Barium Isotopic Composition of Mainstream Silicon Carbides from Murchison: Constraints for s-Process Nucleosynthesis in AGB Stars

We present barium, carbon, and silicon isotopic compositions of 38 acid-cleaned presolar SiC grains from Murchison. Comparison with previous data shows that acid washing is highly effective in removing barium contamination. Strong depletions in $\delta$($^{138}$Ba/$^{136}$Ba) values are found, down to $-$400 permil, which can only be modeled with a flatter $^{13}$C profile within the $^{13}$C pocket than is normally used. The dependence of $\delta$($^{138}$Ba/$^{136}$Ba) predictions on the distribution of $^{13}$C within the pocket in AGB models allows us to probe the $^{13}$C profile within the $^{13}$C pocket and the pocket mass in asymptotic giant branch (AGB) stars. In addition, we provide constraints on the $^{22}$Ne$(\alpha,n)^{25}$Mg rate in the stellar temperature regime relevant to AGB stars, based on $\delta$($^{134}$Ba/$^{136}$Ba) values of mainstream grains. We found two nominally mainstream grains with strongly negative $\delta$($^{134}$Ba/$^{136}$Ba) values that cannot be explained by any of the current AGB model calculations. Instead, such negative values are consistent with the intermediate neutron capture process ($i$-process), which is activated by the Very Late Thermal Pulse (VLTP) during the post-AGB phase and characterized by a neutron density much higher than the $s$-process. These two grains may have condensed around post-AGB stars. Finally, we report abundances of two $p$-process isotopes, $^{130}$Ba and $^{132}$Ba, in single SiC grains. These isotopes are destroyed in the $s$-process in AGB stars. By comparing their abundances with respect to that of $^{135}$Ba, we conclude that there is no measurable decay of $^{135}$Cs ($t_{1/2}$= 2.3 Ma) to $^{135}$Ba in individual SiC grains, indicating condensation of barium, but not cesium into SiC grains before $^{135}$Cs decayed.


INTRODUCTION
Presolar silicon carbides (SiC grains) are pristine microcrystals that condensed in carbonrich stellar winds and/or explosions (Lodders & Fegley 1995), were ejected into the interstellar medium preserving their nucleosynthetic origin, transported to the protosolar nebula, incorporated in meteorite parent bodies, and delivered to Earth in meteorites, where they were discovered over 25 years ago via their exotic isotopic signatures (Bernatowicz et al. 1987;Zinner et al. 1987;Lewis et al. 1990). Extensive studies of isotopic anomalies of light elements (A<56) in presolar SiC grains by Secondary Ion Mass Spectrometry (SIMS) confirmed that different types of SiC grains have different types of parent stars, with the majority of them (mainstream grains) originating from low-mass Asymptotic Giant Branch (AGB) stars (Hoppe et al. 1994;Zinner 2004;Clayton & Nittler 2004;Davis 2011). AGB stars are the astrophysical source of the main s-process component Arlandini et al. 1999;Bisterzo et al. 2011).
Previous isotopic measurements of heavy elements (strontium, ruthenium, zirconium, molybdenum and barium) in single mainstream grains by Resonance Ionization Mass Spectrometry (RIMS) showed clear s-process signatures (Nicolussi et al. 1997(Nicolussi et al. , 1998Savina et al. 2003aSavina et al. , 2004Barzyk et al. 2007a), providing constraints on free parameters in AGB nucleosynthesis calculations. Mainstream SiC grains condensed in the envelope outflows of AGB stars prior to solar system formation. The measurement of abundance anomalies in their isotopic compositions allows the study of s-process nucleosynthesis in individual stars at a level of precision unavailable to spectroscopic observations. Barium isotopic compositions in presolar grains have been measured in SiC aggregates (Ott & Begemann 1990;Zinner et al. 1991;Prombo et al. 1993;Jennings et al. 2002) and in single mainstream grains (Jennings et al. 2002;Savina et al. 2003a; Barzyk et al. 2007a;Marhas et al. 2007;Ávila et al. 2013). With the exception of very large mainstream grains (7-58 μm, Ávila et al. 2013), s-process barium isotopic patterns were found, although there exist systematic differences between single grain data and model predictions (Lugaro et al. 2003a). In addition, it was found that mainstream SiC grains with smaller sizes tend to contain higher barium concentrations and a barium isotopic signature more strongly enriched in s-process isotopes (Zinner et al. 1991;Marhas et al. 2007;Ávila et al. 2013), supporting the view that barium was implanted in the grains. However, the implantation model proposed by Verchovsky et al. (2004) predicts that the 'G-component' (pure s-process barium isotopes made in the helium intershell), defined by Lewis et al. (1990), should be implanted more efficiently into larger SiC grains at high energy compared to the 'N-component' (initial barium isotopes present in the convective envelope, low energy), which is the opposite of the observed trend. More importantly, the implantation scenario would have cesium coimplanted with barium into SiC grains due to their similar ionic radii, masses, and ionization potentials. The fact that there is no detectable radiogenic 135 Ba from 135 Cs decay in SiC aggregates (Lugaro et al. 2003a) challenged the implantation scenario for the refractory element barium. Barzyk et al. (2007a)

found solar system barium contamination in Murchison mainstream
SiC grains and proposed that the contamination was caused by aqueous alteration on the Murchison parent body. Clean samples are therefore a prerequisite in order to study the sizedependent trend of barium isotopic compositions in mainstream SiC grains.
While other heavy elements can suffer from severe isobaric interferences and can only be measured by RIMS, barium isotopes are the most abundant stable nuclides in the atomic mass region around A = 140 and are only interfered by the isotopes of xenon, which is present in presolar SiC in extremely low concentrations. Thus, barium isotopes can be measured using SIMS and Thermal Ionization Mass Spectrometry (TIMS). The previous barium isotopic data on SiC aggregates by TIMS and on single SiC grains by a high spatial resolution SIMS instrument, the NanoSIMS, however, suffered from potential molecular interferences, which complicated the interpretation of the results (Ávila et al. 2013). Early SIMS measurements on SiC aggregates and recent Sensitive High Resolution Ion Microprobe-Reverse Geometry (SHRIMP-RG) measurements on large mainstream SiC grains used energy filtering to suppress molecular interferences in the barium mass region (Zinner et al. 1991;Ávila et al. 2013). A few grains were analyzed using RIMS by Jennings et al. (2002) and Savina et al. (2003a). The precision of RIMS barium isotopic data obtained subsequently by Barzyk et al. (2007a) was limited by low counting statistics since they aimed to do multielement measurements sequentially in presolar grains, with barium measurements completed after much of the grain material had been consumed in measuring molybdenum and zirconium isotopic compositions. In the present work, we report barium isotopic measurements of 61 individual acid-washed presolar SiC grains from Murchison, out of which only 40 had high enough barium concentrations to give adequate counting statistics for a reliable estimate of isotopic ratios. The carbon and silicon isotopic compositions of most of these grains were subsequently measured by NanoSIMS.
One of the aims of this work was to minimize the Solar System barium contamination in Murchison mainstream SiC grains found by Barzyk et al. (2007a). We succeeded in making barium measurements in grains free from Solar System barium contamination, providing more powerful constraints for s-process nucleosynthesis in AGB stars.

EXPERIMENTAL METHODS
The SiC separation and mounting methods were described by Levine et al. (2009).
Briefly, we followed the standard procedure  to extract presolar SiC grains from the Murchison meteorite. Grains were treated additionally with concentrated H 2 SO 4 at 200°C for 12 h, HClO 4 at 195°C for 4 h, a solution of 10 M HF and 1.2 M HCl at 20°C for 12 h, and HClO 4 at 180°C for 3 h in order to remove any parent-body or terrestrial contamination. The SiC grains used in this study are from the KJG series and were separated in size by sedimentation ; they are typically 1−3 μm in diameter. Some KJG grains were deposited on a high purity gold foil from a water-isopropanol suspension and pressed into the gold foil with a sapphire disk. SiC grains were identified on the mount with secondary electron and energy dispersive X-ray images prior to RIMS analysis. Most of the grains are well separated from each other (more than 20 μm apart).
Barium isotopic compositions of the presolar SiC grains were measured on the CHARISMA instrument at Argonne National Laboratory using the experimental methods described by Savina et al. (2003aSavina et al. ( , 2003b. The two-color resonance ionization scheme used for barium in this study was that of Barzyk et al. (2007aBarzyk et al. ( , 2007b, which is different from the one used by Savina et al. (2003aSavina et al. ( , 2003b. According to the saturation curves reported in Fig. 2 of Barzyk et al. (2007b), both resonance and ionization transitions are well saturated with the beam intensities used in this study. We used a rastered ~1 μm-size UV desorption laser (351 nm) to release material with a raster size sufficient to desorb from the complete grain. Rastered areas are 10×10 to 20×20 μm in size and are thus smaller than the distances between grains in almost all cases. Postanalysis imaging with a scanning electron microscope verified that no more than one grain was analyzed at a time. Twenty-one of the 61 grains analyzed had so little barium that the desorbing laser fluence required to produce a signal was damaging the gold mount and resulted in significant backgrounds due to secondary ions.
Updates to CHARISMA since the previous studies include new Ti:sapphire lasers that produce much more powerful beams with broader bandwidth in order to diminish the effect of isotope shifts, which had previously limited the precision of RIMS measurements (Isselhardt et al. 2011). Additionally, more powerful beams suppress odd/even isotope effects and therefore yield smaller isotopic fractionation between odd and even mass isotopes (Fairbank et al. 1997;Wunderlich et al. 1992Wunderlich et al. , 1993. After RIMS analysis, carbon and silicon isotopic compositions in the 40 grains were determined with the Cameca NanoSIMS 50 at Washington University, by rastering a primary Cs + beam over each grain and simultaneously collecting secondary ions of 12 C − , 13 C − , 28 Si − , 29 Si − , and 30 Si − . One of the 40 grains was determined to be of type AB and another to be of type Z (Hoppe et al. 1994(Hoppe et al. , 1997. Of the remaining 38 grains, 24 grains were determined to be mainstream. The remaining 14 grains were completely consumed during the RIMS measurement and could not be classified by NanoSIMS; these are grouped as mainstream grains for purposes of discussion since >90 % of SiC grains are mainstream (Hoppe et al. 1994). Carbon, silicon and barium isotope ratios of the 38 mainstream and unclassified grains are reported in Table 1. All uncertainties are reported as 2σ, and include both counting errors and external reproducibility.
Carbon isotopic data are reported as ratios; silicon and barium isotopic data are given as δ-values, defined as deviation in parts per thousand from isotope ratios measured in samples relative to standards (e.g., δ 134 Ba = [( 134 Ba/ 136 Ba) grain /( 134 Ba/ 136 Ba) standard −1)×1000]. For NanoSIMS measurements of carbon and silicon isotopes, terrestrial SiC aggregates were used as standards and measured in between every ten grain measurements in order to monitor and correct for instrumental drift.  Si are chosen as the reference isotopes for carbon and silicon, respectively. The uncertainties of NanoSIMS data are calculated by including errors from both counting statistics and the overall scatter on the measured standards. For the RIMS study, terrestrial BaTiO 3 was measured as a standard on each day prior to grain measurements. In previous RIMS studies (e.g., Barzyk et al. 2007a), data uncertainties were underestimated, as only uncertainties resulted from counting statistics were taken into consideration. More reliable data are obtained in this study by using Isoplot software (Ludwig 2012) to calculate Mean Square Weighted Deviations (MSWDs) of standard measurements in order to estimate uncertainties beyond Poisson statistics related to instabilities of the instrument and laser beams.
No long-time drifting is found and MSWDs are close to unity, which demonstrates that instrumental instabilities are not a significant source of uncertainty. The error calculation equation for barium isotope ratios used in this study is given by where i Ba grain , etc. are the number of atoms counted.

RESULTS
A partial section of the chart of the nuclides in the xenon-lanthanum region is shown in Fig. 1. Although 134 Ba and 136 Ba both are pure s-process isotopes shielded by their stable xenon isobars, their relative abundances produced during AGB nucleosynthesis may deviate from the solar ratio because of the branch point at 134 Cs. The stellar βdecay rate of 134 Cs has a strong temperature dependence (Takahashi & Yokoi 1987, TY87 hereafter), increasing almost two orders of magnitude as the temperature rises to ~3×10 8 K during thermal pulses (TPs). Despite its shorter half-life at higher stellar temperature, the relatively high peak neutron density during TPs due to the marginal activation of the 22 Ne neutron source (n n = ~10 9 cm -3 , compared to 10 7 −10 8 cm -3 for 13 C neutron source) can increase the neutron capture rate of 134 Cs above its β − decay rate, such that 135 Cs production is favored over 134 Ba production. Once 135 Cs is produced, it is stable (t 1/2 = 2.3 Ma) on the timescale of the s-process in AGB stars (t ~ 20 ka, Gallino et al. 1998) and continues to undergo neutron capture to form unstable 136 Cs (t 1/2 = 13 days), almost all of which decays to 136 Ba. Thus high neutron fluxes partially bypass both 134 Ba and 135 Ba, and accumulate 136 Ba. As shown in Fig. 1, these two branches in the s-process path in the cesium-barium region join at 136 Ba, so little branching effect is seen for 137 Ba or 138 Ba (Lugaro et al. 2003a).
Barium isotopic compositions of the 38 grains, along with carbon and silicon data when available, are reported in Table 1, including 24 mainstream and 14 unclassified grains. Errors in Table 1 are given as 2σ. Seven mainstream grains had a significant number of counts of the rare p-only isotopes 130 Ba and 132 Ba (0.1 % abundance each in terrestrial barium, as shown in Fig. 1), so we summed the counts of 130 Ba and 132 Ba in order to reduce the uncertainties in δ-values. The data are reported as δ 130+132 Ba in Table 1 and are used to discuss AGB nucleosynthesis models and barium condensation into SiC grains around AGB stars.
All barium data are plotted in Fig. 2. In general, unclassified grains have relatively small error bars because we consumed the grains in their entirety and therefore had more barium counts. Nearly all of the grains have strongly negative δ 135 Ba values (< −400 ‰) with the exception of two unclassified grains (G379 and G89, which are shown as a blue triangle and a grey dot, respectively, in Fig. 2). The barium isotopic compositions of the mainstream and unclassified grains generally agree with previous studies and with AGB model predictions (see below) with a few exceptions. Two mainstream grains (G244 and G232, shown in red in Fig. 2 and highlighted in Table 1) have strongly negative δ 134 Ba values in comparison to both previous studies and AGB model predictions, and might require a different stellar source such as post-AGB stars (see discussion in Section 4.5).
We chose mainstream and unclassified grains from this and previous studies whose 2σ errors in δ 135 Ba were less than 160 ‰ (the number is chosen to include most of the grains from this and previous studies while excluding the ones with relatively large uncertainties) and plotted them in Fig. 3. This criterion was used to eliminate the effects of varying useful yields, and of different amounts of grains consumed and analyzed in these measurements. More importantly, grain data with less uncertainty allows us to derive more stringent constraints on stellar model predictions. Six of the 38 mainstream and unclassified grains from this study have 2σ errors greater than 160 ‰ and are therefore excluded in Fig. 3. Because the error is linear in the δ-value (Equation 1), larger errors can result from fewer counts and/or higher δ-values. This could introduce a selection bias against grains with higher δ-values. Only one of the six excluded grains (G89) plots significantly outside the cluster of known mainstream grains. This grain is unclassified as shown in Table 1; thus the criterion excludes grains primarily on the basis of lower barium counts and causes little or no selection bias with respect to isotope ratios.

Solar System Barium Contamination in SiC Grains
The isotopic composition of presolar SiC grains contaminated with Solar System material is indistinguishable from the 'N-component' in AGB stellar envelopes when making comparisons between grain data and model predictions based on stars starting with near-solar metallicity (initial isotopic composition variations of model predictions are within ±200 ‰, Bisterzo et al. 2011). Models with low mean neutron exposures can account for near-solar barium isotopic compositions in the absence of contamination. Barzyk et al. (2007a) did multielement/multi-isotope analysis of single SiC grains and found that five of 23 Murchison grains were contaminated with Solar System barium. Marhas et al. (2007) imaged the spatial distribution of carbon, silicon and barium signals for each presolar SiC grain with NanoSIMS and found barium-rich rims around or on the surfaces of some of the grains and therefore excluded such grains from their study. Empirically, based on the Marhas study and our work, it appears likely that mainstream grains with δ 135 Ba values above −400 ‰ are contaminated with Solar System barium. Three of the "uncontaminated" grains of Barzyk et al. (2007a) have δ 135 Ba values above −400 ‰, but all three have large analytical uncertainties (>±160 ‰).
The barium isotopic compositions of Murchison mainstream SiC grains from this study are compared with data from previous studies in Fig. 3 using the selection criterion of 2σ(δ 135 Ba) < 160 ‰ as described above. Eight of the 15 grains measured by Savina et al. (2003a;purple squares in Fig. 3) show δ 135 Ba above −400 ‰ indicating probable solar system barium contamination. In comparison, all our selected mainstream and unclassified grains except G379 show strongly negative δ 135 Ba (below −400 ‰), in good agreement with the known uncontaminated grains from previous studies. The unclassified grain, G379, was the largest grain on the mount (3×7 μm), and had almost solar barium isotopic composition (δ 135 Ba = −47±42 ‰).
We were not able to determine carbon or silicon isotope ratios for this grain and therefore cannot classify it.
Two of the nine mainstream SiC grains from Indarch meteorite in Fig. 3 have δ 135 Ba values greater than −400 ‰ (Jennings et al., 2002). Indarch is an enstatite chondrite, which has mineralogical indicators of formation in a reduced environment (Keil 1968). It was argued that Indarch SiC grains are likely to be less contaminated due to the lack of aqueous alteration on the parent body (Barzyk et al. 2007a). However, a recent paper has argued that enstatite chondrites formed under conditions similar to those of other chondrites and were then exposed to a hydrogen-poor, and carbon-, sulfur-rich gaseous reservoirs (Lehner et al. 2013 Ávila et al. (2013) reported close-to-solar barium isotopic composition for 12 large SiC grains (7−58 μm). Contamination cannot be excluded in their study, especially considering the extremely low barium concentrations in these large grains. Barium contamination in presolar SiC grains is therefore likely caused by both laboratory chemical procedures and alteration/metamorphism on the parent body. Our data indicates that the acid cleaning procedure used in this study effectively removes surface-sited terrestrial and/or parent-body contamination.

Comparison with Previous Data in Single SiC Grains
Barium isotopic compositions in mainstream SiC grains from this study are less scattered than those from previous studies due to improved stability of laser beams and the instrument. For instance, the new data tend to form a straight line on the δ 137 Ba versus δ 135 Ba plot with all values in good agreement with grain aggregate results as shown in Fig. 4a. Implantation of cesium into grains by the NanoSIMS in the Marhas et al. (2007) study resulted in interference with 134 Ba + due to the formation of 133 CsH + ions even at high vacuum conditions. If SiC grains were bombarded with a Cs + beam prior to RIMS measurements (e.g., Savina et al. 2003a), the tail of a large 133 Cs secondary ion peak that is not completely suppressed in RIMS extends to mass 134 and causes an interference. Since NanoSIMS analysis was done last in this study, there is no cesium interference in our RIMS spectra.

Torino Postprocessing AGB Model
An in-depth description of the postprocessing AGB model calculations adopted here is given by Gallino et al. (1998). The profile of the main neutron source, 13 C(α,n) 16 O, i.e., the distribution of 13 C mass fraction with mass in a one-dimensional model, is poorly constrained by theory. This is caused by several uncertainties affecting AGB stellar models, most importantly, those related to the treatment of convective instabilities. In the current postprocessing model calculations, a 13 C pocket with a decreasing distribution profile of 13 C and 14 N is therefore artificially introduced. A schematic graph of the 13 C pocket is given in Fig. 1 of Gallino et al. (1998). All the parameters of this 13 C pocket are listed in Table 2. It is subdivided into three zones (I, II and III) with fixed mass fractions of 13 C, X( 13 C) and 14 N, X( 14 N) in each zone to allow the AGB model to reproduce the s-process main component in the solar system (A>90) based on the so-called mean neutron exposure (τ 0 ) (Clayton 1968;Gallino et al. 1998). It was later shown that the nucleosynthesis predictions obtained by using zoned or constant 13 C profiles provide comparable s-process element distributions. For instance, the s-process index [hs/ls] obtained for different metallicities (Z) is almost the same for zoned and constant 13 C profiles (Busso et al. 2001). [hs/ls] is the log of the ratio of heavy-s (barium-peak) to light-s (zirconium-peak) elements, divided by the same ratio in the Solar System.
The 13 C pocket structure is fixed in the calculations and the parameters of the 13 C pocket in the standard (ST) case are given in Table 2 for reference. The naming of the ST case derives from the fact that the Solar System s-process pattern is best reproduced by averaging 1.5 M  and 3 M  AGB model yields of the ST case at half-solar metallicity (Arlandini et al. 1999 U2: 1.81). Since the total mass of the 13 C pocket is also fixed in the calculations, the values obtained by multiplying different mass fractions of 13 C and 14 N by the constant 13 C pocket mass therefore simply correspond to different amounts of 13 C and 14 N for the s-process nucleosynthesis calculations. We call this model 'Three-zone' hereafter to distinguish it from the model calculations with another 13 C profile discussed in the following sections, in which, only Zone-II contains 13 C. The current postprocessing AGB models have been updated with the most recent cross-section measurements for the entire nucleosynthesis network (see KADoNiS 1 ).
Recommended solar abundances by Lodders et al. (2009) are adopted for initial input in model calculations. For r-mostly isotopes, their initial abundances are higher than solar values (e.g., 135 Ba) in half solar metallicity calculations because of consideration of Galactic Chemical Evolution (GCE) ).

FRUITY Model
In this work, we also consider the AGB nucleosynthesis calculations from the FRUITY database (FRANEC Repository of Updated Isotopic Tables & Yields) 2 . Details of these models are given by Cristallo et al. (2009;2011). In particular, while the 13 C pocket is introduced in the calculations as a free parameter in the Torino postprocessing models, it self-consistently forms after Third Dredge-Up (TDU) episodes in the FRUITY models (see Cristallo et al. 2009 for details). Cristallo et al. (2011) pointed out that the weighted average 13 C efficiency in FRUITY is comparable to the ST case in the Three-zone postprocessing model for a Z  , 2 M  AGB star.
One significant difference between the two AGB model calculations is that the mass of the 13 C in the pocket is constant after each TDU episode in the Torino model, whereas it varies in the FRUITY calculations following the natural shrinking of the helium intershell region. Compared to the KADoNiS database used in the Torino postprocessing calculations, a list of neutron capture rates from Bao et al. (2000) is adopted in FRUITY (see Cristallo et al. 2009 for more detail). For the initial input, recommended solar abundances by Lodders (2003) are adopted in FRUITY.

Barium Isotopic Compositions of Mainstream Grains versus AGB Model Calculations
In this section, we discuss the effect of stellar masses and metallicities on barium isotope ratios in nucleosynthesis calculations of 1.5 M  to 3 M  AGB stars with close-to-solar metallicity. The 2 M  , 0.5 Z  AGB Three-zone model is chosen as representative for comparison with mainstream grain data in this study.

Effects of Mass and Metallicity of AGB Stars on Barium Isotope Ratios
Previous studies concluded that mainstream SiC grains came from AGB stars of about 1.5−3 M  with close-to-solar metallicities (Hoppe et al. 1994;Zinner 2004;Barzyk et al. 2007a).
The major effect of AGB star progenitor mass is the increasing contribution from the minor neutron source, 22 Ne(α,n) 25 Mg, with increasing initial mass. This affects the s-process isotopic pattern due to more effective activation of neutron-capture channels at various branch points (e.g., 134 Cs as discussed above; see also Käppeler et al. 2011). In Fig. 5 (Lodders & Fegley 1995).
As discussed in Straniero et al. (2003), the peak temperature within the convective zone powered by a TP depends on core mass and initial metallicity. It is also mildly affected by the erosion of the hydrogen-rich envelope caused by mass loss. For AGB stars with close-to-solar metallicity, the core mass is quite similar in the models with initial masses ranging between 1.5 and 2.5 M  . Therefore, in such AGB stars the peak temperature within the convective zone powered by a TP is similar (e.g., Straniero et al. 2003). The 2 M  , 0.5 Z  model predictions for δ 137 Ba versus δ 135 Ba ( Fig. 4a) are grouped on a nearly straight line, in agreement with the single grain and the aggregate data. The 3 M  AGB model with lower-than-solar metallicity needs to be considered separately, since it is characterized by a larger core mass and higher peak temperature at the bottom of the TPs. In Fig. 4b, the upward bending at the tail of the 3 M  model calculations for δ 137 Ba values reveals this effect. It is caused by the opening of the branching at 136 Cs at higher stellar temperature ( Fig. 1), which results in reduced 136 Ba production and therefore increased δ 137 Ba values compared to 2 M  model predictions. Because of their similar core mass, the 1.5 M  , 0.5 Z  model predictions are quite similar to 2 M  , 0.5 Z  ones, as shown in Figs. 5a, c. We will consider the predictions from the 2 M  AGB model as representative of low mass AGB stars.
The 13 C neutron source is primary; the amount of 13 C depends on the number of protons mixed into the helium intershell that are captured by primary 12 C generated in the TP by partial helium burning. In general, the higher the metallicity, the lower the core mass and, in turn, the lower the peak temperature during a TP (Straniero et al. 2003). In AGB stars of lower-than-solar metallicity, the convective envelope starts with less oxygen and thus becomes carbon-rich after fewer TPs than a solar metallicity star, as the carbon-rich phase in the 2 M  , 0.5 Z  model ( Fig.   5c) is longer (i.e., extends over more TPs) than that in the 2 M  , Z  model (Fig. 5d); this can also be seen in Fig. 6 for FRUITY predictions with different metallicities. We compared the grain data with Z  and 0.5 Z  model predictions and chose 0.5 Z  as representative because the carbon-rich phase of the 0.5 Z  model is more extended and better covers the range of the grain data. We therefore compare grain data with the Torino postprocessing model predictions of 2 M  , 0.5 Z  AGB stars, but it does not necessarily mean that all the grains only came from these AGB stars. The model predictions are shown with a range of 13 C efficiencies from D3 to U2 cases. We assume no contribution to 135 Ba from decay of 135 Cs, a subject discussed in Section 4.6.

δ 134 Ba versus δ 135 Ba
The variation of δ 134 Ba values in presolar SiC grains shown in Fig

δ 137 Ba versus δ 135 Ba
Mainstream SiC grains form a straight line in the plot of δ 137 Ba versus δ 135 Ba ( Fig. 4) while the predictions show an upward bending toward the end of AGB phase due to partial activation of 22 Ne neutron source at stellar temperatures above 3×10 8 K (Lugaro et al. 2003a), especially in 3 M  , 0.5 Z  AGB stars as shown in Fig. 4b. We calculated the linear regression line of all grain data, including mainstream, AB and Z grain, using Orthogonal Distance Regression (ODR) fit 3 in Igor software with 95 % confidence shown as grey area in Fig. 4a. The linear fitting of single SiC grains is in excellent agreement with that of SiC aggregate data whose uncertainty is negligible (Prombo et al. 1993). Despite differences between samples and techniques, the general agreement points towards a systematic offset of the model predictions.
Indeed, the discrepancy could be solved by increasing the 137 Ba cross-section by 30 % (Gallino et al. 1997). Koehler et al. (1998), however, remeasured this cross-section and confirmed the previous value. One alternative is that this discrepancy could result from the present uncertainties

δ 138 Ba versus δ 135 Ba
The neutron-magic isotope 138 Ba (N = 82) acts as a bottleneck in the s-process path due to its extremely small neutron-capture cross-section. Its abundance strongly depends on the strength of the major neutron source 13 C(α,n) 16 O during interpulse periods in AGB stars at 8 keV (~10 8 K, Gallino et al. 1998;Lugaro et al., 2003a). In Fig. 7, six out of the 61 grains measured in this study show a strong depletion of 138 Ba (< −400 ‰). As shown in Fig. 3 In this section, we (1) summarize existing observational constraints on the 13 C pocket, (2) discuss effects of 13 C pocket profile and 13 C pocket mass on model predictions of δ 138 Ba and demonstrate the necessity of a smaller 13 C pocket with a flat 13 C profile to explain δ 138 Ba < −400 ‰ measured in some of the acid-cleaned mainstream grains, and (3) explore possible physical mechanisms that could flatten a 13 C profile in the pocket.

Previous Constraints on the 13 C Pocket
Historically, three lines of evidence provided constraints on the range of mean neutron exposures in the 13 C pocket in AGB stars: the solar system s-process pattern, spectroscopic which could be due to, e.g., different initial masses (e.g., Gallino et al. 1998), or different stellar rotational velocities of AGB stars (Herwig et al. 2003;Siess et al. 2004;Piersanti et al. 2013) 4 .
Spectroscopic observations from AGB stars mainly provide elemental abundances; only a few isotopic ratios are available, such as 12 C/ 13 C and 14 N/ 15 N (e.g., Hedrosa et al. 2013 (Barzyk et al. 2007a). More recent isotopic ratios measured in clean SiC grains with high precision (see Section 2) provide a unique and much improved tool to constrain AGB model calculations.

How to Reach δ 138 Ba < −400 ‰ in AGB Model Calculations
Several branches of the s-process path in the cesium-barium region join at 136 Ba, so production of 137 Ba and 138 Ba is little affected by those branchings (Lugaro et al. 2003a). We find

Zone-II AGB Model Calculations
We did postprocessing AGB model calculations with single-zone 13 C pockets based on the Three-zone 13 C pocket in the ST case with the parameters listed in  Even lower δ 138 Ba values can be obtained by decreasing the mass of Zone-II by a factor of 2.5, from 5.2×10 −4 M  (Zone-II in Table 3) to 2.1×10 −4 M  (Zone-II_d2.5 in Table 3) as shown in Fig. 7d. As a matter of fact, ST is the only case in which δ 138 Ba for a reduced mass of Zone-II-only 13 C pocket decreases to more negative values; in the U2 and U1.3 cases, they remain almost the same, and in the D3 to D1.5 cases, the predictions are much closer to the solar system value due to less 'G-component' barium produced by the s-process in the helium intershell. Moreover, we did Zone-II calculations with a finer grid of 13 C efficiencies around the ST case to search for lower δ 138 Ba values and failed. Thus, the effect of reducing Zone-II mass cannot be compensated by varying 13 C efficiencies. All the grains can be well matched by model calculations with the reduced Zone-II mass (defined as Zone-II_d2.5 model in Table 3), except the unclassified grain G260 (Fig. 7). We therefore adopted 2.1×10 −4 M  as the lower limit for Zone-II 13 C pocket mass. Good agreement is maintained for δ 134 Ba and δ 137 Ba versus δ 135 Ba plots. Based on these calculations, a smaller 13 C pocket with a Zone-II-only 13 C profile in the 13 C efficiency range of D3−U1.3 is needed to match barium isotopic composition in all the presolar grains from this study.
As noted in Section 4.1.1, high precision isotopic data from presolar grains provide unique constraints on AGB models.  Table 4, and from 0.11 to 0.32 in the corresponding Three-zone models (not shown). The values of spectroscopic observables such as [hs/ls] are also largely unaffected by the 13 C profile within the 13 C pocket and the pocket mass; [hs/ls] values range from −0.50 to 0.62 in Zone-II models, and −0.55 to 0.35 in Three-zone models.
We also did postprocessing calculations with smaller Three-zone 13 C pockets; the results in the ST case are shown in Table 2. Similar to Zone-II models, the Three-zone models predict that δ 138 Ba values decrease with decreasing mass of the Three-zone 13 C pocket down to Three-zone_d2.5 model shown in Fig. 7b Zone-II_d2.5 models can match all the grain data for δ 138 Ba. Since it is highly likely that different 13 C pockets exist in the parent AGB stars of mainstream grains, we point out the fact that the Zone-II 13 C pockets are only required to explain the grains with δ 138 Ba < −400 ‰.

Flattening the Distribution of 13 C in the 13 C Pocket of AGB Stars
In our calculations a smaller 13 C pocket with a flat 13 C abundance profile provides better agreement with the grain data. There are several possible mixing mechanisms that could yield flat 13 C profiles. Rotation-induced mixing may lead to a partial mixing of 13 C and 14 N within a 13 C pocket (Herwig et al. 2003;Siess et al. 2004;Piersanti et al. 2013 and references therein). In calculations provide an indication that rotation may have an important influence, at present it seems not to be the mechanism responsible for δ 138 Ba values below −400 ‰, which is observed in ~10 % of the mainstream grains in this study and previous studies.
The existence of a nonnegligible magnetic field could potentially affect the shape and size of the 13 C pocket. Indeed, magnetic buoyancy has been proposed as an alternative mechanism for forming the 13 C pocket in low-mass AGB stars (Busso et al. 2012). Since this mechanism may continue to operate during the interpulse period, the resulting mixing might produce larger and more flattened 13 C pockets. Such a possibility deserves further investigation.

Effects of Flatter 13 C Pockets on Some Other s-Process Isotopes
We compared Zone-II and Three-zone model predictions in a 2 M  , 0.5 Z  AGB star for isotopes of other elements, and found that δ( 88 Sr/ 86 Sr), δ( 138 Ba/ 136 Ba) and δ( 208 Pb/ 206 Pb) at the three s-process peaks are extremely sensitive to the 13 C pocket profile and the pocket mass. The sensitivity is caused by the extremely small MACS values at 30 keV of neutron-magic 88 Sr (6.13±0.11 mb), 138 Ba (4.00±0.20 mb) and 208 Pb (0.36±0.03 mb). By comparing Zone-II model predictions to previous presolar grain data for other elements, we observe that better agreement is obtained for zirconium isotopes, δ( 92 Zr/ 94 Zr) in particular. A detailed comparison of the grain data with the Zone-II models for zirconium isotopes will be given elsewhere. Good agreement remains for molybdenum and ruthenium isotopes and for δ( 87 Sr/ 86 Sr) (Nicolussi et al. 1997(Nicolussi et al. , 1998Savina et al. 2004;Barzyk et al. 2007a). The U2 case for Zone-II calculations yields δ( 90 Zr/ 94 Zr) values that are too negative and δ 138 Ba values that are too positive to match the grain data (see Fig. 7 for Ba), so this case can therefore be safely excluded from our discussion. and from 1400 ‰ to 1050 ‰ for the ST case. We plan to measure correlated strontium and barium isotope abundances in acid-cleaned mainstream SiC grains to better constrain the nuclear and the 13 C pocket uncertainties in the future.

Extra-Mixing Processes during Red Giant Branch (RGB) & AGB Phases
The 12 C/ 13 C ratios range from 30 to 97 ( 12 C/ 13 C  = 89) in the 24 mainstream SiC grains from this study, and are plotted versus δ 135 Ba in Fig. 8. δ 135 Ba is chosen because it is relatively independent of model parameters and nuclear input uncertainties (e.g., uncertainty in the 22 Ne(α,n) 25 Mg rate) in nucleosynthesis calculations. The grain data are consistent with both model calculations (Torino postprocessing and FRUITY) except two grains with 12 C/ 13 C ~30 that cannot be matched by the model predictions during carbon-rich AGB pulses. In addition, no mainstream SiC grains with 12 C/ 13 C >100 are found in this study, although both models predict that 50 % of the stellar mass is lost when 12 C/ 13 C >100.
Mixing in AGB stellar models currently lacks an appropriate treatment along boundaries of convective regions. Of particular interest is the nonconvective mixing (extra-mixing) at the base of the convective envelope during the RGB and, possibly, the AGB phase. This mechanism should be able to mix hydrogen-burning processed material ( 13 C-rich and, eventually, 14 N-rich) with the convective envelope. When this material is transported back to the surface, lower surface 12 C/ 13 C values are attained. An extra-mixing process during the RGB phase is considered in Torino models (initial 12 C/ 13 C = 12), but not in FRUITY models (initial 12 C/ 13 C ~ 23). Neither of the two models considers an extra-mixing process during the AGB phase, such as cool bottom processing (CBP) (Nollett et al. 2003 and references therein). The RGB extra-mixing process is required to attain a 12 C/ 13 C ratio between 30 and 60 during the AGB phase. Lower values can be attained if extra-mixing is also at work during the AGB phase. Note, however, that around 10 % of mainstream SiC grains studied so far have 10 < 12 C/ 13 C < 30 (WUSTL Presolar Database 5 , Hynes & Gyngard 2009). These values could be attained by hypothesizing the activation of CBP, or alternatively, the occurrence of proton ingestion episodes at the end of the AGB phase (see Section 4.5). By introducing the RGB extra-mixing process in the FRUITY calculations, the 12 C/ 13 C ratios in 2 M  , 0.7 Z  calculations, for instance, would be reduced by roughly a factor of two, in better agreement with the grain data. In addition, only 0.4 % of mainstream SiC grains have 12 C/ 13 C >100 (WUSTL Presolar Database), while the Torino models yield larger values ( 12 C/ 13 C ~150) for close-to-solar metallicity AGB stars. However, if lower initial carbon isotope ratios are adopted ( 12 C/ 13 C = 8, still compatible with observations, e.g., Fig. 5a

Constraints on the 22 Ne(α,n) 25 Mg Rate From δ 134 Ba Values in Mainstream Grains
In this section we consider uncertainties in the nuclear inputs at the 134 (Harris 1981;Shibata et al. 2002;Koning et al. 2005). In both Torino postprocessing and FRUITY model calculations, the MACS values from Patronis et al. (2004) are adopted.
As noted above, the β − decay rate of 134 Cs is highly sensitive to stellar temperature, increasing by a factor of ~65 as the temperature rises from 1×10 8 K to 3×10 8 K during TPs (TY87 Table). The β − decay rate of 134 Cs adopted in both AGB model calculations is based on the TY87  Goriely (1999)

Effects of 22 Ne(α,n) 25 Mg Rates on δ 134 Ba Values
The experimentally determined reaction rate of 22 Ne(α,n) 25 Mg has large uncertainties at low energy due to the possibility of low-energy resonances (Wiescher et al. 2012). Different recommended 22 Ne(α,n) 25 Mg rates with different orders of magnitude uncertainties have been reported in the literature ( Caughlan & Fowler 1988: 1.86;Käppeler et al. 1994: 9.09;Angulo et al. 1999: 4.06;Jaeger et al. 2001: 2.69;Longland et al. 2012: 3.36; for T = 3×10 8 K and a rate unit of ×10 −11 cm 3 mol −1 s −1 ). For instance, the uncertainty reported by Angulo et al. (1999) is up to 60 (in the same units) for the rate at T = 3×10 8 K. Note that 22 Ne(α,n) 25 Mg is marginally activated around 3×10 8 K, which corresponds to the maximum temperature reached at the base of the convective shell generated by TPs in 2 M  AGB stars  and K94 cases, respectively, in Fig. 9; the highest δ 134 Ba value in 2×K94 LL predictions is about the same as 2×K94 case (δ 134 Ba is shifted to around −100 ‰ for the last TP). In summary, decreasing the 134 Cs β − decay rate to its lower limit shifts our constraints to ¼×K94−½×K94 rates. Due to the uncertainty in the 134 Cs β − decay rate resulting from the existence of many lowlying states with unknown log ft decay strengths (Goriely 1999), a more conservative constraint on the 22 Ne(α,n) 25 Mg rate is from ¼×K94 to K94. As the K94 rate is 4.14×10 −11 cm 3 mol −1 s −1 at 3×10 8 K, the median rate corresponds to 2.07×10 −11 cm 3 mol −1 s −1 with an uncertainty of a factor of two, which is in agreement with the experimental determination by Jaeger et al. (2001) and the recent evaluation by Longland et al. (2012). In addition, the grain data in the plot of δ 137 Ba versus δ 135 Ba in Fig. 4 also suggests a lower than the K94 rate as discussed earlier, in good agreement with the constraints from the δ 134 Ba values in grains.
The lowest δ 134 Ba value that can be achieved in Zone-II model calculations is around −100 ‰. In FRUITY model calculations, the set of 22 Ne(α,n) 25 Mg rates from Jaeger et al. (2001) is adopted. The rate reported by Jaeger et al. (2001) is a factor of two lower than the K94 rate at T = 3×10 8 K. As can be seen in Fig. 6 from the fact that the major neutron source 13 C is not completely burned during radiative conditions due to the relatively low stellar temperature at this stage; later on, the leftover 13 C burns convectively during the TP phase at higher temperature, which leads to a higher neutron density and therefore a negative δ 134 Ba value as neutron capture is favored over 134 Cs decay.
Predicted negative δ 134 Ba values in FRUITY disappear when the metallicity increases to 1.5 Z  for 2 M  because TDU efficiency decreases with increasing metallicity; less of the processed material in the helium intershell is brought up to mix with the convective envelope. In 3 M  FRUITY calculations with close-to-solar metallicities, the absence of this phenomenon is due to the effective radiative burning of 13 C during the interpulse phase because of increased stellar temperature at higher stellar mass. To conclude, equal production or overproduction of 134 Ba compared to 136 Ba is a signature of s-process nucleosynthesis in AGB stars, and two nominally mainstream grains identified in this study that lie far below the predictions (G244 and G232 in Table 1 and Figs. 6 & 9) likely do not come from AGB stars. Their possible stellar origin is discussed in the next section.

Negative δ 134 Ba Values: Signature of the i-Process in Post-AGB Stars
The intermediate neutron capture process (i-process) was first introduced by Cowan & Rose (1977) for evolved red giant stars. The typical neutron densities in the i-process are 10 15 −10 16 neutrons cm -3 with the main neutron source being 13 C(α,n) 16 O, where 13 C is formed by the ingestion of hydrogen in helium-burning conditions. Rapid burning of ingested hydrogen causes neutron fluxes higher than typical s-process densities (~10 7 −10 8 neutrons cm -3 ), but much less than required for the r-process (~>10 20 neutrons cm -3 , e.g., Thielemann et al. 2011 and references therein).
After the loss of their envelope during the AGB phase, remnant stars continue their evolution along the post-AGB track (e.g., Werner & Herwig 2006). The chemical and physical conditions of their progenitor AGB stars are therefore recorded in their surface and in the helium intershell. Based on simulation results, Iben (1984) concluded that ~10 % of the stars leaving the AGB stage undergo a VLTP and become born-again AGB stars during their post-AGB evolution.
Sakurai's object is one of the two observed objects that have undergone a VLTP with 12 C/ 13 C ≈ 4±1 on the surface (Pavlenko et al. 2004). The abundances of 28 elements of Sakurai's object have been determined by Asplund et al. (1999), and show an enhancement at the first s-process peak. With the guidance of the observational data, Herwig et al. (2011) investigated hydrogen ingestion nucleosynthesis during a VLTP event for Sakurai's object. In brief, a small amount of hydrogen remaining on the surface of an AGB star is convectively mixed into the heliumburning zone underneath to form a 13 C neutron source during the VLTP. A neutron density around 10 15 −10 16 cm -3 is generated, which is significantly higher than that in an AGB 13 C pocket, and the resulting nucleosynthesis gives rise to an elemental abundance pattern that matches that of Sakurai's object. Figure 10 shows the result for barium isotopes, along with grains G244 and G232. The model assumes that there is no significant s-process contribution during the AGB phase, in agreement with the observation of Sakurai's object. One-dimensional hydrostatic models predict that the helium intershell immediately splits into two different zones by the energy generated by the hydrogen ingested. In contrast, Herwig et al. (2011) assumed a delay of few hours for the occurrence of the split after the ingestion event, allowing the i-process to be activated to reproduce the observed anomalous abundances. The split time is therefore considered as a free parameter in the model. Two cases reproducing the general trend of Sakurai's object observations are shown in Fig. 10, with delay times of 800 and 1200 minutes respectively.
The model was developed to explain Sakurai's object with many specific assumptions regarding this star, which is not the parent star of these two grains. The discussion below is therefore only qualitative. Uncertainties in the reaction rates affecting the neutron production are also considered and presented as two cases with the split time set at 1000 minutes. The weighted average barium isotopic compositions of the helium-burning zone start at solar composition in the three-isotope plots at t = 0 and evolve to negative values with time. Unstable cesium isotopes are produced in great abundance during the rapid burning process. Assuming that grains were formed over a period of a few years, we consider that the shorter-lived isotopes of cesium, 136 Cs and 138 Cs (half-lives shown in Fig. 1), decay to barium prior to grain formation. The half-life of 134 Cs is reduced from the terrestrial value of 2.1 years to 3.8 days at 3×10  A range of 12 C/ 13 C ratios is predicted in the VLTP model calculations, which could explain the carbon isotopic compositions of the two grains as shown in Fig. 11 (model predictions of δ 134 Ba are shown as the case of complete 134 Cs decay in Fig. 10). The helium-burning zone starts with pure 12 C (the weighted average 12 C/ 13 C ratio of the zone at t = 0 is 10 8 ).
The 12 C/ 13 C ratio quickly drops to solar value (~700 min after t = 0) before the zone-split in the model calculations. The final 12 C/ 13 C ratio of the helium-burning zone is also affected by the reaction rates of 13 C(α,n) 16 O and 14 N(n,p) 14 C as shown in Fig. 11, 2× 13 C(α,n) 16 O and 2× 14 N(n,p) 14 C, respectively, but it mainly depends on the hydrodynamic properties of the hydrogen ingestion event. The

Barium in Mainstream SiC Grains: Condensation or Implantation?
The unstable nuclide 135 Cs, sitting along the main s-process path, decays to 135 Ba with a half-life of 2 Ma. Cesium is such a volatile element that it remains in the gas phase when SiC grains condense (Lodders & Fegley 1995 (Prombo et al. 1993). Compared to the aggregate measurements, a single grain study allows unambiguous determination of each grain and provides information about its formation history.
We report the δ 130+132 Ba values of seven single SiC grains in Table 1. Five grains have no carbon or silicon isotope measurement results due to complete consumption during RIMS analysis.
Barium isotopic compositions of all seven grains show s-process signatures and are all well within the range of barium isotope ratios of other mainstream SiC grains. It is therefore safe to consider these grains as mainstream for the following discussion. In Fig. 12 SiC grains contradicts the implantation scenario for barium (Verchovsky et al. 2003(Verchovsky et al. , 2004. Implantation efficiency depends on ionic radius and ionization potential. Barium and cesium have similar ionic radii and more importantly, the ionization potential of cesium is lower than that of barium, which makes cesium easier to be ionized and implanted into SiC grains compared to barium. Unless the implantation model is able to decouple barium from cesium, it is likely that barium condensed into mainstream SiC grains in the AGB circumstellar envelope. 3. Zone-II model calculations predict a large enhancement of 88 Sr, in contrast to presolar SiC grain data from Nicolussi et al. (1998). The fact that most grains in that study lie significantly closer to solar values than any of the model predictions indicates that the grains may be highly contaminated. Strontium isotope measurements in acid-cleaned grains can test the degree of strontium contamination in the previous study. Therefore, correlated strontium and barium isotope measurements in acid-cleaned grains will be done in order to better constrain the 13 C profile within the 13 C pocket and the pocket mass.