Constraining the 6.05 MeV $0^{+}$ and 6.13 MeV $3^{-}$ Cascade Transitions in the ${}^{12}\mathrm{C}(\alpha ,\gamma ){}^{16}\mathrm{O}$ Reaction Using the Asymptotic Normalization Coefficients

The ${}^{12}\mathrm{C}(\alpha ,\gamma ){}^{16}\mathrm{O}$ reaction plays a fundamental role in astrophysics and needs to be known with accuracy better than 10%. Cascade $\gamma$ transitions through the excited states of ${}^{16}\mathrm{O}$ are contributing to the uncertainty. We constrained the contribution of the $0^{+}$ (6.05 MeV) and $3^{-}$ (6.13 MeV) cascade transitions by measuring the asymptotic normalization coefficients for these states using the $\alpha$-transfer reaction ${}^6\mathrm{L}\mathrm{i}({}^{12}\mathrm{C},d{)}^{16}\mathrm{O}$ at sub-Coulomb energy. The contribution of the $0{}^{+}$ and $3^{-}$ cascade transitions at 300 keV is found to be $1.96\pm 0.3$ and $0.12\pm 0.04\mathrm{keV}\mathrm{b}$ for destructive interference of the direct and resonance capture and $4.36\pm 0.45$ and $1.44\pm 0.12\mathrm{keV}\mathrm{b}$ for constructive interference, respectively. The combined contribution of the $0^{+}$ and $3^{-}$ cascade transitions to the ${}^{12}\mathrm{C}(\alpha ,\gamma {)}^{16}\mathrm{O}$ reaction cross section at 300 keV does not exceed 4%. Significant uncertainties have been dramatically reduced.

Figure 1

$\mathrm{\Delta }E$ vs $E$ scatter plot with the cut on the deuterons using the 9 MeV data for a pin detector at 30° in the laboratory frame.

Figure 2

Spectrum of deuterons from the ${}^6\mathrm{L}\mathrm{i}({}^{12}\mathrm{C},d{)}^{16}\mathrm{O}$ reaction. The ${}^{12}\mathrm{C}$ effective beam energy is 8.7 MeV (energy in the middle of the ${}^6\mathrm{Li}$ target), and the deuteron scattering angle is 119° in the center of mass.

Figure 3

Experimental data and DWBA cross section as a function of center-of-mass angle for the $0^{+}$ at 6.05 MeV (a), $3^{-}$ at 6.13 MeV (b), $2^{+}$ at 6.92 MeV (c), and $1^{-}$ at 7.12 MeV (d) in ${}^{16}\mathrm{O}$ using the ${}^{12}\mathrm{C}$ beam energies of 5 MeV (dashed-dotted line), 7 MeV (solid line), and 9 MeV (dashed line).

Figure 4

$S$ factor of the ${}^{12}\mathrm{C}(\alpha ,\gamma ){}^{16}\mathrm{O}$ reaction for the direct $E2$ transition to the subthreshold $0^{+}$ at 6.05 MeV (a) and for the $3^{-}$ at 6.13 MeV (b) without interference (solid line) and assuming constructive (dashed line) or destructive (dash-dotted line) interference with the $2^{+}$ at 6.92 MeV and the $1^{-}$ state at 7.12 MeV in ${}^{16}\mathrm{O}$, respectively. The shaded bands correspond to one standard deviation uncertainties.