Investigation of direct capture in the ${}^{23}\mathrm{Na}(p,\gamma ){}^{24}\mathrm{Mg}$ reaction

The ${}^{23}\mathrm{Na}(p,\gamma ){}^{24}\mathrm{Mg}$ reaction plays an important role in the nucleosynthesis of elements in the hot bottom burning environment of asymptotic giant branch stars by providing a breakout path from the NeNa cycle to the MgAl cycle. At temperatures above $\approx 0.06$ GK, the underlying nuclear reaction contributions to the rate are primarily narrow resonances, but at lower temperatures direct and broad resonance tail contributions come to dominate. While there have been recent studies to improve the uncertainties associated with these narrow resonances, little attention has been paid to the nonresonant component. In this work, experimental measurements are reported over the energy range from 0.5 and 1.05 MeV proton beam energy, with a focus on studying the off-resonance region of the cross section. Several transitions were observed where two broad resonances dominate the energy range and whose low energy tails contribute strongly to the low-energy, nonresonant, cross section. In addition, a clear signature of direct capture has been observed for the first time in the ${}^{23}\mathrm{Na}(p,\gamma ){}^{24}\mathrm{Mg}$ reaction.

Figure 1

Fractional contribution of the individual DC transitions to the total DC $S$ factor. Only those transitions that contribute more than 5% to the total are shown. The calculations used the $C^{2}S$ values from Garrett et al. [16].

Figure 2

The top panel shows a comparison between close geometry (red) and far geometry (254 mm, green) measurements of the $\gamma$-ray full-energy peak detection efficiency. Both data sets include the effects of summing, but in the far geometry they are negligible. The solid lines indicate empirical fits to the data where the summing is either negligible (far geometry) or has been corrected as described in Imbriani et al. [25]. The total efficiency is indicated by the dashed grey line. The bottom panel gives the residuals between the measured and simulated full-energy peak efficiencies (with summing) in close geometry.

Figure 3

An off-resonance $\gamma$-ray spectrum at $E_p=879$ keV. The strongest transitions (the full-energy peak and two escape peaks where they are visible) of the ${}^{23}\mathrm{Na}(p,\gamma ){}^{24}\mathrm{Mg}$ reaction are indicated. A strong background peak is observed from the ${}^{19}\mathrm{F}(p,\alpha \gamma ){}^{16}\mathrm{O}$ reaction, owing to fluorine contamination in the Ta backing. Transitions are indicated for both primary resonance (R) or direct capture (DC) to different final states (indicated by the number of the excited state for low lying states) as well as the strong secondary transition from the first excited state to the ground state (GS). The line at 511 keV due to electron-positron annihilation is also indicated.

Figure 4

$R$-matrix fit to the ${}^{23}\mathrm{Na}(p,p_0){}^{23}\mathrm{Na}$ data of Baumann et al. [28] and the ${}^{23}\mathrm{Na}(p,{\gamma }_{0,1,3}){}^{24}\mathrm{Mg}$ data of the present work. The solid red lines indicate the best fit (50% quantile) while the dashed red lines indicate the 16% and 84% quantiles found from the MCMC analysis. The “n” in the two upmost plots are normalization factors; see Table 1.

Figure 5

As in Fig. 4, except for the ${}^{23}\mathrm{Na}(p,{\gamma }_{2,4,5,8.44\text{MeV},10.37\text{MeV}}){}^{24}\mathrm{Mg}$ cross sections.

Figure 6

$R$-matrix components used to fit the primary transition to the $E_x=10.73$ MeV final state in the ${}^{23}\mathrm{Na}(p,\gamma ){}^{24}\mathrm{Mg}$ reaction. The experimental data indicate the presence of both a single resonance, corresponding to the unbound state at $E_x=12.53$ MeV, and external capture, modeled using $E1$ multipolarity only.

Figure 7

Relative contributions to the total reaction rate over the low temperature region where the rate is dominated by the DC and resonances at $E_p=133$, 251, and 309 keV.

Figure 8

Comparison of different total nonresonant $S$-factor calculations for the ${}^{23}\mathrm{Na}(p,\gamma ){}^{24}\mathrm{Mg}$ reaction. The jezebel calculations from this work (red solid line) and those of Hale et al. [42] (grey dashed line) represent potential model DC $S$ factors. That of Iliadis et al. [39] (black dashed-dotted line) is a “nonresonant” calculation including both DC and the tails of higher energy broad resonances. The resonant-only $S$ factor from the $R$-matrix calculation is also shown (green double-dotted-dashed line), as well as the sum of it and the jezebel calculation (yellow dash-dash-dotted line).

Figure 9

Comparisons of the hard sphere external capture $S$ factors calculated from azure2 using the upper limit ANCs determined in this work, given in Table 1 (red line), to the direct capture $S$ factors calculated using $C^{2}S$ values from Garrett et al. [16] and the potential model code jezebel (blue dashed-dotted line).

Figure 10

As Fig. 9 but for the transition to the fifth excited state in ${}^{24}\mathrm{Mg}$ and the two high lying bound states at $E_x=8.44$ and 10.73 MeV.

Figure 11

Ratio of the present rate (blue solid line) and uncertainty (blue shaded region) to the median of Marshall et al. [46]. The relative uncertainties of Marshall et al. [46] (gray shaded region) are also shown for comparison.