Absolute cross section of the ${}^{12}\mathrm{C}(p,\gamma ){}^{13}\mathrm{N}$ reaction

Solar neutrino measurements have recently reached a level of sensitivity such that CNO fluxes can now be experimentally determined. While these first measurements are still only sensitive to the higher energy neutrinos resulting from the ${\beta }^{+}$ decays of ${}^{15}\mathrm{O}$ produced by the ${}^{14}\mathrm{N}(p,\gamma ){}^{15}\mathrm{O}$ reaction, future measurements will work towards detection of neutrinos from the ${\beta }^{+}$ decay of ${}^{13}\mathrm{N}$ from the ${}^{12}\mathrm{C}(p,\gamma ){}^{13}\mathrm{N}$ reaction. This paper reports on a recent measurement of the ${}^{12}\mathrm{C}(p,\gamma ){}^{13}\mathrm{N}$ reaction covering a broad laboratory energy range between 1.0 and 2.5 MeV. The measurement was made to better determine the overall normalization of the absolute cross section and to explore the interference effects between the two broad, overlapping resonances at proton energies of 0.460 and 1.689 MeV and the direct capture to the ground state of ${}^{13}\mathrm{N}$ in the framework of a multichannel $R$-matrix analysis. This work takes into account previous radiative capture as well as elastic ${}^{12}\mathrm{C}(p,p){}^{12}\mathrm{C}$ scattering data, making uncertainty estimations using a Bayesian framework, to determine a reliable extrapolation of the low energy $S$ factor towards the stellar energy range of CNO hydrogen burning. These new experimental results, and a detailed investigation of the past literature data, suggest that the resonant component of the cross section should be 30% lower than previously accepted.

Figure 1

The efficiency of the experimental setup for a detector located at ${55}^{\circ }$ based on the decay lines of a ${}^{56}\mathrm{Co}$ source and the $\gamma$ transitions of the ${}^{27}\mathrm{Al}(p,\gamma ){}^{28}\mathrm{Si}$ resonance at $E_p=992\mathrm{keV}$, measured at a distance of 259 mm between target and detector.

Figure 2

All ${}^{12}\mathrm{C}(p,\gamma ){}^{13}\mathrm{N}$ measurements were acquired with the same target. Ten stability checks were performed over the course of the experiments by making repeated measurements at $E_p=1700\mathrm{keV}$ to monitor target degradation. A linear degradation was found as a function of deposited charge. The total loss after two weeks and a median current of about $100\mu \mathrm{A}$ was less $\approx 30\%$.

Figure 3

The $\gamma$-ray spectrum from the ${}^{12}\mathrm{C}(p,\gamma ){}^{13}\mathrm{N}$ reaction at three different proton beam energies. The shift in $\gamma$-ray energy with the beam energy clearly demonstrates a mechanism based on the broad resonance at $E_p=1.6\mathrm{MeV}$ interfering with the underlying direct capture population of the ground state. Single escape (s.e.) and double escape (d.e.) peaks are indicated.

Figure 4

Excitation functions of the ${}^{12}\mathrm{C}(p,\gamma ){}^{13}\mathrm{N}$ reaction at ${\theta }_{\gamma }=0$ and ${55}^{\circ }$ compared to the $R$-matrix fit described in Sec 5. Note that the absolute cross section of the excitation functions was obtained by normalizing to the $R$-matrix cross section, whose scale is constrained by the lower energy data.

Figure 5

Angular distributions of the ${}^{12}\mathrm{C}(p,\gamma ){}^{13}\mathrm{N}$ reaction of this work (blue squares) compared to the previous work of Burtebaev et al. [37] (black circles) and Young et al. [31] (green diamonds) and the $R$-matrix fit described in Sec. 5. Where indicated, the cross section scale should be multiplied by the indicated factor. Note that the absolute cross section for the angular distributions was obtained by normalizing to the $R$-matrix cross section.

Figure 6

(a) 511 keV $\gamma$-ray yield observed after activation of the thick ${}^{12}\mathrm{C}$ target, sampled once per minute. The data were well reproduced, as shown by the residual plot (b), by a single decay curve, yielding a half-life for ${}^{13}\mathrm{N}$ of 9.928(38) minutes, in good agreement with the accepted value of 9.965(4) minutes [44].

Figure 7

Comparison of thick-target yield measurements from an unpublished work by Fowler (which is given in Seagrave [24]), Seagrave [24], Hinds and Barker [34], and the present work. The experimental values are compared with the integrated cross section of the $R$-matrix fit described in Sec. 5, where the uncertainty band for the $R$-matrix calculation corresponds to the uncertainty in the srim stopping power. Except for the data of Hinds and Barker [34], the absolute scale of the previous data agrees well with the present work when their systematic uncertainties are considered. To better facilitate comparison, the yields of the data sets are scaled by a multiplicative factor as indicated. All of the data have a consistent energy dependence.

Figure 8

Comparison of the ${}^{12}\mathrm{C}(p,p){}^{12}\mathrm{C}$ data of Meyer et al. [38] with the $R$-matrix fit of the present work.

Figure 9

Comparison of the angle integrated ${}^{12}\mathrm{C}(p,\gamma ){}^{13}\mathrm{N}$ radiative capture $S$ factors for the data of [30], Hinds and Barker [34] and Burtebaev et al. [37] (renormalized as discussed in the text). The angle integrated $S$ factor from the present data, obtained by a Legendre fit to the measured angular distributions, is also shown for comparison. The solid red line represents the best $R$-matrix fit while the dashed red lines indicate the 68% confidence interval obtained from the Bayesian uncertainty analysis using the brick code [54].

Figure 10

Comparison of the present $R$-matrix fit with the data of Rolfs and Azuma [32], renormalized as discussed in the text. The data were not included in the present fit because of a discrepancy with their energy calibration above $\approx 1\mathrm{MeV}$.

Figure 11

Comparison of the extrapolated $R$-matrix fit and uncertainty from the analysis described in Sec. 5, where the bands correspond to a 68% confidence level, with the low energy data of Hall and Fowler [26], Bailey and Stratton [27], and Lamb and Hester [28], with no adjusted normalization. The lowest energy data of Vogl [30], renormalized as described in the text, are also shown for comparison.

Figure 12

Ratio of the rates for the ${}^{12}\mathrm{C}(p,\gamma ){}^{13}\mathrm{N}$ reaction for previous calculations by Li et al. [35] and NACRE II [5] and the present work to that of the NACRE compilation [60]. The uncertainty band of the present work corresponds to a 68% confidence interval.