Density dependence of the symmetry energy and the nuclear equation of state : A dynamical and statistical model perspective

The density dependence of the symmetry energy is a key unknown in the equation of state of isospin asymmetric nuclear matter. Due to its direct relevance to the structure and stability of systems as diverse as the neutron stars and neutron-rich nuclei, there is a significant interest in determining this quantity. Theoretical studies based on microscopic “ab-initio” calculations predict a variety of different forms of the density dependence of the symmetry energy. Experimental measurements on the density dependence of the symmetry energy are still scarce and difficult. In the past, we have tried to study the density dependence of the symmetry energy using the fragment isotope yields in multifragmentation reactions compared to the dynamical model Antisymmetrized Molecular Dynamic (AMD). We have now investigated the same using the statistical model approach of multifragmentation reaction. We report here a comparison between the two approaches along with several other independent studies to obtain a constraint on the form of the density dependence of the symmetry energy. We also report several predictions that follow from the constraint. Fig. 1 shows the density dependence of the symmetry energy obtained from the statistical model approach using the experimentally measured isoscaling parameter (red and green circle symbols) [1]. The green solid curve corresponds to the density dependence of the symmetry energy obtained from the Gogny-AS interaction in our previous analysis using the dynamical AMD approach [2,3], assuming the sequential decay effect to be small. The red dashed curve corresponds to the one obtained from an accurately calibrated relativistic mean field interaction, used for describing the Giant Monopole Resonance (GMR) in Zr and Pb, and the IVGDR in Pb by Piekarewicz et al. [4]. The light-blue dashed curve correspond to the one used to explain the isospin diffusion results of NSCL-MSU using the isospin dependent Boltzmann-Uehling-Uhlenbeck (IBUU) model by Tsang et al. [5]. The blue dotdashed curve also corresponds to the one used for explaining the isospin diffusion data of NSCL-MSU by Chen et al. [6], but with the momentum dependence of the interaction included in the IBUU calculation. This dependence has been further modified to include the isospin dependence of the in-medium nucleonnucleon cross-section by Li et al. [7], and is in good agreement with the present study. The shaded region in the figure corresponds to that obtained by constraining the binding energy, neutron skin thickness and isospin analogue state in finite nuclei using the mass formula of Danielewicz [8]. The yellow solid curve corresponds to the parameterization adopted by Heiselberg et al. [9] in their studies on neutron stars. By fitting earlier predictions of the variational calculations by Akmal et al. [10], where the many-body and special relativistic corrections are progressively incorporated, Heiselberg and Hjorth-Jensen obtained a value of Csym = 32 MeV and γ = 0.6, similar to those obtained from the present measurements. A similar result is also obtained from the relativistic Dirac-Brueckner calculation, with Csym = 32.9 MeV and γ = 0.59 [11]. The Dirac-Brueckner is an “ab-initio" calculation based on nucleon-nucleon interaction with Bonn A type potential instead of the AV18 potential used in the variational calculation of Ref. [10]. The density dependence of the symmetry energy has also been studied in the framework of an expanding