On the collectivity of Pygmy Dipole Resonance within schematic TDA and RPA models

Within schematic models based on the Tamm-Dancoff Approximation and the Random-Phase Approximation with separable interactions, we investigate the physical conditions which determine the emergence of the Pygmy Dipole Resonance in the E1 response of atomic nuclei. We find that if some particle-hole excitation manifests a different, weaker residual interaction, an additional mode will appear, with an energy centroid closer to the distance between two major shells and therefore well below the Giant Dipole Resonance. This state, together with Giant Dipole Resonance, exhausts all the transition strength in the Tamm-Dancoff Approximation and all the Energy Weighted Sum Rule in the Random-Phase Approximation. Thus, within our scheme, this mode, which could be associated with the Pygmy Dipole Resonance, is of collective nature. By relating the coupling constants appearing in the separable interaction to the symmetry energy value at and below saturation density we explore the role of the density dependence of the symmetry energy on the low energy dipole response.