Incompressibility in finite nuclei and nuclear matter

The incompressibility (compression modulus) $K_{\rm 0}$ of infinite symmetric nuclear matter at saturation density has become one of the major constraints on mean-field models of nuclear many-body systems as well as of models of high density matter in astrophysical objects and heavy-ion collisions. We present a comprehensive re-analysis of recent data on GMR energies in even-even $^{\rm 112-124}$Sn and $^{\rm 106,100-116}$Cd and earlier data on 58 $\le$ A $\le$ 208 nuclei. The incompressibility of finite nuclei $K_{\rm A}$ is expressed as a leptodermous expansion with volume, surface, isospin and Coulomb coefficients $K_{\rm vol}$, $K_{\rm surf}$, $K_\tau$ and $K_{\rm coul}$. \textit{Assuming} that the volume coefficient $K_{\rm vol}$ is identified with $K_{\rm 0}$, the $K_{\rm coul}$ = -(5.2 $\pm$ 0.7) MeV and the contribution from the curvature term K$_{\rm curv}$A$^{\rm -2/3}$ in the expansion is neglected, compelling evidence is found for $K_{\rm 0}$ to be in the range 250 $ < K_{\rm 0} < $ 315 MeV, the ratio of the surface and volume coefficients $c = K_{\rm surf}/K_{\rm vol}$ to be between -2.4 and -1.6 and $K_{\rm \tau}$ between -840 and -350 MeV. We show that the generally accepted value of $K_{\rm 0}$ = (240 $\pm$ 20) MeV can be obtained from the fits provided $c \sim$ -1, as predicted by the majority of mean-field models. However, the fits are significantly improved if $c$ is allowed to vary, leading to a range of $K_{\rm 0}$, extended to higher values. A self-consistent simple (toy) model has been developed, which shows that the density dependence of the surface diffuseness of a vibrating nucleus plays a major role in determination of the ratio K$_{\rm surf}/K_{\rm vol}$ and yields predictions consistent with our findings.