Astrophysical $S$ factor for the ${}^{15}\mathrm{N}$($p$,$\gamma$)${}^{16}\mathrm{O}$ reaction

The $R$-matrix approach has proved to be very useful in extrapolating the astrophysical factor down to astrophysically relevant energies, since the majority of measurements are not available in this region. However, such an approach has to be critically considered when no complete knowledge of the reaction model is available. To get reliable results in such cases one has to use all the available information from independent sources and, accordingly, fix or constrain variations of the parameters. In this paper we present a thorough $R$-matrix analysis of the ${}^{15}\mathrm{N}(p,\gamma ){}^{16}\mathrm{O}$ reaction, which provides a path from the CN cycle to the CNO bi-cycle and CNO tri-cycle. The measured astrophysical factor for this reaction is dominated by resonant capture through two strong $J^{\pi }=1^{-}$ resonances at $E_R=312$ and $962$ keV and direct capture to the ground state. Recently, a new measurement of the astrophysical factor for the ${}^{15}\mathrm{N}(p,\gamma ){}^{16}\mathrm{O}$ reaction has been published [P. J. LeBlanc et al., Phys. Rev. C 82, 055804 (2010)]. The analysis has been done using the $R$-matrix approach with unconstrained variation of all parameters including the asymptotic normalization coefficient (ANC). The best fit has been obtained for the square of the ANC $C^2=539.2$ fm${}^{-1}$, which exceeds the previously measured value by a factor of $\approx 3$. Here we present a new $R$-matrix analysis of the Notre Dame–LUNA data with the fixed within the experimental uncertainties square of the ANC $C^2=200.34$ fm${}^{-1}$. Rather than varying the ANC we add the contribution from a background resonance that effectively takes into account contributions from higher levels. Altogether we present ten fits, seven unconstrained and three constrained. For the unconstrained fit with the boundary condition $B_c=S_c(E_2)$, where $E_2$ is the energy of the second level, we get $S(0)=39.0\pm 1.1$ keVb and normalized ${\mathrm{\chi ̃}}^2=1.84$, i.e., the result which is similar to LeBlanc et al. From all our fits we get the range $33.1⩽S(0)⩽40.1$ keVb which overlaps with the result of LeBlanc et al. We address also the physical interpretation of the fitting parameters.

Figure 1

The astrophysical $S(E)$ factor for the ${}^{15}\mathrm{N}(p,\gamma ){}^{16}\mathrm{O}$ reaction. The black squares are experimental data from Ref. [21]. The red solid line is our unconstrained fit with the boundary condition $B_c=S_c(E_2=0.956\hspace{0.5em}\mathrm{MeV})$, which takes into account three interfering amplitudes: two $1^{-}$ resonances and a nonresonant term. No background pole is included. The blue dot-dashed line is a similar fit with the boundary condition $B_c=S_c(E_1=0.30872\hspace{0.5em}\mathrm{MeV})$. The ANC is $C=14.154$ fm${}^{-1/2}$ as in all other fits.

Figure 2

The astrophysical $S(E)$ factor for the ${}^{15}\mathrm{N}(p,\gamma ){}^{16}\mathrm{O}$ reaction. The black squares are experimental data from Ref. [21]. The red solid line is our unconstrained $R$-matrix fit $A$(113) with the boundary condition $B_c=S_c(E_2)$, which takes into account four interfering amplitudes: two $1^{-}$ resonances, a nonresonant term, and background resonance at $5.07$ MeV. The blue dot-dashed line is our unconstrained $R$-matrix fit $A$(113) similar to the previous one but with the boundary condition $B_c=S_c(E_1)$. The magenta dashed line is the nonresonant $S(E)$ factor for the ANC $C=14.154$ fm${}^{-1/2}$.

Figure 3

The astrophysical $S(E)$ factor for the ${}^{15}\mathrm{N}(p,\gamma ){}^{16}\mathrm{O}$. The red solid line is the constrained fit $B$(113) with the boundary condition $B_c=S_c(E_2=0.956\hspace{0.5em}\mathrm{MeV})$, the blue dot-dashed line is the constrained fit $B$(113) with the boundary condition $B_c=S_c(E_1=0.30872\hspace{0.5em}\mathrm{MeV})$. The black squares are experimental data from Ref. [21]. The ANC is $C=14.154$ fm${}^{-1/2}$.

Figure 4

The astrophysical $S(E)$ factor for the ${}^{15}\mathrm{N}(p,\gamma ){}^{16}\mathrm{O}$. The band for the astrophysical factor $S(E)$ obtained from the unconstrained fit $A$(113) with the boundary condition $B_c=S_c(E_2)$. The upper and lower limits of the band correspond to the fitting of the experimental data, which deviate by $1\sigma$ up and down from the center, correspondingly. Note that the borders of the band have practically the same particle-reduced width amplitudes ${\gamma }_{\nu \hspace{0.5em}c}$ for the first and second levels. The proton partial width for the second resonance within the band is ${\mathrm{\Gamma ̃}}_{2\hspace{0.5em}p}=82.8\pm 0.6$ keV, the $\alpha$-particle partial width for the second resonance ${\mathrm{\Gamma ̃}}_{2\hspace{0.5em}\alpha }=63.2\pm 0.9$ keV, the radiative width of the second resonance ${\Gamma }_{2\hspace{0.5em}\gamma }=63.6\pm 2.4$ eV, and the radiative width of the background pole ${\Gamma }_{\gamma (\mathrm{BG})}=354.9\pm 23.6$ eV. The black squares are experimental data from Ref. [21].

Figure 5

The astrophysical $S$($E$) factor for the ${}^{15}\mathrm{N}(p,\gamma ){}^{16}\mathrm{O}$ reaction in the low-energy region including the first resonance (70 data points). The black squares are the experimental data from Ref. [21]. The red solid line is the unconstrained fit $A$(70) for the boundary condition $B_c=S_c(E_1=0.30872\hspace{0.5em}\mathrm{MeV})$; for this fit $E_2=1.056$ MeV, ${\gamma }_{2\hspace{0.5em}\gamma (\mathrm{int})}=0.056$ MeVfm${}^{3/2}$, $\hspace{0.5em}{\gamma }_{2\hspace{0.5em}\gamma (\mathrm{ext})}=0.0607+i\hspace{0.5em}0.000059$ MeVfm${}^{3/2}$. The blue dot-dashed line is the constrained fit $B$(70) with the boundary condition $B_c=S_c(E_1=0.30872\hspace{0.5em}\mathrm{MeV})$; for this fit $E_2=1.05$ MeV, ${\gamma }_{2\hspace{0.5em}\gamma (\mathrm{int})}=0.082$ MeVfm${}^{3/2}$, $\hspace{0.5em}{\gamma }_{2\hspace{0.5em}\gamma (\mathrm{ext})}=0.052+i\hspace{0.5em}0.000051$ MeVfm${}^{3/2}$. The ANC is $C=14.154$ fm${}^{-1/2}$.