Narrow Results

In this work, a new functional is introduced to treat pairing correlations in finite many-body systems. Guided by the projected BCS framework, the energy is written as a functional of occupation numbers. It is shown to generalize the BCS approach and to provide an alternative to Variation After Projection framework. Illustrations of the new approach are given for the pairing Hamiltonian for various particle numbers and coupling strengths. In all case, a very good agreement with the exact solution is found.

We propose a local energy density functional for global description of pairing correlations by focusing on the neutron excess dependence. We show the clear correlation between pairing gaps and effective mass parameters as a function of neutron excess. This effect can be taken into account to the density functional by the isovector density dependence in the particle-particle channel.

The exact renormalization group method is applied to many-fermion systems with short-range attractive forces. The strength of the attractive fermion-fermion interaction is determined from the vacuum scattering length. A set of approximate flow equations is derived including fermionic bosonic fluctations. The numerical solutions show a phase transition to a gapped phase. The inclusion of bosonic fluctuations is found to be significant only in the small-gap regime.

We introduce a new class of exactly solvable boson pairing models using the technique of Richardson and Gaudin. Analytical expressions for all energy eigenvalues and the first few energy eigenstates are given. In addition, another solution to Gaudin's equation is also mentioned. A relation with the Calogero–Sutherland model is suggested.

The physical properties of the nuclear shape have been investigated through the charge square radius (<r²>) and the quadrupole (Q₂) and hexadecapole (Q₄) moments of the even–even neutron-rich rare-earth nuclei. The single-particle energies used are those of a deformed Woods–Saxon mean-field. The pairing effects have been included by means of an exact projection method. The model has been tested for the "ordinary" nuclei near the shell closure N = 82 and has correctly reproduced the experimental data and particularly the "Kink" effect. The study has then been extended to the neutron-rich nuclei and has shown a stability of the <r²> and Q₂ results for N≃100 which may be attributed to the existence of a new magic number. On the other hand, a saturation of the prolate shape appears around N = 108 for the elements Nd, Sm and Gd and near N = 102 for the Dy, Er and Yb. These observations could not be confirmed by the investigation of the hexadecapole moment.

The variation of the two-neutron separation energy (S_2N), as a function of the neutron number N, is studied using a microscopic model that includes the pairing effects rigorously within the Fixed-Sharp-BCS method. The model was first tested on "ordinary" nuclei and allowed one to suitably reproduce the experimental data and to confirm the results of previous studies. The model was then applied to the even–even neutron-rich isotopes in the rare-earth region and showed, on the one hand, a relatively important variation of S_2N, when N = 100, that could lead to the assumption of the existence of a new magic number in this region, and on the other hand, a weak variation of S_2N when N > 100. These findings corroborate the previously obtained results for the charge mean square radius and the quadrupole and hexadecapole moments within the same model.

Using HF + BCS method we study light nuclei with nuclear charge in the range 2 ≤ Z ≤ 8 and lying near the neutron drip line. The HF method uses effective Skyrme forces and allows for axial deformations. We find that the neutron drip line forms stability peninsulas at ¹⁸He and ⁴⁰C. These isotopes are found to be stable against one neutron emission and possess the highest known neutron to proton ratio in stable nuclei.

Nuclear level densities of ${}^{207}$Pb and ${}^{89}$Y are calculated using the Lipkin–Nogami (LN) method and Bradeen–Cooper–Schrieffer (BCS) model. It is revealed that the calculated nuclear level densities are highly matched with the experimental data of Oslo group. The excitation energy and entropy are calculated for mentioned nuclei. In the case of two studied nuclei the characteristic of being magic for the number of neutrons or protons causes the decrease of the excitation energy and entropy contribution of magic system at low temperatures.

Gap parameter of Lipkin–Nogami (LN) model is replaced by order parameter of the exact Ginzburg–landau (EGL) theory. Thermodynamic quantities such as energy, entropy and heat capacity for ${}^{96,97}$Mo nuclei are calculated using this modified form of the LN model (MLN). In the LN model, the gap parameter suddenly decreases to zero at critical temperature. This causes singular points in the graph of heat capacity. However, in the MLN method, the order parameter does not become zero at critical temperature and gradually decreases along with the temperature. This causes the singular points, which are predicted in the heat capacity of LN model to be eliminated. Therefore, the heat capacity as a function of temperature becomes continuous and S-shaped, which is qualitatively in agreement with the experimental data.