Evolution of the pygmy dipole resonance in nuclei with neutron excess

The electric dipole excitation of various nuclei is calculated with a Random Phase Approximation phenomenological approach. The evolution of the strength distribution in various groups of isotopes of oxygen, calcium, zirconium, and tin is studied. The neutron excess produces $E1$ strength in the low-energy region. Indexes to measure the collectivity of the excitation are defined. We studied the behavior of proton and neutron transition densities to determine the isoscalar or isovector nature of the excitation. We observed that in medium-heavy nuclei the low-energy $E1$ excitation has characteristics rather different than those exhibited by the giant dipole resonance. This new type of excitation can be identified as a pygmy dipole resonance.

Figure 1

Transition densities for the $3^{-}$ low-lying states of the ${}^{16}\mathrm{O}$, ${}^{40}\mathrm{Ca}$, ${}^{132}\mathrm{Sn}$, and ${}^{208}\mathrm{Pb}$ nuclei. The full lines indicates the proton transition densities, and the dashed lines the neutron transition densities. Here, and in the following figures, we drop the dependence on ω with respect to the definition [Eq. (5)], since each transition density is calculated for a specific value of the excitation energy.

Figure 2

Dipole results for the ${}^{208}\mathrm{Pb}$ nucleus. In panel (a) we show the $B(E1)$ values as a function of the excitation energy. In the other panels, we present the transition densities for the states indicated by the arrows, whose excitation energies are given in the panels. The meaning of the lines in panels (b), (c), and (d) is the same as in Fig. 1.

Figure 3

$B(E1)$ distributions for the oxygen isotopes we have studied. The collectivity indexes of the states indicated by the arrows are given in Table .

Figure 4

Transition densities of some $1^{-}$ states for various oxygen isotopes. The numbers in the panels indicate the excitation energy. Full and dashes lines represent proton and neutron densities, respectively.

Figure 5

Same as in Fig. 3, but for the calcium isotopes we have studied. The collectivity indexes of the states indicated by arrows are given in Table . The meaning of the dashed lines is explained in Fig. 11 and in the related text.

Figure 6

Same as in Fig. 4, but for various calcium isotopes.

Figure 7

Same as in Fig. 3, but for the zirconium isotopes we have studied. The collectivity indexes of the states indicated by arrows are given in Table .

Figure 8

Same as in Fig. 4, but for various zirconium isotopes.

Figure 9

Same as in Fig. 3, but for the tin isotopes we have studied. The collectivity indexes of the states indicated by arrows are given in Table .

Figure 10

Same as in Fig. 4, but for the tin isotopes we have studied.

Figure 11

Ratios ${\mathcal{R}}_{\mathrm{int}}$ between the integrated $B(E1)$ values of the PDR and the global $B(E1)$ strength against the number of neutrons in excess with respect to the doubly magic core of the (a) Ca, (b) Zr, and (c) Sn isotopic chains. The $B(E1)$ values of the PDRs have been obtained as a sum of all the values below the dashed lines indicated in Figs. 5, 7, and 9.

Figure 12

(a) The ratio ${\mathcal{R}}_{\mathrm{int}}$ between the integrated $B(E1)$ values of the PDR and the global $B(E1)$ strength against the difference between the proton ($R_p$) and neutron ($R_n$) root-mean-square radii for, from left to right, $A=108,114,116,128$, and 132 Sn isotopes. (b) The difference $R_n-R_p$ as a function of the mass number $A$.