Presolar Grain Isotopic Ratios as Constraints to Nuclear and Stellar Parameters of Asymptotic Giant Branch Star Nucleosynthesis

It has been known for many years (Harris et al. 1985; Gilroy & Brown 1991) that the isotopic abundances shown by red giant photospheres can be accounted for only if nonconvective mixing mechanisms below the envelope are at play. Their nature has been an important subject of research in stellar physics in recent decades (see, e.g., Wasserburg et al. 1995; Langer et al. 1999; Denissenkov & Tout 2003; Denissenkov & VandenBerg 2003; Herwig et al. 2003; Eggleton et al. 2006, 2008; Cristallo et al. 2009; Busso et al. 2010; Charbonnel & Lagarde 2010; Nucci & Busso 2014; Cristallo et al. 2018, and references therein). Some of these mechanisms must operate also below the TDU in AGB stars in order to let protons from the envelope be mixed into the He-rich layers. They will subsequently produce ¹³C at H-burning reactivation and then neutrons, due to the ¹³C(α, n)¹⁶O reaction (Arlandini et al. 1999; Busso et al. 2001; Cristallo et al. 2009; Bisterzo et al. 2012; Trippella et al. 2014). Later (several 10⁴ yr later, in stars below about 3 M_⊙), a thermal instability occurs (thermal pulse, or TP) in the He shell that lays at the bottom of the He-rich zone below the convective envelope. This instability is due to the combined effects of the gradual compression of these layers, operated by H-shell burning, by the degeneracy of the C-O core and by the natural difficulty of maintaining stability in a thin burning shell (Yoon et al. 2004). An intermediate convective buffer then fills the He-rich region, while the temperature increases rapidly. For low-mass stars (1 ≲ M/M_⊙ ≲ 3), this growth does not achieve values significantly larger than T ≃ 3 × 10⁸ K, but in more massive red giants it can lead to quite higher temperatures (above 3.5 × 10⁸ K). Then the complementary neutron source ²²Ne(α,n)²⁵Mg can be activated: it is restricted at rather high temperature regimes because its Q value is negative (${Q}_{22}^{(\alpha ,n)}$ = −478 keV, see, e.g., Soppera et al. 2011; Adsley et al. 2021). Hence, in low-mass stars (where, as said, T ≲ 3 × 10⁸ K) its effects are not large: they mainly control the freeze-out of reaction branchings for which nuclear parameters depend on T. The cycle of ¹³C(α,n)¹⁶O and ²²Ne(α,n)²⁵Mg activation is repeated several times in AGB stars: actually, repeated exposures have been known for decades to be required to adequately feed the whole distribution from Fe to Pb (Seeger et al. 1965; Kaeppeler et al. 1990a).

In the framework outlined above, previous MHD models for mass circulation in evolved stars (Busso et al. 2007; Nucci & Busso 2014; Trippella et al. 2016; Palmerini et al. 2018) have been applied recently (Vescovi et al. 2020; Busso et al. 2021) to the production of ¹³C and then of neutron-rich elements beyond Fe through slow neutron captures. The basic properties of those recent computations can be understood with reference to Figures 1 and 2. The first one illustrates the occurrence of a TDU episode, with the underlying layers that must be affected by extra-mixing, while the second shows the peculiar distribution of protons (and later of ¹³C and ¹⁴N ) ensuing from this specific mixing scheme. Extensive discussions of the computations that lay behind the formation of reservoirs similar to the one of Figure 2 have been presented in the above-mentioned works.

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As Figure 2 shows, the new distribution has some peculiar characteristics, namely, the abundance of protons always remains very low; so low, actually, that H-burning cannot efficiently proceed to ¹⁴N because it is almost completely consumed by the ¹³C production. This is so everywhere, except in a thin top layer where ¹³C is abundant. By comparison, a purely exponential distribution is shown in Figure 2 (see panel (a)). It presents a rather wide layer in mass where protons are sufficiently abundant to synthesize considerable ¹⁴N concentrations. Since ¹⁴N is an efficient poison, or neutron absorber, the number of neutrons available to be captured by ⁵⁶Fe and its progeny is enhanced in our new models, and so is the neutron exposure per cycle, in the various pulse-interpulse cycles mentioned for AGB stars.

By contrast, in our computations the shortage of ¹⁴N drastically cuts the nuclear channels starting from nitrogen, thus reducing the production of several isotopes in the mass range of 15–30, including ¹⁸O, ¹⁹F, and ²²Ne. The reduced production of the latter, in particular, hampers the efficiency of the ²²Ne(α, n)²⁵Mg source in the TPs, leading to a type of s-processing never achieving high neutron densities, hence, only marginally feeding nuclei at the neutron-rich side of crucial branching points of the s-process, like ⁸⁶Kr, ⁸⁷Rb, ⁹⁶Zr, etc., even during the neutron flows available in the convective buffers accompanying TPs. Another consequence of the reduced ¹⁴N abundance is that the production of ¹⁹F is sharply inhibited, which fact promises to account well for its low abundance in AGB photospheres (Abia et al. 2015, 2019). This last point deserves closer scrutiny, which goes beyond the scope of this paper and is presented by Vescovi et al. (2021).

Our mixing scheme was originally built from an analogy based on the physics of magnetic flux tubes in the Sun (Busso et al. 2007). Solar magnetic structures cross the convective layer emerging from the regions in the tachocline where the dynamo mechanism has its roots and a similar behavior can be at least partially envisaged in evolved stars (Ayres et al. 1981). A substantial difference lies in the fact that stars of Classes IV, III, and II, with spectral types later than K3 stay in the Hertzsprung–Russell diagram, at the cool side of the so-called coronal dividing line (CDL; Linsky & Haisch 1979), i.e., of the border separating bluer, X-ray emitting active stars (where high-energy radiation is due to charged particles trapped in coronal loops) from redder giants not displaying evidence of a magnetized corona in space-borne X-ray observations (Haisch et al. 1991). Following Holzwarth & Schüssler (2001), the dividing line can actually be seen as a border beyond which the large convective envelopes tend to trap the buoyant flux tubes, generally hampering their outward emergence in a corona.

The above scenario is largely accepted; for our purposes, trapping and breaking of magnetic flux tubes in the envelope guarantees the mixing of nucleosynthesis products (Trippella et al. 2016; Vescovi et al. 2020; Busso et al. 2021).

Despite the lack of a real corona as a large-scale structure, observations of magnetic fields at the surface of AGB stars are numerous (see, e.g., the reviews by Vlemmings (2011, 2012), and references therein). In particular, Jordan et al. (2005) reported field values of the order of a kilogauss in central stars of planetary nebulae (PNe), while fields of lower intensity (up to 10 G) were observed through very long baseline interferometry techniques since the 1990s (Kemball & Diamond 1997). More recently, Herpin et al. (2006, 2007) reported measurements of the magnetic Zeeman effect on SiO masers at 2–10 au from the central stars, suggesting that the corresponding fields act as catalysts for dust formation and as collimators for the winds. Fields of similar intensity (from 0.6–10 G) have been observed around C-rich evolved stars (Duthu et al. 2017) and post-AGB objects (Sabin et al. 2015a, 2015b).

MHD models of a dynamo effect in AGB stars and of its consequences on mass outflows have been performed by Pascoli & Lahoche (2008, 2010) and Pascoli (2020). They suggest that, from the point of view of a distant observer, the nebula generated around an AGB star did not show signs of deformation from a spherical shape up to and through the slow superwind phase. By contrast, anisotropic structures were already developing, induced by magnetic sources; these would remain hidden in the innermost regions, creating a bipolar cavity during the early superwind phases. Then, the pre-PNe stage would begin when the fast wind emitted by the core engulfs this cavity and increases the anisotropy of the gas distribution.

The development of anisotropic structures has been suggested earlier by Rosner et al. (1991, 1995), who assumed a change in the topology of the atmospheric magnetic field from large, coherent coronal loops (that appeared at the hot side of the coronal dividing line, CDL) to an open field-line geometry where only small loops would remain (so that the star could move to the cool side of the CDL). The larger surface fraction covered by open field lines would then allow for the escape of a cool, massive stellar wind driven by Alfvén waves. Among subsequent suggestions of persisting links between magnetic activity and AGB winds, one has to mention the work by Soker & Kastner (2003). According to them, magnetic flares can occur in AGB stars, above photospheric cool regions similar to solar dark spots. Their presence locally enhanced both the wind efficiency and the dust formation. Another important confirmation on the presence of magnetic fields in low-mass stars comes from recent white dwarf (WDs) studies. Magnetism in WDs is determined via polarization measurements and the observation of characteristic distortions and shifts of spectral lines due to Zeeman effects. In the past, the estimated frequency of the occurrence of magnetic fields in WDs has been rather low because it was mainly based on discoveries made with low-resolution classification spectroscopy, which is mainly sensitive to field strengths between 1 and 100 MG. In recent years, new high-resolution spectroscopic and spectropolarimetric measurements have shown that at least 20% of WDs host magnetic fields, with typical strengths from a few kilogauss up to about 1000 MG. Recently, Bagnulo & Landstreet (2020) discovered a significant number of new WDs with strong magnetic fields (from 5–200 MG in strength) in the 20 pc volume around the solar system, highlighting again the importance of spectropolarimetry in detecting magnetic fields in these compact objects. This result is in agreement with previous work (Aznar Cuadrado et al. 2004; Kawka & Vennes 2014), confirming the suspicion that magnetism is more a common rather than a rare characteristic of WDs.

For the purposes of this work, we can then recognize that there is ample support in the literature for the existence of open magnetic structures at the AGB surface (relics of the original flux tubes survived to turbulence in the convective layers) where dust formation can occur and even become very efficient. This possibility is broadly in line with the present picture of AGB mass loss, where the traditional idea of a rate steadily increasing with time, up to the final superwind stages where most of the mass would be lost, has been substituted by a very complex scenario, characterized by a non-monotonic growth of the wind efficiency, affected by pulsation and other dynamical phenomena and in which the total mass ejected in final hot superwind stages is possibly reduced by their rather short duration (see, e.g., Höfner & Olofsson 2018, for an updated review)

In our approach, among the dynamical phenomena affecting the winds in the TDU phase there is the formation of regions in which a few magnetic structures, carrying C-rich material outward, would ultimately open, accompanying and possibly guiding gaseous and dusty winds. A schematic representation of this phenomenon is shown as a cartoon in Figure 3.

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The above scenario offers an important tool for the comparison of model results with observations. In particular, C-rich structures, breaking within the wind and mixing there with the ambient composition, could be present at any moment in the evolution of an AGB star. As the material they transport from the He-shell regions is very largely enriched in carbon (about 25% of abundance in mass) tiny traces of it remaining in the wind could be sufficient to permit formation of blobs with a C/O ratio exceeding unity, so that carbon-based dust like SiC can form, even when the general composition of the envelope is O-rich.

In order to account for the above possibility, in Section 6 and in the accompanying figures, we shall compare the isotopic composition measured in presolar SiC grains both with model envelope abundances and with those of a phase (called magnetic wind, or MW) obtained by mixing the envelope with a limited percentage (from 0.5%–5%) of material having the He-shell composition most recently transported outward by flux tubes. This would mimic the effects of magnetic flares or of other instabilities breaking the originally closed loops into open wind structures. A preliminary version of this idea has been presented earlier, in a simplified form, by Busso et al. (2021), where the composition of this C-rich wind phase was roughly approximated through the so-called G-component of the s-processed material (Zinner 1998).

Several nuclei, produced by neutron captures near the magic neutron number N = 50, remarkably depend more or less on the operation of the crucial branching points of the s-process chain at ⁸⁵Kr and at its parent ⁸⁴Kr (see Figure 4, and e.g., the paper by Walter et al. 1986). A recent discussion of this and other branching points on the main component in the light of parameterized AGB models was presented by Bisterzo et al. (2015). In the K1 repository, the 30 keV recommended cross section for ⁸⁴Kr is 33.1 mb. This compares very well with the data listed on the BNL site being close to a sort of average of them. The channel pointing to the isomeric state of ⁸⁵Kr has a branching ratio of 0.586, which means that almost 60% of the flux goes to the isomer (⁸⁵Kr^m , at 305 keV and with t_1/2 ≃ 4.5 hr). This estimate is in line with more recent evaluations by Tessler et al. (2021, submitted); it is however higher than in previous ST choices, e.g., in the previous release of KADONIS (v0.3 or K03, Dillmann et al. 2006), where this branching ratio was suggested to be about 40%. Concerning the unstable ⁸⁵Kr, its ground state has a half-life of about 10.5 yr, sufficient to effectively capture neutrons before the decay, in stellar conditions. Its cross section has only a theoretical estimate and is affected by a high uncertainty (of the order of 50%, typically). The K1 recommended value (73 ± 34 mb) represents again a sort of average of those reported in the BNL repository. A further clarification of all these points is vital because the flow through the isomeric state subsequently feeds primarily (at 80%) the channel passing through ^85,86Rb and then ending up at ^86,87Sr, at the expense of ⁸⁶Kr and ⁸⁷Rb, which feed only ⁸⁸Sr. This implies that the ratio of Sr isotopes shown by presolar grains depends not only on their own cross sections, but also on those of ^84,85Kr, on the branching ratio to ⁸⁵Kr^m and on the decay rates of unstable isotopes through which the flow passes (like ⁸⁵Kr^m itself and ⁸⁶Rb). New cross -section measurements in this mass region would also be important for the understanding of the Rb/Sr ratio in AGB stars and their relatives (see, e.g., Roriz et al. 2020, and references therein). With new facilities for measuring weak interactions in ionized plasmas only a couple of years from being finalized, it will certainly be worth fixing all these issues on experimental grounds.

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The precise values for the cross sections of the Sr isotopes that are fed by slow neutron captures (⁸⁶Sr, ⁸⁷Sr, and ⁸⁸Sr) are crucial for interpreting the isotopic admixtures of Sr itself, as well as their ratios to Ba isotopes, as measured in presolar grains (Liu et al. 2014, 2015, 2018; Stephan et al. 2018). For ⁸⁶Sr, the 30 keV recommended value is 60 mb, again in good agreement with the BNL data and only slightly lower than the previous choice (64 mb) made by K03.

For ⁸⁷Sr, the recommended value is from experimental measurements (Bauer et al. 1991) and the energy dependence is derived from the repositories mentioned above. The choice by K1 (93.8 ± 3.8 mb) is again only slightly smaller than the one of the previous K03 compilation and compatible (somewhat in the higher part of the distribution) with BNL data.

For ⁸⁸Sr, K1 recommends a renormalization of previous weighted averages, from measurements such as those by Koehler et al. (2000) and Kaeppeler et al. (1990b). However, the presence of discrepancies in the published data from the time of flight (TOF) and activation methods is noticed and this is a special case in which new experimental efforts are needed. In this respect, we notice that in the most recent measurements by Katabuchi et al. (2011), the 30 keV reference value (9.4 mb), is much larger than recommended by K1 (6.3 mb). This last datum is then slightly larger than most of those from BNL (that group around 5.2 mb). This is important especially if considered together with the uncertainties already discussed, affecting the previous branching points of the s-chain at ^84,85Kr (see above). One can in particular notice that very similar relative production factors for the isotopes of Sr can be obtained in two different ways, namely, (i) by adopting, in neutron captures on ⁸⁴Kr, the lower value of the branching ratio to ⁸⁵Kr^m (40%) and taking the neutron-capture cross section of ⁸⁸Sr from the K1 recommendations; or (ii) adopting the higher branching ratio to ⁸⁵Kr^m (60%), but then using, for the cross section of ⁸⁸Sr, the measurements by Katabuchi et al. (2011). Our ST choice here will be that of the K1 recommendations, however one has to remember that a lower production of ⁸⁸Sr is possible using either of the previously mentioned choices (whose effects are mimicked in our case v2). It is clear that on these issues an experimental clarification is urgent, as is a proper treatment of the decay rates of ⁸⁵Kr and ⁸⁶Rb in ionized plasmas.

An analysis of the possible uncertainties affecting the reproduction of Zr isotopes in SiC grains has been presented earlier by Lugaro et al. (2003), to which we refer the reader for a general assessment of the problem. In our work, for stable Zr isotopes and for the rather long-lived ⁹³Zr, the K1 recommended cross sections include the measurements of the n_TOF Collaboration (Tagliente et al. 2008a, 2008b, 2010, 2011a, 2011b, 2013). In particular, the values provided for MACS from 5–25 keV were recalculated and normalized to the experimental data at 25 keV. For the unstable ⁹⁵Zr MACS from 5–100 keV are from theoretical calculations. Uncertainties are indicated to be between 25% and 50%. The data reported by BNL are on average larger by 30%. These choices would produce a larger amount of ⁹⁶Zr and a lower contribution to ⁹⁵Mo. We shall mimic this behavior in our case V2 by using there the K1 cross section at the upper limit of its reported error bar.

From ⁹³Zr an interesting reaction branching departs, based on its β⁻decay (early recognized by Bahcall 1962, who indicated it as a source of information on the stellar physical conditions). This decay leads to ⁹³Nb via two channels, feeding directly the ground state (with 27% probability) or passing through its isomer ⁹³Nb^m (73% probably). The resulting half-life is reported in the ENSDF tables ¹⁷ to be 1.6 × 10⁶ yr, hence of the same order as the whole duration of the TP-AGB phase. The ⁹³Zr half-life does not decrease with T below about 4 × 10⁸, according to Takahashi & Yokoi (1987). During the development of TPs, the temperature quickly increases up to 280–300 MK, where ⁹³Zr is strongly produced, so that at the end of the AGB phase its abundance is between one-quarter and one-half of those for the nearby nuclei (⁹²Zr and ⁹⁴Zr). Due to its long half-life, this isotope behaves almost as a stable species during s-processing and ⁹³Nb is largely bypassed. Indeed, normal AGB stars, while showing in their spectra the unstable ⁹⁹Tc, which is shorter lived with respect to ⁹³Zr, are instead Nb-poor. On this basis one would find that AGB stars contribute minimally to the nuclei that immediately descend from ⁹³Nb itself, like the unstable ⁹⁴Nb and its decay daughter ⁹⁴Mo, whose s-process contribution was recently found to be of 1 to a few percent (Stephan et al. 2019; Busso et al. 2021). However, due to the high abundance contrast (⁹²Zr and ⁹⁴Zr are more abundant than ⁹⁴Mo in the Sun by about a factor of 10), even a small leakage from the ⁹³Zr decay to ⁹³Nb (say, 1%) would contribute significantly to the chain ⁹³Nb-⁹⁴Nb-⁹⁴Mo. In summary, although ⁹⁴Mo remains a mainly-p nucleus, it cannot be excluded that it may receive a contribution from the s-process larger than estimated by Busso et al. (2021). On the other hand, the total remaining abundance of ⁹³Zr decays to Nb at the end of the TP-AGB stage. Therefore, when the star is in a binary system exchanging mass between the components (i.e., a so-called Ba star), ⁹³Nb accumulates on the surface of the companion. When this last stage evolves to the AGB stage and undergoes s-processing, in its turn, neutron captures will occur on a material that is Nb enhanced, albeit by limited amounts (less than about 2–3 times the initial abundance), due to dilution in the envelope at the red giant stage. One of these very peculiar evolved stars, rich in both ⁹⁹Tc and ⁹³Nb, was recently observations by Shetye et al. (2020) and defined to be a bi-intrinsic AGB star. In such an object, a peculiar s-processing will occur, more effectively feeding ⁹⁴Mo. Although the effect is probably not large, due to the mentioned dilution, the s-process contribution to ⁹⁴Mo remains quite uncertain for the above reasons. This fact may affect the attempts at accounting for the Mo isotopic admixture of presolar SiC grains, where the ratio ⁹⁴Mo/⁹⁶Mo plays an important role. Experimental clarifications of the nuclear data (decay and cross sections) for this rather peculiar channel (⁹³Zr-⁹³Nb-⁹⁴Nb-⁹⁴Mo) would therefore be welcome.

A further source of uncertainty for the Zr isotopes concerns the β⁻ unstable ⁹⁵Zr. As mentioned, its (n,γ) cross section is still purely theoretical, although an experimental method to determine it (and the ones of other short-living s-process nuclei acting as branching points of the flow) was discussed by Sonnabend et al. (2004) using data of the inverse (γ,n) reactions. According to this discussion and to the one presented by Lugaro et al. (2003), improvements can be expected essentially only from new estimates of the cross section itself, as the decay rate seems to be well determined. In particular, the ground level (a 5/2⁺ state with a half-life of 64.02 days) decays to ⁹⁵Nb and then to ⁹⁵Mo with probably little to no dependence on the temperature conditions, because of the very high energy of the excited states.

The element Mo plays an important role for reconstructing the pollution of the solar system in s-process elements and its heterogeneity, as discussed by Stephan & Davis (2021). The same authors clarify how the trend of isotope ratios in presolar grains is separate from the meteoritic ones, very well defined and characteristic of s-processing environments in AGB stars. Here, the lightest Mo isotope that can receive some (marginal) contribution from s-processing in AGB stars is ⁹⁴Mo. Although we already mentioned that it is basically a p-process nucleus, in evolved low-mass stars it can be fed by neutron captures destroying ⁹²Mo, through the daughter of this last, ⁹³Mo. As it is shown by Figure 5, two channels then lead to ⁹⁴Mo from this source: a direct link via n-captures on ⁹³Mo itself (radioactive via e⁻-captures, with a half-life of 4000y), or an indirect one, through the decay product ⁹³Nb, undergoing n-captures to ⁹⁴Nb. This last then can either decay to ⁹⁴Mo (β⁻-decay, with half-life of 2·10⁴ yr) or participate to further neutron captures, feeding ⁹⁵Nb (half-life 35 days) and ⁹⁶Nb. In this respect, the channel from ⁹⁴Nb merges (on one side) with the already mentioned contribution coming from the initial ⁹³Nb abundance (and/or its refurbishing in the rare case of a bi-intrinsic AGB star). On the other side, it also starts new branching reactions on further unstable Nb isotopes. Concerning heavier Mo isotopes, we must notice that the solar system abundance of ⁹⁶Mo, which is an s-only nucleus, is reproduced very well by present s-process models (Liu et al. 2019; Busso et al. 2021). Improvements in nuclear data would therefore mainly help in clarifying details of the r and p processes, through their contributions to Mo isotopes, which are precisely predicted by presolar grain measurements (Stephan et al. 2019). The recommended cross sections of Mo isotopes from the K1 compilation almost coincide with the values listed in the BNL site (with exclusion of the unstable ⁹⁹Mo, where the BNL value, from the ENDF/B and JEFF collaborations, is higher: 481 against 366 ± 92 mb: the effects of this alternative choice is again tested in our V2 cases). The uncertainty is in any case high, as only theoretical estimates are available. For other Mo isotopes, where experimental data exist, they are however quite old, so that new measurements are required and are actually already planned by the n_TOF collaboration (Guerrero et al. 2013). For that purpose, Liu et al. (2019) stressed the importance to have precise neutron-capture cross sections as a function of the temperature even in energy regions not yet explored by activation techniques. On the other hand, a search for improvements in decay rates may concentrate on the subtle contribution to ⁹⁴Mo from ⁹⁴Nb mentioned above.

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The s-process contribution to the element Ba starts from neutron captures on the stable isotope ¹³³Cs, whose recommended cross section is indicated to be 502 ± 28 mb in K1, in agreement within the error bar with most of the values quoted at the BNL site. In nucleosynthesis models, the production of this nucleus is in its turn affected by the choice one makes for the initial abundance of Xe, whose average concentration in the solar system is unknown (Lodders 2021). When knowledge of tiny details is required, this implies some remaining ambiguity on the absolute levels of production for Ba isotopes.

As Figure 6 shows, after ¹³³Cs, the flux proceeds through a branching point at the radioactive ¹³⁴Cs, where n-captures compete mainly with β⁻decay (laboratory half-life of 2 yr) to excited states of ¹³⁴Ba and, much less effectively, with electron captures to ¹³⁴Xe (half-life of 6.8 · 10⁵ yr). From ¹³⁴Cs, neutron captures feed the longer-lived ¹³⁵Cs, whose half-life would require a specific reanalysis (see below), and then ¹³⁶Cs (half-life of 13.16 days) and ¹³⁷Cs (half-life of 30.07 yr), which are sites of branching points for the s-process path, but whose decay rates remain essentially unchanged for varying temperatures.

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This is not so for ¹³⁴Cs and ¹³⁵Cs. According to the computations (made under conditions of thermodynamic equilibrium) by Takahashi & Yokoi (1987), at 3 · 10⁸K (a temperature rather typical of TPs) the decay rate of ¹³⁴Cs is enhanced with respect to the laboratory by a factor of about 200. However, these phenomenological computations are affected by large uncertainties that could be reduced with better nuclear parameters obtained through more modern approaches. As discussed in Busso et al. (2021), when one assumes the rate of ¹³⁴Cs decay from Takahashi & Yokoi (1987), the galactic production of ¹³⁴Ba and ¹³⁶Ba (two s-only nuclei) does not fit the solar constraints very well. This point will be analyzed separately in next section, as an example of the general need for better weak interactions along the s-path. Also the half-life of ¹³⁵Cs might deserve a specific reanalysis and this would be so in particular if the decay rate of ¹³⁴Cs were to be reconsidered, with a larger production of ¹³⁵Cs itself (see the next two sections). Even following the Takahashi & Yokoi (1987) recommendations, in the temperature and density conditions prevailing in TPs, the half-life of ¹³⁵Cs would be of the order a few hundred years, thus inducing some branching effect in the s-process chain. In our computations, we consider the possibility of a reduced efficiency in the ¹³⁴Cs decay, hence of an increased importance of the ¹³⁵Cs branching, in our V2 cases.

Concerning the cross sections of Ba isotopes themselves, the K1 repository quotes for them rather small uncertainties (always less than 10%) and the Maxwellian-averaged data are either taken directly from recent measurements or renormalized according to Massimi et al. (2010), Lederer et al. (2011).

In the analysis of the nuclear parameters presented so far, we mainly concentrated on cross sections, but occasionally we met crucial decay processes whose accuracy one would like to see improved for clarifying important branching points. We consider here the example of the ¹³⁴Cs decay, affecting the isotopic admixture of Ba and the effectiveness of the branching point at ¹³⁵Cs. As mentioned, the two stable nuclei more heavily affected by the decay at ¹³⁴Cs are ¹³⁴Ba and ¹³⁶Ba, whose status as purely s-process nuclei is not reproduced very well by galactic neutron-capture nucleosynthesis, when the decay rate is taken from Takahashi & Yokoi (1987). See also, for this, Prantzos et al. (2018, 2020).

An illustration of this uncertain behavior is attempted in Figure 7. It shows the production factors from a model AGB star of 2 M_⊙, for a metallicity slightly lower than solar, which represents a sort of average model for galactic s-processing, in the sense that it naturally generates s-elements in solar proportions (see discussion of this point in Trippella et al. 2016; Busso et al. 2021). In particular, the figure shows purely s-nuclei in the atomic mass range from 125 to 150. They are displayed in logarithmic scale, normalized to the average production factor of s-nuclei, which is indicated on the plot. For the computations of the first panel (left) the nuclear parameters of the ST choice discussed above were adopted and it is clear that in such a case there is a discrepancy between ¹³⁴Ba and ¹³⁶Ba that is on average larger than for nearby s-nuclei. To level their production (right panel) one needs to assume that the temperature dependence of the decay rate for ¹³⁴Cs is less steep than suggested by Takahashi & Yokoi (1987) by a rather large factor (in the illustrative example shown, we changed it by a factor of 8). This confirms the finding by Cristallo et al. (2015), who obtained a 5% decrease of ¹³⁴Ba abundance by decreasing the ¹³⁴Cs decay by a factor of three. We notice that a similar leveling would have also been possible by changing upward the cross section of ¹³⁴Cs by more than a factor of two. Although the accepted value is from theoretical calculations, the possibility of such a large variation seems today unlikely.

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Very recently a study by Simonucci et al. (2021, to be submitted) confirmed that the temperature dependence of the ¹³⁴Cs β-decay rate should indeed be less steep than so far assumed. In their approach, the Dirac–Hartree–Fock equations were solved for both the electron phase and the nucleus, modeling the nucleon-nucleon interaction through a relativistic one-body Wood-Saxon potential and factorizing the hadronic and leptonic currents as two non-interacting parts, also including the contributions from bound-state decays. Figure 8 shows that in the temperature range relevant for He-burning in the AGB stages, the new half-life, as obtained by the authors, is longer than in Takahashi & Yokoi (1987) by a factor ranging from 2.5 (near 8–10 keV) to more than 30 (at 30 keV). In particular, the red line includes the contribution to the half-life of ¹³⁴Cs of the atomic electrons, which populate the atomic orbitals according to a Fermi–Dirac (FD) distribution function (Morresi et al. 2018), while in the blue curve the electrons are clamped down to the electronic ground state (zero-temperature, close to laboratory conditions). In the latter case, the half-life is indeed higher as escaping β⁻electrons are prevented by Pauli's exclusion principle from occupying bound orbitals (that are fully occupied) and can only be emitted in the continuum.

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Here one is forced to wait for future measurements of the decay rate in ionized plasmas, hoping that these calculations and the previously mentioned astrophysical suggestions can find in that way an experimental confirmation.

For the scope mentioned in the previous subsection we analyzed in some detail the real possibility that a variation in the rate of the β⁻decay of ¹³⁴Cs and of other radionuclei by factors similar to the one inferred by the new calculations there shown can be measured in ionized plasma environments.

In this framework, the PANDORA project (Plasmas for Astrophysics,Nuclear Decay Observation and Radiation for Archaeometry; Mascali et al. 2017, 2020) aims to measure for the first time β-decay rates in controlled ionized plasmas made of radionuclides, at different plasma densities and temperatures. The innovative PANDORA plasma trap (presently under construction at LNS-INFN) will be able to produce and confine plasmas with electron-ion densities up to 10¹³ cm⁻³ and electron temperatures T_e in the range 0.1–30 keV. These conditions would mimic some important aspects of stellar environments, in particular the charge states. More in detail, the experimental approach consists in determining a direct correlation of the plasma properties with the nuclear decay itself. This will be achieved by simultaneously identifying and discriminating, via a multi-diagnostic system, the γ-ray products following β-decays from unstable isotopes and the photons self-emitted by the plasma (Naselli et al. 2019, 2020). PANDORA has been designed in order to maintain the radionuclides in a dynamical equilibrium for several days or even weeks. Indeed, simulations of the γ-decay detection efficiency (as a function of the radionuclide lifetime and of the effective activity in the plasma volume) show that the minimum measurement time needed to achieve a sufficiently accurate measure (≥ 3 σ) ranges from hours to several days, depending on the isotope under investigation.

Virtual experiments have been performed to study the feasibility of measuring the decay rates of several nuclei in the PANDORA trap and to predict possible enhancements of weak-interaction rates in stellar environments. Among the cases of study there are several nuclei involved in s-process nucleosynthesis, ⁹³Zr, ⁹⁴Nb and ¹³⁴Cs in particular. The charge state distribution in the plasma of these isotopes was investigated, starting from a "stellar-like" scenario and down to the PANDORA density range, by using the FLYCHK (Chung et al. 2005) population kinetics code (Chung et al. 2005). This was done in both local thermodynamic equilibrium (LTE) and non-LTE conditions. FLYCHK can estimate the probability for each ion stage to be populated, together with the mean charge state of the plasma species, according to the atomic-level population distribution and the plasma thermodynamics. Once the radionuclide CDSs have been modeled in non-LTE conditions, the in-plasma β − decay rates of ⁹³Zr, ⁹⁴Nb and ¹³⁴Cs can be estimated. This is illustrated in Figure 9, which shows the behavior of the $\mathrm{log}({\tau }^{* }/{\tau }^{0})$ (namely the log-ratio between the lifetime in stellar conditions and the one in terrestrial conditions τ⁰) for the nuclei studied, as a function of the plasma density ρ_e and of the thermal energy k_B T_e . Calculations show that the plasma temperature affects largely the isotope decay rates and that temperature effects dominate over those related to density. This finding strengthens the credibility of the half-lives in stellar conditions that will be estimated from experimental data taken by PANDORA in its operating density range. The results of this analysis will be discussed in detail for a larger sample of radionuclides in a forthcoming paper.

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In general, the observed properties of the stars becoming C-rich during their AGB evolution match well those expected in the range 1 to 3 M_⊙ (Abia et al. 2020), where the lowest masses pertain to older objects, dynamically distributed similarly to the old disk population (Claussen et al. 1987). There is also a tail for more massive members (up to maybe 4 or 5 M_⊙), which can be identified with very red objects, belonging to the thin disk, as early noticed by Barnbaum et al. (1991). In this rather wide range, the AGB stars that are the main parents of presolar grains have been recently suggested to be about 2 M_⊙, with metallicities close to the solar one (Cristallo et al. 2020). This suggestion, especially for what concerns the metallicity, is in accordance with the ages of presolar SiC grains recovered from the Murchison chondrite, which have been estimated by Heck et al. (2020), based on the cosmogenic ²¹Ne produced inside those solids, from ²¹Na decay. Most (60%) of the grains turned out to be older than the solar system by less than 300 Myr, with a minority showing ages up to 3 Gyr. According to recent estimates of the relations between age and abundances for various representative elements in the galactic disk (Feuillet et al. 2019; Nissen et al. 2020; Sharma et al. 2020) we expect a rather small spread of metallicities across the above-mentioned age span at the galactic radius of the solar system. For these reasons, and in agreement with Cristallo et al. (2020), our comparisons between model predictions and observed isotopic ratios in SiC grains will be restricted to a small spread, −0.15≲ [Fe/H] ≲ 0.1. In those comparisons, the mentioned suggestions lead us to expect that stellar models of about 2 M_⊙ explain most of the SiC-grain constraints.

SiC crystals form the most abundant sample of presolar grains and are among the most precise tools available to constrain s-process nucleosynthesis and the ensuing envelope enrichment in AGB stars. Some dust particles of AGB origin were captured in pristine meteorites, showing us the isotopic composition of the stellar winds where they were formed. The laboratory measurements of the isotopic admixtures for trace heavy elements in such SiC grains offer then diagnostic tools for the composition of evolved red giants that cannot be obtained from their stellar spectroscopy, which is hampered on one hand by variability of the stellar sources and on the other by restriction to elemental abundances in most cases.

Among the families of presolar SiC grains so far recovered, the largest one (more than 16,500 grains, according to Stephan et al. 2020) is that of Main-Stream (hereafter MS) grains. Their isotopic ratios of C, N, and Si, along with traces of s-elements measured in a subset of them, certify their AGB origins (Zinner 2014). So far, Sr, Zr, Mo, or Ba isotopic ratios have been measured only in 214 MS-SiC grains out of the several thousands available in the database maintained at the Washington University in St. Louis (Hynes & Gyngard 2009). For none of the grains the isotopic ratios were measured for all the 4 elements. However, in most cases data about the isotopic mix of Ba and at least of one or two of the other elements are available. Many authors determined the isotopic composition for smaller subsets of the 214 grains using different techniques, e.g., Resonance Ionization Mass Spectrometry or Nano-Secondary Ion Mass Spectrometry.

The results are reported in several papers (Nicolussi et al. 1997, 1998; Jennings et al. 2002; Savina et al. 2003; Barzyk et al. 2007; Marhas et al. 2007; Liu et al. 2014, 2015, 2018; Stephan et al. 2018, 2019). Despite this fragmented scenario, data from the different samples show common trends for the measured values of the isotopic ratios, as shown in Figure 10. There (as well as in any reference to isotopic ratios presented in this work) the data are shown in the so-called δ notation, where, for example:

$$\begin{eqnarray}&&\delta \left(\displaystyle \frac{{}^{135}\mathrm{Ba}}{{}^{136}\mathrm{Ba}}\right)={10}^{3}\cdot \left[\left(\displaystyle \frac{{}^{135}\mathrm{Ba}}{{}^{136}\mathrm{Ba}}\right)/{\left(\displaystyle \frac{{}^{135}\mathrm{Ba}}{{}^{136}\mathrm{Ba}}\right)}_{\odot }-1\right]\end{eqnarray} \tag{ 1 }$$

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The mentioned concordant collective behaviors are maintained despite the different techniques adopted and the different research groups involved, thus revealing a remarkable strength in the constraints posed by MS-SiC data to s-process nucleosynthesis. In addition to the MS samples, the cited authors determined the isotope ratios for Sr, Zr, Mo, and Ba also in a few tens of further presolar SiC grains, which have been classified as U (unknown), due to lack of clear data about C or N isotopic ratios. Since the heavy element isotopic ratios shown by grains of the U group are in agreement with the ones from the MS-SiC family, it is reasonable to believe that they are of AGB origins, too. We shall therefore use both the grain samples for the comparison with nucleosynthesis predictions of our models. For an easier reading, in the figures showing the comparisons we represent the SiC isotopic ratio data with gray dots, independently of the authors who actually did the measurements. The only distinction we maintain is the one between the MS and U families, which are plotted by filled and empty dots, respectively.

We start the comparisons of model predictions with measurements in presolar SiC grains by considering the relative behavior of the isotopes of two crucial elements (Sr and Ba), characterizing the neutron flow at the neutron magic numbers N = 50 and N = 82.

Figure 11 illustrates the case of the measured isotopic ratios ⁸⁸Sr/⁸⁶Sr and ¹³⁵Ba/¹³⁶Ba, as compared to predictions from our 2 M_⊙ models and reveals some interesting details. In general, the area covered by the measurements and the one covered by the models overlap largely (see especially the panel at the right side). This indicates that the quality of the nuclear parameters, as adopted after the discussion of Section 3, is already rather satisfactory. However, important improvements are possible. Let us analyze this situation in some detail. As shown by the left panel of Figure 11, models for the C-rich phases in the envelopes tend to produce intermediate delta values for the ratio ¹³⁵Ba/¹³⁶Ba, not suited to cover the wide area of the measurements. This problem was common in previous approaches in the literature (see, e.g., Lugaro et al. 2018). When instead we consider the magnetized winds discussed in the text (see Section 2.2), their compositions sample a wider distribution of ¹³⁵Ba/¹³⁶Ba ratios, and cover increasing portions of the measurement area for increasing amounts of the He-shell material added in them. They can actually account for these SiC data almost completely, as shown in the right panel of Figure 11. This fact shows the relevance of this wind component, not considered so far in the literature. Anyhow, further improvements in the model predictions might be welcome if one wants to refer also to 3 M_⊙ models. Their behavior with respect to the experimental constraints is illustrated in Figure 12 for the same cases chosen in the previous plot. As is clear, with our ST choice of the nuclear parameters these models should be excluded. Indeed, when they cover the whole range of ¹³⁵Ba/¹³⁶Ba data (see full lines with heavy dots in Figure 12), their Sr isotopic ratio in the ordinate extends at too high δ-values, where there are no measurements. This is due to the higher efficiency, in these more massive models (due to the higher pulse temperature) of the reaction branch that from ⁸⁵Kr feeds ⁸⁶Kr, ⁸⁷Rb, and ⁸⁸Sr (see Section 3.1). Can we exclude completely, on this basis, that SiC grains be formed, at least in part, by stars more massive than 2 M_⊙ at the adopted metallicities? The previous analysis by Lugaro et al. (2018), on the basis of the same troubles with the Sr versus Ba plot, concluded in this way and suggested to move to higher-than-solar metallicities (where the reduced neutron exposure feeds ⁸⁸Sr less efficiently). We do not really know if this is a solution, but we think that such a conclusion might be too drastic, in the light of our previous discussion of Sr cross sections (see Section 3.2). Indeed, with our test choice V2 of the parameters (where,for the cross section of ⁸⁸Sr,we adopted the value measured by Katabuchi et al. (2011), larger by 30% with respect to the K1 recommendations) the situation changes considerably, as shown in Figure 13. As is illustrated there, in this case the range of predictions from 3 M_⊙ models for the ⁸⁸Sr/⁸⁶Sr ratio shrinks sharply. The right panel of Figure 13 would represent now a quite good reproduction of experimental data, without invoking excessively high metallicities, which we consider unlikely for presolar grains (in this case the fits of Figure 11 would not be modified largely). We leave for the moment this possible indication as a warning on the ⁸⁸Sr neutron-capture cross section.

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We can also notice that the ¹³⁵Ba/¹³⁶Ba ratios (and in general those for Ba isotopes of higher atomic mass) are sensitive to further uncertain nuclear parameters, namely the (n,γ) cross sections for the radioactive Cs isotopes ¹³⁴Cs and ¹³⁵Cs, for which only theoretical values exist, and their β⁻decay rates. In addition to the already discussed case of ¹³⁴Cs, indeed (see Section 4), also the case of ¹³⁵Cs is worth a dedicated study, as its dependence on temperature is large above 2 · 10⁸ K and might well be different than assumed here from Takahashi & Yokoi (1987). A slightly longer half-life (so far estimated to be of the order of a few hundred years in TP conditions) would modify the ratio ¹³⁵Ba/¹³⁶Ba in a complex way, with the effect of further stretching the area of the measurements covered by envelope and wind models with respect to what is obtained in Figures 11–13. Similar effects would be induced by variations in the ¹³⁵Cs neutron-capture cross section.

Another crucial test for nucleosynthesis models and their nuclear parameters emerges from the δ-values of ⁸⁸Sr/⁸⁶Sr, when plotted as a function of the ones for ¹³⁸Ba/¹³⁶Ba. Figure 14 shows the situation for a 2 M_⊙ star: here we illustrate the composition of the envelope and that of magnetized winds (with 2% to 5% of He-shell material admixed). Even in this 2 M_⊙ case, in our estimates a considerable fraction of the TPs (full dots) would produce in the winds values of ⁸⁸Sr/⁸⁶Sr that are higher than measured (see in particular the right panel). This fact may, again, play in favor of a larger ⁸⁸Sr neutron-capture cross section than recommended in the K1 database, as illustrated in Figure 15, where the shrinking in the dispersion of the ordinate is obtained by adopting the already mentioned choice V2 for the nuclear parameters.

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Even more evident in Figures 14 and 15 is that there is some excess of ¹³⁸Ba in the models. This nucleus is at the N = 82 magic number and is fed effectively when the neutrons available are abundant. However, this is not the case in the computations shown here, which refer to solar-like metallicities (hence to stars with high contents of Fe), with rather small neutron exposures. A solution, especially for the 2 M_⊙ case, would be to accept that magnetized winds be generated with even larger fractions of He-shell matter than displayed in the figures, as the model curves perform a U-turn that reaches down to lower and lower values of the ¹³⁸Ba/¹³⁶Ba ratio for increasing mixing fractions. This solution, however, would be restricted to the 2 M_⊙ models, as those for the 3 M_⊙ cases with high mixing ratios tend to produce values excessively positive in the Sr isotopic ratio in the ordinate, even in the test case v2. This is indeed illustrated in Figure 16. The best compromises for the 3 M_⊙ cases seem to be limited to 1%–2% of mixing of He-shell matter in the winds. Alternatives exist, but must invoke further changes in the nuclear parameters. As mentioned, these might affect the decay rates of ¹³⁵Cs and the cross sections of radioactive Cs isotopes.

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On the other hand, the nuclear data on barium neutron-capture cross sections are satisfactorily precise at large enough temperatures (T ∼ 300 MK), but are rather loosely constrained at lower energies (around 8 keV, i.e., the energy characterizing neutron release in the ¹³C pocket). Also in this case, refinements in the experimental data are urgently needed.

A comparison of model predictions with the measured δ-values for Sr isotopes is shown in Figures 17 and 18, adopting for illustration a 2 and a 3 M_⊙ model, respectively. We refer directly to the cases V2, discussed in the previous section, because the excess production of ⁸⁸Sr there identified can be avoided. As is made clear in Figure 18, in a 3 M_⊙ star the envelope alone (left panel) does not offer a good reproduction of the measurements, while referring to the magnetized winds allows for a much better agreement, which improves when the percentage of He-shell material in the outflows increases (central and right panels).

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The cases for 2 M_⊙ stellar models (Figure 17) look similar, but with a lower proportion of the area containing experimental data covered by models. It can be noticed that the large spread in the measured data for ⁸⁷Sr is not well matched by model curves. Actually, such a spread can best be interpreted as a consequence of a range of temperature values (i.e., of stellar masses), in which the complex path leading to ⁸⁷Sr varies. In this respect, the unique value for the branching ratio at ⁸⁴Kr to the isomer of ⁸⁵Kr, which we are forced to assume from the K1 repository, in the absence of different data, is certainly an oversimplified assumption (see Section 3.1). Here we clearly need new nuclear physics measurements on the chains departing from Kr isotopes, in this case the one proceeding through ⁸⁵Kr^m , ⁸⁵Rb, ⁸⁶Rb and ^86,87Sr. In general, therefore, the evidence provided so far by SiC grains points to the needs of revisions in the nuclear parameters discussed in Sections 3.1 and 3.2.

Concerning the various Ba isotopes, Figures 19 and 20 illustrate a synthesis of what can be obtained for them in representative cases among those discussed so far. In particular, we plot there situations pertaining to magnetized winds, with 5% admixture of material from the He shell for 2 M_⊙ models and 2% for 3 M_⊙ ones. This latter case seems to represent a rough average condition permitting to reproduce sufficiently well the three-isotope plots of barium, including the constraints from ¹³⁴Ba, ¹³⁷Ba and ¹³⁸Ba. Despite the good accord, these plots confirm that small revisions in the decay rates or in the cross sections for Cs isotopes, leading to wider spreads in ¹³⁷Ba and ¹³⁸Ba, would be welcome.

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The isotopic ratios involving Zr, as measured in SiC grains, appear to pose much smaller problems to AGB models and the nuclear parameters seem to be in this case sufficiently good. This is so also for isotopes that were so far quoted by other groups as being problematic, like ⁹²Zr (Lugaro et al. 2014) and which are instead compatible with our magnetized winds. Also for the Zr ratios, therefore, including this wind component largely increases the mutual compatibility of data and models, improving considerably over previously published analyses. We show here, in Figure 21 and in Figure 22 a couple of synthetic, representative cases, involving ⁹⁰Zr/⁹⁴Zr, ⁹²Zr/⁹⁴Zr, and ⁹⁶Zr/⁹⁴Zr.

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Figure 21 presents the data for the ⁹⁰Zr/⁹⁴Zr ratio as a function of ⁹⁶Zr/⁹⁴Zr, compared to the outputs from a choice of our models for the winds. Panels at the left and at the center refer to 2 M_⊙ models with 1% and 2% admixtures of He-shell materials, respectively, while the right panel represents the case of 3 M_⊙ models at 0.5% dilution. The three plots together show how the measurements can be well accounted for by the ensemble of our models, with 2 M_⊙ cases explaining better the vertical spread and 3 M_⊙ ones complementing them for the horizontal distribution. No special change in the nuclear parameters seems necessary.

Figure 22 then illustrates the situation with ⁹²Zr. Previous attempts in the literature (Lugaro et al. 2014) concluded that δ-values of ⁹²Zr in excess of −50 could not be explained by AGB models. Here, instead, we see that 2 M_⊙ models are sufficient to account for the data, provided we considered various He-shell admixtures. Grains with higher δ values for ⁹²Zr/⁹⁴Zr are explained by progressively higher percentages of mixing. In the plots provided we consider the composition of the winds for increasing He-shell admixtures, where grains with progressively higher δ values for ⁹²Zr/⁹⁴Zr are found. In this respect we recall the present complex picture of AGB winds illustrated by Höfner & Olofsson (2018), as enhanced mass-loss rates and dust production in phases preceding the final superwinds are probably needed to support our model. Flaring activity might actually lead to such enhanced mass-loss rates and dust formation (Soker & Kastner 2003).

Concerning Mo, Stephan et al. (2019) showed how its isotopic mix seems to derive from a combination of essentially only two components, one purely s-process controlled, the second of isotopically solar composition. In this condition, as is known, the data in three-isotope plots distribute along almost straight diagonal lines. Beyond this simple trend, however, there are tiny details, from which new lessons can be learned.

Models for the C-rich envelopes tend to represent rather exstreme admixtures, very close to the s-process end-member (see the discussion in Liu et al. 2019) The composition of the magnetized winds, instead, allows us to cover larger and larger portions of the area filled by the data, for increasing efficiencies of He-shell matter mixed. This is so because in the winds, through the mechanism sketched in Figure 3, one samples small portions of the C-rich component transported by magnetized structures and large portions of the envelope, even in phases preceding the final ones of the C-star formation. As mentioned, small but important details reveal, even here, that something needs to be improved. Let us review these details, starting with inspection of Figures 23–26, all representing the envelop composition (left panels) and winds of 2 M_⊙ models, enriched at the 0.5% and the 2% in C-rich material (central and right panels, respectively). We choose the ratio ⁹⁴Mo/⁹⁶Mo as a common abscissa and the first evidence is that the models at the TDU episodes (full dots) crowd at δ values for ⁹⁴Mo which are slightly, but clearly, too low with respect to the measured data. Some increase in ⁹⁴Mo would solve the problem and we believe that this is an indication either of a smaller value of its neutron-capture cross section than recommended in the K1 repository, or of the need to invoke for ⁹⁴Mo the subtle contributions from the chain passing through ⁹³Nb and ⁹⁴Nb. Another evidence is shown by the heaviest isotopes ⁹⁷Mo, ⁹⁸Mo and ¹⁰⁰Mo, see Figures 26–28; the last two comparing the data with 3 M_⊙ models, the previous one with 2 M_⊙ models. Here, the straight diagonal line represents an average trend, over which some significant vertical spread exist, at different levels for the different isotopes. This behavior is not accounted for with our ST choice of the nuclear parameters, but can instead be interpreted by referring to our test case V2. Among the changes introduced there (see Section 6.3), we enhanced the cross section of ⁹⁵Zr to its upper limit allowed by the K1 recommendations, to take into account the suggestions for a larger cross section contained in the BNL repository. We also adopted, for ⁹⁹Mo, the larger value provided by the same BNL site. Figures 27 and 28 show the effects of these changes on the plots involving ⁹⁸Mo and ¹⁰⁰Mo, in our 3 M_⊙ models. In particular, Figure 27 shows the effects on ⁹⁸Mo, of the first of the mentioned changes. Increasing the (n,γ) cross section of ⁹⁵Zr has the effect of feeding more efficiently the chain ⁹⁶Zr-⁹⁷Zr-⁹⁷Nb-⁹⁷Mo, at the expense of ⁹⁶Mo. As a consequence, the latter is slightly reduced. Since it is at the denominator of the isotopic ratio in the ordinate, its decrease induces higher values of the latter. Indeed, as the figure shows, in 3 M_⊙ models one has originally a downward spread (left panel) in the model trend, worsening the fit with respect to the 2 M_⊙ cases, while with the V2 choice the direction of the spread is reverted. Should one slightly correct ⁹⁴Mo in the way already discussed, this would result in fitting quite well even the small dispersion of the observed points. A very similar change is induced by the test case V2 on ¹⁰⁰Mo, this time being the direct effect of the larger cross section adopted for ⁹⁹Mo, as illustrated in Figure 28.

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We also recall that in the case in which the abundance of a nuclide in grains is extremely poor, thus also making very small a connected isotopic ratio, a minimum sample pollution (e.g., if a grain is coated by a very thin residue of solar matter) could affect the measured value in such a way to mimic a dilution of the progenitor stellar winds with further unpolluted solar material (Lugaro et al. 2017). This fact may account for the composition of grains that show Mo isotopic admixtures close to the solar values.