Semi-infinite Nuclear Matter Research Articles

Hartree–Fock Calculations in Semi-Infinite Matter with Gogny Interactions

Hartree–Fock equations in semi-infinite nuclear matter for finite range Gogny interactions are presented together with a detailed numerical scheme to solve them. The value of the surface energy is then extracted and given for standard Gogny interactions.

Surface energy coefficient of an N2LO Skyrme energy functional: A semiclassical extended Thomas-Fermi approach

We generalize to N2LO Skyrme functionals the semi-classical approach of Grammaticos and Voros in order to calculate the Extended Thomas Fermi expressions of the new densities and currents appearing at the N2LO level. Within a one dimensional symmetric semi infinite nuclear matter model and using a simple Fermi-like density profile, we obtain an easy-to-use formula for the surface energy including the contributions of the central, density-dependent and spin-orbit terms up to $\hbar^2$. Such a formula can be easily used as a first attempt to constrain the surface properties of new N2LO Skyrme parametrizations. The N2LO parametrization tested in this paper is shown to exhibit a shift (compared to a full Hartree-Fock calculation) which is quantitatively similar to the one obtained with the traditional Skyrme parametrizations.

The relationship of the neutron skin thickness to the symmetry energy and its slope

Abstract The neutron skin thickness of asymmetric semi-infinite nuclear matter (δRM) is shown to be a function of the Coulomb energy (Vc), the asymmetry energy coefficient (J), the slope (L) of the asymmetry energy, and the incompressibility coefficient (K), in addition to the Fermi momentum (kF) and the asymmetry parameter (I = (N − Z)/A). The relational formula is derived on the basis of the Hugenholtz–Van Hove theorem in the mean-field (MF) approximation for nuclear matter. Using the formula as a guide, the neutron skin thickness (δR) in 208Pb is examined in the MF models. The apparently spurious linear correlation between L and δR appears through the model-dependent correlation of L with J that is included in the main components of the formula.

Constraining the surface properties of effective Skyrme interactions

The purpose of this study is threefold: first, to identify a scheme for the determination of the surface energy coefficient a_surf that offers the best compromise between robustness, precision, and numerical efficiency; second, to analyze the correlation between values for a_surf and the characteristic energies of the fission barrier of Pu240; and third, to lay out a procedure how the deformation properties of the Skyrme energy density functional (EDF) can be constrained during the parameter fit. There are several frequently used possibilities to define and calculate the surface energy coefficient a_surf of effective interactions. The most direct access is provided by the model system of semi-infinite nuclear matter, but a_surf can also be extracted from the systematics of binding energies of finite nuclei. Calculations can be carried out either self-consistently (HF), which incorporates quantal shell effects, or in one of the semi-classical Extended Thomas-Fermi (ETF) or Modified Thomas-Fermi (MTF) approximations. The surface energy coefficient of 76 parameterizations of the Skyrme EDF have been calculated. Values obtained with the HF, ETF and MTF methods are not identical, but differ by fairly constant systematic offsets. Despite having some drawbacks compared to the other methods studied here, the MTF approach provides sufficiently precise values for a_surf such that it can be used as a very robust constraint on surface properties during a parameter fit at negligible additional cost. Through the construction of a series of eight parameterizations SLy5s1-SLy5s8 of the standard Skyrme EDF with systematically varied a_surf value it is shown how to arrive at a fit with realistic deformation properties. [Note: The abstract has been abbreviated because of length restrictions imposed by the arXiv. See the paper for the full abstract.]

Optimizing relativistic energy density functionals: covariance analysis

The stability of model parameters for a class of relativistic energy density functionals, characterized by contact (point-coupling) effective inter-nucleon interactions and density-dependent coupling parameters, is analyzed using methods of statistical analysis. A set of pseudo-observables in infinite and semi-infinite nuclear matter (SINM) is used to define a quality measure χ2 for subsequent analysis. Uncertainties of model parameters and correlation coefficients between parameters are computed, and the eigenvectors and eigenvalues of the matrix of second derivatives of χ2 at the minimum are evaluated. Using these quantities, the stability of the density functional in nuclear matter is examined, and weakly and strongly constrained combinations of parameters are determined. In addition, uncertainties of observables that are not included in the calculation of χ2 are analyzed: binding energy of asymmetric nuclear matter, surface thickness of SINM, binding energies and charge radii of finite nuclei.

Thermal properties of the nuclear surface

The thermal evolution of a few thermodynamic properties of the nuclear surface like its thermodynamic potential energy, entropy and the symmetry free energy are examined for both semi-infinite nuclear matter and finite nuclei. The Thomas-Fermi model is employed. Three Skyrme interactions, namely, SkM$^*$, SLy4 and SK255 are used for the calculations to gauge the dependence of the nuclear surface properties on the energy density functionals. For finite nuclei, the surface observables are computed from a global liquid-drop inspired fit of the energies and free energies of a host of nuclei covering the entire periodic table. The hot nuclear system is modelled in a subtracted Thomas-Fermi framework. Compared to semi-infinite nuclear matter, substantial changes in the surface symmetry energy of finite nuclei are indicated; surface thermodynamic potential energies for the two systems are, however, not too different. Analytic expressions to fit the temperature and asymmetry dependence of the surface thermodynamic potential of semi-infinite nuclear matter and the temperature dependence of the surface free energy of finite nuclei are given.

Relativistic Description of Semi-Infinite Nuclear Matter with Vacuum Polarization

We study the property of semi-infinite nuclear matter using the relativistic mean field (RMF) model with vacuum polarization. The vacuum polarization effect is calculated by the derivative expansion method, which has been shown to be a good approximation in rigorous numerical calculations. The Lagrangian density is written in terms of positive energy nucleon states together with the meson fields and the derivative meson terms for the vacuum polarization. We find that the vacuum polarization makes both the surface thickness and surface energy coefficient larger, and therefore, it is important to describe the empirical nuclear surface properties. Moreover, the vacuum polarization smoothens the oscillatory behavior of the nuclear density observed inside the nuclear surface in the RMF model.

Finite- to zero-range relativistic mean-field interactions

We study the relation between the finite-range (meson-exchange) and zero-range (point-coupling) representations of effective nuclear interactions in the relativistic mean-field framework. Starting from the phenomenological interaction DD-ME2 with density-dependent meson-nucleon couplings, we construct a family of point-coupling effective interactions for different values of the strength parameter of the isoscalar-scalar derivative term. In the meson-exchange picture this corresponds to different values of the σ-meson mass. The parameters of the isoscalar-scalar and isovector-vector channels of the point-coupling interactions are adjusted to nuclear matter and ground-state properties of finite nuclei. By comparing results for infinite and semi-infinite nuclear matter, ground-state masses, charge radii, and collective excitations, we discuss constraints on the parameters of phenomenological point-coupling relativistic effective interaction.

Surface behaviour of the pairing gap in a slab of nuclear matter

The surface behaviour of the pairing gap previously studied for semi-infinite nuclear matter is analyzed in the slab geometry. The gap-shape function is calculated in two cases: (a) pairing with the Gogny force in a hard-wall potential and (b) pairing with the separable Paris interaction in a Saxon-Woods mean-field potential. It is shown that the surface features are preserved in the case of slab geometry, being almost independent of the width of the slab. It is also demonstrated that the surface enhancement is strengthened as the absolute value of chemical potential $|\mu|$ decreases which simulates the approach to the nucleon drip line.

Local-potential approximation for the Brueckner G matrix and problem of optimally choosing model subspace

The validity of the local-potential approximation, which was proposed previously for the singlet-pairing problem in semi-infinite nuclear matter, is investigated in the Bethe-Goldstone equation for the Brueckner G matrix. For this purpose, use is made of the method developed earlier for solving this equation for a planar slab of nuclear matter in the case of a separable form of NN interaction. The 1 S 0 singlet and the 3 S 1+3 D 1 triplet channel are considered. The complete two-particle Hilbert space is split into a model and the complementary subspace that are separated by an energy E 0. The two-particle propagator is calculated precisely in the first subspace, and the local-potential approximation is used in the second subspace. With an eye to subsequently employing the G matrix to calculate the Landau-Migdal amplitude, the total two-particle energy is fixed at E=2μ, where μ is the chemical potential of the system under consideration. Specific numerical calculations are performed at μ=−8 MeV. The applicability of the local-potential approximation is investigated versus the cutoff energy E 0. It is shown that, in either channel being considered, the accuracy of the local-potential approximation is rather high for E 0≥10 MeV.

Isospin dependence of the nuclear surface tension

Using the Thomas-Fermi nuclear model we have constructed a standard dimensionless function (a simple hyperbola) which can be used to estimate the dependence of the surface tension $\ensuremath{\gamma}$ on the relative neutron excess $\ensuremath{\delta}$ for a variety of models, using only the models' properties calculated for standard (semi-infinite) nuclear matter with $\ensuremath{\delta}=0.$ We add some comments clarifying the relation between the Gibbs and the droplet model definitions, ${\ensuremath{\gamma}}_{\ensuremath{\mu}}$ and ${\ensuremath{\gamma}}_{e},$ of the surface tension.

Modified Derivative Scalar Coupling Model in Semi-infinite Nuclear Matter**The project supported in part by National Natural Science Foundation of China, and Major State Basic Research Developing Program with Grant No. G2000077400

Semi-infinite nuclear matter has been investigated in relativistic Thomas–Fermi and Hartree approximations based on the modified derivative scalar coupling model. Our results show that the spin-orbit potential has been improved by the tensor coupling. However, the surface tension and the surface thickness become considerably small with increasing of the tensor coupling constant. The effects of the -meson mass on the spin-orbit potential, on the surface tension, and on the surface thickness have also been discussed.

Microscopic calculation of a pairing gap in semi-infinite nuclear matter

The pairing gap in semi-infinite nuclear matter has been calculated microscopically by solving the gap equation for a nonlocal interaction with the aid of the method proposed by V.A. Khodel, A.V. Khodel, and J.W. Clark [Nucl. Phys. A 598, 390 (1996)] for the case of infinite nuclear matter. The calculation employs the effective pairing interaction obtained previously for semi-infinite geometry on the basis of the separable 3×3 representation of the Paris nucleon-nucleon potential. The gap found in this way changes sharply in the surface region, where it has a pronounced maximum. The dependence of the surface effect on the chemical potential of nuclear matter has been investigated.

Surface behaviour of the pairing gap in semi-infinite nuclear matter

The 1S 0 -pairing gap in semi-infinite nuclear matter is evaluated microscopically using the effective pairing interaction recently derived in the coordinate representation from a separable form of the Paris NN-potential. Instead of a direct iterative solution of the gap equation, a new method proposed by V. A. Khodel, V. V. Khodel and J. W. Clark was used which simplifies the procedure significantly. The gap Δ obtained in our calculations exhibits a strong variation in the surface region with a pronounced maximum near the surface.

Surface properties of nuclear pairing with the Gogny force in a simplified model

Surface properties of neutron-neutron (T=1) pairing in semi-infinite nuclear matter in a hard wall potential are investigated in BCS approximation using the Gogny force. Surface enhancement of the gap function, pairing tensor and correlation energy density is put into evidence.

Nuclear surface properties in relativistic effective field theory

We perform Hartree calculations of symmetric and asymmetric semi-infinite nuclear matter in the framework of relativistic models based on effective hadronic field theories as recently proposed in the literature. In addition to the conventional cubic and quartic scalar self-interactions, the extended models incorporate a quartic vector self-interaction, scalar-vector non-linearities and tensor couplings of the vector mesons. We investigate the implications of these terms on nuclear surface properties such as the surface energy coefficient, surface thickness, surface stiffness coefficient, neutron skin thickness and the spin-orbit force.

Semiclassical treatment of asymmetric semi-infinite nuclear matter: surface and curvature properties in relativistic and non-relativistic models

Surface and curvature properties of asymmetric semi-infinite nuclear matter are studied to beyond the proton drip. Using the semiclassical extended Thomas-Fermi method with corrections of order h ̷ 2 , the calculations are performed in the non-relativistic and relativistic mean field theories (Skyrme forces and non-linear σ-ω parametrizations). First we discuss the bulk equilibrium between the nuclear and drip phases. Next we analyse the asymmetric surface as a function of the bulk neutron excess. We examine local quantities related to the density profiles and, for two definitions of the bulk reference energy, the surface and curvature energy coefficients. The calculation of the curvature energy is carefully treated. The sensitivity of the nuclear surface to the relativistic effects is investigated. Mass formulae useful for arbitrary neutron excess are discussed, and their limit at small asymmetries is compared with the liquid droplet model mass formula.

Microscopic study of the effective pairing interaction in semi-infinite nuclear matter with the separable NN-interaction

The problem of pairing in semi-infinite nuclear matter is considered microscopically. Equations for the effective pairing interaction in the semi-infinite matter for the separable form of the NN-interaction are derived and solved directly, without use of any form of local approximation. Numerical solution is carried out for a separable representation of the Paris NN-potential. A version of the local approximation, the “local potential approximation”, is suggested which works sufficiently well even in the surface region. The effective pairing interaction obtained in our calculations reveals a strong variation in the surface region changing from a strong attraction outside the nuclear matter to almost zero value inside. Thus, it confirms the idea of the surface nature of nuclear pairing.

Variational Wigner-Kirkwood approach to relativistic mean field theory

The recently developed variational Wigner-Kirkwood approach is extended to the relativistic mean field theory for finite nuclei. A numerical application to the calculation of the surface energy coefficient in semi-infinite nuclear matter is presented. The new method is contrasted with the standard density functional theory and the fully quantal approach. {copyright} {ital 1997} {ital The American Physical Society}

Skyrme representation of a relativistic spin-orbit field

We find that a spin-orbit force with the conventional Skyrme form reproduces satisfactorily the isospin dependence of the spin-orbit field generated in relativistic Hartree calculations on semi-infinite nuclear matter with different neutron excesses. We also examine a generalization of the Skyrme-type energy-density functional that leads to still better agreement with the relativistic Hartree calculations. {copyright} {ital 1997} {ital The American Physical Society}