Roles of tensor force and pairing correlation in two-proton radioactivity of halo nuclei

The tensor force and pairing correlation effects on the two-proton radioactivity of^(18)Mg and^(20)Si with a pronounced two-proton halo are explored in the framework of spherical Skyrme-Hartree-Fock-Bogoliubov theory.It is shown that the halo sizes are enhanced with the increase in the strength of the tensor force and pairing correlation.Furthermore,the increasing halo sizes lead to the enhancement of diproton emission.Then,the tensor force is found to have a small influence on the two-proton decay energies,and the two-proton decay energies calculated with strong surface pairing are smaller than those with weak mixed pairing.Because the two-proton decay energies are relat-ively large,the predicted order of magnitude of half-lives within the effective liquid drop model is not sensitive to the decay energy variation caused by the tensor force and pairing correlation,which has a value of approximately 10^(-18)s.

Effects of the tensor force on low-energy heavy-ion fusion reactions:a mini review

In recent several years,the tensor force,one of the most important components of the nucleon-nucleon force,has been implemented in time-dependent density functional theories and it has been found to influence many aspects of low-energy heavy-ion reactions,such as dissipation dynamics,sub-barrier fusions,and low-lying vibration states of colliding partners.Especially,the effects of tensor force on fusion reactions have been investigated from the internuclear potential to fusion crosssections systematically.In this work,we present a mini review on the recent progress on this topic.Considering the recent progress of low-energy reaction theories,we will also mention more possible effects of the tensor force on reaction dynamics.

Effect of tensor force on the density dependence of symmetry energy within the BHF framework

The effect of tensor force on the density dependence of nuclear symmetry energy has been investigated within the framework of the Brueckner-Hartree-Fock （BHF） approach. It is shown that the tensor force manifests its effect via the tensor 3SD1 channel. The density dependence of symmetry energy Esym turns out to be determined essentially by the tensor force from the π meson and p meson exchanges via the 3SD1 coupled channel. Increasing the strength of the tensor component due to the p-meson exchange tends to enhance the repulsion of the equation of state of symmetric nuclear matter and leads to the reduction of symmetry energy. The present results confirm the dominant role played by the tensor force in determining nuclear symmetry energy and its density dependence within the microscopic BHF framework.

Exploring effects of tensor force and its strength via neutron drops

The tensor-force effects on the evolution of spin-orbit spittings in neutron drops are investigated within the framework of the relativistic Hartree-Fock theory.For a fair comparison on the pure mean-field level,the results of the relativistic Brueckner-Hartree-Fock calculation with the Bonn A interaction are adopted as meta-data.Through a quantitative analysis,we certify that theπ-pscudovector(π-PV)coupling affects the evolutionary trend through the embedded tensor force.The strength of the tensor force is explored by enlarging the strength fr of theπ-PV coupling.It is found that weakening the density dependence of fn is slightly better than enlarging it with a factor.We thus provide a semiquantitative support for the renormalization persistency of the tensor force within the frame-work of density functional theory.This will serve as important guidance for further development of relativistic effective interactions with particular focus on the tensor force.

Relativistic interpretation of the nature of the nuclear tensor force

The spin-dependent nature of the nuclear tensor force is studied in detail within the relativistic HartreeFock approach. The relativistic formalism for the tensor force is supplemented with an additional Lorentz-invariant tensor formalism in the a-scalar channel, so as to take into account almost fully the nature of the tensor force brought about by the Fock diagrams in realistic nuclei. Specifically, the tensor sum rules are tested for the spin and pseudospin partners with and without nodes, to further understand the nature of the tensor force within the relativistic model. It is shown that the interference between the two components of nucleon spinors causes distinct violations of the tensor sum rules in realistic nuclei, mainly due to the opposite signs on the κ quantities of the upper and lower components, as well as the nodal difference. However, the sum rules can be precisely reproduced if the same radial wave functions are taken for the spin/pseudo-spin partners in addition to neglecting the lower/upper components,revealing clearly the nature of the tensor force.

Contributions of optimized tensor interactions on the binding energies of nuclei

The tensor parts of Skyrme interactions are constrained from the collective charge-exchange spin-dipole and Gamow-Teller excitation energies in 90Zr and 208Pb,together with the isotopic dependence of energy splitting between proton h11=2 and g7=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 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.

Stress and fabric in granular material

It has been well recognized that, due to anisotropic packing structure of granular material, the true stress in a specimen is different from the applied stress. However, very few research efforts have been focused on quantifying the relationship between the true stress and applied stress. In this paper, we derive an explicit relationship among applied stress tensor, material-fabric tensor, and force-fabric tensor; and we propose a relationship between the true stress tensor and the applied stress tensor. The validity of this derived relationship is examined by using the discrete element simulation results for granular material under biaxial and triaxial loading con- ditions.

Dynamical effects of spin-dependent interactions in low- and intermediate-energy heavy-ion reactions

It is well known that noncentral nuclear forces, such as the spin-orbital coupling and the tensor force, play important roles in understanding many interesting features of nuclear structures. However, their dynamical effects in nuclear reactions are poorly known because only the spin-averaged observables are normally studied both experimentally and theoretically. Realizing that spin-sensitive observables in nuclear reactions may convey useful information about the in-medium properties of noncentral nuclear interactions, besides earlier studies using the time-dependent Hartree-Fock approach to understand the effects of spin-orbital coupling on the threshold energy and spin polarization in fusion reactions, some efforts have been made recently to explore the dynamical effects of noncentral nuclear forces in intermediate-energy heavy-ion collisions using transport models. The focus of these studies has been on investigating signatures of the density and isospin dependence of the form factor in the spin-dependent single-nucleon potential. Interestingly, some useful probes were identified in the model studies but so far there are still no data to compare with. In this brief review, we summarize the main physics motivations as well as the recent progress in understanding the spin dynamics and identifying spin-sensitive observables in heavy-ion reactions at intermediate energies. We hope the interesting, important, and new physics potentials identified in the spin dynamics of heavy-ion collisions will stimulate more experimental work in this direction.

Relativistic Hartree-Fock model and its recent progress on the description of nuclear structure

Regarding the stage progress on the relativistic Hartree-Fock(RHF)model achieved recently,we review the extensive developments of the model itself,including the descriptions of axially deformed unstable nuclei and nuclear spin-isospin excitations,which shows that a complete RHF framework is now available for exploring the tensor force effects in both ground state and excited states of unstable nuclei.Meanwhile,the recent RHF descriptions of the pseudo-spin symmetry restoration and the new magicity are also reviewed.It shows that the Fock terms,particularly theρ-tensor coupling and naturally introduced tensor force components,bring about significant improvements in maintaining the delicate in-medium balance of nuclear attractions and repulsions,and uniformly interpreting the emergence of new magicity inCa.The revealed microscopic mechanisms not only deepen our understanding on the properties of nuclear structure,but also help to guide the further development of the effective nuclear force.

Many-body correlations in shell model effective interactions derived by ab initio methods and the V_(low-k) approach

We investigate many-body correlations caused by two- and three-body （2-, 3bd） forces. Shell-model effective interactions derived from ab initio methods （coupled-cluster method, no-core shell model） are adopted. Vlow-k potentials, based on many-body perturbation theory, are also tested, especially for their cut-off dependence. We compare the central, tensor and spin-orbit interactions from microscopic theory to the fitted interactions. After the inclusion of the three-body force, the matrix elements become fairly close to those fitted directly to experimental data. Calculations of neutron-rich oxygen isotopes are performed, to clarify the effects of 3bd forces, tensor, and spin-orbit interactions on the nuclear binding and excitation energies. We find that the 3bd force can influence the binding energies greatly, which also determines the drip line position, while its effect on excitation energies is not very pronounced. The spin-orbit force, which is part of the 2bd force, can affect the shell structure explicitly, at least for neutron-rich systems.