γwidth of the unbound 5.17 MeV level ofO14and the ‘‘hot’’ CNO cycle

We have measured the γ-ray branching ratio ${\Gamma }_{\gamma }$/Γ of the unbound 5.17 MeV first excited state in ${}_{}{}^{14}{}_{}{}^{}\mathrm{O}$ using the ${}_{}{}^{12}{}_{}{}^{}\mathrm{C}$ ${(}^3$He,nγ) reaction at ${\mathit{E}}_{3_{\mathrm{He}}}$=9.35 MeV. We detected coincidences between $n_1$ neutrons populating the ${}_{}{}^{14}{}_{}{}^{}\mathrm{O}$ state and 5.17 MeV γ rays from its decay, and obtained ${\Gamma }_{\gamma }$/Γ=(7.2±3.5)×${10}^{\mathrm{-}5}$. When combined with the known total width Γ=38.1±1.8 keV, our branching ratio corresponds to ${\Gamma }_{\gamma }$=2.7±1.3 eV. We use this γ-ray width, together with previously measured nonresonant ${}_{}{}^{13}{}_{}{}^{}\mathrm{C}$(p,${\gamma }_1$) data, to find the ${}_{}{}^{13}{}_{}{}^{}\mathrm{N}$(p,γ) reaction rate at astrophysical energies, thereby determining the stellar environments where hydrogen burning through ‘‘hot’’ or β-limited CNO cycle can occur.