Nuclear Matter Binding Research Articles

Surface properties for Ne, Na, Mg, Al, and Si isotopes in the coherent density fluctuation model using the relativistic mean-field densities

We have systematically studied the surface properties, such as symmetry energy, neutron pressure, and symmetry energy curvature coefficient for Ne, Na, Mg, Al, and Si nuclei from the proton to neutron drip lines. The coherent density fluctuation model (CDFM) is used to estimate these quantities taking the relativistic mean-field densities as inputs. The Brückner energy density functional is taken for the nuclear matter binding energy and local density approximation is applied for its conversion to coordinate space. The symmetry energy again decomposed to the volume and surface components within the liquid drop model formalism to the volume and surface parts separately. Before calculating the surface properties of finite nuclei, the calculated bulk properties are compared with the experimental data, whenever available. The NL3* parameter set with the Bardeen–Cooper–Schrieffer (BCS) pairing approach in an axially deformed framework is used to take care of the pairing correlation when needed. The deformed density is converted to its spherical equivalent with a two-Gaussian fitting, which is used as an input for the calculation of weight function in the CDFM approximation. With the help of the symmetry energy, the isotopes 29F, 28Ne, 29,30Na, and 31,35,36Mg are considered to be within the island of inversion (Han et al. Phys. Lett. B, 772, 529 (2017). doi:10.1016/j.physletb.2017.07.007). Although we get large symmetry energies corresponding to a few neutron numbers for this isotopic chain as expected, an irregular trend appears for all these considered nuclei. The possible reason behind this abnormal behavior of symmetry energy for these lighter mass nuclei is also included in the discussion, which gives a direction for future analysis.

Nuclear response functions with low-momentum interactions

Linear density response functions for nuclear matter are calculated with separable 1S0 and 3S1 2-nucleon interactions obtained from experimental phase-shifts by inverse scattering techniques. It has been shown that results of nuclear matter binding energy Brueckner calculations with these potentials agree closely with those using realistic Bonn interactions. It has also been shown that low-momentum versions give binding energies (nearly) independent of momentum cut-offs $\Lambda \geq \sim 2$ fm-1. Shown here is that the $\Lambda = 2$ rank-one version of the separable interaction (for the S-states) yields a near agreement between a second-order and an all-order (Brueckner) binding energy calculation. Second order calculations of response functions are then made with this version using a method that Kwong and Bonitz used for the Coulomb gas. It is based on time-evolving two-time Green’s functions by Kadanoff-Baym (KB) equations. The effect of correlations, going beyond the conventional HF+RPA method, is included by “dressing” the Green’s function propagators with time-dependent complex self-energies in addition to the real HF-field. It would be of interest to see if the calculated response is also, like the binding energy, independent of cut-offs larger than $\sim 2$ fm-1. That would require using separable potentials of a higher rank, not accommodated by the present computer program. Some results with a 1.5 fm-1 cut-off are however also shown below together with those with a 2.0 fm-1 cut-off. The KB-equations are time-evolved until the nuclear medium is fully correlated. The time-dependent density response to an external perturbation is then registered and Fourier transformed to obtain energy-dependent density response functions. Most previous calculations have been made with in-medium effective (e.g., Gogny or Skyrme) interactions. The medium dependence is in the present calculations contained in the second-order terms.

Temperature of projectile like fragments in heavy ion collisions

A model in which a projectile like fragment can be simply regarded as a remnant after removal of some part of the projectile leads to an excited fragment. This excitation energy can be calculated with a Hamiltonian that gives correct nuclear matter binding, compressibility and density distribution in finite nuclei. In heavy ion collisions the model produces a dependence of excitation energy on impact parameter which appears to be correct but the magnitude of the excitation energy falls short. It is argued that dynamic effects left out in the model will increase this magnitude. The model can be directly extended to include dynamics but at the expense of increased computation. For many calculations for observables, a temperature is an easier tool to use rather than an excitation energy. Hence temperature dependences on impact parameter in heavy ion collisions are displayed.

Nuclear equation of state in a relativistic independent quark model with chiral symmetry and dependence on quark masses

We have calculated the properties of nuclear matter in a self-consistent manner with a quark-meson coupling mechanism incorporating the structure of nucleons in vacuum through a relativistic potential model; where the dominant confining interaction for the free independent quarks inside a nucleon is represented by a phenomenologically average potential in equally mixed scalar-vector harmonic form. Corrections due to spurious center of mass motion as well as those due to other residual interactions, such as the one gluon exchange at short distances and quark-pion coupling arising out of chiral symmetry restoration, have been considered in a perturbative manner to obtain the nucleon mass in vacuum. The nucleon-nucleon interaction in nuclear matter is then realized by introducing additional quark couplings to $\ensuremath{\sigma}$ and $\ensuremath{\omega}$ mesons through mean field approximations. The relevant parameters of the interaction are obtained self-consistently while realizing the saturation properties such as the binding energy, pressure, and compressibility of the nuclear matter. We also discuss some implications of chiral symmetry in nuclear matter along with the nucleon and nuclear $\ensuremath{\sigma}$ term and the sensitivity of nuclear matter binding energy with variations in the light quark mass.

VIRIAL EXPANSION OF THE NUCLEAR EQUATION OF STATE

We study the equation of state (EOS) of nuclear matter as function of density. We expand the energy per particle (E/A) of symmetric infinite nuclear matter in powers of the density to take into account 2, 3, …, N-body forces. New EOS are proposed by fitting ground state properties of nuclear matter (binding energy, compressibility and pressure) and assuming that at high densities a second-order phase transition to the quark–gluon plasma (QGP) occurs. The latter phase transition is due to symmetry breaking at high density from nuclear matter (locally color white) to the QGP (globally color white). In the simplest implementation of a second-order phase transition we calculate the critical exponent δ by using Landau's theory of phase transition. We find δ = 3. Refining the properties of the EOS near the critical point gives δ = 5 in agreement with experimental results. We also discuss some scenarios for the EOS at finite temperatures.

Clustering at the nuclear surface and symmetry energy

With in the energy density functional formalism a phenomenological theory of nuclei is developed which incorporates clustering at the nuclear surface in a general form. This explains the large values of symmetry energy extracted recently at low values of nuclear matter density. It is shown that the nuclear matter binding energy per nucleon (B/A), in the neighbourhood of zero density, must approach its value at the saturation density. The parameters of the theory are mainly constrained from the binding energies and root mean square radii of 376 spherical nuclei as well as the large values of the recently extracted symmetry energy at low densities. Importance of quartic term in symmetry energy is demonstrated. It is shown that it originates due to clustering as well as due to contribution of three-nucleon interaction in the state-of-the-art equation of state of neutron matter at and below the saturation densities. It is found that clustering significantly reduces the neutron skin thickness in nuclei.

EFFECTS OF IN-MEDIUM MODIFICATION OF WEAKLY INTERACTING LIGHT BOSON MASS IN NEUTRON STARS

The effects of the presence of weakly interacting light boson (WILB) in neutron star matter have been revisited. Direct checking based on the experimental range of symmetric nuclear matter binding energy1 and the fact that the presence of this boson should give no observed effect on the crust properties of neutron star matter, shows that the characteristic scale of WILB [Formula: see text] should be ≤2 GeV-2. The recent observational data with significant low neutron stars radii2 and the recent largest pulsar which has been precisely measured, i.e. J1903+0327 (Ref. 3) indicate that in-medium modification of WILB mass in neutron stars cannot be neglected.

The effect of higher partial waves ( [formula omitted]) on the nuclear matter binding energy and saturation density

The available phenomenological nucleon–nucleon ( NN) potentials usually fit the experimental phase shifts up to a maximum partial wave (energy). In this work we intend to use the lowest order constrained variational (LOCV) method to calculate the binding energy and the saturation density of nuclear matter (NM) up to the different maximum partial waves ( J max ), and investigate their effects on the saturation properties of NM. The NN potentials such as the Reid86, the Δ – Reid 68 , the Reid93 and the Av 18 interactions are used to calculate the saturation energy–density curve of NM with different J max . It is shown that the higher partial waves have sizable effect on the properties of NM, and it is not possible to ignore them simply. Finally, the contribution of various channel energies in terms of J max are presented and discussed, and once again concluded that the inclusion of channel dependence of correlation functions are more important than to parameterize them to reduce the amount of computational procedure or to take into account the many-body cluster terms.

Core nucleus polarization in Λ hypernuclei

The response of the core nucleus to the \ensuremath{\Lambda} in a hypernucleus is studied with a local density approximation. This reproduces the energies and radii of the core nuclei as well as the \ensuremath{\Lambda}-single particle (s.p.) energies quite well. The polarizing effect of the \ensuremath{\Lambda} depends on the core response through an ``effective'' compression modulus ${K}_{A}$ of the nucleus. For a certain class of energy functional, ${K}_{A}$ is found to be almost independent of the compression modulus $K$ of the infinite nuclear matter. This indeed is a surprising result, and varies with the Hartree-Fock calculations with effective interactions. Reasons for this discrepancy are carefully examined. We consider values of $K$ in the range $\ensuremath{\approx}100\text{\ensuremath{-}}400$ MeV. Furthermore, the polarizing effects also depend critically on $D(\ensuremath{\rho})$, the \ensuremath{\Lambda} binding in nuclear matter at density $\ensuremath{\rho}$. For only a direct $\ensuremath{\Lambda}N$ force: $D\ensuremath{\propto}\ensuremath{\rho}$ and the core nucleus contracts giving rise to relatively larger core polarization. However, for a ``saturating'' $D(\ensuremath{\rho})$ (with a maximum at ${\ensuremath{\rho}}_{m}<{\ensuremath{\rho}}_{0}$, where ${\ensuremath{\rho}}_{0}$ is the nuclear matter equilibrium density), which is required to fit the s.p. data, the $s$-shell hypernuclei binding energies and the low energy $\ensuremath{\Lambda}p$ scattering data, which results from a $\ensuremath{\Lambda}N$ force (including exchange) and $\ensuremath{\Lambda}\mathit{NN}$ forces, there may be an expansion of the nucleus with nucleons flowing from the interior to the surface. This is shown to reduce the core polarization effects substantially (for ${\ensuremath{\rho}}_{m}$ in the neighborhood of ${\ensuremath{\rho}}_{0}$). The resulting changes in root mean square radius and core energy depend on $A$, but are mostly very small, justifying their general neglect.

Approximate analytical solution for nuclear matter in a mean-field Walecka model and Coester line behavior

We study nuclear matter, at the mean-field approximation, by considering as equal the values of the scalar and the vector density in the Walecka model, which is a very reasonable approximation up to the nuclear matter saturation density. It turns out that the model has an analytical solution for the scalar and vector couplings as functions only of the nuclear matter density and binding energy. The nuclear matter properties are very close to the original version of the model. This solution allows us to show that the correlation between the binding energy and the saturation density is Coester line like. The liquid-gas phase transition is also studied and the critical and flash temperatures are again very similar to the original ones.

The Coester line in relativistic mean field nuclear matter

The Walecka model contains essentially two parameters that are associated with the Lorentz scalar (S) and vector (V) interactions. These parameters are related to a two-body interaction consisting of S and V, imposing the condition that the two-body binding energy is fixed. We have obtained a set of different values for the nuclear matter binding energies at equilibrium densities. We investigated the existence of a linear correlation between $B_N$ and $\rho_0$, claimed to be universal for nonrelativistic systems and usually known as the Coester line, and found an approximate linear correlation only if $V?S$ remains constant. It is shown that the relativistic content of the model, which is related to the strength of $V?S$, is responsible for the shift of the Coester line to the empirical region of nuclear matter saturation.

SELF-CONSISTENT TREATMENT OF THE SELF-ENERGY IN NUCLEAR MATTER

The influence of hole–hole propagation in addition to the conventional particle–particle propagation on the energy per nucleon and the momentum distribution is investigated. The results are compared to the Brueckner–Hartree–Fock (BHF) calculations with a continuous choice and a conventional choice for the single-particle spectrum. Also, the structure of nucleon self-energy in nuclear matter is evaluated. All the off-shell self-energies are calculated self-consistently. Using the self-consistent self-energy, the hole and particle self-consistent spectral functions including the particle–particle and hole–hole ladder contributions in nuclear matter are calculated using realistic NN interactions. We found that the hole–hole ladder brought about non-negligible contributions to the nuclear matter binding energy per nucleon.

Chiral approach to nuclear matter: Role of explicit short-range NN-terms

We extend a recent chiral approach to nuclear matter by including the most general (momentum-independent) NN-contact interaction. Iterating this two-parameter contact-vertex with itself and with one-pion exchange the emerging energy per particle exhausts all terms possible up-to-and-including fourth order in the small momentum expansion. The equation of state of pure neutron matter, $\bar E_n(k_n)$, can be reproduced very well up to quite high neutron densities of $\rho_n=0.5\fmd$ by adjusting the strength of a repulsive $nn$-contact interaction. Binding and saturation of isospin-symmetric nuclear matter is a generic feature of our perturbative calculation. Fixing the maximum binding energy per particle to $-\bar E(k_{f0})= 15.3 $MeV we find that any possible equilibrium density $\rho_0$ lies below $\rho_0^{\rm max}=0.191\fmd$. The additional constraint from the neutron matter equation of state leads however to a somewhat too low saturation density of $\rho_0 =0.134 \fmd$. We also investigate the effects of the NN-contact interaction on the complex single-particle potential $U(p,k_f)+i W(p,k_f)$. We find that the effective nucleon mass at the Fermi-surface is bounded from below by $M^*(k_{f0}) \geq 1.4 M$. This property keeps the critical temperature of the liquid-gas phase transition at somewhat too high values $T_c \geq 21 $MeV. The downward bending of the asymmetry energy $A(k_f)$ above nuclear matter saturation density is a generic feature of the approximation to fourth order. Altogether, there is within this complete fourth-order calculation no "magic" set of adjustable short-range parameters with which one could reproduce simultaneously and accurately all semi-empirical properties of nuclear matter.

Uncertainty of hyperon couplings and the electrochemical potential in neutron star matter

Uncertainty of the hyperon couplings, in particular, that of the $\Sigma^-$, in dense matter raises the question of the behavior of the electrochemical potential in neutron star matter, which is crucial to the possible presence of the kaon condensed phase. We show that regardless of this uncertainty, the $\Lambda$ hyperon, whose coupling can be constrained by its binding in nuclear matter and other observations, also aided by the $\Xi^-$, introduce a saturation of the electrochemical potential just as the $\Sigma^-$ would otherwise do, which tends to mitigate against kaon condensation. The maximum possible mass of neutron stars appears to be $\sim 1.5 \msun$ independent of the uncertainties.

Field redefinitions at finite density

The apparent dependence of nuclear matter observables on off-shell properties of the two-nucleon potential is re-examined in the context of effective field theory (EFT). Finite density (thermodynamic) observables are invariant under field redefinitions, which extends the well-known theorem about the invariance of S-matrix elements. Simple examples demonstrate how field redefinitions can shift contributions between purely off-shell two-body interactions and many-body forces, leaving both scattering and finite-density observables unchanged. If only the transformed two-body potentials are kept, however, the nuclear matter binding curves will depend on the off-shell part (generating “Coester bands”). The correspondence between field redefinitions and unitary transformations, which have traditionally been used to generate “phase-equivalent” nucleon–nucleon potentials, is also demonstrated.

Status of the Brueckner-Hartree-Fock approximation to the nuclear matter binding energy with the Paris potential.

A new calculation of the binding energy of nuclear matter in the Brueckner-Hartree-Fock approximation with the Paris potential using the standard and continuous choices of single particle energies is presented, paying special attention to the numerical accuracy and higher partial waves. Comparison with other calculations is made and the accuracy of the state of the art for the Brueckner-Hartree-Fock method is assessed.

High-Density Matter in the Chiral Sigma Model

We present a field theoretical equation of state for nuclear matter and neutron-rich matter in β equilibrium using the chiral sigma model. The model includes an isoscalar vector field generated dynamically and reproduces the empirical values of the nuclear matter saturation density and binding energy and the isospin symmetry coefficient for asymmetric nuclear matter. The energy per nucleon of nuclear matter as predicted by our calculation is in very good agreement, up to about a density of 4n s (n s =nucleon number density for saturating nuclear matter), with the estimates inferred from heavy-ion collision data. An astrophysical application, relating to neutron star structure, is presented

Relativistic mean-field calculations of Lambda and Sigma hypernuclei.

Single-particle spectra of {Lambda} and {Sigma} hypernuclei are calculated within a relativistic mean-field theory. The hyperon couplings used are compatible with the {Lambda} binding in saturated nuclear matter, neutron-star masses and experimental data on {Lambda} levels in hypernuclei. Special attention is devoted to the spin-orbit potential for the hyperons and the influence of the {rho}-meson field (isospin dependent interaction).

Binding in nuclear matter of hyperons and charmed baryons

A separable potential model for the hyperon-nucleon interaction in conjunction with the Bethe-Goldstone equation for nuclear matter is used for the evaluation of the binding energy of Λ. The results obtained under the various conditions are presented.

Effective nuclear interaction: Dirac versus conventional approach

The effective interaction in nuclear matter ( G-matrix) is calculated, in the relativistic (Dirac-Brueckner-Hartree-Fock) as well as in the conventional Brueckner-Hartree-Fock approach. Starting point is a nucleon-nucleon potential which, in addition to single-meson exchange, contains delta-isobar box diagrams with ππ- and πϱ-exchange, leading to Pauli blocking and dispersive effects. The implications for nuclear matter binding and Landau parameters are investigated.