New ²⁶P($p$,$\gamma$)²⁷S thermonuclear reaction rate and its astrophysical implication in rp-process

The rapid proton capture process, hereafter rp-process, typically occurs when hydrogen fuel is ignited under highly degenerate conditions in explosive events on the surface of compact objects like white dwarfs (novae) (Truran, 1982, 1990; Shankar et al., 1992; Starrfield et al., 1993) and neutron stars (X-ray bursts) (Wallace & Woosley, 1981; Taam, 1985; Taam et al., 1993; Lewin et al., 1993). During this process, the consecutive proton captures onto stable nuclei can produce nuclei far away from the stability line, even approaching the proton drip line (José et al., 2010). Naturally, this process competes with the $\beta^{+}$-decays and reverse photodisintegration reactions. For a given nucleus, if proton capture on it is inhibited by the strong reverse reaction rate and subsequently has to wait for a slower $\beta^{+}$-decay, the nucleus is often termed a waiting point nucleus (Schatz et al., 1998).

In the nuclide chart, the nucleus ²⁷S is located on the proton-rich side, far beyond the valley of stability. During the rp-process, ²⁷S is synthesized via two successive proton captures on the waiting-point nuclei ²⁵Si which is characterized by a ($p$,$\gamma$)-($\gamma$,$p$) equilibrium between ²⁵Si and ²⁶P. However, regarding the identification of the waiting-point nuclei ²⁵Si, it is actually affected by the net reaction flow leaking out of the equilibrium through ²⁶P($p$,$\gamma$)²⁷S. From the principle of detailed balance, we know that the forward and reverse rates for an arbitrary reaction can be mutually converted via a Q-value-dependent exponential term. Therefore, accurate ²⁶P($p$,$\gamma$)²⁷S rates and the nuclear masses of ²⁶P and ²⁷S are of great importance for determining the degree to which ²⁵Si is a waiting-point nucleus and thus a better understanding of rp-process reaction path in the region around ²⁶P.

In Parikh et al. (2013), by using nucleosynthesis calculations it is shown that the nuclear mass of ²⁷S is essential to quantify the abundance flows proceeding the waiting-point nuclei ²⁵Si. However, it also pointed out that the nuclear mass uncertainty of ²⁷S from AME2003(Audi et al., 2003) used in their calculation cannot reach the expected accuracy. Regarding the issue of how much the actual leakage from ²⁶P occurs via ²⁶P($p$,$\gamma$)²⁷S, it is determined by the degree of competition between the forward reaction ²⁶P($p$,$\gamma$)²⁷S and the reverse process ²⁷S($\gamma$,$p$)²⁶P under the astrophysical condition of X-ray burst. Therefore, the accurate treatment of forward and reverse reaction rates for ²⁶P($p$,$\gamma$)²⁷S is crucial for a better understanding of the rp-process nucleosynthesis path in the region of nuclei with mass number A = 26-27.

Concerning the reaction ²⁶P($p$,$\gamma$)²⁷S, all of the three currently existing rates are from the theoretical calculation by Hauser-Feshbach statistical model (Rauscher & Thielemann, 2000; Cyburt et al., 2010). However, the statistical model is only suitable for the case of the high energy level density in compound nuclei near the proton threshold (Rauscher & Thielemann, 2000). Therefore, both the shell model prediction of ²⁷S and the measured mirror states in ²⁷Na indicate the statistical model is not a good choice for the prediction of the ²⁶P($p$,$\gamma$)²⁷S rate. The respective photodisintegration rate can be derived from the forward rate if the reaction Q-value is available (Herndl et al., 1995; Iliadis, 2015).

The nuclear masses of the involved nuclei ²⁶P and ²⁷S used in the investigation by Parikh are taken from AME2003 (Audi et al., 2003), where the corresponding masses are evaluated as 10970 keV and 17540 keV, both with an uncertainty of 200 keV. Differing from the case of the ²⁶P mass remaining constant as in AME2003, the nuclear mass of ²⁷S varies from 17030(400) keV to 17490(400) keV in several different recent versions of atomic mass evaluations (Wang et al., 2012, 2017, 2021). Because all of these are theoretically deduced from trends in the mass surface, it is difficult to tell which one can represent the actual mass of ²⁷S. Thanks to the recent $\beta$-decay spectroscopy experiment of ²⁷S, a more precise ²⁷S mass excess of 17678(77) keV was proposed based on the measured $\beta$-delayed two-proton energy and the Coulomb displacement energy relations (Sun et al., 2020). The new mass not only affects sensitively the reverse reaction rate of ²⁶P($p$,$\gamma$)²⁷S via an exponential term of exp(-Q/kT) which exists between the forward and reverse reaction rates, but it also has an important influence in the forward reaction rate since the new ²⁷S mass will lead to a huge alteration of the resonance contribution. In this work, we reevaluate the ²⁶P($p$,$\gamma$)²⁷S reaction rate using the new mass of the involved nuclei, in a combination of shell model predictions of the energy structure of ²⁷S. Meanwhile, we also explore the impact of the new rates on the nucleosynthesis path in the rp-process, and the role of nuclear physics uncertainties in abundance prediction.

The paper is structured as follows. Section 2 introduces the shell-model calculation on the energy level structure of ²⁷S and the astrophysical reaction rate calculation. We investigate the effect of new rates for ²⁶P($p$,$\gamma$)²⁷S to the rp-process nucleosynthesis in Section 3. The conclusions are discussed in Section 4.

The energy structure of compound nucleus ²⁷S remains unclear currently. In light of its mirror nucleus ²⁷Na with a low energy level density, it is worth investigating the reaction rate of ²⁶P($p$,$\gamma$)²⁷S, especially the contributions from the resonances near threshold. Below we will make a detailed study on this reaction based on the shell model and resonant reaction theory.

In order to explore the impact of the new forward and reverse rates for ²⁶P($p$,$\gamma$)²⁷S on the rp-process, we make use of the one-zone post-processing nucleosynthesis code ppn, a branch of the NuGrid framework (Herwig et al., 2008; Denissenkov et al., 2014). Both the initial composition of the accreted material($X\rm^{0}_{H}$= 0.735, $X\rm^{0}_{{}^{4}He}$= 0.245, $X\rm^{0}_{{}^{14}O}$ = 0.007 and $X\rm^{0}_{{}^{15}O}$ = 0.013) and the profile used in the simulation are taken from Koike et al. (2004). We perform two simulations with identical nuclear physics and model inputs, but different forward and reverse reaction rates for ²⁶P($p$,$\gamma$)²⁷S. In the first run, we use the ²⁶P($p$,$\gamma$)²⁷S rates from Cyburt et al. (Cyburt et al., 2010) (labeled as ths8 in JINA REACLIB), while the newly obtained forward and reverse rates for this reaction are used in the second run. The calculated results show that the new rates can sensitively affect the abundance ratio of ²⁷S/²⁶P in X-ray burst. It can be seen clearly in Fig.6 that the obtained ²⁷S/²⁶P ratio using the new rates is overall smaller than that adopting the ths8 rates by about a factor of 10. The reason for this is twofold. On one side, this is because the new ²⁶P($p$,$\gamma$)²⁷S rate is much smaller than the ths8 rate, resulting in less direct production of ²⁷S. On the other side, the new $S_{p}$ of 581 keV is also smaller than that of 719 keV from ths8, implying a relatively stronger level of photodisintegration which prevents the synthesis of ²⁷S. Fig.7 shows the final isotope abundance distributions for the case using the new rates(panel (a)) and that adopting ths8 rates(panel (b)), respectively. It is clear that the accumulated ²⁷S abundance in Fig.7 (a) is smaller than that in Fig.7 (b). In other words, for the case of adoption of our new rates, the reaction flow through the branch of ²⁶P($p$,$\gamma$)²⁷S($\beta^{+}$,$\nu$)²⁷P is much smaller compared with the case of using old rates. In order to investigate whether the uncertainty of 214 keV of the new $S_{p}$ will change this conclusion, we also perform additional two runs in which only the photodisintegration rates are changed by taking into account the uncertainty of the new $S_{p}$ for ²⁷S, while all other rates remain the same as those used in the second run. It is known that the uncertainty of the new $S_{p}$ is determined by the uncertainties from the nuclear masses of ²⁶P and ²⁷S. Thus, the first run is for the case considering ${S_{p}}$ upper limit (marked as $\overline{S_{p}}$), which corresponds to the case of adopting the upper limit of the ²⁶P mass and the lower limit of the ²⁷S mass in the process of calculating the respective photodisintegration rate of ²⁶P($\gamma$,$p$) and ²⁷S($\gamma$,$p$). Similarly, the second run is for the case of using the ${S_{p}}$ lower limit (marked as $\underline{S_{p}}$), in which the lower limit of the ²⁶P mass and the upper limit of the ²⁷S mass are used to derive the respective reverse rates. Our calculations show the final accumulated material on ²⁷S through ²⁶P($p$,$\gamma$)²⁷S for both cases of $\overline{S_{p}}$ and $\underline{S_{p}}$ decreases relative to the case of using $S_{p}$ – by about 25% and 39%, respectively. The production of ²⁷S is reduced when using the former due to the enhanced ²⁶P($\gamma$,$p$)²⁵Si rate from assuming the upper limit of the ²⁶P mass, which effectively inhibits the reaction flow passing through ²⁶P. In the case of the latter, using the lower limit of ${S_{p}}$ results in a stronger photodisintegration rate of ²⁷S($\gamma$,$p$)²⁶P, directly preventing the reaction flow to ²⁷S.

In light of the variation of the accumulated material on ²⁷S when assuming different forward and reverse rates of ²⁶P($p$,$\gamma$)²⁷S, we compare the decayed elemental abundances(i.e., accounting for the complete contribution from the radioactive decay of all unstable isotopes) calculated using the new rates with those calculated using the ths8 rates. The results are plotted in Fig.8 in which the solid red squares are the abundances calculated using our new rates and the blue triangle corresponds to that of adopting ths8 rates. It can be seen that using the new rates did not result in a significant change in elemental abundance. The largest difference is seen for aluminum (²⁷Al), with a decrease of 7.1%.

In this study, we employ the shell-model to predict the energy structure information of ²⁷S. In addition, using the updated proton threshold determined by the new ²⁷S mass, we recalculate the thermonuclear reaction rates for the ²⁶P($p$,$\gamma$)²⁷S reaction, which is important in the rp-process. The calculated result shows the direct contribution dominates this reaction rate in comparison to the resonance contribution. Both our new rates for forward and reverse reactions are different from the other three rates from JINA REACLIB. The ratio of the old rates to our new rate can even be up to a factor of 5 orders of magnitude at the temperature of 0.1 GK. At the stellar conditions relevant to this study (0.4 GK<$T_{9}$ <1.35 GK), the typical variation of reaction rate is in the order of 2 to 4. We investigate the effect of the new thermonuclear reaction rates on the nucleosynthesis in rp-process using the ppn post-processing code. It is found that the ratio of isotope abundances of ²⁷S/²⁶P when adopting the new rates is smaller by a factor of 10 than that using the ths8 rates from the JINA database. In addition, the accumulated material on the ²⁶P nucleus is larger than that on ²⁷S during the whole rp-process episode, which is markedly different from the result of using the old rates of ²⁶P($p$,$\gamma$)²⁷S. For the flow reaching branch point nucleus ²⁶P, the reaction chain of ²⁶P($p$,$\gamma$)²⁷S($\beta^{+}$,$\nu$)²⁷P competes with the $\beta^{+}$ branch of ²⁶P($\beta^{+}$,$\nu$)²⁶Si($p$,$\gamma$)²⁷P. Our calculations confirm that ²⁶P($p$,$\gamma$)²⁷S($\beta^{+}$,$\nu$)²⁷P is not the major path for the synthesis of ²⁷P after adopting our new forward and reverse rates. The adoption of the new reaction rates for ²⁶P($p$,$\gamma$)²⁷S only reduces the final production of aluminum by 7.1%, and has no discernible impact on the yield of other elements.