Applicability of the Hartree-Fock (HF) based random phase approximation (RPA) with several Skyrme effective interactions to the description of the isoscalar giant monopole (ISGMR) and the isoscalar giant dipole resonance (ISGDR) in 90Zr, 116Sn, 144Sm and 208Pb nuclei has been investigated. The existing Skyrme interactions SL1, SkM*, SGII, Sly4 and Sk255 were used. Hartree-Fock description of the ground state properties of all nuclei of interest was obtained using these Skyrme interactions. Transition strength distributions for the ISGMR and the ISGDR in nuclei of interest were calculated using coordinate space representation for the RPA in the Green’s function formalism with discretized continuum. A method of projecting out the spurious state contribution from the transition strength distribution and the transition density of the ISGDR was employed to eliminate spurious state mixing, due to a not fully selfconsistent description of the particle-hole interaction within the RPA. Differential cross sections of 240 MeV alpha-particles inelastic scattering on all nuclei of interest were calculated using the folding model within the distorted wave Born approximation (DWBA). Optical potentials were obtained by folding HF ground state densities with a alpha-nucleon density dependent Gaussian interaction. Parameters of the interaction were obtained by fitting experimental angular distribution of alpha-nucleus elastic scattering. The inelastic differential cross sections were calculated using both collective and microscopic transition densities. Possible underestimations of the energy weighted sum rule for the case of the ISGDR are reported. An alternative description for the ISGDR in nuclei based on the Fermi liquid drop model (FLDM) with the collisional Fermi surface distortion was investigated. The FLDM dispersion relation was obtained from the linearized Landau-Vlasov equation. Centroid energies, E0 and E1, and widths, gamma-0 and gamma-1, of the ISGMR and ISGDR, respectively, were calculated as functions of the damping parameter using appropriate boundary conditions. Comparison of the theoretical ratios of the ISGDR and ISGMR centroid energies, E1/E0, to the experimental values resulted in a damping parameter equal to 0.5, however, systematic overestimation of energy of the ISGMR and ISGDR by 2.0-2.5 MeV was observed. The applicability of the HF-RPA to the description for the ISGDR in nuclei is confirmed.

In this manuscript, some relations between theoretical nuclear physics and compact stars, which are known to be excellent tools to test matter under extreme conditions, are studied. Most of these links are performed within theoretical modelling which are used to describe both nuclei and nuclear systems in astrophysics. Self-consistent approaches unifying the description of isolated nuclei, dilute nuclei in a gas of light particles, and uniform matter, are presented in this manuscript and employed to understand the physics of compact stars. The manuscript is organized as follow: The first chapter is a general introduction to impact stars and supernovae physics, as well as to nuclear physics. In the second chapter, various aspects of the modeling of the in-medium nucleon-nucleon interaction are presented, such as, extension of the Skyrme interaction in the spin and spin-isospin channels, the properties of the V(low k) interaction and the low-density properties of nuclear matter. In the third chapter, the superfluid properties of dense matter are investigated, either through an iso-vector pairing interaction design to reproduce microscopic calculations in nuclear matter, or through the impact of superfluidity on derivatives of the EOS, such as the incompressibility and the symmetry energy, and finally, is presented the use of pairing vibration to study the properties of pairing. The questions of the in-medium effective mass and its impact for the dynamics of core-collapse supernovae, and the level density in nuclei are presented in the chapter 4. The microscopic modelling of the crust of neutron stars, as derived from band theory, and including superfluidity, is shown in chapter 5. In chapter 6, some links between the properties of nuclei, and neutron stars are presented: the properties of overflowing nuclei at the neutron drip, the correlation between the curst-core properties and nuclear empirical quantities, and the stiffness of the nuclear EOS, are shown. Finally, conclusions are presented in chapter 7.

Les approches basées sur les modèles de champ moyen reproduisent avec succès certaines propriétés nucléaires comme les masses et les rayons, dans le cadre des théories de la fonctionnelle de la densité pour l'énergie (EDF). Cependant, plusieurs corrélations complexes sont absentes dans les théories de champ moyen et un certain nombre d'observables liées aux propriétés à une particule et collectives des systèmes nucléoniques ne peuvent pas être prédites avec précision. La nécessité de fournir une description précise des données disponibles ainsi que des prévisions fiables dans les régions exotiques de la carte nucléaire motive l'utilisation de modèles plus sophistiqués, qui vont au-delà du champ moyen. Des corrélations et des corrections d'ordre supérieur (au-delà du premier ordre, qui représente l'approximation de champ moyen) sont introduites dans ces modèles. Un aspect crucial dans ces calculs est le choix de l'interaction efficace qui doit être utilisée quand on va au-delà du premier ordre (les interactions efficaces existantes sont généralement ajustées avec des calculs de champ moyen). Dans la première partie de la thèse, nous traitons l'équation d'état de la matière nucléaire, évaluée jusqu'au deuxième ordre avec la force phénoménologique de Skyrme. Nous analysons la divergence ultraviolette qui est liée à la portée nulle de l'interaction et nous introduisons des interactions régularisées de type Skyrme qui peuvent être utilisées au deuxième ordre. Des procédures de régularisation avec un cutoff et des techniques de régularisation dimensionnelle sont analysées et appliquées. Dans le cas de la régularisation dimensionnelle, des connexions sont naturellement établies entre le cadre EDF et des techniques employées dans les théories de champ effectives. Dans la deuxième partie de la thèse, nous vérifions si les interactions régularisées introduites pour la matière nucléaire peuvent être utilisées également pour les noyaux finis. A titre d'illustration, cette analyse est effectuée dans le modèle de couplage particule vibration, qui représente un exemple de modèle qui va au-delà de l'approximation de champ moyen, où une divergence ultraviolette apparaît si des forces de portée nulle sont utilisées. Ces premières applications suggèrent plusieurs directions à explorer pour fournir à plus long terme des interactions régularisées qui sont bien adaptés pour les calculs au-delà du champ moyen pour les noyaux finis. Les conclusions et des perspectives sont illustrées à la fin du manuscrit. / Mean-field approaches successfully reproduce nuclear bulk properties like masses and radii within the Energy Density Functional (EDF) framework. However, complex correlations are missing in mean-field theories and several observables related to single-particle and collective nuclear properties cannot be predicted accurately. The necessity to provide a precise description of the available data as well as reliable predictions in the exotic regions of the nuclear chart motivates the use of more sophisticated beyond-mean-field models. Correlations and higher-order corrections (beyond the leading mean-field order) are introduced. A crucial aspect in these calculations is the choice of the effective interaction to be used when one goes beyond the leading order (available effective interactions are commonly adjusted at the mean-field level). In the first part, we deal with the equation of state of nuclear matter evaluated up to the second order with the phenomenological Skyrme force. We analyze the ultraviolet divergence that is related to the zero range of the interaction and we introduce Skyrme-type regularized interactions that can be used at second order for matter. Cutoff regularization and dimensional regularization techniques are explored and applied. In the latter case, connections are naturally established between the EDF framework and some techniques employed in Effective Field Theories. In the second part, we check whether the regularized interactions introduced for nuclear matter can be employed also for finite nuclei. As an illustration, this analysis is performed within the particle-vibration model that represents an example of beyond mean-field models where an ultraviolet divergence appears if zero-range forces are used. These first applications suggest several directions to be explored to finally provide regularized interactions that are specially tailored for beyond-mean-field calculations for finite nuclei. Conclusions and perspectives are finally illustrated.