Properties Of Asymmetric Nuclear Matter Research Articles

Constraining the symmetry energy from fragment yields

Methods of extraction of the symmetry energy coefficient to temperature ratio from isobaric and isotopic yields of fragments produced in Fermi-energy heavy-ion collisions are discussed. Consistent results are obtained when the hot fragmenting source is well characterized and its excitation energy and isotopic composition are properly taken into account. The results, independent of the mass number of the detected fragments, suggest that their fate is decided very early in the reaction. A comparison to the Statistical Multifragmentation Model (SMM) predictions is also presented. The effort to understand the properties of asymmetric nuclear matter both at normal densities and at densities away from the saturation density has stimulated a growing interest in isospin effects in nuclear reactions. The isotopic distribution of fragments produced in multifragmentation events in central collisions at intermediate energies can probe properties of low density nuclear matter. Different methods to extract the “symmetry energy” coefficient from yield ratios have been proposed. Among those, we focus on three seemingly different approaches: isoscaling [1–4], m-scaling and isobaric yield ratio method [5], and we show that consistent experimental results can be extracted provided that the properties of the hot fragmenting source, such as its isospin composition and excitation energy, are properly taken into account. We remark that experimentally whether the equilibrium process, occurring during the multifragmentation, takes place at constant pressure or volume (freeze-out hypothesis) is not determined. This ambiguity casts some doubts on the derived quantity, i.e. symmetry energy or symmetry enthalpy. Keeping in mind this ambiguity we will refer to the experimentally extracted quantity as the symmetry energy.

Nuclear matter properties, phenomenological theory of clustering at the nuclear surface, and symmetry energy

We present a phenomenological theory of nuclei that incorporates clustering at the nuclear surface in a general form. The theory explains the recently extracted large symmetry energy by Natowitz et al., at low densities of nuclear matter and is fully consistent with the static properties of nuclei. In a phenomenological way, clusters of all sizes and shapes along with medium modifications are included. Symmetric nuclear matter properties are discussed in detail. Arguments are given that lead to an equation of state of nuclear matter consistent with clustering in the low-density region. We also discuss properties of asymmetric nuclear matter. Because of clustering, an interesting interpretation of the equation of state of asymmetric nuclear matter emerges. As a framework, an extended version of Thomas-Fermi theory is adopted for nuclei which also contain phenomenological pairing and Wigner contributions. This theory connects the nuclear matter equation of state, which incorporates clustering at low densities, with clustering in nuclei at the nuclear surface. Calculations are performed for various equations of state of nuclear matter. We consider measured binding energies of 2149 nuclei for $N$, $Z$ \ensuremath{\geqslant} 8. The importance of the quartic term in symmetry energy is demonstrated at and below the saturation density of nuclear matter. It is shown that it is largely related to the use of, ab initio, a realistic equation of state of neutron matter, particularly the contribution arising from the three neutron interactions and somewhat to clustering. Reasons for these are discussed. Because of clustering the neutron skin thickness in nuclei is found to reduce significantly. The developed theory predicts situations and regimes to be explored both theoretically and experimentally.

Temperature and density dependence of asymmetric nuclear matter and protoneutron star properties within an extended relativistic mean field model

The effect of temperature and density dependence of the asymmetric nuclear matter properties is studied within the extended relativistic mean field (ERMF) model, which includes the contribution from the self and mixed interaction terms by using different parametrizations obtained by varying the neutron skin thickness $\Delta$r and $\omega$-meson self-coupling ($\zeta$). We observed that the symmetry energy and its slope and incompressibility coefficients decrease with increasing temperatures up to saturation densities. The ERMF parametrizations were employed to obtain a new set of equations of state (EOS) of the protoneutron star (PNS) with and without inclusion of hyperons. In our calculations, in comparison with cold compact stars, we obtained that the gravitational mass of the protoneutron star with and without hyperons increased by $\sim 0.4M_{\odot}$ and its radius increased by $\sim 3$km. Whereas in case of the rotating PNS, the mass shedding limit decreased with increasing temperature, and this suggested that the keplerian frequency of the PNS, at T = 10 MeV should be smaller by $14-20%$ for the EOS with hyperon, as compared to the keplerian frequency of a cold compact star.

Density slope of the nuclear symmetry energy from the neutron skin thickness of heavy nuclei

Expressing explicitly the parameters of the standard Skyrme interaction in terms of the macroscopic properties of asymmetric nuclear matter, we show in the Skyrme-Hartree-Fock approach that unambiguous correlations exist between observables of finite nuclei and nuclear matter properties. We find that existing data on neutron skin thickness $\ensuremath{\Delta}{r}_{\mathit{np}}$ of Sn isotopes give an important constraint on the symmetry energy ${E}_{\mathrm{sym}}({\ensuremath{\rho}}_{0})$ and its density slope $L$ at saturation density ${\ensuremath{\rho}}_{0}$. Combining these constraints with those from recent analyses of isospin diffusion and the double neutron/proton ratio in heavy-ion collisions at intermediate energies leads to a more stringent limit on $L$ approximately independent of ${E}_{\mathrm{sym}}({\ensuremath{\rho}}_{0})$. The implication of these new constraints on the $\ensuremath{\Delta}{r}_{\mathit{np}}$ of $^{208}\mathrm{Pb}$ as well as the core-crust transition density and pressure in neutron stars is discussed.

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 ( $ \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 $ \alpha^{2}_{}$ law, etc., are examined. The symmetry energy is reduced by about 7MeV when we use $ \tilde{{v}}_{{14}}^{}$ instead of its non-relativistic version, i.e. the AV14interaction. The results are compared with other many-body calculations.

Isospin- and momentum-dependent effective interactions for the baryon octet and the properties of hybrid stars

The isospin- and momentum-dependent MDI interaction, which has been extensively used in intermediate-energy heavy-ion reactions to study the properties of asymmetric nuclear matter, is extended to include the nucleon-hyperon and hyperon-hyperon interactions by assuming same density, momentum and isospin dependence as for the nucleon-nucleon interaction. The parameters in these interactions are determined from the empirical hyperon single-particle potentials in symmetric nuclear matter at saturation density. The extended MDI interaction is then used to study in the mean-field approximation the equation of state of hypernuclear matter and also the properties of hybrid stars by including the phase transition from the hypernuclear matter to the quark matter at high densities. In particular, the effects of attractive and repulsive $\Sigma$N interactions and different values of symmetry energies on the hybrid star properties are investigated.

EOS and Single Particle Properties of Asymmetric Nuclear Matter

We have investigated the equation of state (EOS) and single particle (s.p.) properties of asymmetric nuclear matter within the framework of the Brueckner-Bethe-Goldstone approach. We have discussed particularly the effect of microscopic three-body forces (TBF). It is shown that the TBF affects significantly the predicted properties of nuclear matter at high densities.

Isospin constraints on the parametric coupling model for nuclear matter

We make use of isospin constraints to study the parametric coupling model and the properties of asymmetric nuclear matter. Besides the usual constraints for nuclear matter---the effective nucleon mass and the incompressibility at saturation density---and the neutron star constraints---maximum mass and radius---we study the properties related to the symmetry energy. These properties constrain the parameters of the model to a small range. We apply our results to study thermodynamic instabilities in the liquid-gas phase transition as well as neutron star configurations.

Neutron star cooling: A challenge to the nuclear mean field

The two recent density-dependent versions of the finite-range M3Y interaction (CDM3Y$n$ and M3Y-P$n$) have been probed against the bulk properties of asymmetric nuclear matter (NM) in the nonrelativistic Hartree Fock (HF) formalism. The same HF study has also been done with the famous Skyrme (SLy4) and Gogny (D1S and D1N) interactions which were well tested in the nuclear structure calculations. Our HF results are compared with those given by other many-body calculations like the Dirac-Brueckner Hartree-Fock approach or ab-initio variational calculation using free nucleon-nucleon interaction, and by both the nonrelativistic and relativistic mean-field studies using different model parameters. Although the two considered density-dependent versions of the M3Y interaction were proven to be quite realistic in the nuclear structure or reaction studies, they give two distinct behaviors of the NM symmetry energy at high densities, like the Asy-soft and Asy-stiff scenarios found earlier with other mean-field interactions. As a consequence, we obtain two different behaviors of the proton fraction in the $\beta$-equilibrium which in turn can imply two drastically different mechanisms for the neutron star cooling. While some preference of the Asy-stiff scenario was found based on predictions of the latest microscopic many-body calculations or empirical NM pressure and isospin diffusion data deduced from heavy-ion collisions, a consistent mean-field description of nuclear structure database is more often given by some Asy-soft type interaction like the Gogny or M3Y-P$n$ ones. Such a dilemma poses an interesting challenge to the modern mean-field approaches.

Isobaric incompressibility of isospin asymmetric nuclear matter

The isospin dependence of the saturation properties of asymmetric nuclear matter, particularly the incompressibility ${K}_{\ensuremath{\infty}}(X)={K}_{\ensuremath{\infty}}+{K}_{\ensuremath{\tau}}{X}^{2}+O({X}^{4})$ at saturation density, is systematically studied using density-dependent M3Y interaction. ${K}_{\ensuremath{\tau}}$ characterizes the isospin dependence of the incompressibility at saturation density ${\ensuremath{\rho}}_{0}$. The approximate expression ${K}_{\mathrm{asy}}\ensuremath{\approx}{K}_{\mathrm{sym}}\ensuremath{-}6L$ is often used for ${K}_{\ensuremath{\tau}}$ where $L$ and ${K}_{\mathrm{sym}}$ represent the slope and curvature parameters of the symmetry energy at ${\ensuremath{\rho}}_{0}$, respectively. It can be expressed accurately as ${K}_{\ensuremath{\tau}}={K}_{\mathrm{sym}}\ensuremath{-}6L\ensuremath{-}({Q}_{0}/{K}_{\ensuremath{\infty}})L$, where ${Q}_{0}$ is the third-order-derivative parameter of symmetric nuclear matter at ${\ensuremath{\rho}}_{0}$. The results of this addendum to [Phys. Rev. C 80, 011305(R) (2009)] indicate that the ${Q}_{0}$ contribution to ${K}_{\ensuremath{\tau}}$ is not insignificant.

Nuclear matter and neutron matter for an improved quark mass density-dependent model with ρ mesons

A new improved quark mass density-dependent model including u, d quarks, σ mesons, ω mesons and ρ mesons is presented. By employing this model, the properties of nuclear matter, neutron matter and neutron star are studied. After introducing the isovector ρ mesons, the isospin effect for asymmetric nuclear matter is considered. Under mean field approximation, we find that our model can describe the properties of asymmetric nuclear matter, in particular the neutron matter, successfully. The results given by the new improved quark mass density-dependent model and by the quark–meson coupling model are compared.

EFFECTS OF THE ω MESON SELF COUPLING ON THE THERMAL PROPERTIES OF ASYMMETRIC NUCLEAR MATTER

Effects of the ω meson self coupling (OMSC) on the thermal properties of asymmetric nuclear matter (ANM) are studied within the framework of relativistic mean field (RMF) model that includes contributions of all possible mixed interactions among meson fields involved up to quartic order. In particular, we study the mechanical and chemical instabilities (spinodal), as well as the liquid-gas phase transition (binodal) at finite temperature. It is found that the onset of spinodal instabilities and the binodal curve are only marginally affected by variation of the OMSC parameter, whereas the binodal curve shows a strong correlation to the symmetry energy. Comparison with other ERMF parameter sets is also performed.

Properties of asymmetric nuclear matter in different approaches

Properties of asymmetric nuclear matter are derived from various many-body approaches. This includes phenomenological ones like the Skyrme Hartree-Fock and relativistic mean field approaches, which are adjusted to fit properties of nuclei, as well as more microscopic attempts like the Brueckner-Hartree-Fock approximation, a self-consistent Greens function method and the so-called ${V}_{\mathrm{low}\ensuremath{-}k}$ approach, which are based on realistic nucleon-nucleon interactions which reproduce the nucleon-nucleon phase shifts. These microscopic approaches are supplemented by a density-dependent contact interaction to achieve the empirical saturation property of symmetric nuclear matter. The predictions of all these approaches are discussed for nuclear matter at high densities in $\ensuremath{\beta}$-equilibrium. Special attention is paid to behavior of the isovector component of the effective mass in neutron-rich matter.

Neutron star properties in density-dependent relativistic Hartree-Fock theory

With the equations of state provided by the newly developed density dependent relativistic Hartree-Fock (DDRHF) theory for hadronic matter, the properties of the static and $\beta$-equilibrium neutron stars without hyperons are studied for the first time, and compared to the predictions of the relativistic mean field (RMF) models and recent observational data. The influences of Fock terms on properties of asymmetric nuclear matter at high densities are discussed in details. Because of the significant contributions from the $\sigma$- and $\omega$-exchange terms to the symmetry energy, large proton fractions in neutron stars are predicted by the DDRHF calculations, which strongly affect the cooling process of the star. The critical mass about 1.45 $M_\odot$, close to the limit 1.5 $M_\odot$ determined by the modern soft X-ray data analysis, is obtained by DDRHF with the effective interactions PKO2 and PKO3 for the occurrence of direct Urca process in neutron stars. The maximum masses of neutron stars given by the DDRHF calculations lie between 2.45 M$_\odot$ and 2.49 M$_\odot$, which are in reasonable agreement with high pulsar mass $2.08 \pm 0.19 M_\odot$ from PSR B1516+02B. It is also found that the mass-radius relations of neutron stars determined by DDRHF are consistent with the observational data from thermal radiation measurement in the isolated neutron star RX J1856, QPOs frequency limits in LMXBs 4U 0614+09 and 4U 1636-536, and redshift determined in LMXBs EXO 0748-676.

Determination of nuclear symmetry energy in the Cornwall-Jackiw-Tomboulis approach

Within the Cornwall-Jackiw-Tomboulis (CJT) approach a general formalism is established for the study of asymmetric nuclear matter (ANM) described by the four-nucleon interactions. Restricting ourselves to the double-bubble approximation (DBA), we determine the bulk properties of ANM, in particular, the density dependence of the nuclear symmetry energy, which is in good agreement with data of recent analyses.

Effects of isospin and momentum dependent interactions on thermal properties of asymmetric nuclear matter

Thermal properties of asymmetric nuclear matter are studied within a self-consistent thermal model using an isospin and momentum-dependent interaction (MDI) constrained by the isospin diffusion data in heavy-ion collisions, a momentum-independent interaction (MID), and an isoscalar momentum-dependent interaction (eMDYI). In particular, we study the temperature dependence of the isospin-dependent bulk and single-particle properties, the mechanical and chemical instabilities, and liquid-gas phase transition in hot asymmetric nuclear matter. Our results indicate that the temperature dependence of the equation of state and the symmetry energy are not so sensitive to the momentum dependence of the interaction. The symmetry energy at fixed density is found to generally decrease with temperature and for the MDI interaction the decrement is essentially due to the potential part. It is further shown that only the low momentum part of the single-particle potential and the nucleon effective mass increases significantly with temperature for the momentum-dependent interactions. For the MDI interaction, the low momentum part of the symmetry potential is significantly reduced with increasing temperature. For the mechanical and chemical instabilities as well as the liquid-gas phase transition in hot asymmetric nuclear matter, our results indicate that the boundaries of these instabilities and the phase-coexistence region generally shrink with increasing temperature and are sensitive to the density dependence of the symmetry energy and the isospin and momentum dependence of the nuclear interaction, especially at higher temperatures.

Isospin-dependent properties of asymmetric nuclear matter in relativistic mean field models

Using various relativistic mean-field models, including nonlinear ones with meson field self-interactions, models with density-dependent meson-nucleon couplings, and point-coupling models without meson fields, we have studied the isospin-dependent bulk and single-particle properties of asymmetric nuclear matter. In particular, we have determined the density dependence of nuclear symmetry energy from these different relativistic mean-field models and compared the results with the constraints recently extracted from analyses of experimental data on isospin diffusion and isotopic scaling in intermediate energy heavy-ion collisions as well as from measured isotopic dependence of the giant monopole resonances in even-$A$ Sn isotopes. Among the 23 parameter sets in the relativistic mean-field model that are commonly used for nuclear structure studies, only a few are found to give symmetry energies that are consistent with the empirical constraints. We have also studied the nuclear symmetry potential and the isospin splitting of the nucleon effective mass in isospin asymmetric nuclear matter. We find that both the momentum dependence of the nuclear symmetry potential at fixed baryon density and the isospin splitting of the nucleon effective mass in neutron-rich nuclear matter depend not only on the nuclear interactions but also on the definition of the nucleon optical potential.

Equation of state for neutron-rich matter with self-consistent Green function approach

The properties of asymmetric nuclear matter have been investigated in the framework of the self-consistent Green function approach at zero temperature. Results of the total energy per nucleon as a function of the density and asymmetry parameter are presented for two modern realistic nucleon–nucleon interactions: Nijmegen II and CD-Bonn. For comparison purposes, the same calculations are performed for Brueckner–Hartree–Fock approximation. Also we have compared our results with the predictions of other theoretical models especially the Dirac–Brueckner–Hartree–Fock approach. The self-consistent Green's function approach leads to a stiffer equation of state as compared to the Brueckner–Hartree–Fock approximation. This effect increases with density.

Relativistic mean field model based on realistic nuclear forces

In order to predict properties of asymmetric nuclear matter, we construct a relativistic mean field (RMF) model consisting of one-meson exchange (OME) terms and point coupling (PC) terms. In order to determine the density dependent parameters of this model, we use properties of isospin symmetric nuclear matter in combination with the information on nucleon-nucleon scattering data, which are given in the form of the density dependent G-matrix derived from Brueckner calculations based on the Tamagaki potential. We show that the medium- and long-range components of this G-matrix can be described reasonably well by our effective OME interaction. In order to take into account the short-range part of the nucleon-nucleon interaction, which cannot be described well in this manner, a point coupling term is added. Its analytical form is taken from a model based on chiral perturbation theory. It contains only one additional parameter, which does not depend on the density. It is, together with the parameters of the OME potentials adjusted to the equation of state of symmetric nuclear matter. We apply this model for the investigation of asymmetric nuclear matter and find that the results for the symmetry energy as well as for the equation of state of pure neutron matter are in good agreement with either experimental data or with presently adopted theoretical predictions. In order to test the model at higher density, we use its equation of state for an investigation of properties of neutron stars.

Momentum, density, and isospin dependence of symmetric and asymmetric nuclear matter properties

Properties of symmetric and asymmetric nuclear matter have been investigated in the relativistic Dirac-Brueckner-Hartree-Fock approach based on projection techniques using the Bonn A potential. The momentum, density, and isospin dependence of the optical potentials and nucleon effective masses are studied. It turns out that the isovector optical potential depends sensitively on density and momentum, but is almost insensitive to the isospin asymmetry. Furthermore, the Dirac mass ${m}_{D}^{*}$ and the nonrelativistic mass ${m}_{\mathit{NR}}^{*}$ which parametrizes the energy dependence of the single particle spectrum, are both determined from relativistic Dirac-Brueckner-Hartree-Fock calculations. The nonrelativistic mass shows a characteristic peak structure at momenta slightly above the Fermi momentum ${k}_{F}$. The relativistic Dirac mass shows a proton-neutron mass splitting of ${m}_{D,n}^{*}<{m}_{D,p}^{*}$ in isospin asymmetric nuclear matter. However, the nonrelativistic mass has a reversed mass splitting ${m}_{\mathit{NR},n}^{*}>{m}_{\mathit{NR},p}^{*}$ which is in agreement with the results from nonrelativistic calculations.