Theoretical study of the $\alpha +d\to {}^6\mathrm{Li}+\gamma$ astrophysical capture process in a three-body model. II. Reaction rates and primordial abundance

The astrophysical $S$ factor and reaction rate of the direct capture process $\alpha +d\to {}^6\mathrm{Li}+\gamma$, as well as the abundance of the ${}^6\mathrm{Li}$ element, are estimated in a three-body model. The initial state is factorized into the deuteron bound state and the $\alpha +d$ scattering state. The final nucleus ${}^6\mathrm{Li}$($1^{+}$) is described as a three-body bound state $\alpha +n+p$ in the hyperspherical Lagrange-mesh method. Corrections to the asymptotics of the overlap integral in the $S$ and $D$ waves have been done for the $E2 S$ factor. The isospin forbidden $E1 S$ factor is calculated from the initial isosinglet states to the small isotriplet components of the final ${}^6\mathrm{Li}$($1^{+}$) bound state. It is shown that the three-body model is able to reproduce the newest experimental data of the LUNA Collaboration for the astrophysical $S$ factor and the reaction rates within the experimental error bars. The estimated ${}^6\mathrm{Li}/\mathrm{H}$ abundance ratio of $(0.67\pm 0.01)\times {10}^{-14}$ is in a very good agreement with the recent measurement $(0.80\pm 0.18)\times {10}^{-14}$ of the LUNA Collaboration.

Figure 1

Convergence of the ${}^6\mathrm{Li}$ ground-state energy with respect to $K_{\max }$ within models A and B.

Figure 2

Partial $E1$ astrophysical $S$ factors of the direct $\alpha +d\to {}^6\mathrm{Li}+\gamma$ capture process within model A (see text).

Figure 3

Overlap integral with the initial three-body and the corrected (at $R_0=5.5\mathrm{fm}$) asymptotics within model A.

Figure 4

Partial $E2$ astrophysical $S$ factors of the direct $\alpha +d\to {}^6\mathrm{Li}+\gamma$ capture process within model A with the corrected asymptotics of the overlap integral.

Figure 5

Relative contributions of the $E1$ and $E2$ astrophysical $S$ factors of the direct $\alpha +d\to {}^6\mathrm{Li}+\gamma$ capture process within model A in comparison with available experimental data.

Figure 6

Theoretical astrophysical $S$ factor for the direct $\alpha +d\to {}^6\mathrm{Li}+\gamma$ capture process within models A and B in comparison with available experimental data.

Figure 7

Reaction rates of the direct $\alpha +d\to {}^6\mathrm{Li}+\gamma$ capture process within models A and B normalized to the NACRE 1999 experimental data.