Contributions of optimized tensor interactions on the binding energies of nuclei

Contributions of optimized tensor interactions on the binding energies of nuclei 1 first-author https://orcid.org/0000-0003-0174-4135 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 2 3 1 4 College of Physics, Sichuan University, Chengdu 610065, China College of Physics, Sichuan University, Chengdu 610065, China RIKEN Nishina Center, Wako 351-0198, Japan RIKEN Nishina Center, Wako 351-0198, Japan Center for Mathematics and Physics, University of Aizu, Aizu-Wakamatsu, Fukushima 965-8560, Japan Center for Mathematics and Physics, University of Aizu, Aizu-Wakamatsu, Fukushima 965-8560, Japan China institute of atomic energy, Beijing 102413, China China institute of atomic energy, Beijing 102413, China The tensor parts of Skyrme interactions are constrained from the collective charge-exchange Spin-Dipole (SD) and Gamow-Teller (GT) excitation energies in 90 Zr and 208 Pb, together with the isotopic dependence of energy splitting between proton h 11/2 and g 7/2 single-particle orbits along the Z =50 isotopes. With the optimized tensor interactions, the binding energies of spherical or weakly deformed nuclei with A =54 228 are studied systematically. The present results show that the global effect of tensor interaction is attractive, and systematically increases the binding energies of all these nuclei and makes the nuclei more bound. The root mean squared deviation (RMSD) of the calculated binding energies from the experimental values is significantly improved by the optimized tensor interactions and the contribution of the tensor interaction to the binding energy is estimated. Tensor force Binding energy Gamow-Teller transition Spin-Dipole transition Single-particle energy differences. 1 Introduction 1 Nuclear energy density functionals (EDFs) such as Skyrme, Gogny, and relativistic mean field models (RMF, RHF) have achieved great success for describing the properties of nuclear ground states as well as excited states in the last few decades [ 1 1 - 4 4 1-4 ]. By fitting to a set of experimental data, primarily saturation properties of nuclear matter, binding energies, and charge radii of some closed shell nuclei, EDFs can predict the properties of ground-states and excited-state, i.e., excitation energies and transition strengths of both low-lying collective states and giant resonances along the wide region of nuclear chart. Tensor force is an important component of the bare nucleon-nucleon interaction [ 5 5 ]; specifically, it was recognized historically as an essential constituent to make the deuteron bound [ 6 6 ]. However, its relevance for EDF has been discarded for a long time after the early achievement of Skyrme inter actions in the 1970s. This implies that in earlier implementations of density functional theory (DFT), based on effective interactions such as Skyrme’s, the tensor force was either considered of trivial importance or ignored for the purpose of simplicity. Diversely, the Skyrme EDFs show explicit contributions of tensor interactions. The contributions are demonstrated in the spin-orbit potential as the spin-current density with those from the momentum dependent terms of the central force. As these terms vanish in spin-orbit saturated nuclei, and many nuclei used in the fits belong to these nuclei, most of the early Skyrme parametrizations ignored not only the tensor terms but also the spin-current terms [ 7 7 ]. The revival of tensor terms occurred in the beginning of the 21st century with the development of new radioactive ion beam facilities, by which more neutron-rich nuclei have been studied and new phenomena have been observed with respect to the evolution of the spin-orbit splitting with neutron excess [ 8 8 ]. Afterwards, the importance of an effective neutron-proton tensor force was suggested to explain this evolution [ 9 9 ]. Inclusion of tensor terms could produce more qualitatively experimental data within the Gogny [ 10 10 ], Skyrme [ 7 7 , 11 11 - 15 15 11-15 ], and relativistic frameworks [ 16 16 - 19 19 16-19 ] (see also a review article ref. [ 20 20 ]). But until recently, the strength of the tensor force is not well determined and the global effect of the tensor interaction for the nuclei is obscure. There are two ways to determine the tensor terms in Skyrme EDF. First is to add the tensor terms on top of existing parameter sets and study experimental data of spin-orbit splitting and giant resonances [ 13 13 - 15 15 13-15 , 21 21 - 24 24 21-24 ]. The second way is to optimize the central part of Skyrme EDF for a given set of tensor terms. In the optimized process, the parameters of central part of EDF are fitted for a protocol of simultaneous reproduction of nuclear bulk properties such as binding energies and radii of finite nuclei and the empirical characteristics of infinite nuclear matter. For example, Tij EDFs belong to this category. [ 7 7 , 11 11 , 25 25 ]. We refer to the first and the second categories as "adiabatic EDF with tensor terms", and "variational EDF with tensor terms", respectively. By employing empirical excitation energies of Gamow-Teller (GT) and charge-exchange Spin-Dipole (SD) states in 90 Zr and 208 Pb and the evolution of energy splitting between proton h 11/2 and g 7/2 single-particle orbits along the Z =50 isotopes, our strategy starts firstly from some existing parameter sets and optimizes the tensor terms. Here, we propose optimized tensor interactions for the adiabatic EDFs. We then apply these EDF+optimized tensor terms for the calculations of binding energies of 22 isotopic chains from Ca to Ra to study the contribution of the tensor interactions to the binding energies. Further, we choose a few variational EDFs that have similar tensor terms to the optimized tensor terms of adiabatic EDFs and calculate the evolution of energy splitting between proton h 11/2 and g 7/2 states and the binding energies of the 22 isotopic chains. In comparison with the adiabatic EDF and the variational EDF, we explore not only the role of optimized tensor terms for nuclear structure calculations but also the difference between the two approaches. This paper is organized as follows. In Sect. 2, we briefly describe the formulas of Skyrme EDF that will be used in the calculations. In Sect. 3, the strengths of the tensor terms are optimized by empirical main peak energies of GT and SD transitions in 90 Zr and 208 Pb and the evolution of energy splitting between proton h 11/2 and g 7/2 single-particle orbits along the Z =50 isotopes is examined. Furthermore, the optimized adiabatic and variational EDFs are applied to study the binding energies of 22 isotopic chains by using Hartree-Fock (HF) and Hartrer-Fock-Bogolyubov (HFB) models in Sect. 4. The summary is stated in Sect. 5. 2 Formulism 1 The zero-range two-body tensor force was originally proposed by Skyrme [ 26 26 , 27 27 ]: <math display="block" id="M1"><mtable columnalign="left"><mtr><mtd><msup><mi>V</mi><mi>T</mi></msup><mo>=</mo><mfrac><mi>T</mi><mn>2</mn></mfrac><mo>{</mo><mo stretchy="false">[</mo><mo stretchy="false">(</mo><msub><mi>σ</mi><mstyle mathvariant="bold" mathsize="normal"><mn>1</mn></mstyle></msub><mo>⋅</mo><msup><mstyle mathvariant="bold" mathsize="normal"><mi>k</mi></mstyle><mo>'</mo></msup><mstyle mathvariant="bold" mathsize="normal"><mo stretchy="false">)</mo><mo stretchy="false">(</mo></mstyle><msub><mi>σ</mi><mstyle mathvariant="bold" mathsize="normal"><mn>2</mn></mstyle></msub><mo>⋅</mo><msup><mstyle mathvariant="bold" mathsize="normal"><mi>k</mi></mstyle><mo>'</mo></msup><mstyle mathvariant="bold" mathsize="normal"><mo stretchy="false">)</mo></mstyle><mo>−</mo><mfrac><mstyle mathvariant="bold" mathsize="normal"><mn>1</mn></mstyle><mstyle mathvariant="bold" mathsize="normal"><mn>3</mn></mstyle></mfrac><mrow><mo>(</mo><mrow><msub><mi>σ</mi><mstyle mathvariant="bold" mathsize="normal"><mn>1</mn></mstyle></msub><mo>⋅</mo><msub><mi>σ</mi><mstyle mathvariant="bold" mathsize="normal"><mn>2</mn></mstyle></msub></mrow><mo>)</mo></mrow><msup><mstyle mathvariant="bold" mathsize="normal"><mi>k</mi></mstyle><mrow><mo>'</mo><mstyle mathvariant="bold" mathsize="normal"><mn>2</mn></mstyle></mrow></msup><mstyle mathvariant="bold" mathsize="normal"><mo stretchy="false">]</mo></mstyle><mi>δ</mi><mstyle mathvariant="bold" mathsize="normal"><mo stretchy="false">(</mo><mi>r</mi><mo stretchy="false">)</mo></mstyle></mtd></mtr><mtr><mtd><mo>+</mo><mi>δ</mi><mo stretchy="false">(</mo><mi>r</mi><mo stretchy="false">)</mo><mo stretchy="false">[</mo><mo stretchy="false">(</mo><msub><mi>σ</mi><mn>1</mn></msub><mo>⋅</mo><mi>k</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msub><mi>σ</mi><mn>2</mn></msub><mo>⋅</mo><mi>k</mi><mo stretchy="false">)</mo><mo>−</mo><mfrac><mn>1</mn><mn>3</mn></mfrac><mrow><mo>(</mo><mrow><msub><mi>σ</mi><mn>1</mn></msub><mo>⋅</mo><msub><mi>σ</mi><mn>2</mn></msub></mrow><mo>)</mo></mrow><msup><mi>k</mi><mn>2</mn></msup><mo stretchy="false">]</mo><mo>}</mo></mtd></mtr><mtr><mtd><mo>+</mo><mfrac><mi>U</mi><mn>2</mn></mfrac><mo>{</mo><mrow><mo>(</mo><mrow><msub><mi>σ</mi><mn>1</mn></msub><mo>⋅</mo><msup><mi>k</mi><mo>'</mo></msup></mrow><mo>)</mo></mrow><mi>δ</mi><mo stretchy="false">(</mo><mi>r</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msub><mi>σ</mi><mn>2</mn></msub><mo>⋅</mo><mi>k</mi><mo stretchy="false">)</mo><mo>+</mo><mrow><mo>(</mo><mrow><msub><mi>σ</mi><mn>2</mn></msub><mo>⋅</mo><msup><mi>k</mi><mo>'</mo></msup></mrow><mo>)</mo></mrow><mi>δ</mi><mo stretchy="false">(</mo><mi>r</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msub><mi>σ</mi><mn>1</mn></msub><mo>⋅</mo><mi>k</mi><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mo>−</mo><mfrac><mn>2</mn><mn>3</mn></mfrac><mrow><mo>[</mo><mrow><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><msup><mi>k</mi><mo>'</mo></msup><mo>⋅</mo><mi>δ</mi><mo stretchy="false">(</mo><mi>r</mi><mo stretchy="false">)</mo><mi>k</mi></mrow><mo>]</mo></mrow><mo>}</mo><mn>,</mn></mtd></mtr></mtable></math> where the operator <math id="M2"><mrow><mi>k</mi><mo>=</mo><mo stretchy="false">(</mo><msub><mo>∇</mo><mn>1</mn></msub><mtext> </mtext><mo>−</mo><msub><mo>∇</mo><mn>2</mn></msub><mo stretchy="false">)</mo><mo>/</mo><mn>2</mn><mtext>i</mtext></mrow></math> acts on the right and <math id="M3"><mrow><msup><mtext>k</mtext><mo>'</mo></msup><mo>=</mo><mo>−</mo><mrow><mo>(</mo><mrow><msubsup><mo>∇</mo><mn>1</mn><mo>'</mo></msubsup><mo>−</mo><msubsup><mo>∇</mo><mn>2</mn><mo>'</mo></msubsup></mrow><mo>)</mo></mrow><mo>/</mo><mn>2</mn><mtext>i</mtext></mrow></math> acts on the left. The parameters T and U denote the strengths of time-even (TE) and time-odd (TO) tensor terms, respectively. For spherical symmetry, the tensor contributions to the Skyrme EDF are associated with the spin-orbit density J q , <math display="block" id="M4"><mtable columnalign="left"><mtr><mtd><msub><mi>J</mi><mi>q</mi></msub><mo>=</mo><mfrac><mn>1</mn><mrow><mn>4</mn><mi>π</mi><msup><mi>r</mi><mn>3</mn></msup></mrow></mfrac><mstyle displaystyle="true"><munder><mo>∑</mo><mrow><mi>i</mi><mo>∈</mo><mi>q</mi></mrow></munder></mstyle><mo stretchy="false">(</mo><mn>2</mn><msub><mi>j</mi><mi>i</mi></msub><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo><msubsup><mi>v</mi><mi>i</mi><mn>2</mn></msubsup></mtd></mtr><mtr><mtd><mo>×</mo><mrow><mo>[</mo><mrow><msub><mi>j</mi><mi>i</mi></msub><mo stretchy="false">(</mo><msub><mi>j</mi><mi>i</mi></msub><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo><mo>−</mo><msub><mi>l</mi><mi>i</mi></msub><mo stretchy="false">(</mo><msub><mi>l</mi><mi>i</mi></msub><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo><mo>−</mo><mn>3</mn><mo>/</mo><mn>4</mn></mrow><mo>]</mo></mrow><msubsup><mi>R</mi><mi>i</mi><mn>2</mn></msubsup><mo stretchy="false">(</mo><mi>r</mi><mo stretchy="false">)</mo><mn>,</mn></mtd></mtr></mtable></math> where q =0(1) denotes neutrons (protons), and <math id="M5"><mrow><msubsup><mi>v</mi><mi>i</mi><mn>2</mn></msubsup></mrow></math> is the occupation probability of state i , which takes value 1 or 0 for occupied or unoccupied states, respectively, in the closed shell nuclei. The spin-orbit density part of the EDF is given by [ 7 7 , 28 28 ]: <math display="block" id="M6"><mrow><mi>Δ</mi><mi>H</mi><mo>=</mo><mstyle displaystyle="true"><munder><mo>∑</mo><mrow><mi>t</mi><mo>=</mo><mn>0,1</mn></mrow></munder></mstyle><mfrac><mn>1</mn><mn>2</mn></mfrac><msubsup><mi>C</mi><mi>t</mi><mi>J</mi></msubsup><msubsup><mi>J</mi><mi>t</mi><mn>2</mn></msubsup><mn>,</mn></mrow></math> where the isoscalar and isovector spin-orbit densities are defined by J t =0 = J n + J p and J t =1 = J n - J p . The coefficients <math id="M7"><mrow><msubsup><mi>C</mi><mi>t</mi><mi>J</mi></msubsup></mrow></math> receive contributions from both the nonlocal central terms and the tensor terms, namely, <math display="block" id="M8"><mrow><msubsup><mi>C</mi><mi>t</mi><mi>J</mi></msubsup><mo>=</mo><msubsup><mi>A</mi><mi>t</mi><mi>J</mi></msubsup><mo>+</mo><msubsup><mi>B</mi><mi>t</mi><mi>J</mi></msubsup><mn>,</mn></mrow></math> with <math display="block" id="M9"><mrow><msubsup><mi>A</mi><mn>0</mn><mi>J</mi></msubsup><mo>=</mo><mfrac><mn>1</mn><mn>8</mn></mfrac><msub><mi>t</mi><mn>1</mn></msub><mo stretchy="false">(</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>−</mo><msub><mi>x</mi><mn>1</mn></msub><mo stretchy="false">)</mo><mo>−</mo><mfrac><mn>1</mn><mn>8</mn></mfrac><msub><mi>t</mi><mn>2</mn></msub><mo stretchy="false">(</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>+</mo><msub><mi>x</mi><mn>2</mn></msub><mo stretchy="false">)</mo><mn>,</mn></mrow></math> <math display="block" id="M10"><mrow><msubsup><mi>A</mi><mn>1</mn><mi>J</mi></msubsup><mo>=</mo><mfrac><mn>1</mn><mrow><mn>16</mn></mrow></mfrac><mo stretchy="false">(</mo><msub><mi>t</mi><mn>1</mn></msub><mo>−</mo><msub><mi>t</mi><mn>2</mn></msub><mo stretchy="false">)</mo><mn>,</mn></mrow></math> <math display="block" id="M11"><mrow><msubsup><mi>B</mi><mn>0</mn><mi>J</mi></msubsup><mo>=</mo><mfrac><mn>5</mn><mrow><mn>48</mn></mrow></mfrac><mo stretchy="false">(</mo><mi>T</mi><mo>+</mo><mn>3</mn><mi>U</mi><mo stretchy="false">)</mo><mn>,</mn></mrow></math> <math display="block" id="M12"><mrow><msubsup><mi>B</mi><mn>1</mn><mi>J</mi></msubsup><mo>=</mo><mfrac><mn>5</mn><mrow><mn>48</mn></mrow></mfrac><mo stretchy="false">(</mo><mi>U</mi><mo>−</mo><mi>T</mi><mo stretchy="false">)</mo><mn>.</mn></mrow></math> Equation (3) can be represented in the notation used in Ref. [ 29 29 ] as: <math display="block" id="M13"><mrow><mi>Δ</mi><mi>H</mi><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mi>α</mi><mo stretchy="false">(</mo><msubsup><mi>J</mi><mtext>n</mtext><mn>2</mn></msubsup><mo>+</mo><msubsup><mi>J</mi><mtext>p</mtext><mn>2</mn></msubsup><mo stretchy="false">)</mo><mo>+</mo><mi>β</mi><msub><mi>J</mi><mtext>n</mtext></msub><msub><mi>J</mi><mtext>p</mtext></msub></mrow></math> with <math display="block" id="M14"><mrow><mi>α</mi><mo>=</mo><msubsup><mi>C</mi><mn>0</mn><mi>J</mi></msubsup><mo>+</mo><msubsup><mi>C</mi><mn>1</mn><mi>J</mi></msubsup><mn>,</mn></mrow></math> <math display="block" id="M15"><mrow><mi>β</mi><mo>=</mo><msubsup><mi>C</mi><mn>0</mn><mi>J</mi></msubsup><mo>−</mo><msubsup><mi>C</mi><mn>1</mn><mi>J</mi></msubsup><mn>.</mn></mrow></math> J q is almost negligible in the spin-orbit saturated nuclei, i.e., in the nuclei with both spin-orbit partners filled. The contributions of the J 2 tensor terms to the HF spin-orbit potential tur n out to be <math display="block" id="M16"><mrow><mi>Δ</mi><msubsup><mi>U</mi><mrow><mtext>SO</mtext></mrow><mrow><mo stretchy="false">(</mo><mi>q</mi><mo stretchy="false">)</mo></mrow></msubsup><mo>=</mo><mi>α</mi><mfrac><mrow><msub><mi>J</mi><mi>q</mi></msub></mrow><mi>r</mi></mfrac><mo>+</mo><mi>β</mi><mfrac><mrow><msub><mi>J</mi><mrow><msup><mi>q</mi><mo>'</mo></msup></mrow></msub></mrow><mi>r</mi></mfrac><mn>,</mn></mrow></math> where q ’=1- q . The contributions of tensor forces to Eq. (12) are <math display="block" id="M17"><mrow><msub><mi>α</mi><mtext>T</mtext></msub><mo>=</mo><mfrac><mn>5</mn><mrow><mn>12</mn></mrow></mfrac><mi>U</mi><mn>,</mn></mrow></math> <math display="block" id="M18"><mrow><msub><mi>β</mi><mtext>T</mtext></msub><mo>=</mo><mfrac><mn>5</mn><mrow><mn>24</mn></mrow></mfrac><mo stretchy="false">(</mo><mi>T</mi><mo>+</mo><mi>U</mi><mo stretchy="false">)</mo><mn>.</mn></mrow></math> Employing the Skyrme EDF, the self-consistent charge-exchange proton-neutron Random Phase Approximation ( pn RPA) model is quite an effective method for studying the charge-exchange excitations, such as the GT and charge exchange SD states. The operators for GT transitions are defined as <math display="block" id="M19"><mrow><msub><mover accent="true"><mi>O</mi><mo>^</mo></mover><mrow><mtext>GT</mtext><mo>±</mo></mrow></msub><mo>=</mo><mstyle displaystyle="true"><munder><mo>∑</mo><mrow><mi>i</mi><mi>m</mi></mrow></munder></mstyle><msubsup><mi>t</mi><mo>±</mo><mi>i</mi></msubsup><msubsup><mi>σ</mi><mi>m</mi><mi>i</mi></msubsup></mrow></math> and the spin-dependent charge-exchange SD operators are given by <math display="block" id="M20"><mtable columnalign="left"><mtr><mtd><msubsup><mover accent="true"><mi>O</mi><mo>^</mo></mover><mrow><mtext>SD</mtext><mo>±</mo></mrow><mi>J</mi></msubsup><mo>=</mo><mstyle displaystyle="true"><munder><mo>∑</mo><mrow><mi>i</mi><mi>M</mi></mrow></munder></mstyle><msubsup><mi>t</mi><mo>±</mo><mi>i</mi></msubsup><msub><mi>r</mi><mi>i</mi></msub><msup><mrow><mo stretchy="false">(</mo><msubsup><mi>Y</mi><mrow><mi>l</mi><mo>=</mo><mn>1</mn></mrow><mi>i</mi></msubsup><mo stretchy="false">(</mo><mover accent="true"><mi>r</mi><mo>^</mo></mover><mo stretchy="false">)</mo><mo>×</mo><msup><mi>σ</mi><mi>i</mi></msup><mo stretchy="false">)</mo></mrow><mrow><mi>J</mi><mi>M</mi></mrow></msup><mn>,</mn></mtd></mtr><mtr><mtd><mtext> </mtext><mtext>with </mtext><msup><mi>J</mi><mi>π</mi></msup><mo>=</mo><msup><mn>0</mn><mo>−</mo></msup><msup><mn>,1</mn><mo>−</mo></msup><msup><mn>,2</mn><mo>−</mo></msup></mtd></mtr></mtable></math> in terms of the standard isospin operators, t ± =( t x ± it y ). In the EDF+ pn RPA calculations, the HF equation is solved first in coordinate space with a large reference radius extending up to 20 fm with a mesh step of 0.1 fm. In HF calculations, the single-particle wave functions of the occupied and unoccupied states are obtained by using a harmonic oscillator basis in which the maximum major quantum number N max =10,12 for 90 Zr and 208 Pb, respectively. In the charge exchange pn RPA calculations, all the residual interactions including the central, tensor, and two-body spin-orbit interaction, are included to make the calculation fully self-consistent. 3 Constraint for the tensor force 1 Here, we try to optimize the tensor interaction or choose the existing parameter sets fitted with tensor interaction by using several empirical results. We choose the empirical results from both the shell structure and collective transition of nuclei. For the ground states, we adopt the data of the evolution of energy splitting between 1 h 11/2 and 1 g 7/2 proton single-particle states along Z =50 isotopes, [ 8 8 ] which is widely used to constrain the tensor interaction [ 13 13 , 14 14 ]. From the collective excitations, it was reported that the GT and SD transitions can also provide a strong constraint for the tensor interaction [ 21 21 - 24 24 21-24 ]. 3.1 Constraint the tensor force by the adiabatic approach 2 In this study, we added the tensor forces to the existing Skyrme interactions, SGII and SkO′, which give a better description of the charge-exchange transitions of nuclei. The density functional SGII presented in Ref. [ 30 30 ] was adjusted to provide a reasonable value for the spin-isospin channel of Landau-Migdal parameter <math id="M21"><mrow><msubsup><mi>g</mi><mn>0</mn><mo>'</mo></msubsup><mo>=</mo><mn>0.503</mn></mrow></math> with the effective mass 0.79. Bender et al. [ 31 31 , 32 32 ] examined several Skyrme parameter sets and found that the SkO′ is quite promising for studying the GT resonance and β -decay half-lives. The SkO′ interaction gives the Landau-Migdal parameter <math id="M22"><mrow><msubsup><mi>g</mi><mn>0</mn><mo>'</mo></msubsup><mo>=</mo><mn>0.79</mn></mrow></math> and the effective mass 0.896. We calculate the GT and charge-exchange SD excitations in 90 Zr and 208 Pb by employing SkO′ interaction plus tensor forces in Eq. (1). In the optimizing process, we calculate the centroid energy, which is defined as the ratio between energy-weighted sum rule strength m (1) and non-energy-weighted sum rule strength m (0), i.e., m (1)/ m (0) [ 24 24 ]. These centroid energies labeled as E th are then compared with the experimental result E exp . In this study, the excitation energies are calculated with respect to the ground states of the mother nuclei. The difference <math display="block" id="M23"><mrow><mi>Δ</mi><mi>E</mi><mo>=</mo><mo>|</mo><msub><mi>E</mi><mrow><mtext>th</mtext></mrow></msub><mo>−</mo><msub><mi>E</mi><mrow><mtext>exp</mtext></mrow></msub><mo>|</mo></mrow></math> is treated as a criterion applied to optimize the tensor parameter sets. As shown in Fig. 1 Fig. 1 , the GT and SD transitions are calculated by adopting tensor terms with a sufficiently wide range; T =(-400→800) MeVfm 5 and U =(-800→250) MeVfm 5 . Within this range, the tensor parameters T and U are constrained by requiring Δ E 2.0 MeV. The allowed range of T varies from approximately 200–700 MeVfm 5 , which is mainly constrained by the SD 1 - centroid energy in 208 Pb. Furthermore, the allowed range of the time-odd tensor strength U is negative varying from approximately -20–-800 MeVfm 5 . The value -800 MeVfm 5 is observed because the RPA equation collapses with U value beyond this border. As the tensor force shifts the GT main peak downward, the line with red-squares labelled by "GT Pb higher" implies that when the strengths T and U go beyond this line, the GT main peak energy in 208 Pb becomes lower than the E exp -2.0 = 17.2 MeV. Alternately, the line with black-triangle labelled by "GT Zr lower’’ indicates that the strengths of T and U are sufficiently strong to shift the GT main peak energy upward to E exp +2.0 = 17.6 MeV at this border and will exceed the criterium when T and U go beyond this line. The wide allowed range of ( T , U ) is because the adopted interaction SkO′ has a properly large <math id="M24"><mrow><msubsup><mi>g</mi><mn>0</mn><mo>'</mo></msubsup></mrow></math> value, which makes reproducing the main peak energies of GT and SD more efficient. The optimal values of tensor terms on top of SGII were studied in Ref. [ 24 24 ] in terms of GT and SD excitation energies and are referred to henceforth. In addition to the main peak energies of GT and SD states, we also study the evolution of spin-orbit splitting of Z =50 isotopes, which is highlighted in the certain tensor force studies [ 9 9 , 13 13 , 14 14 ]. In addition to the central and two-body spin-orbit terms of Skyrme interaction, the tensor terms contribute to the spin-orbit splitting which modifies the SPE remarkably [ 29 29 ]. Thus, the energy splitting along an isotopic or isotonic chain may produce strong constraint for the strengths of tensor terms. Hence, the proton SPE splitting between 1 h 11/2 and 1 g 7/2 of Sn isotopes with N varying from 56–82 has been calculated to further constrain the strength of tensor terms. In our calculation, the Hartree-Fock Bogolyubov (HFB) model described in Ref. [ 33 33 , 34 34 ] was applied, solving the Skyrme HFB equations in coordinate space under the spherical symmetry. The zero-range surface pairing interaction is used: <math display="block" id="M25"><mrow><mi>V</mi><mo>=</mo><msub><mi>V</mi><mn>0</mn></msub><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><mfrac><mrow><mi>ρ</mi><mo stretchy="false">(</mo><mfrac><mrow><msub><mi>r</mi><mn>1</mn></msub><mo>+</mo><msub><mi>r</mi><mn>2</mn></msub></mrow><mn>2</mn></mfrac><mo stretchy="false">)</mo></mrow><mrow><msub><mi>ρ</mi><mn>0</mn></msub></mrow></mfrac><mo stretchy="false">)</mo><mi>δ</mi><mo stretchy="false">(</mo><msub><mi>r</mi><mn>1</mn></msub><mo>−</mo><msub><mi>r</mi><mn>2</mn></msub><mo stretchy="false">)</mo><mn>,</mn></mrow></math> where the density parameter ρ 0 is set to be 0.16 fm -3 and the strength parameter V 0 is determined to reproduce the empirical value of neutron pairing gap calculated by the three-point index Δ(3)=( B ( N , Z )-2 B ( N +1, Z )+ B ( N +2, Z ))/2. The evolution of proton SPE splitting between 1 h 11/2 and 1 g 7/2 states is calculated in HFB model adopting the Skyrme EDFs with and without tensor interactions and shown in Fig. 2 Fig. 2 . In the calculations, SGII and SkO′ are adopted. The strengths of tensor interactions are set to be ( T , U ) = (500, -320) and (500, -480), respectively, for SkO′ and SGII. It is observed that the presently optimized density functionals with tensor terms SkO′+(500, -320) and SGII+ (500, -480) reproduce the empirical data. For a quantitative study of optimizations of tensor interactions, we examine the following criterion for the energy difference δ = ε 1 h 11/2 - ε 1 g 7/2 , i.e., the RMSD between calculated and empirical energy differences; <math display="block" id="M26"><mrow><mtext>RMSD</mtext><mo stretchy="false">(</mo><mo>|</mo><msub><mi>δ</mi><mrow><mtext>th</mtext></mrow></msub><mo>−</mo><msub><mi>δ</mi><mrow><mtext>exp</mtext></mrow></msub><mo>|</mo><mo stretchy="false">)</mo><mo>=</mo><msqrt><mrow><mfrac><mn>1</mn><mrow><msub><mi>N</mi><mtext>D</mtext></msub></mrow></mfrac><mstyle displaystyle="true"><munder><mo>∑</mo><mi>N</mi></munder></mstyle><mo>|</mo><msub><mi>δ</mi><mrow><mtext>th</mtext></mrow></msub><mo stretchy="false">(</mo><mi>N</mi><mo stretchy="false">)</mo><mo>−</mo><msub><mi>δ</mi><mrow><mtext>exp</mtext></mrow></msub><mo stretchy="false">(</mo><mi>N</mi><msup><mrow><mo stretchy="false">)</mo><mo>|</mo></mrow><mn>2</mn></msup></mrow></msqrt></mrow></math> where N D is the number of data points. For the calculated results in Fig. 2 Fig. 2 , RMSD values are 1.67, 0.65, 3.24, and 0.47 for SkO’, SkO’+tensor, SGII, and SGII+tensor, respectively. To constrain T and U values, we set RMSD to be less than 1 MeV. The optimized parameter region for T and U is shown in Fig. 3 Fig. 3 for SkO’ and Fig. 4 Fig. 4 for SGII, respectively. As observed in the figures, the isotope dependence of SPE splittings provides a strict constraint for the strength of time-odd tensor term U . The range of U is further constrained to be from -150 to -450 MeVfm 5 and from -350 to -550 MeVfm 5 for SkO’ and SGII, respectively. The tensor interactions within this range produce attractive effects on the binding energy of ground state, which can be qualitatively understood from Eqs. (9), (13), and (14). 3.2 The variational EDFs with the tensor terms 2 It was reported in Ref.[ 24 24 ] that some variational EDFs T21, T32, T43, and T54 in Tij family can well reproduce the GT and SD empirical data in 90 Zr and 208 Pb. The evolution of proton SPE splitting between 1 h 11/2 and 1 g 7/2 states is also calculated in HFB model adopting the four parameter sets from Tij family and is observed in Fig. 5 Fig. 5 . For the calculated results in Fig. 5 Fig. 5 , RMSD values for T21 is 1.08 MeV, while they are 1.52, 1.81, and 2.17 MeV for T32, T43, and T54, respectively. While the agreement of Tij family is not as efficient as those of the adiabatic EDFs, SkO’+tensor, and SGII+tensor, it should be noticed that the best T21 among Tij family has similar tensor terms as the two adiabatic EDFs as listed in Table 1 Table 1 . According to Eq. (3), the three sets of EDFs with tensor parameters, SkO’+, SGII+, and T21 will produce an attractive effect for the binding energy as the α T values are negative and larger than the corresponding positive β T values. The tensor contributions α T and β T (MeVfm 5 ) to the spin-orbit potential (12) are given by Eqs. (13) and (14) 1 1 1 1 SkO’+ 1 1 SGII+ 1 1 T21 1 1 T32 1 1 T43 1 1 T54 1 1 T (MeVfm 5 ) 1 1 500.00 1 1 500.00 1 1 476.95 1 1 613.06 1 1 590.60 1 1 727.35 1 1 U (MeVfm 5 ) 1 1 -320.00 1 1 -480.00 1 1 -369.36 1 1 -231.53 1 1 -147.48 1 1 -8.36 1 1 α T (MeVfm 5 ) 1 1 -133.33 1 1 -162.5 1 1 -153.90 1 1 -96.47 1 1 -61.45 1 1 -3.48 1 1 β T (MeVfm 5 ) 1 1 37.5 1 1 4.17 1 1 22.42 1 1 79.48 1 1 93.32 1 1 147.79 4 The optimized tensor terms on the binding energies 1 Nuclear binding energy, i.e., the nuclear mass is important for the study of nuclear reactions such as the r -process synthesis of nuclei [ 35 35 ] as well as the β -delayed multi-nucleon emission. Furthermore, it is of great importance for the studies of the superheavy nuclei and the drip line nuclei [ 36 36 , 37 37 ]. In recent years, many efforts have been devoted to accurately measuring the nuclear mass of unstable nuclei [ 38 38 - 42 42 38-42 ]. Theoretically, there are some successful macroscopic and microscopic models of nuclear mass [ 43 43 - 53 53 43-53 ], which can provide accurate mass tables, but the tensor interaction was not included in these models. A Skyrme EDF UNEDF2 [ 54 54 ], which includes the tensor terms and optimizes both the ground state properties and shell structure, provides reasonable binding energies of heavy nuclei, in which the tensor interaction also produces an attractive effect. In this section, we apply the optimized Skyrme EDFs SKO’ and T21 without and with tensor terms for the binding energies of 22 isotopic chains from Ca to Ra. Although some nuclei in the study might be deformed, we adopt the HFB model with spherical symmetry. Thus, the nuclei studied are selected to be spherical or at most weakly deformed with the deformation parameter β 2 < 0.3. The binding energies of the Sn isotopes are calculated with SkO′, SGII and T21 with and without tensor terms listed in Table 1. The differences of the binding energies between the theoretical and experimental values are illustrated in Fig. 6 Fig. 6 . As was shown in Fig. 6 Fig. 6 , both the SkO′ and T21 without tensor terms give less bound of the order of approximately 5–10 MeV than the experimental data. SGII gives the overbindings for all Sn isotopes and SGII+(500, -480) does not contribute to this feature of binding energies. With the tensor interaction included in the calculations, both SkO′ +tensor and T21 can reproduce the experimental data within a few MeV difference for all the Sn isotopes. We apply further SkO′ and T21 with and without tensor interaction for the binding energies of 22 isotopic chains of spherical and weakly deformed medium heavy and heavy nuclei. The difference between the calculated binding energies and the corresponding experimental values [ 55 55 ] are shown in Figs. 7 Figs. 7 and 8 8 for SKO′ and T21, respectively. Inclusion of the tensor interaction in both SKO′ and T21 increases the total binding energies systematically and improves the results. The RMSD for the calculated binding energies with respect to the experimental values is calculated as follows <math display="block" id="M27"><mrow><mtext>RMSD</mtext><mo stretchy="false">(</mo><mi>B</mi><mo stretchy="false">)</mo><mo>=</mo><msqrt><mrow><mfrac><mn>1</mn><mrow><msub><mi>N</mi><mrow><mtext>tot</mtext></mrow></msub></mrow></mfrac><mstyle displaystyle="true"><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mrow><msub><mi>N</mi><mrow><mtext>tot</mtext></mrow></msub></mrow></munderover></mstyle><msup><mrow><mo stretchy="false">(</mo><mi>B</mi><msub><mrow><mo stretchy="false">(</mo><msub><mi>N</mi><mi>i</mi></msub><mn>,</mn><msub><mi>Z</mi><mi>i</mi></msub><mo stretchy="false">)</mo></mrow><mrow><mtext>th</mtext><mtext>.</mtext></mrow></msub><mo>−</mo><mi>B</mi><msub><mrow><mo stretchy="false">(</mo><msub><mi>N</mi><mi>i</mi></msub><mn>,</mn><msub><mi>Z</mi><mi>i</mi></msub><mo stretchy="false">)</mo></mrow><mrow><mtext>exp</mtext><mtext>.</mtext></mrow></msub><mo stretchy="false">)</mo></mrow><mn>2</mn></msup></mrow></msqrt><mn>,</mn></mrow></math> where N tot is the number of nuclei. For SKO′, the RMSD( B ) for all of the calculated nuclei shown in Fig. 7 Fig. 7 are 8.97 MeV and 3.99 MeV for SkO′ without and with tensor terms, respectively. For T21, as shown in Fig. 8 Fig. 8 , the RMSD( B ) are approximately 13 MeV and 6.2 MeV without and with tensor interaction. The two EDFs with tensor terms give good global agreements of the binding energies and although the two EDFs are optimized by different strategic approaches, the tensor contributions to the binding energies are quite similar, i.e., approximately 5 MeV for SKO’+Tensor and 6.8 MeV for T21. 5 Summary 1 We adopted the GT and charge-exchange SD centroid energies and the evolution of proton SPE difference between 1 h 11/2 and 1 g 7/2 orbits of Sn isotopes as the constraints to choose appropriate parameter sets of tensor terms. We first implemented the adiabatic approach and optimized the tensor terms T and U on top of existing parameter sets SkO’ and SGII. The time-even tensor parameter T is well controlled by the energies of GT and SD giant resonances. Further, a strong constraint on the time-odd tensor term U comes mainly from the SPE splitting between proton h 11/2 and g 7/2 orbits along the Z =50 isotopes. For the adiabatic EDFs, the constraints provide a small region of the ( T , U ) parameters both for SGII and SkO’; T =(400-600)MeVfm 5 and U =(-350 to -550)MeVfm 5 for SGII and T =(250 to 600)MeVfm 5 and U =(-150 to -450)MeVfm 5 for SkO’. The optimized adiabatic parameter sets suggest T21 EDF as the best candidate for nuclear structure calculations among Tij family, as T21 has very similar tensor terms to the adiabatic EDFs, SkO’+tensor and SGII+tensor. Because of these constraints, the tensor interaction provides an attractive effect for the binding energies of nuclei, which is consistent with the primary purpose of introducing tensor interaction to make deuteron and light nuclei bound [ 57 57 , 58 58 ]. Qualitatively, the Skyrme interaction with tensor terms, α T < 0 and β T > 0, acts similar to the realistic interactions in the proton-neutron channel and the isospin dependence of spin-orbit coupling [ 12 12 , 20 20 , 59 59 ]. The EDFs SkO’ and T21 with and without the tensor terms are further applied for the study of binding energies of 22 isotopic chains from Ca to Ra. Though the SkO’+tensor and T21 EDFs are obtained in different methodologies, i.e., the central terms are optimized without and with tensor terms, the RMSD deviation of binding energies are approximately 3.99 MeV and 6.2 MeV, respectively. Furthermore, the similar average contributions of the tensor terms are evaluated for the binding energies to be approximately 5 and 6.8 MeV for SKO′ and T21, respectively, which originated from a negative strength of time-odd tensor term U . 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(Science Press), Shanghai Institute of Applied Physics, the Chinese Academy of Sciences, ChineseNuclear Society and Springer Nature Singapore Pte Ltd. 2020 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/s41365-020-0727-7 http://dx.doi.org/10.1007/s41365-020-0727-7 . 2020 Nuclear Science and Techniques 2 31