Low-energy isovector and isoscalar dipole response in neutron-rich nuclei

The self-consistent random-phase approximation, based on the framework of relativistic energy density functionals, is employed in the study of isovector and isoscalar dipole response in ${}^{68}\mathrm{Ni},{}^{132}\mathrm{Sn}$, and ${}^{208}\mathrm{Pb}$. The evolution of pygmy dipole states (PDSs) in the region of low excitation energies is analyzed as a function of the density dependence of the symmetry energy for a set of relativistic effective interactions. The occurrence of PDSs is predicted in the response to both the isovector and the isoscalar dipole operators, and its strength is enhanced with the increase in the symmetry energy at saturation and the slope of the symmetry energy. In both channels, the PDS exhausts a relatively small fraction of the energy-weighted sum rule but a much larger percentage of the inverse energy-weighted sum rule. For the isovector dipole operator, the reduced transition probability $B\left(E1\right)$ of the PDSs is generally small because of pronounced cancellation of neutron and proton partial contributions. The isoscalar-reduced transition amplitude is predominantly determined by neutron particle-hole configurations, most of which add coherently, and this results in a collective response of the PDSs to the isoscalar dipole operator.

Figure 1

The RRPA isovector (left column) and isoscalar (right column) dipole strength distributions in ${}^{68}\mathrm{Ni},{}^{132}\mathrm{Sn}$, and ${}^{208}\mathrm{Pb}$, calculated with a set of five effective interactions that differ in the value of the symmetry energy at saturation ($a_4$) and the corresponding slope parameter $L$.

Figure 2

Predictions for the electric dipole polarizability Eq. (7) of ${}^{208}\mathrm{Pb}$, calculated with a set of five effective interactions that differ in the value of the symmetry energy at saturation ($a_4$) and the corresponding slope parameter $L$ (circles), and the effective interaction DD-ME2 (star) [57]. The shaded area delineates the experimental constraint ${\alpha }_D=(20.1\pm 0.6){\mathrm{fm}}^3$ [20].

Figure 3

Moments $m_0,m_1$, and $m_{-1}$ of the isovector [panels (a)–(f)] and isoscalar [panels (g)–(i)] dipole strength distributions of ${}^{68}\mathrm{Ni}$. Absolute values of the moments in the low-energy PDS region below $E<12\mathrm{MeV}$ (left column) and the percentages of the total moments exhausted by the PDS (right column) as a function of the slope parameter $L$ of the six effective interactions used in this paper.

Figure 4

Partial contributions of neutron and proton configurations to the isovector- (left) and isoscalar- (right) reduced transition amplitude of the PDS state at 7.81-MeV excitation energy in ${}^{132}\mathrm{Sn}$ for the effective interaction with $a_4=32$ and $K_{\mathrm{nm}}=250\mathrm{MeV}$ as functions of the unperturbed energy of the particle-hole configurations. Only amplitudes with a magnitude larger than $\approx 0.01\mathrm{fm}$ (isovector) and $\approx 0.1{\mathrm{fm}}^3$ (isoscalar) are shown in the figure.

Figure 5

Partial contributions of neutron and proton configurations to the isovector GDR state at 15.24-MeV excitation energy in ${}^{132}\mathrm{Sn}$ for the effective interaction with $a_4=32$ and $K_{\mathrm{nm}}=250\mathrm{MeV}$ as functions of the unperturbed energy of the particle-hole configurations. Only amplitudes with a magnitude larger than $\approx 0.01\mathrm{fm}$ are shown in the figure.