Isoscalar giant resonances in ${}^{28}\mathrm{Si}$ and the mass dependence of nuclear compressibility

The giant resonance region from $8\mathrm{MeV}<E_x<55\mathrm{}\mathrm{MeV}$ in ${}^{28}\mathrm{Si}$ has been studied with inelastic scattering of 240 MeV α particles at small angles including 0°. Strength corresponding to $81\pm 10\%,$ $68\pm 9\%,$ and $15\pm 4\%$ of the isoscalar $E0,$ $E2,$ and $E1\mathrm{}$ sum rules, respectively, was identified with centroids of $21.25\pm 0.38\mathrm{MeV},$ $18.54\pm 0.25\mathrm{MeV},$ $19.15\pm 0.60\mathrm{MeV},$ and rms widths of $6.4\pm 0.6\mathrm{MeV},$ $4.7\pm 0.6\mathrm{MeV},$ and $6.9\pm 0.7\mathrm{MeV}.$ The mass dependence of the compression modulus of finite nuclei is shown to be reasonably well reproduced from $A=24\mathrm{}$ to 208 in relativistic mean field calculations with the NLC interaction having $K_{\mathrm{nm}}=225\mathrm{}\mathrm{MeV}$ and in nonrelativistic calculations with the RATP interaction having $K_{\mathrm{nm}}=240\mathrm{}\mathrm{MeV}.$