Skyrme Energy Density Functionals Research Articles

Nuclear Matter Equation of State in the Brueckner–Hartree–Fock Approach and Standard Skyrme Energy Density Functionals

The equation of state of asymmetric nuclear matter as well as the neutron and proton effective masses and their partial-wave and spin–isospin decomposition are analyzed within the Brueckner–Hartree–Fock approach. Theoretical uncertainties for all these quantities are estimated by using several phase-shift-equivalent nucleon–nucleon forces together with two types of three-nucleon forces, phenomenological and microscopic. It is shown that the choice of the three-nucleon force plays an important role above saturation density, leading to different density dependencies of the energy per particle. These results are compared to the standard form of the Skyrme energy density functional, and we find that it is not possible to reproduce the BHF predictions in the (S,T) channels in symmetric and neutron matter above saturation density, already at the level of the two-body interaction, and even more including the three-body interaction.

Impact of choices for center-of-mass correction energy on the surface energy of Skyrme energy density functionals

Impact of choices for center-of-mass correction energy on the surface energy of Skyrme energy density functionals

Isoscalar giant resonances of <inline-formula><tex-math id="M1">\begin{document}$^{{\bf{18}}}_{{\boldsymbol{\Lambda\Lambda}}}{\bf{O}}$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="4-20231531_M1.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="4-20231531_M1.png"/></alternatives></inline-formula> in relativistic approach

The interactions between hyperon-nucleon and hyperon-hyperon have been an important topic in strangeness nuclear physics, which play an important role in understanding the properties of hypernuclei and equation of state of strangeness nuclear matter. It is very difficult to perform a direct scattering experiment of the nucleon and hyperon because the short lifetime of the hyperon. Therefore, the hyperon-nucleon interaction and the hyperon-hyperon interaction have been mainly investigated experimentally by <inline-formula><tex-math id="M4">\begin{document}$\gamma$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="4-20231531_M4.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="4-20231531_M4.png"/></alternatives></inline-formula> spectroscopy of single-<inline-formula><tex-math id="M5">\begin{document}$\Lambda$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="4-20231531_M5.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="4-20231531_M5.png"/></alternatives></inline-formula> hypernuclei or double-<inline-formula><tex-math id="M6">\begin{document}$\Lambda$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="4-20231531_M6.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="4-20231531_M6.png"/></alternatives></inline-formula> hypernuclei. There are also many theoretical methods developed to describe the properties of hypernuclei. Most of these models focus mostly on the ground state properties of hypernuclei, and have given exciting results in producing the banding energy, the energy of single-particle levels, deformations, and other properties of hypernuclei. Only a few researches adopting Skyrme energy density functionals is devoted to the study of the collective excitation properties of hypernuclei. In present work, we have extended the relativistic mean field and relativistic random phase approximation theories to study the collective excitation properties of hypernuclei, and use the methods to study the isoscalar collective excited state properties of double <inline-formula><tex-math id="M7">\begin{document}$\Lambda$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="4-20231531_M7.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="4-20231531_M7.png"/></alternatives></inline-formula> hypernuclei. First, the effect of <inline-formula><tex-math id="M8">\begin{document}$\Lambda$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="4-20231531_M8.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="4-20231531_M8.png"/></alternatives></inline-formula> hyperons on the single-particle energy of <sup>16</sup>O and <inline-formula><tex-math id="M9">\begin{document}$^{18}_{\Lambda\Lambda}{\rm{O}}$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="4-20231531_M9.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="4-20231531_M9.png"/></alternatives></inline-formula> are discussed in the relativistic mean field theory, the calculations are performed within TM1 parameter set and related hyperon-nucleon interaction, and hyperon-hyperon interaction. We find that it gives a larger attractive effect on the <inline-formula><tex-math id="M10">\begin{document}${{\mathrm{s}}}_{1/2}$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="4-20231531_M10.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="4-20231531_M10.png"/></alternatives></inline-formula> state of proton and neutron, while gives a weaker attractive effect on the state around Fermi surface. The self-consistent relativistic random phase approximation is used to study the collectively excited state properties of hypernucleus <inline-formula><tex-math id="M11">\begin{document}$^{18}_{\Lambda\Lambda}{\rm{O}}$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="4-20231531_M11.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="4-20231531_M11.png"/></alternatives></inline-formula>. The isoscalar giant monopole resonance and quadrupole resonance are calculated and analysed in detail, we pay more attention to the effect of the inclusion of <inline-formula><tex-math id="M12">\begin{document}$\Lambda$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="4-20231531_M12.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="4-20231531_M12.png"/></alternatives></inline-formula> hyperons on the properties of giant resonances. Comparing with the strength distributions of <sup>16</sup>O, changes of response function of <inline-formula><tex-math id="M13">\begin{document}$^{18}_{\Lambda\Lambda}{\rm{O}}$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="4-20231531_M13.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="4-20231531_M13.png"/></alternatives></inline-formula> are evidently found both on the isoscalar giant monopole resonance and quadrupole resonance. It is shown that the difference comes mainly from the change of Hartree energy of particle-hole configuration and the contribution of the excitations of <inline-formula><tex-math id="M14">\begin{document}$\Lambda$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="4-20231531_M14.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="4-20231531_M14.png"/></alternatives></inline-formula> hyperons. We find that the hyperon-hyperon residual interactions have small effect on the monopole resonance function and quadrupole response function in the low-energy region, and have almost no effect on the response functions in the high-energy region.

Enhanced moments of inertia for rotation in neutron-rich nuclei

Ground-state moments of inertia (MoI) are investigated for approximately 1700 even–even nuclei from the proton drip line to the neutron drip up to Z=120 and N=184. The cranked Hartree–Fock–Bogoliubov equation is solved in the coordinate space using the Skyrme and pairing energy density functionals. This model well describes the available experimental data of the first excited Iπ=2+ state energy of more than 300 nuclides possessing appreciable deformation. The MoI is found to greatly increase near the drip line, whereas the deformation is found to be not as strong as estimated by the empirical relation when the pairing functional including the isovector-density dependence is adopted. The currently used pairing functionals produce large uncertainties in the prediction of E(21+) values near the neutron drip line.

Finite-temperature calculation of the isovector giant dipole resonance in Sn120 using self-consistent inputs

Background: Although microscopic models are frequently used to investigate giant resonances built on the ground state of a nucleus, similar calculations at nonzero temperature are limited. Approaches based on the classical thermal shape-fluctuation model (TSFM) are often utilized to analyze the properties of giant dipole resonance at finite temperature. However, TSFM fails to predict the correct trend of resonance width over a wide range of temperature starting from zero.Purpose: To study the isovector giant dipole resonance (IVGDR) at finite temperature, we present an improved version of TSFM, where driving potential and IVGDR centroid energies are obtained self-consistently using the nuclear energy density functional formalism. The temperature dependence of IVGDR width in $^{120}\mathrm{Sn}$ is calculated to benchmark the proposed method.Method: Nuclear free energy surfaces and entropy surfaces are simulated at different temperatures for two different parametrizations of Skyrme energy density functionals. Moreover, IVGDR centroid energies are calculated by incorporating nuclear deformations extracted from the corresponding self-consistent densities. Subsequently, IVGDR widths are obtained within TSFM. Also, we demonstrate the role of pairing fluctuation in improving the low temperature behavior of IVGDR width.Result: We found good overall agreement of our results with the measured IVGDR widths. Specifically, except at a very low temperature, unprecedented accuracy has been achieved in TSFM. Calculated IVGDR widths are shown to be robust against the choices of driving potential and Skyrme parametrization.Conclusion: The present paper provides guidance on using TSFM based calculations in predicting the IVGDR width over a broad range of temperature. For a more consistent calculation, pairing fluctuations are required to be included as additional degrees of freedom.

Skyrme pseudopotentials at next-to-next-to-leading order: Construction of local densities and first symmetry-breaking calculations

There is an ongoing quest to improve on the spectroscopic quality of nuclear energy density functionals (EDFs) of the Skyrme type through extensions of its traditional form. One direction for such activities is the inclusion of terms of higher order in gradients in the EDF. We report on exploratory symmetry-breaking calculations performed for an extension of the Skyrme EDF that includes central terms with four gradients at next-to-next-to-leading order (N2LO) and for which the high-quality parametrization SN2LO1 has been constructed recently [P. Becker et al, Phys. Rev. C 96, 044330 (2017)]. Up to now, the investigation of such functionals with higher-order terms was limited to infinite matter and spherically symmetric configurations of singly- and doubly-magic nuclei. We address here nuclei and phenomena that require us to consider axial and non-axial deformation, both for reflection-symmetric and also reflection-asymmetric shapes, as well as the breaking of time-reversal invariance. Achieving these calculations demanded a number of formal developments. These all resulted from the formulation of the N2LO EDF requiring the introduction of new local densities with additional gradients that are not present in the EDF at NLO. Their choice is not unique, but can differ in the way the gradients are coupled. While designing a numerical implementation of N2LO EDFs in Cartesian 3d coordinate-space representation, we have developed a novel definition and a new unifying notation for normal and pair densities that contain gradients at arbitrary order. The resulting scheme resolves several issues with some of the choices that have been made for local densities in the past, in particular when breaking time-reversal symmetry. Guided by general practical considerations, we propose an alternative form of the N2LO contribution to the Skyrme EDF that is built from a different set of densities.

Microscopic origin of reflection-asymmetric nuclear shapes

Background: The presence of nuclear ground states with stable reflection-asymmetric shapes is supported by rich experimental evidence. Theoretical surveys of odd-multipolarity deformations predict the existence of pear-shaped isotopes in several fairly localized regions of the nuclear landscape in the vicinity of near-lying single-particle shells with $\Delta\ell=\Delta j=3$. Purpose: We analyze the role of isoscalar, isovector, neutron-proton, neutron-neutron, and proton-proton multipole interaction energies in inducing the onset of reflection-asymmetric ground-state deformations. Methods: The calculations are performed in the framework of axial reflection-asymmetric Hartree-Fock-Bogoliubov theory using two Skyrme energy density functionals and density-dependent pairing force. Results: We show that reflection-asymmetric ground-state shapes of atomic nuclei are driven by the odd-multipolarity neutron-proton (or isoscalar) part of the nuclear interaction energy. This result is consistent with the particle-vibration picture, in which the main driver of octupole instability is the isoscalar octupole-octupole interaction giving rise to large $E3$ polarizability.

Isospin Symmetry Breaking Effects on the Mass-Radius Relation of a Neutron Star

Isospin symmetry breaking effects on the mass-radius relation of a cold, non-accreting neutron star are studied on the basis of two Skyrme Energy Density Functionals (EDFs). One functional contains isospin symmetry breaking terms other than those typically included in Skyrme EDFs while its counterpart is of standard form. Both functionals are based on the same fitting protocol except for the observables and pseudo-observables sensitive to the isospin symmetry breaking channel. The quality of those functionals is similar in the description of terrestrial observables but choosing either of them has a non-negligible effect on the mass-radius relation and tidal deformability of a neutron star. Further investigations are needed to clarify the effects of isospin symmetry breaking on these and other observables of neutron stars that are, and will become, available.

Finite-amplitude method for collective inertia in spontaneous fission

Background: Microscopic description of spontaneous fission is one of the most challenging subjects in nuclear physics. It is necessary to evaluate the collective potential and the collective inertia along a fission path for a description of quantum tunneling in spontaneous or low-energy fission. In past studies of the fission dynamics based on nuclear energy density functional (EDF) theory, the collective inertia has been evaluated with the cranking approximation, which neglects dynamical residual effects. Purpose: The purpose is to provide a reliable and efficient method to include dynamical residual effects in the collective inertia for fission dynamics. Methods: We use the local quasiparticle random-phase approximation (LQRPA) to evaluate the collective inertia along a fission path obtained by the constrained Hartree-Fock-Bogoliubov method with the Skyrme EDF. The finite-amplitude method (FAM) with a contour integration technique enables us to efficiently compute the collective inertia in a large model space. Results: We evaluate the FAM-QRPA collective inertia along a symmetric fission path in $^{240}$Pu and $^{256}$Fm. The FAM-QRPA inertia is significantly larger than the one of the cranking approximation, and shows pronounced peaks around the ground state and the fission isomer. This is due to dynamical residual effects. Conclusions: To describe the spontaneous or low-energy fission, we provide a reliable and efficient method to construct the collective inertia with dynamical residual effects that have been neglected in most of EDF-based works in the past. We show the importance of dynamical residual effects to the collective inertia. This work will be a starting point for a systematic study of fission dynamics in heavy and superheavy nuclei to microscopically describe the nuclear large-amplitude collective motions.

M1 resonance in Pb208 within the self-consistent phonon-coupling model

The main goal of the paper is to investigate theoretically the experimentally observed fragmentation of the isovector $M1$ resonance in $^{208}$Pb within a self-consistent model based on an energy-density functional (EDF) of the Skyrme type. This fragmentation (spread of the $M1$ strength) is not reproduced in a conventional one-particle--one-hole ($1p1h$) random-phase approximation (RPA) and thus has to be investigated in the framework of more complicated models. However, previously applied models of this type were not self-consistent. In the present work, we use a recently developed renormalized version of the self-consistent time blocking approximation (RenTBA) in which the $1p1h\otimes$phonon configurations are included on top of the RPA $1p1h$ configurations. We have determined several sets of the parameters of the modified Skyrme EDF fitted within the RenTBA and RPA and have found the necessary condition of producing the fragmentation of the $M1$ resonance in $^{208}$Pb in our model. We present also the results of the RenTBA and RPA calculations for the first excited states of the natural parity modes in $^{208}$Pb obtained with these modified parametrizations.

Trajectory in 2D plot of isoscalar and isovector densities of Ca48 and Pb208 , and symmetry energy

We propose a new two-dimensional (2D) plotting method of isoscalar-isovector (IS-IV) densities of $^{48}\mathrm{Ca}$ and $^{208}\mathrm{Pb}$ to extract not only empirical symmetry energy coefficients $J, L$, and ${K}_{\mathrm{sym}}$ but also the asymmetric isospin term ${K}_{\ensuremath{\tau}}$ in the nuclear incompressibility. We adopted several modern relativistic and nonrelativistic energy density functional (EDF) for the theoretical calculations of IS and IV densities to compare with experimental neutron and proton density distributions determined by recent proton elastic scatterings. We found very strong correlations between the trajectory of the 2D density at the density $\ensuremath{\rho}=0.1\phantom{\rule{4pt}{0ex}}{\mathrm{fm}}^{\ensuremath{-}3}$ and the symmetry energy coefficients $J, L$, and also ${K}_{\ensuremath{\tau}}$. For ${K}_{\mathrm{sym}}$, the Skyrme EDF SAMi-J family shows a strong correlation in the trajectory of 2D density, while the RMF Lagrangian DDME-J family shows a peculiar quadratic curve and does not provide the experimental value. The symmetry energy term ${K}_{\ensuremath{\tau}}$ of nuclear incompressibility is also discussed in term of the slope of 2D trajectory.

Generalized Skyrme random-phase approximation for nuclear resonances: Different trends for electric and magnetic modes

We discuss major differences between electric and magnetic excitations in nuclei appearing in self-consistent calculations based on Skyrme energy-density functionals (EDFs). For this we calculate collective low- and high-lying electric and magnetic excitations in $^{208}\mathrm{Pb}$ within a self-consistent Skyrme EDF approach using the random-phase approximation (RPA) and a more sophisticated particle-hole plus phonon-coupling model, coined the time-blocking approximation (TBA). Tools of analysis are Landau-Migdal parameters for bulk properties and the RPA and TBA results for finite nuclei. We show that the interplay between the effective mass and the effective particle-hole interaction, well known in the Landau-Migdal theory, renders the final results rather independent of the effective mass by virtue of the ``backflow effect.'' It explains the success of self-consistent calculations of electric transitions in such approaches. This effect, however, is absent in the magnetic case and leads to higher fluctuations in the results. It calls for further developments of the Skyrme functional in the spin channel.

Gamow-Teller Strength in ^{48}Ca and ^{78}Ni with the Charge-Exchange Subtracted Second Random-Phase Approximation.

We develop a fully self-consistent subtracted second random-phase approximation for charge-exchange processes with Skyrme energy-density functionals. As a first application, we study Gamow-Teller excitations in the doubly magic nucleus ^{48}Ca, the lightest double-β emitter that could be used in an experiment, and in ^{78}Ni, the single-beta-decay rate of which is known. The amount of Gamow-Teller strength below 20 or 30MeV is considerably smaller than in other energy-density-functional calculations and agrees better with experiment in ^{48}Ca, as does the beta-decay rate in ^{78}Ni. These important results, obtained without adhoc quenching factors, are due to the presence of two-particle-two-hole configurations. Their density progressively increases with excitation energy, leading to a long high-energy tail in the spectrum, a fact that may have implications for the computation of nuclear matrix elements for neutrinoless double-β decay in the same framework.

Proton and neutron skins and symmetry energy of mirror nuclei

The neutron skin of nuclei is an important fundamental property, but its accurate measurement faces many challenges. Inspired by charge symmetry of nuclear forces, the neutron skin of a neutron-rich nucleus is related to the difference between the charge radii of the corresponding mirror nuclei. We investigate this relation within the framework of the Hartree-Fock-Bogoliubov method with Skyrme interactions. Predictions for proton skins are also made for several mirror pairs in the middle mass range. For the first time the correlation between the thickness of the neutron skin and the characteristics related with the density dependence of the nuclear symmetry energy is investigated simultaneously for nuclei and their corresponding mirror partners. As an example, the Ni isotopic chain with mass number A=48−60 is considered. These quantities are calculated within the coherent density fluctuation model using Brueckner and Skyrme energy-density functionals for isospin asymmetric nuclear matter with two Skyrme-type effective interactions, SkM* and SLy4. Results are also presented for the symmetry energy as a function of A for a family of mirror pairs from selected chains of nuclei with Z=20, N=14, and N=50. The evolution curves show a similar behavior crossing at the N=Z nucleus in each chain and a smooth growing deviation when N≠Z starts. Comparison of our results for the radii and skins with those from the calculations based on high-precision chiral forces is made.

Fully self-consistent calculation of β-decay half-lives within Skyrme energy density functional * *Supported by the National Natural Science Foundation of China (11975096, 11805280, 11775014, 11635003, 11161130520), National Basic Research Program of China (2010CB832903), European Commission's 7th Framework Programme (Fp7-PEOPLE-2010-IRSES) under Grant Agreement Project (269131), and Continuous Basic Scientific Research Project (WDJC-2019-13)

-decay half-lives of some magic and semi-magic nuclei have been studied in a fully self-consistent Skyrme Hartree-Fock (HF) plus charge-exchange random phase approximation (RPA). The self-consistency is addressed, in that the same Skyrme energy density functional is adopted in the calculation of ground states and Gamow-Teller excited states. First, the impact of terms on the -decay half-lives is investigated by using the SGII interaction, revealing a large influence. Subsequently, numerical calculations are performed for the selected nuclei with Skyrme energy density functionals SGII, LNS, SKX, and SAMi. Finally, comparisons to available experimental data and predictions of different theoretical models are discussed.

Origin of the evolution of spin-orbit and pseudospin-orbit splittings in neutron drops

We found the same staggering pattern in the evolution of the spin-orbit (SO) and pseudospin-orbit (PSO) splittings in neutron drops obtained by the energy-density functional (EDF) theory using SAMi-T and SLy5 parametrizations, with and without the tensor terms respectively. This staggering evolution is similar to the results recently obtained by the relativistic Brueckner-Hartree-Fock (RBHF) theory with the Bonn A potential, which was claimed to be a tensor effect. In this work, we present an intuitive but essential explanation on the origin of this staggering evolution in the Skyrme EDF theory. We found that, for both the SO and PSO partners, the energy splittings mainly originate from the SO potential. The staggering evolution of both the SO and PSO splittings is mainly due to the drastic change of density while neutrons fill different single-particle orbits. The key to determine the specific staggering pattern is the strength of the neutron density gradient term and the SO density term in the SO potential, not necessarily the tensor force.

Global description of β− decay with the axially deformed Skyrme finite-amplitude method: Extension to odd-mass and odd-odd nuclei

We use the finite amplitude method (FAM), an efficient implementation of the quasiparticle random phase approximation, to compute beta-decay rates with Skyrme energy-density functionals for 3983 nuclei, essentially all the medium-mass and heavy isotopes on the neutron rich side of stability. We employ an extension of the FAM that treats odd-mass and odd-odd nuclear ground states in the equal filling approximation. Our rates are in reasonable agreement both with experimental data where available and with rates from other global calculations.

Model nuclear energy density functionals derived from ab initio calculations

We present the first application of a new approach, proposed in (2016 J. Phys. G: Nucl. Part. Phys. 43 04LT01) to derive coupling constants of the Skyrme energy density functional (EDF) from ab initio Hamiltonian. By perturbing the ab initio Hamiltonian with several functional generators defining the Skyrme EDF, we create a set of metadata that is then used to constrain the coupling constants of the functional. We use statistical analysis to obtain such an ab initio-equivalent Skyrme EDF. We find that the resulting functional describes properties of atomic nuclei and infinite nuclear matter quite poorly. This may point to the necessity of building up the ab initio-equivalent functionals from more sophisticated generators. However, we also indicate that the current precision of the ab initio calculations may be insufficient for deriving meaningful nuclear EDFs.

Beyond the proton drip line: Bayesian analysis of proton-emitting nuclei

The limits of the nuclear landscape are determined by nuclear binding energies. Beyond the proton drip lines, where the separation energy becomes negative, there is not enough binding energy to prevent protons from escaping the nucleus. Predicting properties of unstable nuclear states in the vast territory of proton emitters poses an appreciable challenge for nuclear theory as it often involves far extrapolations. In addition, significant discrepancies between nuclear models in the proton-rich territory call for quantified predictions. With the help of Bayesian methodology, we mix a family of nuclear mass models corrected with statistical emulators trained on the experimental mass measurements, in the proton-rich region of the nuclear chart. Separation energies were computed within nuclear density functional theory using several Skyrme and Gogny energy density functionals. We also considered mass predictions based on two models used in astrophysical studies. Quantified predictions were obtained for each model using Bayesian Gaussian processes trained on separation-energy residuals and combined via Bayesian model averaging. We obtained a good agreement between averaged predictions of statistically corrected models and experiment. In particular, we quantified model results for one- and two-proton separation energies and derived probabilities of proton emission. This information enabled us to produce a quantified landscape of proton-rich nuclei. The most promising candidates for two-proton decay studies have been identified. The methodology used in this work has broad applications to model-based extrapolations of various nuclear observables. It also provides a reliable uncertainty quantification of theoretical predictions.

Breakdown of the tensor component in the Skyrme energy density functional

The tensor force, as an important component of strong nuclear force, generates a variety of intriguing effects ranging from few-body systems to neutron stars. It is responsible for the nucleon-nucleon correlation beyond mean-field approximation, and is accordingly proved to play no role in the standard Skyrme energy density functionals in the present work. Therefore, the Skyrme's original tensor interaction that is extensively-employed presently is invalid. As an alternative strategy, we introduced a central interaction, i.e., the $\bm{\sigma }_{1}\cdot \bm{\sigma }_{2}$ term, to improve the description of experimental single-particle structure, and to address its effect, we established two Skyrme interactions IMP1 and IMP2 complemented by the calibrated charge-violating interactions. The central $\bm{\sigma }_{1}\cdot \bm{\sigma }_{2}$ interaction turns out to substantially improve the description of shell evolution in Sn isotopes and $N=82$ isotones.