We review recent studies of the evolution of collective excitations in atomic nuclei far from the valley of β-stability. Collective degrees of freedom govern essential aspects of nuclear structure, and for several decades the study of collective modes such as rotations and vibrations has played a vital role in our understanding of complex properties of nuclei. The multipole response of unstable nuclei and the possible occurrence of new exotic modes of excitation in weakly bound nuclear systems, present a rapidly growing field of research, but only few experimental studies of these phenomena have been reported so far. Valuable data on the evolution of the low-energy dipole response in unstable neutron-rich nuclei have been gathered in recent experiments, but the available information is not sufficient to determine the nature of observed excitations. Even in stable nuclei various modes of giant collective oscillations had been predicted by theory years before they were observed, and for that reason it is very important to perform detailed theoretical studies of the evolution of collective modes of excitation in nuclei far from stability. We therefore discuss the modern theoretical tools that have been developed in recent years for the description of collective excitations in weakly bound nuclei. The review focuses on the applications of these models to studies of the evolution of low-energy dipole modes from stable nuclei to systems near the particle emission threshold, to analyses of various isoscalar modes, those for which data are already available, as well as those that could be observed in future experiments, to a description of charge-exchange modes and their evolution in neutron-rich nuclei, and to studies of the role of exotic low-energy modes in astrophysical processes.

Correlations between the behavior of the nuclear symmetry energy, the neutron skins, and the percentage of energy-weighted sum rule (EWSR) exhausted by the pygmy dipole resonance (PDR) in ${}^{68}\mathrm{Ni}$ and ${}^{132}\mathrm{Sn}$ are investigated by using different random phase approximation (RPA) models for the dipole response, based on a representative set of Skyrme effective forces plus meson-exchange effective Lagrangians. A comparison with the experimental data has allowed us to constrain the value of the derivative of the symmetry energy at saturation. The neutron skin radius is deduced under this constraint.

The information on the symmetry energy and its density dependence is deduced by comparing the available data on the electric dipole polarizability ${\alpha }_D$ of ${}^{68}\text{Ni}$, ${}^{120}\text{Sn}$, and ${}^{208}\text{Pb}$ with the predictions of the random-phase approximation, using a representative set of nuclear energy density functionals. The calculated values of ${\alpha }_D$ are used to validate different correlations involving ${\alpha }_D$, the symmetry energy at the saturation density $J$, the corresponding slope parameter $L$, and the neutron skin thickness $\mathrm{\Delta }{r}_{np}$, as suggested by the droplet model. A subset of models that reproduce simultaneously the measured polarizabilities in ${}^{68}\text{Ni}$, ${}^{120}\text{Sn}$, and ${}^{208}\text{Pb}$ are employed to predict the values of the symmetry energy parameters at saturation density and $\mathrm{\Delta }{r}_{np}$. The resulting intervals are $J=30–35$ MeV, $L=20–66$ MeV; and the values for $\mathrm{\Delta }{r}_{np}$ in ${}^{68}\text{Ni}$, ${}^{120}\text{Sn}$, and ${}^{208}\text{Pb}$ are in the ranges 0.15–0.19, 0.12–0.16, and 0.13–0.19 fm, respectively. The strong correlation between the electric dipole polarizabilities of two nuclei is instrumental to predict the values of electric dipole polarizabilities in other nuclei.

The recent high-resolution measurement of the electric dipole ($E1$) polarizability ${\alpha }_{\mathrm{D}}$ in ${}^{208}$Pb [A. Tamii et al., Phys. Rev. Lett. 107, 062502 (2011)] provides a unique constraint on the neutron-skin thickness of this nucleus. The neutron-skin thickness $r_{\mathrm{skin}}$ of ${}^{208}$Pb is a quantity of critical importance for our understanding of a variety of nuclear and astrophysical phenomena. To assess the model dependence of the correlation between ${\alpha }_{\mathrm{D}}$ and $r_{\mathrm{skin}}$, we carry out systematic calculations for ${}^{208}$Pb, ${}^{132}$Sn, and ${}^{48}$Ca based on the nuclear density functional theory using both nonrelativistic and relativistic energy density functionals. Our analysis indicates that whereas individual models exhibit a linear dependence between ${\alpha }_{\mathrm{D}}$ and $r_{\mathrm{skin}}$, this correlation is not universal when one combines predictions from a host of different models. By averaging over these model predictions, we provide estimates with associated systematic errors for $r_{\mathrm{skin}}$ and ${\alpha }_{\mathrm{D}}$ for the nuclei under consideration. We conclude that precise measurements of $r_{\mathrm{skin}}$ in both ${}^{48}$Ca and ${}^{208}$Pb—combined with the recent measurement of ${\alpha }_{\mathrm{D}}$—should significantly constrain the isovector sector of the nuclear energy density functional.