Indirect Measurement of the $^{23}\textbf{Na}(p,\gamma)^{24}$Mg Direct Capture Reaction Rate via ($^3$He,d) Spectroscopy

TL;DR

This study measures the $^{23}$Na($^3$He,d)$^{24}$Mg transfer reaction to indirectly determine the $^{23}$Na(p,$ extgamma$)$^{24}$Mg reaction rate, providing a more accurate astrophysical S factor for stellar models.

Contribution

The paper introduces a new transfer reaction measurement to constrain the direct capture reaction rate of $^{23}$Na(p,$ extgamma$)$^{24}$Mg, improving previous estimates with Bayesian uncertainty analysis.

Findings

New spectroscopic factors for $^{24}$Mg states in 7-12 MeV range.

Calculated astrophysical S factor and reaction rate at stellar energies.

The new reaction rate is 43% smaller at low temperatures compared to previous data.

Abstract

The cross section of the $^{23}\text{Na}(p,\gamma)^{24}\text{Mg}$ reaction is dominated by direct capture at low energies relevant for stellar burning. Such cross sections can be constrained using spectroscopic factors($C^2S$) or asymptotic normalization coefficients(ANCs) from transfer reactions. In this work, the $^{23}\text{Na}(^3\text{He},d)^{24}\text{Mg}$ reaction was measured at $E_{lab}=21$ MeV to extract spectroscopic factors for $^{24}\text{Mg}$ states with excitation energies in $E_x=7\sim12~$MeV using the Enge split-pole spectrograph at the Triangle Universities Nuclear Laboratory. A new non-resonant astrophysical S factor and the direct capture reaction rate for the $^{23}\text{Na}(p,\gamma)$ reaction are calculated and presented based on this measurement. The new rate at $T<0.04$ GK is 43$\%$ smaller than in previous studies. Rigorous treatments of uncertainties are presented using a Bayesian Markov Chain Monte Carlo (MCMC) method. Sources of uncertainties for computing the direct capture cross section are also discussed in detail.