We introduce a new relativistic energy density functional constrained by the ground state properties of atomic nuclei along with the isoscalar giant monopole resonance energy and dipole polarizability in ${}^{208}\mathrm{Pb}$. A unified framework of the relativistic Hartree-Bogoliubov model and random phase approximation based on the relativistic density-dependent point coupling interaction is established in order to determine the DD-PCX parametrization by ${\chi }^2$ minimization. This procedure is supplemented with the covariance analysis in order to estimate statistical uncertainties in the model parameters and observables. The effective interaction DD-PCX accurately describes the nuclear ground state properties including the neutron-skin thickness, as well as the isoscalar giant monopole resonance excitation energies and dipole polarizabilities. The implementation of the experimental data on nuclear excitations allows constraining the symmetry energy close to the saturation density, and the incompressibility of nuclear matter by using genuine observables on finite nuclei in the ${\chi }^2$ minimization protocol, rather than using pseudo-observables on the nuclear matter, or by relying on the ground state properties only, as it has been customary in the previous studies.

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.

A method is introduced that establishes relations between properties of collective excitations in finite nuclei and the phase transition density ${n}_{t}$ and pressure ${P}_{t}$ at the inner edge separating the liquid core and the solid crust of a neutron star. A theoretical framework that includes the thermodynamic method, relativistic nuclear energy density functionals, and the quasiparticle random-phase approximation is employed in a self-consistent calculation of $({n}_{t},{P}_{t})$ and collective excitations in nuclei. Covariance analysis shows that properties of charge-exchange dipole transitions, isovector giant dipole and quadrupole resonances, and pygmy dipole transitions are correlated with the core-crust transition density and pressure. A set of relativistic nuclear energy density functionals, characterized by systematic variation of the density dependence of the symmetry energy of nuclear matter, is used to constrain possible values for $({n}_{t},{P}_{t})$. By comparing the calculated excitation energies of giant resonances, energy-weighted pygmy dipole strength, and dipole polarizability with available data, we obtain the weighted average values: ${n}_{t}=0.0955\ifmmode\pm\else\textpm\fi{}0.0007$ ${\mathrm{fm}}^{$-${}3}$ and ${P}_{t}=0.59\ifmmode\pm\else\textpm\fi{}0.05$ MeV ${\mathrm{fm}}^{$-${}3}$. This approach crucially depends on experimental results for collective excitations in nuclei and, therefore, accurate measurements are necessary to further constrain the structure of the crust of neutron stars.

$\beta$-decay properties of nuclei are investigated within the relativistic nuclear energy density functional framework by varying the temperature and density, conditions relevant to the final stages of stellar evolution. Both thermal and nuclear pairing effects are taken into account in the description of nuclear properties and in the finite-temperature proton-neutron relativistic quasiparticle random-phase approximation (FT-PNRQRPA) to calculate the relevant allowed and first-forbidden transitions in the $\beta$ decay. The temperature and density effects are studied on the $\beta$-decay half-lives at temperatures $T=0$–$1.5\mathrm{MeV}$ and at densities $\rho Y_e={10}^7\mathrm{g}/{\mathrm{cm}}^3$ and ${10}^9\mathrm{g}/{\mathrm{cm}}^3$. The relevant Gamow-Teller transitions are also investigated for Ti, Fe, Cd, and Sn isotopic chains at finite-temperatures. We find that the $\beta$-decay half-lives increase with increasing density $\rho Y_e$, whereas half-lives generally decrease with increasing temperature. It is shown that the temperature effects decrease the half-lives considerably in nuclei with longer half-lives at zero temperature, while only slight changes for nuclei with short half-lives are obtained. We also show the importance of including the de-excitation transitions in the calculation of the $\beta$-decay half-lives at finite-temperatures. Comparing the FT-PNQRPA results with the shell-model calculations for $pf$-shell nuclei, a reasonable agreement is obtained for the temperature dependence of $\beta$-decay rates. Finally, large-scale calculations of $\beta$-decay half-lives are performed at temperatures $T_9\left(\text{K}\right)=5$ and $T_9\left(\text{K}\right)=10$ and densities $\rho Y_e={10}^7$ and ${10}^9\mathrm{g}/{\mathrm{cm}}^3$ for even-even nuclei in the range $8\le Z\le 82$, relevant for astrophysical nucleosynthesis mechanisms.