Isoscalar and Isovector Splitting of Pygmy Dipole Structures

The electric-dipole response of ${}^{140}\mathrm{Ce}$ is investigated using the fully consistent relativistic quasiparticle random phase approximation. By analyzing the isospin structure of the $E1$ response, it is shown that the low-energy (pygmy) strength separates into two segments with different isospin character. The more pronounced pygmy structure at lower energy is composed of predominantly isoscalar states with surface-peaked transition densities. At somewhat higher energy the calculated $E1$ strength is primarily of isovector character, as expected for the low-energy tail of the giant dipole resonance. The results are in qualitative agreement with those obtained in recent ($\gamma$, ${\gamma }^{\prime }$) and ($\alpha$, ${\alpha }^{\prime }\gamma$) experiments, and provide a simple explanation for the splitting of low-energy $E1$ strength into two groups of states with different isospin structure and radial dependence of the corresponding transition densities.

Figure 1

The total $\mathrm{RHB}+\mathrm{RQRPA}$ $E1$ transition strength for ${}^{140}\mathrm{Ce}$ (solid curve). The dotted, dashed, and dash-dotted curves connect states with predominant (at least 70%, 80%, and 90%, respectively) isoscalar (IS) components, identified by analyzing the corresponding proton and neutron transition densities over the radial interval [0, 8] fm. The vertical arrow denotes the one-neutron separation energy $S_n=9.2\mathrm{MeV}$.

Figure 2

Same as in Fig. 1, but the dotted, dashed, and dash-dotted curves are determined from the radial dependence of the neutron and proton transition densities in the surface region: $4.25<r<8\mathrm{fm}$.

Figure 3

The RQRPA neutron and proton transition densities for the peaks at 8.4 MeV and 15.1 MeV excitation energy in ${}^{140}\mathrm{Ce}$.

Figure 4

The RQRPA neutron and proton transition matrix elements for the state at 8.4 MeV in ${}^{140}\mathrm{Ce}$, as a function of the unperturbed energy of the corresponding $2qp$ configurations.

Figure 5

The total $\mathrm{RHB}+\mathrm{RQRPA}$ $E1$ transition strength for ${}^{140}\mathrm{Ce}$ in comparison with calculation using experimental neutron single-particle energies.