Systematic $R$-matrix analysis of the ${}^{13}\mathrm{C}(p,\gamma ){}^{14}\mathrm{N}$ capture reaction

Background: The proton capture reaction ${}^{13}\mathrm{C}\left(p,\gamma \right){}^{14}\mathrm{N}$ is an important reaction in the CNO cycle during hydrogen burning in stars with mass greater than the mass of the Sun. It also occurs in astrophysical sites such as red giant stars: the asymptotic giant branch (AGB) stars. The low energy astrophysical $S$ factor of this reaction is dominated by a resonance state at an excitation energy of around 8.06 MeV $\left(J^{\pi }=1^{-},T=1\right)$ in ${}^{14}\mathrm{N}$. The other significant contributions come from the low energy tail of the broad resonance with $J^{\pi }=0^{-},T=1$ at an excitation of 8.78 MeV and the direct capture process.

Purpose: Measurements of the low energy astrophysical $S$ factor of the radiative capture reaction ${}^{13}\mathrm{C}\left(p,\gamma \right){}^{14}\mathrm{N}$ reported extrapolated values of $S\left(0\right)$ that differ by about $30\%$. Subsequent $R$-matrix analysis and potential model calculations also yielded significantly different values for $S\left(0\right)$. The present work intends to look into the discrepancy through a detailed $R$-matrix analysis with emphasis on the associated uncertainties.

Method: A systematic reanalysis of the available decay data following the capture to the $J^{\pi }=1^{-},T=1$ resonance state of ${}^{14}\mathrm{N}$ around 8.06 MeV excitation had been performed within the framework of the $R$-matrix method. A simultaneous analysis of the ${}^{13}\mathrm{C}\left(p,p_0\right)$ data, measured over a similar energy range, was carried out with the capture data. The data for the ground state decay of the broad resonance state $\left(J^{\pi }=0^{-},T=1\right)$ around 8.78 MeV excitations was included as well. The external capture model along with the background poles to simulate the internal capture contribution were used to estimate the direct capture contribution. The asymptotic normalization constants (ANCs) for all states were extracted from the capture data. The multichannel, multilevel $R$-matrix code azure2 was used for the calculation.

Results: The values of the astrophysical $S$ factor at zero relative energy, resulting from the present analysis, are found to be consistent within the error bars for the two sets of capture data used. However, it is found from the fits to the elastic scattering data that the position of the $J^{\pi }=1^{-},T=1$ resonance state is uncertain by about 0.6 keV, preferring an excitation energy value of 8.062 MeV. Also the extracted ANC values for the states of ${}^{14}\mathrm{N}$ corroborate the values from the transfer reaction studies. The reaction rates from the present calculation are about $10–15\%$ lower than the values of the NACRE II compilation but compare well with those from NACRE I.

Conclusion: The precise energy of the $J^{\pi }=1^{-},T=1$ resonance level around 8.06 MeV in ${}^{14}\mathrm{N}$ must be determined. Further measurements around and below 100 keV with precision are necessary to reduce the uncertainty in the $S$-factor value at zero relative energy.

Figure 1

Level scheme of ${}^{14}\mathrm{N}$ from Ref. [18]. The vertical arrows denote the transitions considered in the $R$-matrix calculation. The solid arrows indicate $E1$ transitions while the dashed arrows indicate $M1/E2$ transitions.

Figure 2

$R$-matrix fit to the $S\left(E\right)$ data for the $R,DC\to 0.0$ transition. The solid red line gives the total $S\left(E\right)$ for this transition and solid green line gives the contribution from resonance R2. The dashed-dotted line describes the R5 resonant component while the dotted line describes the direct capture component. (a) shows the fit to Data Set I and (b) the fit to Data Set II (see text).

Figure 3

Fits to the elastic scattering data at (a) ${90}^{\circ }$ (and b) ${120}^{\circ }$. The solid line resulted with the parameters from the fit to Data Set I and the dashed line with parameters from the fit to Data Set II.

Figure 4

The low energy astrophysical $S$ factor for the ${}^{13}\mathrm{C}\left(p,\gamma \right){}^{14}{\mathrm{N}}_{gs}$ reaction. The data points are from King et al. [6] (black $•)$, Hester et al. [10] (blue $\diamond )$, Woodbury et al. [7] (orange $\Delta )$, Hebbard et al. [9] (green $\oplus )$, and Genard et al. [11] (violet $☆)$. The solid curve denotes the $R$-matrix calculation with a parameter set (Set 1) fitting and the dashed curve represents calculation with parameters (Set 2) fitting Data Set II.

Figure 5

The astrophysical $S$ factors for the captures to the first four excited states of ${}^{14}\mathrm{N}$ in the ${}^{13}\mathrm{C}\left(p,\gamma \right){}^{14}\mathrm{N}$ reaction. The red solid lines in all four panels denote the total $S$ factor from the fits while the green lines represent the contributions of the $1^{-}$ resonance state. The dotted lines in the panels are the calculated direct capture contributions to the states. The dashed-dotted lines in the lower panels (b) and (d) show the contributions of the $0^{-}$ resonance state in the captures to the second and fourth excited states of ${}^{14}\mathrm{N}$.

Figure 6

The upper panel (a) shows the $S$ factor for the capture to the fifth excited state $(1^{-}$, 5.69 MeV) with the corresponding best-fit curve (red solid line). The contributions of the $1^{-}$ resonance are given by the green solid curve and of the $0^{-}$ resonances by the dashed-dotted curve (multiplied by 10). The direct capture contribution is shown by the dotted curve. The lower panel shows the available data and the fit to it assuming only a direct capture mechanism.

Figure 7

Reaction rate ratio of the present reaction rate to the rate compiled in NACRE II [29] as a function of temperature for the ${}^{13}\mathrm{C}\left(p,\gamma \right){}^{14}\mathrm{N}$ capture reaction. The shaded region corresponds to estimated uncertainties of the $S$-factor values.