Incompressibility in finite nuclei and nuclear matter

The incompressibility (compression modulus) ${K}_{0}$ 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. It is usually extracted from data on the giant monopole resonance (GMR) or calculated using theoretical models. We present a comprehensive reanalysis of recent data on GMR energies in even-even ${}^{112--124}$Sn and ${}^{106,100--116}$Cd and earlier data on $58\ensuremath{\le}A\ensuremath{\le}208$ nuclei. The incompressibility of finite nuclei ${K}_{A}$ is calculated from experimental GMR energies and expressed in terms of ${A}^{\ensuremath{-}1/3}$ and the asymmetry parameter $\ensuremath{\beta}=(N\ensuremath{-}Z)/A$ as a leptodermous expansion with volume, surface, isospin, and Coulomb coefficients ${K}_{\mathrm{vol}}$, ${K}_{\mathrm{surf}}$, ${K}_{\ensuremath{\tau}}$, and ${K}_{\mathrm{Coul}}$. Only data consistent with the scaling approximation, leading to a fast converging leptodermous expansion, with negligible higher-order-term contributions to ${K}_{A}$, were used in the present analysis. Assuming that the volume coefficient ${K}_{\mathrm{vol}}$ is identified with ${K}_{0}$, the ${K}_{\mathrm{Coul}}=\ensuremath{-}(5.2\ifmmode\pm\else\textpm\fi{}0.7)$ MeV and the contribution from the curvature term ${K}_{\mathrm{curv}}{A}^{\ensuremath{-}2/3}$ in the expansion is neglected, compelling evidence is found for ${K}_{0}$ to be in the range 250 $<{K}_{0}<$ 315 MeV, the ratio of the surface and volume coefficients $c={K}_{\mathrm{surf}}/{K}_{\mathrm{vol}}$ to be between $\ensuremath{-}2.4$ and $\ensuremath{-}1.6$ and ${K}_{\ensuremath{\tau}}$ between $\ensuremath{-}840$ and $\ensuremath{-}350$ MeV. In addition, estimation of the volume and surface parts of the isospin coefficient ${K}_{\ensuremath{\tau}}$, ${K}_{\ensuremath{\tau},v}$, and ${K}_{\ensuremath{\tau},s}$, is presented. We show that the generally accepted value of ${K}_{0}$ = (240 $\ifmmode\pm\else\textpm\fi{}$ 20) MeV can be obtained from the fits provided $c\ensuremath{\sim}\ensuremath{-}1$, 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 ${K}_{0}$, extended to higher values. The results demonstrate the importance of nuclear surface properties in determination of ${K}_{0}$ from fits to the leptodermous expansion of ${K}_{A}$. 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 ${K}_{\mathrm{surf}}/{K}_{\mathrm{vol}}$ and yields predictions consistent with our findings.

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