Energies of the ground state and first excited 0+0^{+} state in an exactly solvable pairing model

Several approximations are tested by calculating the ground-state energy and the energy of the first excited $0^{+}$ state using an exactly solvable model with two symmetric levels interacting via a pairing force. They are the BCS approximation (BCS), Lipkin - Nogami (LN) method, random-phase approximation (RPA), quasiparticle RPA (QRPA), the renormalized RPA (RRPA), and renormalized QRPA (RQRPA). It is shown that, in the strong-coupling regime, the QRPA which neglects the scattering term of the model Hamiltonian offers the best fit to the exact solutions. A recipe is proposed using the RRPA and RQRPA in combination with the pairing gap given by the LN method. Applying this recipe, it is shown that the normal-superfluid phase transition is avoided, and a reasonably good description for both of the ground-state energy and the energy of the first excited $0^{+}$ state is achieved.Comment: 18 pages, 4 figure

Superfluid-normal phase transition in finite systems and its effect on damping of hot giant resonances

Thermal fluctuations of quasiparticle number are included making use of the secondary Bogolyubov's transformation, which turns quasiparticles operators into modified-quasiparticle ones. This restores the unitarity relation for the generalized single-particle density operator, which is violated within the Hartree-Fock-Bogolyubov (HFB) theory at finite temperature. The resulting theory is called the modified HFB (MHFB) theory, whose limit of a constant pairing interaction yields the modified BCS (MBCS) theory. Within the MBCS theory, the pairing gap never collapses at finite temperature T as it does within the BCS theory, but decreases monotonously with increasing T. It is demonstrated that this non-vanishing thermal pairing is the reason why the width of the giant dipole resonance (GDR) does not increase with T up to T around 1 MeV. At higher T, when the thermal pairing is small, the GDR width starts to increase with T. The calculations within the phonon-damping model yield the results in good agreement with the most recent experimental systematic for the GDR width as a function of T. A similar effect, which causes a small GDR width at low T, is also seen after thermal pairing is included in the thermal fluctuation model.Comment: Invited lecture at the Predeal international summer school in nuclear physics on ``Collective motion and phase transitions in nuclear systems'', 28 August - 9 September, 2006, Predeal, Romania; 18 pages, 3 figures; to be published by World Scientific in the proceedings of this schoo