Collectivity of the pygmy dipole resonance within schematic Tamm-Dancoff approximation and random-phase approximation models

Within schematic models based on the Tamm-Dancoff approximation and the random-phase approximation with separable interactions, we investigate the physical conditions that may determine the emergence of the pygmy dipole resonance in the $E1$ response of atomic nuclei. By introducing a generalization of the Brown-Bolsterli schematic model with a density-dependent particle-hole residual interaction, we find that an additional mode will be affected by the interaction, whose energy centroid is closer to the distance between two major shells and therefore well below the giant dipole resonance (GDR). This state, together with the GDR, 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 density dependence of the symmetry energy on the low-energy dipole response.

Figure 1

The GDR and PDR energy centroids as a function of the ratio ${\lambda }_2/{\lambda }_1$. The black thick lines refer to the TDA while the red lines to RPA calculations. For ${}^{68}\mathrm{Ni}$ [(a) and (b)] the solid lines correspond to $N_e=6$; the dashed lines correspond to $N_e=12$. (b) For ${}^{132}\mathrm{Sn}$ [(c) and (d)] the solid lines correspond to $N_e=12$; the dashed lines correspond to $N_e=32$. In (b) and (d) the horizontal blue line indicates the unperturbed energy value.

Figure 2

(a) The EWSR fraction exhausted by GDR in RPA calculations for ${}^{68}\mathrm{Ni}$. $N_e=6$ (red solid lines) and $N_e=12$ (blue dashed lines). (b) The EWSR fraction exhausted by PDR in RPA calculations for ${}^{68}\mathrm{Ni}$. $N_e=6$ (red solid lines) and $N_e=12$ (blue dashed lines). (c) The EWSR fraction exhausted by GDR in RPA calculations for ${}^{132}\mathrm{Sn}$. $N_e=6$ (orange solid lines) and $N_e=12$ (blue dashed lines). (d) The EWSR fraction exhausted by PDR in RPA calculations for ${}^{132}\mathrm{Sn}$. $N_e=6$ (orange solid lines) and $N_e=12$ (green dashed lines).

Figure 3

The GDR energy centroids as a function of the ratios ${\lambda }_2/{\lambda }_1,{\lambda }_3/{\lambda }_1$ in the case of ${}^{132}\mathrm{Sn}$ and $N_e=13.5,({\lambda }_2>{\lambda }_3)$.

Figure 4

The EWSR fraction ascribed to GDR as a function of the ratios ${\lambda }_2/{\lambda }_1,{\lambda }_3/{\lambda }_1$ in the case of ${}^{132}\mathrm{Sn}$ and $N_e=13.5,({\lambda }_2>{\lambda }_3)$.

Figure 5

The PDR energy centroids as a function of the ratios ${\lambda }_2/{\lambda }_1,{\lambda }_3/{\lambda }_1$ in the case of ${}^{132}\mathrm{Sn}$ and $N_e=13.5,({\lambda }_2>{\lambda }_3)$.

Figure 6

The EWSR fraction ascribed to PDR as a function of the ratios ${\lambda }_2/{\lambda }_1,{\lambda }_3/{\lambda }_1$ in the case of ${}^{132}\mathrm{Sn}$ and $N_e=13.5,({\lambda }_2>{\lambda }_3)$.

Figure 7

The ratio $\lambda \left(\rho \right)/\lambda \left({\rho }_0\right)$ as a function of density for asystiff EOS (black solid lines), asysuperstiff EOS (blue dot-dashed lines) and asysoft EOS (red dashed lines). The inset: the trapezoidal distribution of neutron (black solid line) and proton (black dashed line) densities for ${}^{132}\mathrm{Sn}$ considered in the calculations.