New determination of the astrophysical $S_{E1}$ factor of the ${}^{12}\mathrm{C}(\alpha ,\gamma ){}^{16}\mathrm{O}$ reaction via the ${}^{12}\mathrm{C}({}^{11}\mathrm{B},{}^7\mathrm{Li}){}^{16}\mathrm{O}$ transfer reaction

The ${}^{12}\mathrm{C}(\alpha ,\gamma ){}^{16}\mathrm{O}$ reaction substantially influences the abundance ratios of the main isotopes of carbon and oxygen and hence has been regarded as the holy grail in nuclear astrophysics. Its importance lies in the fact that it significantly affects the yield of key elements and that it plays a crucial role in the study of the mass gap of black holes. This reaction's cross section is greatly influenced by the subthreshold state 7.117-MeV $1^{-}$ of ${}^{16}\mathrm{O}$, which is challenging to determine. To study such states the $\alpha$-cluster transfer reaction can be helpful. In this work, the angular distribution of the ${}^{12}\mathrm{C}({}^{11}\mathrm{B},{}^7\mathrm{Li}){}^{16}\mathrm{O}$ reaction was measured, leading to the 7.117-MeV $1^{-}$ state at $E_{{}^{11}\mathrm{B}}\left(\mathrm{lab}\right)=50$ MeV. By using the finite-range distorted-wave Born approximation and coupled-reaction-channel analysis, we obtained the asymptotic normalization coefficient (ANC) to be $(2.54\pm 0.40)\times {10}^{28}$ ${\mathrm{fm}}^{-1}$ and the reduced $\alpha$ width to be $4.92\pm 0.77$ keV at a channel radius of 6.5 fm. Then, by using the $R$-matrix code azure, we calculated the astrophysical $S$ factor of the ${}^{12}\mathrm{C}(\alpha ,\gamma ){}^{16}\mathrm{O}$ reaction and found that the astrophysical $S_{E1}$ (300) factor of the ground-state transitions is $55.3\pm 9.0$ keV b. This value is lower than what was found in previous works, indicating that the $S$ factor is highly sensitive to the ANC.

Figure 1

Focal-plane position spectrum of ${}^7\mathrm{Li}$ at ${\theta }_{\mathrm{lab}}={10}^{\circ }$. (a) Two-dimensional spectrum of energy versus $p_x$. The ${}^7\mathrm{Li}$ events are marked by a solid red rectangle. (b) One-dimensional spectrum of $p_x$.

Figure 2

Angular distribution of the ${}^{12}\mathrm{C}({}^{11}\mathrm{B},{}^7\mathrm{Li}){}^{16}\mathrm{O}$ reaction leading to the 7.117-MeV $1^{-}$ state of ${}^{16}\mathrm{O}$. The red line represents the FRDWBA calculation summed to the compound nuclear component.

Figure 3

Comparison of $\alpha$-particle ANCs of the 7.117-MeV $1^{-}$ state of ${}^{16}\mathrm{O}$ with previous works. Brune $etal$. [17] and Belhout $etal$. [19] used the ${}^{12}\mathrm{C}({}^6\mathrm{Li},\mathrm{d}){}^{16}\mathrm{O}$ reaction to get the ANCs, while Avila $etal$. [18] used the ${}^6\mathrm{Li}({}^{12}\mathrm{C},d){}^{16}\mathrm{O}$ reaction and Oulebsir $etal$. [20] used the ${}^{12}\mathrm{C}({}^7\mathrm{Li},t){}^{16}\mathrm{O}$ reaction. The other ANC values were deduced from the $R$-matrix analysis [3, 13, 36].

Figure 4

The comparison of $R$-matrix calculations of $E1S$ factor with experimental data [39, 40, 41, 42, 43, 44, 45]. The solid line represents the best fit using our ANC of the 7.117-MeV $1^{-}$ state. The dashed lines represents the upper and lower limits.

Figure 5

The comparison of the present $S_{E1}$(300) factor to previous experimental data [13, 17, 20, 36, 42, 43, 44, 46]. The gray shadow represents the compilation value of NACRE II [9]. The blue dot-dashed line is the value in deBoer et al. [10].