Nguyen Duy Ly¹, Nguyen Ngoc Duy^,2,*, Nguyen Kim Uyen², Vinh N T Pham³¹Faculty of Fundamental Sciences, Vanlang University, Ho Chi Minh City 700000, Vietnam
²Department of Physics, Sungkyunkwan University, Suwon 16419, Republic of Korea
³Department of Physics, Ho Chi Minh City University of Education, Ho Chi Minh City 700000, Vietnam

First author contact: ^*Author to whom any correspondence should be addressed.
Received:2021-05-26Revised:2021-08-5Accepted:2021-08-16Online:2021-09-03


Abstract
We examined the conditions of neutron density (n) and temperature (T₉) required for the N = 50, 82, and 126 isotopes to be waiting points (WP) in the r-process. The nuclear mass based on experimental data presented in the AME2020 database (AME and AME ± Δ) and that predicted using FRDM, WS4, DZ10, and KTUY models were employed in our estimations. We found that the conditions required by the N = 50 WP significantly overlap with those required by the N = 82 ones, except for the WS4 model. In addition, the upper (or lower) bounds of the n − T₉ conditions based on the models are different from each other due to the deviations in the two-neutron separation energies. The standard deviations in the nuclear mass of 108 isotopes in the three N = 50, 82, and 126 groups are about rms = 0.192 and 0.434 MeV for the pairs of KTUY-AME and WS4-KTUY models, respectively. We found that these mass uncertainties result in a large discrepancy in the n_n − T₉ conditions, leading to significant differences in the conditions for simultaneously appearing all the three peaks in the r-process abundance. The newly updated FRDM and WS4 calculations can give the overall conditions for the appearance of all the peaks but vice versa for their old versions in a previous study. The change in the final r-process isotopic abundance due to the mass uncertainty is from a few factors to three orders of magnitude. Therefore, accurate nuclear masses of the r-process key nuclei, especially for ⁷⁶Fe, ⁸¹Cu, ¹²⁷Rh, ¹³²Cd, ¹⁹²Dy, and ¹⁹⁷Tm, are highly recommended to be measured in radioactive-ion beam facilities for a better understanding of the r-process evolution.
Keywords： neutron capture;r-process;nuclear mass;isotopic abundance;precise mass measurement


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Nguyen Duy Ly, Nguyen Ngoc Duy, Nguyen Kim Uyen, Vinh N T Pham. Examination of n - T₉ conditions required by N = 50, 82, 126 waiting points in r-process. Communications in Theoretical Physics, 2021, 73(10): 105301- doi:10.1088/1572-9494/ac1dad

1. Introduction

The synthesis of elements beyond iron is believed to mainly proceed via the rapid neutron-capture process (r-process) [1, 2]. Analyses on the galactic elemental evolution [3–6] indicate that elemental abundance of the Solar system with three peaks at A ≈ 80 (Ge-Kr), 130 (Te-Xe), and 195 (Os-Pt) is a result of the r-process [7–10]. Indeed, about half of heavy elements, up to thorium, are produced in this process via a combination of the (n, γ), (γ, n) reactions, β-decay, and fission. Once the neutron capture is faster than the radioactive decay (mainly via β⁻-decay) of the exotic seed nuclei, neutron-rich isotopes are enriched, leading to an expansion of the nuclear landscape far from the β stability. The competition between the (n, γ) − (γ, n) reaction and β⁻-decay fluxes strongly impacts the isotopic abundance and reaction path of nucleosynthesis. A freeze-out of the neutron capture flux can occur when the (n, γ) − (γ, n) equilibrium is established under specific stellar conditions of neutron density and temperature [11–13], leading to the enrichment of long-lived isotopes, which are referred as waiting points (WP). In the equilibrium, the so-called WP nuclei await the β decay to shift the reaction path along the r-process towards the stability [14–16]. Since the WP, whose neutron numbers are in the vicinity of neutron shell-closures at N = 50, 82, 126, have relatively long lifetimes, their enrichment during the freeze-out results in the corresponding peaks at A = 80, 130, 195 in the r-process abundance. Notice that the three peaks in the Solar-system r-process are thought not to be produced under the same set of stellar conditions [11, 17, 18]. Hence, studies on the astrophysical conditions for the equilibrium and β-decay half-lives of extremely neutron-rich nuclei are important for a better understanding of the scenarios of the r-process.

Since the behaviors of the r-process strongly depend on the (n, γ), (γ, n), and β-decay rates [19–22], both nuclear properties (e.g. mass, spin, half-life, etc) of unstable isotopes far from stability and astrophysical conditions are important inputs for r-process calculations. For instance, in the study of r-process matter flow, Niu et al [23] indicated that reaction paths from ₂₀Ca to ₅₀Sn can be expanded up to 14 (or 4) neutrons from N = 82 if the β-decay half-lives based on the RHFB+QRPA model [24, 25] (or measurements [26, 27]) were employed in the calculations. Recently, some new developments for predicting the β-decay rate (or half-life) have been conducted, which are helpful for r-process studies. For example, a large-scale shell model including Gamow-Teller and first-forbidden transitions developed by Zhi et al [28] and the interacting shell model using large spaces of single-particle valence and many-body Hamiltonians proposed by Kostensalo et al [29] are emerging as good candidates for calculating the half-lives of heavy nuclei. The Hartree–Fock calculation including forbidden transitions with Skyrme interaction and pairing correlations can also well reproduce measured half-lives of neutron-rich nuclei with A ≈ 190 [30]. In other studies [10, 31–33], it was found that the determination of the neutron density and temperature conditions for the (γ, n) − (n, γ) equilibrium establishment or WP approximation strongly depends on neutron and/or two-neutron separation energies of neutron-rich nuclei. In particular, the deviations in two-neutron separation energies of ⁷⁸Ni and ⁸⁴Zn due to the differences among the RMF [34], FRDM [35], and WS [36, 37] mass models lead to a large change, from 10²³ to 10²⁶ cm⁻³ at 1 GigaKelvin (GK), in the neutron density required by the N = 50 WP [33]. The same phenomenon is also observed for the N = 82 isotones [33]. The results in these studies indicate that the (n, γ) − (γ, n) equilibrium, the waiting-point approximation for the N = 50, 82, 126 nuclei, and subsequently isotopic abundance are very sensitive to nuclear mass. Unfortunately, the nuclear masses and half-lives of these WP are mostly predicted using theoretical models.

Recently, experimental studies on the nuclear mass have been expanded with the help of modern radioactive ion beam facilities [38–41]. Several theoretical works have been developed to improve accuracy in the mass predictions [34–36, 42–44]. For instance, Wang et al proposed a semi-empirical formula (namely WS) based on the macroscopic-microscopic method [35] with the isospin dependence and Skyrme energy density function being taken into account to predict the nuclear mass of isotopes up to A ≈ 312 [36, 37]. The neutron separation energies estimated by using this model differ from those measured in laboratories by an average standard deviation of 0.332 MeV for 1988 nuclei [36]. In another work, the Skyrme–Hartree–Fock–Bogoliubov (HFB) formula was developed by Goriely et al to predict the nuclear mass with a standard deviation of 0.581 MeV in a comparison with experimental data [42]. Still in a theoretical study, the nuclear masses of isotopes calculated using the FRDM formula newly updated by Moller et al in 2019 differ from measured data by a standard error of only 0.4948 MeV [44]. On the other hand, available experimental data until October 2020, including uncertainty, of the nuclear mass have been evaluated by Wang et al, which were reported in the AME2020 database in March 2021 [45]. In that work, the reliably experimental and extrapolated data have been reviewed to provide the last updated nuclear mass data.

As aforementioned, the stellar conditions estimated in the r-process waiting-point approximation strongly depend on the masses of neutron-rich isotopes, especially for relatively long-lived nuclei. Therefore, in this study, we employed reliable nuclear mass in the newly updated AME2020 database to examine the neutron density and temperature (n_n – T₉) conditions required by the N = 50, 82 WP. Notice that the neutron- and two-neutron separation energies of the N = 126 WP and their neighbors are still not available in AME2020. The conditions are also investigated using the updated nuclear mass calculations based on the FRDM [44], Weizsacker–Skyrme (WS4) [46], Duflo-Zuker (DZ10) [47, 48], and Koura–Tachibana–Uno–Yamada (KTUY) [49] models for all the considered N = 50, 82, and 126 WP. The mass uncertainty due to different models is examined to suggest the best prediction for further studies. The stellar conditions associated with the different mass data sets (FRDM, AME2020 with and without uncertainty, WS4, DZ10, and KTUY) are compared to each other to determine the lower and upper bounds of the n_n – T₉ conditions for r-process simulations. Subsequently, we investigated the impact of the uncertainty in the n_n – T₉ conditions on the r-process isotopic abundance.

The present paper is constructed as follows. The theoretical framework for the WP approximation is presented in section 2. The results of the stellar conditions based on the updated mass data from AME2020, FRDM, WS4, DZ10, and KTUY calculations are discussed in section 3. The summary of this work is given in section 4.

2. Theoretical framework

The (n, γ) − (γ, n) equilibrium can be established under high neutron density (n_n ≥ 10²⁰) and temperature (T ≥ 10⁹) conditions. Under such scenario, the abundance ratio of (Z, A + 1) and (Z, A) isotopes is calculated by the Saha relation as [11–13](1)$\begin{eqnarray}\begin{array}{l}\displaystyle \frac{Y(Z,A+1)}{Y(Z,A)}={n}_{n}{\left(\displaystyle \frac{2\pi {{\hslash }}^{2}}{{ukT}}\right)}^{3/2}\\ \displaystyle \frac{G(Z,A+1)}{2G(Z,A)}{\left(\displaystyle \frac{A+1}{A}\right)}^{3/2}\\ \times \,\exp \left[\displaystyle \frac{{S}_{n}(Z,A+1)}{{kT}}\right],\end{array}\end{eqnarray}$where Y(Z, A), u, G(Z, A), and S_n are the abundance of the (Z, A) nuclide, atomic mass unit, temperature-dependent partition function [50] of the (Z, A) nuclide, and one-neutron separation energy; k and ℏ denote Boltzmann and reduced Planck constants, respectively. Due to the equilibrium, the r-process is paused at WP, leading to peaks established at A ≈ 80, 130, and 195 (N = 50, 82, and 126). At the highest abundance for each element, the abundance ratio is approximately equal to 1. By ignoring small differences in the partition functions and mass numbers of two neighboring isotopes, the relation in equation (1) can be rewritten as [33](2)$\begin{eqnarray}\displaystyle \frac{Y(Z,A+1)}{Y(Z,A)}=\exp \left[\displaystyle \frac{{S}_{n}(Z,A+1)-{S}_{n}^{0}({n}_{n},T)}{{kT}}\right]\,,\end{eqnarray}$where the average neutron-separation energy ${S}_{n}^{0}({n}_{n},T)$ is calculated by(3)$\begin{eqnarray}\begin{array}{l}{S}_{n}^{0}({n}_{n},T)\\ =\ {T}_{9}\left[2.79+\displaystyle \frac{1.5\mathrm{log}{T}_{9}-\mathrm{log}\left({n}_{n}/{10}^{20}\right)}{5.04}\right]\,\,\mathrm{MeV}.\end{array}\end{eqnarray}$In equation (3), T₉ is the temperature T in the unit of GigaKelvin (GK). Notice that the average neutron-separation energies ${S}_{n}^{0}({n}_{n},T)$ of all the nuclei at the highest abundance are almost the same.

Since all the WP have even neutron numbers (even-N) due to the paring effect on neutron binding, the abundance ratio of two neighboring even-N isotopes can be estimated by(4)$\begin{eqnarray}\displaystyle \frac{Y(Z,A)}{Y(Z,A-2)}=\exp \left[\displaystyle \frac{{S}_{2n}(Z,A)-2{S}_{n}^{0}({n}_{n},T)}{{kT}}\right]\,,\end{eqnarray}$where S_2n(Z, A) is the two-neutron separation energy of the isotope (Z, A). Obviously, the abundance of the (Z, A) even-N nuclide is increased with increasing neutron number N when Y(Z, A)/Y(Z, A − 2) > 1 until ${S}_{2n}\lt 2{S}_{n}^{0}$, from this point the abundance is decreasing with N when Y(Z, A)/Y(Z, A − 2) < 1. As a result, abundance peaks locate at the nuclei whose two-neutron separation energies fulfill the following condition [33](5)$\begin{eqnarray}{S}_{2n}(Z,A+2)\leqslant 2{S}_{n}^{0}({n}_{n},T)\leqslant {S}_{2n}(Z,A)\,.\end{eqnarray}$By taking equation (3) together with equation (5), the n_n – T₉ conditions for a (Z, A) nuclide to be a WP can be determined. Obviously, the formalisms above indicate that the estimation of the n_n – T₉ conditions strongly depends on the accuracy of the nuclear mass. Hence, in this study, the neutron- and/or two-neutron separation energies taken from the newly updated AME2020 database [45] and nuclear mass predicted using the last updated FRDM, WS4, DZ10, and KTUY models are utilized for the WP approximation. Notice that the two-neutron separation energy of the (Z, A) nuclide can be deduced in terms of the mass excesses, M(Z, A) and M(Z, A − 2) (in MeV), as(6)$\begin{eqnarray}\begin{array}{l}{S}_{2n}(Z,A)=M(Z,A-2)\\ \,-M(Z,A)+2\times {{\rm{\Delta }}}_{n},\end{array}\end{eqnarray}$with Δ_n = 8.071 MeV being the neutron mass excess. The deviation ΔS_2n in the two-neutron separation energy due to the mass-excess uncertainty is determined using the error propagation rule in statistics [51]. Subsequently, the average neutron separation ${S}_{n}^{0}({n}_{n},T)$ in equation (5) is selected in the range of(7)$\begin{eqnarray}\begin{array}{l}[{S}_{2n}-{\rm{\Delta }}{S}_{2n}](Z,A+2)\leqslant 2{S}_{n}^{0}({n}_{n},T)\\ \leqslant [{S}_{2n}+{\rm{\Delta }}{S}_{2n}](Z,A)\,.\end{array}\end{eqnarray}$By putting the values of ${S}_{n}^{0}({n}_{n},T)$ calculated in equation (7) into (3), the upper and lower bounds of the n_n − T₉ condition in the case of ΔS_2n being taken into account (for the AME ± Δ data set) are determined.

3. Results and discussion

We considered the waiting-point approximation for twelve extremely neutron-rich nuclei with N = 50, 82, and 126 which are ⁷⁶Fe, ⁷⁷Co, ⁷⁸Ni, ⁷⁹Cu, ¹²⁷Rh, ¹²⁸Pd, ¹²⁹Ag, ¹³⁰Cd, ¹⁹²Dy, ¹⁹³Ho, ¹⁹⁴Er, and ¹⁹⁵Tm. The nuclear masses of these isotopes are still very uncertain. In table 1, we list the two-neutron separation energies of the investigated WP (Z, A) and their even-N neighboring isotopes (Z, A + 2), which were employed for the examination of the n_n − T₉ conditions required by the considered WP. Figure 1 shows the n_n – T₉ conditions, which are based on the six mass data sets [AME2020 (AME), AME2020 with uncertainty (AME ± Δ), FRDM, WS4, DZ10, and KTUY] required by the N = 50 [panels (A)–(F)] and N = 82 [panels (G)–(L)] WP. It is clear that the upper and lower bounds of temperature (or neutron density) at a given neutron density (or temperature) required by the heavier isotones are higher (or lower) than those required by the lighter ones. For example, with the AME mass data, the required neutron density is ranging in n_n = 2.5 × 10²¹–3.1 × 10²⁶ cm⁻³ for ⁷⁹Cu while it is n_n = 9 × 10²⁵–9 × 10²⁹ cm⁻³ for ⁷⁶Fe at T = 2 GK, as can be seen in panel (A). Still with the AME data set, panel (G) shows that the condition is ranging in n_n = 8.5 × 10²⁴–5.4 × 10²⁸ cm⁻³ for ¹³⁰Cd but n_n = 10²⁶–3.8 × 10²⁹ cm⁻³ for ¹²⁸Pd at T = 3 GK. The same behaviors are also observed for the AME ± Δ, FRDM, WS4, DZ10, and KTUY mass models. This phenomenon can be understood by the greater two-neutron separation energies (S_2n) of the heavier elements. For instance, the two-neutron separation energy is S_2n = 9.69 (or 5.65) MeV for ⁷⁹Cu (or ⁸¹Cu) but it is S_2n = 6.07 (or 2.9) MeV for ⁷⁶Fe (or ⁷⁸Fe). Subsequently, the average neutron separation energies for ⁷⁶Fe and ⁷⁹Cu are $1.45\leqslant {S}_{n}^{0}{({n}_{n},{T}_{9})}_{{\rm{Fe}}}\leqslant 3.035\,\mathrm{MeV}$ and $2.825\leqslant {S}_{n}^{0}{({n}_{n},{T}_{9})}_{{\rm{Cu}}}\leqslant 4.845\,\mathrm{MeV}$, respectively.

Figure 1.

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Figure 1.n_n − T₉ conditions needed for [panels (A)–(C)] ⁷⁶Fe, ⁷⁷Co, ⁷⁸Ni, ⁷⁹Cu and [panels (D)–(F)] ¹²⁷Rh, ¹²⁸Pd, ¹²⁹Ag, ¹³⁰Cd to be the N = 50 and N = 82 WP, respectively, in the r-process. The conditions were estimated using two-neutron separation energies, S_2n, from AME2020, AME2020 with uncertainty ΔS_2n [(AME ± Δ)], and FRDM, WS4, DZ10, and KTUY calculations. The grey shaded areas in panels (A)–(C), (E), and (F) [or (G)–(L)] denote the overall conditions for all the investigated N = 50 [or N = 82] WP. Yellow shaded are in panel (D) indicates the separation between conditions required by ⁷⁶Fe and ⁷⁹Cu.



Table 1.
Table 1.Two-neutron separation energies (S_2n, in MeV) of the ⁷⁶Fe, ⁷⁷Co, ⁷⁸Ni, ⁷⁹Cu, ¹²⁷Rh, ¹²⁸Pd, ¹²⁹Ag, ¹³⁰Cd, ¹⁹²Dy, ¹⁹³Ho, ¹⁹⁴Er, ¹⁹⁵Tm critical waiting points (WPs) [(Z, A) nuclei], and their even-N neighbors [(Z, A + 2) nuclei]. The separation energy values calculated using the FRDM, WS4, DZ10, and KTUY models taken from [44, 46, 48, 49], respectively, are compared to available experimental data obtained from the AME2020 database [45]. The half-lives of the isotopes are taken from the NUBASE2020 database [52] or new QRPA calculations in [44].

WPs	[44, 52]	AME20		FRDM		WS4		DZ10		KTUY
AXN	T1/2(ms)	Z, A	Z, A + 2	Z, A	Z, A + 2	Z, A	Z, A + 2	Z, A	Z, A + 2	Z, A	Z, A + 2
76Fe50	3.0	6.07 ± 0.78	2.90 ± 1.30	6.66	2.74	5.43	2.45	7.87	2.46	6.85	3.01
77Co50	15.0	7.49 ± 0.72	4.49 ± 1.16	7.41	3.59	7.23	3.32	8.74	3.48	7.94	3.79
78Ni50	122.2	8.83 ± 0.50	4.50 ± 0.72	8.58	4.43	8.83	4.26	9.48	4.45	9.33	4.45
79Cu50	241.3	9.69 ± 0.10	5.65 ± 0.32	9.60	5.56	9.66	5.64	10.00	5.58	10.15	5.61
127Rh82	28.0	8.04 ± 0.78	3.23 ± 1.15	9.15	3.55	7.96	3.68	8.66	4.05	8.80	3.89
128Pd82	35.0	8.74 ± 0.64	4.48 ± 0.58	9.49	4.38	8.62	4.31	9.16	4.68	9.48	4.48
129Ag82	49.9	9.36 ± 0.45	5.02 ± 0.64	10.17	5.00	9.41	4.90	9.70	5.22	10.13	5.12
130Cd82	126.8	10.02 ± 0.02	5.49 ± 0.06	10.87	5.60	10.23	5.47	10.46	5.63	10.80	5.68
192Dy126	0.02	N/A	N/A	6.21	2.49	5.49	2.45	5.99	2.57	6.54	2.48
193Ho126	0.02	N/A	N/A	6.54	2.97	5.75	3.07	6.48	3.05	6.97	3.52
194Er126	0.05	N/A	N/A	6.97	3.39	6.02	3.60	6.97	3.52	7.47	3.34
195Tm126	0.04	N/A	N/A	7.42	3.77	6.38	4.07	7.48	3.97	7.92	3.80

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The shaded areas in the panels (A)–(F) or (G)–(L) of figure 1 are shown for the overall n_n − T₉ conditions based on the AME, AME ± Δ, FRDM, WS4, DZ10, and KTUY nuclear masses for the N = 50 or N = 82 WP of interest. In the panels (A)–(F), the upper curves were determined using the two-neutron separation energy of the lightest N = 50 WP, S_2n(⁷⁶ Fe); the lower curves were deduced using S_2n(⁸¹ Cu), which is the two-neutron separation energy of the N = 50 + 2 isotope of the heaviest N = 50 WP (⁷⁹Cu), except for the WS4 model in the panel (D). Similarly to the N = 50 chain, the upper and lower curves for the N = 82 WP were deduced using the two-neutron separation energies of ¹²⁷Rh (N = 82) and ¹³²Cd (N = 82 + 2), respectively. For the exception with the WS4 calculations, S_2n(⁸¹ Cu) = 5.6406 MeV is larger than S_2n(⁷⁶ Fe) = 5.4335 MeV [refer to panel (D) of figure 2], leading to the separation between the condition bands required by ⁷⁶Fe and ⁷⁹Cu to be WP. In other words, there is no n_n − T₉ condition for both ⁷⁶Fe and ⁷⁹Cu to simultaneously be WP. This exception is interesting because it clearly shows the sensitivity of the WP n_n − T₉ conditions to the mass models.

Figure 2.

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Figure 2.Two-neutron separation energies S_2n/2 (in MeV) based on the AME2020 database and FRDM, WS4, DZ10, KTUY mass models of the even-N isotopes considered in the present study. The grey shaded areas denote the average neutron separation energies, S⁰_n, which result in the overall n_n − T₉ conditions required by N = 50 or 82 waiting points. Yellow shaded area in panel (D) indicates the S⁰_n value which does not fulfill the conditions required by both ⁷⁶Fe and ⁷⁹Cu to be simultaneous waiting points.


By taking equation (3) together with equation (5), it is clear that the shaded areas in figure 1 correspond to those in figure 2, which indicate the differences between two-neutron separation energies of even-N (Z, A) and (Z, A + 2) isotopes of each element. On the other hand, the neutron-density ranges required by the N = 82 WP are much broader than those necessary for the N = 50 ones for all the nuclear mass models of interest. This large deviation is due to the greater difference between the two-neutron separation energies S_2n(Z, A) and S_2n(Z, A + 2) of the lightest N (N = 50, 82) and N + 2 (52, 84) isotopes of the heaviest WP, respectively. For instance, based on AME, we have ${S}_{2n}{(}_{26}^{76}$ Fe ${}_{50})-{S}_{2n}{(}_{29}^{81}$ Cu₅₂) = 6.07–5.65 = 0.42 MeV for the N = 50 WP but ${S}_{2n}{(}_{45}^{127}$ Rh ${}_{82})-{S}_{2n}{(}_{48}^{132}$ Cd₈₄) = 8.04–5.49 = 2.55 MeV for the N = 82 ones, leading to a factor of 6 for the average neutron energy (S⁰_n) band of the N = 82 broader than that of the N = 50 WP [see panels (A) and (G) of figure 2] at T = 2 GK. Since the neutron density n_n logarithmically depends on S⁰_n, the factor of 6 can result in the much narrower n_n − T₉ band (n_n = 9 × 10²⁵–3 × 10²⁶ cm⁻³ at 2 GK) for the N = 50 WP [panel (A) of figure 1] compared to that (n_n = 3 ×10²³–5 × 10²⁶ cm⁻³ at 2 GK) for the N = 82 WP [panel (G) of figure 1].

Similarly to the N = 50 and 82 nuclei, the n_n − T₉ conditions for the N = 126 WP of ¹⁶²Dy, ¹⁶³Ho, ¹⁶⁴Er, and ¹⁶⁵Tm are determined by taking two-neutron separation energies S_2n of ¹⁶²Dy and ¹⁶⁷Tm for the upper and lower bounds, respectively. The results of the conditions [upper panels (A)–(D)] based on the energies [lower panels (E)–(H)] calculated using the FRDM, WS4, DZ10, and KTUY models are shown in figure 3. Notice that the results based on AME cannot be considered due to the lack of experimental data for these nuclei. The results show that S_2n values of the N = 126 nuclei are about 1 MeV lower than those of the N = 50 and 82 ones. This leads to the higher n_n – T₉ conditions for the N = 126 WP. For instance, the upper limit of neutron densities required by the N = 126 isotopes are n_n = 10²⁴ and n_n = 10²⁹ (cm⁻³) at T = 1 and 2.5 GK, respectively, which are about 1–4 orders of magnitude higher than those required by the N = 50 and 82 WP. On the other hand, the lower and upper limits of the n_n − T₉ conditions are also very uncertain, which may vary by 1–2 orders of magnitude due to the uncertainties in the two-neutron separation energies based on different mass models. Moreover, it can be seen in panels (A)–(D) that the conditions based on the WS4 and DZ10 models are mostly similar to each other but different from those based on the FRDM and KTUY methods. These results can be understood by the differences in S_2ns of the investigated isotopes, as shown in panels (E)–(H).

Figure 3.

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Figure 3.Comparisons of the overall n_n – T₉ conditions required by the N = 126 waiting points [upper panels] and associated S_2n/2 (in MeV) based on the FRDM, WS4, DZ10, and KTUY mass models of N = 126 neighboring even-N isotopes [lower panels]. The shaded areas in the lower panels indicates the ${S}_{2n}^{0}$ values, which result in the shaded areas in the conditions in the upper panels.


The results and discussion above obviously indicate that the n_n − T₉ conditions are sensitive to the mass predictions. Hence, it is worth studying the significant difference among the models to examine the influence of mass predictions on the overall conditions required by the WP. In figure 4, we show the differences in the two-neutron separation energies among the nuclear mass models [AME, FRDM, WS4, DZ10, KTUY] for 108 investigated isotopes, which are shown in figures 2 and 3. The difference is defined by the standard deviation, rms, which is given by(8)$\begin{eqnarray}\begin{array}{rcl}{rms} & = & \sqrt{\frac{1}{n}\sum {\left({S}_{2n}^{i}-{S}_{2n}^{\mathrm{AME}}\right)}^{2}}\,\,\,\,\mathrm{or}\\ {rms} & = & \sqrt{\frac{1}{n}\sum {\left({S}_{2n}^{i}-{S}_{2n}^{\mathrm{KTUY}}\right)}^{2}}\,.\end{array}\end{eqnarray}$

Figure 4.

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Figure 4.Differences in the two-neutron separation energies between the FRDM, WS4, DZ10, KTUY calculations and experimental data AME2020 [panels (A)–(D)] and between FRDM, WS4, DZ10 and KTUY calculations [panels (E)–(G)]. The dashed lines are to guide the eyes.


The results show that the WS4 calculations give a larger deviation (rms = 0.254 MeV) [panel (B)] compared to the FRDM, DZ10, and KTUY computations in the comparisons to experimental data (AME). In contrast, the KTUY prediction is close to the AME2020 data since the deviation is only rms = 0.192 MeV between them [panel (D)]. This result suggests that the KTUY model is reliable to be employed for the investigations of the ${n}_{n}-{T}_{9}$ conditions required by the N = 50, 82, and 126 WP. Since the KTUY calculation is closer to experimental data, we examined the differences between the others and KTUY, as can be seen in panels (E)–(G). It was found that the KTUY prediction is closer to the DZ10 calculation than to the FRDM estimation. The largest deviation of rms = 0.434 MeV in panel (F) indicates that it is significantly different between the WS4 and KTUY predictions. On the other hand, the comparisons show that the deviations among the predictions based on different models are increased with increasing neutron numbers, which correspond to the charge asymmetry factor I > 0.315. In other words, the uncertainty in mass calculations for extremely neutron-rich isotopes with I > 0.315 is still very large, which is from a few factors to one order of magnitude.

Although with a small deviation of rms ≤ 0.254 MeV, the mass uncertainty strongly impacts the overall n_n – T₉ conditions required by the N = 50, 82, and 126 WP, as can be seen in figure 5. For instance, in the case of the N = 50 WP, it can be seen in the left panel that the n_n – T₉ band based on the AME data set (vertical hatched band) is very narrow while it is much broader if the AME ± Δ or the other mass data (i.e. FRDM, DZ10, etc) are taken into the calculations. The deviation between the n_n – T₉ bandwidths can be understood by the uncertainty, which results in the large difference between the AME and AME ± Δ S⁰_n-bands [see panels (A) and (B) of figure 2], of the two-neutron separation energies of the concerned nuclei. The difference between these bandwidths is found to be ${S}_{n}^{0}[{\rm{AME}}\pm {\rm{\Delta }}]-{S}_{n}^{0}[{\rm{AME}}]\,=0.76-0.21$$=0.55\,\mathrm{MeV}$ due to the uncertainties of ΔS_2n(⁷⁶ Fe) = 0.78 and ΔS_2n(⁸¹ Cu) = 0.32 MeV in the two-neutron separation energies of ⁷⁶Fe and ⁸¹Cu, respectively. As a result, the bandwidth of the neutron density required by the N = 50 WP is extended from 9.2 × 10²⁵–3.1 × 10²⁶ (AME, vertically hatched band) to 9.5 × 10²⁴–7.8 × 10²⁶ cm⁻³ (AME ± Δ, horizontally hatched band) at T = 2 GK. In other words, the current uncertainties of about 12% and 5% in the experimental two-neutron separation energies of ⁷⁶Fe and ⁸¹Cu lead to a large deviation by a factor of 35 in the n_n – T₉ conditions required by the N = 50 WP at 2 GK. For the FRDM data, the S⁰_n bandwidth is 0.34 MeV, which is larger than that (0.21 MeV) of the AME-S⁰_n band [see panels (A) and (C) of figure 2], leading to a difference of one order of magnitude between the n_n – T₉ bandwidths based on the AME (vertically hatched band) and FRDM (blue shaded are) data sets. Still in the left panel, it is clear that the DZ10 model results in the widest range of the conditions while the KTUY method can give a range comparable with that based on the AME mass data.

Figure 5.

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Figure 5.Comparisons of the overall n_n – T₉ conditions required by N = 50 [left panel], N = 82 [middle panel], and N = 126 waiting points based on various mass models (AME, AME ± Δ, FRDM, WS4, DZ10, and KTUY). The overlapped areas indicate the same n_n − T₉ conditions which are determined by all the mass models. Notice that the WS4 results are not included in the case of N = 50 because there is no conditions for both Fe and Cu to be simultaneous waiting points as discussed in previous sections.


For the N = 82 WP (middle panel of figure 5), the n_n − T₉ condition are also uncertain because of the deviation in the two-neutron separation energies of ¹²⁷Rh (N = 82) and ¹³²Cd (N = 82 + 2). It can be seen in the middle panel of figure 5 that the n_n − T₉ bandwidths based on AME, AME ± Δ, and FRDM are 2.8 × 10²³–4.7 × 10²⁶ (green shaded band), 3.5 × 10²²–5.1 × 10²⁶ (vertically hatched band), and 1.2 × 10²²–3.2 × 10²⁶ (yellow shaded area) cm⁻³ at T = 2 GK, respectively. These n_n − T₉ bandwidths correspond to the ${S}_{n}^{0}({n}_{n},{T}_{9})$ bandwidths of 1.3, 1.7, and 1.8 MeV in the panels (G), (H), and (I) of figure 2, respectively. The results show that the uncertainty in the S_2n energies of the even-N isotopes can lead to a change of one order of magnitude in the lower thresholds of the neutron density required for ¹²⁷Rh, ¹²⁸Pd, ¹²⁹Ag, and ¹³⁰Cd (N = 82) to be the r-process WP. Besides, it was found that the pairs of AME-WS4 and DZ10-KTUY calculations are in good agreement with each other due to almost the same S⁰_n values, as can be seen in panels (G)–(J) and (K)–(L) of figure 2, respectively. On the other hand, in the right panel of figure 5, we show the conditions required by the N = 126 WP, which were calculated using FRDM (vertically hatched band), WS4 (purple shaded area), DZ10 (yellow shaded area), and KTUY (horizontally hatched band) models. The results indicate that the condition based on the WS4 model is narrowest while that based on the KTUY is widest. These results confirm that the estimations of the stellar conditions for the r-process WP are very sensitive to the mass models.

Because the WP calculations strongly depend on mass models as discussed above, it is necessary to determine the general n_n − T₉ conditions required by the WP regardless of a specific mass model. By taking whole the conditions calculated with the various models, we can determine the mixed areas in figure 5, which indicate the general conditions so that all the mass models are satisfied. The conditions are summarized in table 2. It is clear that the N = 126 isotopes required much higher neutron density and temperature, up to two orders of magnitude, to be the r-process WP. These results are consistent with those obtained by Xu et al in [33].


Table 2.
Table 2.Summary of the n_n − T₉ conditions, which are overlaps calculated by all the mass models (AME, AME ± Δ, FRDM, WS4, DZ10, KTUY), for the N = 50 and 82 waiting points; and by the FRDM, WS4, DZ10, KTUY models for the N = 126 nuclei. Notice that neutron density n_n is in the unit of cm⁻³.

T9 (in GK)	1	1.5	2	2.5	3
nn (N = 50)	[1.5–7.5] × 1019	[0.3–1.4] × 1024	[0.8–2.4] × 1026	[1.8–5.4] × 1027	[1.6–3.6] × 1028
nn (N = 82)	[0.0002–5.5] × 1019	[0.0004–1.5] × 1024	[0.004–2.1] × 1026	[0.03–4.5] × 1027	[0.05–2.5] × 1028
nn (N = 126)	[0.006–7.5] × 1023	[0.004–1] × 1027	[0.05–3.3] × 1028	[0.08–2.4] × 1029	[0.06–1] × 1030

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Since the total half-lives of the N = 50 (∑T_1/2(⁷⁶Fe − ⁷⁹Cu) ≈ 382 ms [52]), N = 82 (∑T_1/2(¹²⁷Rh − ¹³⁰Cd) ≈ 240 ms [52]), and N = 126 (∑T_1/2(¹⁹²Dy − ¹⁹⁵Tm) ≈ 130 ms [44]) are about 752 ms, the distribution of isotopes in the first (N = 50 or A ≈ 80), second (N = 82 or A ≈ 130), and third (N = 126 or A ≈ 195) peaks in the r-process abundance during the β⁻-decay waiting time of 0.752 s strongly depends on the n_n − T₉ conditions. The conditions for all the peaks simultaneously appearing in the waiting time can be determined by taking the overlaps of the n_n − T₉ condition bands required by the N = 50, N = 82, and N = 126 WP, as shown in figure 6. It is found that the n_n − T₉ conditions needed for the N = 50 isotopes (horizontally hatched bands) to be the WP also fulfill the requirement for the N = 82 ones for all the models. In other words, when the temperature and neutron density of stellar environments (i.e. the bubble of high entropy neutrino-driven winds, neutron merging stars, etc) satisfy the yellow shaded areas, both the A ≈ 80 and A ≈ 130 peaks are simultaneously produced in the r-process. This result is similar to that obtained in a previous study conducted by Xu et al [33] with the HFB-17, RMF, and previous versions of the FRDM [35] and WS [36] mass models. We found that the three peaks can appear at the same time if the FRDM and KTUY models [panel (C) and (F)] are employed in the calculations. This finding contradicts the calculations [33] using the HFB-17, WS, and the previous version of the FRDM [35] models. Besides, the conditions for the simultaneous appearance of the peaks are very sensitive in the case of the DZ10 model as can be seen in panel (E). If the neutron density is decreased by a factor of 3, there is no appearance of the N = 126 peak.

Figure 6.

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Figure 6.The overall n_n − T₉ conditions required for both A = 80, A = 130, and/or A = 195 peaks simultaneously appearing in the r-process abundance. The yellow shaded, vertically hatched, and/or horizontally hatched bands indicate the conditions required for the N = 50 N = 82, and/or N = 126 isotones to be waiting points, respectively, which were calculated using the AME, AME ± Δ, FRDM, WS4, DZ10, and KTUY data sets.


To evaluate the impact of the n_n − T₉ condition differences due to the mass uncertainty on the nucleosynthesis via the r-process, we calculated the r-process abundance of isotopes using the n_n − T₉ conditions, which were shown in figure 6, as the input of the calculation. The other necessary inputs are determined as follows: the β-decay rates of WP are obtained from the NUBASE2020 database [52] or in [44]; the (n, γ), and (γ, n) reaction rates of the WP are calculated using the TALYS code [53] based on the Hauser-Fesbach model [50]; and the β-decay and β-delayed neutron emission rates are taken from [35]. Since the astrophysical sites for the r-process are still very complicated, we assumed that the r-process occurs in a high-entropy neutron-driven wind, which is popularly believed to be a suitable site for the main r-process [54–57]. For the computation, we employed the calculation method which was well described in [58] by using the r-Java computer code [58].

The estimated r-process abundances based on each mass model are shown in figure 7. Although these abundances are much (more than 3–4 orders of magnitude) lower, the positions of the peaks are consistent with those of the Solar system, which are adopted from [59]. Notice that the lower abundance of the simulations can be understood by the short r-process timescale, which is a few milliseconds [10] compared to 4.6 billion years of the Solar system [60]. The left panel shows that there is no appearance of the third peak since the conditions are only overlapped for the N = 50 and 82 WP, as can be seen in panels (A) and (E) of figure 6. A very faint peak around A ≈ 190 (the third peak) clearly reflects the narrow overlapped band of the condition in the case of the DZ10 model. On the other hand, the nucleosyntheses based on the AME and DZ10 models of the isotopes beyond the second peak are also different from each other. The AME abundance is much lower than that based on the DZ10 mass data. It was found that the small deviation of rms < 0.216 MeV in the AME and DZ10 two-neutron separation energies (S_2n) can lead to a large discrepancy, up to two orders of magnitude, in the abundances. For the FRDM and KTUY cases, the overlapped conditions in panels (C) and (F) of figure 6 clearly result in the simultaneous appearances of all the three peaks, as can be seen in the right panel. Besides, the KTUY abundance is higher than that based on the FRDM for the A < 190, and vice versa for the A > 190 nuclei. Moreover, their abundances are sensitive to the nuclear mass of the investigated WP. We found that the deviation of rms = 0.281 in the two-neutron separation energies based on the FRDM and KTUY models results in a difference up to three and two orders of magnitude in the abundances of the isotopes in the ranges of A = 110–140 and A = 200–250, respectively. Obviously, the r-process calculation is still very uncertain with the current predicted nuclear mass. The results of the present study indicate that the mass uncertainty should be reduced to be, at least, smaller than 0.216 MeV. Unfortunately, there are large discrepancies in both theoretical calculations and experimental data, e.g. results reported in [61] and [45]. Therefore, precise mass measurements for extremely neutron-rich nuclei of ⁷⁶Fe, ⁷⁷Co, ⁷⁸Ni, ⁷⁹Cu, ¹²⁷Rh, ¹²⁸Pd, ¹²⁹Ag, ¹³⁰Cd, ¹⁹²Dy, ¹⁹³Ho, ¹⁹⁴Er, ¹⁹⁵Tm, and their even-N neighboring isotopes are necessary for a better understanding of the r-process properties (i.e. reaction paths, isotopic abundance, etc).

Figure 7.

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Figure 7.The r-process abundances of isotopes calculated using the n_n − T₉ conditions based on the AME, DZ10, FRDM, and KTUY mass data sets. Notice that the calculated abundances are upscaled by a factor of 1000 to normalize with the A = 130 peak of the Solar abundance which is taken from [59] (black dots with error bars). The yellow shaded areas indicate the differences among the abundances based on the different models. The thin lines are to guide the eyes at the A ≈ 80, 130, and 190 peaks.

4. Conclusion

In the present study, we estimated the neutron density and temperature conditions required by the WP, which distribute in the A = 80, 130, and 195 peaks of the r-process abundance, using the newest updated two-neutron separation energies from the AME2020 database (with [AME ± Δ] and without uncertainty [AME]) and from calculations based on the FRDM, WS4, DZ10, and KTUY models. The findings in this work are different from those investigated using the RMF, HFB, and WS mass models in a previous work. The results indicate that the n – T₉ conditions based on the AME data are much narrower than those estimated using the AME ± Δ and FRDM mass data. The FRDM and KTUY calculations are in a good agreement for the n_n – T₉ condition approximation. The stellar condition ranging from the upper to lower bounds for the N = 50 isotones is mostly smaller and overlaps with that for the N = 82 ones when the AME, AME ± Δ, FRDM, DZ10, and KTUY models are employed. Subsequently, with these models, the n – T₉ conditions necessary for the N = 50 WP fulfill the conditions to simultaneously produce both A = 80 and A = 130 peaks in the r-process abundance. In contrast, the results based on the WS4 are different from those of the others for the simultaneous appearances of all the three peaks. Moreover, the conditions required by the N = 126 isotopes are much higher than and mostly out of those required by the N = 50 and 82 ones. On the other hand, the systematic study on the two-neutron separation energy indicates that the KTUY model is closer to experimental data in the AME2020 database than to the others. Hence, the KTUY model is reasonable to be employed for the mass predictions. The results show that the current mass uncertainties of ⁷⁶Fe and ⁸¹Cu, ¹²⁷Rh, ¹³²Cd, ¹⁹²Dy, and ¹⁹⁷Tm strongly impact the determination of the n – T₉ conditions at the (n, γ)–(γ, n) equilibrium for the N = 50, 82, and 126 WP, leading to a large discrepancy in the r-process abundance. Therefore, precise mass measurements for these nuclei are necessary to reduce the uncertainty in the r-process WP approximation. Finally, this work provides useful information for further studies on the r-process.

Acknowledgments

This work was supported by the National Research Foundation of Korea (NRF) Grant funded by the Korean Ministry of Education, Science, and Technology (No. NRF-2020R1C1C1006029).

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