Finite-range Effective Interaction Research Articles

Neutron-proton effective mass splitting in terms of symmetry energy and its density slope

Using a simple density-dependent finite-range effective interaction having Yukawa form, the density dependence of isoscalar and isovector effective masses is studied. The isovector effective mass is found to be different for different pairs of like and unlike nucleons. Using HVH theorem, the neutron-proton effective mass splitting is represented in terms of symmetry energy and its density slope. It is again observed that the neutron-proton effective mass splitting has got a positive value when isoscalar effective mass is greater than the isovector effective mass and has a negative value for the opposite case. Furthermore, the neutron-proton effective mass splitting is found to have a linear dependence on asymmetry β. The second-order symmetry potential has a vital role in the determination of density slope of symmetry energy but it does not have any contribution on neutron-proton effective mass splitting. The finite-range effective interaction is compared with the SLy2, SKM, f−, f0, and f+ forms of interactions.

DIRHB—A relativistic self-consistent mean-field framework for atomic nuclei

The DIRHB package consists of three Fortran computer codes for the calculation of the ground-state properties of even–even atomic nuclei using the framework of relativistic self-consistent mean-field models. Each code corresponds to a particular choice of spatial symmetry: the DIRHBS, DIRHBZ and DIRHBT codes are used to calculate nuclei with spherical symmetry, axially symmetric quadrupole deformation, and triaxial quadrupole shapes, respectively. Reflection symmetry is assumed in all three cases. The latest relativistic nuclear energy density functionals are implemented in the codes, thus enabling efficient and accurate calculations over the entire nuclide chart. Program summaryProgram title: DIRHB package (codes DIRHBS, DIRHBZ and DIRHBT)Catalogue identifier: AESN_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AESN_v1_0.htmlProgram obtainable from: CPC Program Library, Queen’s University, Belfast, N. IrelandLicensing provisions: noneNo. of lines in distributed program, including test data, etc.: 44516No. of bytes in distributed program, including test data, etc.: 317161Distribution format: tar.gzProgramming language: Fortran 77.Computer: All computers.Operating system: All operating systems. The Makefiles are specific for a Unix OS and have to be modified for Windows.RAM: Depends on the imposed symmetry and number of oscillator shells. For the triaxial test case it takes 300 MB.Classification: 17.22.Nature of problem:Ground-state properties of even–even open-shell nuclei can be calculated using the framework of self-consistent mean-field models based on relativistic energy density functionals. The structure of arbitrary heavy nuclei with spherical symmetry, axially symmetric quadrupole deformation, and triaxial quadrupole shapes, is modeled using the latest zero- and finite-range relativistic effective interactions. The particle–particle channel of the effective inter-nucleon interaction is described by a separable finite-range pairing force.Solution method:The current implementation of the model computes the mean-field solution of the nuclear many-body problem for even–even open-shell spherical and quadrupole deformed nuclei. The codes are used to solve the stationary relativistic Hartree–Bogoliubov equations in a self-consistent iteration scheme. At each iteration the matrix elements of the equations are updated using the modified Broyden method or the linear mixing method. The single-nucleon wave functions are expanded in a basis of spherical, axially symmetric or triaxial harmonic oscillator, depending on the assumed symmetry of the nuclear shape. For calculations that constrain the shape to specific values of the deformation parameters, the augmented Lagrangian method is used.Restrictions:Time-reversal and reflection symmetries are assumed. Open-shell even–even spherical and quadrupole deformed nuclei are considered.Running time:Depends on the imposed symmetry and number of oscillator shells. For the test cases it runs from few seconds (spherical) up to few hours (triaxial).

Coulomb and spin-orbit interactions in random-phase approximation calculations

We present a fully self-consistent computational framework composed by Hartree-Fock plus random-phase approximation where the spin-orbit and Coulomb terms of the interaction are included in both steps of the calculations. We study the effects of these terms of the interaction on the random-phase approximation calculations, where they are usually neglected. We carry out our investigation of excited states in spherical nuclei of oxygen, calcium, nickel, zirconium, tin, and lead isotope chains. We use finite-range effective nucleon-nucleon interactions of Gogny type. The size of the effects we find is, usually, of a few hundred keV. There are not simple approximations which can be used to simulate these effects since they strongly depend on all the variables related to the excited states, angular momentum, parity, excitation energy, isoscalar, and isovector characters. Even the Slater approximation developed to account for the Coulomb exchange terms in the Hartree-Fock approximation is not valid in random-phase approximation calculations.

Role of symmetry potential in nuclear symmetry energy and its density slope parameter

Using a density dependent finite-range effective interaction of Yukawa form the nuclear mean field in asymmetric nuclear matter is expanded in terms of power series of asymmetry β ( = ρ n − ρ p ρ ) as u τ ( k , ρ , β ) = u 0 ( k , ρ ) ± u sym , 1 ( k , ρ ) β + u sym , 2 ( ρ ) β 2 . The behavior of nuclear symmetry potential u sym , 1 ( k , ρ ) around the Fermi momentum k f is found to be connected to the density dependence of symmetry energy E sym ( ρ ) and nucleon effective mass m 0 ⁎ m ( k = k f , ρ ) in symmetric nuclear matter. Two different trends of momentum dependence for nuclear symmetry potential is observed depending on the choice of strength parameters of exchange interaction, but at Fermi momentum it is found to be independent of the choice of parameters. The nuclear symmetry energy E sym ( ρ ) and its slope L ( ρ ) are expressed analytically in terms of nuclear mean field in isospin asymmetric nuclear matter using the same interaction. We find that the second order nuclear symmetry potential u sym , 2 ( ρ ) cannot be neglected while calculating the density slope of symmetry energy L ( ρ ) as well as the nuclear mean field in extremely neutron (proton) rich nuclear matter.

An effective Nuclear Model: from Nuclear Matter to Finite Nuclei

The momentum and density dependence of mean fields in symmetric and asymmetric nuclear matter are analysed using the simple density dependent finite range effective interaction containing a single Gaussian term alongwith the zero-range terms. Within the formalism developed, it is possible to reproduce the various diverging predictions on the momentum and density dependence of isovector part of the mean field in asymmetric matter. The finite nucleus calculation is formulated for the simple Gaussian interaction in the framework of quasilocal density functional theory. The prediction of energies and charge radii of the interaction for the spherical nuclei compares well with the results of other effective theories.

Neutron–proton effective mass splitting and thermal evolution in neutron-rich matter

The thermal evolution of properties of neutron-rich asymmetric nuclear matter such as entropy density, internal energy density, free energy density and pressure is studied in the non-relativistic mean field theory using finite range effective interactions. In this framework, the thermal evolution of nuclear matter properties is directly connected to the neutron and proton effective mass properties. Depending on the magnitude of neutron–proton effective mass splittings, two distinct behaviours in the thermal evolution of nuclear matter properties are noticed.

Proton radioactivity with a Yukawa effective interaction

The half-lives of proton radioactivity of proton emitters are investigated theoretically. Proton-nucleus interaction potentials are obtained by folding the densities of the daughter nuclei with a finite-range effective nucleon-nucleon interaction having Yukawa form. The Wood-Saxon density distributions for the nuclei used in calculating the nuclear as well as the Coulomb interaction potentials are predictions of the interaction. The quantum mechanical tunneling probability is calculated within the WKB framework. These calculations provide reasonable estimates for the observed proton radioactivity lifetimes. The effects of neutron-proton effective mass splitting in neutron-rich asymmetric matter as well as the nuclear matter incompressibility on the decay probability are investigated.

Evolution of the nuclear properties in the long chain of the Ni isotopes

The work presents systematic theoretical investigation of even-even isotopes of Ni (Z = 28) from the proton-excess 48Ni up to the neutron-excess 78Ni. To describe the ground-state properties of these nuclei, we employed the self-consistent approach that makes use of the Skyrme functional and the pairing correlations. The random-phase approximation using the finite-range two-body effective interaction is applied for the description of the energy spectra and the electromagnetic transition rates. A comprehensive comparison with the experiment is carried out, including the values of binding energies and the values of the one and two-nucleon separation energies as well as the energies of the excited states and the probabilities of electromagnetic transitions that are available by now.

Microscopic calculation ofPu240scission with a finite-range effective force

Hartree-Fock-Bogoliubov calculations of hot fission in $^{240}\mathrm{Pu}$ have been performed with a newly implemented code that uses the D1S finite-range effective interaction. The hot-scission line is identified in the quadrupole-octupole-moment coordinate space. Fission-fragment shapes are extracted from the calculations. A benchmark calculation for $^{226}\mathrm{Th}$ is obtained and compared with results in the literature. In addition, technical aspects of the use of HFB calculations for fission studies are examined in detail. In particular, the identification of scission configurations, the sensitivity of near-scission calculations to the choice of collective coordinates in the HFB iterations, and the formalism for the adjustment of collective-variable constraints are discussed. The power of the constraint-adjustment algorithm is illustrated with calculations near the critical scission configurations with up to seven simultaneous constraints.

Temperature dependence of the nuclear symmetry energy and equation of state of charge neutral n + p + e + μ matter in beta equilibrium

The temperature and density dependence of the nuclear symmetry energy is studied in the nonrelativistic mean field theory by using a density-dependent finite range effective interaction. The temperature evolution of the interaction part of symmetry energy is decided by the nature of the finite range exchange interactions acting between a pair of like and unlike nucleons, an area which is less understood. In view of this, two cases corresponding to different strengths of exchange interaction between two like nucleons are considered to examine their influence on the temperature dependence of the nuclear symmetry energy. The symmetry energy obtained as a function of temperature and density is used to study the temperature dependence of leptonic fractions, proton fraction and equation of state of charge neutral n + p + e + μ matter under β-equilibrium for the two different cases.

Gaussian matrix elements in a cylindrical harmonic oscillator basis

We derive a formalism, the separation method, for the efficient and accurate calculation of two-body matrix elements for a Gaussian potential in the cylindrical harmonic-oscillator basis. This formalism is of critical importance for Hartree–Fock and Hartree–Fock–Bogoliubov calculations in deformed nuclei using realistic, finite-range effective interactions between nucleons. The results given here are also relevant for microscopic many-body calculations in atomic and molecular physics, as the formalism can be applied to other types of interactions beyond the Gaussian form. The derivation is presented in great detail to emphasize the methodology, which relies on generating functions. The resulting analytical expressions for the Gaussian matrix elements are checked for speed and accuracy as a function of the number of oscillator shells and against direct numerical integration.

Density dependence of the symmetry free energy of hot nuclei

The density and excitation energy dependence of symmetry energy and symmetry free energy for finite nuclei are calculated microscopically in a microcanonical framework, taking into account thermal and expansion effects. A finite-range momentum and density-dependent two-body effective interaction is employed for this purpose. The role of mass, isospin, and equation of state (EOS) on these quantities is also investigated; our calculated results are in consonance with the available experimental data.

Nuclear mean field and equation of state of asymmetric nuclear matter

Neutron and proton mean field properties and equation of state of highly isospin asymmetric dense nuclear matter are studied using simple finite range effective interactions. The density dependence of nuclear symmetry energy is constrained on the basis of an approximate universal high density behaviour of asymmetric contribution to the nucleonic part of energy density of infinite n + p + e + μ matter in beta-equilibrium. Recently discovered astrophysical bounds from neutron star mass measurements and cooling phenomenology as well as informations coming from optical model analysis of nucleon–nucleus scattering data at intermediate energies, transport model analysis of flow data in heavy ion collisions and monopole mode of vibrations in finite nuclei have also been used to constrain the interaction parameters involved.

Nuclear response functions in homogeneous matter with finite range effective interactions

The question of nuclear response functions in a homogeneous medium is examined. A general method for calculating response functions in the random-phase approximation with exchange is presented. The method is applicable for finite-range nuclear interactions. Examples are shown in the case of symmetric nuclear matter described by a Gogny interaction. It is found that the convergence of the results with respect to the multipole truncation is quite fast. Various approximation schemes such as the Landau approximation, or the Landau approximation for the exchange terms only, are discussed in comparison with the exact results.

Momentum and density dependence of the isospin part of nuclear mean field and equation of state of asymmetric nuclear matter

Momentum and density dependence of the isospin part of nuclear mean field u τ ( k , ρ ) which is still, in part, the open problem of the old Lane potential is analysed using density dependent finite range effective interactions. The behaviour of u τ ( k = k f , ρ ) around the Fermi momentum k f is found to be related to the density dependence of nuclear symmetry energy J τ ( ρ ) and nucleon effective mass M ∗ ( k = k f , ρ ) / M in symmetric nuclear matter. The momentum dependence of u τ ( k , ρ ) is separated out in terms of a simple functional u τ e x ( k , ρ ) which vanishes at k = k f and involves only the finite range parts of the exchange interactions between pairs of like and unlike nucleons. Depending on the choice of the parameters of these exchange interactions two conflicting trends of momentum dependence are noticed which lead to two opposite types of splitting of neutron and proton effective masses. The equation of state of asymmetric nuclear matter and the high density behaviour of nuclear symmetry energy J τ ( ρ ) are studied by constraining the additional parameters involved on the basis that pure neutron matter should not be predicted to be bound by any reasonable nuclear interaction. Emphasis is also given on the need of experimental data sensitive to the differences between neutron and proton transport properties in highly asymmetric dense nuclear matter and its analysis to constrain the high density behaviour of nuclear symmetry energy as well as to resolve the controversy on the two opposite types of splitting of neutron and proton effective masses.

Mean-field calculations with Gaussian expansion method(Mean-field approaches with finite-range effective interactions,YITP Workshop on New Developments in Nuclear Self-Consistent Mean-Field Theories (MF05))

Mean-field calculations with Gaussian expansion method(Mean-field approaches with finite-range effective interactions,YITP Workshop on New Developments in Nuclear Self-Consistent Mean-Field Theories (MF05))

Nuclear liquid-gas phase transition and supernovae evolution

It is shown that the large density fluctuations appearing at the onset of the first order nuclear liquid-gas phase transition can play an important role in the supernovae evolution. Due to these fluctuations, the neutrino gas may be trapped inside a thin layer of matter near the proto-neutron star surface. The resulting increase of pressure may induce strong particle ejection a few hundred milliseconds after the bounce of the collapse, contributing to the revival of the shock wave. The HartreeFock+RPA scheme, with a finite-range nucleon-nucleon effective interaction, is employed to estimate the effects of the neutrino trapping due to the strong density fluctuations, and to discuss qualitatively the consequences of the suggested new scenario.

Collective excitations far from the valley of stability

Collective Vibrations of nuclei far from the valley of stability are investigated within the quasi-particle random phase approximation based on relativistic Hartree-Bogoliubov theory with non-linear meson couplings. Pairing correlations are described in by a finite range effective particle-particle interaction of Gogny type. The quasi-particle random phase equations are solved in the canonical basis and collective strength distributions are discussed.

Momentum dependence of the mean field and equation of state of nuclear matter

The momentum and density dependence of the mean field and other properties of nuclear matter are studied with finite-range effective interactions having different functional forms by evaluating the single-particle momentum distribution function self-consistently. In these calculations relativistic effects and supraluminous behaviour of nuclear matter are also taken into account. The equation of states obtained from these effective interactions are used to study liquid–gas phase transition in nuclear matter.

Nuclear equation of state with finite-range effectiveinteractions

The equation of state of nuclear matter has been studied byusing finite-range phenomenological effective interactionshaving Gaussian and Yukawa functional form. The single-particlemomentum distribution function has been evaluatedself-consistently to obtain the single-particle energy, energydensity and chemical potential at a thermal equilibrium. Byusing this self-consistent procedure the free-energy density,pressure, entropy density, effective mass and incompressibilityof nuclear matter have also been calculated. The possibletemperature, momentum and density dependence of thesequantities, particularly the mean field, effective mass andincompressibility, have been discussed. The phase transition toquark-gluon plasma has been studied for a zeroth-order bagmodel expression of quark pressure. The influence of thefunctional form of the effective interaction on these studieshas also been discussed.