Astrophysical $S$ factor and reaction rate of the direct ${}^{12}\mathrm{C}(p,\gamma {)}^{13}\mathrm{N}$ capture process within a potential model approach

Abstract

The astrophysical direct nuclear capture reaction ${}^{12}\mathrm{C}(p,\gamma {)}^{13}\mathrm{N}$ is studied within the framework of a potential model. Parameters of the nuclear $p{-}^{12}$C interaction potentials of the Woods-Saxon form are adjusted to reproduce experimental $p{-}^{12}$C scattering phase shifts, as well as the binding energies and empirical values of the asymptotic normalization coefficient (ANC) for the ${}^{13}$N(1/2${}^{-}$) ground state from the literature. The reaction rates are found to be very sensitive to the description of the value of the ANC of the ${}^{13}$N($1/2^{-}$) ground state and width of the ${}^{13}$N($1/2^{+}$) resonance at the $E_x=2.365$ MeV excitation energy. The potential model, which yields the ANC value of 1.63 fm${}^{-1/2}$ for the ${}^{13}$N($1/2^{-}$) ground state and a value $\Gamma$=39 keV for the ${}^{13}$N($1/2^{+}$) resonance width, is able to reproduce the astrophysical $S$ factor in the energy interval up to 2 MeV, the empirical values of the reaction rates in the temperature region up to $T=10^{10}$ K of the LUNA Collaboration and the results of the R-matrix fit. The astrophysical factor $S(0)=1.35$ keV b is found using the asymptotic expansion method of D. Baye. The obtained value is in a good agreement with the Solar Fusion II result. At the same time, the calculated value of 1.44 keV b of the astrophysical $S$ factor at the Solar Gamow energy is consistent with the result of the R-matrix fit of $S(25\mathrm{keV})=1.48\pm 0.09$ keV b by Kettner {\it et al.}, but slightly less than the result of $S(25\mathrm{keV})=1.53\pm 0.06$ keV b the LUNA Collaboration.

Comments: 7 pages, 5 figures