Reaction rates for the $s$-process neutron source ${}^{22}$Ne $+$ $\alpha$

The ${}^{22}$Ne($\alpha$,$n$)${}^{25}$Mg reaction is an important source of neutrons for the $s$-process. In massive stars responsible for the weak component of the $s$-process, ${}^{22}$Ne($\alpha$,$n$)${}^{25}$Mg is the dominant source of neutrons, both during core helium burning and in carbon-shell burning. For the main $s$-process component produced in asymptotic giant branch (AGB) stars, the ${}^{13}$C($\alpha$,$n$)${}^{16}$O reaction is the dominant source of neutrons operating during the interpulse period, with the ${}^{22}$Ne$+\alpha$ source affecting mainly the $s$-process branchings during a thermal pulse. Rate uncertainties in the competing ${}^{22}$Ne($\alpha$,$n$)${}^{25}$Mg and ${}^{22}$Ne($\alpha$,$\gamma$)${}^{26}$Mg reactions result in large variations of $s$-process nucleosynthesis. Here, we present up-to-date and statistically rigorous ${}^{22}$Ne$+\alpha$ reaction rates using recent experimental results and Monte Carlo sampling. Our new rates are used in postprocessing nucleosynthesis calculations both for massive stars and AGB stars. We demonstrate that the nucleosynthesis uncertainties arising from the new rates are dramatically reduced in comparison to previously published results, but several ambiguities in the present data must still be addressed. Recommendations for further study to resolve these issues are provided.

Figure 1

Dimensionless reduced $\alpha$-particle widths of unbound states from Ref. [36] and references therein (see also [22]). Also plotted is the Porter-Thomas distribution that best fits these data at small values. It is apparent from the figure that states with large $\alpha$-particle spectroscopic factors are not represented by the Porter-Thomas distribution. These levels most likely have an $\alpha$-particle cluster structure and would be populated preferentially in transfer measurements, such as the (${}^6$Li,$d$) measurements of Refs. [44, 56]. Also shown in green and blue are the normalized spectroscopic factors measured in Ref. [44]. The values normalized using the $E_r^{\text{lab}}=1434$ keV resonance are shown in blue, while those normalized to the $E_r^{\text{lab}}=831$ keV resonance are shown in green. Clearly, the normalizations are vastly different, and the spectroscopic factors obtained using the $E_r^{\text{lab}}=831$ keV resonance as a normalization reference appear to be too high, as shown in the figure inset, which displays the same information but on an expanded scale. See text for more detail.

Figure 2

Reaction rate probability densities for the ${}^{22}$Ne($\alpha$,$\gamma$)${}^{26}$Mg reaction at various stellar temperatures. In each panel, the red histogram represents the Monte Carlo results, while the solid line shows the log-normal approximation. Note that the solid line is not a fit to the histogram but was calculated from the log-normal parameters $\mu$ and $\sigma$ (Table ), which in turn were determined from Eq. (12). It is apparent that the log-normal approximation to the reaction rates holds in the temperature range of the $s$-process (near 0.3 GK).

Figure 3

Reaction rate probability densities for the ${}^{22}$Ne($\alpha ,n$)${}^{25}$Mg reaction. (See the caption to Fig. 2.)

Figure 4

The uncertainty bands for the ${}^{22}$Ne($\alpha ,\gamma$)${}^{26}$Mg reaction. The uncertainties are the result of upper limit resonance contributions and of resonance strength uncertainties. The color densities represent the present reaction rate probability densities normalized to our recommended rate. The thick and thin black lines represent the 68% and 95% uncertainties, respectively. The dashed blue lines represent the literature rates from Ref. [18], with the thick and thin lines denoting the recommended rate and rate limits, respectively, normalized to our recommended rate. Values below unity (dotted line) indicate that the rates are lower than the present recommended rate. The relevant temperatures for helium- and carbon-shell burning have been added as red bars with the labels “He” and “,C” respectively.

Figure 5

The probability densities for the ${}^{22}$Ne($\alpha ,n$)${}^{25}$Mg reaction in comparison to those presented by Ref. [19]. See caption of Fig. 4 for an explanation.

Figure 6

Uncertainty bands of the reaction rate ratio, $N_A{\langle \sigma v\rangle }_{(\alpha ,n)}/N_A{\langle \sigma v\rangle }_{(\alpha ,\gamma )}$. The solid (black) lines represent the present reaction rate ratio, while the dashed (blue) lines represent the ratio of rates from Ref. [19] for ${}^{22}$Ne($\alpha$,$n$)${}^{25}$Mg and from Ref. [18] for ${}^{22}$Ne($\alpha$,$\gamma$)${}^{26}$Mg. The recommended ratio (the center line in each set) was calculated by dividing the recommended ${}^{22}$Ne($\alpha$,$n$)${}^{25}$Mg reaction rate by that of the ${}^{22}$Ne($\alpha$,$\gamma$)${}^{26}$Mg reaction at each temperature. To obtain the uncertainty bands for the rate ratio, the high rate for ${}^{22}$Ne($\alpha$,$\gamma$)${}^{26}$Mg was divided by the low rate for ${}^{22}$Ne($\alpha$,$n$)${}^{25}$Mg, and vice versa. Values greater than unity indicate that more neutrons (and ${}^{25}$Mg) are produced than $\gamma$ rays (and ${}^{26}$Mg) per $\alpha$-particle capture. The temperatures relevant in helium- and carbon-shell burning are represented by red bars and are marked with “He” and “C,” respectively.

Figure 7

Ratio of final abundances resulting from the new recommended ${}^{22}$Ne($\alpha$,$\gamma$)${}^{26}$Mg and ${}^{22}$Ne($\alpha$,$n$)${}^{25}$Mg rates to those obtained from the old recommended rates. Points above unity (red line) represent a net increase in abundance. (a) At the end of core He burning in a $25M_{\odot }$ star, the most significant abundances affected by the new rates are those of ${}^{26}$Mg and the $p$-nuclei ${}^{74}$Se, ${}^{78}$Kr, and ${}^{84}$Sr. (b) For AGB stars, higher mass nuclei are produced in larger quantities, as evidenced by the “g” particle that monitors neutron captures above molybdenum.

Figure 8

Comparison of abundance variations vs mass number arising from ${}^{22}$Ne$+\alpha$ rate uncertainties presented in the literature (Ref. [18] for the ${}^{22}$Ne($\alpha$,$\gamma$)${}^{26}$Mg reaction rate and Ref. [19] for the ${}^{22}$Ne($\alpha$,$n$)${}^{25}$Mg reaction rate) and from the current ${}^{22}$Ne$+\alpha$ rate uncertainties. Abundance changes based on the present and previous rate uncertainties are shown as thick black bars and thin red bars, respectively, for (a) massive stars and (b) AGB stars.