New Experimental ²³Na(α, p)²⁶Mg Reaction Rate for Massive Star and Type Ia Supernova Models

The ${}^{23}\mathrm{Na}{\left(\alpha ,p\right)}^{26}\mathrm{Mg}$ reaction rate has been the subject of a number of nuclear astrophysics studies in the last decade. During carbon fusion in massive stars, ²³Na is produced directly by one of the main fusion channels, ¹²C ${\left({}^{12}{\rm{C}},p\right)}^{23}$ Na. Sodium made in this phase will provide the bulk of the sodium ejected by core-collapse supernovae, which are an important source of sodium in the Universe (e.g., Timmes et al. 1995; Woosley et al. 2002; Kobayashi et al. 2011; Pignatari et al. 2016; Sukhbold et al. 2016; Limongi & Chieffi 2003; Chieffi & Limongi 2013). In these conditions, the ²³Na(p,α)²⁰Ne is the main destruction channel for ²³Na. However, among other reactions ${}^{23}\mathrm{Na}{\left(\alpha ,p\right)}^{26}\mathrm{Mg}$ is also fully activated and needs to be considered to calculate stellar abundances. For instance, Iliadis et al. (2011) showed that a factor of 10 increase in the ${}^{23}\mathrm{Na}{\left(\alpha ,p\right)}^{26}\mathrm{Mg}$ reaction rate would increase the ²⁶Al production by a factor of 3, acting as a source of protons for the ²⁵Mg ${\left(p,\gamma \right)}^{26}$ Al reaction. ²⁶Al is a key radioisotope for astronomy. Its decay signature has been measured in the interstellar-medium by γ-ray telescopes COMPTEL (Chen et al. 1995) and INTEGRAL (Diehl et al. 2013), and its abundance derived from meteoritic material is used as a fundamental diagnostic to study the formation of the solar system (Lugaro et al. 2018, and references therein). The reaction has also been identified as affecting the production of ²¹Ne, ²³Na, ²⁶Mg, ²⁹Si, ⁴³Ca, and ⁴⁷Ti, in Type Ia supernovae (Bravo & Martínez-Pinedo 2012), with most of the abundances changing by a factor of 0.12–2 when the ${}^{23}\mathrm{Na}{\left(\alpha ,p\right)}^{26}\mathrm{Mg}$ reaction rate is increased or decreased by a factor of 10, and with ²⁶Mg and ⁴³Ca changing by over a factor of 2. The impact on ²³Na has also been identified in the deflagration-detonation transition supernova (DDT-SN) model of Parikh et al. (2013b), where an increase of the ${}^{23}\mathrm{Na}{\left(\alpha ,p\right)}^{26}\mathrm{Mg}$ rate by a factor of 10 yields a change in abundance by a factor of 0.47.

The reaction has been subjected to renewed experimental interest recently, due to identified weaknesses in early experimental data and uncertainties in the application of statistical Hauser-Feshbach models (Iliadis et al. 2011; Parikh et al. 2013a). There now exist three new independent measurements of the reaction cross section utilizing novel techniques to solve the issues of early experiments. Two of these experiments (Almaraz-Calderon et al. 2015; Tomlinson et al. 2015) are only capable of detecting a small angular range of the emitted protons, necessitating an assumption on the variation of the reaction's cross section with respect to angle (angular distribution) in order to calculate a total cross section for astrophysical purposes. The two measurements assumed different distributions. The third measurement (Howard et al. 2015) has sufficient angular coverage to directly measure the angular distribution of the protons. In the present paper we show that when corrected for the angular distributions presented, these measurements are all consistent to within 30% with one another in the energy range E_cm = 1.7–3.0 MeV.

The angular distributions have been obtained from the data taken at Aarhus University (Howard et al. 2015), which can be applied to the other data sets to eliminate angular distribution assumptions, reducing systematic uncertainties on their cross-section measurement, and ensuring consistency across the three measurements. These new angular-distribution-corrected measurements have been combined to obtain a new recommended experimental astrophysical reaction rate, with a significantly reduced uncertainty. The impact of this rate has been modeled in massive stars and Type Ia supernova.

A schematic of the ${}^{23}\mathrm{Na}{\left(\alpha ,p\right)}^{26}\mathrm{Mg}$ reaction is shown in Figure 1: a ²³Na ion reacts with an α particle (⁴He) producing a compound ²⁷Al nucleus in an excited state. This excited compound nucleus rapidly decays into ²⁶Mg by emitting a proton. If ²⁶Mg is produced in its ground state the protons are called p₀ protons, and if in its first excited state p₁ protons.

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The earliest measurement of the ${}^{23}\mathrm{Na}{\left(\alpha ,p\right)}^{26}\mathrm{Mg}$ reaction was by Kuperus (1964) who measured individual resonance strengths (ω γ) of narrow resonances between E_α = 1–3.3 MeV, producing ²⁶Mg in its ground state (p₀ protons). This was followed up by Whitmire & Davids (1974) who measured resonance strengths producing ²⁶Mg in its first excited state (p₁ protons) as well as the ground state. They additionally calculated a reaction rate based on their data. Because Whitmire & Davids did not reassess the p₀ resonance strengths at the energies already published by Kuperus, the combined work which incorporates both p₀ and p₁ cross sections (derived from the published resonance strengths) will be treated as one data set in the subsequent discussion. Both experiments were in forward kinematics, impinging a ⁴He beam on a NaCl target.

The experiments involved observation of individual resonances, and therefore the targets used were thin (≈25 μg cm⁻²) and measurements were taken with small increases in beam energy. Thus, the beam current of the incoming α particles was fairly high (≈400 nA) in order to collect sufficient statistics. This unfortunately led to degradation of the targets due to heating of the target and the low melting point of NaCl of 801°C. This effect was discussed by Whitmire & Davids (1974), but was not quantified, leading to a large systematic uncertainty in the data. Subsequently reaction rate compilations such as REACLIB (Cybert et al. 2010) therefore used cross sections and reaction rates predicted from statistical Hauser-Feshbach (HF) models.

A detailed sensitivity study of ²⁶Al production was performed in 2011 by Iliadis et al. (2011), which identified the ${}^{23}\mathrm{Na}{\left(\alpha ,p\right)}^{26}\mathrm{Mg}$ reaction as having a significant effect on ²⁶Al production in massive-star models (a factor of 10 in the rate affects ²⁶Al production by a factor of 3). The experimental limitations and reliance on statistical models were also highlighted, including limitations on the applicability of HF to low-mass, alpha-induced, reactions such as the ${}^{23}\mathrm{Na}{\left(\alpha ,p\right)}^{26}\mathrm{Mg}$ rate (Mohr 2015). A comparison of HF to experimental reaction rates by Parikh et al. (2013a) noted a large number of values which exceed the traditional factor of 2 reliability of HF. Based on the study, further experimental work was strongly advised.

The first modern measurement on the ${}^{23}\mathrm{Na}{\left(\alpha ,p\right)}^{26}\mathrm{Mg}$ reaction was performed by Almaraz-Calderon et al. (2014, 2015) at Argonne National Laboratory (ANL) near Chicago, USA. This measurement was performed in inverse kinematics, utilizing a ²³Na beam and a cryogenic ⁴He gas target. The use of inverse kinematics avoided the problems of target deterioration observed by Kuperus and Whitmire & Davids. Due to the nature of the experimental setup only protons emitted at angles of θ_cm ≥ 160° were measured. In order to obtain a full cross section, the angular distribution of the protons was assumed to be the same as the ²⁷Al(α, p)³⁰Si reaction. The initial results indicated cross sections much higher than expected from Kuperus and Whitmire & Davids or HF models, but these data were subsequently re-analyzed producing cross sections consistent with the HF cross sections.

Three additional experiments on the ${}^{23}\mathrm{Na}{\left(\alpha ,p\right)}^{26}\mathrm{Mg}$ reaction were performed soon after the first result, the first, by Howard et al. (2015) at Aarhus University in Denmark, in forward kinematics. By utilizing a much lower beam intensity (200–500 ppA) than Kuperus and Whitmire & Davids while also monitoring Rutherford scattered alpha particles to monitor target deterioration, the large source of uncertainty from the target stoichiometry was eliminated. This experiment was also able to directly measure the proton angular distributions, which are presented in the next section.

The second experiment, by Tomlinson et al. (2015), was performed in inverse kinematics at TRIUMF in Canada. This experiment had a very similar experimental setup to that of Almaraz-Calderon et al. (2014), but with a room-temperature target and a broader angular coverage and assuming an isotropic angular distribution. Both Howard et al. (2015) and Tomlinson et al. (2015) measured cross sections consistent with those of the NON-SMOKER HF model (Rauscher & Thielemann 2000).

The third experiment was performed by Avila et al. (2016), also at ANL. This measurement utilized an active target measurement to directly measure the total cross section of the ${}^{23}\mathrm{Na}{\left(\alpha ,p\right)}^{26}\mathrm{Mg}$ reaction, in inverse kinematics, covering an energy range of 2–6 MeV in the center-of-mass frame. The cross sections of this experiment were also consistent with those of NON-SMOKER.

The published cross sections for the five data sets are shown in Figure 2. All four recent measurements of the ${}^{23}\mathrm{Na}{\left(\alpha ,p\right)}^{26}\mathrm{Mg}$ reaction have reasonably consistent results, with Tomlinson et al. (2015) and Howard et al. (2015) agreeing within 50%. The data from Almaraz-Calderon et al. (2015) can be seen to be systematically lower by a factor 2.5 from the other two data sets. The newly analyzed angular distribution data from AU-2015 can be applied to TRIUMF-2015 and ANL-2015 to eliminate the discrepancy between the three data sets, as detailed in the following.

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The angular distributions of the emitted protons produced by the ${}^{23}\mathrm{Na}{\left(\alpha ,p\right)}^{26}\mathrm{Mg}$ reaction were measured at Aarhus University, Denmark as part of the direct measurement of the cross sections, and the experimental setup is described in detail by Howard et al. (2015). The detector array comprises of two detectors covering the laboratory angular range of 60°–120°, 140°–164°. Protons are unambiguously identified at E_cm > 1.9 MeV, while p₁ protons below this energy were only measured at 140°–164°.

For the higher-energy measurements a full angular distribution was obtained by fitting a sum of the first four even Legendre polynomials:

$$\begin{eqnarray}&&\displaystyle \frac{d\sigma }{d{\rm{\Omega }}}=a+{{bP}}_{2}(\cos (\theta ))+{{cP}}_{4}(\cos (\theta ))+{{dP}}_{6}(\cos (\theta ))\end{eqnarray} \tag{ 1 }$$

which can then be integrated over all angles to obtain a total cross section. These distributions are symmetric around 90°, which is expected for a reaction involving an intermediate compound nucleus, but a conservative uncertainty of 20% was introduced nevertheless. For the lowest energies where the p₁ protons could only be measured at high angles, an isotropic distribution was assumed. This adds a systematic uncertainty of 30% to the cross sections at E_cm < 1.9 MeV. A full account of the uncertainty budget is detailed by Howard et al. (2015).

All energies were additionally fitted to isotropic distributions using only the high-angle data. This allows evaluation of the systematic uncertainty on the low-energy points discussed above, and application of the distributions from the Howard data to the TRIUMF-2015 and ANL-2015 data, by applying the ratio of measured to isotropic cross sections to their isotropic cross sections. For ANL-2015 the total cross sections were calculated from their differential cross sections, rather than applying the angular distribution assumed in the paper (Almaraz-Calderon et al. 2015), a comparison of the effect of the different angular distributions for ANL-2015 is shown in Figure 3. All evaluated cross sections using the measured distributions (σ) and the cross sections assuming an isotropic distribution (σ^ISO) are shown in Table 1, where it can be seen that on average the cross sections are 30% higher than under an isotropic approximation. Although the angular distribution impact should depend on the incident beam energy, the thick targets employed by all the measurements averages out these effects. This can be noted by the cross-section impact having no significant energy dependence, with the exception of the point at E_cm = 1831 keV. This factor of 4.88 increase at E_cm = 1831 keV is due to a strong resonance at 1800 keV producing a nearly pure ℓ = 1 distribution that drops off strongly at angles above 120°. The angular distributions are shown in Figure 4. For the subsequent analysis an energy-independent increase of 30% over the isotropic assumption is used to correct the data from ANL-2015 and TRIUMF-2015.

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Table 1. Angle-integrated Cross Sections and Their Total Errors, Compared to Cross Sections Assuming an Isotropic Angular Distribution (σ^ISO)

Ecm (keV)	σp0 (mb)	${\sigma }_{p0}^{\mathrm{ISO}}$ (mb)	${\sigma }_{p0}/{\sigma }_{p0}^{\mathrm{ISO}}$	σp1 (mb)	${\sigma }_{p1}^{\mathrm{ISO}}$ (mb)	${\sigma }_{p1}/{\sigma }_{p1}^{\mathrm{ISO}}$
1744	0.05(1)	0.04	1.5(4)	$0.06{(}_{-2}^{+2})$	⋯	⋯
1831	0.11(2)	0.02	4.9(14)	$0.20{(}_{-6}^{+6})$	⋯	⋯
1998	0.09(2)	0.09	1.1(2)	$0.26{(}_{-6}^{+5})$	0.17	1.6(3)
2071	0.22(4)	0.17	1.3(3)	$0.56{(}_{-12}^{+11})$	0.43	1.3(3)
2139	0.34(7)	0.20	1.7(4)	$2.80{(}_{-62}^{+56})$	1.69	1.7(4)
2328	0.28(6)	0.22	1.2(3)	$1.58{(}_{-35}^{+32})$	1.30	1.2(3)
2400	0.62(12)	0.34	1.8(4)	$1.52{(}_{-33}^{+30})$	1.45	1.0(2)
2469	1.76(35)	1.19	1.5(3)	$3.05{(}_{-67}^{+61})$	2.61	1.2(2)

Note. The error budget is discussed in detail elsewhere (Howard et al. 2015).

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With the three recent cross-section measurements we obtain a new experimental reaction rate which can be used in preference to the Hauser-Feshbach (HF) reaction rate, removing the uncertainties associated with the models. Before combining the three measurements, we apply the angular distributions discussed above to Almaraz-Calderon et al. (2015) and Tomlinson et al. (2015), reducing uncertainties associated with the assumptions of angular distributions. These corrections result in a 30% increase over the isotropic distribution assumed by Tomlinson et al. (2015), and a 250% increase compared to the non-isotropic ²⁷Al ${\left(\alpha ,p\right)}^{30}$ Si angular distribution assumed by Almaraz-Calderon et al. (2015), which is substantially more than their quoted 50% systematic error. These corrected cross sections are shown in Figure 5, and full agreement within error between the three data sets is now observed. Agreement with HF at high energies is very good, but the experimental cross section deviates below HF around E_cm = 1.38 MeV due to the stricter upper limit determined by TRIUMF-2015.

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Each of the cross sections was then integrated to obtain a reaction rate using exp2rate (Rauscher 2003). This code takes as input center-of-mass energies (E_cm), their cross sections (σ), and the associated cross-section uncertainties. It then converts these cross sections into astrophysical S factors before numerically integrating to obtain the reaction rate with errors. The reaction rate is given tabulated over a range of temperatures. These three individual tabulated rates are then combined by taking a weighted average of the three. Combining the rates in this way avoids issues with the different cross-section energies measured in the three experiments. For the data point at E_cm = 1.38 MeV, only the more constrained limit by TRIUMF-2015 is taken into account, being used for the upper limit and recommended reaction rate, and a cross section of 0 at this energy is used for the lower limit of the combined reaction rate. The new experimental reaction rate and its uncertainty is shown in Figure 6, and tabulated in Table 2 The rate agrees within uncertainty with the HF predicted rate, but deviates to below the HF rate as temperatures under 1.2 GK, due to the upper limit of the TRIUMF-2015 data point at E_cm = 1.38 MeV.

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For most astrophysical models, a large reaction database such as REACLIB (Cybert et al. 2010) is used. In these databases the reaction rates are stored as coefficients to the function:

$$\begin{eqnarray}&&\lambda =\exp \left[{a}_{0}+\displaystyle \sum _{i=1}^{5}{a}_{i}{T}_{9}^{\tfrac{2i-5}{3}}+{a}_{6}\mathrm{ln}({T}_{9})\right],\end{eqnarray} \tag{ 2 }$$

which accurately captures the temperature dependence of most reaction rates. The combined tabulated reaction rates were fitted using the Computation Infrastructure for Nuclear Astrophysics (CINA) system (Smith et al. 2006), and the coefficients for Equation (2) are tabulated in Table 3

The reaction rate for the reverse reaction ²⁶Mg ${\left(p,\alpha \right)}^{23}$ Na can also be related to the ${}^{23}\mathrm{Na}{\left(\alpha ,p\right)}^{26}\mathrm{Mg}$ reaction rate, and thus can also be computed from the experimental data and the partition functions (G), which are tabulated by Rauscher & Thielemann (2000). This reverse rate was also fitted to Equation (2) using CINA. The coefficients are tabulated in Table 4, with the unspecified coefficients unchanged from Table 3.

The impact of this new experimental rate in the C-burning convective shell is investigated using the NuGrid nucleosynthesis post-processing toolkit (Pignatari & Herwig 2012). An extended network of reaction rates is used at given stellar conditions, following consistently the production and destruction of isotopic abundances over time. For a detailed description of the nuclear reaction network and reaction rates used, we refer to Pignatari et al. (2016). Iliadis et al. (2011) performed the sensitivity study of the ${}^{23}\mathrm{Na}{\left(\alpha ,p\right)}^{26}\mathrm{Mg}$ rate in two massive-star models of initial mass 25M_⊙ and 60M_⊙, respectively. Therefore, we selected two massive-star models of the same mass from the Pignatari et al. (2016) stellar set. The stellar structure is produced by GENEC (Eggenberger et al. 2008), with initial metallicity Z = 0.02. For this work, we used the software ppn to investigate the variation of isotopic abundances in single trajectories extracted from C-burning regions of the star.

The trajectories (T, ρ as a function of time) and initial abundances were taken from the deepest area of the C-burning convective shell. For the 60M_⊙ star, we extracted the trajectory at mass coordinate 3.5M_⊙, between 1.3 and 0.12 years before collapse.

The temperature ranges from 1.1 to 1.3 GK and the density 6.0 × 10⁴–8.6 × 10⁴ g cm⁻³. With this trajectory we used a network of 1100 isotopes with changes made solely to the ${}^{23}\mathrm{Na}{\left(\alpha ,p\right)}^{26}\mathrm{Mg}$ reaction and its inverse, the rest of the network setup was identical to that used by Pignatari et al. (2016).

For the impact of the new rate, ppn simulations are presented for eight rates: the original REACLIB rate, factors of 2 and 10 increase and decrease as upper and lower limits, the new experiment rate, and the upper and lower limits for this rate. The absolute differences of elemental abundances for elements with Z = 5–25 compared to REACLIB is small, as would be expected with the similar cross sections, and so the data are plotted relative to REACLIB in Figure 7, with the shaded regions corresponding to the uncertainty of the rates.

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The uncertainty on the REACLIB rates is generally expected to be anywhere from a factor of 2 to 10. STARLIB, which provides uncertainties on all the rates, gives a factor of 10 uncertainty on the ${}^{23}\mathrm{Na}{\left(\alpha ,p\right)}^{26}\mathrm{Mg}$ rate (Sallaska et al. 2013). These uncertainties correspond to a change in ²⁶Al of (+27%,−14%) for a factor of 2 and (+128%, −29%) for a factor of 10, and a change in ²³Na of (+17%, −23%) for a factor of 2 and (+34%, −70%) for a factor of 10. In contrast, the uncertainties on the present experimental rate produce a change in both ²⁶Al and ²³Na of (+7%, −7%). In general the uncertainties on the abundances of the elements is constrained to within 5% with the experimental rate, compared to approximately 20% with the NON-SMOKER rates. The impact on intermediate-mass elements is negligible. However it should be noted that the largest contribution to the ²⁶Al yield comes from explosive burning and the H-burning shells in massive stars, and so the contribution of the C convective shell to the total yield is negligible (Limongi & Chieffi 2006).

The impact of our new ${}^{23}\mathrm{Na}{\left(\alpha ,p\right)}^{26}\mathrm{Mg}$ rate was also examined using a 2D Chandrasekhar-mass deflagration-to-detonation transition Type Ia supernova explosion model (Parikh et al. 2013b). As shown in Figure 8, the ²³Na variation due to the uncertainty of our rate is within 15%, and within 5% for all other species.

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A new combined experimental reaction rate for the ${}^{23}\mathrm{Na}{\left(\alpha ,p\right)}^{26}\mathrm{Mg}$ reaction has been calculated from three recent independent measurements (Almaraz-Calderon et al. 2015; Howard et al. 2015; Tomlinson et al. 2015), and utilizing directly measured angular distributions for the observed cross section to reduce systematic uncertainty. Impact on astrophysical scenarios has been evaluated in the carbon-burning convective shell in massive stars, and in Type Ia supernovae. The rate is found to be in good agreement with the statistical reaction rate previously used, but the uncertainty has been reduced from greater than a factor of 2 to 30%. This uncertainty is dominated by uncertainties on the differential cross-section measurements themselves, and a conservative assumption on front-back asymmetry in the angular distributions which was not possible to observe. The abundance of ²³Na is increased by a factor of 1.1, and ²⁶Al is unchanged from HF predictions. Both abundances are constrained to within 7%, this is compared to to an uncertainty of (+128%, −29%) for ²³Na, and (+34%, −70%) for ²⁶Al, using the factor of 10 uncertainty given by STARLIB (Sallaska et al. 2013). While the overall yield of ²⁶Al has not been impacted by this work, the uncertainty on the ${}^{23}\mathrm{Na}{\left(\alpha ,p\right)}^{26}\mathrm{Mg}$ rate has been significantly reduced.

We acknowledge financial support from the European Research Council under ERC Starting grant LOBENA, No. 307447 and from the UK Science & Technology Facilities Council under grant No. ST/J000124/1. This work benefited from support by the National Science Foundation under grant No. PHY-1430152 (JINA Center for the Evolution of the Elements). N. Hubbard would like to expect gratitude to the University of York High Performance Computing team for providing access to YARCC (the York Advanced Research Computing Cluster) on which the NuGrid calculations were performed. M. Pignatari acknowledges significant support to NuGrid from the STFC (through the University of Hull's Consolidated Grant ST/R000840/1), and access to viper, the University of Hull High Performance Computing Facility. M. Pignatari acknowledges the support from the "Lendulet-2014" Programme of the Hungarian Academy of Sciences (Hungary) and the ERC Consolidator Grant (Hungary) funding scheme (Project RADIOSTAR, G.A. n. 424560). This article is based upon work from the ChETEC COST Action (CA16117), supported by COST (European Cooperation in Science and Technology), and the UK network BRIDGCE.

Software: ppn, mppnp, nugridpy (Pignatari & Herwig 2012), CINA (Smith et al. 2006), exp2rate (Rauscher 2003), DDT Model (Parikh et al. 2013b).