Determining a Skyrme-type effective interaction from realistic two-nucleon interaction

The Variational Monte Carlo (VMC) method is employed to determine characteristics of symmetric and asymmetric nuclearmatter. The realistic Urbana v14 nucleon-nucleon interaction potential of Lagaris and Pandharipande was used in the VMC calculations with addition of a phenomenological density-dependent term to simulate many-body interactions. A new Skyrme parameter set SKaan-U14 is found to consistently reproduce the characteristics of the nuclear matter obtained from VMC calculations. The properties of symmetric and asymmetric nuclear matter are calculated by the new Skyrme parameter set. The results obtained by using the new Skyrme parameter set are compared with results obtained by different Skyrme parameter sets in the literature.     

Spin Instability in Nuclear Matter and the Skyrme Interaction

In the present work the problem of spin instability of nuclear matter with the Skyrme interaction is analyzed. The three-body part of the interaction is replaced by using a density dependent potential which is a modification to that given previously by Dabrowski. The symmetry energies are calculated and their rearrangement corrections. Good agreement is obtained with previous calculations.     

Neutron skin effect of some Mo isotopes in pre-equilibrium reactions

The neutron skin effect has been investigated for even isotopes of molybdenum at 25.6 MeV ^{94 − 100}Mo(p, xn) reaction using the geometry-dependent hybrid model of pre-equilibrium nuclear reactions. Here the initial neutron/proton exciton numbers were calculated from the neutron/ proton densities obtained from an effective nucleon–nucleon interaction of the Skyrme type. Initial exciton numbers from different radii of even Mo isotopes were used to obtain the corresponding neutron emission spectra. In this investigation the calculated results are compared with the experimental data as also with each other. The results using central densities in the geometry-dependent hybrid model are in better agreement with the experimental data.     

Thermodynamic properties of superconducting nuclear matter

Phase diagrams of superconducting nuclear matter are calculated by solving a set of finite temperature gap equations, using several Skyrme effective interactions. Our results indicate that nuclear matter may have a superconducting phase in a small region with density near one half of the normal nuclear matter density and temperature k_BT ≲ 1.4 MeV. Our calculation is based on a finite temperature Green's function method with an abnormal pair cutoff approximation. The same approximation is employed in deriving the internal energy, entropy and chemical potential of superconducting nuclear matter. In this way, its equation of state is obtained, and compared with that of normal nuclear matter. The energy gap of superconducting nuclear matter is found to depend rather sensitively on both density and temperature. This dependence is analysed in terms of the Skyrme interaction parameters. The correlation effect on chemical potential is found to be important at high density, and its inclusion is essential in determining the equation of state of superconducting nuclear matter.     

CALCULATION OF WIDTHS,SPECTRAL FUNCTION AND HOLE OCCUPATION NUMBERS OF SINGLE-PARTICLE STATES

The single-particle widths, the spectral function as well as the hole occupation numbers are calculated by means of the complex off-shell single-particle potential obtained from the Skyrme interactions in nuclear matter. The calculated results are reasonable and comparable to some extent with the experimental data available.     

Semi-microscopic optical potential calculation by the nuclear matter approach

In this paper the semi-microscopic nuclear matter approach has been introduced to calculate the microscopic optical potential. The first- and second-order mass operators in asymmetric nuclear matter have been derived with Skyrme effective interactions and the real and imaginary parts of the optical potential for finite nuclei have been obtained by applying a local density approximation. Five Skyrme interactions II–VI have been used and compared with the experimental data to determine how well these Skyrme interaction function for our purposes. Our results obtained in this simple way are to some extent comparable with those obtained with the “nuclear matter” and “nuclear structure” approaches without adjusting the parameters of the Skyrme interactions.     

The effective interaction for unbound nucleons and the optical potential

The effective interaction for an unbound nucleon in a complex nucleus is studied with particular emphasis upon the relationship of the unbound state interaction with the interaction for bound states. A successful unified treatment of unbound and bound states is given, and the unbound state effective interaction is related to the standard “optical model”. The attractive force for an unbound nucleon is shown to be of an exchange type and is best represented by a non-local, energy-independent potential which can be calculated, at least for not too high incident nucleon energy, from the attractive interaction between bound nucleons in a nucleus. We relate the unbound state potential to the bound state effective force as calculated by Negele using Hartree-Fock theory.

A non-local potential used by Elton, Webb, and Barrett in a successful analysis of electron scattering is shown, with minor modification, to work remarkably well in an analysis of elastic scattering of protons. Excellent agreement with experiment is obtained for the differential cross section and the polarization of protons with an incident energy up to 60 MeV scattered from ⁴⁰Ca, ⁵⁸Ni, ¹²⁰Sn and ²⁰⁸Pb. The real potential parameters are shown to be independent of the incident energy and physically reasonable; the imaginary potential parameters agree with expectations from Fermi gas and collective models. Bound state energy levels are also calculated with no readjustment of the real potential parameters in order to check the consistency of the interaction with the bound states, and the results are in good agreement with Negele.

The interaction of Skyrme as modified by Vautherin and Brink is also used in an analysis of elastic proton scattering. Good results are obtained for the differential cross section and the polarization for 20 MeV protons scattered from ⁵⁸Ni. At higher incident proton energy, and for larger nuclei, the results are poorer.     

Potential energy surface and formation of superheavy nuclei with the Skyrme energy-density functional

With the Skyrme energy-density functional theory, the nucleus–nucleus potential is calculated and the potential energy surface is obtained with different effective forces for accurately estimating the formation cross sections of superheavy nuclei in massive fusion reactions. The width and height of the potential pocket are influenced by the Skyrme effective forces SkM, SkM^*, SkP, SIII, Ska, and SLy4, which correspond to the different equations of state for the isospin symmetry nuclear matter. It is found that the nucleus–nucleus potential is associated with the collision orientation and Skyrme forces. A more repulsive nuclear potential is pronounced with increasing the incompressible modulus of nuclear matter, which hinders the formation of superheavy nuclei. The available data in the fusion-evaporation reaction of ⁴⁸Ca+²³⁸U are nicely reproduced with the SkM^* parameter by implementing the potential into the dinuclear system model.     

Potential of interaction between nuclei and nucleon-density distribution in nuclei

Nuclear-interaction potentials that are calculated by using Skyrme forces within the extended Thomas-Fermi approximation and Hartree-Fock-Bardeen-Cooper-Schrieffer theory are studied in detail. It is shown that the nuclear component of the potential simulating the interaction between nuclei grows with increasing number of neutrons in colliding isotopes and with increasing diffuseness parameter of the density distribution in interacting nuclei. An increase in the diffuseness parameter of the density distribution in interacting nuclei leads to a decrease in the height of the barrier between the nuclei and to an increase in the depth of the capture well and in the fusion cross section. It is shown that the diffuseness parameter calculated for the nuclear component of the potential at large distance between interacting nuclei by using Skyrme forces exceeds the diffuseness parameter of the nucleon-density distribution in these nuclei by a factor of about 1.5. Realistic values of the diffuseness parameter of nuclear interaction between medium-mass and heavy nuclei fall within the range a ≈ 0.75–0.90 fm.     

Comparison between global phenomenological and microscopic optical potentials for proton as projectile below 100 MeV

For 112 target nuclei (52 elements) with proton as projectile, we calculate the reaction cross sections and elastic scattering angular distributions, as well as the X2 values for 16 kinds of proton optical model potentials: two sets of phenomenological global optical potentials and the microscopic optical potentials proposed by Shen et al for 14 sets of Skyrme force parameters: GSI-6, SBJS, SKM, SGI-Ⅱ, SKa-b, SG01-Ⅱ.We find that for obtaining the proton microscopic optical potential based on the nuclear matter approach with Skyrme force, SGI, SKa and SKb are the three sets of optimal Skyrme force parameters.     

SEMI-MICROSCOPIC OPTICAL POTENTIAL CALCULATION BY THE NUCLEAR MATTER APPROACH——I. SYMMETRIC NUCLEAR MATTER

In this paper semi-microscopic nuclear matter approach is introduced to calculatethe microscopic optical potential. The first and second order mass operator in sym-metric nuclear matter is derived with Skyrme effective interactions and the real andimaginary part of the optical potential for finite nuclei is obtained by applying alocal density approximation. The five kinds of the different parameters of Skyrmeinteractions Ⅱ-Ⅵ are used and compared with the experimental data to study how wellthese Skyrme interactions can work for our purposes. Our results obtained in such asimple way seem to be to some extent comparable with those obtained with "nuclearmatter approach" and "nuclear structure approach" without adjusting the parame-ters of the Skyrme interactions so far.     

Effect of tensor nucleon-nucleon forces on the nucleon-nucleus optical potential

In the approximation of unpolarized nuclear matter, the optical potential for nucleon-nucleus scattering is calculated on the basis of the effective Skyrme interaction with allowance for tensor nucleon-nucleon forces. It is shown that the tensor Skyrme forces make a significant contribution to the imaginary part of the optical potential. The effect of tensor nucleon-nucleon forces on the radial distribution of the imaginary part of the optical potential is investigated by considering the example of elastic neutron scattering by ⁴⁰Ca nuclei at scattering energies of about a few tens of MeV.     

Nuclear matter equation of state and three-body forces

The energy per particle, symmetry energy, pressure, and free energy are calculated for symmetric nuclear matter using BHF approach with modern nucleon-nucleon CD-Bonn, Nijm1, Argonne v₁₈, and Reid 93 potentials. To obtain saturation in nuclear matter we add three-body interaction terms which are equivalent to a density-dependent two-nucleon interaction a la Skyrme force. Good agreement is obtained in comparison with previous theoretical estimates and experimental data.     

The neutron halo in heavy nuclei calculated with the Gogny force

The proton and neutron density distributions, one- and two-neutron separation energies and radii of nuclei for which neutron halos are experimentally observed, are calculated using the self-consistent Hartree-Fock-Bogoliubov method with the effective interaction of Gogny. Halo factors are evaluated assuming hydrogen-like antiproton wave functions. The factors agree well with experimental data. They are close to those obtained with Skyrme forces and with the relativistic mean-field approach.     

Spinodal instabilities of asymmetric nuclear matter within the Brueckner–Hartree–Fock approach

We study the spinodal instabilities of asymmetric nuclear matter at finite temperature within the microscopic Brueckner–Hartree–Fock (BHF) approximation using the realistic Argonne V18 nucleon–nucleon potential plus a three-body force of Urbana type. Our results are compared with those obtained with the Skyrme force SLy230a and the relativistic mean field models NL3 and TW. We find that BHF predicts a larger spinodal region. This result is a direct consequence of the fact that our Brueckner calculation predicts a larger critical temperature and saturation density of symmetric nuclear matter than the Skyrme and relativistic mean field ones. We find that the instability is always dominated by total density fluctuations, in agreement with previous results of other authors. We study also the restoration of the isospin symmetry in the liquid phase, i.e., the so-called isospin distillation or fragmentation effect, finding that its efficiency increases with increasing proton fraction and decreases as temperature and density increase. In general, we find that the Brueckner results are comparable to those obtained with the Skyrme and the relativistic mean field models, although the restoration of isospin symmetry is not so efficient in this case.     

Nuclear interface energy at finite temperatures

The thermodynamic properties at finite temperatures of the plane interface between two phases of nuclear matter in equilibrium are examined theoretically, and explored numerically. The microscopic hamiltonian, the Skyrme I′ interaction, is used in the Thomas-Fermi approximation to obtain the finite-temperature extensions of earlier zero-temperature results which used the Hartree-Fock and Thomas-Fermi methods. Approximate analytic fits are given to the χ_i (proton fraction on the dense-matter side) dependence of the critical temperature, and to the T and χ_i dependences of the surface thermodynamic potentials, the density of surface neutrons, the surface entropy and the neutron and proton chemical potentials at phase equilibrium. These fits are an ingredient in a compressible liquid-drop nuclear model, the basis of an equation of state for hot, dense matter needed in certain astrophysical applications.The liquid-drop model is used here to construct an isolated, low-T nucleus, whose properties are compared with the original zero-T Hartree-Fock calculations which lead to the Skyrme I interaction, and with other mass formulae. The low-temperature expansion of the surface energy is compared with that obtained in other calculations. The nuclear level density at the Fermi surface, related to the low-T expansion of the entropy of the whole nucleus, is also discussed.     

Self-consistent calculations for highly excited compound nuclei

A method for self-consistently calculating average nuclear properties at high excitation energies is derived. Calculations using this formalism have been performed for a double-magic and a deformed nucleus employing the Skyrme force for the nucleon-nucleon interaction. The effects of excitation energy on the nuclear structure are discussed and a comparison of self-consistently calculated level densities with those obtained in the usual statistical method is made. The calculations for the deformed nucleus show the transition from the deformed to a spherical shape with increasing excitation.     

Generator coordinate calculations of giant resonances with the Skyrme interaction

The Skyrme interaction is shown to lead to significant simplifications in generator coordinate calculations. As an illustration, giant resonances are calculated using pure oscillator wave functions. We present results for monopole, dipole and quadrupole isoscalar and isovector modes using two different Skyrme forces, SIII and SIV. A good agreement with available experimental data is obtained.     

Energy dependent potential in nuclear collisions

Energy dependence of the real part of the nucleus-nucleus potential is studied using a modified Seyler-Blanchard two-body effective interaction which contains density dependence along with the momentum dependence. The nucleus-nucleus potential is evaluated in the proximity picture of Blocki et al. The calculated energy dependence of the ion-ion potential compares well with the phenomenology. The various sets of the Skyrme force have also been used to study this energy dependence and widely different results are obtained for the different sets. Heavy-ion fusion excitation functions have also been calculated using the energy-dependent potential in the effective mass approximation. It is observed that the high-energy fusion cross sections are significantly suppressed when calculated with the energy-dependent ion-ion interaction. The important deep-inelastic observables are found to be insensitive to the energy dependence of the potential.