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 lowenergy 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 Ni and 132 Sn have been 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 recent high-resolution measurement of the electric dipole (E1) polarizability (alphad) in 208Pb [Phys. Rev. Lett. 107, 062502 (2011)] provides a unique constraint on the neutron-skin thickness of this nucleus. The neutron-skin thickness (rskin) of 208Pb 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 alphad and rskin, we carry out systematic calculations for 208Pb, 132Sn, and 48Ca based on the nuclear density functional theory (DFT) using both non-relativistic and relativistic energy density functionals (EDFs). Our analysis indicates that whereas individual models exhibit a linear dependence between alphad and rskin, 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 rskin and alphad for the nuclei under consideration. We conclude that precise measurements of rskin in both 48Ca and 208Pb---combined with the recent measurement of alphad---should significantly constrain the isovector sector of the nuclear energy density functional.Comment: Manuscript contains 5 pages, 2 figures, and 1 table. Submitted to Physical Review Letter

The possible constraints on the poorly determined symmetry part of the effective nuclear Hamiltonians or effective energy functionals, i.e., the so-called symmetry energy S(ρ), are very much under debate. In the present work, we show that the value of the symmetry energy associated with Skyrme functionals, at densities ρ around 0.1 fm −3 , is strongly correlated with the value of the centroid of the Giant Dipole Resonance (GDR) in spherical nuclei. Consequently, the experimental value of the GDR in, e.g., 208 Pb can be used as a constraint on the symmetry energy, leading to 23.3 MeV < S(ρ = 0.1 fm −3 ) < 24.9 MeV.

The information on the symmetry energy and its density dependence is deduced by comparing the available data on the electric dipole polarizability αD of 68 Ni, 120 Sn, and 208 Pb with the predictions of Random Phase Approximation, using a representative set of nuclear energy density functionals.The calculated values of αD are used to validate different correlations involving αD, the symmetry energy at the saturation density J, corresponding slope parameter L and the neutron skin thickness ∆rnp, as suggested by the Droplet Model. A subset of models that reproduce simultaneously the measured polarizabilities in 68 Ni, 120 Sn, and 208 Pb are employed to predict the values of symmetry energy parameters at saturation density and ∆rnp. The resulting intervals are: J = 30-35 MeV, L = 20-66 MeV; and the values for ∆rnp in 68 Ni, 120 Sn, and 208 Pb are in the ranges: 0.15-0.19 fm, 0.12-0.16 fm, 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.

We study the electric dipole polarizability α D in 208 Pb based on the predictions of a large and representative set of relativistic and nonrelativistic nuclear mean-field models. We adopt the droplet model as a guide to better understand the correlations between α D and other isovector observables. Insights from the droplet model suggest that the product of α D and the nuclear symmetry energy at saturation density J is much better correlated with the neutron skin thickness r np of 208 Pb than the polarizability alone. Correlations of α D J with r np and with the symmetry energy slope parameter L suggest that α D J is a strong isovector indicator. Hence, we explore the possibility of constraining the isovector sector of the nuclear energy density functional by comparing our theoretical predictions against measurements of both α D and the parity-violating asymmetry in 208 Pb. We find that the recent experimental determination of α D in 208 Pb in combination with the range for the symmetry energy at saturation density J = [31 ± (2) est ] MeV suggests r np ( 208 Pb) = 0.165 ± (0.009) expt ± (0.013) theor ± (0.021) est fm and L = 43 ± (6) expt ± (8) theor ± (12) est MeV.

The nuclear incompressibility K ∞ is deduced from measurements of the Isoscalar Giant Monopole Resonance (ISGMR) in medium-heavy nuclei, and the resulting value turns out to be model dependent. Since the considered nuclei have neutron excess, it has been suggested that the model dependence is due to the different behaviour of the symmetry energy in different models. To clarify this issue, we make a systematic and careful analysis based on new Skyrme forces which span a wide range of values for K ∞ , for the value of the symmetry energy at saturation and for its density dependence. By calculating, in a fully self-consistent fashion, the ISGMR centroid energy in 208 Pb we reach, for the first time within the non-relativistic framework, three important conclusions:(i) the monopole energy, and consequently the deduced value of K ∞ , depend on a well defined parameter related to the shape of the symmetry energy curve and called K sym ; (ii) Skyrme forces of the type of SLy4 predict K ∞ around 230 MeV, in agreement with the Gogny force (previous estimates using Skyrme interactions having been plagued by lack of full self-consistency); (iii) it is possible to build forces which predict K ∞ around 250 MeV, although part of this increase is due to our poor knowledge of the density dependence and effective mass.

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