1 Introduction

The null β^-decay half-lives of the very neutron-rich nuclei play an important role in the r-process nucleosynthesis, which governs the synthesis of nearly half of the elements heavier than iron [1-3]. The r-process occurs in neutron rich sites and involves the continuous and rapid transfer of nuclei to the neutron-rich region through neutron capture, until the neutron capture rates become comparable to the ones of β^--decay. The β^--decay increases the proton number of neutron-rich nuclei and enables their conversion into heavier elements. The final abundance of any stable nuclei depends strongly on the competition between the β-decay and the neutron capture rates. Therefore, the β-decay half-life is an essential input for the r-process calculations.

As the r-process nuclei are in the neutron-rich area, most of them are beyond the scope of the current experiment; therefore, the r-process calculations must rely on theoretical estimations. Various theoretical models have been developed for this purpose. Basically, the various β decays can be understood by utilizing the time-dependent perturbation theory [4]. Macroscopic gross theory, which adopts the sum rules of the β decay strength function as well as the single particle energy distribution, can estimate the β decay half-lives for the entire nuclide chart [5]. Among the most well-known microscopic approaches, the nuclear shell-model includes all the correlations consistently and accurately reproduces the experimental β decay half-lives of the waiting-point nuclei at N = 50 [6], 82[7, 8], and 126 [9]. Furthermore, the roles played by Gamow-Teller and first-forbidden transitions have also been investigated in the large-scale shell model calculations [10]. However, their applicability is limited to medium-heavy nuclei or those localized around magic numbers, owing to their large configuration space. Another useful microscopic approach is the proton-neutron quasi-particle random phase approximation (pn-QRPA) [11-13]. pn-QRPA models can be used for global calculations, which are based on realistic nucleon-nucleon interactions[14-16] or effective interactions such as the relativistic interaction [17-19], non-relativistic zero-range Skyrme interaction [20-22], and finite-range Gogny interaction [23]. In the QRPA calculations, the isoscalar pairing interaction [17, 20] and the tensor interaction [21] were shown to provide strong impact on the low-lying excited states and subsequently the β-decay half-life.

Inspired by the shell model calculations performed to determine the role played by the tensor force in elucidating the new magic numbers in the exotic neutron-rich nuclei [24-26], significant attention has been paid to the role played by the tensor terms (especially the zero-range one) in nuclear many-body calculations. These studies show that zero-range tensor interactions can improve the description of ground-state properties, such as single-particle energies [27, 28], shell evolutions[29, 30] binding energies [31, 32] and the stability of superheavy nuclei [33, 34]. Moreover, such terms also strongly impact the collective excitations associated with the non-charge exchange [35-37] and the spin-isospin excitations associated with charge-exchange, such as the Gamow-Teller (GT) transitions [38, 39], Spin-Dipole transitions[40-42], and the β-decay half-life [21]. After noticing the above effects induced by the zero-range tensor interactions in the non-relativistic calculations, such effects were also investigated in the relativistic density functional calculations by separating out tensor force components; these studies show improvements in the descriptions of ground state properties derived from tensor force [43-45]. However, for the QRPA calculations based on the relativistic mean field (in which the tensor interaction was not included due to the exclusion of the Fock term), the effects of tensor force on the spin-isospin excitations [46] and β-decay half-life were not observed [17, 19]. When the tensor force was taken into account by including the Fock term, the effect of the tensor force on the GT transition was still negligible [47, 48]. Meanwhile, studies on finite-range tensor terms have also been performed using Gogny interactions to elucidate the exotic neutron-rich nuclei. These studies have shown that the inclusion of the tensor force improves the single-particle energies [49-51] and collective spin-isospin excitation energies [52, 53]; however, the effect of the tensor force on the β decay half-life has not been studied so far.

In this work, the HF+RPA model based on Gogny force has been applied to study the impact of the finite-range tensor terms on the β decay of magic and semimagic nuclei of (³⁴Si, ^68,78Ni and ¹³²Sn). The aim was to determine if the finite-range tensor force can improve the β-decay half-life; this has been reported previously in the DFT+QRPA model that is based on the zero-range tensor force; however, this result has not been obtained using the RMF+QRPA model.

The remainder of this paper is organized as follows. In Sect. 2, we briefly present the HF+RPA model and the β decay theory. In Sect. 3, some details of the HF+RPA calculations as well as parameters of the Gogny interactions including finite-range tensor terms are provided. In Sect. 4, the β^--decay half-lives and the Q and log ft values are predicted, analyzed, and compared to the experimental data. The conclusions are given in Sect. 5.

2 Formulism and Method

The β decay of the presently studied nuclei in this work is dominated by the GT transition, in which the transition operator is defined as:

where the isospin operator, $t_{\pm }=\frac{1}{2}(t_x+i{t}_y)$. In the RPA model, the creation operator for these collective excited states can be expressed as:

where the coherent coefficients X and Y denote the forward and backward amplitudes, respectively, which may be obtained by solving the RPA equation.

${\mathcal{A}}_{qk\mathrm{,}{J}^{\pi }M}^{†}\equiv {[a_q^{†}{a}_k]}_{J^{\pi }M}$

In the above formula, a and a represent the standard single-particle creation and annihilation operators, respectively. Meanwhile, J and π denote the total angular momentum and parity, respectively, for the transitions.

After diagonalizing the RPA equation, we can obtain the GT strength:

where ωm is the excitation energy of the m^th GT state with respect to the ground state of the mother nucleus.

Once the GT states have been obtained from the HF+RPA calculations, the GT-type β^-decay half-life can be calculated using the expression [20]:

where D=6163.4± 3.8 s (e.g., see Ref. [54]), and the ratio of the axial-vector and vector coupling constants, g_A≡G_A/G_V=1.26, where G_A is the axial-vector coupling constant, G_V is the vector coupling constant. By assuming a quenching factor of 0.7 [55, 56], g_A was set at 0.882. Here, the sum runs over all 1⁺ states within the β decay energy window Q=Δ_nH-ωm >0 MeV, with Δ_nH=0.78227 MeV denoting the mass difference between the neutron and hydrogen atom.

The log ftm value for each allowed state, which is related to the GT decay with ΔJπ=1⁺, can be obtained [20]:

3 Details of the Calculations

In this section, we provide details of the HF+RPA calculations and the parameters.

The HF equation has been solved using the spherical-harmonic oscillator basis expansion method with the main quantum number cut-off being set at N_max=6, 8, and 10 for ³⁴Si, ^68,78Ni, and ¹³²Sn, respectively. For each nucleus, the harmonic oscillator width ℏω has been chosen according to an interpolation formula [57]

In our present RPA calculations, all the central and tensor terms have been included. The residual two-body spin-orbit interaction is excluded in the RPA, because the RPA correlations of the spin-orbit interaction produce a trivial effect on the GT transition. Furthermore, the self-consistency is well kept in the present HF+RPA model. The only input required by our self-consistent HF+RPA approach is the Gogny-like interactions that include tensor terms. The original Gogny interaction was proposed to exhibit the following form [58, 59]:

It is composed of central and spin-orbit terms. The central interaction consists of two distinct Gaussians with ranges μ₁ and μ₂. ${\widehat{P}}_{\sigma }$ and ${\widehat{P}}_{\tau }$ denote the standard spin and isospin exchange operators, respectively.

Following the work of Refs. [49, 50, 60], we included the tensor terms of the Gaussian form:

where μ_T corresponds to the longest range of the central terms, V_T1 and V_T2 are the two parameters, and ${\widehat{S}}_{12}$ is the usual tensor operator [52].

We performed HF+RPA calculations with the well-known D1S [61] and D1M [62] interactions. All tensor parameters based on Eq.(8), including D1ST2a [49], D1ST2b [49], D1ST2c [50], D1MT2a [63], D1MT2c [53, 63], and D1MTd [51], were employed to calculate the relevant quantities (e.g., half-lives t_1/2, Q value, and log ft) of ¹³²Sn, ^68,78Ni, and ³⁴Si. After the calculations, we found that compared to the standard D1S or D1M parameters, the above-mentioned parameters do not improve the description of the properties associated with β decay of these nuclei and fail to reproduce the half-lives. Therefore, the tensor parameters must be tuned to accurately reproduce the β decay properties of these nuclei.

Following the procedure adopted in Refs. [49, 50], we tune the values of V_T1 and V_T2 by reproducing the 8.8 MeV experimental splitting between the neutrons of the 1f_7/2 and 1f_5/2 states in ⁴⁸Ca [64], together with the experimental half-lives of doubly magic and semimagic nuclei. Using this procedure, two sets of parameters (D1STx and D1MTx) were obtained, as shown in Table 1. It should be noted that the tensor terms are just added perturbatively on top of the existing force, and the present strengths of the tensor terms are adjusted with only a few properties that are considered. For a wider global use, one needs to produce effective interactions with the tensor force by using a full variational procedure to fit the tensor and the central terms on equal footing; this procedure should take into account more quantities.

4 Results and Discussion

In this section, we reveal the β decay half-lives of doubly magic and semi-magic nuclei of ¹³²Sn, ^68,78Ni, and ³⁴Si, as well as their Q and log ft values. We focus on the differences between the results calculated with and without the tensor force.

The β-decay half-lives of ³⁴Si, ^68,78Ni, and ¹³²Sn are shown inFig. 1 and have been compared with experimental data [65, 66]. The half-lives calculated with D1S and D1STx are presented in panel (a), whereas those calculated with D1M and D1MTx are shown in panel (b). The arrows denote the infinite half-lives. The results calculated with D1S and D1M systematically overestimate the experimental data of ⁶⁸Ni, ³⁴Si, and ⁷⁸Ni, and fail to predict the β-decay of ¹³²Sn. However, when the tensor forces D1MTx and D1STx are included, the half-lives are reduced systematically, the calculations can predict the β decay for ¹³²Sn, and other significant improvements can be observed. With D1MTx, the half-lives of ⁶⁸Ni, ³⁴Si, and ⁷⁸Ni are reduced and close to the experimental values; however, for ¹³²Sn, the half-life is still larger than its experimental counterpart. With D1STx, the half-lives of ⁶⁸Ni, ³⁴Si, and ⁷⁸Ni agree well with experimental data, and the half-life of ¹³²Sn was closer to the experimental value.

Fig. 1.Full-screen View

(Color online) β-decay half-lives of ³⁴Si, ^68,78Ni, and ¹³²Sn obtained by performing calculations with D1S and D1STx (a), as well as D1M and D1MTx (b) interactions. The results calculated without the tensor force are labeled using red diamond-dashed lines, while those obtained with the tensor force are labeled using blue diamond-dash-dot lines. The experimental data have been obtained from Refs. citeAudi,NNDC, which are labeled using black circle lines with error bars. The arrows denote infinite half-lives.

The β decay Q values of the four nuclei, together with the experimental data, are displayed inFig. 2 to understand the origin of this systematic reduction in the half-lives that is caused by the tensor force. In the figure, the Q values of the branches with the largest branching ratios are selected for comparison with the experimental Q values. The decay scheme of ⁷⁸Ni has not been identified experimentally; the experimental $Q_{{\beta }^{-}}$ value [65, 66] is shown instead in the panels. It can be seen that the Q values for the four nuclei are enhanced by the tensor force as it shifts the low-energy GT states downwards. As observed in panel (a), with D1STx, the Q values for ⁶⁸Ni and ⁷⁸Ni increased by approximately 1 MeV; for ³⁴Si, the Q value is enhanced slightly. Meanwhile, for ¹³²Sn, β decay becomes possible, although the theoretical Q value is underestimated. In general, with the inclusion of the tensor force, the Q values for ⁶⁸Ni and ³⁴Si can be accurately reproduced, while the Q value for ¹³²Sn is still approximately 1 MeV smaller than the experimental value. As observed in panel (b), with D1MTx, the Q values of ^68,78Ni are enhanced significantly; however, this increase is less significant for ³⁴Si.

Fig. 2.Full-screen View

(Color online) Same as Fig.1, but for the Q values of the β decay of ¹³²Sn, ⁶⁸Ni, ³⁴Si and ⁷⁸Ni.

Furthermore, the log ft values of the four nuclei, which are related to the GT strength of the decay with the largest branching ratio (Fig. 3), have been compared with the experimental data. As observed in panel (a), when the tensor force is absent, the log ft values of ⁶⁸Ni and ³⁴Si are much larger than the experimental values. After the tensor force D1STx is included, the log ft values of ⁶⁸Ni and ³⁴Si are drastically reduced, which implies that the tensor force can shift the strength to the low energy GT states and further shift these states downwards to lower excitation energies. As observed in panel (b), with D1M and D1MTx, the log ft values are reasonably reproduced; however, for ³⁴Si, D1M tends to largely overestimate the experimental value.

Fig. 3.Full-screen View

(Color online) The same as Fig. 2, but for log ft. The experimental data of log ft values of ¹³²Sn, ⁶⁸Ni, and ³⁴Si were taken from Refs. citeAudi,NNDC. The experimental reference value of log ft value for ⁷⁸Ni [65, 66] is evaluated from its experimental $Q_{{\beta }^{-}}$ value.

Based on the above-mentioned results of the Q and log ft values, the effects of the tensor force on the β-decay half-life have been understood thoroughly. In ¹³²Sn, the tensor force shifts The low-energy GT state to an energy lower than the β decay threshold; hence, β decay is possible. However, the Q values obtained by D1STx and D1MTx are lower than the experimental values by approximately 1.2 and 1.6 MeV, respectively, while the calculated half-lives are still greater than their experimental values. In the case of ⁶⁸Ni, its half-life is accurately reproduced with D1STx as both the $Q_{{\beta }^{-}}$ and log ft values are also reproduced well; however, with D1MTx, the half-life is underestimated slightly because the Q value is somewhat overestimated. In ³⁴Si, owing to the large log ft value (weak GT strength), the calculation performed without tensor force overestimates the half-life, whereas the inclusion of the tensor force circumvents this problem. In ⁷⁸Ni, as the tensor force enhances the Q value, the half-life is reproduced well.

5 Conclusion

In this study, we added a finite-range tensor force to the self-consistent HF+RPA model by employing the Gogny interaction to investigate the impact of the tensor force on the β decay half-lives of magic and semi-magic nuclei of ¹³²Sn, ^68,78Ni, and ³⁴Si. The tensor force was added on the existing Gogny interactions D1S and D1M, and two parameter sets were adopted: D1STx and D1MTx. Our calculations show that the β-decay half-live calculations for the four nuclei are significantly and systematically improved with the inclusion of the tensor force, which considerably reduces the half-lives by shifting the low-energy GT state downward and increasing the strength of these transitions. We also determine the Q and log ft values, which provide details of how the tensor force reduces the half-lives. These results suggest that the tensor force decreases half-lives naturally as calculations for both the Q and log ft values are improved. The global calculations of the β decay half-lives, including the finite-range tensor force, need to be performed in future studies. This is because the present HF+RPA model is only valid for semi-magic and magic nuclei, which are close to the shells and spherical in shape. Consequently, the pairing interaction and the quasi-particle approximation should be added to the open-shell nuclei, while the deformation effect should be included.

Moreover, this work, together with that of Ref. [21], shows that for both the nonrelativistic effective Gogny and Skyrme interactions, the tensor force can reduce the β decay half-lives; however, this effect was not observed in the calculations performed using the conventional relativistic effective interaction [19]. Similar phenomenon was observed in the spin-isospin excitations of nuclei [38, 39, 46, 53]. This comparison indicates that strengths of the tensor force in the spin-isospin channel between the relativistic and nonrelativistic effective interactions are different; this is because the relativistic one was constrained based only on the ground state properties. As shown in Ref. [31], it is hard to determine the strength of the tensor force only by using the ground state properties because central and spin-orbit interactions may also contribute to these differences. In fact, one can attain a good description of some ground-state properties, such as mass, even without including the tensor force [67, 68]; however, when dealing with single-particle energies[29, 30] or spin-isospin excitations [21, 41], the tensor force is essential because it provides a strong constraint for the tensor force [32, 42].