Microscopic determination of the nuclear incompressibility within the non-relativistic framework

The nuclear incompressibility $K_\infty$ is deduced from measurements of the Isoscalar Giant Monopole Resonance (ISGMR) in medium-heavy nuclei, and the resulting value turns out to be model dependent. Since the considered nuclei have neutron excess, it has been suggested that the model dependence is due to the different behaviour of the symmetry energy in different models. To clarify this issue, we make a systematic and careful analysis based on new Skyrme forces which span a wide range of values for $K_\infty$, for the value of the symmetry energy at saturation and for its density dependence. By calculating, in a fully self-consistent fashion, the ISGMR centroid energy in $^{208}$Pb we reach, for the first time within the non-relativistic framework, three important conclusions: (i) the monopole energy, and consequently the deduced value of $K_\infty$, depend on a well defined parameter related to the shape of the symmetry energy curve and called $K_{sym}$; (ii) Skyrme forces of the type of SLy4 predict $K_\infty$ around 230 MeV, in agreement with the Gogny force (previous estimates using Skyrme interactions having been plagued by lack of full self-consistency); (iii) it is possible to build forces which predict $K_\infty$ around 250 MeV, although part of this increase is due to our poor knowledge of the density dependence and effective mass.