Microscopic determination of the nuclear incompressibility within the nonrelativistic 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 behavior 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 ${}^{208}\mathrm{Pb}$, we reach 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 $K_{\mathit{sym}}$; (ii) Skyrme forces of the type of SLy4 predict $K_{\infty }$ around $230\mathrm{MeV}$, in agreement with the Gogny force (previous estimates using Skyrme interactions having been plagued by a lack of full self-consistency); (iii) it is possible to build forces which predict $K_{\infty }$ around $250\mathrm{MeV}$, although part of this increase is due to our poor knowledge of the density dependence and effective mass.

Figure 1

Density dependence of the symmetry energy for one of the set of forces ($\alpha =1∕6$ and $K_{\infty }=240\mathrm{MeV}$). The symmetry energy $J$ is varied from $26$ to $40\mathrm{MeV}$.

Figure 2

Difference between experimental binding energies (left) or experimental charge radii (right) with the predictions of forces characterized by $\alpha =1∕6$ and by different values of $K_{\infty }$ and $J$, for typical spherical nuclei. Note that the binding energies of ${}^{40,48}\mathrm{Ca}$, ${}^{56}\mathrm{Ni}$, ${}^{132}\mathrm{Sn}$, and ${}^{208}\mathrm{Pb}$, as well as the charge radii of ${}^{40,48}\mathrm{Ca}$, ${}^{56}\mathrm{Ni}$, and ${}^{208}\mathrm{Pb}$ are used in the fit of the force parameters.

Figure 3

Same as Fig. 2 in the case of the forces with $\alpha =0.3563$.

Figure 4

The ${}^{208}\mathrm{Pb}$ ISGMR centroid energy $E_{-1}$ calculated with the Skyrme parameter sets with $\alpha =1∕6$, as a function of $J$. The different symbols correspond to the values of $K_{\infty }$ (see inset). Lines are numerical fits and are simply intended to guide the eye. The area delimited by the two horizontal lines correspond to the experimental value.

Figure 5

Same as Fig. 4, for the Skyrme parameter sets with $\alpha =0.3563$.