Constraining the Nuclear Equation of State at Subsaturation Densities

Only one-third of the nucleons in ${}^{208}\mathrm{Pb}$ occupy the saturation density area. Consequently, nuclear observables related to the average properties of nuclei, such as masses or radii, constrain the equation of state not at the saturation density but rather around the so-called crossing density, localized close to the mean value of the density of nuclei: $\rho \simeq 0.11{\mathrm{fm}}^{-3}$. This provides an explanation for the empirical fact that several equation of state quantities calculated with various functionals cross at a density significantly lower than the saturation one. The third derivative $M$ of the energy per unit of volume at the crossing density is constrained by the giant monopole resonance measurements in an isotopic chain rather than the incompressibility at saturation density. The giant monopole resonance measurements provide $M=1100\pm 70\mathrm{MeV}$ (6% uncertainty), whose extrapolation gives $K_{\infty }=230\pm 40\mathrm{MeV}$ (17% uncertainty).

Figure 1

Matter density of ${}^{208}\mathrm{Pb}$ calculated with the HF approach using the SLy4 functional with different X axis scales. Top: representation taking into account the nucleons’ distribution in the nucleus. Bottom: usual representation.

Figure 2

EOS incompressibility calculated with various functionals, showing the crossing density around $0.7{\rho }_0\simeq 0.11{\mathrm{fm}}^{-3}$.

Figure 3

Centroid of the GMR in ${}^{208}\mathrm{Pb}$ and ${}^{120}\mathrm{Sn}$ calculated with the CHF method vs the value of $M$ for various functionals. The experimental values for ${}^{208}\mathrm{Pb}$ and ${}^{120}\mathrm{Sn}$ are taken from Refs. 21, 27, respectively, with respective error bars of $\pm 200$ and $\pm 100\mathrm{keV}$.

Figure 4

Positive or negative relative contribution of the kinetic (circles), central (squares), finite range (diamonds) and density (triangles) terms to the equation of state (a), its first (b), second (c) and third (d) derivatives at the crossing density, calculated for 36 Skyrme functionals.