The $E2$ component of the ${}^{12}\mathrm{C}(\alpha ,\gamma ){}^{16}\mathrm{O}$ cross section is investigated in two ways: by a microscopic cluster model, and by $R$-matrix fits. The $\alpha {+}^{12}\mathrm{C}$ microscopic calculation is performed in the framework of the generator coordinate method (GCM) by including all ${}^{12}\mathrm{C}$ states ($T=0$) within the $p$ shell. Using different nucleon-nucleon interactions we find $S_{E2}(300\hspace{0.5em}\mathrm{keV})\approx 50$ keV $·$ b for ground-state transitions. We also study cascade transitions to the $0_2^{+}$ and $2_1^{+}$ excited states of ${}^{16}\mathrm{O}$. Then the $S$-factor is analyzed in the phenomenological $R$-matrix theory. We show that the background term plays a crucial role, and cannot be determined without ambiguity. Using the experimental phase shifts and capture cross sections, only an upper limit on the extrapolated $S$ factor can be obtained [$S_{E2}(300\hspace{0.5em}\mathrm{keV})<190$ keV $·$ b]. To constrain the $R$-matrix analysis, we use the GCM asymptotic normalization constant (ANC) of the $2_1^{+}$ level, well known to be a cluster state. This procedure strongly reduces the uncertainties on the $R$-matrix fit, and we end up with a recommended value of $S_{E2}(300\hspace{0.5em}\mathrm{keV})=42\pm 2$ keV $·$ b. We show that ANC values derived from indirect methods are not consistent with the ${}^{12}\mathrm{C}(\alpha ,\gamma ){}^{16}\mathrm{O}$ cascade transitions to the $2_1^{+}$ state, and suggest that a remeasurement of this cross section is desirable.

• Received 10 April 2008

DOI:https://doi.org/10.1103/PhysRevC.78.015808