Astrophysical ${}^3\mathrm{He}(\alpha ,\gamma ){}^7\mathrm{Be}$ and ${}^3\mathrm{H}(\alpha ,\gamma ){}^7\mathrm{Li}$ direct capture reactions in a potential-model approach

The astrophysical ${}^3\mathrm{He}(\alpha ,\gamma ){}^7\mathrm{Be}$ and ${}^3\mathrm{H}(\alpha ,\gamma ){}^7\mathrm{Li}$ direct capture processes are studied in the framework of the two-body model with potentials of a simple Gaussian form, which describe correctly the phase shifts in the $s$, $p$, $d$, and $f$ waves, as well as the binding energy and the asymptotic normalization constant of the ground $p_{3/2}$ and the first excited $p_{1/2}$ bound states. It is shown that the $E1$ transition from the initial $s$ wave to the final $p$ waves is strongly dominant in both capture reactions. On this basis the $s$-wave potential parameters are adjusted to reproduce the new data of the LUNA Collaboration around 100 keV and the newest data at the Gamov peak estimated with the help of the observed neutrino fluxes from the sun, $S_{34}\left({23}_{-5}^{+6}\mathrm{keV}\right)=0.548\pm 0.054$ keV b for the astrophysical $S$ factor of the capture process ${}^3\mathrm{He}(\alpha ,\gamma ){}^7\mathrm{Be}$. The resulting model describes well the astrophysical $S$ factor in the low-energy big-bang nucleosynthesis region of 180–400 keV; however, it has a tendency to underestimate the data above 0.5 MeV. The energy dependence of the $S$ factor is mostly consistent with the data and the results of the no-core shell model with continuum, but substantially different from the fermionic molecular dynamics model predictions. Two-body potentials, adjusted for the properties of the ${}^7\mathrm{Be}$ nucleus, ${}^3\mathrm{He}+\alpha$ elastic scattering data, and the astrophysical $S$ factor of the ${}^3\mathrm{He}(\alpha ,\gamma ){}^7\mathrm{Be}$ direct capture reaction, are able to reproduce the properties of the ${}^7\mathrm{Li}$ nucleus, the binding energies of the ground $3/2^{-}$ and first excited $1/2^{-}$ states, and phase shifts of the ${}^3\mathrm{H}+\alpha$ elastic scattering in partial waves. Most importantly, these potential models can successfully describe both absolute value and energy dependence of the existing experimental data for the mirror astrophysical ${}^3\mathrm{H}(\alpha ,\gamma ){}^7\mathrm{Li}$ capture reaction without any additional adjustment of the parameters.

Figure 1

$p$-wave phase-shift description of ${}^3\mathrm{He}+\alpha$ (a) and ${}^3\mathrm{H}+\alpha$ (b) scattering within different potential models in comparison with available data.

Figure 2

$s$-wave phase-shift description of ${}^3\mathrm{He}+\alpha$ (a) and ${}^3\mathrm{H}+\alpha$ (b) scattering within different potential models in comparison with available data.

Figure 3

Contributions of the partial $E1$ (a), $E2$ (b), and $M1$ (c) components to the astrophysical $S$ factor for the ${}^3\mathrm{He}(\alpha ,\gamma ){}^7\mathrm{Be}$ capture process resulting from calculations with the $V_{M1}$ potential.

Figure 4

Contributions of the total $E1$, $E2$, and $M1$ transitions to the astrophysical $S$ factor for the ${}^3\mathrm{He}(\alpha ,\gamma ){}^7\mathrm{Be}$ reaction estimated using the $V_{M1}$ potential.

Figure 5

Convergence of the astrophysical $S$ factor for the ${}^3\mathrm{He}(\alpha ,\gamma ){}^7\mathrm{Be}$ reaction with respect to the number of integration points with fixed value of $h=0.05$ fm estimated with the $V_{M1}$ potential.

Figure 6

(a) Astrophysical $S$ factor for the ${}^3\mathrm{He}(\alpha ,\gamma ){}^7\mathrm{Be}$ synthesis reaction, estimated using different potential models in comparison with available experimental data and ab initio calculations. Panel (b) highlights the low-energy region.

Figure 7

Contributions of the $E1$, $E2$, and $M1$ transitions to the astrophysical $S$ factor for the ${}^3\mathrm{H}(\alpha ,\gamma ){}^7\mathrm{Li}$ synthesis reaction calculated with the $V_{M1}$ potential.

Figure 8

(a) Astrophysical $S$ factor for the ${}^3\mathrm{H}(\alpha ,\gamma ){}^7\mathrm{Li}$ synthesis reaction calculated with different potentials in comparison with available experimental data and ab initio calculations. Panel (b) highlights the low-energy region.