Impact of finite-range tensor terms in the Gogny force on the β decay of magic nuclei

Impact of finite-range tensor terms in the Gogny force on the β decay of magic nuclei 1 first-author 2 3 1 corresp Corresponding author, bclphy@scu.edu.cn Corresponding author, bclphy@scu.edu.cn bclphy@scu.edu.cn https://orcid.org/0000-0002-2032-0306 College of Physics, Sichuan University, Chengdu 610065, China College of Physics, Sichuan University, Chengdu 610065, China Institute of Modern Physics, Chinese Academy of Sciences, Lanzhou 730000, China Institute of Modern Physics, Chinese Academy of Sciences, Lanzhou 730000, China University of Chinese Academy of Sciences, Beijing 100049, China University of Chinese Academy of Sciences, Beijing 100049, China Effects of finite-range tensor force on β decay of magic and semimagic nuclei of 34 Si, 68,78 Ni, and 132 Sn have been investigated using the self-consistent Hartree-Fock plus random phase approximation model. The tensor force shifts the low-lying Gamow-Teller states downward and systematically improves the calculations of Q and log ft values. Consequently, it systematically reduces the deviations between the theoretical and experimental data and significantly improves the calculation of β -decay half-lives. This effect is similar to that of zero-range tensor force. Finite-range tensor force Gogny force β decay Magic nuclei Half-life Q value log ft value 1 Introduction 1 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 1 - 3 3 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 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 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 6 ], 82[ 7 7 , 8 8 ], and 126 [ 9 9 ]. Furthermore, the roles played by Gamow-Teller and first-forbidden transitions have also been investigated in the large-scale shell model calculations [ 10 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 11 - 13 13 11-13 ]. pn-QRPA models can be used for global calculations, which are based on realistic nucleon-nucleon interactions[ 14 14 - 16 16 14-16 ] or effective interactions such as the relativistic interaction [ 17 17 - 19 19 17-19 ], non-relativistic zero-range Skyrme interaction [ 20 20 - 22 22 20-22 ], and finite-range Gogny interaction [ 23 23 ]. In the QRPA calculations, the isoscalar pairing interaction [ 17 17 , 20 20 ] and the tensor interaction [ 21 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 24 - 26 26 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 27 , 28 28 ], shell evolutions[ 29 29 , 30 30 ] binding energies [ 31 31 , 32 32 ] and the stability of superheavy nuclei [ 33 33 , 34 34 ]. Moreover, such terms also strongly impact the collective excitations associated with the non-charge exchange [ 35 35 - 37 37 35-37 ] and the spin-isospin excitations associated with charge-exchange, such as the Gamow-Teller (GT) transitions [ 38 38 , 39 39 ], Spin-Dipole transitions[ 40 40 - 42 42 40-42 ], and the β -decay half-life [ 21 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 43 - 45 45 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 46 ] and β -decay half-life were not observed [ 17 17 , 19 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 47 , 48 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 49 - 51 51 49-51 ] and collective spin-isospin excitation energies [ 52 52 , 53 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 ( 34 Si, 68,78 Ni and 132 Sn). 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 1 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: <math display="block" id="M1"><mrow><msup><mi>O</mi><mrow><mtext>GT</mtext><mo>±</mo></mrow></msup><mo>=</mo><mstyle displaystyle="true"><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>A</mi></munderover></mstyle><msubsup><mi>t</mi><mo>±</mo><mi>i</mi></msubsup><msup><mi>σ</mi><mi>i</mi></msup><mn>,</mn></mrow></math> where the isospin operator, <math id="M2"><mrow><msub><mi>t</mi><mo>±</mo></msub><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo stretchy="false">(</mo><msub><mi>t</mi><mi>x</mi></msub><mo>+</mo><mi>i</mi><msub><mi>t</mi><mi>y</mi></msub><mo stretchy="false">)</mo></mrow></math> . In the RPA model, the creation operator for these collective excited states can be expressed as: <math display="block" id="M3"><mrow><msubsup><mi>Q</mi><mrow><msup><mi>J</mi><mi>π</mi></msup><mi>M</mi></mrow><mrow><mi>m</mi><mo>†</mo></mrow></msubsup><mo>=</mo><mstyle displaystyle="true"><munder><mo>∑</mo><mrow><mi>q</mi><mi>k</mi></mrow></munder></mstyle><mo stretchy="false">[</mo><msubsup><mi>X</mi><mrow><mi>q</mi><mi>k</mi></mrow><mi>m</mi></msubsup><msubsup><mi mathvariant="script">A</mi><mrow><mi>q</mi><mi>k</mi><mn>,</mn><msup><mi>J</mi><mi>π</mi></msup><mi>M</mi></mrow><mo>†</mo></msubsup><mo>−</mo><msubsup><mi>Y</mi><mrow><mi>q</mi><mi>k</mi></mrow><mi>m</mi></msubsup><msub><mover accent="true"><mi mathvariant="script">A</mi><mo>˜</mo></mover><mrow><mi>q</mi><mi>k</mi><mn>,</mn><msup><mi>J</mi><mi>π</mi></msup><mi>M</mi></mrow></msub><mo stretchy="false">]</mo><mn>,</mn></mrow></math> where the coherent coefficients X and Y denote the forward and backward amplitudes, respectively, which may be obtained by solving the RPA equation. <math id="M4"><mrow><msubsup><mi mathvariant="script">A</mi><mrow><mi>q</mi><mi>k</mi><mn>,</mn><msup><mi>J</mi><mi>π</mi></msup><mi>M</mi></mrow><mo>†</mo></msubsup><mo>≡</mo><msub><mrow><mo stretchy="false">[</mo><msubsup><mi>a</mi><mi>q</mi><mo>†</mo></msubsup><msub><mi>a</mi><mi>k</mi></msub><mo stretchy="false">]</mo></mrow><mrow><msup><mi>J</mi><mi>π</mi></msup><mi>M</mi></mrow></msub></mrow></math> 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: <math display="block" id="M5"><mrow><msubsup><mi>B</mi><mrow><msubsup><mn>1</mn><mi>m</mi><mo>+</mo></msubsup></mrow><mrow><mtext>GT</mtext><mo>±</mo></mrow></msubsup><mo stretchy="false">(</mo><msub><mi>ω</mi><mi>m</mi></msub><mo stretchy="false">)</mo><mo>=</mo><mo>|</mo><mo>〈</mo><msubsup><mn>1</mn><mi>m</mi><mo>+</mo></msubsup><mo>|</mo><mo>|</mo><msup><mi>O</mi><mrow><mtext>GT</mtext><mo>±</mo></mrow></msup><mo>|</mo><mo>|</mo><mn>0</mn><mo>〉</mo><msup><mo>|</mo><mn>2</mn></msup><mn>,</mn></mrow></math> where ω m 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 20 ]: <math display="block" id="M6"><mrow><msub><mi>T</mi><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msub><mo>=</mo><mfrac><mi>D</mi><mrow><msubsup><mi>g</mi><mi>A</mi><mn>2</mn></msubsup><mstyle displaystyle="true"><munder><mo>∑</mo><mi>m</mi></munder></mstyle><msubsup><mi>B</mi><mi>m</mi><mrow><mtext>GT</mtext><mo>−</mo></mrow></msubsup><msub><mi>f</mi><mn>0</mn></msub><mo stretchy="false">(</mo><mi>Z</mi><mn>,</mn><mi>A</mi><mn>,</mn><msub><mi>ω</mi><mi>m</mi></msub><mo stretchy="false">)</mo></mrow></mfrac><mn>,</mn></mrow></math> where D =6163.4± 3.8 s (e.g., see Ref. [ 54 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 55 , 56 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 - ω m > 0 MeV, with Δ nH =0.78227 MeV denoting the mass difference between the neutron and hydrogen atom. The log ft m value for each allowed state, which is related to the GT decay with Δ J π =1 + , can be obtained [ 20 20 ]: <math display="block" id="M7"><mrow><mi>log</mi><mi>f</mi><msub><mi>t</mi><mi>m</mi></msub><mo>=</mo><msub><mrow><mtext>log</mtext></mrow><mrow><mn>10</mn></mrow></msub><mrow><mo>(</mo><mrow><mfrac><mi>D</mi><mrow><msubsup><mi>g</mi><mi>A</mi><mn>2</mn></msubsup><msubsup><mi>B</mi><mrow><msubsup><mn>1</mn><mi>m</mi><mo>+</mo></msubsup></mrow><mrow><mtext>GT</mtext><mo>−</mo></mrow></msubsup></mrow></mfrac></mrow><mo>)</mo></mrow><mn>.</mn></mrow></math> 3 Details of the Calculations 1 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 34 Si, 68,78 Ni, and 132 Sn, respectively. For each nucleus, the harmonic oscillator width ℏω has been chosen according to an interpolation formula [ 57 57 ] <math display="block" id="M8"><mrow><mi>ℏ</mi><mi>ω</mi><mo>=</mo><mo stretchy="false">(</mo><mn>0.0002</mn><msup><mi>A</mi><mn>2</mn></msup><mo>−</mo><mn>0.1</mn><mi>A</mi><mo>+</mo><mn>21.1</mn><mo stretchy="false">)</mo><mtext>MeV</mtext><mn>.</mn></mrow></math> 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 58 , 59 59 ]: <math display="block" id="M9"><mtable columnalign="left"><mtr><mtd><msub><mi>V</mi><mrow><mtext>eff</mtext></mrow></msub><mo stretchy="false">(</mo><mn>1,2</mn><mo stretchy="false">)</mo><mo>=</mo><mstyle displaystyle="true"><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mn>2</mn></munderover></mstyle><mtext>exp</mtext><mo stretchy="false">[</mo><mfrac><mrow><mo>−</mo><msup><mrow><mo stretchy="false">(</mo><msub><mstyle mathvariant="bold" mathsize="normal"><mi>r</mi></mstyle><mn>1</mn></msub><mo>−</mo><msub><mstyle mathvariant="bold" mathsize="normal"><mi>r</mi></mstyle><mn>2</mn></msub><mo stretchy="false">)</mo></mrow><mn>2</mn></msup></mrow><mrow><msubsup><mi>μ</mi><mi>i</mi><mn>2</mn></msubsup></mrow></mfrac><mo stretchy="false">]</mo></mtd></mtr><mtr><mtd><mo>×</mo><mo stretchy="false">(</mo><msub><mi>W</mi><mi>i</mi></msub><mo>+</mo><msub><mi>B</mi><mi>i</mi></msub><msub><mover accent="true"><mi>P</mi><mo>^</mo></mover><mi>σ</mi></msub><mo>−</mo><msub><mi>H</mi><mi>i</mi></msub><msub><mover accent="true"><mi>P</mi><mo>^</mo></mover><mi>τ</mi></msub><mo>−</mo><msub><mi>M</mi><mi>i</mi></msub><msub><mover accent="true"><mi>P</mi><mo>^</mo></mover><mi>σ</mi></msub><msub><mover accent="true"><mi>P</mi><mo>^</mo></mover><mi>τ</mi></msub><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mo>+</mo><mi>i</mi><msub><mi>W</mi><mrow><mi>L</mi><mi>S</mi></mrow></msub><mo stretchy="false">(</mo><msub><mi>σ</mi><mn>1</mn></msub><mo>+</mo><msub><mi>σ</mi><mn>2</mn></msub><mo stretchy="false">)</mo><mo>⋅</mo><msub><mover accent="true"><mo>∇</mo><mo stretchy="true">→</mo></mover><mrow><mn>12</mn></mrow></msub><mo>×</mo><mi>δ</mi><mo stretchy="false">(</mo><msub><mstyle mathvariant="bold" mathsize="normal"><mi>r</mi></mstyle><mn>1</mn></msub><mo>−</mo><msub><mstyle mathvariant="bold" mathsize="normal"><mi>r</mi></mstyle><mn>2</mn></msub><mo stretchy="false">)</mo><msub><mover accent="true"><mo>∇</mo><mo stretchy="true">←</mo></mover><mrow><mn>12</mn></mrow></msub></mtd></mtr><mtr><mtd><mo>+</mo><msub><mi>t</mi><mn>3</mn></msub><mo stretchy="false">(</mo><mn>1</mn><mo>+</mo><msub><mi>x</mi><mn>0</mn></msub><msub><mover accent="true"><mi>P</mi><mo>^</mo></mover><mi>σ</mi></msub><mo stretchy="false">)</mo><mi>δ</mi><mo stretchy="false">(</mo><msub><mstyle mathvariant="bold" mathsize="normal"><mi>r</mi></mstyle><mn>1</mn></msub><mo>−</mo><msub><mstyle mathvariant="bold" mathsize="normal"><mi>r</mi></mstyle><mn>2</mn></msub><mo stretchy="false">)</mo><msup><mrow><mo stretchy="false">[</mo><mi>ρ</mi><mo stretchy="false">(</mo><mfrac><mrow><msub><mstyle mathvariant="bold" mathsize="normal"><mi>r</mi></mstyle><mn>1</mn></msub><mo>+</mo><msub><mstyle mathvariant="bold" mathsize="normal"><mi>r</mi></mstyle><mn>2</mn></msub></mrow><mn>2</mn></mfrac><mo stretchy="false">)</mo><mo stretchy="false">]</mo></mrow><mi>α</mi></msup></mtd></mtr><mtr><mtd><mo>+</mo><mo stretchy="false">(</mo><mfrac><mrow><mn>1</mn><mo>−</mo><msub><mi>τ</mi><mrow><mn>1</mn><mi>z</mi></mrow></msub></mrow><mn>2</mn></mfrac><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mfrac><mrow><mn>1</mn><mo>−</mo><msub><mi>τ</mi><mrow><mn>2</mn><mi>z</mi></mrow></msub></mrow><mn>2</mn></mfrac><mo stretchy="false">)</mo><mfrac><mrow><msup><mi>e</mi><mn>2</mn></msup></mrow><mrow><mo>|</mo><msub><mstyle mathvariant="bold" mathsize="normal"><mi>r</mi></mstyle><mn>1</mn></msub><mo>−</mo><msub><mstyle mathvariant="bold" mathsize="normal"><mi>r</mi></mstyle><mn>2</mn></msub><mo>|</mo></mrow></mfrac><mn>.</mn></mtd></mtr></mtable></math> It is composed of central and spin-orbit terms. The central interaction consists of two distinct Gaussians with ranges μ 1 and μ 2 . <math id="M10"><mrow><msub><mover accent="true"><mi>P</mi><mo>^</mo></mover><mi>σ</mi></msub></mrow></math> and <math id="M11"><mrow><msub><mover accent="true"><mi>P</mi><mo>^</mo></mover><mi>τ</mi></msub></mrow></math> denote the standard spin and isospin exchange operators, respectively. Following the work of Refs. [ 49 49 , 50 50 , 60 60 ], we included the tensor terms of the Gaussian form: <math display="block" id="M12"><mtable columnalign="left"><mtr><mtd><msub><mi>V</mi><mrow><mtext>tensor</mtext></mrow></msub><mo stretchy="false">(</mo><mn>1,2</mn><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><msub><mi>V</mi><mrow><mtext>T1</mtext></mrow></msub><mo>+</mo><msub><mi>V</mi><mrow><mtext>T2</mtext></mrow></msub><msub><mover accent="true"><mi>P</mi><mo>^</mo></mover><mi>τ</mi></msub><mo stretchy="false">)</mo><msub><mover accent="true"><mi>S</mi><mo>^</mo></mover><mrow><mn>12</mn></mrow></msub><mtext>exp</mtext><mo stretchy="false">[</mo><mo>−</mo><mfrac><mrow><msup><mrow><mo stretchy="false">(</mo><msub><mstyle mathvariant="bold" mathsize="normal"><mi>r</mi></mstyle><mn>1</mn></msub><mo>−</mo><msub><mstyle mathvariant="bold" mathsize="normal"><mi>r</mi></mstyle><mn>2</mn></msub><mo stretchy="false">)</mo></mrow><mn>2</mn></msup></mrow><mrow><msubsup><mi>μ</mi><mtext>T</mtext><mn>2</mn></msubsup></mrow></mfrac><mo stretchy="false">]</mo></mtd></mtr><mtr><mtd><mo>=</mo><mo stretchy="false">[</mo><mo stretchy="false">(</mo><msub><mi>V</mi><mrow><mtext>T1</mtext></mrow></msub><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><msub><mi>V</mi><mrow><mtext>T2</mtext></mrow></msub><mo stretchy="false">)</mo><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><msub><mi>V</mi><mrow><mtext>T2</mtext></mrow></msub><msub><mi>τ</mi><mn>1</mn></msub><mo>⋅</mo><msub><mi>τ</mi><mn>2</mn></msub><mo stretchy="false">]</mo></mtd></mtr><mtr><mtd><mo>×</mo><msub><mover accent="true"><mi>S</mi><mo>^</mo></mover><mrow><mn>12</mn></mrow></msub><mtext>exp</mtext><mo stretchy="false">[</mo><mo>−</mo><mfrac><mrow><msup><mrow><mo stretchy="false">(</mo><msub><mstyle mathvariant="bold" mathsize="normal"><mi>r</mi></mstyle><mn>1</mn></msub><mo>−</mo><msub><mstyle mathvariant="bold" mathsize="normal"><mi>r</mi></mstyle><mn>2</mn></msub><mo stretchy="false">)</mo></mrow><mn>2</mn></msup></mrow><mrow><msubsup><mi>μ</mi><mtext>T</mtext><mn>2</mn></msubsup></mrow></mfrac><mo stretchy="false">]</mo><mn>,</mn></mtd></mtr></mtable></math> where μ T corresponds to the longest range of the central terms, V T1 and V T2 are the two parameters, and <math id="M13"><mrow><msub><mover accent="true"><mi>S</mi><mo>^</mo></mover><mrow><mn>12</mn></mrow></msub></mrow></math> is the usual tensor operator [ 52 52 ]. We performed HF+RPA calculations with the well-known D1S [ 61 61 ] and D1M [ 62 62 ] interactions. All tensor parameters based on Eq.(8), including D1ST2a [ 49 49 ], D1ST2b [ 49 49 ], D1ST2c [ 50 50 ], D1MT2a [ 63 63 ], D1MT2c [ 53 53 , 63 63 ], and D1MTd [ 51 51 ], were employed to calculate the relevant quantities (e.g., half-lives t 1/2 , Q value, and log ft ) of 132 Sn, 68,78 Ni, and 34 Si. 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 49 , 50 50 ], we tune the values of V T1 and V T2 by reproducing the 8.8 MeV experimental splitting between the neutrons of the 1 f 7/2 and 1 f 5/2 states in 48 Ca [ 64 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 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. 1 1 1 1 V T1 (MeV) 1 1 V T2 (MeV) 1 1 μ T (fm) 1 1 D1STx 1 1 -155 1 1 75 1 1 1.2 1 1 D1MTx 1 1 -210 1 1 160 1 1 1.0 4 Results and Discussion 1 In this section, we reveal the β decay half-lives of doubly magic and semi-magic nuclei of 132 Sn, 68,78 Ni, and 34 Si, 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 34 Si, 68,78 Ni, and 132 Sn are shown in Fig. 1 Fig. 1 and have been compared with experimental data [ 65 65 , 66 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 68 Ni, 34 Si, and 78 Ni, and fail to predict the β -decay of 132 Sn. However, when the tensor forces D1MTx and D1STx are included, the half-lives are reduced systematically, the calculations can predict the β decay for 132 Sn, and other significant improvements can be observed. With D1MTx, the half-lives of 68 Ni, 34 Si, and 78 Ni are reduced and close to the experimental values; however, for 132 Sn, the half-life is still larger than its experimental counterpart. With D1STx, the half-lives of 68 Ni, 34 Si, and 78 Ni agree well with experimental data, and the half-life of 132 Sn was closer to the experimental value. The β decay Q values of the four nuclei, together with the experimental data, are displayed in Fig. 2 Fig. 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 78 Ni has not been identified experimentally; the experimental <math id="M14"><mrow><msub><mi>Q</mi><mrow><msup><mi>β</mi><mo>−</mo></msup></mrow></msub></mrow></math> value [ 65 65 , 66 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 68 Ni and 78 Ni increased by approximately 1 MeV; for 34 Si, the Q value is enhanced slightly. Meanwhile, for 132 Sn, β decay becomes possible, although the theoretical Q value is underestimated. In general, with the inclusion of the tensor force, the Q values for 68 Ni and 34 Si can be accurately reproduced, while the Q value for 132 Sn is still approximately 1 MeV smaller than the experimental value. As observed in panel (b), with D1MTx, the Q values of 68,78 Ni are enhanced significantly; however, this increase is less significant for 34 Si. 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 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 68 Ni and 34 Si are much larger than the experimental values. After the tensor force D1STx is included, the log ft values of 68 Ni and 34 Si 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 34 Si, D1M tends to largely overestimate the experimental 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 132 Sn, 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 68 Ni, its half-life is accurately reproduced with D1STx as both the <math id="M16"><mrow><msub><mi>Q</mi><mrow><msup><mi>β</mi><mo>−</mo></msup></mrow></msub></mrow></math> 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 34 Si, 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 78 Ni, as the tensor force enhances the Q value, the half-life is reproduced well. 5 Conclusion 1 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 132 Sn, 68,78 Ni, and 34 Si. 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 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 19 ]. Similar phenomenon was observed in the spin-isospin ex citations of nuclei [ 38 38 , 39 39 , 46 46 , 53 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 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 67 , 68 68 ]; however, when dealing with single-particle energies[ 29 29 , 30 30 ] or spin-isospin excitations [ 21 21 , 41 41 ], the tensor force is essential because it provides a strong constraint for the tensor force [ 32 32 , 42 42 ]. References E.M. Burbidge , G.R. Burbidge , W.A. Fowler et al. , Synthesis of the elements in stars . Rev. Mod. Phys . 29 , 547 ( 1957 ). doi: 10.1103/RevModPhys.29.547 http://doi.org/10.1103/RevModPhys.29.547 Synthesis of the elements in stars J.J. Cowan , F.-K. Thielemann , J.W. Truran , The R-process and nucleochronology , Phys. Rep . 208 , 267 ( 1991 ). doi: 10.1016/0370-1573(91)90070-3 http://doi.org/10.1016/0370-1573(91)90070-3 The R-process and nucleochronology H.-L. Liu , D.-D. Han , P. Ji et al. , Reaction rate weighted multilayer nuclear reaction network , Chinese Phys. Lett . 37 , 112601 ( 2020 ), doi: 10.1088/0256-307X/37/11/112601 http://doi.org/10.1088/0256-307X/37/11/112601 Reaction rate weighted multilayer nuclear reaction network M. Ji , C. Xu , Quantum anti-zeno effect in nuclear β decay . Chin. Phys. Lett . 38 , 032301 ( 2021 ), doi: 10.1088/0256-307X/38/3/032301 http://doi.org/10.1088/0256-307X/38/3/032301 Quantum anti-zeno effect in nuclear β decay K. Takahashi , M. Yamada , Gross theory of nuclear β-decay . Prog. Theor. Phys . 41 , 1470 ( 1969 ). doi: 10.1143/PTP.41.1470; http://doi.org/10.1143/PTP.41.1470; Gross theory of nuclear β-decay T. Tachibana , M. Yamada , and Y. Yoshida , Improvement of the Gross Theory of β-Decay. II: One-Particle Strength Function , Prog. Theor. Phys . 84 , 641 ( 1990 ). doi: 10.1143/ptp/84.4.641 http://doi.org/10.1143/ptp/84.4.641 Improvement of the Gross Theory of β-Decay. II: One-Particle Strength Function K. Langanke , G. Martínez-Pinedo , Shell-model calculations of stellar weak interaction rates: II. Weak rates for nuclei in the mass range A=45-61 in supernovae environments . Nucl. Phys. A 673 , 481 ( 2000 ). doi: 10.1016/S0375-9474(00)00131-7 http://doi.org/10.1016/S0375-9474(00)00131-7 Shell-model calculations of stellar weak interaction rates: II. Weak rates for nuclei in the mass range A=45-61 in supernovae environments G. Martínez-Pinedo , K. Langanke , Shell-model half-lives for N=82 nuclei and their implications for the r process . Phys. Rev. Lett . 83 , 4502 ( 1999 ). doi: 10.1103/PhysRevLett.83.4502 http://doi.org/10.1103/PhysRevLett.83.4502 Shell-model half-lives for N=82 nuclei and their implications for the r process J. J. Cuenca-García , G. Martínez-Pinedo , K. Langanke et al. , Shell model half-lives for r-process N=82 nuclei . Eur. Phys. J. A 34 , 99 ( 2007 ). doi: 10.1140/epja/i2007-10477-3 http://doi.org/10.1140/epja/i2007-10477-3 Shell model half-lives for r-process N=82 nuclei T. Suzuki , T. Yoshida , T. Kajino et al. , β decays of isotones with neutron magic number of N=126 and r-process nucleosynthesis . Phys. Rev. C 85 , 015802 ( 2012 ). doi: 10.1103/PhysRevC.85.015802 http://doi.org/10.1103/PhysRevC.85.015802 β decays of isotones with neutron magic number of N=126 and r-process nucleosynthesis Q. Zhi , E. Caurier , J.J. Cuenca-García et al. , Shell-model half-lives including first-forbidden contributions for r-process waiting-point nuclei . Phys. Rev. C 87 , 025803 ( 2013 ). doi: 10.1103/PhysRevC.87.025803 http://doi.org/10.1103/PhysRevC.87.025803 Shell-model half-lives including first-forbidden contributions for r-process waiting-point nuclei J. Krumlinde , P. Möller , Calculation of Gamow-Teller β-strength functions in the rubidium region in the RPA approximation with Nilsson-model wave functions . Nucl. Phys. A 417 , 419 ( 1984 ). doi: 10.1016/0375-9474(84)90406-8 http://doi.org/10.1016/0375-9474(84)90406-8 Calculation of Gamow-Teller β-strength functions in the rubidium region in the RPA approximation with Nilsson-model wave functions P. Möller , J. Randrup , New developments in the calculation of β-strength functions . Nucl. Phys. A 514 , 1 ( 1990 ). doi: 10.1016/0375-9474(90)90330-O http://doi.org/10.1016/0375-9474(90)90330-O New developments in the calculation of β-strength functions I. N. Borzov , S. Goriely , and J. M. Pearson , Microscopic calculations of β-decay characteristics near the A=130 r-process peak . Nucl. Phys. A 621 , 307 ( 1997 ). doi: 10.1016/S0375-9474(97)00260-1 http://doi.org/10.1016/S0375-9474(97)00260-1 Microscopic calculations of β-decay characteristics near the A=130 r-process peak D.L. Fang , B.A. Brown , and T. Suzuki , Investigating β-decay properties of spherical nuclei along the possible r-process path . Phys. Rev. C 88 , 034304 ( 2013 ). doi: 10.1103/PhysRevC.88.034304 http://doi.org/10.1103/PhysRevC.88.034304 Investigating β-decay properties of spherical nuclei along the possible r-process path D.D. Ni , Z.Z. Ren , β-decay rates of neutron-rich Zr and Mo isotopes in the deformed quasiparticle random-phase approximation with realistic interactions . Phys. Rev. C 89 , 064320 ( 2014 ). doi: 10.1103/PhysRevC.89.064320 http://doi.org/10.1103/PhysRevC.89.064320 β-decay rates of neutron-rich Zr and Mo isotopes in the deformed quasiparticle random-phase approximation with realistic interactions D.D. Ni , Z.Z. Ren , β+/EC decay rates of deformed neutron-deficient nuclei in the deformed QRPA with realistic interactions . Phys. Lett. B 744 , 22 ( 2015 ). doi: 10.1016/j.physletb.2015.03.025 http://doi.org/10.1016/j.physletb.2015.03.025 β+/EC decay rates of deformed neutron-deficient nuclei in the deformed QRPA with realistic interactions T. Nikšić , T. Marketin , D. Vretenar et al. , β-decay rates of r-process nuclei in the relativistic quasiparticle random phase approximation . Phys. Rev. C 71 , 014308 ( 2005 ). doi: 10.1103/PhysRevC.71.014308 http://doi.org/10.1103/PhysRevC.71.014308 β-decay rates of r-process nuclei in the relativistic quasiparticle random phase approximation T. Marketin , D. Vretenar , P. Ring , Calculation of β-decay rates in a relativistic model with momentum-dependent self-energies . Phys. Rev. C 75 , 024304 ( 2007 ). doi: 10.1103/PhysRevC.75.024304 http://doi.org/10.1103/PhysRevC.75.024304 Calculation of β-decay rates in a relativistic model with momentum-dependent self-energies Z.M. Niu , Y.F. Niu , H.Z. Liang et al. , β-decay half-lives of neutron-rich nuclei and matter flow in the r-process . Phys. Lett. B 723 , 172 ( 2013 ). doi: 10.1016/j.physletb.2013.04.048 http://doi.org/10.1016/j.physletb.2013.04.048 β-decay half-lives of neutron-rich nuclei and matter flow in the r-process J. Engel , M. Bender , J. Dobaczewski et al. , β decay rates of r-process waiting-point nuclei in a self-consistent approach . Phys. Rev. C 60 , 014302 ( 1999 ). doi: 10.1103/PhysRevC.60.014302 http://doi.org/10.1103/PhysRevC.60.014302 β decay rates of r-process waiting-point nuclei in a self-consistent approach F. Minato , C.L. Bai , Impact of tensor force on β decay of magic and semimagic nuclei . Phys. Rev. Lett . 110 , 122501 ( 2013 ). doi: 10.1103/PhysRevLett.110.122501 http://doi.org/10.1103/PhysRevLett.110.122501 Impact of tensor force on β decay of magic and semimagic nuclei M.T. Mustonen , J. Engel , Global description of β-decay in even-even nuclei with the axially-deformed Skyrme finite-amplitude method . Phys. Rev. C 93 , 014304 ( 2016 ). doi: 10.1103/PhysRevC.93.014304 http://doi.org/10.1103/PhysRevC.93.014304 Global description of β-decay in even-even nuclei with the axially-deformed Skyrme finite-amplitude method M. Martini , S. Peru , S. Goriely , Gamow-Teller strength in deformed nuclei within the self-consistent charge-exchange quasiparticle random-phase approximation with the Gogny force . Phys. Rev. C 89 , 044306 ( 2014 ). doi: 10.1103/PhysRevC.89.044306 http://doi.org/10.1103/PhysRevC.89.044306 . Gamow-Teller strength in deformed nuclei within the self-consistent charge-exchange quasiparticle random-phase approximation with the Gogny force T. Otsuka , R. Fujimoto , Y. Utsuno et al. , Magic numbers in exotic nuclei and spin-isospin properties of the NN interaction . Phys. Rev. Lett . 87 , 082502 ( 2001 ). doi: 10.1103/PhysRevLett.87.082502 http://doi.org/10.1103/PhysRevLett.87.082502 Magic numbers in exotic nuclei and spin-isospin properties of the NN interaction T. Otsuka , T. Suzuki , R. Fujimoto et al. , Evolution of nuclear shells due to the tensor force . Phys. Rev. Lett . 95 , 232502 ( 2005 ). doi: 10.1103/PhysRevLett.95.232502 http://doi.org/10.1103/PhysRevLett.95.232502 Evolution of nuclear shells due to the tensor force T. Otsuka , T. Matsuo , D. Abe , Mean field with tensor force and shell structure of exotic nuclei . Phys. Rev. Lett . 97 , 162501 ( 2006 ). doi: 10.1103/PhysRevLett.97.162501 http://doi.org/10.1103/PhysRevLett.97.162501 Mean field with tensor force and shell structure of exotic nuclei B. A. Brown , T. Duguet , T. Otsuka et al. , Tensor interaction contributions to single-particle energies . Phys. Rev. C 74 , 061303(R) ( 2006 ). doi: 10.1103/PhysRevC.74.061303 http://doi.org/10.1103/PhysRevC.74.061303 Tensor interaction contributions to single-particle energies M. Grasso , Z. Y. Ma , E. Khan et al. , Evolution of the proton sd states in neutron-rich Ca isotopes , Phys. Rev. C 76 , 044319 ( 2007 ). doi: 10.1103/PhysRevC.76.044319 http://doi.org/10.1103/PhysRevC.76.044319 Evolution of the proton sd states in neutron-rich Ca isotopes G. Colò , H. Sagawa , S. Fracasso et al. , Spin–orbit splitting and the tensor component of the Skyrme interaction . Phys. Lett. B 646 , 227 ( 2007 ). doi: 10.1016/j.physletb.2007.01.033 http://doi.org/10.1016/j.physletb.2007.01.033 Spin–orbit splitting and the tensor component of the Skyrme interaction D.M. Brink , Fl. Stancu , Evolution of nuclear shells with the Skyrme density dependent interaction . Phys. Rev. C 75 , 064311 ( 2007 ). doi: 10.1103/PhysRevC.75.064311 http://doi.org/10.1103/PhysRevC.75.064311 Evolution of nuclear shells with the Skyrme density dependent interaction T. Lesinski , M. Bender , K. Bennaceur et al. , Tensor part of the Skyrme energy density functional: Spherical nuclei . Phys. Rev. C 76 , 014312 ( 2007 ). doi: 10.1103/PhysRevC.76.014312 http://doi.org/10.1103/PhysRevC.76.014312 Tensor part of the Skyrme energy density functional: Spherical nuclei D. Wu , C.-L. Bai , H. Sagawa et al. , Contributions of optimized tensor interactions on the binding energyies . Nucl. Sci. Tech 31 , 14 ( 2020 ). doi: 10.1007/s41365-020-0727-7 http://doi.org/10.1007/s41365-020-0727-7 Contributions of optimized tensor interactions on the binding energyies E. B. Suckling , P. D. Stevenson , The effect of the tensor force on the predicted stability of superheavy nuclei . Europhys. Lett . 90 , 12001 ( 2010 ). doi: 10.1209/0295-5075/90/12001 http://doi.org/10.1209/0295-5075/90/12001 The effect of the tensor force on the predicted stability of superheavy nuclei X.-R. Zhou , H. Sagawa , Effect of tensor interaction on heavy and superheavy nuclei . Mod. Phys. Lett. A 25 , 1809 ( 2010 ). doi: 10.1142/S0217732310000381 http://doi.org/10.1142/S0217732310000381 Effect of tensor interaction on heavy and superheavy nuclei L.-G. Cao , G. Colò , H. Sagawa et al. , Effects of the tensor force on the multipole response in finite nuclei . Phys. Rev. C 80 , 064304 ( 2009 ). doi: 10.1103/PhysRevC.80.064304 http://doi.org/10.1103/PhysRevC.80.064304 Effects of the tensor force on the multipole response in finite nuclei D. Davesne , M. Martini , K. Bennaceur et al. , Nuclear response for the Skyrme effective interaction with zero-range tensor terms . Phys. Rev. C 80 , 024314 ( 2009 ). doi: 10.1103/PhysRevC.80.024314 http://doi.org/10.1103/PhysRevC.80.024314 Nuclear response for the Skyrme effective interaction with zero-range tensor terms L.-G. Cao , H. Sagawa , and G. Colò , Effects of tensor correlations on low-lying collective states in finite nuclei . Phys. Rev. C 83 , 034324 ( 2011 ). doi: 10.1103/PhysRevC.83.034324 http://doi.org/10.1103/PhysRevC.83.034324 Effects of tensor correlations on low-lying collective states in finite nuclei C.L. Bai , H. Sagawa , H.Q. Zhang et al. , Effect of tensor correlations on Gamow–Teller states in 90Zr and 208Pb . Phys. Lett. B 675 , 28 ( 2009 ). doi: 10.1016/j.physletb.2009.03.077 http://doi.org/10.1016/j.physletb.2009.03.077 Effect of tensor correlations on Gamow–Teller states in 90Zr and 208Pb C.L. Bai , H.Q. Zhang , X.Z. Zhang et al. , Quenching of Gamow-Teller strength due to tensor correlations in 90Zr and 208Pb . Phys. Rev. C 79 , 041301(R) ( 2009 ). doi: 10.1103/PhysRevC.79.041301 http://doi.org/10.1103/PhysRevC.79.041301 Quenching of Gamow-Teller strength due to tensor correlations in 90Zr and 208Pb C.L. Bai , H.Q. Zhang , X.Z. Zhang et al. , Effect of the tensor force on charge-exchange spin-dependent multipole excitations . Chin. Phys. Lett . 27 , 102101 ( 2010 ). doi: 10.1088/0256-307X/27/10/102101 http://doi.org/10.1088/0256-307X/27/10/102101 Effect of the tensor force on charge-exchange spin-dependent multipole excitations C.L. Bai , H.Q. Zhang , H. Sagawa et al. , Effect of the tensor force on the charge exchange spin-dipole excitations of 208Pb . Phys. Rev. Lett . 105 , 072501 ( 2010 ). doi: 10.1103/PhysRevLett.105.072501 http://doi.org/10.1103/PhysRevLett.105.072501 Effect of the tensor force on the charge exchange spin-dipole excitations of 208Pb C.L. Bai , H.Q. Zhang , H. Sagawa et al. , Spin-isospin excitations as quantitative constraints for the tensor force . Phys. Rev. C 83 , 054316 ( 2011 ). doi: 10.1103/PhysRevC.83.054316 http://doi.org/10.1103/PhysRevC.83.054316 Spin-isospin excitations as quantitative constraints for the tensor force L.J. Jiang , S. Yang , B.Y. Sun et al. , Nuclear tensor interaction in a covariant energy density functional . Phys. Rev. C 91 , 034326 ( 2015 ). doi: 10.1103/PhysRevC.91.034326 http://doi.org/10.1103/PhysRevC.91.034326 Nuclear tensor interaction in a covariant energy density functional Z.-H. Wang , Q. Zhao , H.Z. Liang et al. , Quantitative analysis of tensor effects in the relativistic Hartree-Fock theory . Phys. Rev. C 98 , 034313 ( 2018 ). doi: 10.1103/PhysRevC.98.034313 http://doi.org/10.1103/PhysRevC.98.034313 Quantitative analysis of tensor effects in the relativistic Hartree-Fock theory J. Geng , J. Xiang , B.Y. Sun et al. , Relativistic Hartree-Fock model for axially deformed nuclei . Phys. Rev. C 101 , 064302 ( 2020 ). doi: 10.1103/PhysRevC.101.064302 http://doi.org/10.1103/PhysRevC.101.064302 Relativistic Hartree-Fock model for axially deformed nuclei T. Marketin , E. Litvinova , D. Vretenar et al. , Fragmentation of spin-dipole strength in 90Zr and 208Pb . Phys. Lett. B 706 , 477 ( 2012 ). doi: 10.1016/j.physletb.2011.11.050 http://doi.org/10.1016/j.physletb.2011.11.050 Fragmentation of spin-dipole strength in 90Zr and 208Pb H.Z. Liang , N.V. Giai , J. Meng , Spin-isospin resonances: A self-consistent covariant description . Phys.Rev.Lett . 101 , 122502 ( 2008 ). doi: 10.1103/PhysRevLett.101.12250 http://doi.org/10.1103/PhysRevLett.101.12250 Spin-isospin resonances: A self-consistent covariant description Z.H. Wang , T. Naito , H. Z. Liang et al. , Self-consistent random-phase approximation based on the relativistic Hartree-Fock theory: Role of ρ-tensor coupling . Phys.Rev. C 101 , 064306 ( 2020 ). doi: 10.1103/PhysRevC.101.064306 http://doi.org/10.1103/PhysRevC.101.064306 Self-consistent random-phase approximation based on the relativistic Hartree-Fock theory: Role of ρ-tensor coupling M. Anguiano , M. Grasso , G. Cò , V. De Donno et al. , Tensor and tensor-isospin terms in the effective Gogny interaction . Phys. Rev. C 86 , 054302 ( 2012 ). doi: 10.1103/PhysRevC.86.054302 http://doi.org/10.1103/PhysRevC.86.054302 Tensor and tensor-isospin terms in the effective Gogny interaction M. Grasso , M. Anguiano , Tensor parameters in Skyrme and Gogny effective interactions: Trends from a ground-state-focused study , Phys. Rev. C 88 , 054328 ( 2013 ). doi: 10.1103/PhysRevC.88.054328 http://doi.org/10.1103/PhysRevC.88.054328 Tensor parameters in Skyrme and Gogny effective interactions: Trends from a ground-state-focused study G. Cò , M. Anguiano , V. De Donno et al. , Matter distribution and spin-orbit force in spherical nuclei . Phys. Rev. C 97 , 034313 ( 2018 ). doi: 10.1103/PhysRevC.97.034313 http://doi.org/10.1103/PhysRevC.97.034313 Matter distribution and spin-orbit force in spherical nuclei M. Anguiano , G. Cò , V. De Donno , and A. M. Lallena , Tensor effective interaction in self-consistent random-phase approximation calculations . Phys. Rev. C 83 , 064306 ( 2011 ). doi: 10.1103/PhysRevC.83.064306 http://doi.org/10.1103/PhysRevC.83.064306 Tensor effective interaction in self-consistent random-phase approximation calculations V. De Donno , G. Cò , M. Anguiano , and A. M. Lallena , Charge-exchange excitations with finite-range interactions including tensor terms . Phys. Rev. C 90 , 024326 ( 2014 ). doi: 10.1103/PhysRevC.90.024326 http://doi.org/10.1103/PhysRevC.90.024326 Charge-exchange excitations with finite-range interactions including tensor terms I. N. Borzov , S. Goriely , Weak interaction rates of neutron-rich nuclei and the r-process nucleosynthesis . Phys. Rev. C 62 , 035501 ( 2000 ). doi: 10.1103/PhysRevC.62.035501 http://doi.org/10.1103/PhysRevC.62.035501 Weak interaction rates of neutron-rich nuclei and the r-process nucleosynthesis C. Gaarde , in Proceedings of the Niels Bohr Centennial Conference , edited by R. Broglia , G. Hagemann , and B. Herskind ( North-Holland , Amsterdam , 1985 ), p. 449 . E. Caurier , G. Martínez-Pinedo , F. Nowacki et al. , The shell model as a unified view of nuclear structure . Rev. Mod. Phys . 77 , 427 ( 2005 ). doi: 10.1103/RevModPhys.77.427 http://doi.org/10.1103/RevModPhys.77.427 The shell model as a unified view of nuclear structure M. Kleban , B. Nerlo-Pomorska , J.F. Berger et al. , Global properties of spherical nuclei obtained from Hartree-Fock-Bogoliubov calculations with the Gogny force , Phys. Rev. C 65 , 024309 ( 2002 ). doi: 10.1103/PhysRevC.65.024309 http://doi.org/10.1103/PhysRevC.65.024309 Global properties of spherical nuclei obtained from Hartree-Fock-Bogoliubov calculations with the Gogny force J. Decharge , M. Girod , D. Gogny , Self consistent calculations and quadrupole moments of even Sm isotopes . Phys. Lett. B 55 , 361 ( 1975 ). doi: 10.1016/0370-2693(75)90359-7 http://doi.org/10.1016/0370-2693(75)90359-7 Self consistent calculations and quadrupole moments of even Sm isotopes D. Gogny , in Nuclear Self-Consistent Fields , edited by G. Ripka and M. Porneuf ( North-Holland , Amsterdam , 1975 ). Nuclear Self-Consistent Fields N. Onishi , J. W. Negele , Two-body and three-body effective interactions in nuclei . Nucl. Phys. A 301 , 336 ( 1978 ). doi: 10.1016/0375-9474(78)90266-X http://doi.org/10.1016/0375-9474(78)90266-X Two-body and three-body effective interactions in nuclei J.F. Berger , M. Girod , D. Gogny , Microscopic analysis of collective dynamics in low energy fission . Nucl. Phys. A 428 , 23 ( 1984 ). doi: 10.1016/0375-9474(84)90240-9 http://doi.org/10.1016/0375-9474(84)90240-9 Microscopic analysis of collective dynamics in low energy fission S. Goriely , S. Hilaire , M. Girod et al. , First Gogny-Hartree-Fock-Bogoliubov nuclear mass model . Phys. Rev. Lett . 102 , 242501 ( 2009 ). doi: 10.1103/PhysRevLett.102.242501 http://doi.org/10.1103/PhysRevLett.102.242501 First Gogny-Hartree-Fock-Bogoliubov nuclear mass model M. Anguiano , A.M. Lallena , G. Cò , V. De Donno et al. , Gogny interactions with tensor terms . Eur. Phys. J. A 52 , 183 ( 2016 ). doi: 10.1140/epja/i2016-16183-1 http://doi.org/10.1140/epja/i2016-16183-1 Gogny interactions with tensor terms P.D. Cottle , K. W. Kemper , Single-neutron energies near N=28 and the absence of the N=34 subshell closure in the Ti isotopes , Phys. Rev. C 78 , 037304 ( 2008 ). doi: 10.1103/PhysRevC.78.037304 http://doi.org/10.1103/PhysRevC.78.037304 Single-neutron energies near N=28 and the absence of the N=34 subshell closure in the Ti isotopes G. Audi , F.G. Kondev , M. Wang et al. , The NUBASE2016 evaluation of nuclear properties . Chin. Phys. C 41 , 030001 ( 2017 ). doi: 10.1088/1674-1137/41/3/030001 http://doi.org/10.1088/1674-1137/41/3/030001 The NUBASE2016 evaluation of nuclear properties National Nuclear Data Center , " NuDat 2.8 Database ", https://www.nndc.bnl.gov/nudat2 https://www.nndc.bnl.gov/nudat2 NuDat 2.8 Database S. Goriely , N. Chamel , J.M. Pearson , Further explorations of Skyrme-Hartree-Fock-Bogoliubov mass formulas. XIII. The 2012 atomic mass evaluation and the symmetry coefficient , Phys. Rev. C 88 , 024308 ( 2013 ). doi: 10.1103/PhysRevC.88.024308 http://doi.org/10.1103/PhysRevC.88.024308 Further explorations of Skyrme-Hartree-Fock-Bogoliubov mass formulas. XIII. The 2012 atomic mass evaluation and the symmetry coefficient S. Goriely , N. Chamel , J. M. Pearson , Further explorations of Skyrme-Hartree-Fock-Bogoliubov mass formulas. XVI. Inclusion of self-energy effects in pairing , Phys. Rev. C 93 , 034337 ( 2016 ). doi: 10.1103/PhysRevC.93.034337 http://doi.org/10.1103/PhysRevC.93.034337 Further explorations of Skyrme-Hartree-Fock-Bogoliubov mass formulas. XVI. Inclusion of self-energy effects in pairing All authors contributed to the study conception and design. Material preparation, data collection and analysis were performed by Da-Zhuang Chen, Dong-Liang Fang and Chun-Lin Bai. The first draft of the manuscript was written by Da-Zhuang Chen and all authors commented on previous versions of the manuscript. All authors read and approved the final manuscript. 10.1007/s41365-021-00908-9 This work was supported by the National Natural Science Foundation of China (Nos. 11575120 and 11822504) and the Science Specialty Program of Sichuan University (No. 2020SCUNL210) . 2021-04-15 2021-05-17 2021-05-26 2021-07-16 2021-07-01 2021-08-05 © China Science Publishing & Media Ltd. (Science Press), Shanghai Institute of Applied Physics, the Chinese Academy of Sciences, ChineseNuclear Society and Springer Nature Singapore Pte Ltd. 2021 This is a post-peer-review, pre-copyedit version of an article published in Nuclear Science and Techniques. The final authenticated version is available online at: http://dx.doi.org/10.1007/s41365-021-00908-9 http://dx.doi.org/10.1007/s41365-021-00908-9 . 2021 Nuclear Science and Techniques 07 32