Reactions for Weak s-process

The ²²Ne(α, n)²⁵Mg reaction is active at T ~ 0.2 GK and 1.0 GK in He and C burning shells, respectively. This reaction competes with ²²Ne(α, γ)²⁶Mg, which consumes ²²Ne without releasing neutrons. In these shells, ¹⁶O is the most abundant isotope and acts as the main neutron poison through ¹⁶O(n, γ)¹⁷O. Fortunately, the neutrons absorbed by ¹⁶O can be released again via ¹⁷O(α, n)²⁰Ne. Therefore, the availability of neutrons for the ws-process is determined by the (α, n)/(α, γ) ratio for both ²²Ne+α and ¹⁷O+α reactions.

In Fig. 1, we show the (α, n)/(α, γ) ratios for the ²²Ne+α (top panel) and ¹⁷O+α (bottom panel) reactions as a function of the temperature. In the top panel, the (α, n)/(α, γ) ratio for ²²Ne+α, as recommended by Wiescher et al. [40], is observed to be 1.2 to 2.0 times higher than the values provided by REACLIB below 1.5 GK, a range typically associated with He and C shell burning. Notably, this enhancement increases dramatically, reaching several tens of times above 1.5 GK.

Fig. 1Full-screen View

(Color online) The (α, n)/(α, γ) ratio as a function of temperature for ²²Ne+α (top) and ¹⁷O+α (bottom) reactions

In the bottom panel, the (α, n)/(α, γ) ratio for the ¹⁷O+α reaction suggested by Best et al. [39] is similar to REACLIB below 0.7 GK, where only He burns, but this ratio rapidly increases to several tens of times in the C, Ne, and O layers. With these updated reaction rates, we anticipate an increase in neutron release from ²²Ne while reducing neutron consumption by ¹⁶O. Consequently, the yields of the ws-process isotopes were significantly enhanced.

Input Physics in MESA

To achieve the convergence of the model structures within approximately 10%, a nuclear network comprising at least 127 isotopes should be included [43]. In this study, we utilized a more extensive nuclear network (mesa_161.net) that incorporates additional neutron-rich isotopes. Table 2 lists all the isotopes in mesa_161.net. We adopt a metallicity of and assume solar metallicity ratios based on the work of Anders and Grevesse [44].

We enhanced both the temporal and spatial resolutions to ensure numerical convergence. The mass resolution is critical for accurately capturing changes in the stellar structure [43, 45]. The parameter max_dq controls the maximum fractional mass of a cell in the model, and we set max_dq=5d-4, which resulted in over 3,500 cells in the model. We adopted a minimum diffusion coefficient of D_min = 10^-2 cm² s^-1 to ensure that the global mixing timescale () was significantly longer than the lifetimes of the stellar models. This allows us to neglect the effects of global mixing and smooth local composition gradients [45].

After C burning, the core structure becomes more complex because of multi-shell burning, with the central entropy being significantly influenced by shell burning (see Xin et al. [42]). To achieve finer granularity during evolution, we imposed limits on the changes in the logarithm of the central density and temperature. Specifically, we set and . Additionally, we restrict the change in the mass fraction of isotopes with dX_nuc_drop_limit=3 × 10^-2 and tightening this limit to dX_nuc_drop_limit_at_high_T=10^-2 when log T_c > 9.45.

Post-processing Calculation with WinNet

The detailed nucleosynthesis in the stellar models was computed in post-processing using an extensive nuclear reaction network code, WinNet [38]. The network consists of approximately 2000 isotopes from neutron and proton to thorium (Z = 90). The reaction rates of , , , (α, n), (α, p), , and their inverse reactions from the JINA REACLIB database [31] were included. Theoretical weak rates from Langanke and Martínez-Pinedo [46], electron chemical potentials from Timmes and Arnett [47], and screening corrections from Kravchuk and Yakovlev [48] were used.

For each stellar model, we mapped the initial composition and time evolution of the temperature and density from the MESA simulation on the trajectories. The nucleosynthesis calculation of these trajectories was performed until the onset of iron core collapse. The region inside the steepest density jump is expected to eventually collapse into a neutron star and does not contribute to the yield of the ws-process nucleosynthesis. The steepest density jump occurs at the most active burning shell and has been defined by Xin et al. [42]. We briefly describe the MESA result in Sect. 3.

Evolution of Massive Stars

In Fig. 2, we show the central temperature against the central density for the evolution of stars with M (ZAMS) = 15, 20, 25, and 30 M_⊙. The gray dashed lines show the ignition lines of C, Ne, O, and Si burning, where the energy generation rate by nuclear burning equals the energy loss rate by neutrino emissions. In the region to the left of the black line, stars are dynamically unstable owing to electron-positron pair creation (indicated as “pair instability”) (Ohkubo et al. [49]), general relativistic effects (“GR instability”) (see, for example, Osaki [50]), and the photo-disintegration of matter in nuclear statistical equilibrium (NSE) at Y_e = 0.5 (“photodisintegration”) (Ohkubo et al. [49]). After core He burning, the mass fraction of ¹²C in the center is smaller for larger M (ZAMS). Only the star with M (ZAMS) = 15 M_⊙ can ignite convective C burning in the center, as it has sufficient fuel with X(¹²C) ~ 0.2. In contrast, other models with larger M (ZAMS) undergo contraction because the neutrino energy loss rate exceeds the energy production rate of C burning. After Si burning, the star with M (ZAMS) = 15 M_⊙ exhibits distinct behavior compared to other models because shell Si burning is energetic. However, the effects of shell Si burning are not the focus of this study and will be discussed in future works.

Fig. 2Full-screen View

(Color online) The central temperature against the central density for the evolution of stars with M (ZAMS) = 15, 20, 25, and 30 M_⊙. The gray dashed lines show the ignition lines of C, Ne, O, and Si burning, where the energy generation rate by nuclear burning equals the energy loss rate by neutrino emissions. In the region to the left of the black line, stars are dynamically unstable owing to electron-positron pair creation (indicated as “pair instability”) (Ohkubo et al. [49]), general relativistic effects (“GR instability”) (see, for example, Osaki [50]), and the photo-disintegration of matter in nuclear statistical equilibrium (NSE) at Y_e = 0.5 (“photo-disintegration”) (Ohkubo et al. [49])

More massive stars eject more materials but explode less frequently [51]. Considering the combined effects of ejected masses and event frequencies, stars with M (ZAMS) = 25 M_⊙ are regarded as the most significant contributors to the chemical enrichment of galaxies [21, 52]. Therefore, we select the model of M (ZAMS) = 25 M_⊙ as a typical example for discussing stellar nucleosynthesis.

In Fig. 3, we present the Kippenhahn diagram for the star with M(ZAMS) = 25 M_⊙, tracking its evolution from H burning to Fe core collapse. The central temperature reaches approximately 0.2 GK at τ = t_final-t = 10^5.6 yr. Here, t is the time from the ZAMS and t_final denotes the time at the final stage of evolution, which is defined as the moment when the infall speed of the Fe core reaches 1000 km s^-1. The orange line indicates the isotherm at T = 0.2 GK.

Fig. 3Full-screen View

(Color online) The Kippenhahn diagram of the star with M(ZAMS) = 25 M_⊙. The inner part of M_r = 0 - 14 M_⊙ is shown. Blue, gray, and pink represent the convection, overshoot, and semiconvection regions, respectively. The orange line is the isotherm line of 0.2 GK, and the red line shows the location of M_r=1.84 M_⊙. Between these two lines, the hatched region indicates where the ws-process is considered

The ws-process is assumed to occur interior to this isotherm. After core He burning, this region extends to M_r ~ 6.0 M_⊙, and the He and CO core masses are 7.8 and 4.96 M_⊙, respectively. C burning ignites off-center at yr, nearly three years before the collapse. After τ=10^-3 yr (10 h before collapse), shell C burning merges with shell O burning at M_r = 1.84 M_⊙, marking the location of the highest energy generation rate, as indicated by the red line. The inner part of this region is predicted to form a proto-neutron star (PNS), whereas the outer layers are ejected. Therefore, this study considers only the ws-process isotopes produced in the hatched region.

Figure 4 illustrates the mass distributions of the main isotopes at . The ws-process region extends from the Si/O interface to the bottom of the He burning shell, primarily composed of ¹⁶O, ²⁸Si, ²⁰Ne, ¹²C, and ⁴He. The mass fraction of the iron-group elements ranges from 10^-4 to 10^-5. Additionally, the neutron excess, expressed as , of the ws-process region was observed to be 10^-3–10^-4, as shown in Fig. 5. In the interior to M_r=1.84 M_⊙, the neutron excess increases rapidly toward the center, reaching η ~ 0.2 at the center. This jump is primarily attributed to the reactions during O burning, including ¹⁶O(¹⁶O, n)³¹S and the weak interactions such as ³⁰P(e⁺, ν)³⁰S, ³³S(e^-, ν)³³P, ³⁵Cl(e^-, ν)³⁵S, and ³⁷Ar(e^-, ν)³⁷Cl [53].

Fig. 4Full-screen View

(Color online) The mass distribution of the main isotopes at t=t_final from MESA. The two grey regions indicate where M_⊙ and GK, respectively

Fig. 5Full-screen View

(Color online) The mass distribution of the Y_e (red) and logρ (blue) at t=t_final. The two black lines indicate where M_r = 1.84 M_⊙ and T = 0.2 GK, respectively

The Mass Cut

In Fig. 5, prominent jumps in both the density and Y_e are observed near the mass coordinate M_r = 1.84 M_⊙. This layer corresponds to the base of shell O burning, which is the layer of peak energy generation. To measure the strength of shell burning, we use(1)where U and V are defined in earlier studies [54-57].

As explained in detail by Xin et al. [42], U is related to the degree of the density jump, and V/U is the pressure gradient against M_r. The mass coordinate at which V/U reaches its maximum is represented as M(V/U_max). The relationship between V/U and the strength of shell burning is straightforward. When shell O burning is more energetic, it produces higher energy to prevent the contraction and even to cause the expansion of outer layers, which makes the gradients of entropy and pressure against Mr (i.e., V/U) larger.

Figure 6 shows the distribution of log V/U against M_r at for each model. We note that M(V/U_max) coincides with M₄, that is, M_r at a specific entropy of s=4 erg g^-1K^-1, which was previously used for the mass cut that would divide the inner proto-neutron star (PNS) and the outer ejecta in the explosion [58-61]. In the present study, we adopt M(V/U_max) as the mass cut, because it is the location of the steepest gradients of pressure and density [42]. The core masses and M(V/U_max) values for our models are listed in Table 3.

Fig. 6Full-screen View

(Color online) The mass distribution of log V/U at t=t_final. The black diamonds represent the location of M₄, which is defined at the mass coordinate of specific entropy s=4 erg g^-1K^-1)

The Nucleosynthesis in the 25 M_⊙ model

In this section, we present the results of post-process nucleosynthesis and discuss the effects of the updated reaction rates.

We selected a zone at M_r = 2.3 M_⊙ as a representative example to reveal the change in the mass fraction X(i) of isotope i occurring during each burning stage after updating the (α, n) reaction rates. X(Ga-Zr) and X(Nb-Th) are the cumulative mass fractions of isotopes from Ga to Zr (Z = 31 - 40) and Nb to Th (Z>41-90), respectively. The reaction rate recipes used in Fig. 7(a–d) correspond to cases 1–4 listed in Table 1. The chemical evolution of the main isotopes is depicted in the top panel of Fig. 7. The changes in X(n) and ⁴He for the default rates are displayed in the bottom panel of Fig. 7(a), while in Fig. 7(b)–(d), the value is normalized by X_def(i) in Fig. 7(a) to emphasize the effect of these reaction rates. Table 4 lists the total mass fraction of the “Ga-Zr” elements in the initial abundance (X_ini), after He burning (X_He-b), C burning (X_C-b), and Ne burning (X_Ne-b). The data are visualized in Fig. 8 by the ratios of ΔX to X_ini, where ΔX is the change in the mass fraction in each burning stage.

Fig. 7Full-screen View

(Color online) Chemical evolution in the OSi shell (M_r=2.3 M_⊙). In the top panels of (a)–(d), changes in the mass fractions X(i) of ⁴He, ¹²C, ¹⁶O, and ²⁰Ne show the burning stages. X(Ga - Zr) and X(Nb - Th) are the cumulative mass fractions of the s-process isotopes from Ga to Zr (Z = 31 - 40) and Nb to Th (Z=41-90), respectively. In the bottom panel, the changes in X_def(i) of neutrons and ⁴He for the default rates are shown in (a). In (b)–(d) shown are the changes in the ratios X(i)/X_def(i) between the “new” and default rates for n and ⁴He

Fig. 8Full-screen View

(Color online) The total mass fraction of the ws-process isotopes from A=31-40 after He, C, and Ne burning. All values are normalized by the initial abundance, X_ini=7.87 × 10^-8. The total value is written above each pillar, and the fractional contribution of each burning stage is written in the center of the box

Overall, these new reaction rates significantly alter the production of the “Ga-Zr” elements rather than the “Nb-Th” elements. Thus, we focus on the “Ga-Zr” elements in this section. The initial value of X(Ga-Zr) was . Enhancements in X(Ga-Zr) are observed four times at the end of He burning, the beginning of C burning, the end of C burning and Ne burning. These coincide with the neutron peaks and ⁴He production in Fig. 7.

After Ne burning, the total enhancement of X(Ga-Zr) is estimated by the ratio of ()/X_ini, where and X_ini are listed in Table 4. Compared with X_ini, X(Ga-Zr) increases by a factor of 6.56, 23.77, 31.58, and 113.62 for (a)–(b), respectively. The forthcoming O burning will not enhance but rather reduce their abundance because of the increased destruction at high temperatures [15].

When the default rates are used as in Fig. 7(a), the “Ga-Zr” elements are mainly synthesized during the He (51%) and C (41%) burning stages (see Fig. 8). Only 8% of them are synthesized during the Ne burning stage because the main neutron source isotope ²²Ne is almost exhausted during this stage.

With the new ¹⁷O+α reaction rates in Fig. 7(c), more than 93% of the “Ga-Zr” elements are synthesized during the He and C burning stages, which is similar to (a). The final X (Ga-Zr) is enhanced by a factor of 4.81 compared to the default rates. Because both new ¹⁷O(α, n)²⁰Ne and ¹⁷O(α, γ)²¹Ne reaction rates are lower than the default ones at temperatures below 0.7 GK (see Sect. 5.1), X(¹⁷O) reaches a higher level at the end of He burning with the new rates. When the temperature exceeds 0.7 GK, the ratio of (α, n)/(α, γ) increases significantly. Therefore, the new ¹⁷O+α reaction rates significantly enhanced the production of the “Ga-Zr” elements at all stages, although only slightly alter their contribution percentages.

Comparing Fig. 7(a) and (b), the production of the “Ga-Zr” elements is enhanced by a factor of 23.8 using the new ²²Ne+α reaction rates. This is smaller than the increase in using the new ¹⁷O+α reaction rates. Because the new ²²Ne+α rates are smaller than those in REACLIB (see Fig. 14), ²²Ne is not exhausted until the core collapse. The (α, n)/(α, γ) ratio of the new rates is 10 times higher than that of the default rates when the temperature exceeds 1.5 GK. A significant increase in neutron levels was observed six months before the explosion. As a result, almost 89% of the “Ga - Zr” elements are synthesized during C and Ne burning.

In Fig. 7(d), both the new ¹⁷O+α and new ²²Ne+α reaction rates are updated. The production of the “Ga-Zr” elements was enhanced by more than one order of magnitude. However, the contributions of He and Ne burning are only 10% and 18%, respectively, and most of the “Ga-Zr” elements are synthesized during the C burning stage, which should alter the isotope composition of the “Ga-Zr" elements. By comparing (a, c) with (b, d), we can also determine whether Ne burning contributes significantly to the ws-process, depending on the ²²Ne(α, γ)²⁵Mg reaction rate.

Figure 9 shows the abundance distributions of the s-process elements. We observe two distinct bumps in the abundance of both “Ga-Zr” and “Nb-Th” in the region of M_r = 2.2–5.9 M_⊙. The first bump, located at M_r = 2.0–3.6 M_⊙, corresponds to the C, Ne, and O burning shells, while the second bump, found at M_r = 5.0–5.9 M_⊙, is associated with shell He burning. Between these two bumps, X (Ga-Zr) decreases owing to low α production in the unburned regions.

Fig. 9Full-screen View

(Color online) The abundance distributions of the ws-process elements at M_r = 1.84 - 7.3 M_⊙ in the He core of the M(ZAMS)=25 M_⊙ star. The solid lines symbolize the “Ga-Zr" elements, while the dashed lines symbolize the heavier elements

In Fig. 10, the distribution of ²¹Ne is similar to the “Ga-Zr” elements, while ²²Ne displays an opposite trend. Compared to the unburned shells, more ⁴He is produced in the burning shells, which can consume ²²Ne and release more neutrons. The new ²²Ne+α reaction rates enhanced X (Ga-Zr) only in the first bump, whereas the new ¹⁷O+α reaction rates positively affected X (Ga-Zr) across all the ws-process regions. Additionally, the new ¹⁷O+α reaction rates increase X (Nb–Th) by 50%, unlike the ²²Ne+α reaction rates, because these elements are produced during the He burning stage. Notably, the sharp peak observed at M_r = 1.84–2.0 M_⊙ remains unaffected by both the ²²Ne+α and ¹⁷O+α reaction rates, as these “Ga - Zr” elements are generated through the NSE process.

Fig. 10Full-screen View

(Color online) The distribution of the mass fractions of the main isotopes at obtained from the post-processing calculation. Note that mixing is not considered in the post-processing calculation because WinNet is a one-zone code. A comparison between Fig. 10 and Fig. 4 is presented in Sect. 5.2

In this section, we follow the variation of the ws-process isotopes throughout the stellar evolution history and their mass distribution at the final stage for various reaction networks. We found that both the new ²²Ne+α and ¹⁷O+α reaction rates increased the production of ws-process isotopes.

(1) The new ¹⁷O+α reaction rates only increase neutron density by ~3 times during He and C burning stages. In contrast, the new ²²Ne+α reaction rates increase the neutron density by several tens of times during the C and Ne burning.

(2) The new ¹⁷O+α reaction rates doesn’t vary the contribution in each burning stage. In contrast, the new ²²Ne+α reaction rates significantly increase the contribution in the C and Ne burning stages, but decrease that in the He burning stage.

(3) Before the explosion, the ws-process isotopes are primarily concentrated in the burning shells, with their abundances decreasing in the outer layers of the CO core. This is because there is no C burning in the outer layers of the CO core, and thus, little ⁴He is released.

The Integrated Yields

As indicated in Fig. 7, the ws-process isotopes produced between the mass cut and the top of the He burning shell may contribute to the overall enhancement. We integrated all isotopes in this region rather than the entire star. We assumed that the modifications of the ws-process yields in explosive nucleosynthesis can be ignored and that all radioactive isotopes decay into stable ones after the explosion. Such approximations are reasonable for investigating the sensitivity of the reaction rates.

In Fig. 11 and 12, we show the ratios between the yields with the new reaction rates (Yield_new) and those with the default reaction rates (Yield_def) for M(ZAMS) = 15, 20, 25, and 30 M_⊙, respectively.

Fig. 11Full-screen View

(Color online) The ratios of isotopic yields (from C to Zr) between new rates and default rates for M(ZAMS) = 15, 20, 25 and 30 M_⊙, respectively

Fig. 12Full-screen View

(Color online) Same as Fig. 11, but for elements from Mo to Bi

(1) From C to Zn: With the new ²²Ne+α reaction rates, we observe the increase in the yields of several neutron-rich isotopes of Ne, Mg, Si, S, Ar, and Ca, particularly those in ²⁵Mg, ^{29, 30}Si, ³⁶S, ⁴⁰Ar, and ⁴⁶Ca. The significant decrease in the ²²Ne(α, γ)²⁶Mg reaction rate at GK (during Ne burning) reduces the yield of ²⁶Mg. As a result, a greater amount of ²²Ne was converted to ²⁵Mg (see Sect. 5.1), leading to a significant increase in neutron production during Ne burning. Consequently, the yields of ^{21, 22}Ne increase owing to neutron capture on ²⁰Ne. Similarly, some ^{29, 30}Si and most of the rare isotopes ³⁶S, ⁴⁰Ar, and ⁴⁶Ca are also produced in the Ne burning shell [21]. In contrast, the new ¹⁷O+α reaction rates do not increase the number of neutrons in the Ne shell, thus leaving the yields of those isotopes unchanged. The iron peak isotopes remain unaffected by both the new ¹⁷O+α and ²²Ne+α reaction rates, as they are primarily produced by the NSE process at M_⊙, which is affected by Y_e. However, Y_e is only altered by weak interactions.

(2) From Ga to Zr: The isotopes in this range are most significantly affected by the new reaction rates. The effect of the new ¹⁷O+α rates differs significantly from that of the new ²²Ne+α reaction rates. For the same element, the increases owing to the new ¹⁷O+α reaction rates in the isotopic yields are similar. However, for the new ²²Ne+α reaction rates, the yields of some isotopes with fewer neutrons were reduced. With enriched neutrons, the isotopic yields increase rapidly. As M(ZAMS) increases, the enhancement due to the new ¹⁷O+α reaction rates also increases, whereas the enhancement due to the new ¹⁷O+α reaction rates is not significantly affected by M(ZAMS).

(3) From Mo to Bi The isotopes in this range were not significantly altered by the new reaction rates. The new ²²Ne+α reaction rates increase the yields of only a few isotopes for M(ZAMS) = 15 M_⊙ models. The number of such isotopes increases only slightly for more massive models.

In principle, the abundance of elements in the solar system arises from the cumulative contributions of numerous generations of stars with varying metallicities. Typically, stars in the range contribute more than 90% to the solar abundance [14]. It is general to use the production factors (PFs) to see which element is contributed significantly by a generation of stars. The PF of element i is defined as(2)where denotes the solar mass fraction of element i. runs over all elements and represents the yield of element i averaged by the initial mass function (IMF) from Salpeter [62] with γ=-2.35.

Figure 13 shows the production factors of elements from Cu to Zr. Using the default and new ²²Ne+α reaction rates, most elements are underproduced (green lines). However, when the new ¹⁷O+α reaction rates are included, the PFs from Zn to Rb increase by more than 0.5 dex. The contribution of stars with 0.1 to the solar abundance should be limited, as approximately 50% of the solar abundance is produced by stars with . Nevertheless, accounting for both the new reaction rates leads to the overproduction of Ga, Ge, As, and Se. Therefore, it is worth calculating the models with 0.5 and and verifying whether the predictions concerning these elements align with observational data. Given the considerable uncertainties involved, it is essential to enhance the measurement accuracy of the (α,n) reaction rate, particularly for the ¹⁷O+α reactions.

Fig. 13Full-screen View

(Color online) The production factors of the elements from Cu to Zr. Each element was integrated from 15 to 30 M_⊙ with the Salpeter IMF with γ=-2.35

Comparisons with Results from Other Reaction Rates

In Fig. 14(a), we compare the ²²Ne+α and ¹⁷O+α reaction rates used in JINA REACLIB with those reported in different references. As mentioned in Sect. 2.1, the (α, n)/(α, γ) ratio of W23 increases quickly above 1.5 GK because the ²²Ne(α, γ)²⁶Mg reaction rate decreases significantly. Compared with the default rates, the rates in W23 enhance the contribution of Ne burning in the ws-process (see Fig. 8). Related enhancement is also observed in the abundance of “Ga-Zr” elements in the region of M_r = 2.2–3.5 M_⊙ in Fig. 15.

Fig. 14Full-screen View

(Color online) Top: ²²Ne(α, γ)²⁶Mg and ²²Ne(α, n)²⁵Mg reaction rates from different references. The gray dashed line shows the default rates in JINA REACLIB from Longland et al. [19]. The red and blue lines represent the reaction rates reported by Adsley et al. [63] and Wiescher et al. [40]. Bottom: ¹⁷O(α, γ)²¹Ne and ¹⁷O(α, n)²⁰Ne reaction rates from different references. The gray dashed lines show the default rates in JINA REACLIB, which are from CF88 and NACRE. The red and green lines represent the reaction rates reported by Best et al. [39] and Hammache et al. [64]. To show the difference, all the rates shown here are normalized by the default rate in JINA REACLIB

Fig. 15Full-screen View

(Color online) Same as Fig. 9, but only for the abundances of “Ga-Zr”’ elements. The effects of ²²Ne+α reaction rates in different references are compared in the left panel, while the effects of ¹⁷O+α reaction rates in different references are compared in the right panel

Because A21 only provides reaction rates below 1.25 GK, in this trial, we switched to the rate from Longland et al. [19] at T > 1.25 GK. In Fig. 14(b), A21 suggests a lower ²²Ne(α, n)²⁵Mg reaction rate at GK. Consequently, the production of the “Ga-Zr” elements is reduced in both core and shell He burning, as shown in Fig. 15.

In Fig. 14(c) and (d), H24 suggests a similar ¹⁷O(α, γ)²¹Ne reaction rate with the rate in B13 and a higher ¹⁷Ne(α, n)²⁰Ne reaction rate at T < 0.7 GK. At T > 1.0 GK, we also switch to the rate in B13. As a result, the production of the “Ga-Zr” elements during core He burning was enhanced compared with the model with the B13 rate in Fig. 15.

In conclusion, the ²²Ne+α reaction rates reported in different studies lead to variations in the yields of the ws-process elements by a factor of approximately 3–5. In contrast, the ¹⁷O+α reaction rates published in the literature produce order-of-magnitude differences in the ws-process elemental yields.

The Effect of Mixing

Nucleosynthesis in each zone was calculated separately because WinNet is a one-zone code. Thus, the effect of convective mixing was not considered. Mixing affects our results in two main aspects. As seen in Fig. 10, ²²Ne and ¹⁷O are exhausted only in the burning shells in the CO core because the abundance of α particles is quite small in the unburned shells. In Fig. 4 in the region of M_r = 2.5–5.1 M_⊙, mixing can transport ²²Ne and ¹⁷O from unburned shells to the burning shells. Consequently, more neutrons should be released in the MESA calculation. Similarly, the mass fraction of ¹⁷O in Fig. 4 is quite low.

Convective mixing can also affect the locations of the C, Ne, O, and Si burning shells. With mixing, more fresh fuel was transported from the outer layers to the bottom of the burning shells. This would make the life of the burning shells longer. Thus, the bottom of shell O burning located at M_r = 1.84 M_⊙ in Fig. 4, would move to M_r = 2.10 M_⊙ in Fig. 10. Similarly, the bases of the C and Ne shells are at M_r = 2.0 M_⊙ would shift to M_r = 2.6 and 2.7 M_⊙, respectively without mixing in WinNet.

The Effect of Explosion

As mentioned in Sect. 3.2, we assume that the mass cut is located at M(V/U_max) and that the region of contributes to the chemical enrichment. We also assume that the ws-process isotopes produced in explosive nucleosynthesis are destroyed by the shock during the explosion. Thus, we did not calculate the explosive nucleosynthesis. The results from Tur et al. [15] show that explosive burning would reduce the ws-process isotopes by less than 15%. Limongi and Chieffi [14] mentioned that for the isotopes of ⁷⁰Zn, ⁷⁶Ge, ^{74, 77, 82}Se, ⁷⁸Kr, ⁸⁷Rb, and ⁸⁴Sr, more than 50% of the yields are produced during explosive burning. These isotopes should be significantly changed by the shock wave. Because the exact explosion mechanism of core-collapse supernovae is not well understood, the explosion energy and choice of mass cut would also affect the final yields of these isotopes.

Other Effects

The ws-process occurs mainly during the He, C, and Ne burning phases. The physical processes that affect these burning phases may also affect the ws-process yields, such as reaction rates, convection, rotation, and magnetic fields [65]. Tur et al. [15] showed that a 15% change in the 3 α and ¹²C(α, γ)¹⁶O reaction rates may change the yields of the ws-process isotopes by more than a factor of 2. Limongi and Chieffi [16] presented many rotating massive star models, including ws-process nucleosynthesis. Their models involved M(ZAMS) = 13–120 M_⊙ and metallicity of . They found the interplay between the He core and the H burning shell, which is triggered by rotation-induced instabilities, enhances the products of CNO (especially for ¹⁴N) and produces more neutrons. Consequently, the ws-process should be significantly enhanced in rotating models.