Constraining the 6.05 MeV<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:msup><mml:mrow><mml:mn>0</mml:mn></mml:mrow><mml:mrow><mml:mo>+</mml:mo></mml:mrow></mml:msup></mml:mrow></mml:math>and 6.13 MeV<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:msup><mml:mrow><mml:mn>3</mml:mn></mml:mrow><mml:mrow><mml:mo>−</mml:mo></mml:mrow></mml:msup></mml:mrow></mml:math>Cascade Transitions in the<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:mmultiscripts><mml:mrow><mml:mi mathvariant="normal">C</mml:mi></mml:mrow><mml:mprescripts/><mml:none/><mml:mrow><mml:mn>12</mml:mn></mml:mrow></mml:mmultiscripts><mml:mo stretchy="false">(</mml:mo><mml:mi>α</mml:mi><mml:mo>,</mml:mo><mml:mi>γ</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mmultiscripts><mml:mrow><mml:mi mathvariant="normal">O</mml:mi></mml:mrow><mml:mprescripts/><mml:none/><mml:mrow><mml:mn>16</mml:mn></mml:mrow></mml:mmultiscripts></mml:mrow></mml:math>Reaction Using the Asymptotic Normalization Coefficients

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

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