Cross section measurement of ${}^{14}\mathrm{N}(p,\gamma ){}^{15}\mathrm{O}$ in the CNO cycle

Background: The CNO cycle is the main energy source in stars more massive than our sun; it defines the energy production and the cycle time that lead to the lifetime of massive stars, and it is an important tool for the determination of the age of globular clusters. In our sun about 1.6% of the total solar neutrino flux comes from the CNO cycle. The largest uncertainty in the prediction of this CNO flux from the standard solar model comes from the uncertainty in the ${}^{14}\mathrm{N}(p,\gamma ){}^{15}\mathrm{O}$ reaction rate; thus, the determination of the cross section at astrophysical temperatures is of great interest.

Purpose: The total cross section of the ${}^{14}\mathrm{N}(p,\gamma ){}^{15}\mathrm{O}$ reaction has large contributions from the transitions to the $E_x=6.79\mathrm{MeV}$ excited state and the ground state of ${}^{15}\mathrm{O}$. The $E_x=6.79\mathrm{MeV}$ transition is dominated by radiative direct capture, while the ground state is a complex mixture of direct and resonance capture components and the interferences between them. Recent studies have concentrated on cross-section measurements at very low energies, but broad resonances at higher energy may also play a role. A single measurement has been made that covers a broad higher-energy range but it has large uncertainties stemming from uncorrected summing effects. Furthermore, the extrapolations of the cross section vary significantly depending on the data sets considered. Thus, new direct measurements have been made to improve the previous high-energy studies and to better constrain the extrapolation.

Methods: Measurements were performed at the low-energy accelerator facilities of the nuclear science laboratory at the University of Notre Dame. The cross section was measured over the proton energy range from $E_p=0.7$ to 3.6 MeV for both the ground state and the $E_x=6.79\mathrm{MeV}$ transitions at ${\theta }_{\text{lab}}=0^{\circ }$, ${45}^{\circ }$, ${90}^{\circ }$, ${135}^{\circ }$, and ${150}^{\circ }$. Both TiN and implanted-${}^{14}\mathrm{N}$ targets were utilized. $\gamma$ rays were detected by using an array of high-purity germanium detectors.

Results: The excitation function as well as angular distributions of the two transitions were measured. A multichannel $R$-matrix analysis was performed with the present data and is compared with previous measurements. The analysis covers a wide energy range so that the contributions from broad resonances and direct capture can be better constrained.

Conclusion: The astrophysical $S$ factors of the $E_x=6.79\mathrm{MeV}$ and the ground-state transitions were extrapolated to low energies with the newly measured differential-cross-section data. Based on the present work, the extrapolations yield $S_{6.79}\left(0\right)=1.29\pm 0.04\left(\mathrm{stat}\right)\pm 0.09\left(\mathrm{syst}\right)\mathrm{keV}\mathrm{b}$ and $S_{\text{g.s.}}\left(0\right)=0.42\pm 0.04\left(\mathrm{stat}\right)\mathrm{keV}\mathrm{b}$. While significant improvement and consistency is found in modeling the $E_x=6.79\mathrm{MeV}$ transition, large inconsistencies in both the $R$-matrix fitting and the low-energy data are reaffirmed for the ground-state transition. Reflecting this, a systematic uncertainty of ${}_{-0.19}^{+0.09}\mathrm{keV}\mathrm{b}$ is recommended for the ground-state transition.

Figure 1

Level scheme of the ${}^{15}\mathrm{O}$ compound nucleus. The beam (laboratory frame) and resonance (center-of-mass frame) energies and corresponding excitation energies of the important states are given.

Figure 2

(a) Detector setups of the angular distribution measurement and (b) of the clover detector.

Figure 3

Peak efficiency for the clover setup at 1 cm. The data shown are summing corrected measurements. The data are ${}^{137}\mathrm{Cs}$ and ${}^{60}\mathrm{Co}$ sources (solid circles), ${}^{56}\mathrm{Co}$ source (open circles), ${}^{14}\mathrm{N}(p,\gamma ){}^{15}\mathrm{O}$ reaction (open squares), ${}^{27}\mathrm{Al}(p,\gamma ){}^{28}\mathrm{Si}$ reaction at 1 cm (up triangles), at 20 cm but scaled to match 1 cm measurements (down triangles), SEP (diamonds), and DEP (crosses) measurements. The dotted lines are fits to the data.

Figure 4

$Q_2$ measurements using the ${}^{16}\mathrm{O}(p,{\gamma }_{0.50}){}^{17}\mathrm{F}$ reaction. The dotted lines are fits to the data using $Y=a_0\{1-Q_2{P}_2[cos\left(\theta \right)]\}$. The relative yields from two different setups are scaled to be better presented in one plot. The measurements are the clover setup at 5 cm (diamonds) and the angular distribution setup (circles).

Figure 5

$S$ factor of the $E_x=6.79\mathrm{MeV}$ primary transition. The data shown are from Ref. [9] (diamonds), Ref. [12] (triangles), Ref. [11] (circles), and the present measurement (crosses). Note that the statistical uncertainties on several of the present measurements are smaller than the symbols.

Figure 6

$S$ factor of the ground-state transition. The data shown are from Ref. [9] (diamonds), Ref. [12] (triangles), Ref. [11] (circles), and the present measurement (crosses). Note that the statistical uncertainties on several of the present measurements are smaller than the symbols.

Figure 7

The $a_1$ and $a_2$ coefficients of the $E_x=6.79\mathrm{MeV}$ primary transition. The data shown are the present data (squares) from Legendre polynomial fits and those of Ref. [9] (circles) where only the $a_2$ term was considered. The dotted lines represent calculations from an $R$-matrix fit. Note that the $a_2$ coefficients from the $R$-matrix fit are systematically lower than both measurements but are in significantly better agreement with those of the current measurement. The reason for the large disagreement between the $R$-matrix fit and the data of Ref. [9] is unknown. Sometimes, the present experimental values are above the $R$-matrix predictions because there are weak resonance transitions over this energy region that have not been included.

Figure 8

The $a_1$ and $a_2$ coefficients resulting from the angular analysis of the ground-state transition. The data shown are from Ref. [9] (circles) and polynomial fits of the present data (diamonds). The dashed lines represent calculations from an $R$-matrix fit. The blue solid lines represent external-capture calculations based on the $R$-matrix fit.

Figure 9

Ratio between the $E_x=6.79\mathrm{MeV}$ transition yield measured at ${\theta }_{\text{lab}}={45}^{\circ }$ and ${135}^{\circ }$. The dotted line represents the ratio if only $E1$ external capture is included. The solid line represents the inclusion of $E1$ and $E2$ external capture and the interference between them. The data in the vicinity of the narrow resonances have been excluded.

Figure 10

$R$-matrix fit to the differential $S$ factors of the $E_x=6.79\mathrm{MeV}$ primary transition. The resonance at $E_R=2.31\mathrm{MeV}$ was included in the fit but does not contribute significantly to the low-energy $S$ factor.

Figure 11

$R$-matrix fit to the differential $S$ factors from the ground-state transition.

Figure 12

$R$-matrix fit to the differential $S$ factors obtained by using the clover detector setup at ${\theta }_{\text{lab}}={45}^{\circ }$. The transition to the state at $E_x=6.79\mathrm{MeV}$ and the ground-state transition are shown in panels (a) and (b), respectively.

Figure 13

$R$-matrix calculation of the $E_x=6.79\mathrm{MeV}S$ factor using the parameters from the $R$-matrix fit (solid) together with the $R$-matrix fit from Ref. [16] (dotted), and case (ii) from Ref. [4] where the data sets were limited to $E_{\mathrm{c}.\mathrm{m}.}<1.2\mathrm{MeV}$ (dashed). The data are those from the angular distribution measurements of the present work that were fit with Eq. (6) to obtain angle-integrated cross sections that have then been transformed to $S$ factors. The inset shows the extrapolation to zero energy.

Figure 14

Extrapolation of the $R$-matrix fit to the higher-energy data of this work (red solid line). The fit is compared to the low-energy data of Refs. [11, 12, 15] and previous $R$-matrix extrapolations from Refs. [4, 12, 16]. The large inconsistencies between both the different data sets and the $R$-matrix extrapolations leads to a large systematic uncertainty in the value of $S\left(0\right)$ for the ground-state transition.

Figure 15

$R$-matrix calculation of the ground-state $S$ factor using the parameters from the best fit (solid) together with the fit from Ref. [16] (dotted) and Ref. [4] (dash-dot).