New ²⁶P(p, γ)²⁷S Thermonuclear Reaction Rate and Its Astrophysical Implications in the rp-process

The rapid proton capture process, hereafter the 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; Lewin et al. 1993; Taam 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 β⁺-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 β⁺-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 nucleus ²⁵Si, which is characterized by a (p, γ)–(γ,p) equilibrium between ²⁵Si and ²⁶P. However, regarding the identification of the waiting-point nucleus ²⁵Si, it is actually affected by the net reaction flow leaking out of the equilibrium through ²⁶P(p, γ)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, γ)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 the 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 from the waiting-point nucleus ²⁵Si. However, it also pointed out that the nuclear mass uncertainty of ²⁷S from AME2003 (Audi et al. 2003), hereafter AME2003 used in their calculation cannot reach the expected accuracy. Regarding the issue of how much the actual leakage from ²⁶P occurs via ²⁶P(p, γ)²⁷S, it is determined by the degree of competition between the forward reaction ²⁶P(p, γ)²⁷S and the reverse process ²⁷S(γ,p)²⁶P under the astrophysical condition of an X-ray burst. Therefore, the accurate treatment of forward and reverse reaction rates for ²⁶P(p, γ)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, γ)²⁷S, all of the three currently existing rates are from the theoretical calculation by the 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 that the statistical model is not a good choice for the prediction of the ²⁶P(p, γ)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, where the corresponding masses are evaluated as 10,970 and 17,540 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 17,030(400) to 17,490(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 β-decay spectroscopy experiment of ²⁷S, a more precise ²⁷S mass excess of 17,678(77) keV was proposed based on the measured β-delayed two-proton energy and the Coulomb displacement energy relations (Sun et al. 2020). The new mass not only sensitively affects the reverse reaction rate of ²⁶P(p, γ)²⁷S via an exponential term of exp(–Q/kT), which exists between the forward and reverse reaction rates, it also has an important influence on 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, γ)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, γ)²⁷S on the rp-process nucleosynthesis in Section 3. The conclusions are discussed in Section 4.

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

2.1. Shell Model Calculation

Regarding the ²⁷S energy structure, the only existing information is on the ground state (Shamsuzzoha Basunia 2011). Despite this, the energy levels of its mirror nucleus ²⁷Na have been obtained from the experiment of the β-decay of ²⁷Ne by Tripathi et al. (2006); the spin and γ transition probabilities of these excited states of interest were not given. Therefore, we have to choose a shell model to obtain the necessary energy structure information used to calculate the resonant reaction rates. We calculate the ²⁷S energy levels using the KSHELL shell model code (Shimizu et al. 2019) and the sd-shell model space involving the π0d_5/2, π0d_3/2, π1s_1/2, ν0d_5/2, ν0d_3/2, and ν1s_1/2 valence orbits. Here π denotes proton, and ν denotes neutron. The new isospin-breaking USDC interaction was used in the present work (Magilligan & Brown 2020). The calculated results on the energy levels of ²⁷S and the mirror nucleus ²⁷Na are shown in Figure 1. For comparison, the available experimental energy levels of this pair of mirror nuclei are added as well. We can see that the predicted low energy levels of ²⁷Na by the shell model basically match the experimental result. In addition, the critical information used to calculate the resonant reaction rates, such as reduced transition probabilities B(M1) and B(E2) and the spectroscopic factor of ²⁶P(p, γ)²⁷S, are also obtained through the shell model.

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2.2. Reaction Rate Calculation

The thermonuclear ²⁶P(p, γ)²⁷S reaction rate is the incoherent sum of all resonant and nonresonant capture contributions. It is well known that only the resonances located in the energy window of astrophysical interest (called the Gamow window) contribute significantly to the reaction rate. For a narrow isolated resonance, the resonant reaction rate can be calculated using the expression (Herndl et al. 1995; He et al. 2017)

$$\begin{eqnarray}\begin{array}{rcl}{N}_{{\rm{A}}}{\left\langle \sigma \nu \right\rangle }_{r} & = & 1.5394\times {10}^{11}\times {(\mu {T}_{9})}^{-3/2}\times \omega \gamma \\ & & \times \,\exp \left(-\displaystyle \frac{11.605{E}_{r}}{{T}_{9}}\right)({\mathrm{cm}}^{3}\,{{\rm{s}}}^{-1}\,{\mathrm{mol}}^{-1}),\end{array}\end{eqnarray} \tag{ 1 }$$

where N_A is Avogadro's constant, the reduced mass μ is given by A_T A_p /(A_p + A_T ), and A_p = 1 and A_T = 26 are the mass numbers of the proton and ²⁶P, respectively. Here T₉ is the temperature in units of gigakelvins, and both the resonance energy E_r and resonance strength ω γ are given in units of MeV. Here the resonance strength is defined as

$$\begin{eqnarray}&&\omega \gamma =\frac{2{J}_{r}+1}{(2{J}_{p}+1)(2{J}_{T}+1)}\frac{{{\rm{\Gamma }}}_{p}\times {{\rm{\Gamma }}}_{\gamma }}{{{\rm{\Gamma }}}_{\mathrm{tot}}},\end{eqnarray} \tag{ 2 }$$

where J_r is the spin of the resonance, J_p = 1/2 is the spin of a proton, and J_T is the spin of ²⁶P. The Γ_p and Γ_γ are the proton- and γ-decay width, respectively. The total width Γ_tot of the resonance is thought to be the sum of its proton- (Γ_p ) and γ-decay (Γ_γ ) widths, Γ_tot = Γ_p + Γ_γ . The proton width can be expressed as (Herndl et al. 1995)

$$\begin{eqnarray}&&{{\rm{\Gamma }}}_{p}={C}^{2}S\times {{\rm{\Gamma }}}_{{sp}},\end{eqnarray} \tag{ 3 }$$

with C² S denoting the corresponding spectroscopic factor for a particular state, and Γ_sp being the single-particle width, which can be obtained by using

$$\begin{eqnarray}&&{{\rm{\Gamma }}}_{{sp}}=\displaystyle \frac{3{{\hslash }}^{2}}{\mu {R}^{2}}{P}_{l}(E),\end{eqnarray} \tag{ 4 }$$

where $R={r}_{0}\times (1+{{\rm{A}}}_{T}^{\tfrac{1}{3}})$ fm is the nuclear channel radius, and P_l is the penetrability factor,

$$\begin{eqnarray}&&{P}_{l}(E)=\displaystyle \frac{{kR}}{{F}_{l}^{2}(E)+{G}_{l}^{2}(E)},\end{eqnarray} \tag{ 5 }$$

where k is the wavenumber, R is the channel radius, and F_l and G_l are the standard Coulomb functions (Hou et al. 2015).

The γ-decay widths are obtained from electromagnetic reduced transition probabilities B(ΩL; J_i → J_f ) (where Ω stands for electric or magnetic), which carry the nuclear structure information of the resonance states and the final bound states. The reduced transition probabilities were computed within the framework of the shell model. The corresponding γ-decay widths for the most contributed transitions (M1 and E2) can be expressed as (Herndl et al. 1995)

$$\begin{eqnarray}\begin{array}{r}\begin{array}{rcl}{{\rm{\Gamma }}}_{M1}[{\rm{e}}{\rm{V}}] & = & 1.16\times {10}^{-2}{E}_{\gamma }^{3}[{\rm{M}}{\rm{e}}{\rm{V}}]B(M1)[{\mu }_{N}^{2}]\,{\rm{a}}{\rm{n}}{\rm{d}}\\ {{\rm{\Gamma }}}_{E2}[{\rm{e}}{\rm{V}}] & = & 8.13\times {10}^{-7}{E}_{\gamma }^{5}[{\rm{M}}{\rm{e}}{\rm{V}}]B(E2)[{e}^{2}{fm}^{4}].\end{array}\end{array}\end{eqnarray} \tag{ 6 }$$

In a stellar plasma, the low-lying excited states of the target nucleus are thermally populated and might have considerable influence on the real astrophysical reaction rate. Thus, the rate contribution from the first excited state of 164 keV in ²⁶P is also taken into consideration because of its low excitation energy. In this work, the mass excesses of ²⁶P and ²⁷S are taken from AME2020 (Wang et al. 2021) and Sun et al. (2019), where the specific values are 10,970(200) and 17,678(77) keV, respectively. Then, the proton separation energy (S_p ) is fixed to be 581(214) keV. The four resonances closest to the proton threshold are schematically plotted in Figure 2. We calculate the individual reaction rate contribution for the above four resonances and plot them separately in Figure 3. Since the contribution from the resonance at 2.86 MeV(5/2⁺) is negligible, we here consider three states of ²⁷S at the resonance energies E_r = 1.104 MeV(1/2⁺), 1.597 MeV(9/2⁺), and 1.777 MeV(7/2⁺). All of the relevant information for the three states is summarized in Table 1; the upper part of the table is for ground-state capture, and the lower part is for capture on the first excited state in ²⁶P. The fifth through seventh columns refer to the spectroscopic factors of the corresponding 0d_5/2, 1s_1/2, and 0d_3/2 orbits.

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Table 1. Parameters for the Present ²⁶P(p, γ)²⁷S Resonant Rate Calculation

Ex (MeV)	Er (MeV)	Jπ	l	C2 S5/2	C2 S1/2	C2 S3/2	Γγ (eV)	Γp (eV)	ωγ (MeV)
1.685	1.104	1/2+	2	0.0861			0.000346	26.676	4.939 × 10−11
2.178	1.597	9/2+	2	0.0028		0.3667	0.001578	1375.825	1.127 × 10−9
2.358	1.777	7/2+	0	0.0269	0.0002	0.217	0.002893	36270.718	1.653 × 10−9
1.685	0.940	1/2+	0		0.1190	0.0137	0.000346	606.16	1.152 × 10−10
2.178	1.430	9/2+					0.001578
2.358	1.613	7/2+	2	0.0118			0.002893	45.294	3.858 × 10−9

Note. Listed are the excitation energy E_x , center-of-mass resonance energy E_r , spin and parity J^π , spectroscopic factors C² S, γ-decay width Γ_γ , proton-decay width Γ_p , and resonance strength ω γ. The upper part is for ground-state capture; the lower part is for capture on the first excited state in ²⁶P.

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For the nonresonant contribution, it is directly related to the effective astrophysical S factor (S_eff) in the energy range of the Gamow window via the expression (Sun et al. 2019)

$$\begin{eqnarray}\begin{array}{rcl}{N}_{{\rm{A}}}{\left\langle \sigma \nu \right\rangle }_{\mathrm{dc}} & = & 7.8327\times {10}^{9}\times {\left(\displaystyle \frac{{Z}_{p}{Z}_{T}}{\mu {T}_{9}^{2}}\right)}^{1/3}\times {S}_{\mathrm{eff}}\\ & & \times \,\exp \left[-4.2487{\left(\displaystyle \frac{{Z}_{p}^{2}{Z}_{T}^{2}\mu }{{T}_{9}}\right)}^{1/3}\right]({\mathrm{cm}}^{3}\,{{\rm{s}}}^{-1}\,{\mathrm{mol}}^{-1}),\end{array}\end{eqnarray} \tag{ 7 }$$

where S_eff can be parameterized by the formula

$$\begin{eqnarray}&&{S}_{\mathrm{eff}}\approx S(0)\left[1+0.09807{\left(\displaystyle \frac{{T}_{9}}{{Z}_{p}^{2}{Z}_{T}^{2}\mu }\right)}^{1/3}\right],\end{eqnarray} \tag{ 8 }$$

and S(0) is the astrophysical S factor at zero energy in units of MeV·b.

The S(0) for direct capture into the ground state of ²⁷S has been calculated with a RADCAP code (Bertulani 2003) by using a Woods–Saxon nuclear potential (central + spin–orbit) and a Coulomb potential of a uniform charge distribution. The nuclear potential parameters were determined by matching the bound-state energy. The spectroscopic factors used for the direct capture calculation are taken from the shell model calculation, and the obtained S(0) are 33.67 and 0.68 keV·b, which corresponds to the direct capture from the ground state and first excited state of ²⁶P, respectively. Our uncertainty of S(0) for direct capture is set to be 41%, which includes not only an assumed uncertainty of 40% as in Downen et al. (2022) but also the contribution due to the uncertainty of the Q-value.

In principle, the total reaction rate is the sum of the capture rate on all thermally excited states in the target nucleus weighted with their individual population factors (Schatz et al. 2005):

$$\begin{eqnarray}\begin{array}{rcl}{N}_{{\rm{A}}}\left\langle \sigma \nu \right\rangle & = & \displaystyle \sum _{i}({N}_{{\rm{A}}}{\left\langle \sigma \nu \right\rangle }_{\mathrm{res}\ i}+{N}_{{\rm{A}}}{\left\langle \sigma \nu \right\rangle }_{{dc}\ i})\\ & & \times \,\displaystyle \frac{(2{J}_{i}+1){e}^{-{E}_{i}/{kT}}}{{\sum }_{n}(2{J}_{n}+1){e}^{-{E}_{n}/{kT}}}.\end{array}\end{eqnarray} \tag{ 9 }$$

In this work, we consider only capture on the ground state and the first excited state in ²⁶P. The ²⁶P ground state is known to have spin and parity 3⁺. The spin for the experimentally known first excited state at 164.4 keV has not been determined unambiguously, but we assign a spin of 1⁺ based on the level structure of the ²⁶Na mirror and our shell model calculations.

We calculate the direct and total contributions of the three resonances in which the thermalization effect on the reaction rate is considered, as shown in Figure 4. The solid red line is for the direct component, with the narrow red shaded band for its 41% uncertainty. The solid blue line is for the total contribution of the three resonances, and the blue shaded band is the corresponding uncertainty of the resonant rate from the resonance energy uncertainty of 271 keV, which produced in quadrature the uncertainty of S_p and that of the resonance state energy in ²⁷S. Here the uncertainty of the ²⁷S resonance state energy is assumed to be 166 keV, which is the maximum energy difference between the shell model prediction and the experimental measurements for an arbitrary state below 2.5 MeV in ²⁷Na. We can clearly see that the direct capture makes the most important contribution to the total ²⁶P(p, γ)²⁷S reaction rate, while the contributions from resonances are negligible because the resonance energies of the three resonances are all larger than 1 MeV. The weighted reaction rate is obtained by summing the direct contribution and the resonance contributions after the thermalization correction, as shown in Table 2. The new rate in units of cm³ mol⁻¹ s⁻¹ can be well fitted (less than 0.33% error in 0.01–10 GK) by the following analytic expression in the standard seven-parameter format of REACLIB:

$$\begin{eqnarray}\begin{array}{rcl}{N}_{{\rm{A}}}\left\langle \sigma \nu \right\rangle & = & \exp (21.4303-0.00292768{T}_{9}^{-1}\\ & & -25.0636{T}_{9}^{-1/3}-\,1.69602{T}_{9}^{1/3}+0.0866331{T}_{9}\\ & & -0.00201917{T}_{9}^{5/3}-0.106906\mathrm{ln}{T}_{9}).\end{array}\end{eqnarray} \tag{ 10 }$$

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Table 2. Direct, Resonant, and Total Reaction Rates for ²⁶P(p, γ)²⁷S Based on the Present Work (in Units of cm³ mol⁻¹ s⁻¹)

T9 (GK)	Ground State		First Excited State		Weighted Reaction Rate
	${N}_{{\rm{A}}}{\left\langle \sigma \nu \right\rangle }_{\mathrm{nr}}$	${N}_{{\rm{A}}}{\left\langle \sigma \nu \right\rangle }_{\mathrm{res}}$	${N}_{{\rm{A}}}{\left\langle \sigma \nu \right\rangle }_{\mathrm{nr}}$	${N}_{{\rm{A}}}{\left\langle \sigma \nu \right\rangle }_{\mathrm{res}}$	${N}_{{\rm{A}}}{\left\langle \sigma \nu \right\rangle }_{\mathrm{total}}$
0.1	4.09 × 10−15	5.81 × 10−54	8.25 × 10−17	2.50 × 10−45	4.09 × 10−15
0.2	2.18 × 10−10	1.36 × 10−26	4.39 × 10−12	4.30 × 10−22	2.18 × 10−10
0.3	4.14 × 10−08	1.39 × 10−17	8.34 × 10−10	1.84 × 10−14	4.14 × 10−08
0.4	1.12 × 10−06	3.91 × 10−13	2.25 × 10−08	1.06 × 10−10	1.12 × 10−06
0.5	1.15 × 10−05	1.69 × 10−10	2.32 × 10−07	1.78 × 10−08	1.14 × 10−05
0.6	6.80 × 10−05	9.24 × 10−09	1.37 × 10−06	5.13 × 10−07	6.68 × 10−05
0.7	2.80 × 10−04	1.56 × 10−07	5.63 × 10−06	5.47 × 10−06	2.73 × 10−04
0.8	8.94 × 10−04	1.27 × 10−06	1.80 × 10−05	3.15 × 10−05	8.63 × 10−04
0.9	2.38 × 10−03	6.47 × 10−06	4.80 × 10−05	1.20 × 10−04	2.28 × 10−03
1.0	5.54 × 10−03	2.38 × 10−05	1.12 × 10−04	3.48 × 10−04	5.26 × 10−03
1.5	1.07 × 10−01	1.44 × 10−03	2.15 × 10−03	8.40 × 10−03	9.79 × 10−02
2.0	6.77 × 10−01	1.40 × 10−02	1.36 × 10−02	4.75 × 10−02	6.02 × 10−01
2.5	2.50 × 10+00	5.79 × 10−02	5.03 × 10−02	1.49 × 10−01	2.16 × 10+00
3.0	6.71 × 10+00	1.49 × 10−01	1.35 × 10−01	3.31 × 10−01	5.67 × 10+00
3.5	1.47 × 10+01	2.86 × 10−01	2.96 × 10−01	5.84 × 10−01	1.23 × 10+01
4.0	2.80 × 10+01	4.58 × 10−01	5.64 × 10−01	8.83 × 10−01	2.28 × 10+01
5.0	7.65 × 10+01	8.48 × 10−01	1.54 × 10+00	1.52 × 10+00	6.05 × 10+01

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For the ²⁶P(p, γ)²⁷S reaction, we know that its reverse reaction rate (also called photodisintegration rate) can be calculated directly by the expression (Iliadis 2015)

$$\begin{eqnarray}&&{\lambda }_{(\gamma ,p)}=\displaystyle \frac{2{G}_{f}}{{G}_{i}}{\left(\displaystyle \frac{\mu {kT}}{2\pi {{\hslash }}^{2}}\right)}^{3/2}\exp \left(-\displaystyle \frac{{Q}_{(p,\gamma )}}{{kT}}\right)\left\langle \sigma v\right\rangle ,\end{eqnarray} \tag{ 11 }$$

where G_i and G_f are the partition functions of the initial and final nuclei. As mentioned above, the ratio of the (γ, p) reaction rate to the (p, γ) rate depends exponentially on Q_{(p, γ)} and is therefore very sensitive to nuclear masses. Using the new Q-value determined by the new ²⁷S mass, we recalculate the ²⁷S(γ,p)²⁶P rate. In Figure 5, we plot the new reaction rate of ²⁶P(p, γ)²⁷S and the uncertainties. The other versions (rath, rpsm, ths8) collected in JINA REACLIB are also added for comparison. Forward reaction rates are shown in panel (a), while reverse rates are shown in panel (b). It can be seen that the newly obtained forward rate is smaller than all three of the previous rates. For reverse process ²⁷S(γ,p)²⁶P, the new rate is also smaller than those from rpsm and ths8 over the temperature range of 0.2–2 GK but larger than the rath rate, which is due to the rather large Q-value of 1.452 MeV.

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In order to explore the impact of the new forward and reverse rates for ²⁶P(p, γ)²⁷S on the rp-process, we make use of the one-zone postprocessing nucleosynthesis code ppn, a branch of the NuGrid framework (Herwig et al. 2009; Denissenkov et al. 2014). Both the initial composition of the accreted material (X⁰ _H = 0.735, ${X}_{{}^{4}\mathrm{He}}^{0}$ = 0.245, ${X}_{{}^{14}{\rm{O}}}^{0}$ = 0.007, and ${X}_{{}^{15}{\rm{O}}}^{0}$ = 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, γ)²⁷S. In the first run, we use the ²⁶P(p, γ)²⁷S rates from 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 bursts. It can be clearly seen in Figure 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, γ)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. Figure 7 shows the final isotope abundance distributions for the cases using the new rates (panel (a)) and adopting the ths8 rates (panel (b)). It is clear that the accumulated ²⁷S abundance in Figure 7(a) is smaller than that in Figure 7(b). In other words, for the case of adoption of our new rates, the reaction flow through the branch of ²⁶P(p, γ)²⁷S(β⁺,ν)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 an 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 the 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(γ,p) and ²⁷S(γ,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 that the final accumulated material on ²⁷S through ²⁶P(p, γ)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(γ,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(γ,p)²⁶P, directly preventing the reaction flow to ²⁷S.

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In light of the variation of the accumulated material on ²⁷S when assuming different forward and reverse rates of ²⁶P(p, γ)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 Figure 8, in which the red squares are the abundances calculated using our new rates, and the blue triangles correspond to that adopting the 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%.

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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, γ)²⁷S reaction, which is important in the rp-process. The calculated result shows that the direct contribution dominates this reaction rate in comparison to the resonance contribution. Both of 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 a temperature of 0.1 GK. At the stellar conditions relevant to this study (0.4 GK < T₉ < 1.35 GK), the typical variation of the reaction rate is on the order of 2–4. We investigate the effect of the new thermonuclear reaction rates on the nucleosynthesis in the rp-process using the ppn postprocessing 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, γ)²⁷S. For the flow reaching the branch point nucleus ²⁶P, the reaction chain of ²⁶P(p, γ)²⁷S(β⁺,ν)²⁷P competes with the β⁺ branch of ²⁶P(β⁺,ν)²⁶Si(p, γ)²⁷P. Our calculations confirm that ²⁶P(p, γ)²⁷S(β⁺,ν)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, γ)²⁷S only reduces the final production of aluminum by 7.1% and has no discernible impact on the yield of other elements.

This work was financially supported by National Key R&D Program of China grant No. 2022YFA1603300, Strategic Priority Research Program of the Chinese Academy of Sciences grant No. XDB34020204, and the Youth Innovation Promotion Association of the Chinese Academy of Sciences under grant No. 2019406, as well as in part by the National Science Foundation under grant No. OISE-1927130 (IReNA), the National Natural Science Foundation of China under grant No. 12205340, and the Gansu Natural Science Foundation under grant No. 22JR5RA123. This research was made possible using the computing resources of the Gansu Advanced Computing Center. C.A.B. acknowledges support by U.S. DOE grant DE-FG02-08ER41533. M.P. and T.T. acknowledge support from STFC (through the University of Hull's consolidated grant ST/R000840/1) and access to viper, the University of Hull HPC Facility. We acknowledge the support of NuGrid from the National Science Foundation (NSF, USA) under grant No. PHY-1430152 (JINA Center for the Evolution of the Elements). M.P. thanks the "Lendulet-2014" Program of the Hungarian Academy of Sciences (Hungary); the ERC Consolidator Grant funding scheme (Project RADIOSTAR, G.A.n. 724560, Hungary); the ChETEC COST Action (CA16117), supported by the European Cooperation in Science and Technology; and the IReNA network supported by NSF AccelNet. M.P. acknowledges support from the ChETEC-INFRA project funded by the European Union's Horizon 2020 Research and Innovation program (grant agreement No. 101008324).