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.     

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.     

A relativistic approach to the equation of state of asymmetric nuclear matter

The properties of asymmetric nuclear matter for a wide range of densities and asymmetric parameters are investigated within the lowest-order-constrained variational (LOCV) method by employing the relativistic Hamiltonian with a potential which has been fitted relativistically to N-N phase shifts ( [(v)\tilde]₁₄ \tilde{{v}}_{{14}}^{} and to the AV₁₄interaction. Like our previous work on symmetric nuclear matter, the boost interaction corrections as well as the relativistic one-body and two-body kinetic corrections are calculated. The various properties of asymmetric nuclear matter such as the symmetry energy, the saturation energy and the validity of the a² \alpha^{2}_{} law, etc., are examined. The symmetry energy is reduced by about 7MeV when we use [(v)\tilde]₁₄ \tilde{{v}}_{{14}}^{} instead of its non-relativistic version, i.e. the AV₁₄interaction. The results are compared with other many-body calculations.     

Equation of state of dense nuclear matter

An equation of state for cold nuclear matter for the region of densities ρ_nm−4ρ_nm, where ρ_nm is empirical nuclear-matter density, is constructed. We begin from the detailed calculation of Day and Wiringa for the two-body interactions; these give a saturation density of ∼2ρ_nm. This density is brought down to ρ_nm by the addition of relativistic corrections. Additional binding is obtained from three-body forces. A reasonable picture is obtained with the Day-Wiringa compression modulus for the two-body calculation, but the picture can be further improved by choosing this to be smaller.Our equation of state is similar to that of Friedman and Pandharipande in the region of nuclear matter denstiy ρ_nm, but, due to higher-order terms in the loop correction, is substantially softer at high density. Basically what happens is that the many-body effects saturate with increasing density, leaving only the two-body interactions.With this equation of state, prompt supernova explosions are very powerful when the compression modulus of neutron-rich matter (twice as many neutrons as protons) is ∼150 MeV, which corresponds to K_nm ∼ 190 MeV for symmetric nuclear matter.Analysis shows that hot nuclear matter formed in heavy ion collisions demands a very stiff equation of state. We understand this as arising from the strong velocity dependence in the real part of the optical model potential which follows chiefly from the Lorentz character of the interactions, the vector mean field growing with increasing density and the scalar one decreasing. This gives a substantial repulsive contribution to the energy per particle and produces a stiff effective equation of state for several hundred MeV heavy-ion collisions. With increasing degree of equilibration the magnitude of the repulsive energy decreases since equilibration decreases the effective momentum. Given the strong velocity dependence in the interaction, the hot equation of state can be reconciled with the cool one.     

Stability of the fragments and thermalization at the peak centre-of-mass energy

We simulated the central reactions of nearly symmetric and asymmetric systems, for energies at which maximum production of intermediate mass fragments (IMFs) occurred (E_c.m.^peakE_{\rm c.m.}^{\rm peak}). This study was carried out using hard EOS along with Cugnon cross-section employing MSTB method for clusterization. We studied the various properties of fragments. The stability of fragments was checked through persistence coefficient and gain term. The information about the thermalization and stopping in heavy-ion collisions was obtained via relative momentum, anisotropy ratio and rapidity distribution. We found that for a complete stopping of incoming nuclei very heavy systems are required. The mass dependence of various quantities (such as average and maximum central density, collision dynamics as well as the time zone for hot and dense nuclear matter) was also presented. In all cases (i.e., average and maximum central density, collision dynamics as well as the time zone for hot and dense nuclear matter) a power-law dependence was obtained.     

Finite nuclei to nuclear matter: a leptodermous approach

The liquid drop model (LDM) expansions of energy and incompressibility of finite nuclei are studied in an analytical model using Skyrme-like effective interactions to examine, whether such expansions provide an unambiguous way to go from finite nuclei to nuclear matter, and thereby can yield the saturation properties of the latter, from nuclear masses. We show that the energy expansion is not unique in the sense that, its coefficients do not necessarily correspond to the ground state of nuclear matter and hence, the mass formulas based on it are not equipped to yield saturation properties. The defect is attributed to its use of liquid drop without any reference to particles as its basis, which is classical in nature. It does not possess an essential property of an interacting many-fermion system namely, the single particle property, in particular the Fermi state. It is shown that, the defect is repaired in the infinite nuclear matter model by the use of generalized Hugenholtz–Van Hove theorem of many-body theory. So this model uses infinite nuclear matter with well defined quantum mechanical attributes for its basis. The resulting expansion has the coefficients which are at the ground state of nuclear matter. Thus a well defined path from finite nuclei to nuclear matter is found out. Then using this model, the saturation density 0.1620 fm⁻³ and binding energy per nucleon of nuclear matter 16.108 MeV are determined from the masses of all known nuclei. The corresponding radius constant r₀ equal to 1.138 fm thus determined, agrees quite well with that obtained from electron scattering data, leading to the resolution of the so-called ‘r₀-paradox’. Finally a well defined and stable value of 288±20 MeV for the incompressibility of nuclear matter K_∞ is extracted from the same set of masses and a nuclear equation of state is thus obtained.     

Nuclear symmetry energy in terms of single-nucleon potential and its effect on the proton fraction of <Emphasis Type="Italic">β</Emphasis>-stable <Emphasis Type="Italic">npeμ</Emphasis> matter

Momentum and density dependence of single-nucleon potential u_τ (k, ρ, β) is analyzed using a density dependent finite range effective interaction of the Yukawa form. Depending on the choice of the strength parameters of exchange interaction, two different trends of the momentum dependence of nuclear symmetry potential are noticed which lead to two opposite types of neutron and proton effective mass splitting. The 2nd-order and 4th-order symmetry energy of isospin asymmetric nuclear matter are expressed analytically in terms of the single-nucleon potential. Two distinct behavior of the density dependence of 2nd-order and 4th-order symmetry energy are observed depending on neutron and proton effective mass splitting. It is also found that the 4th-order symmetry energy has a significant contribution towards the proton fraction of β-stable npeμ matter at high densities.     

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.     

Properties of nuclear matter using the velocity-dependent effective potential ofs-wave interaction

Two forms of a velocity-dependent effective potential are used. The binding energy and the incompressibility of the nuclear matter are calculated. These are found to be in a good agreement with those obtained by others and with the experimental results. The single-particle potential, the effective massM ^* and the nuclear surface energy are also discussed and compared with those obtained by the others.     

Nuclear physics in colourful worlds: Quantumchromodynamics and nuclear binding

When quantumchromodynamics (QCD) is generalized from SU(3) to an SU(N_c) gauge theory, where N_c is the number of colours, it depends on only two parameters: N_c and the bare quark mass m_q. A more general understanding of nuclear physics can be achieved by considering what it would be like in worlds with the number of colours different from 3, and bare quark masses different from the “empirical” ones. Such an investigation can be carried out within a framework of meson-exchange interactions. The empirical binding energy of nuclear matter results from a very near cancellation between attractive and repulsive terms which are two orders of magnitude larger and may be expected to depend sensitively on the parameters of QCD. It is indeed found that our world is wedged into a small corner of the two-dimensional manifold of m_q versus N_c. If the number of colours were decreased by one, or the bare quark masses raised by more than 20%, nuclear matter would become unbound. By tracing the origin of this state of affairs, one obtains a clearer picture of the relative importance of various effects on the behaviour of the bulk nuclear matter. In particular, correlations like those embodied in the Coester band of saturation points appear to have a broader degree of validity than is implied by fits to the actual physical world only.     

Density Dependence of Nuclear Matter Symmetry Energy: VMC Calculations

Physics of Atomic Nuclei - A Variational Monte-Carlo (VMC) method is employed to investigate the density dependence of the symmetry energy of isospin asymmetric nuclear matter. The realistic Urbana...     

Model nuclear matter calculations in several Brueckner and Green's function theories

Model nuclear matter calculations are performed using two versions of the Brueckner theory (standard lowest order theory and a version with self-consistent “physical” onshell insertions in particle lines) and three Green's function theories (Λ ₀₀,Λ ₁₀ and Galitskii's ladder approximation). Ground state properties are derived using a strongs-wave nonlocal separable nucleon-nucleon potential. We investigate the differences between the results obtained using different theories, stemming from different treatment of the exclusion principle and of dispersive effects of the medium. The effect of the off-shell self-consistency in theΛ ₁₀ theory is found to be important.     

Variational calculations of asymmetric nuclear matter

We report on variational calculations of the energy E(ρ, β) of asymmetric nuclear matter having ? = ?_n + ?_p = 0.05 to 0.35 fm^?3, and β = (?_n ? ?_p/g9 = 0 to 1. The nuclear h used in this work consists of a realistic two-nucleon interaction, called v₁₄, that fits the available nucleon-nucleon scattering data up to 425 MeV, and a phenomenological three nucleon interaction adjusted to reproduce the empirical properties of symmetric nuclear matter. The variational many-body theory of symmetric nuclear matter is extended to treat matter with neutron excess. Numerical and analytic studies of the β-dependence of various contributions to the nuclear matter energy show that at ? < 0.35 fm^?3 the β⁴ terms are very small, and that the interaction energy EI(ρ, β) defined as E(ρ, β) ? T_F(ρ, β), where T_F is the Fermi-gas energy, is well approximated by EI₀(?) + β²EI₂(ρ). The calculated symmetry energy at equilibrium density is 30 MeV and it increases from 15 to 38 MeV as ? increases from 0.05 to 0.35 fm^?3.     

The exchange of effective mesons between nucleons in nuclei

A study is made of the nuclear matter G-matrix. A method is developed which allows the determination of a local G-operator. This method is applied to analyse the correlations of the G-matrix, i.e. G without the bare interaction V, in terms of meson-exchange-type interactions. For a strong potential (REID) these effective mesons have masses which exceed 800 MeV implying very short-ranged correlations. For a weaker potential these correlations are somewhat longer ranged. The dominant feature of these G-operator correlations is found to be a strong scalar isoscalar attractive interaction which provides the binding of the nuclear system. By comparing with exact results for Landau parameters and binding energies it is shown that the approximation made with this local operator is excellent. A weak density dependence of the correlations is found and the structure of the Landau parameter ? is discussed. Finally it is proposed that this G-operator is a suitable object to study finite nuclei.     

The effect of density dependent Av18 effective interaction on the ground state properties of heavy closed shell nuclei

The channel-dependent Argonne Av₁₈ effective two-body interactions (CDEI) which are generated through the lowest order constrained variational (LOCV) calculation for asymmetric nuclear matter with the charge-dependent Av₁₈ bare nucleon–nucleon potential are used to calculate the ground state properties of heavy closed shell nuclei such as ⁴⁸Ca, ⁹⁰Zr, ¹²⁰Sn and ²⁰⁸Pb. The harmonic oscillator basis, and the local density approximation (LDA) are applied to create the relative and the center of mass dependent effective two-body potential. We get more binding with respect to the similar calculation with the Reid types potentials. It is tried to omit the LDA and perform full calculation with the Av₁₈CDEI for light nuclei. The results indicate that the LDA works quite well. It is also shown that in case of heavy closed shell nuclei and unlike our previous report with Reid68Day   interaction, the contributions of higher partial waves (J>2$J>2$) are very important for the calculations with Av₁₈ potential and we get reasonable agreement between our calculated binding energies and RMS radii, with those predicated by the others methods, and the experimental data. Finally, the various aspects of channel and density dependent two-body effective interactions are discussed.     

Ground state properties of nuclear matter

The ground state properties of nuclear matter are calculated in theΛ ₁₁-approximation¹. A nucleon-nucleon interaction of the Yamaguchi-type and thes-wave part ofTabakin's potential have been considered. In both cases too large values for the density of nuclear matter and the binding energy per nucleon are found. The momentum distribution turns out to be very small for momenta larger than the Fermi momentum.     

Abnormal nuclear matter and many-body forces

Qualitative aspects of quantum corrections to the Lee-Wick abnormal nuclear matter are studied in terms of many-body forces in the normal nuclear matter implied by the σ-model Lagrangian field theory. Using a simplified model for the scalar meson self-energy in the nuclear medium and restricting to a set of graphs which in non-relativistic normal nuclear matter reduces to the well-known random phase approximation (RPA), we have found that an abnormal nuclear state can be bound or unbound depending upon whether strongly attractive multi-body forces are present or absent in the normal matter. This is in support of our previous result obtained heuristically from some general considerations of quantum corrections. A strongly bound abnormal matter with an equilibrium density of a few times the normal nuclear matter density ρ₀ can be formed if large attractive manybody forces can be accommodated in the normal nuclear matter. However if one accepts the present status of theories of nuclear matter binding energy in which no attractive many-body forces are called for, then the abnormal state can occur only at large densities (perhaps 8 to 10 times ρ₀) and is expected to be unbound by several hundred MeV per particle.     

The properties of nuclear matter at zero and finite temperatures

The properties of nuclear matter are studied in the frame of the Brueckner theory. The Brueckner-Hartree-Fock approximation plus two-body density-dependent Skyrme potential which is equivalent to three-body interaction are used. Various modern nucleon-nucleon potentials are used in the framework of the Brueckner-Hartree-Fock approximation, e.g.: CD-Bonn potential, Nijm1 potential, and Reid 93 potential. These modern nucleon-nucleon potentials fit the deuteron properties and are phase shifts equivalent. The equation of state at T = 0, pressure at T = 0, 8, and 12 MeV, free energy at T = 8 and 12 MeV, nuclear matter incompressibility, and the symmetry energy calculation are presented. The hot properties of nuclear matter are calculated using T ²-approximation method at low temperatures. Good agreement is obtained in comparison with previous theoretical estimates and experimental data, especially at low densities.