Asymptotic normalization coefficients from transfer reactions and $R$-matrix analysis of direct capture in the ${}^{22}\mathrm{Ne}(p,\gamma ){}^{23}\mathrm{Na}$ reaction

The ${}^{22}\mathrm{Ne}(p,\gamma ){}^{23}\mathrm{Na}$ reaction in the Ne-Na cycle plays an important role in the production of the only stable sodium isotope ${}^{23}\mathrm{Na}$. This nucleus is processed by the Ne-Na cycle during hot bottom burning (HBB) in the asymptotic giant branch (AGB) stage of low metallicity intermediate mass stars $(4M_{\odot }\le M\le 6M_{\odot })$. Recent measurements have addressed the uncertainty in the thermonuclear reaction rate of this reaction at relevant astrophysical energies through the identification of low lying resonances at $E_p$ = 71, 105, 156.2, 189.5, and 259.7 keV. In addition, precise measurements of the low energy behavior of nonresonant capture have been performed and the contribution of the subthreshold resonance at 8664 keV excitation in ${}^{23}\mathrm{Na}$ has been established. Here we present a systematic $R$-matrix analysis of direct capture to the bound states and the decay of the subthreshold resonance at 8664 keV to the ground state of ${}^{23}\mathrm{Na}$. A finite-range distorted-wave Born approximation (FRDWBA) calculation has been performed for the ${}^{22}\mathrm{Ne}({}^3\mathrm{He},d){}^{23}\mathrm{Na}$ transfer reaction data to extract the asymptotic normalization coefficients (ANCs) required to estimate the nonresonant capture cross sections or astrophysical $S$-factor values in the $R$-matrix analysis. Simultaneous $R$-matrix analysis constrained with ANCs from transfer calculations reproduced the astrophysical $S$-factor data over a wide energy window. The value $S_{tot}^{DC}\left(0\right)=48.8\pm 9.5\mathrm{keV}\mathrm{b}$ compares well with the result of Ferraro et al. and has a lower uncertainty. The resultant thermonuclear reaction is slightly larger in the $0.1\le T\le 0.2$ GK temperature range but otherwise in agreement with Ferraro et al.

Figure 1

Level scheme of ${}^{23}\mathrm{Na}$. The black box in the figure corresponds to Gamow window for $T_9=0.1$.

Figure 2

Transfer angular distributions fitted with FRDWBA model calculation.

Figure 3

DWBA fit to angular distribution data of Power et al. [20] for the state $E_x=8664\mathrm{keV}$. The line represents calculated cross section

Figure 4

Variation of spectroscopic factor with single particle ANC for 8664 MeV (top panel) and 440 MeV (lower panel) states.

Figure 5

Variation of ANC [$C$ (${\mathrm{fm}}^{-1/2}$)] as a function of SPANC [$b$ (${\mathrm{fm}}^{-1/2}$)] for 440 keV (top panel) and 8664 keV (bottom panel) states.

Figure 6

Variation of ANC [$C$ (${\mathrm{fm}}^{-1/2}$)] as a function of binding energy for 8664 keV (top panel) and 6917 keV (bottom panel) states for fixed geometry parameters of the bound state potential.

Figure 7

Angular distribution data and DWBA fits to the 151 keV resonance state of ${}^{23}\mathrm{Na}$.

Figure 8

$R$-matrix fit to the $S$-factor curve for direct capture to six bound states of ${}^{23}\mathrm{Na}$. The red solid curves are the $R$-matrix fits with the contribution of background poles while the dashed curves represent the calculation without the background poles. The panel showing the $S$-factor curve for the DC $\to 8664\mathrm{keV}$ subthreshold state also includes the $R$-matrix fit (green dashed curve) with ANC value fixed from the transfer reaction calculation. The solid curve in this panel depicts the $R$-matrix fit with ANC value of $166{\mathrm{fm}}^{-1/2}$ for the state.

Figure 9

$R$-matrix fit to the data of direct capture to the ground state of ${}^{23}\mathrm{Na}$.

Figure 10

$R$-matrix fit to the data of total direct capture in ${}^{23}\mathrm{Na}$.

Figure 11

Ratio of reaction rate from present calculation to the STARLIB rate for direct and resonant captures in the ${}^{22}\mathrm{Ne}(p,\gamma ){}^{23}\mathrm{Na}$ reaction. The solid curve represents the ratio with the total capture rate of the reaction.