© The Authors 2025

1. Introduction

At the end of their lives, low-mass stars (1 M_⊙ ≲ M ≲ 8 M_⊙) evolve to the asymptotic giant branch (AGB, Iben & Renzini 1983; Busso et al. 1999; Herwig 2004; Karakas & Lattanzio 2014). AGB stars are characterised by a degenerate C-O core, a H and a He shell, and an extensive, convective, H-rich envelope. The two shells are separated by a thin He-rich layer called the intershell. The energy production occurs in cycles: a relatively long phase of H burning is periodically (every ∼10³–10⁵ years, depending on the stellar mass) interrupted by a thermal instability of the He shell, which is called a thermal pulse (TP, Schwarzschild & Härm 1965; Weigert 1966). The rapid release of a substantial amount of energy during a TP causes the development of a convective region in the intershell, hereafter referred to as a ‘convective pulse’, and the star to expand from the He shell outward. This expansion allows the envelope to extend downwards after the extinction of the convective pulse, potentially penetrating the intershell region and carrying material from there to the surface. The resulting mixing episodes are collectively referred to as the third dredge-up (TDU). The intershell material carried to the stellar surface contains a large amount of carbon produced by partial He burning; therefore, the TDU can convert the surface layers of the AGB star to evolve from being oxygen-rich to being carbon-rich (C/O ≥ 1, Wallerstein & Knapp 1998). The cycle described above is repeated several times (tens to hundreds of times, depending on the initial mass) before the envelope is completely eroded by mass loss, and the C-O core of the star is left as a white dwarf.

In the thermally pulsing AGB phase, conditions are suitable for (α,n) neutron-source reactions to be initiated in the intershell. The ²²Ne(α, n)²⁵Mg reaction (Cameron 1960; Truran & Iben 1977) is activated for temperatures greater than 300 MK during the convective pulses of AGB stars with an initial mass greater than ∼3 M_⊙ and results in a high maximum neutron density (∼10¹³ n cm⁻³, e.g. Fishlock et al. 2014) but a low total amount of free neutrons. The ¹³C(α, n)¹⁶O reaction (initially reported by Hollowell & Iben 1988) is activated during the H-burning phase, at T ∼ 90 MK and results in a high total amount of free neutrons but a low maximum neutron density (∼10⁷ n cm⁻³, Straniero et al. 1995). The captures of the resulting free neutrons drive slow neutron captures; that is, the s process (Burbidge et al. 1957; Käppeler et al. 2011; Lugaro et al. 2023b), which produces roughly half of the abundances of elements heavier than iron (up to lead) in the Universe (see review by Lugaro et al. 2023b).

During the s process, by definition, the probability of neutron capture on an unstable isotope is lower than its decay; therefore, the main neutron-capture path follows the valley of β-stability. However, depending on the temperature, density, and neutron density, the neutron-capture rate of some unstable isotope may compete with their decay rate. These unstable isotopes are known as branching points on the s-process path (e.g. Bisterzo et al. 2015; Lugaro & Chieffi 2018). The branching factor, f_β, is defined as:

$$\begin{array}{c}\hfill f_{\beta }=1-f_{\mathrm{n}}=\frac{{\lambda }_{\beta }}{{\lambda }_{\beta }+{\lambda }_{\mathrm{n}}},\end{array}$$(1)

where f_n is the branching factor of the neutron capture channel, λ_β is the β-decay rate (λ_β = ln2 $t_{1/2}^{-1}$, where t_1/2 is the half-life), and λ_n = N_n ⟨σv⟩ is the neutron-capture rate given by the neutron density, N_n, the relative velocity of neutrons and target nuclei, v, and the neutron capture cross section, σ. The ⟨σv⟩ can be approximated as the product of the Maxwellian-average cross section (MACS or σ_M) and the thermal velocity, v_th.

The calculation of f_β is not trivial because not only N_n varies; λ_β and σ_M can also change with T and ρ. Neutron-capture rates on a nucleus ground state do not significantly vary with temperature; however, to obtain neutron-capture rates in a stellar environment, in addition to laboratory ground-state cross sections, the theoretical relative contribution of excited states (Rauscher 2012), generally described using a stellar enhancement factor (SEF), needs to be included. Also, laboratory β-decay rates can be largely modified at stellar temperatures due to the thermal population of low-lying excited states, which may have different β-decay lifetimes (e.g. Reifarth et al. 2018a). In addition, temperature and electron density change the population of electronic states, providing the possibility of the bound-state decay (Bahcall 1961). Electron-capture rates are also influenced by temperature and electron density: the likelihood of a bound electron near the nucleus decreases with temperature, while the probability of a free electron near the nucleus increases with electron density (e.g. Takahashi & Yokoi 1987). Accounting for these effects is essential for the accurate modelling of the s process in AGB stars.

Here, we present a major upgrade of nuclear reaction rates of the Monash nucleosynthesis code. We replaced the constant radioactive-decay and electron-capture reactions with T- and ρ-dependent tables. The tables were based on the compilation of NETGEN (nuclear NETwork GENerator, Jorissen & Goriely 2001; Aikawa et al. 2005; Xu et al. 2013). Most other sets of models – NuGrid (Nucleosynthesis Grid, e.g. Pignatari & Herwig 2012), FUNS (FUll Network Stellar, e.g. Straniero et al. 2006), and STAREVOL (e.g. Siess 2006) – have already implemented T- and ρ-dependent decay rates and provided the resulting yields and abundances. However, the thorough analysis that we present here regarding the impact of using variable weak rates on the activation of branching points and the implications on the comparison with observations, including the interpretation of stardust grain data, is still missing from the literature. To our knowledge, recently only Bisterzo et al. (2015) have presented a detailed analysis of branching points, but from the point of view of the impact of the uncertainties of the neutron source reactions. Furthermore, we include new tabulated neutron-capture rates for 92 reactions using re-evaluated experimental MACSs from the ASTRAL database (ASTrophysical Rate and rAw data Library, Reifarth et al. 2018b; Vescovi & Reifarth 2021; Vescovi et al. 2023) and SEFs from KaDoNiS (Karlsruhe Astrophysical Database of Nucleosynthesis in Stars, Dillmann et al. 2006). We provide the first database of surface abundances and stellar yields of the isotopes heavier than iron from the Monash models.

The paper is organised as follows. We start with a description of our methods and of the seven stellar evolutionary models used for the post-processing nucleosynthesis calculations (Sect. 2). In Sect. 3, we present the impact of nuclear input changes on nucleosynthesis predictions and discuss the results including a comparison to other studies and a selection of isotopic ratios measured in presolar stardust silicon carbide (SiC) grains from AGB stars. The conclusions are provided in Sect. 4.

2. Methods and models

We provide predictions for stellar abundances from AGB stars, using the Monash post-processing nucleosynthesis code (Cannon 1993), which calculates the changes in the abundances of currently 328 nuclear species due to convective mixing and nuclear burning on the basis of detailed stellar structure calculated by the Monash evolution code (Lattanzio 1986; Karakas 2014). In summary, the current nuclear reaction network of the nucleosynthesis code includes 2351 reactions, most of which are based on the JINA reaclib database (Cyburt et al. 2010), where reaction rates are expressed as a function of temperature T₉ (in units of 10⁹ Kelvin) using a seven-parameter (a_0 − 6) analytical formula (Thielemann et al. 1986):

$$\begin{array}{c}\hfill \lambda =\exp [a_0+{\displaystyle \sum _{\mathrm{i}=1}^5}{a}_{\mathrm{i}}{T}_9^{\frac{2\mathrm{i}-5}{3}}+a_6\ln T_9].\end{array}$$(2)

While the JINA reaclib database is advantageous for building a large network, it has two major disadvantages: (1) For some experimental rates, there can be a variation of up to 10%–20% of the fit from the actual rate. (2) The β-decay and electron-capture rates are constant; that is, they do not depend on temperature and electron density. For these reasons, a new version of the nucleosynthesis code was developed by van Raai et al. (2012), which includes a routine that allows us to use specific reaction rates as tabulated values instead of the reaclib fit, and here it was extended to decay rates.

2.1. Stellar models

As input for post-processing nucleosynthesis calculations, we used a selection from the stellar structure evolution models presented in Karakas (2014) and Karakas & Lugaro (2016). We selected M = 2, 3, 4 M_⊙, Y = 0.28 stellar models of solar metallicity (Z = 0.014, based on the solar abundances from Asplund et al. 2009), M = 3, 4 M_⊙, Y = 0.26 stellar models at half-solar metallicity (Z = 0.007), and M = 3, 4 M_⊙, Y = 0.30 stellar models at double-solar metallicity (Z = 0.03)¹. Input physics and methodology of the stellar structure models are described in full in the papers mentioned above. In summary, mass loss was not included in the red giant branch (RGB) phase and the mass-loss rate of Vassiliadis & Wood (1993) was used in the AGB phase. The models used the mixing length theory of convection with a mixing-length parameter α = 1.86. The convective overshoot was not included in the calculations except for the 3 M_⊙Z = 0.03 model, where an overshoot parameter (N_ov = 1) was used which resulted in a C-rich star.

Table 1 shows some relevant properties of the calculated AGB models. (1) The total number of TDUs increases with increasing initial stellar mass. (2) The number of TPs during which the envelope is C-rich depends also on the mass, the mass loss, and the activation of hot bottom burning (i.e. H-burning at the base of the envelope) in the 4 M_⊙ stars. (3) The maximum temperature reached during TPs, which controls the activation of the ²²Ne(α, n)²⁵Mg neutron source increases with increasing stellar mass and decreasing metallicity.

2.2. Post-processing calculations

The operation of the ¹³C(α, n)¹⁶O neutron source requires the formation of a partial mixing zone (PMZ), where protons from the convective envelope are mixed with the intershell matter and captured by the abundant ¹²C, resulting in the so-called ¹³C pocket (Straniero et al. 1997; Gallino et al. 1998; Goriely & Mowlavi 2000). Our method of including the ¹³C pocket is described in detail in Karakas & Lugaro (2016). In summary, protons are partially mixed at the end of each TDU down to a mass extent in the intershell denoted by M_PMZ, using an exponential profile. At the top of the intershell (corresponding to the base of the envelope during the TDU), the abundance of protons X_p is that of the envelope, while at M_PMZ in the intershell it is X_p = 10⁻⁴. Below this mass point, X_p = 0. Different M_PMZ values as functions of the stellar mass are defined and justified by Karakas & Lugaro (2016) and Buntain et al. (2017). Here, we used the ‘standard’ values reported in Table 1.

We ran post-processing nucleosynthesis calculations with three different inputs (see Table 2) for the nuclear network. The original set (hereafter Set 0) contains nuclear reactions mostly from the JINA reaclib database. The neutron-source reactions ¹³C(α, n)¹⁶O and ²²Ne(α, n)²⁵Mg are from Heil et al. (2008a) and Iliadis et al. (2010), respectively. For neutron capture reactions, Set 0 uses the ka02 labelled fits of JINA reaclib, which are based on the KaDoNiS database, except for Zr isotopes, whose rates are from Lugaro et al. (2014a), and for a few other isotopes, whose rates are taken directly from KaDoNiS. These were selected because the JINA fit did not accurately cover the KaDoNiS data. This is the same network used by Karakas & Lugaro (2016), except that in Set 0 we updated the temperature dependence of the β-decay rate of ¹³⁴Cs from Takahashi & Yokoi (1987) used in Karakas & Lugaro (2016) to Li et al. (2021) and of ¹⁸¹Hf from Lugaro et al. (2014b), which was included as terrestrial constant in Karakas & Lugaro (2016).

In the second set (hereafter Set 1), we implemented the temperature and density dependence of β-decays and electron captures in the network using the default (most recent) rates from the NETGEN library. Below mass number A = 56, the rates are based on Horiguchi et al. (1996), except for the β⁻-decay rate of ⁴⁸Ca, which is based on Langanke & Martínez-Pinedo (2000), and the electron-capture rates of ⁷Be and ³⁶Cl, which are based on Caughlan & Fowler (1988) and Oda et al. (1994), respectively. Above A = 56, the rates are based on the nominal rates of Goriely (1999), while the reactions not included in Goriely (1999) are based on Takahashi & Yokoi (1987), except for the β⁻-decay rate of the excited ⁸⁵Kr isomer, which is based on Horiguchi et al. (1996). In addition, we extend the reaction list by including the bound-state β decay of the terrestrial stable ²⁰⁵Tl based on Goriely (1999). Most of our decay tables contain the values in a grid of 250 temperature points between 1 MK and 1 GK on a logarithmic scale. For the rates that also depend on the electron density, the NETGEN tables consist of data in the same temperature grid as above for four electron densities: 1, 3, 9.5 or 10 (depending on the reference), and 30 × 10²⁶ cm⁻³, except for ³⁶Cl and ⁴⁸Ca, which include data for 11 electron density points between 6 × 10²⁴ and 6 × 10³³ cm⁻³. We tested the possible effect of the choice of temperature grid on the M = 3 M_⊙, Z = 0.014 model. We found that the application of a looser grid (125 points) or a finer grid (300 points) results in the same abundances within 1%, except for ²⁰⁴Pb, ²⁰⁵Pb, and ²⁰⁵Tl, which changed by less than 10%.

In the third set (hereafter Set 2), we upgraded the neutron-capture rates of 92 reactions using re-evaluated experimental MACSs from the ASTRAL database and SEFs from KaDoNiS v0.3. The current version (v0.2) of ASTRAL includes MACSs for 122 isotopes based on time-of-flight measurements, which have been re-evaluated by taking into account the new recommended value for the gold differential cross section (Reifarth et al. 2018b) and 50 isotopes whose experimental cross section data are based on activation measurements (Vescovi et al. 2023). The neutron-capture rates at each temperature are calculated as λ_n = (N_A σ_M v_th) × SEF, where N_A = 6.02 × 10²³ is the Avogadro constant.

2.3. Surface abundances, stellar yields, and δ values

The nucleosynthesis code provides us with the time-evolution of normalised number abundances, $Y_{\mathrm{k}}^{\mathrm{i}}$, of each isotope i at the surface of the model stars. Using these outputs, we can calculate the total stellar yield, M_i, as the total mass of each isotope, i, expressed in unit of solar mass, ejected into the interstellar medium (ISM) during the entire life of the star. To do this, we used an approximate formula, which sums over all the TPs (instead integrating over the entire lifetime of the star) the mass lost for each isotope in-between each TP (identified by the index, k) according to:

$$\begin{array}{c}\hfill M_{\mathrm{i}}=\left[{\displaystyle \sum _{k=1}^{\mathit{final}}}(M_{\mathrm{k}-1}-M_{\mathrm{k}})X_{\mathrm{k}}^{\mathrm{i}}\right]+(M_{\mathrm{final}}-M_{\mathrm{core}})X_{\mathrm{final}}^{\mathrm{i}},\end{array}$$(3)

where M_k is the mass of the model star at the end of TP k, $X_{\mathrm{k}}^{\mathrm{i}}$ = $Y_{\mathrm{k}}^{\mathrm{i}}{m}_{\mathrm{i}}$ is the surface abundance of isotope i in the mass fraction at the end of same TP k (where m_i is the atomic mass of isotope i), and M_final and M_core are, respectively, the total mass and the core mass at the end of the computed evolution (i.e. when the code stops converging). The second term of the equation was added to remove the full envelope, assuming that no more TDU happens. We provide $Y_{\mathrm{k}}^{\mathrm{i}}$ and M_i for all our models as Online Material. We note that this approximated formula works here because the models we consider do not have any mass loss on the RGB and we tested that our Set 0 yields are within 2% of those published by Karakas & Lugaro (2016).

While stellar yields are an important input for the Galactic chemical evolution (GCE) calculations (e.g. Côté et al. 2019; Trueman et al. 2022), isotopic abundance ratios are needed to interpret the composition of presolar stardust grains, such as SiC grains found in meteorites. Most of the SiC grains condensed in carbon-rich AGB stellar winds and were ejected into the ISM (Zinner 2014). The grains were transported to the protosolar nebula, trapped within the parent bodies of meteorites, which delivered them to Earth where they can be extracted and analysed in the laboratory (e.g. Bernatowicz et al. 1987; Lewis et al. 1990). The Presolar Grain Database (PGD, Hynes & Gyngard 2009; Stephan et al. 2024) contains all the available isotope data for single presolar stardust SiC grains. To compare to the SiC grain data, we calculated the isotopic ratios of each element in the form of δ values: that is, per mil variations with respect to the solar ratio as:

$$\begin{array}{c}\hfill {\delta }_k{{}^{i}E}_j=(\frac{(Y_{\mathrm{k}}^{\mathrm{i}}/Y_{\mathrm{k}}^{\mathrm{j}})}{{(Y_{\mathrm{k}}^{\mathrm{i}}/Y_{\mathrm{k}}^{\mathrm{i}})}_{\odot }}-1)\times 1000,\end{array}$$(4)

where i, j represents the isotopes of element E at the end of TP k. The solar abundances in this case are also based on Asplund et al. (2009).

3. Results and discussion

3.1. Comparison of Set 1 to Set 0

In this section, we compare the predictions of the Set 0 and Set 1 models using the final surface normalised number abundances, $Y_{\mathrm{final}}^{\mathrm{i}}$, of the model stars. There are 30 isotopes whose abundance changed by at least 10% in at least one Set 1 stellar model compared to the corresponding Set 0 stellar model. The list of these isotopes and their Y_{Set 1, final}ⁱ/Y_{Set 0, final}ⁱ ratio are shown in Table A.1. The abundances of individual isotopes, in addition to the nuclear input, depend on the characteristics of the stellar models (e.g. temperature, electron and neutron density, number of mixing episodes) and therefore the effects of the new rates can differ greatly between models. In the following, we present the most relevant isotope cases.

Table A.2 highlights the relevant information for each case, including the branching isotopes, the decay modes and decay rates of the corresponding isotopes, the direction in which the abundance of each isotope has changed in the Set 1 models compared to Set 0 models, and the relevance of each case.

3.1.1. ⁶⁴Ni

The s-process production of ⁶⁴Ni depends on the operation of the ⁶³Ni and ⁶⁴Cu branching points. ⁶³Ni suffers β⁻ decay with a laboratory half-life of 99.8 yr producing ⁶³Cu. According to NETGEN, the β⁻-decay rate of ⁶³Ni increases by roughly an order of magnitude from the terrestrial value under the conditions that prevail in the intershell (see Table A.2). As is shown in Eq. (1), the probability of neutron capture decreases with increasing decay rates; therefore, we obtain a lower ⁶⁴Ni abundance from this channel in Set 1 models than in Set 0 models. The isotope ⁶⁴Cu, with a half-life of 12.8 hours, can both β⁻ and β⁺ decay, where the latter channel is twice as fast compared to the β⁻ under terrestrial conditions. While the rate of the β⁻ channel of ⁶⁴Cu slightly increases with T, the β⁺-decay rate is instead reduced by about a third at 300 MK. The probability of the β⁺ channel also decreases with electron number density. These effects also contribute to the decrease in the amount of ⁶⁴Ni in Set 1 models. In total, we obtain 1% to 12% less ⁶⁴Ni in the final stage of the Set 1 models compared to Set 0. For nickel, a large number of SiC grain data are available following the work of Trappitsch et al. (2018); therefore, we compare the predictions of our models with these data in Section 3.3.1.

3.1.2. ^80, 81Kr

The s-process production of the stable ⁸⁰Kr and the unstable ⁸¹Kr with a half-life of 229 kyr depends on the operation of the ⁷⁹Se and ⁸⁰Br branching points. ⁷⁹Se suffers β⁻ decay with a laboratory half-life of 297 kyr producing ⁷⁹Br. Neutron captures on ⁷⁹Br produce unstable ⁸⁰Br with a half-life of 17.7 minutes. ⁸⁰Br can both β⁻ and β⁺ decay, the former channel being almost ten times faster than the latter and resulting in the production of ⁸⁰Kr. From ⁸⁰Kr, ⁸¹Kr can be produced by neutron captures, in addition to its amount also depending on its own electron capture rate. Comparing the final surface abundances of the Set 1 models with those of Set 0, we obtain 1.94–4.65 times more ⁸⁰Kr and 7.23–344.43 times more ⁸¹Kr. We note that the abundance of ⁸²Kr also increases slightly, by 2–15%. All these changes are mainly due to the operation of the ⁷⁹Se branching point, whose decay rate increases by orders of magnitude under stellar conditions (Table A.2). We note that a measurement-based T- and ρ-dependent half-life of ⁷⁹Se is available (Klay & Käppeler 1988). The half-life from NETGEN is within 10% of the half-life of Klay & Käppeler (1988) over the whole temperature range.

Krypton is a noble gas. It does not condense into dust, but it can be ionised and implanted into the grains (Verchovsky et al. 2004). In fact, the isotopic composition of trace amounts of Kr in SiC grains shows the signature of mixing of pure s process and solar composition material. Lewis et al. (1994) analysed the isotopic composition of Kr in bulk samples of a million SiC grains from the Murchison KJ series (Amari et al. 1994) and used linear regression to derive the isotopic composition of Kr in the intershell of AGB stars. They found that the isotopic ratios vary with grain size (the ⁸⁰Kr/⁸²Kr ratio decreases with grain size), suggesting that these SiC grains came from stars of different masses and metallicities.

Table 3 shows the ⁸⁰Kr/⁸²Kr ratios in the intershell of the Set 0 and Set 1 models before the last TDU episode. We compare only the solar and double-solar metallicity model predictions with the Lewis et al. (1994) data because the vast majority (> 90%) of SiC grains, the mainstream (MS) grains, are believed to originate in low-mass C-rich AGB stars with around solar (Hoppe et al. 1994), slightly super-solar (Lewis et al. 2013), and double-solar (Lugaro et al. 2020) metallicity. Also, recent results by Cristallo et al. (2020) suggest that the stardust SiC grains predominantly came from AGB stars around M = 2 M_⊙ and Z = 0.014. For most of the Set 0 models, this ratio is an order of magnitude lower than the range of Lewis et al. (1994); for the M = 4 M_⊙Z = 0.014 model, this difference is even greater. In contrast, the ⁸⁰Kr/⁸²Kr ratios derived by the Set 1 models are of the same order of magnitude as the possible intershell ratios; in addition, the predictions of three Set 1 models are within the range observed. We note that a new ⁸⁰Kr(n, γ) cross section is available following the work of Tessler et al. (2021). When testing on M = 2 and 3 M_⊙Z = 0.014 models, the new neutron-capture rate of ⁸⁰Kr based on Tessler et al. (2021) results in slightly higher (up to 13%) ⁸⁰Kr/⁸²Kr ratios and does not change our conclusions above.

3.1.3. ⁹⁴Mo

By definition, stable proton-rich isotopes that cannot be reached by neutron-capture processes are p-only isotopes (or the p-nuclei, Cameron 1957). The p-only ⁹⁴Mo can also be produced via the s process by two consecutive neutron captures on ⁹²Mo and the operation of ⁹³Zr and ⁹⁴Nb branching points. ⁹⁴Mo is the daughter nucleus of ⁹⁴Nb. The latter can be produced by neutron captures on both the initial amount of stable ⁹³Nb and the small fraction of ⁹³Nb that comes from the β⁻-decay of ⁹³Zr. ⁹²Mo is also a p-nucleus and it cannot be produced by the s process under any circumstances. The ^92, 93Mo isotopes are not included in our network; therefore, the ⁹²Mo(n, γ)⁹³Mo(n, γ)⁹⁴Mo channel is also missing. However, we tested that its impact on the abundance of ⁹⁴Mo is typically less than ∼1%. Although the decay rate of ⁹³Zr increases slightly with temperature and electron density, the decay rate of ⁹⁴Nb is six orders of magnitude higher in the intershell region of AGB stars than under terrestrial conditions (see Table A.2). Therefore, while more ⁹⁴Nb is produced in the Set 1 models compared to Set 0, its abundance also decays much faster, leading to a decrease in the final surface abundance by orders of magnitude and a milder increase (at most 53%) in the abundance of ⁹⁴Mo.

Several publications have analysed the isotopic compositions of molybdenum in SiC grains (e.g. Nicolussi et al. 1998; Lugaro et al. 2003; Stephan et al. 2019). As was shown in these previous studies, the isotope ratios observed in SiC grains show the mixtures of pure s-process matter with solar material. In addition, the δ⁹⁴Mo₉₆ values are correlated with the δ⁹²Mo₉₆ values, and the slope of the regression line indicates the presence of a small s-process contribution to ⁹⁴Mo. We calculated the linear regression² of δ values from Barzyk (2007), Liu et al. (2017b, 2019) and Stephan et al. (2019) and we obtained an intercept of δ⁹⁴Mo₉₆ = −961.8‰ ± 1.2‰ at δ⁹²Mo₉₆ = −1000‰, which is consistent with the result of Stephan et al. (2019). We can estimate the s-process contribution, s_i, for ⁹⁴Mo from our models using the following equation:

$$\begin{array}{c}\hfill s_{\mathrm{i}}=\left(\frac{M_{\mathrm{i}}^{\mathrm{wm}}{X}_{\odot }^{{}^{150}\mathrm{Sm}}}{X_{\odot }^i{M}_{{}^{150}\mathrm{Sm}}^{\mathrm{wm}}}\right),\end{array}$$(5)

where $M_{\mathrm{i}}^{\mathrm{wm}}$ is the weighted mean yield (see below) of isotope i obtained by our stellar models normalised to the unbranched s-only isotope ¹⁵⁰Sm, whose solar abundance and neutron capture cross section are known with high precision (Gallino 2023). The weighted mean yields were calculated with a trapezoidal rule quadrature with a constant mass interval:

$$\begin{array}{c}\hfill M_{\mathrm{i}}^{\mathrm{wm}}=\frac{{\displaystyle {\sum }_{\mathrm{j}=0}^{\mathrm{n}-1}}(M_{\mathrm{i},\mathrm{j}}{M}_{\mathrm{j}}^{-2.35}+M_{\mathrm{i},\mathrm{j}+1}{M}_{\mathrm{j}+1}^{-2.35})}{{\displaystyle {\sum }_{\mathrm{j}=0}^{\mathrm{n}-1}}(M_{\mathrm{j}}^{-2.35}+M_{\mathrm{j}+1}^{-2.35})},\end{array}$$(6)

where j indicates the mass in question. The weight is based on the Salpeter initial mass function (IMF, Salpeter 1955), used here to mimic the yields for the entire AGB stellar population. Our prediction for the s-process contribution of ⁹⁴Mo does not change significantly: it slightly increases from 0.034 to 0.04 from Set 0 to Set 1, and both are consistent with ∼0.038 coming from the data regression (see also Stephan et al. 2019).

In addition, we can compare the surface evolution of each model with the grain data. For an easier comparison to model predictions, in Fig. 1 we plot the logarithms of the ratio of the isotopes i and j at each TP k normalised to solar composition using standard astrophysical notation:

Fig. 1. [94Mo/96Mo] vs [92Mo/96Mo] from SiC grain data as dark grey circles MS grains and downward and upward triangles for Y and Z grains, respectively, with 1σ error bars (from Barzyk 2007; Liu et al. 2017b, 2019; Stephan et al. 2019) compared to the surface evolution of Set 0 and Set 1 stellar models of solar (left panel) and of double-solar metallicity (right panel) of different masses (see legend). Circles (Set 0) and triangles (Set 1) on the lines represent the composition after each TDU after the envelope becomes C-rich. The solid black line represents the linear regression from δ94Mo96 vs δ92Mo96, while the dashed black line represents the [94Mo/96Mo] = [92Mo/96Mo] line.

$$\begin{array}{c}\hfill {[}^{i}E{/}^{j}E]=log{(Y^{\mathrm{i}}/Y^{\mathrm{j}})}_{\mathrm{sample}}-log{(Y^{\mathrm{i}}/Y^{\mathrm{j}})}_{\odot }.\end{array}$$(7)

As in the case of ⁸⁰Kr, we only use the predictions of the solar and double-solar metallicity models, but include the few data points available for grains from the minor Y and Z populations, believed to have originated in stars of sub-solar metallicity, because in Fig. 1 these grains plot in the same region as the MS grains. The two 4 M_⊙ models are on the 1:1 line regardless of the choice of the reaction rates, which means that the ⁹⁴Mo production is not relevant in these stars; therefore, they do not represent the majority of the grain parent stars, especially in the case of the solar metallicity model (Lugaro et al. 2018a). On the other hand, the surface evolution of the 2 and 3 M_⊙ Set 1 models closely follows the regression line. Most previous models (e.g. Arlandini et al. 1999; Lugaro et al. 2003; Bisterzo et al. 2014; Liu et al. 2019) were unable to produce enough ⁹⁴Mo to match the grain data, our new models resolve this problem. Models in FRUITY (FUll-Network Repository of Updated Isotopic Tables & Yields, e.g. Cristallo et al. 2009, 2011, 2015) also produce a significant amount of ⁹⁴Mo, actually resulting in values lying even above the line defined by the grains (Vescovi et al. 2020). Therefore, we conclude that the previous mismatch was likely due to inaccurate nuclear input data and/or treatment of the branching points, which was updated in the past decade or so in the new generation of AGB nucleosynthesis models.

3.1.4. ¹⁰⁸Cd

The case of ¹⁰⁸Cd is similar to ⁹⁴Mo: it is a p-nucleus with a small s-process contribution. It can be produced via s process by the β⁻-decay of ¹⁰⁸Ag, which can be produced by neutron captures on the initial amount of ¹⁰⁷Ag and the small fraction that comes from the β⁻-decay of ¹⁰⁷Pd. ¹⁰⁸Ag can both β⁻ and β⁺ decay with a relatively short half-life (2.4 min). The β⁻ channel does not depend significantly on temperature and density and is faster than the β⁺ channel under both stellar and terrestrial conditions (see Table A.2). In contrast, the β⁻-rate of ¹⁰⁷Pd increases by orders of magnitude with temperature, causing the amount of ¹⁰⁸Cd to increase by 2-64% in the Set 1 models with respect to the Set 0 models. We estimate the s-process contribution for ¹⁰⁸Cd according to the method used for ⁹⁴Mo. As in the case of ⁹⁴Mo, our predictions of the two sets are quite close: the estimated s-process contribution of the solar ¹⁰⁸Cd abundance increases slightly from 0.04 to 0.048 from Set 0 to Set 1.

3.1.5. ¹²⁹I

The small s-process production of r-only ¹²⁹I depends on the β⁻ decay of ¹²⁸I. By definition, species that cannot be produced by the s process due to an unstable nucleus that precedes them on the s-process path are traditionally r-only isotopes. In fact, no isotope is completely shielded against s-process production, nuclei on the neutron-rich side of the β-stability valley can have a low s contribution, depending on the activation of branching points (Bisterzo et al. 2015). In addition, ¹²⁹I is a so-called short-lived radionuclide (SLR, a radioactive isotope with a half-life between about 0.1 to 100 million years, see, e.g. Lugaro et al. 2018b); therefore, its amount depends also on its decay. Neither the decay of ¹²⁸I nor that of ¹²⁹I is significantly dependent on the electron density, but the β⁻-decay rate of ¹²⁸I decreases slightly with temperature, while the decay rate of ¹²⁹I increases by orders of magnitude with temperature (see Table A.2). While we produce more ¹²⁹I from this channel in the Set 1 models than in the Set 0 models, its final amount decreases by up to an order of magnitude due to its shorter half-life at stellar temperatures.

The presence of SLRs has been documented in the early Solar System (ESS) from the laboratory analysis of meteorites (e.g. Dauphas & Chaussidon 2011; Davis 2022). These nuclei have now decayed completely, but their ESS abundances can be inferred from measurable excesses in the abundances of their daughter nuclei. Therefore, we cannot determine the solar s-contribution, but we can calculate the so-called abundance production factor P_s; that is, the relative amount produced by the s process compared its stable reference isotope:

$$\begin{array}{c}\hfill P_{\mathrm{s}}=\frac{M_{\mathrm{i}}^{\mathrm{wm}}{m}_{\mathrm{j}}}{M_{\mathrm{j}}^{\mathrm{wm}}{m}_{\mathrm{i}}},\end{array}$$(8)

where i represents the radioactive and j represents the stable isotope of the corresponding element. The ¹²⁹I/¹²⁷I production factors derived by our Set 0 and Set 1 models are 1 × 10⁻⁴ and 2.8 × 10⁻⁴, respectively. These values are similar to the previous Monash result (P_s ≃ 10⁻⁴, Lugaro et al. 2014b), based on AGB models with initial mass between 1.25 and 8.5 M_⊙. These production factors are generally four magnitudes lower than the r-process production factor P(r) = 1.35 (Lugaro et al. 2014b; Côté et al. 2021), which confirms that the ¹²⁹I contribution from the s process is very marginal.

3.1.6. ¹⁵²Gd

The s-process production of ¹⁵²Gd depends on the operation of the ¹⁵¹Sm and ¹⁵²Eu branching points. ¹⁵²Eu can capture electrons and undergo β⁻ decay with a laboratory half-life of 13.3 yr. The rate of electron capture increases with both temperature and electron density, while the rate of the β⁻ channel increases only with temperature. The amount of ¹⁵²Eu produced is determined by the β⁻-decay of ¹⁵¹Sm, whose rate also increases with temperature depending on the electron density (see Table A.2). Comparing the Set 1 models to the Set 0 models, we produce more ¹⁵²Eu, which is more likely to suffer β⁻ decay under stellar conditions than under terrestrial conditions; therefore, we obtain ∼2–28 times more ¹⁵²Gd, depending on the stellar model.

¹⁵²Gd was one of the 35 original p-nuclei (Cameron 1957), but nowadays it is well accepted that it has a large s-process contribution (Arlandini et al. 1999; Bisterzo et al. 2011, 2014), or it is even considered as an s-only isotope (Travaglio et al. 2011). Arlandini et al. (1999) calculated s-process contributions using the arithmetic average yields of 1.5 and 3 M_⊙ half-solar metallicity AGB models from Gallino et al. (1998) and normalised to solar abundances by Anders & Grevesse (1989). Bisterzo et al. (2011) updated their results, taking into account more recent neutron-capture cross sections and using solar abundances from Lodders et al. (2009) to normalise. The s-contribution for ¹⁵²Gd is 0.883 and 0.705 in Arlandini et al. (1999) and Bisterzo et al. (2011), respectively. Bisterzo et al. (2014) obtained the s-components of the solar composition using the GCE code described by Travaglio et al. (2004) and s-process yields from AGB stellar models by Gallino et al. (1998). They predicted 0.865 for the s-contribution of ¹⁵²Gd from low-mass AGB stars. We used the method described in Sect. 3.1.3 to obtain the s-contributions of ¹⁵²Gd. Comparing the predictions of the Set 0 and Set 1 models shows that we need to accurately model the operation of the branching points to understand the nucleosynthesis origin of this isotope. In the Set 0 models, the solar s-contributions of ¹⁵²Gd is only 0.072, which would lead us to conclude that ¹⁵²Gd is produced mainly by the p process. In contrast, Set 1 models predict 0.658 solar s-contributions, which confirm that this isotope predominantly originates from low-mass AGB stars. This prediction is broadly consistent with recent results (Prantzos et al. 2018; Busso et al. 2021) and with the results presented above, especially with Bisterzo et al. (2011), whose method of calculating the s-process contributions was similar to ours, but used only two models. We emphasise that our method gives an approximate estimate, while the Bisterzo et al. (2014) results were based on a full GCE model, which is a more accurate approach because it includes many more models which are also of a much lower metallicity than the ones considered here, and evolves each star according to its mass and therefore lifetime.

3.1.7. ¹⁷⁶Lu and ¹⁷⁶Hf

The β⁻-decay rate of ¹⁷⁶Lu is highly sensitive to temperature during TPs (Takahashi & Yokoi 1987; Klay et al. 1991; Doll et al. 1999) due to the coupling of the very long-lived ground state (t_1/2 = 37.9 Gyr) and the short-lived isomer (t_1/2 = 3.7 h) at 123 keV via a thermal population of mediating states at higher energies, in particular at E = 838.6 keV, since direct transition is forbidden by selection rules. The high T dependence affects the abundance ratio of ¹⁷⁶Lu and its daughter nucleus ¹⁷⁶Hf in AGB stars, which must match the relative solar abundances of these isotopes, since both isotopes are s-only, shielded by ¹⁷⁶Yb against r-process production. In other words, these isotopes must have the same overabundances as ¹⁵⁰Sm; that is, their s-contributions must be close to 1. A minor deviation in the s-contribution (1.05 ± 0.05 for ¹⁷⁶Lu and 1 ± 0.05 for ¹⁷⁶Hf) is acceptable (Heil et al. 2008b) due to the β⁻ decay of ¹⁷⁶Lu in the ISM.

The discrepancy between the theoretical and observed ¹⁷⁶Hf/¹⁷⁶Lu ratio is a long-standing problem that has been addressed in a number of studies over the past decades (e.g. Heil et al. 2008b; Mohr et al. 2009; Cristallo et al. 2010). According to the description of Klay et al. (1991), the branching factor of ¹⁷⁶Lu is a complex function of temperature and neutron density and depends, in particular, on the ratio of the two partial (n,γ) cross sections of ¹⁷⁵Lu to the ground state and to the isomer state of ¹⁷⁶Lu, the lifetime of the first mediating state and the decay branching of the first mediating state towards the isomer. By branching analysis, Heil et al. (2008b) found that a plausible value of the isomeric to total ratio IR of 0.81–0.82 resulted in a ¹⁷⁶Hf/¹⁷⁶Lu ratio consistent with the solar abundances. Less plausible values of 0.8–0.75 implemented in a large set of AGB models (Cristallo et al. 2009) and then in a full GCE model (Prantzos et al. 2020) provided results within 4σ of the solar abundances. However, new experimental data on the nuclear states of ¹⁷⁶Lu (Mohr et al. 2009) led to an overproduction of ¹⁷⁶Lu, which was later confirmed by Cristallo et al. (2010). Current ideas suggest that the solution should be sought in the coupling scheme between the thermally populated states of ¹⁷⁶Lu (Cristallo et al. 2010; Bisterzo et al. 2015).

We calculated the solar s-process contribution for ¹⁷⁶Lu and ¹⁷⁶Hf by Eq. (5) and (6) with Set 1 models, and we strongly overproduced ¹⁷⁶Lu and underproduced ¹⁷⁶Hf: their s-contributions are 2.36 and 0.56, respectively. This is because the NETGEN rate of ¹⁷⁶Lu assumes that the ground state and isomer are in thermal equilibrium under s-process conditions, but this is only true above 300 MK (e.g. Heil et al. 2008b). Interestingly, our results are similar to those obtained by Cristallo et al. (2010, 2.4 and 0.55 for ¹⁷⁶Lu and ¹⁷⁶Hf, respectively) using a 2 M_⊙Z = 0.02 FRANEC model and the β⁻-decay rate of ¹⁷⁶Lu based on parameters of Mohr et al. (2009).

3.1.8. ¹⁸²Hf

The s-process production of ¹⁸²Hf depends on the ¹⁸¹Hf branching point. The β⁻-decay rate of ¹⁸¹Hf is based on Lugaro et al. (2014b) and was already included in the nuclear network of Set 0. In addition to this, the amount of ¹⁸²Hf depends on its own decay. ¹⁸²Hf is a SLR with a terrestrial half-life of 8.8 Myr. Its β⁻-decay rate is T- and ρ-dependent, and it decreases by six orders of magnitude under stellar conditions, according to NETGEN. We obtain 5%–28% less ¹⁸²Hf on the surface at the final stage of the Set 1 models compared to Set 0.

As in the case of ¹²⁹I, we can calculate the production factor of ¹⁸²Hf. We can compare the ¹⁸²Hf/¹⁸⁰Hf production factors derived by individual stellar models with the Monash production factors used by Trueman et al. (2022, Table 6). These authors used two sets of AGB yields and calculated timescales relevant for the birth of the Sun using the OMEGA+ GCE code (Côté et al. 2018) and SLRs heavier than iron produced by the s process. The production factors of Trueman et al. (2022) are slightly higher than our Set 0 results (see Table 4), because their Monash yields came from Karakas & Lugaro (2016), whose models include the terrestrial β⁻-decay rate of ¹⁸¹Hf without any T- and ρ-dependence, which is slightly higher in the stellar interior than under terrestrial conditions. A further small decrease in production factors is also present between the Set 0 and Set 1 models, as expected from the introduction of the ¹⁸²Hf T-dependent decay.

3.1.9. ²⁰⁵Pb

In the case of ²⁰⁵Tl and ²⁰⁵Pb, two processes compete with each other depending on temperature and electron density: the bound-state β decay of ²⁰⁵Tl and the electron capture of ²⁰⁵Pb. While ²⁰⁵Tl is stable under terrestrial conditions, its bound-state decay is active only under stellar conditions, and its rate increases with temperature (Leckenby et al. 2024). In contrast, the rate of electron capture of ²⁰⁵Pb strongly increases with temperature up to 50–100 MK (depending on the electron number density), and then it decreases slightly. According to NETGEN, the rates of these two processes are equal somewhere between 130 MK and 310 MK, the exact temperature of the intersection increasing with increasing electron number density. This means that the ²⁰⁵Tl/²⁰⁵Pb ratio in stellar models strongly depends on the temperature condition (e.g. maximum temperature, average temperature, and their time evolution) of the intershell. We produce less ²⁰⁵Pb and more ²⁰⁵Tl in the Set 1 models compared to Set 0 models because the electron-capture reaction is stronger than the bound-state decay, except for the 4 M_⊙Z = 0.007 model, which is hot enough to overproduce ²⁰⁵Pb, in other words more ²⁰⁵Tl decays to ²⁰⁵Pb than vice versa, and the 4 M_⊙Z = 0.014 model, in which the abundances of both isotopes decrease relative to the Set 0 case, because less ²⁰⁵Tl is produced from the ²⁰⁴Tl(n, γ)²⁰⁵Tl channel. ²⁰⁴Tl is a branching point, it suffers β⁻-decay with a terrestrial half-life of 3.9 yr, which is much shorter in stellar interior (Table A.2).

Using the SLRs and their abundance production factors, we can obtain information on the chronology of the evolution of the ESS, including the time interval between the separation of Solar System material from ISM and the formation of the first solids. This so-called ‘isolation time’ corresponds to the free decay time required for the abundance ratio predicted in the ISM to reach the ESS ratio measured in meteorites (Lugaro et al. 2018b). ²⁰⁵Pb is a unique isotope because among the measurable SLRs heavier than iron, it is the only that cannot be produced by the r process. In addition, its reference isotope ²⁰⁴Pb is also a s-only. Despite this, it has not been extensively used in the literature for calculating the isolation time of the Solar System, because of both the high uncertainty of its ESS ratio (Palk et al. 2018) and the strongly T- and ρ-dependent electron-capture rate, which is not well determined (Mowlavi et al. 1998). We attempted to calculate the isolation time using the ²⁰⁵Pb/²⁰⁴Pb ratio as

$$\begin{array}{c}\hfill T_{\mathrm{iso}}=-\ln \left[\frac{{\left({}^{205}Pb{/}^{204}Pb\right)}_{\mathrm{ESS}}}{{\left({}^{205}Pb{/}^{204}Pb\right)}_{\mathrm{ISM}}}\right]{\tau }_{{}^{205}\mathrm{Pb}},\end{array}$$(9)

where τ_²⁰⁵Pb = 24.5 Myr is the mean life of ²⁰⁵Pb (value is taken from Kondev et al. 2021), (²⁰⁵Pb/²⁰⁴Pb)_ESS = (1.8 ± 0.6) × 10⁻³ (value is taken from Palk et al. 2018, with 1σ error bar), and (²⁰⁵Pb/²⁰⁴Pb)_ISM are the ESS and ISM ratio of ²⁰⁵Pb to ²⁰⁴Pb, respectively. The ISM ratio can be estimated from the following steady-state equation:

$$\begin{array}{c}\hfill {\left({}^{205}Pb{/}^{204}Pb\right)}_{\mathrm{ISM}}=P\times K\times \frac{{\tau }_{{}^{205}\mathrm{Pb}}}{T_{\mathrm{Gal}}},\end{array}$$(10)

where P is the stellar production factor, T_Gal = 8.4 Gyr is the galactic age at the formation of Solar System, and K = 1.6 (min), 2.3 (best), or 5.7 (max) is the GCE parameter that represents the uncertainties coming from galactic evolution (Côté et al. 2019). The obtained ISM ratios in the unit of 10⁻⁴ are 2.4, 3.4, and 8.5, for K = 1.6, 2.3, and 5.7, respectively.

These ISM ratios are lower than the ESS ratio; therefore, we cannot obtain physical – that is, positive – isolation times. It should be noted that we used Eq. (10) with only one production factor calculated from Eq. (6) and (8) from solar metallicity models, while the GCE model of Trueman et al. (2022) handles the production factors of each stellar model. In other words, we used approximate equations; therefore, our method is not as accurate as a full GCE model. However, it is very unlikely that the full GCE models could solve this problem because it has been shown that Eq. (10) reproduces the full GCE models within 50% (Leckenby et al. 2024); therefore, we could obtain a value just above the minimum ESS value (allowed within 1σ) only by applying the maximum value of K and the maximum +50% variation.

This long-standing problem of ²⁰⁵Pb was recently solved by Leckenby et al. (2024), who present new measurement-based rates of the bound-state β decay of ²⁰⁵Tl and the electron capture of ²⁰⁵Pb for a wide range of astrophysical conditions. Using these new rates and a method similar to that described in this work, these authors were able to derive positive isolation times that are consistent with other s-process chronometers.

3.2. Comparison of Set 1 to FRUITY results

With the updated β-decay rates, the isotopic predictions of our models can be compared to those of the FRUITY models, as these were calculated including the T- and ρ-dependence. Because the Z = 0.007 and Z = 0.03 metallicity models are not available in the FRUITY database, we used the Z = 0.014 models for comparison. In general, using the NETGEN rates, most of the abundances of the isotopes discussed in Sect. 3.1 are closer to the FRUITY predictions, as expected. Figure 2 shows the ratio of the final surface abundances normalised to the abundance of ¹⁵⁰Sm of Set 0 and Set 1 relative to FRUITY for isotopes with a mass number greater than A = 56. The highlighted isotopes are those that showed the highest differences between the Set 0 and Set 1 models and were discussed in Sect. 3.1.

Fig. 2. Ratio between the final surface isotopic distributions (normalised to the abundance of 150Sm) of the Set 0 (left panel) and Set 1 (right panel) with respect to the FRUITY calculations of solar metallicity models of different masses, as indicated.

The remaining differences between predictions from our models and FRUITY can result from: (1) Somewhat different nuclear inputs. In fact, FRUITY’s temperature and density-dependent β-decay rates of isotopes heavier than iron are based on Takahashi & Yokoi (1987) with a few exceptions (see Straniero et al. 2006, for details) and use different functions than NETGEN, and therefore Monash code to interpolate from the lowest temperature available in the database to the terrestrial values (log versus linear, respectively). (2) In our post-processing code, the equations that represent the abundance changes include nuclear burning and mixing, whereas the version of the FRANEC code that was used for the FRUITY database combines the calculation of stellar structure and nucleosynthesis, but not mixing. (3) Our method of including the ¹³C pocket differs from that used to produce the FRUITY results, which is based on the time-dependent convective overshoot at the base of the envelope. In addition, the ¹³C pocket is kept constant in our models, while it self-consistently decreases with pulse number in the FRUITY models. (4) Because of different methods and input physics, the total number of TPs differs, our models have more TPs (21, 26, and 22 vs 13, 17, and 9, for 2, 3 and 4 M_⊙ models, respectively). In addition, the maximum temperatures reached by the TPs are also different: while our 2 M_⊙Z = 0.014 model is cooler than the FRUITY model (280 MK versus 291 MK), 3 M_⊙Z = 0.014 models reach quite similar maximum temperatures (302 MK versus 304 MK), our 4 M_⊙Z = 0.014 model reaches higher maximum T (348 MK versus 312 MK) than the FRUITY stars. The effect of temperature difference is most visible at 4 M_⊙, where the branching points are more active and cause a larger fluctuation in the abundances.

3.3. Comparison of Set 2 to Set 1

The extent of the difference between the predictions of Set 2 and Set 1 is affected by how much of the given isotope is produced by each model. This in turn is determined by the features of the model, the total number of neutrons available, and the differences in the nuclear input. The total amount of free neutrons is usually expressed by the time-integrated neutron flux; that is, the neutron exposure,

$$\begin{array}{c}\hfill \tau ={\displaystyle \int N_{\mathrm{n}}}{v}_{\mathrm{th}}\mathrm{dt}.\end{array}$$(11)

The neutron exposure depends on neutron sources and the amount of Fe seed nuclei. Of the two neutron-source reactions, ¹³C(α, n)¹⁶O results in a higher neutron exposure than ²²Ne(α, n)²⁶Mg, because, while only a few percent of ²²Ne nuclei burn, all of the ¹³C nuclei are completely destroyed. While the ¹³C abundance, and thus the number of neutrons produced, does not vary with the metallicity, the amount of Fe seeds increases with increasing metallicity. Therefore, a higher Fe abundance leads to a lower neutron exposure, as there are more Fe seeds to capture neutrons and a smaller number of free neutrons remains for the production of the s-process isotopes. The neutron exposure determines whether the neutron flux can overcome the bottlenecks at stable neutron magic nuclei (neutron number N = 28, 50, 82). Once a bottleneck is bypassed, the neutron-capture flow can reach all isotopes up to the next bottleneck, and the abundances between the bottlenecks are in roughly steady-state equilibrium (Lugaro et al. 2023b):

$$\begin{array}{c}\hfill Y^{\mathrm{i}}{⟨\sigma v⟩}_{\mathrm{i}}\simeq \mathrm{constant}.\end{array}$$(12)

The final surface normalised number abundances, Y_finalⁱ, of 34 isotopes changed by at least 10% in at least one of the Set 2 models with respect to the Set 1 models. The list of these isotopes and their Y_{Set 2, final}ⁱ/Y_{Set 1, final}ⁱ ratio are shown in Table A.3. In the following, we discuss the most relevant cases and in Table 5 we list the ratio of MACSs used in Set 2 and Set 1 for the related isotopes.

3.3.1. ⁶⁴Ni

In the case of ⁶⁴Ni, the ratio between its ASTRAL and KaDoNiS neutron-capture cross sections decreases with temperature (see Table 5). In general, the neutron-capture rate based on ASTRAL is the highest of the two during the ¹³C(α, n) phases, while the rate based on the MACS from KaDoNiS is the higher during the ²²Ne(α, n) phases. In most cases, this results in a 6%–20% reduction in the abundance of ⁶⁴Ni in the Set 2 models compared to the Set 1 models, except for the 4 M_⊙Z = 0.007 model, where there was no difference between the predictions.

Of the five stable Ni isotopes, the most abundant ⁵⁸Ni (which was used as reference isotope for the stardust grain isotopic ratio data) and ⁶⁰Ni are not affected by s-process nucleosynthesis, in contrast to the much rarer ^{61, 62, 64}Ni isotopes (1.14%, 3.63%, and 0.93% of the total Ni abundance, respectively), which can be produced by neutron captures. In Fig. 3, we compare δ⁶⁰Ni₅₈ versus δ⁶⁴Ni₅₈ from Trappitsch et al. (2018) to the predictions of the Set 0 and Set 2 models, since the final surface abundance of ⁶⁴Ni also changes between Set 1 and Set 0 (see Sect. 3.1.1). We see a relatively large reduction (168–724‰, depending on the stellar model and TP) in δ⁶⁴Ni₅₈ from Set 2 to Set 0 during the stellar evolution. The Set 0 models (except for 2 M_⊙Z = 0.014 model) overproduce ⁶⁴Ni relative to grain data at the end of the evolution, while Set 2 models can better cover grain data. We note that the δ⁶⁰Ni₅₈ values do not change between Set 2 and Set 0. Their fluctuation in the stardust grain data is probably due to GCE as the δ⁶⁰Ni₅₈ values do not change significantly in AGB models (Trappitsch et al. 2018).

Fig. 3. δ60Ni58 vs δ64Ni58 from MS SiC grain data (dark grey circles with 1σ error bars from Trappitsch et al. 2018) compared to the surface evolution of Set 0 and Set 2 stellar models of solar (top panel) and of double-solar metallicity (bottom panel) of different masses (see legend). Circles (Set 0) and triangles (Set 2) on the lines represent the composition after each TDU after the envelope becomes C-rich. The dashed black lines represent the solar composition (δ = 0 by definition).

3.3.2. ^{134, 135, 136, 137, 138}Ba

Among the five barium isotopes on the main s-process path, the final surface abundance of ^{134, 136, 137}Ba decreases significantly in the Set 2 models compared to the Set 1 models, consistent with the change in reaction rates (see Table 5). The only exception is the 4 M_⊙Z = 0.007 model, where the final surface abundance of ¹³⁷Ba slightly increases from Set 1 to Set 2, probably due to some effects related to the ¹³⁷Cs branching point.

When we compare the MS SiC grain data to our predictions from the Set 1 and Set 2 models, we conclude the following: (1) The surface abundances of the s-only ¹³⁶Ba used as a reference isotope (as commonly used in stardust grain analysis) are lower in Set 2 models compared to Set 1 models. Following this, the δ¹³⁵Ba₁₃₆ values slightly increase (by 5–11‰), except for the 4 M_⊙Z = 0.03 model, where it increases more significantly (by 26‰). For the same reason, the δ¹³⁴Ba₁₃₆ and δ¹³⁸Ba₁₃₆ values also increase (by 7–38‰ and 55–110‰, respectively), despite the fact that the surface abundances of ¹³⁴Ba also decrease, and the neutron-capture rate of ¹³⁸Ba does not change. (2) The δ¹³⁷Ba₁₃₆ values decrease by 76–100‰ (and by −32‰ in 4 M_⊙Z = 0.03 model) because the cross section of ¹³⁷Ba increase more than the cross section of ¹³⁶Ba (see Table 5). The exception is the 4 M_⊙Z = 0.014 model, where ¹³⁷Ba is overproduced (also relative to stardust grain data) due to activation of branching points at ^{134, 135, 136}Cs (more details in Lugaro et al. 2018a). Vescovi & Reifarth (2021) compared the prediction of the original to the ASTRAL-based neutron-capture dataset of the FRUITY Magnetic models (Vescovi et al. 2020), and found similar results to ours in the cases of δ¹³⁷Ba₁₃₆ and δ¹³⁸Ba₁₃₆.

Comparing our predictions to the grain data, the conclusions of Lugaro et al. (2018a, based on the predictions of the Set 0 models) have not changed significantly. The two main differences are: (1) Set 2 models provide us with a better fit to the bulk of the δ¹³⁷Ba₁₃₆ versus δ¹³⁵Ba₁₃₆ stardust grain data. (2) However, the Z = 0.03 models of Set 2 produce higher δ¹³⁸Ba₁₃₆ than the typical values of SiC grains, which was previously only true for solar metallicity models. According to Lugaro et al. (2018a), the δ¹³⁸Ba₁₃₆ ratios can be reduced by decreasing the M_PMZ parameter. Calculating the M = 3 M_⊙ models with a lower M_PMZ than the standard value (M_PMZ = 5 × 10⁻⁴ M_⊙ instead of 2 × 10⁻³ M_⊙), the predictions of 3 M_⊙Z = 0.03 model become more consistent with the grain data in both Set 1 and Set 2 (Fig. 5, bottom panel), although, as discussed in Lugaro et al. (2018a), more efficient TDU of material with such a composition is needed to reach the most extreme data points. However, the 3 M_⊙Z = 0.014 model still produces higher δ¹³⁸Ba₁₃₆ ratios than the typical values of SiC grains (Fig. 5, top panel).

3.3.3. ¹⁸⁶W

Due to the ASTRAL-based neutron-capture rate of ¹⁸⁶W being lower than the JINA reaclib fit to the KaDoNiS data, the Set 2 models obtain 1.07–1.44 times more ¹⁸⁶W than the Set 1 models at the end of the evolution. Ávila et al. (2012) reported the first tungsten isotopic measurements in SiC grains and compared the isotopic ratios to the predictions of FRUITY and the version of the Monash models³. They found that most models predict lower ¹⁸⁶W/¹⁸⁴W ratios than those measured in the grains. To check whether our new model predictions change this conclusion, we plot the ¹⁸⁶W/¹⁸⁴W versus ¹⁸³W/¹⁸⁴W predictions and data in Fig. 6. The Set 2 models provide us with higher ¹⁸⁶W/¹⁸⁴W ratios than the Set 1 models; in addition, the 4 M_⊙Z = 0.014 model results in a similar ¹⁸⁶W/¹⁸⁴W ratios as the LU-41 grain. However, no models are able to match full composition of LU-41, nor the very high ¹⁸⁶W/¹⁸⁴W ratio shown by the bulk KJB data. It should be noted, however, that this result strongly depends on the neutron-capture rate of the branching point isotope ¹⁸⁵W, for which no direct measurements are available, and current estimates are derived from inverse photodisintegration experiments (Sonnabend et al. 2003; Mohr et al. 2004).

Fig. 4. δ134Ba136 (top row), δ137Ba136 (middle row), and δ138Ba136 (bottom row) vs δ135Ba136 from mainstream SiC grain data (dark grey circles with 1σ error bars from Savina et al. 2003; Barzyk 2007; Marhas et al. 2007; Liu et al. 2014, 2015, 2017a,b, 2019, 2022; Stephan et al. 2018, 2019) compared to the surface evolution of Set 1 and Set 2 stellar models of solar (left panels) and of double-solar metallicity (right panels) of different masses (see legend). Circles (Set 1) and triangles (Set 2) on the lines represent the composition after each TDU after the envelope becomes C-rich. The dashed black lines represent the solar composition (δ = 0 by definition).

Fig. 5. Same as the bottom row of Fig. 4, but including only M = 3 M⊙ models calculated with MPMZ = 5 × 10−4 M⊙.

Fig. 6. 186W/184W vs 183W/184W from single large grains (LU) and the average of small grains (KJB, dark grey circles with 1σ error bars from Ávila et al. 2012) compared to the surface evolution of Set 1 and Set 2 stellar models of solar (top panel) and of double-solar metallicity (bottom panel) of different masses (see legend). Circles (Set 0) and triangles (Set 2) on the lines represent the composition after each TDU after the envelope becomes C-rich. The dashed black lines represent the solar composition (δ = 0 by definition). The solid black line is the regression line from Ávila et al. (2012).

Further constrains on the s-process isotopic composition of W come from other types of meteoritic materials. From analysis of chondrules and matrix of the Allende meteorite, Budde et al. (2016) obtained a slope of +1.25 ± 0.06 (2σ) for the relationship between ¹⁸²W/¹⁸⁴W and ¹⁸³W/¹⁸⁴W (when internally normalised to ¹⁸⁶W/¹⁸⁴W and the solar ratios, see, e.g. Lugaro et al. 2023a, for full details), and Kruijer et al. (2014) obtained +1.41 ± 0.05 from analysis of bulk calcium-aluminium-rich inclusions (CAIs). In Fig. 7, we compare the highest and lowest observed slope, derived when considering the error bars, to the slopes derived by selected Set 2 models. The temperature conditions required to match the observations are found for stellar models in range between the 3 M_⊙Z = 0.014 and the 4 M_⊙Z = 0.03 models. The results do not change significantly when considering Set 1 models. Instead, they change significantly depending on the addition of the abundance of radioactive ¹⁸²Hf to the abundance of ¹⁸²W. In fact, the observed slopes can be matched only if ¹⁸²Hf is fully included (as done in Fig. 7), otherwise all the lines have slopes between 1.8 and 2.4, much higher than observed. This indicated that Hf must have fully condensed together with W in the stardust grains that carried these anomalies into the ESS.

Fig. 7. Linear correlations between ϵ182W (including the 182Hf abundance) and ϵ183W (i.e. ratios internally normalised to 186W/184W and relative to the solar ratios, multiplied by 10 000) derived from the lowest and the upper limit of the observations (solid black lines) and from the abundances calculated at the end of the evolution of Set 2 models of Z = 0.014 (blue lines) and Z = 0.03 (green lines), with stellar mass indicated by corresponding labels and different line types: 2 M⊙ (short-dashed), 3 M⊙ (solid), and 4 M⊙ (long-dashed). We note that the M = 3 M⊙Z = 0.014 line overlaps with the line of the observational upper limit. The dotted lines indicate the solar composition. Addition of s-process material produces negative values, while removal correspond to positive values.

Another interesting result is that when we force, for example, the 3 M_⊙Z = 0.014 model to match the stardust data shown in Fig. 6, by decreasing the abundance of ¹⁸³W by 0.7–0.85 and/or increasing the abundance of ¹⁸⁶W by a factor of 2, the slope always decrease to values outside the observed range (because s-process addition produces negative values). This indicates that the stardust grains that carried the anomalies represented in Fig. 7 are not the same stardust grains plotted in Fig. 6. This may not be surprising given that the plotted single LU grains have very large size (roughly 10–20 μm), and are probably be quite rare, while the bulk grain KJB data point represents a grand average of grains of small size (∼0.5 μm). Instead, the chondrules, matrix, and CAIs data indicate a carrier with the signature of the nucleosynthesis produced in AGB models of mass and metallicity similar to those that match the composition of the single KJG grains, of typical μm size (e.g. Lugaro et al. 2018a). It is also possible that other type of grains than SiC carried the anomalies shown, as already suggested for example by Schönbächler et al. (2005), Reisberg et al. (2009) on the basis of leachates experiments.

4. Conclusions

We have studied how temperature- and density-dependent (instead of constant, terrestrial) decay rates and new neutron-capture cross sections affect s-process nucleosynthesis predictions. We have provided the first database of surface abundances and stellar yields of the isotopes heavier than iron from the Monash models. We used seven stellar structure models (Karakas 2014; Karakas & Lugaro 2016) of low-mass AGB stars, and three sets of the nuclear network (Sets 0, 1, and 2) based on three databases: JINA reaclib (Cyburt et al. 2010), NETGEN (Jorissen & Goriely 2001; Aikawa et al. 2005; Xu et al. 2013), and ASTRAL (Reifarth et al. 2018b; Vescovi & Reifarth 2021; Vescovi et al. 2023). We compared the final surface abundances of the isotopes for each set and discussed the relevant cases. We calculated the solar s-process contribution for the p-only isotopes ⁹⁴Mo, ¹⁰⁸Cd, and ¹⁵²Gd, as well as for the s-only isotopes ¹⁷⁶Lu and ¹⁷⁶Hf, the radioactive-to-stable abundance ratios for the short-lived radionuclides ¹²⁹I, ¹⁸²Hf, and ²⁰⁵Pb, and the isolation time of the Solar System using a steady-state equation and the ²⁰⁵Pb/²⁰⁴Pb ratio. We compared our solar metallicity models with the same FRUITY (Cristallo et al. 2009, 2011, 2015) models, which also use T- and ρ-dependent decay rates. In addition to the above, we compared the predictions of the solar and double-solar metallicity models with MS presolar stardust SiC grain data for the ⁶⁰Ni/⁶⁴Ni, ⁸⁰Kr/⁸²Kr, ⁹⁴Mo/⁹⁶Mo, ^{134, 135, 137, 138}Ba/¹³⁶Ba, and ¹⁸⁶W/¹⁸⁴W ratios.

Our main conclusions are:

• (1)

  Taking into account the T and ρ dependence of the decay rates of the ⁷⁹Se and ⁸⁰Br branching isotopes (Set 1), our predictions become more consistent with the intershell ⁸⁰Kr/⁸²Kr ratios derived by Lewis et al. (1994) from stardust SiC grain data than the Set 0 models (see Table 3).

• (2)

  Taking into account the T and ρ dependence of the decay rates of the ⁹³Zr and ⁹⁴Nb branching isotopes (Set 1), the surface evolution of the 2 and 3 M_⊙ solar and double-solar metallicity models follow closely the regression line of the [⁹⁴Mo/⁹⁶Mo] versus [⁹²Mo/⁹⁶Mo] SiC grain data (see Fig. 1). Our models quantitatively reproduce the minor s-process production of the classical p-only ⁹⁴Mo observed in stardust SiC grains, of around 3–4%.

• (3)

  Taking into account the T and ρ dependence of the decay rates of the ¹⁵¹Sm and ¹⁵²Eu branching isotopes (Set 1), we predict a high, 0.7 solar s-process contribution for the traditionally p-only ¹⁵²Gd, which is consistent with results from previous studies (Arlandini et al. 1999; Bisterzo et al. 2011, 2014).

• (4)

  Our Set 1 models do not reproduce the Solar System ratio of the two s-only isotopes, ¹⁷⁶Lu and ¹⁷⁶Hf, due to the NETGEN rate of ¹⁷⁶Lu incorrectly assuming thermal equilibrium between the long-lived ground state and the short-lived isomer over the whole temperature range. This problem can be solved by using a different T-dependent β-decay rate for ¹⁷⁶Lu (e.g. from Heil et al. 2008b), but we note that the coupling between the ground state and the isomer is still uncertain and needs to be revised.

• (5)

  Taking into account the T and ρ dependence of the decay rates of ²⁰⁵Tl, ²⁰⁴Pb and ²⁰⁵Pb (Set 1), we cannot obtain a positive solution for the isolation time of the Solar System using the ²⁰⁵Pb/²⁰⁴Pb ratio from solar metallicity models, which suggests that the rates of the ²⁰⁵Tl–²⁰⁵Pb decay system need to be revised. This was recently done by Leckenby et al. (2024).

• (6)

  With the new (n,γ) cross sections of ⁶⁴Ni and T- and ρ-dependent decay rates of ⁶³Ni and ⁶⁴Cu (Set 2 models), the δ⁶⁴Ni₅₈ values derived from the solar and double-solar metallicity models are smaller than those of Set 0, and better cover the SiC grain data (see Fig. 3).

• (7)

  With the new (n,γ) cross sections of ^136, 137Ba (Set 2), the solar and double-solar metallicity models provide us with a better fit to the δ¹³⁷Ba₁₃₆ SiC grain data, but produce higher δ¹³⁸Ba₁₃₆ than the typical values of the grains and those of the previous sets (see Fig. 4).

• (8)

  The Set 2 models produce higher ¹⁸⁶W, although they still do not match well the W isotopic composition of large SiC stardust grains well (Fig. 6). However, they do match well the W composition observed in other types of meteoritic materials (Fig. 7).

Future plans involve extending our work based on recent and future measurements of β-decay rates and (n,γ) cross sections. The PANDORA project (Plasmas for Astrophysics, Nuclear Decays Observation and Radiation for Archaeometry, Mascali et al. 2022) will be a possible source of β-decay rates from plasma measurements. The first experimental campaign involves three important isotopes for the s process: ⁹⁴Nb, ¹³⁴Cs, and ¹⁷⁶Lu. The CERN neutron time-of-flight facility, n_TOF (Guerrero et al. 2013) has been providing neutron-capture cross sections for various isotopes for many years. The most recent publications present the (n,γ) cross-section measurements of ⁸⁹Y (Tagliente et al. 2024), ⁹⁴Nb (Balibrea-Correa et al. 2023), ¹⁴⁰Ce (Amaducci et al. 2024), and ²⁰⁴Tl (Casanovas-Hoste et al. 2024) that are relevant for the s process. The effect of the ²²Ne(α, n)²⁵Mg reaction rate and its uncertainties will need to be tested. Direct cross-section measurement of this neutron-source reaction is currently being carried out at the LUNA (Laboratory for Underground Nuclear Astrophysics, Ananna et al. 2024). In addition, we plan to calculate more models with different masses and metallicities to be used as inputs for the GCE of the isotopes of the elements heavier than iron and to carry out further detailed comparisonS with meteoritic data.

Data availability

Surface isotopic abundance and stellar yield data tables of the seven stellar models of low-mass AGB stars for the three nuclear networks are available on Zenodo through the following link: https://zenodo.org/records/14981333.

Acknowledgments

This project has been supported by the Lendület Program LP2023-10 of the Hungarian Academy of Sciences. This work was supported by the NKFIH Excellence Grant TKP2021-NKTA-64. Amanda Karakas was supported by the Australian Research Council Centre of Excellence for All Sky Astrophysics in 3 Dimensions (ASTRO 3D), through project number CE170100013. Balázs Szányi thanks ASTRO 3D for supporting his travel to Monash University. The authors thank Marco Pignatari for his constructive comments throughout this project. The authors thank the anonymous referee for their careful comments, which helped to improve the paper.

References

Appendix A: Additional tables

All Tables

All Figures

Fig. 1. [94Mo/96Mo] vs [92Mo/96Mo] from SiC grain data as dark grey circles MS grains and downward and upward triangles for Y and Z grains, respectively, with 1σ error bars (from Barzyk 2007; Liu et al. 2017b, 2019; Stephan et al. 2019) compared to the surface evolution of Set 0 and Set 1 stellar models of solar (left panel) and of double-solar metallicity (right panel) of different masses (see legend). Circles (Set 0) and triangles (Set 1) on the lines represent the composition after each TDU after the envelope becomes C-rich. The solid black line represents the linear regression from δ94Mo96 vs δ92Mo96, while the dashed black line represents the [94Mo/96Mo] = [92Mo/96Mo] line.

In the text

	Fig. 2. Ratio between the final surface isotopic distributions (normalised to the abundance of 150Sm) of the Set 0 (left panel) and Set 1 (right panel) with respect to the FRUITY calculations of solar metallicity models of different masses, as indicated.
In the text

Fig. 3. δ60Ni58 vs δ64Ni58 from MS SiC grain data (dark grey circles with 1σ error bars from Trappitsch et al. 2018) compared to the surface evolution of Set 0 and Set 2 stellar models of solar (top panel) and of double-solar metallicity (bottom panel) of different masses (see legend). Circles (Set 0) and triangles (Set 2) on the lines represent the composition after each TDU after the envelope becomes C-rich. The dashed black lines represent the solar composition (δ = 0 by definition).

In the text

Fig. 4. δ134Ba136 (top row), δ137Ba136 (middle row), and δ138Ba136 (bottom row) vs δ135Ba136 from mainstream SiC grain data (dark grey circles with 1σ error bars from Savina et al. 2003; Barzyk 2007; Marhas et al. 2007; Liu et al. 2014, 2015, 2017a,b, 2019, 2022; Stephan et al. 2018, 2019) compared to the surface evolution of Set 1 and Set 2 stellar models of solar (left panels) and of double-solar metallicity (right panels) of different masses (see legend). Circles (Set 1) and triangles (Set 2) on the lines represent the composition after each TDU after the envelope becomes C-rich. The dashed black lines represent the solar composition (δ = 0 by definition).

In the text

	Fig. 5. Same as the bottom row of Fig. 4, but including only M = 3 M⊙ models calculated with MPMZ = 5 × 10−4 M⊙.
In the text

Fig. 6. 186W/184W vs 183W/184W from single large grains (LU) and the average of small grains (KJB, dark grey circles with 1σ error bars from Ávila et al. 2012) compared to the surface evolution of Set 1 and Set 2 stellar models of solar (top panel) and of double-solar metallicity (bottom panel) of different masses (see legend). Circles (Set 0) and triangles (Set 2) on the lines represent the composition after each TDU after the envelope becomes C-rich. The dashed black lines represent the solar composition (δ = 0 by definition). The solid black line is the regression line from Ávila et al. (2012).

In the text

Fig. 7. Linear correlations between ϵ182W (including the 182Hf abundance) and ϵ183W (i.e. ratios internally normalised to 186W/184W and relative to the solar ratios, multiplied by 10 000) derived from the lowest and the upper limit of the observations (solid black lines) and from the abundances calculated at the end of the evolution of Set 2 models of Z = 0.014 (blue lines) and Z = 0.03 (green lines), with stellar mass indicated by corresponding labels and different line types: 2 M⊙ (short-dashed), 3 M⊙ (solid), and 4 M⊙ (long-dashed). We note that the M = 3 M⊙Z = 0.014 line overlaps with the line of the observational upper limit. The dotted lines indicate the solar composition. Addition of s-process material produces negative values, while removal correspond to positive values.

In the text