Application of theSp(2,R)model to the nuclear breathing mode

We apply the symplectic shell model to the breathing mode in closed-shell nuclei. The group theory of $\mathrm{Sp}\mathrm{}(2,R)\mathrm{}$ is used to embed the description of the collective monopole excitation in the framework of the oscillator shell model. This provides a microscopic description of the breathing mode by way of a diagonalization of the effective interaction in an appropriate collective subspace. We apply this model to closed-shell nuclei ranging from ${}_{}{}^{56}{}_{}{}^{}\mathrm{Ni}$ to ${}_{}{}^{208}{}_{}{}^{}\mathrm{Pb}$ using various Skyrme-type forces. The results confirm that a force with a nuclear matter incompressibility of about 200 MeV reproduces the experimental breathing mode energies. An evaluation of the calculated wave functions using the formalism of the $\mathrm{Sp}\mathrm{}(2,R)\mathrm{}$ boson picture indicates the importance of two-boson correlations that particularly affect the compression modulus. As an element in our analysis we compare $\mathrm{Sp}\mathrm{}(2,R)\mathrm{}$ model and random phase approximation results. This reveals the strengths and limitations of the symplectic model.