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

The astrophysical direct nuclear capture reaction ¹² C(p, γ)¹³ N is studied within the framework of a potential model. Parameters of the nuclear p-¹²C interaction potentials of the Woods-Saxon form are adjusted to reproduce experimental p-¹²C scattering phase shifts, as well as the binding energies and empirical values of the asymptotic normalization coefficient (ANC) for the ¹³N(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 ¹³N(1/2⁻) ground state and width of the ¹³N(1/2⁺) resonance at the Eₓ=2.365 MeV excitation energy. The potential model, which yields the ANC value of 1.63 fm-1/2 for the ¹³N(1/2⁻) ground state and a value Γ=39 keV for the ¹³N(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=10¹⁰ 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~keV)=1.48 ± 0.09 keV b by Kettner { et al.}, but slightly less than the result of S(25~keV)=1.53 ± 0.06 keV b the LUNA Collaboration.