Trajectory in 2D plot of isoscalar and isovector densities of Ca48 and Pb208 , and symmetry energy

Abstract

We propose a new two-dimensional (2D) plotting method of isoscalar-isovector (IS-IV) densities of $^{48}\mathrm{Ca}$ and $^{208}\mathrm{Pb}$ to extract not only empirical symmetry energy coefficients $J, L$, and ${K}_{\mathrm{sym}}$ but also the asymmetric isospin term ${K}_{\ensuremath{\tau}}$ in the nuclear incompressibility. We adopted several modern relativistic and nonrelativistic energy density functional (EDF) for the theoretical calculations of IS and IV densities to compare with experimental neutron and proton density distributions determined by recent proton elastic scatterings. We found very strong correlations between the trajectory of the 2D density at the density $\ensuremath{\rho}=0.1\phantom{\rule{4pt}{0ex}}{\mathrm{fm}}^{\ensuremath{-}3}$ and the symmetry energy coefficients $J, L$, and also ${K}_{\ensuremath{\tau}}$. For ${K}_{\mathrm{sym}}$, the Skyrme EDF SAMi-J family shows a strong correlation in the trajectory of 2D density, while the RMF Lagrangian DDME-J family shows a peculiar quadratic curve and does not provide the experimental value. The symmetry energy term ${K}_{\ensuremath{\tau}}$ of nuclear incompressibility is also discussed in term of the slope of 2D trajectory.