Narrow Results

The equation of state (EOS) is calculated for uniform nuclear matter at zero and finite temperatures with the variational method. Making use of uncertainty of the three-body nuclear force, adjustable parameters in the nuclear EOS are tuned so that the Thomas-Fermi calculations for β-stable nuclei with the EOS reproduce the empirical data. The calculated nuclear properties imply that larger symmetry energy of the EOS is preferable to reproduce the empirical β-stability line. The expectation value of the nuclear Hamiltonian caused by the 2π-exchange three-body nuclear force is uncertain and related to the symmetry energy.

High density behavior of nuclear symmetry energy is studied on the basis of the stiffest density dependence of asymmetric contribution to energy per nucleon in charge neutral n + p + e + $\mu$ matter under beta equilibrium. The density dependence of nuclear symmetry energy obtained in this way is neither very stiff nor soft at high densities and is found to be in conformity with recent observations of neutron stars.

After recalling basic phenomenological features of isospin asymmetric nuclear matter, we review predictions for the interaction part of the symmetry energy obtained from different microscopic approaches. The predictions are compared to updated constraints extracted from heavy-ion (HI) reaction observables of a recent GSI experiment. The discussion is extended to the neutron skin thickness in ${}^{208}\mathrm{\text{Pb}}$ and its relation to the density derivative of the symmetry energy. We underline the importance of giving proper consideration to the theoretical uncertainties of microscopic predictions in order to guide phenomenological analyses. In the end, we report briefly on preliminary neutron star calculations based on chiral nuclear forces and outline future plans.

The analytical expression of the density-dependent binding energy per nucleon for the relativistic mean field (RMF), the so-called relativistic energy density functional (Relativistic-EDF), is used to obtain the isospin-dependent symmetry energy and its components for the isotopic chain of Sc-, Ti-, V- and Cr-nuclei. The procedure of the coherent density fluctuation model is employed to formulate the Relativistic-EDF and Brüeckner energy density functional (Brüeckner-EDF) at local density. A few signatures of shell and/or sub-shell closure are observed in the symmetry energy and its components, i.e. surface and volume symmetry energy, far from the $\beta$-stable region for odd-$A$ Sc- and V-, and even–even Ti- and Cr- nuclei with nonlinear NL3 and G3 parameter sets. A comparison is made with the results obtained from Relativistic-EDF and Brüeckner-EDF with both NL3 and G3 for the considered isotopic chains. We find Relativistic-EDF outperforms the Brüeckner-EDF in predicting the shell and/or sub-shell closure of neutron-rich isotopes at $N=50$ for these atomic nuclei. Moreover, a relative comparison has been made for the results obtained with the nonlinear NL3 and G3 parameter sets.

Probing dense hadronic matter is thus far an uncharted field of physics. Here we give a brief summary of the highlights of what has been so far accomplished and what will be done in the years ahead by the World Class University III Project at Hanyang University in the endeavor to unravel and elucidate the multifacet of the cold dense baryonic matter existing in the interior of the densest visible stable object in the universe, i.e. neutron stars, strangeness stars and/or quark stars, from a modest and simplified starting point of an effective field theory modeled on the premise of QCD as well as from a gravity dual approach of hQCD. The core of the matter of our research is the possible origin of the ~ 99% of the proton mass that is to be accounted for and how the "vacuum" can be tweaked so that the source of the mass generation can be uncovered by measurements made in terrestrial as well as space laboratories. Some of the issues treated in the program concern what can be done — both theoretically and experimentally — in anticipation of what's to come for basic physics research in Korea.

In this work we discuss points of interest for a general phase diagram of nuclear matter at alternative conditions (such as isospin broken phase and existence of anti-matter state(s)) which are relevant for the description of astrophysics. The inclusion of light isovector spin zero mesons in relativistic models of nuclear systems is discussed and deviation from the usual results for the symmetry energy description is found in agreement with a development done for a non-relativistic description of nuclear matter.

In this preliminary study we select a set of six Skyrme models which present reasonable symmetry energies lying in the range of 28–35 MeV to analyze the behavior of several other bulk properties at zero temperature, as well as the critical temperature parameters. The models are also investigated to see whether they satisfy a stringent constraint recently proposed from heavy-ion experiments.

Symmetry energy coefficients of explicitly isospin asymmetric nuclear matter at variable densities (from 0.5ρ₀ up to 2ρ₀) are studied as generalized screening functions. An extended stability condition for asymmetric nuclear matter is proposed. We find the possibility of obtaining stable asymmetric nuclear matter even in some cases for which the symmetric nuclear matter limit is unstable. Skyrme-type forces are extensively used in analytical expressions of the symmetry energy coefficients derived as generalized screening functions in the four channels of the particle hole interaction producing alternative behaviors at different ρ and b (respectively, the density and the asymmetry coefficient). The spin and spin-isospin coefficients, with corrections to the usual Landau Migdal parameters, indicate the possibility of occurring instabilities with common features depending on the nuclear density and n–p asymmetry. Possible relevance for high energy heavy ions collisions and astrophysical objects is discussed.

Using a generalized mean field theory, we have studied relations among effective meson masses, effective meson–nucleon couplings and equations of state (EOS) in asymmetric nuclear matter. If the effective ω-meson mass becomes smaller at high density, the EOS becomes stiffer. However, if we require that the ω-meson mean field is proportional to baryon density, the effective ω–nucleon coupling also becomes smaller at the same time as the effective ω-meson mass becomes smaller. Consequently, the EOS becomes softer. A similar relation is found for the effective ρ-meson mass and the effective ρ–nucleon coupling. We have also studied relations among effective meson masses, effective meson–nucleon couplings and a radius R of a neutron star. R depends somewhat on the value of the effective ω-meson mass and the effective ω–nucleon coupling.

Heavy Ion Collisions (HIC) represent a unique tool to probe the in-medium nuclear interaction in regions away from saturation. In this work we present a selection of reaction observables in dissipative collisions particularly sensitive to the isovector part of the interaction, i.e.to the symmetry term of the nuclear Equation of State (EoS). At low energies the behavior of the symmetry energy around saturation influences dissipation and fragment production mechanisms. We will first discuss the recently observed Dynamical Dipole Radiation, due to a collective neutron-proton oscillation during the charge equilibration in fusion and deep-inelastic collisions. Important Iso - EOS are stressed. Reactions induced by unstable ¹³²Sn beams appear to be very promising tools to test the sub-saturation Isovector EoS. New Isospin sensitive observables are also presented for deep-inelastic, fragmentation collisions and Isospin equilibration measurements (Imbalance Ratios).

The high density symmetry term can be derived from isospin effects on heavy ion reactions at relativistic energies (few AGeV range), that can even allow a "direct" study of the covariant structure of the isovector interaction in the hadron medium. Rather sensitive observables are proposed from collective flows and from pion/kaon production. The possibility of the transition to a mixed hadron-quark phase, at high baryon and isospin density, is finally suggested. Some signatures could come from an expected "neutron trapping" effect. The importance of studying violent collisions with radioactive beams from low to relativistic energies is finally stressed.

We review a variety of theoretical and experimental investigations aimed at improving our knowledge of the nuclear matter equation of state. Of particular interest are nuclear matter extreme states in terms of density and/or isospin asymmetry. The equation of state of matter with unequal concentrations of protons and neutrons has numerous applications. These include heavy-ion collisions, the physics of rare, short-lived nuclei and, on a dramatically different scale, the physics of neutron stars. The "common denominator" among these (seemingly) very different systems is the symmetry energy, which plays a crucial role in both the formation of the neutron skin in neutron-rich nuclei and the radius of a neutron star (a system 18 orders of magnitude larger and 55 orders of magnitudes heavier). The details of the density dependence of the symmetry energy are not yet sufficiently constrained. Throughout this article, our emphasis will be on the importance of adopting a microscopic approach to the many-body problem, which we believe to be the one with true predictive power.

Within the framework of the improved isospin dependent quantum molecular dynamics (ImIQMD) model, the emission of pion in heavy-ion collisions in the region 1 A GeV as a probe of nuclear symmetry energy at supra-saturation densities is investigated systematically, in which the pion is considered to be mainly produced by the decay of resonances Δ(1232) and N*(1440). The π^-/π⁺ yields are calculated for selected Skyrme parameters SkP, SLy6, Ska and SIII, and also for the cases of different stiffness of symmetry energy with the parameter SLy6. Preliminary results compared with the measured data by the FOPI collaboration favor a hard symmetry energy of the potential term proportional to (ρ/ρ₀)^γs with γ_s = 2.

In this work, we review a few structural properties in finite nuclei and nuclear matter that are sensitive to differences in the symmetry energy, and discuss mechanisms that can enhance the sensitivity to differences in the symmetry energy with the relativistic mean-field model. Emphasis has been placed on the establishment of the relationship between the deexcitation energy of superdeformed secondary minima and the density dependence of the symmetry energy.

The status of relativistic nuclear many-body calculations of nuclear systems to be built up in terms of protons and neutrons is reviewed. In detail, relativistic effects on several aspects of nuclear matter such as the effective mass, saturation mechanism, and the symmetry energy are considered. This review will especially focus on isospin asymmetric issues, since these aspects are of high interest in astrophysical and nuclear structure studies. Furthermore, from the experimental side these aspects are experiencing an additional boost from a new generation of radioactive beam facilities, e.g., the future GSI facility FAIR in Germany or SPIRAL2 at GANIL/France. Finally, the prospects of studying finite nuclei in microscopic calculations which are based on realistic NN interactions by including relativistic effects in calculations of low momentum interactions are discussed.

The elliptic flow in collisions of neutron-rich heavy-ion systems at intermediate energies emerges as an observable sensitive to the strength of the symmetry energy at supra-saturation densities. First results obtained by comparing ratios or differences of neutron and hydrogen flows with predictions of transport models favor a moderately soft to linear density dependence, consistent with ab initio nuclear-matter theories. Comprehensive data sets of high accuracy can be expected to improve our knowledge of the equation of state of asymmetric nuclear matter.

We analyze theoretically the neutron skin thickness in nuclei and its correlation with the symmetry energy by using semiclassical and mean field approaches together with nuclear effective interactions. Semiclassical approaches reveal that the neutron skin thickness in nuclei is formed by a combination of bulk and surface contributions. To investigate the neutron skin thickness predicted by mean field models, we fit the corresponding densities by two-parameter Fermi distributions. Using these parametrized densities, we study the neutron skin thickness as well as its bulk and surface contributions in ²⁰⁸Pb and in Zr isotopes, where the influence of shell effects along the isotopic chain is discussed.

A phenomenological momentum dependent interaction (MDI) is considered to describe the equation of state (EOS) for isospin asymmetric nuclear matter (ANM), where the density dependence of the nuclear symmetry is the basic input. In this interaction, the symmetry energy shows soft dependence of density. Within the nonrelativistic mean field approach we calculate the nuclear matter fourth-order symmetry energy E_{sym, 4}(ρ). Our result shows that the value of E_{sym, 4}(ρ) at normal nuclear matter density ρ₀( = 0.161 fm^-3) is less than 1 MeV conforming the empirical parabolic approximation to the EOS of ANM at ρ₀. Then the higher-order effects of the isospin asymmetry on the saturation density ρsat(β), binding energy per nucleon K_sat(β) and isobaric incompressibility K_sat(β) of ANM is being studied, where is the isospin asymmetry. We have found that the fourth-order isospin asymmetry β cannot be neglected, while calculating these quantities. Hence the second-order K_{sat, 2} parameter basically characterizes the isospin dependence of the incompressibility of ANM at saturation density.

The nuclear symmetry energy figures crucially in the structure of asymmetric nuclei and, more importantly, in the equation of state (EoS) of compact stars. At present it is almost totally unknown, both experimentally and theoretically, in the density regime appropriate for the interior of neutron stars. Basing on a strong-coupled structure of dense baryonic matter encoded in the skyrmion crystal approach with a topology change and resorting to the notion of generalized hidden local symmetry in hadronic interactions, we address a variety of hitherto unexplored issues of nuclear interactions associated with the symmetry energy, i.e., kaon condensation and hyperons, possible topology change in dense matter, nuclear tensor forces, conformal symmetry, chiral symmetry, etc., in the EoS of dense compact-star matter. One of the surprising results coming from HLS structure that is distinct from what is given by standard phenomenological approaches is that at high density, baryonic matter is driven by renormalization group flow to the "dilaton-limit fixed point" constrained by "mended symmetries". We further propose how to formulate kaon condensation and hyperons in compact-star matter in a framework anchored on a single effective Lagrangian by treating hyperons as the Callan–Klebanov kaon-skyrmion bound states simulated on crystal lattice. This formulation suggests that hyperons can figure in the stellar matter — if at all — when or after kaons condense, in contrast to the standard phenomenological approaches where the hyperons appear as the first strangeness degree of freedom in matter, thereby suppressing or delaying kaon condensation. In our simplified description of the stellar structure in terms of symmetry energies, which is compatible with that of the 1.97 solar mass star, kaon condensation plays a role of "doorway state" to strange quark matter.

A major goal of contemporary nuclear physics is to improve our knowledge of the nuclear matter equation of state (EoS). In particular, the EoS of isospin-asymmetric matter (that is, with unequal concentrations of protons and neutrons), is not well understood, mostly due to our limited knowledge of the symmetry energy. The latter reduces the binding energy in a nucleus with unequal number of protons and neutrons, and is crucial for understanding nuclear stability. We will review experimental, phenomenological and theoretical facts about the symmetry energy, with particular emphasis on analyzing how the various components of the nuclear force, in a microscopic model, contribute to this important quantity.

We suggest a particular procedure of derivation of the beta-stability line, the nuclear Coulomb energy and the isotopic symmetry energy. The approach is based on the analysis of the A-dependency of the shift of neutron–proton chemical potentials Δλ = λ_n-λ_p. We observe the nonmonotonic (sawtooth) shape of the β-stability line as a function of mass number A and establish the relation of local maxima of the β-stability line to the mass numbers of the double-closed shells. The behavior of the symmetry energy coefficient b_sym(A) at fixed neutron excess Y = N-Z is analyzed. We show that, qualitatively, b_sym(A) has canyon-like behavior for a given value of Y. The width and the position of the bottom of such "canyon" depend on neutron excess.