Stellar ($n,\gamma$) cross section of $^{23}$Na

TL;DR

This study measured the neutron capture cross section of $^{23}$Na at specific energies, finding lower values than previous reports, which impacts models of stellar nucleosynthesis.

Contribution

The paper provides new, more accurate measurements of the $^{23}$Na($n,\gamma$)$^{24}$Na cross section at astrophysically relevant energies, revising previous data and improving nucleosynthesis models.

Findings

Measured cross sections at $kT=5.1$ and 25 keV are significantly lower than literature values.

Revised the radiative width of the first $^{23}$Na resonance.

Established an improved set of Maxwellian average cross sections.

Abstract

The cross section of the $^{23}$Na($n, \gamma$)$^{24}$Na reaction has been measured via the activation method at the Karlsruhe 3.7 MV Van de Graaff accelerator. NaCl samples were exposed to quasistellar neutron spectra at $kT=5.1$ and 25 keV produced via the $^{18}$O($p, n$)$^{18}$F and $^{7}$Li($p, n$)$^{7}$Be reactions, respectively. The derived capture cross sections $\langle\sigma\rangle_{\rm kT=5 keV}=9.1\pm0.3$ mb and $\langle\sigma\rangle_{\rm kT=25 keV}=2.03 \pm 0.05$ mb are significantly lower than reported in literature. These results were used to substantially revise the radiative width of the first $^{23}$Na resonance and to establish an improved set of Maxwellian average cross sections. The implications of the lower capture cross section for current models of $s$-process nucleosynthesis are discussed.

Tables

Table 1. Table 1: Sample characteristics

Sample	Diameter (mm)	Sample massa (mg)		Mass of Au foils (mg)
		NaCl	Na	upstream	downstream
Na-1	6	54.06	21.27	16.49	16.40
Na-2	8	82.67	32.52	27.92	28.47
Na-3	10	88.28	34.73	45.36	43.54
Na-4	10	232.4	91.42	43.92	44.07
Na-5	10	307.6	121.0	43.80	42.98
Na-6	15	500.1	196.7	97.76	98.47

Table 2. Table 2: Irradiation, transfer, and measuring times

Sample	$t_b$	$t_w$	$t_m$
	(h)	(min)	(h)
Na-1	16.3	34.7	8.56
Na-2	21.0	26.8	10.0
Na-3	24.7	33.0	5.00
Na-4	13.8	7.20	72.7
Na-5	17.6	3.00	21.6
Na-6	19.6	4.00	22.2

Table 3. Table 3: Decay characteristics and absolute γ 𝛾 \gamma efficiencies

Nucleus	t1/2	$E_{\gamma }$ (keV)	$I_{\gamma }$ (%)	${ϵ}_{\gamma }$ (%)
				$kT=5.1$ keV	25 keV
24Na	14.9590$\pm$0.0012 h	1368.63	100.00$\pm$0.0111Uncertainties always less than 0.05 mg.	6.94$\pm$0.10	0.048$\pm$0.001
198Au	2.69517$\pm$0.00021 d	411.80	95.58$\pm$0.12222Ref. Chu02	20.6$\pm$0.3	0.224$\pm$0.01

Table 4. Table 4: Measured ( n , γ 𝑛 𝛾 n,\gamma ) cross sections of 23 Na (in mb)

$kT=5.1$ keV		$kT=25$ keV
Sample	${\sigma }_{exp}$$\pm$syst$\pm$stat	Sample	${\sigma }_{exp}$$\pm$syst$\pm$stat
Na-4	9.03$\pm$0.28$\pm$0.26	Na-1	2.05$\pm$0.06$\pm$0.02
Na-5	8.94$\pm$0.27$\pm$0.14	Na-2	2.02$\pm$0.05$\pm$0.01
Na-6	9.33$\pm$0.29$\pm$0.14	Na-3	2.03$\pm$0.05$\pm$0.02
Average	9.10$\pm$0.28$\pm$0.11		2.03$\pm$0.05$\pm$0.01

Table 5. Table 5: Systematic uncertainties (in %) of the measured ( n , γ 𝑛 𝛾 n,\gamma ) cross section of 23 Na

Source	$kT=5.1$ keV		$kT=25$ keV
	Au	Na	Au	Na
Time and decay factors	0.5	0.5	0.6	0.5
Au cross section	2.0	N/A	1.4	N/A
Sample atoms	0.1	0.1	0.2	0.1
Self absorption	N/A	N/A	0.1	0.1
Detector peak efficiency	1.5	1.5	1.5	1.5
Intensity of $\gamma$-decay branch	0.1	0.1	0.1	0.1
Neutron flux ${\Phi }_T$	N/A	3.8	N/A	2.1
Total		3.1		2.7

Table 6. Table 6: Measured cross sections and calculated spectrum-averaged values

Neutron spectrum	Spectrum-averaged cross sections (mb)
(keV)	ENDF/B-VII.1	Experimental value	SAMMY best-fit
25.3 10-6 (thermal)	528	541$\pm$3a	542
Resonance integral	312.3	311$\pm$10b	300
5.1 (q-MB)	12.5	9.10$\pm$0.30	9.07
25 (q-MB)	2.16	2.03$\pm$0.05	2.00

Table 7. Table 7: Maxwellian averaged cross sections (mb) of the 23 Na( n , γ 𝑛 𝛾 n,\gamma ) 24 Na reaction

$kT$ (keV)	5	10	15	20	25	30	40	50	60	80	100
This work	10.1 $\pm$ 0.4	4.12	2.83	2.30	2.00 $\pm$ 0.05	1.80	1.55	1.40	1.30	1.16	1.05
KADoNiS-0.3a	14	5.2	3.4	2.7	2.2	2.1$\pm$ 0.2	1.7	1.5	1.4	1.3	1.2
ENDF/B-VII.1 DPK14	13.9	5.24	3.29	2.51	2.09	1.83$\pm$0.2	1.52	1.35	1.24	1.09	0.99

Equations

$$A={\Phi }_{tot}N\sigma f_b,$$

$$f_b=\frac{{\int }_0^{t_b}\Phi (t)e^{-\lambda (t_b-t)}dt}{{\int }_0^{t_b}\Phi (t)}dt.$$

$$C=A{K}_{\gamma }{ϵ}_{\gamma }{I}_{\gamma }{e}^{-\lambda t_w}(1-e^{-\lambda t_m}).$$

$$⟨\sigma {⟩}_{kT}=\frac{⟨\sigma v⟩}{v_T}=\frac{2}{\sqrt{\pi }}\frac{{\int }_0^{\infty }\sigma (E_n)E_n{e}^{-E_n/kT}d{E}_n}{{\int }_0^{\infty }{E}_n{e}^{-E_n/kT}d{E}_n}$$

Full text

neutron activation, stellar ($n,\gamma$) cross section, $s$ process

I Introduction

Sodium represents a neutron poison for the slow neutron capture ($s$) process, particularly in massive stars with more than about eight solar masses ($>$8 M⊙) RGB93 . The $s$ process in massive stars is particularly efficient in producing species in the mass range 60$<$A$<$90, forming the weak $s$-process component in the inventory of the solar abundances. In addition to its importance for the neutron balance in massive stars, the ($n,\gamma$) cross section of 23Na is also needed to follow the production of sodium in low and intermediate mass asymptotic giant branch (AGB) stars. In these stars, the $main$ $s$ process contributes most of the $s$ abundances in the solar system from Zr to Pb, and the $strong$ $s$-process adds to the Pb/Bi abundances at the termination point of the $s$-process reaction path GAB98 ; KGB11 .

The weak $s$ process in massive stars begins at the end of convective core He-burning ($T_{8}>2.5$), where 22Ne($\alpha,n$)25Mg operates as the principal neutron source. During that period, sodium is only marginally produced by neutron captures on 22Ne. During the subsequent convective C-shell burning, sodium is efficiently made via the 12C(${}^{12}{\rm C},p$)23Na channel although most of the protons (and sodium) are consumed by 23Na($p,\alpha$)20Ne reactions ThA85 . Nevertheless, the C-burning layers of massive stars ejected in the subsequent supernova (SN) are one of the major sources of sodium in the Galaxy WHW02 ; KUN06 , together with stellar winds from AGB stars (e.g. LMJ11 ). In the convective C-burning shell, neutrons are mainly released via 22Ne($\alpha,n$)25Mg reactions as 22Ne is present in the ashes of the convective He-burning core and $\alpha$ particles are liberated in 12C(${}^{12}{\rm C},\alpha$)20Ne reactions (e.g. RBG91a ).

In the weak $s$ process most of the neutrons are captured by abundant light isotopes, which act as neutron poisons, and only a small fraction is available for captures on iron seed nuclei to feed heavy isotope nucleosynthesis. At solar metallicity, more than 70% of the available neutrons are captured by neutron poisons in the He-burning core, and more than 90% in the C-burning shell. For this reason, it is extremely important to quantify the neutron capture rates of light isotopes such as sodium to evaluate the impact of neutron poisons in the weak $s$ process.

Another relevant source for the production of sodium are thermally pulsing low-mass (e.g., Mow99 ; GoM00 ; CPS11 ) and massive AGB stars Kar10 , where the $s$ process is related to the He shell burning stage of evolution. In a first episode, neutrons are produced by 13C($\alpha,n$)16O reactions during the interpulse phase between He shell flashes at temperatures of $T_{8}$=0.9 ($kT$=8 keV) BGW99 . The mixing of protons with the top layer of the He shell, required to provide the necessary 13C for neutron production, has the additional effect of activating the NeNa cycle in the partial mixing zone GoM00 , which then continues in the H-burning shell throughout the interpulse phase Mow99 . A second, weaker neutron exposure takes place during the He shell flash at higher temperatures of $T_{8}$=2.6 ($kT$=23 keV) when the 22Ne($\alpha,n$)25Mg source is marginally activated. As the He flash engulfs the ashes of the H burning shell, further 23Na might be produced by neutron captures on the abundant 22Ne during this second phase of $s$-processing in thermally pulsing AGB stars. Recent studies by Cristallo et al. CGS06 and Bisterzo et al. BGS10 confirm that neutron capture production of primary sodium is particularly efficient in low-mass AGB stars of low metallicity.

Intermediate-mass AGB models experience hot hydrogen burning (HBB), which modifies the Na abundance on the stellar surface depending on the attained temperature and on the interplay with the efficiency for third dredge up VeD11 ; SCP14 .

Despite of its relevance for nuclear astrophysics, the Maxwellian averaged neutron capture cross section (MACS) of 23Na is rather uncertain MAM78b . In this work, we present new experimental data for 23Na measured in quasi-stellar neutron spectra at thermal energies of $kT=5.1$ and 25 keV. Appropriate spectra have been obtained via the 18O($p,n$)18F and 7Li($p,n$)7Be reactions to simulate stellar temperature conditions relevant to $s$-process nucleosynthesis. The experimental details and results of the activation measurements are given in Secs. II and III. In Sec. IV, MACS values are derived for the full range of $s$-process temperatures on the basis of the present results. The implications of these data for the $s$-process abundances are discussed for massive stars as well as for AGB stars.

II Experiment

Similar to many light nuclei, the 23Na($n,\gamma$)24Na cross section is difficult to study experimentally given the high ratio of scattering to capture cross sections. In such cases, neutrons scattered on the sample and subsequently captured in or near the detector can induce a large background when measuring prompt capture gammas with the time of flight (TOF) technique KWG00 . These difficulties can be avoided with the activation method, because the induced activity of the product nucleus is counted only after the irradiation in a low background environment. Therefore, the activation technique is well suited to measure ($n,\gamma$) cross sections of light nuclei with greater precision than reported previously from TOF measurements.

The experiment was carried out by a series of repeated irradiations with a set of different samples and by variation of the relevant activation parameters. In this way, corrections concerning the dimensions of the samples, self absorption, and the decay during irradiations could be constrained and the determination of systematic uncertainties improved.

II.1 Samples

The samples for the individual runs were prepared from NaCl (99.99% pure) pressed into cylindrical pellets 6, 8, 10, and 15 mm in diameter with varying thicknesses. As NaCl is hygroscopic, care was taken to be sure that the water content of the material gave a negligible contribution to the mass. This was verified by heating a quantity of the NaCl at 250 ∘C for two hours, showing that the sample mass before and after heating differed by less than 0.01%.

The sodium cross section was measured relative to that of gold, which is commonly used as a reference in activation measurements. Gold foils 0.03 mm in thickness were cut to the proper diameters and fixed to the front and back of the samples during irradiation. The sample masses, as well as those of the respective gold foils, are given in Table 1.

II.2 Neutron irradiations

All neutron irradiations were performed at the Karlsruhe 3.7 MV Van de Graaff accelerator. The 7Li($p,n$)7Be reaction was utilized to obtain a thermal neutron spectrum at $kT=25$ keV BeK80 ; RaK88 ; LKM12 ; FFK12 . The target was produced by evaporation of 30 $\mu$m of lithium onto a water-cooled copper backing and was transported to the beamline in an argon atmosphere to prevent oxidation. The required proton beam energy of $E_{p}=1912$ keV was adjusted 31 keV above the reaction threshold at 1881 keV. Due to the kinematics of the 7Li($p,n$)7Be reaction, neutrons are emitted in a forward cone with an opening angle of 120∘. As the Maxwellian neutron distribution is obtained by angular integration over the entire forward cone, care was taken to be sure that the samples of varying diameters always extended through the full solid angle.

Correspondingly, the 18O($p,n$)18F reaction has been used to reproduce a thermal neutron distribution at $kT=5.1$ keV HDJ05 as shown in Fig. 1. The target consisted of a 0.2-mm-thick tantalum disk with an oxide layer about 2 $\mu$m in thickness, which was produced by electrolysis of water with 95% enrichment in 18O. The tantalum disk was glued onto a 1-mm-thick water-cooled copper backing. The glue was suited for vacuum applications and was particularly selected to obtain good heat contact with the copper backing. Also in this case, the proper proton energy of $E_{p}=2582$ keV was calibrated against the nearby reaction threshold at $E_{p}=2574$ keV, and care was taken that the sample extended through the full forward cone of the neutron beam.

The neutron flux during the runs at $kT=25$ keV was recorded by a lithium glass monitor, mounted on the beam axis at a distance of 831 mm from the target. Due to the much lower neutron yield for the irradiations at $kT=5.1$ keV, the lithium glass monitor was replaced by a more sensitive ZnS detector. The irradiation times, $t_{b}$, for each run are listed in Table 2. The proton beam current was typically 100 $\mu$A in all irradiations.

II.3 Measurement of induced activity

A high purity (HP) Ge detector of 39 cm3 was utilized to measure the induced activities in the sodium samples and gold foils following the $kT=25$ keV runs. The efficiency of the detector as a function of energy was determined with several calibrated sources to an accuracy of 1.5%. A relatively large source-detector distance of 76 mm reduced the effect of coincidence summing. To reduce natural backgrounds, the detector was shielded with 50 mm lead and 5 mm copper. The measuring times, $t_{m}$, as well as the waiting times between irradiation and the $\gamma$ counting, $t_{w}$, are listed in Table 2.

Due to the much lower neutron flux reached during the irradiations with the 18O target, the weak induced activities had to be determined with a more sensitive setup consisting of two HPGe clover detectors with the sample centered between the detectors in very close geometry as described in detail by Dababneh et al. DPA04 . Again, the photo-peak efficiency was determined using several calibrated sources. In this case, the small source-detector distance required the application of corrections for summing effects and for the sample size. These, along with self absorption corrections, were determined by Monte Carlo simulations with the GEANT4 toolkit Gea03 using a precise modeling of the apparatus. The 1369 keV $\gamma$-ray from the 24Na decay was chosen for the analysis and appeared prominently above background. Figure 2 shows the 1369 keV line from the 6 mm NaCl sample after activation in the $kT=25$ keV spectrum. It should be noted that a short-lived (20 ms) isomeric state exists in 24Na, which has a very small probability to decay directly to 24Mg without producing a 1369 keV $\gamma$-ray. Endt et al. EBF98 quote this probability to be 0.05%, thus it was considered to have a negligible effect on the present measurement.

III Analysis and Results

The process to extract a cross section from the activation data follows the approach of Ref. BeK80 . The number of activated nuclei is defined as

[TABLE]

where $\Phi_{tot}=\int\Phi(t)dt$ represents the time integrated neutron flux, $N$ is the number of sample atoms, and $\sigma$ the cross section in cm2. The correction

[TABLE]

for decays of the produced nuclei during the irradiation phase was determined from the time-dependent yield recorded by the before mentioned lithium glass or zinc sulfide monitors, which was recorded as counts per 60 s intervals. The time integrated neutron flux was determined from the activity of the gold foils. The counts registered in the HPGe detectors are expressed as a function of the induced activity,

[TABLE]

where $K_{\gamma}$ is the total correction factor for self absorption, summing, and sample size effects, $\epsilon_{\gamma}$ the peak efficiency of the HPGe detector, and $I_{\gamma}$ the absolute gamma intensity per decay of the activated nuclei. These quantities are given in Table 3.

The adopted half-lives of 24Na and 198Au were 14.9590$\pm$0.0012 h EBF98 and 2.69517$\pm$0.00021 d Chu02 , respectively.

The 197Au cross sections that are needed for the determination of $\Phi_{tot}$ were determined in the following way: The cross sections averaged over the experimental spectra as well as the respective Maxwellian average spectra at $kT=5$ and 25 keV were calculated using the evaluated gold cross section from the JEFF-3.2 data library JEF14 . These results were then corrected for the 1.1 and 1.2% differences found with respect to the recommended MACS values in the KADoNiS-1.0 compilation DPK14 . The adopted spectrum-averaged gold cross sections are $\sigma_{exp}=637\pm 13$ mb for $kT=25$ keV and $\sigma_{exp}=1922\pm 70$ mb for $kT=5.1$ keV.

The experimental results of the activations are summarized in Table 4. Statistical and systematic errors are quoted separately. The statistical errors arise from the counting of the induced activities and from the corrections applied as a result of the Monte Carlo simulations. Sources of systematic error are listed in Table 5.

IV Astrophysical Implications

IV.1 Maxwellian Averaged Cross Sections

As neutrons are quickly thermalized in a stellar environment, the stellar spectra can be described by a Maxwell-Boltzmann distribution. The MACS of an isotope exposed to such a neutron distribution is

[TABLE]

BBK00 . While the experimental spectra closely reproduce Maxwellian distributions, the small differences from a true MACS must still be accounted for. This is done by folding the experimental neutron distribution with the capture cross section provided by a database to yield a normalization, which is then applied to produce a true MACS calculated from the evaluated energy-dependent cross section. This method supplies a sufficiently accurate approximation for the MACS in the local energy region of the measurement, but is prone to increasing uncertainty in extrapolation to cover the full thermal energy region of stellar $s$-process scenarios from $kT\approx$8-90 keV.

The MACS values for 23Na at $kT=5$ and 25 keV are influenced by a broad s-wave resonance at 2.8 keV, a smaller p-wave resonance at 35 keV, and a broad p-wave resonance at 53 keV. For the present analysis, the ($n,\gamma$) cross section was modeled by means of the R-matrix code SAMMY Lar06 , starting with resonance energies and widths of all resonances up to 500 keV listed in the ENDF/B-VII.1 data library CHO11 .

In addition to the resonance terms, an s-wave direct radiative capture (DRC) component in the 23Na cross section was included by a set of optical model DRC calculations MOI95 , using the spectroscopic factors of Ref. TNB04 . The contribution of the DRC component to the measured spectrum-averaged cross section is rather small but it is important for obtaining a consistent reproduction of the thermal cross section data. In fact, a DRC component of 170 mb at thermal energy with a 1/v energy dependence is required to reproduce the thermal capture cross section as well as to match the resonance integral.

The resulting energy-dependent cross section was then folded with the experimental neutron spectra to produce cross sections that could be directly compared with the activation results. It turned out that it was sufficient to reduce only the radiation width of the resonance at 2.8 keV by 35% in order to reproduce the measured cross sections, while leaving the widths of the other resonances fixed. As shown in Table 6, the measured cross sections and the corresponding values obtained with this best-fit solution agreed to better than 2%. Without any further changes, the thermal cross section FRB14 and also the resonance integral Mug06 could be reproduced by this best fit as well.

The uncertainty of the radiative width for the 2.8 keV resonance was determined by variation of the SAMMY fit to match the uncertainty of the measured cross section values, yielding $\Gamma_{\gamma}=245\pm 8$ meV. This corresponds to an improvement by a factor of three compared to the (optimistic) 10% uncertainty quoted in the early work by Friesenhahn et al. FLF68 .

With the corrected strength of the 2.8-keV resonance, the best-fit solution for the capture cross section has been used to establish the updated set of MACS values for 23Na. The effect of the reduced and more accurate $\Gamma_{\gamma}$ value of the 2.8 keV resonance is illustrated in Table 7 by the comparison of these results with the data set in the KADoNiS v0.3 compilation DPK09 and with the evaluated cross section from the ENDF/B-VII.1 data library CHO11 , which shows a significant reduction in the MACS values especially at thermal energies below 30 keV. The effect of the revised radiative width of the 2.8 keV resonance is most pronounced for the MACS values at $kT=5$ and 10 keV, which are lower by $\sim 30$ and 20%, respectively, whereas it reduces the 25-keV MACS only by $\sim 4$%.

The uncertainties quoted for thermal energies of $kT=5$ and 25 keV could directly be taken from the experimental values, but slightly larger uncertainties have to be considered otherwise to account for the different weights of the involved resonances, which are changing with thermal energy. Below 30 keV this additional uncertainty is less than 3%, because only the first resonance had to be corrected to reproduce the 25-keV cross section. Towards higher values of $kT$, where an increasing number of resonances is contributing, this uncertainty is expected to grow and may reach 10% at $kT=100$ keV.

The correction of these MACS values for the effect of thermally excited nuclear states, the so-called stellar enhancement factor, is negligible over the entire range of $s$-process temperatures RMD11 .

IV.2 Massive stars

The $s$ process in massive stars is known to produce most of the $s$ isotopes in the solar system between Fe and Sr (see KGB11 and references therein). In the convective He core, the neutron exposure starts to increase only in the last phase, close to He exhaustion, when the temperature is high enough to efficiently burn 22Ne via 22Ne($\alpha,n$)25Mg. The 22Ne available at the end of the He core phase is given by the initial abundance of the CNO nuclei. As CNO elements are converted to 14N in the previous H-burning core, 14N is converted to 18O via the reaction channel 14N($\alpha,\gamma$)18F($\beta^{+}$)18O at the beginning of the He-burning core and then to 22Ne by $\alpha$-captures when the temperature exceeds $T_{8}$ = 2.5. At the point of He exhaustion the most abundant isotopes are 16O, 12C, 20,22Ne and 25,26Mg, where the final 12C and 16O abundances are defined by the 12C($\alpha$,$\gamma$)16O reaction. In He core conditions, 23Na is produced by the neutron capture channel 22Ne($n,\gamma$)23Ne($\beta^{-}$)23Na, and it is depleted via 23Na($n,\gamma$)24Na.

In the convective C shell the neutron exposure starts to increase during C ignition at the bottom of the shell, where neutrons are mainly produced again by the 22Ne($\alpha,n$)25Mg reaction. Typical temperatures at the bottom of the C shell are T$\approx$ 1 GK, almost constant during the major part of the shell development (e.g., TEM07 ; PGH10 ). In the last day(s) before the SN, temperatures at the base of the C shell may increase due to thermal instabilities in the deeper O-burning layers, and if the C shell is still fully convective, C-shell nucleosynthesis will be revived PGH10 . At the end of the convective C-burning shell the most abundant isotopes are 16O, 20Ne, 23Na and 24Mg. Sodium is mainly produced via the C-burning reaction 12C(12C, $p$)23Na and marginally via 22Ne($p,\gamma$)23Na and 22Ne($n,\gamma$)23Ne($\beta^{-}$)23Na. The strongest sodium depletion reaction is 23Na($p,\alpha$)20Ne, with smaller contributions from 23Na($p,\gamma$)24Mg and 23Na($n,\gamma$)24Na.

The impact of the new 23Na($n,\gamma$)24Na cross section on the weak $s$-process distribution was studied with the NuGrid post-processing code MPPNP HDF08 for a full 25 M*⊙* stellar model of solar metallicity PHW13 . The stellar structure was previously calculated using the GENEC stellar evolution code EMM08 .

By the end of the core He burning phase the $s$-process abundance distribution between 56Fe and 100Mo was found to be rather insensitive to the MACS values for 23Na($n,\gamma$)24Na. Although the MACS at 25 keV (25$-$30 keV is the temperature range of the $s$ process during core He$-$burning) is about 10% lower compared to the previous rate DPK09 , the final 23Na overabundance increases by only a few % and the effect on the $s$ abundances between Fe and Sr is limited to about 1%. This is explained by the fact that in He core conditions the 23Na production is marginal, and its abundance coupled with the low MACS implies that the neutron poisoning effect of 23Na during core He burning is low.

The final $s$-abundance distribution at the end of C shell burning between 56Fe and 100Mo obtained with the new 23Na($n,\gamma$)24Na MACS is compared in Fig. 4 with the distribution based on the previous rate DPK09 . At this point, the entire isotopic distribution is affected with variations in the order of 5%. At 90 keV thermal energy (typical for the C-burning phase) the new MACS of 23Na is lower by 13% compared to the previous rate of Bao et al. DPK09 . While the effect on the final overabundance of 23Na increases only by about 1%, the effect of 23Na as an important neutron poison in the C shell becomes evident by the propagation effect beyond iron.

Interestingly, the neutron-rich isotopes 70Zn, 76Ge, and 82Se, which are traditionally considered to be of $r$-only origin, are affected by the new MACS of 23Na as much as most of the $s$-only isotopes (e.g., 70Ge and 76Se).

The reduced neutron poison effect of the lower sodium MACS leads to an enhancement of the neutron density, thus increasing the neutron capture probability in the $s$-process branchings. By far the strongest change is obtained for the branching point at 79Se, where neutron capture on the unstable isotope 79Se becomes more probable. As a consequence, the 80Kr/82Kr ratio is reduced by about 3%. The changes in the 85Kr branching affect mostly the final abundances of 86Kr and - by the later decay of 85Kr - of 85Rb, rather than those of the related $s$-only isotopes 86,87Sr. Beyond the abundance peak around Sr, the $s$-process production in massive stars becomes marginal, and the current MACS of 23Na has a negligible effect.

In the model used in this work the C shell is not convective during the last day before the SN. In models, where the C shell stays convective, the neutron density rises from a few 1011 up to a few 1012 because all the residual 22Ne is consumed in ($\alpha,n$) reactions at the final increase of the C-burning temperature PGH10 . In this case, the higher neutron density will lead to a correspondingly larger modification of the abundance pattern in the $s$-process branchings.

It is interesting to note that the higher $s$-process efficiency found with the reduced MACS data for 23Na is partly compensated by the effect of revised ($n,\gamma$) data for the Ne HPU14 and Mg MKB12 isotopes. Accordingly, we confirm the conclusion of Heil et al. HPU14 for the weak component, i.e. that ”the reproduction of the $s$ abundances in the solar system is far from being settled.” Accordingly, further improvements of the neutron capture cross sections for heavy species along the $s$-process path and for light neutron poisons are fundamental for constraining $s$-process nucleosynthesis predictions in massive stars.

IV.3 AGB stars

There are essentially two mechanisms for sodium production in AGB stars. At solar metallicities, sodium is produced primarily during H shell burning where the mixing of protons with the He shell gives rise not only to the formation of a 13C pocket (where neutrons are produced via the 13C($\alpha,n$)16O reaction), but also to related 14N and 23Na pockets, thus activating the NeNa cycle in the latter mixing zone Mow99 ; GoM00 . Under these conditions, neutron reactions on sodium are of minor importance.

At low metallicities, however, large amounts of primary 22Ne are synthesized by conversion of primary 12C into 14N during H burning, which is then transformed during He burning by the sequence 14N($\alpha,\gamma$)18F($\beta^{+}\nu$)18O($\alpha,\gamma$)22Ne Mow99 ; GBH06 ; HGB07 . This 22Ne contributes significantly to the primary production of light isotopes, as 23Na (via 22Ne($n,\gamma$)23Ne($\beta^{-}$)23Na) and 24Mg (via 23Na($n,\gamma$)24Na($\beta^{-}$)24Mg). Accordingly, 22Ne and 23Na are - together with 12C, 14N, and 16O - major neutron poisons in the 13C pocket. As shown in the nucleosynthesis studies of Cristallo et al. CSG09 neutron captures on 22Ne account for about 50% of the total sodium production at very low metallicity ($Z=0.0001$). For higher metallicities this effect decreases and becomes negligible at [Fe/H]\geq$$-1.

With the larger neutron exposures in stars of low metallicity, which are characteristic of the strong $s$-process component, the highly abundant Ne and Na are either acting as seeds for the reaction flow (enhancing the $s$-process production up to Pb/Bi HGB07 ) or as neutron poisons, depending on the efficiency for neutron production in the 13C pocket.

The effect of the present MACS is illustrated in Fig. 5 for the case of a 1.5 M⊙ star with a $\sim$200 times lower metallicity compared to solar (Z=0.0001). For efficient 13C-pockets (e.g. for the standard 13C pocket (ST) adopted in Ref. GAB98 ) the Ne-Na abundances are acting as neutron seeds and are contributing to the $s$-process production up to Pb/Bi. With the smaller MACS for 23Na these contributions are reduced, resulting in the relative reduction of the $s$-distribution indicated in the upper panel of Fig. 5. In less efficient 13C pockets (ST/12) the role of 23Na as a neutron poison becomes dominant as shown in the lower panel. Due to the smaller MACS, more free neutrons are now available for the $s$-process and are leading to an increase of the $s$-distribution.

One of the major uncertainties of the $s$ process in low-mass AGB stars is related to the mixing mechanisms that model the 13C-pocket. A clear answer to the properties involved in such mixing, possibly resulting from the interplay between different physical processes in stellar interiors (e.g., overshooting, semi-convection, rotation, magnetic fields, see review by Herwig Her05 and Refs. SGC06 ; KaL14 ; NuB14 ) has not been reached yet, thus leaving the structure of the 13C-pocket a persisting problem.

Depending on the shape and extension of the 13C pocket, the impact of light neutron poisons may affect the $s$ distribution in different ways. Because the 13C pocket is artificially introduced in our post-process AGB models, the impact of the new 23Na MACS could be explored by adopting different shapes and sizes of the 13C pocket according to recent theoretical and observational indications. From the results obtained in these tests, the $s$-distribution was affected by less than  $\sim$5%, independent of the assumptions for the 13C pocket. Therefore, the improved accuracy of the present MACS provides significant constraints for the neutron poisoning effect of 23Na in AGB stars.

V Summary

The 23Na($n,\gamma$)24Na cross section has been measured at the Karlsruhe Van de Graaff accelerator in quasi-stellar thermal neutron spectra at $kT=5.1$ and 25 keV. The resulting Maxwellian averaged cross sections of $\langle\sigma\rangle_{\rm kT=5keV}=9.1\pm 0.3$ mb and $\langle\sigma\rangle_{\rm kT=25keV}=2.03\pm 0.05$ mb are significantly smaller compared to the recommended values of the KADoNiS-v0.3 compilation DPK09 . After reducing the radiative width of the prominent s-wave resonance at 2.8 keV by 35%, the measured cross sections were found perfectly compatible with the set of resonance data in the ENDF/B-II.1 library.

With this modification, Maxwellian averaged cross sections in the relevant range of thermal energies between $kT=5-100$ keV were derived using the energy dependence obtained by an R-matrix calculation with the SAMMY code Lar06 . The effect of the present cross section on the $s$ process abundances in massive stars (weak $s$ process) is quite small during the He core burning phase, but becomes significant during the carbon-shell burning phase where 23Na is synthesized in increasing quantities via the 12C(${}^{12}{\rm C},p$)23Na reaction. The impact of the present MACS measurement has been investigated within a massive star model. It was found that the new lower MACS causes a propagation effect over the entire weak $s$-process distribution, with a general abundance increase of about 5%.

Acknowledgements

The authors are thankful to D. Roller, E.-P. Knaetsch, W. Seith, and the entire Van de Graaff group for their support during the measurements. EU would also like to acknowledge the support of JINA (Joint Institute for Nuclear Astrophysics), University of Notre Dame, Notre Dame, IN, USA. SB acknowledges financial support from JINA (Joint Institute for Nuclear Astrophysics, University of Notre Dame, IN) and KIT (Karlsruhe Institute of Technology, Karlsruhe, Germany). MP acknowledges support from NuGrid by NSF grant PHY 09-22648 (Joint Institute for Nuclear Astrophysics, JINA), NSF grant PHY-1430152 (JINA Center for the Evolution of the Elements), and EU MIRG-CT-2006-046520. He also appreciates support from the ”Lendulet-2014” Programme of the Hungarian Academy of Sciences (Hungary) and from SNF (Switzerland). MP also acknowledges PRACE, through its Distributed Extreme Computing Initiative, for resource allocations on Sisu (CSC, Finland), Archer (EPCC, UK), and Beskow (KTH, Sweden) and the support of STFC’s DiRAC High Performance Computing Facilities; DiRAC is part of the National E-infrastructure. Ongoing resource allocations on the University of Hull’s High Performance Computing Facility - viper - are gratefully acknowledged. CL acknowledges support from the Science and Technology Facilities Council UK (ST/M006085/1).