The breathing-mode giant monopole resonance and the surface compressibility in the relativistic mean-field theory

The breathing-mode isoscalar giant monopole resonance (GMR) is investigated using the generator coordinate method within the relativistic mean-field (RMF) theory. Employing the Lagrangian models of the nonlinear-$\sigma$ model (NL$\sigma$), the scalar-vector interaction model (SVI) and the $\sigma$-$\omega$ coupling model (SIGO), we show that each Lagrangian model exhibits a distinctly different GMR response. Consequently, Lagrangian models yield a different value of the GMR energy for a given value of the nuclear matter incompressibility $K_\infty$. It is shown that this effect arises largely from a different value of the surface incompressibility $K_{surf}$ inherent to each Lagrangian model, thus giving rise to the ratio $K_{surf}/K_\infty$ which depends upon the Lagrangian model used. This is attributed to a difference in the density dependence of the meson masses and hence to the density dependence of the nuclear interaction amongst various Lagrangian models. The sensitivity of the GMR energy to the Lagrangian model used and thus emergence of a multitude of GMR energies for a given value of $K_\infty$ renders the method of extracting $K_\infty$ on the basis of interpolation amongst forces as inappropriate. As a remedy, the need to 'calibrate' the density dependence of the nuclear interaction in the RMF theory is proposed.