Abstract:
A canonical description of the thermodynamical pairing properties of small systems is achieved by using the Variation After Projection approach at finite temperature. The minimization of the free energy is made by a direct evaluation of the energy and full diagonalization of the entropy. We use the Richardson – pairing model whose exact solution allows to study the reliability of different approaches. We show that the Projection After Variation approach, that is usually performed at zero temperature with rather good success, provides a quite poor description at finite temperature. On the contrary, the Variation After Projection applied at finite temperature provides a perfect reproduction of the exact canonical properties of odd or even systems from very low to high temperature.

Abstract:
A model combining self-consistent mean-field and shell-model techniques is used to study the competition between particle like and proton-neutron pairing correlations in fp-shell even-even self-conjugate nuclei. Results obtained using constant two-body pairing interactions as well as more sophisticated interactions are presented and discussed. The standard BCS calculations are systematically compared with more refined approaches including correlation effects beyond the independent quasi-particle approach. The competition between proton-neutron correlations in the isoscalar and isovector channels is also analyzed, as well as their dependence on the deformation properties. Besides the expected role of the spin-orbit interaction and particle number conservation, it is shown that deformation leads to a reduction of the pairing correlations. This reduction originates from the change of the single-particle spectrum and from a quenching of the residual pairing matrix elements. The competition between isoscalar and isovector pairing in the deuteron transfer is finally addressed. Although a strong dependence the isovector pairing correlations with respect to nuclear deformation is observed, they always dominate over the isoscalar ones.

Abstract:
The isoscalar proton-neutron pairing and isovector pairing, including both isovector proton-neutron pairing and like-particle pairing, are treated in a formalism which conserves exactly the particle number and the isospin. The formalism is designed for self-conjugate (N=Z) systems of nucleons moving in axially deformed mean fields and interacting through the most general isovector and isoscalar pairing interactions. The ground state of these systems is described by a superposition of two types of condensates, i.e., condensates of isovector quartets, built by two isovector pairs coupled to the total isospin T=0, and condensates of isoscalar proton-neutron pairs. The comparison with the exact solutions of realistic isovector-isoscalar pairing Hamiltonians shows that this ansatz for the ground state is able to describe with high precision the pairing correlation energies. It is also shown that, at variance with the majority of Hartree-Fock-Bogoliubov calculations, in the present formalism the isovector and isoscalar pairing correlations coexist for any pairing interactions. The competition between the isovector and isoscalar proton-neutron pairing correlations is studied for N=Z nuclei with the valence nucleons moving in the $sd$ and $pf$ shells and in the major shell above $^{100}$Sn. We find that in these nuclei the isovector pairing prevail over the isoscalar pairing, especially for heavier nuclei. However, the isoscalar proton-neutron correlations are significant in all nuclei and they always coexist with the isovector pairing correlations.

Abstract:
Second RPA calculations with a Skyrme force are performed to describe both high- and low-lying excited states in $^{16}$O. The coupling between 1 particle-1 hole and 2 particle-2 hole as well as that between 2 particle-2 hole configurations among themselves are fully taken into account and the residual interaction is never neglected, not resorting therefore to a generally used approximate scheme where only the first kind of coupling is considered. The issue of the rearrangement terms in the matrix elements beyond standard RPA is addressed and discussed. As a general feature of second RPA results, a several-MeV shift of the strength distribution to lower energies is systematically found with respect to RPA distributions. A much more important fragmentation of the strength is also naturally provided by second RPA due to the huge number of 2 particle-2 hole configurations. A better description of the excitation energies of the low-lying 0$^+$ and 2$^+$ states is obtained with second RPA with respect to RPA.

Abstract:
Low-energy dipole excitations are analyzed for the stable isotopes $^{40}$Ca and $^{48}$Ca in the framework of the Skyrme-second random-phase approximation. The corresponding random-phase approximation calculations provide a negligible strength distribution for both nuclei in the energy region from 5 to 10 MeV. The inclusion and the coupling of 2 particle-2 hole configurations in the second random-phase approximation lead to an appreciable dipole response at low energies for the neutron-rich nucleus $^{48}$Ca. The presence of a neutron skin in the nucleus $^{48}$Ca would suggest the interpretation of the low-lying response in terms of a pygmy excitation. The composition of the excitation modes (content of 1 particle-1 hole and 2 particle-2 hole configurations), their transition densities and their collectivity (number and coherence of the different contributions) are analyzed. This analysis indicates that, in general, these excitations cannot be clearly interpreted in terms of oscillations of the neutron skin against the core with the exception of the peak with the largest $B(E1)$ value, which is located at 9.09 MeV. For this peak the neutron transition density dominates and the neutron and proton transition densities oscillate out of phase in the internal part of the nucleus leading to a strong mixing of isoscalar and isovector components. Therefore, this state shows some features usually associated to pygmy resonances.

Abstract:
We make use of a subtraction procedure, introduced to overcome double--counting problems in beyond--mean--field theories, in the second random--phase--approximation (SRPA) for the first time. This procedure guarantees the stability of SRPA (so that all excitation energies are real). We show that the method fits perfectly into nuclear density--functional theory. We illustrate applications to the monopole and quadrupole response and to low--lying $0^+$ and $2^+$ states in the nucleus $^{16}$O. We show that the subtraction procedure leads to: (i) results that are weakly cutoff dependent; (ii) a considerable reduction of the SRPA downwards shift with respect to the random--phase approximation (RPA) spectra (systematically found in all previous applications). This implementation of the SRPA model will allow a reliable analysis of the effects of 2 particle--2 hole configurations ($2p2h$) on the excitation spectra of medium--mass and heavy nuclei.

Abstract:
We carry out a microscopic analysis of the ground and excited states of the Na_8 metal cluster within the jellium model. We perform a series of configuration interaction calculations on a Hartree-Fock basis and construct eigenstates of the Hamiltonian which carry up to 4-particle 4-hole components. Based on the analysis of the dipole transition strengths, we single out those states which can be interpreted as the collective dipole plasmon and its double excitations. These modes are found to possess a high degree of harmonicity, deviations from the harmonic limit remaining, however, of the order of 10%.

Abstract:
We study the nature of the low-lying dipole strength in neutron-rich nuclei, often associated to the Pygmy Dipole Resonance. The states are described within the Hartree-Fock plus RPA formalism, using different parametrizations of the Skyrme interaction. We show how the information from combined reactions processes involving the Coulomb and different mixtures of isoscalar and isovector nuclear interactions can provide a clue to reveal the characteristic features of these states.

Abstract:
We present an extension of the random--phase approximation (RPA) where the RPA phonons are used as building blocks to construct the excited states. In our model, that we call double RPA (DRPA), we include up to two RPA phonons. This is an approximate and simplified way, with respect to the full second random--phase approximation (SRPA), to extend the RPA by including two particle--two hole configurations. Some limitations of the standard SRPA model, related to the violation of the stability condition, are not encountered in the DRPA. We also verify in this work that the energy--weighted sum rules are satisfied. The DRPA is applied to low--energy modes and giant resonances in the nucleus $^{16}$O. We show that the model (i) produces a global downwards shift of the energies with respect to the RPA spectra; (ii) provides a shift that is however strongly reduced compared to that generated by the standard SRPA. This model represents an alternative way of correcting for the SRPA anomalous energy shift, compared to a recently developed extension of the SRPA, where a subtraction procedure is applied. The DRPA provides results in good agreeement with the experimental energies, with the exception of those low--lying states that have a dominant two particle--two hole nature. For describing such states, higher--order calculations are needed.

Abstract:
We present the first applications of the second random-phase-approximation model with the finite-range Gogny interaction. We discuss the advantages of using such an interaction in this type of calculations where 2 particle-2 hole configurations are included. The results found in the present work confirm the well known general features of the second random-phase approximation spectra: we find a large shift, several MeV, of the response centroids to lower energies with respect to the corresponding random-phase-approximation values. As known, these results indicate that the effects of the 1 particle-1 hole/2 particle-2 hole and 2 particle-2 hole/2 particle-2 hole couplings are important. It has been found that the changes of the strength distributions with respect to the standard random-phase-approximation results are particularly large in the present case. This important effect is due to some large neutron-proton matrix elements of the interaction and indicates that these matrix elements (which do not contribute in the mean-field calculations employed in the conventional fit procedures of the force parameters) should be carefully constrained to perform calculations