Isoscalar giant resonances in the Sn nuclei and implications for the asymmetry term in the nuclear-matter incompressibility

We have investigated the isoscalar giant resonances in the Sn isotopes using inelastic scattering of $386$-MeV $\ensuremath{\alpha}$ particles at extremely forward angles, including ${0}^{\ifmmode^\circ\else\textdegree\fi{}}$. We have obtained completely ``background-free'' inelastic-scattering spectra for the Sn isotopes over the angular range ${0}^{\ifmmode^\circ\else\textdegree\fi{}}$--${9}^{\ifmmode^\circ\else\textdegree\fi{}}$ and up to an excitation energy of $31.5$ MeV. The strength distributions for various multipoles were extracted by a multipole decomposition analysis based on the expected angular distributions of the respective multipoles. We find that the centroid energies of the isoscalar giant monopole resonance (ISGMR) in the Sn isotopes are significantly lower than the theoretical predictions. In addition, based on the ISGMR results, a value of ${K}_{\ensuremath{\tau}}=\ensuremath{-}550\ifmmode\pm\else\textpm\fi{}100$ MeV is obtained for the asymmetry term in the nuclear incompressibility. Constraints on interactions employed in nuclear structure calculations are discussed on the basis of the experimentally obtained values for ${K}_{\ensuremath{\infty}}$ and ${K}_{\ensuremath{\tau}}$.