Pairing Correlations and Phase Transitions in Mesoscopic Systems

Pairing correlations and phase transitions in mesoscopic or small systems are studied through out this dissertation. We start our discussion by showing the importance of short range correlations and their role in forming bound Cooper pairs. For a model Hamiltonian, we solved the Schr¨odinger equation in the harmonic oscillator basis analytically, the concept of self consistency is used to get the whole energy spectrum. Using variational methods applied to a trial wave function, we derived the BCS equations, which again should be solved self consistently with particle number to produce the total energy. Some examples of BCS calculations in realistic case like in the Sn isotopes are shown. Various approximations such as one level, two levels and five levels systems are discussed. In the five levels model calculations, we compare our results with the previous works by other authors. We also find a good agreement with the experimental data. We extend our BCS calculations by adding the three body interaction term. This additional term is unlikely to improve our results compared to the experiment. In a separate work, using numerical and analytical methods implemented for different models we conduct a systematic study of thermodynamic properties of pairing correlations in mesoscopic nuclear systems. Various quantities are calculated and analyzed using the exact solution of pairing. An in-depth comparison of canonical, grand canonical, and microcanonical ensemble is conducted. The nature of the pairing phase transition in a small system is of particular interest. We discuss the onset of discontinuities in the thermodynamic variables, fluctuations, and evolution of zeros of the canonical and grand canonical partition functions in the complex plane. The behavior of the Invariant Correlational Entropy is also studied in the transitional region of interest. The change in the character of the phase transition due to the presence of magnetic field is discussed along with studies of superconducting thermodynamics.