantum-mechanical four-body boundary-corrected continuum-intermediate-state approximation. For the relative motion of heavy particles in the initial and final states, the QBCIS-4B and BCIS-4B methods utilize the full Coulomb wave functions and the associated logarithmic phases, respectively. The other difference is that the BCIS-4B method employs the forward-angle simplification of the eikonal type in the arguments of plane waves as per (13), whereas the QBCIS-4B method uses directly ${\vec{k}}_i·{\vec{r}}_i+{\vec{k}}_f·{\vec{r}}_f$ without resorting to the simplification (13).

Figure 3

Total cross sections (in ${\mathrm{cm}}^2$) as a function of the laboratory incident energy $E$ (keV/amu). Curves in a are for the ${}^7{\mathrm{Li}}^{2+}+{}^4{\mathrm{He}}^{+}\to {}^7{\mathrm{Li}}^{+}+{}^4{\mathrm{He}}^{+}$ collisions. All the results (theoretical and experimental) for this process are multiplied by 10. The experimental data are from [10] ($\blacksquare$). Curves in b are for the ${}^4{\mathrm{He}}^{+}+{}^4\mathrm{He}\to {}^4\mathrm{He}+{}^4{\mathrm{He}}^{+}$ collisions. The experimental data are from Ref. [11] ($\blacksquare$), Ref. [16] ($\square$), Ref. [14] ($\circ$), Ref. [13] ($▿$), Ref. [22] ($▾$), and Ref. [23] ($△$). Curves in c are for the ${}^4{\mathrm{He}}^{+}+\mathrm{H}\to {}^4\mathrm{He}+{\mathrm{H}}^{+}$ collisions. All the results (theoretical and experimental) for this process are divided by 100. The experimental data are from Ref. [19] ($•$), Ref. [15] ($\blacksquare$), and Ref. [21] ($▽$). The solid curves represent the total cross sections in the BCIS-4B method (present computation). The dotted curves display the results from the CDW-4B method [43]. The dashed curves depict the results from the CB1-4B method [41, 42]. All the computations refer to the post forms and have been carried out by using the two-parameter wave function of Silverman et al. [59] for the ${\text{Li}}^{+}\left(1s^2\right)$ and $\text{He}\left(1s^2\right)$ in the exit channels and the RHF wave function [53] for the helium target in the entrance channels.

Figure 4

The data near curve a show the differential cross sections as a function of scattering angle $\theta \equiv {\theta }_{\mathrm{lab}}\left(\mathrm{m}\mathrm{rad}\right)$ in the laboratory frame of reference at the incident energy $E=300\text{keV}/\mathrm{amu}$ for single-electron capture in the ${}^3{\mathrm{He}}^{+}+{}^4\mathrm{He}\to {}^3\mathrm{He}+{}^4{\mathrm{He}}^{+}$ collisions. The solid curve represents the theoretical results obtained by using the post form of the BCIS-4B method (present computations). The ground state of $\mathrm{He}\left(1s^2\right)$ in the exit channel is described by means of the two-parameter wave function of Silverman et al. [59]. The RHF wave function for the helium target in the entrance channel has been used. The experimental data are from [9] ($\blacksquare$). The data near curve b are the same as for curve a, except for the incident energy of $E=630\text{keV}/\mathrm{amu}$. Both the theoretical and experimental results at this energy are divided by a factor of 10.