Abstract:
Excitation energies and transition probabilities of the first $2^+$ excitations in even tin and lead isotopes as well as the quadrupole moments of odd neighbors of these isotopes are calculated within the self-consistent Theory of Finite Fermi Systems based on the Energy Density Functional by Fayans et al. The effect of the density dependence of the effective pairing interaction is analyzed in detail by comparing results obtained with volume and surface pairing. The effect is found to be noticeable. For example, the $2^+$-energies are systematically higher at 200-300 keV for the volume paring as compared with the surface pairing case. But on the average both models reasonably agree with the data. Quadrupole moments of odd-neutron nuclei are very sensitive to the single-particle energy of the state $\lambda$ under consideration due to the Bogolyubov factor ($u^2_{\lambda}-v^2_{\lambda}$). A reasonable agreement with experiment for the quadrupole moments has been obtained for the most part of odd nuclei considered. The method used gives a reliable possibility to predict quadrupole moments of unstable odd nuclei including very neutron rich ones.

Abstract:
Role of pairing correlation for emergence of low-frequency vibrational excitations in neutron drip line nuclei is discussed paying special attention to neutrons with small orbital angular momentum l. Self-consistent pairing correlation in the Hartree-Fock-Bogoliubov (HFB) theory causes the change of the spatial structure of the quasiparticle wave functions; "the pairing anti-halo effect" in the lower component and "the broadening effect" in the upper component. The resultant spatial distribution of the two-quasiparticle states among low-l neutrons, "the broad localization", brings about qualitatively new aspects, especially the large transition strength of the low-frequency vibrational excitations in nuclei close to the neutron drip line. By performing HFB plus quasiparticle random phase approximation (QRPA) calculation for the first 2+ states in neutron rich Ni isotopes, the unique role of self-consistent pairing correlation is pointed out.

Abstract:
We discuss effects of pairing correlation on quasi-particle resonance. We analyze in detail how the width of low-lying quasi-particle resonance is governed by the pairing correlation in the neutron drip-line nuclei. We consider the 46Si + n system to discuss low-lying p wave quasi-particle resonance. Solving the Hartree-Fock-Bogoliubov equation in the coordinate space with scattering boundary condition, we calculate the phase shift, the elastic cross section, the resonance width and the resonance energy. We found that the pairing correlation has an effect to reduce the width of quasi-particle resonance which originates from a particle-like orbit in weakly bound nuclei.

Abstract:
The empirical pairing gaps derived from four different odd-even mass staggering formulas are compared. By performing single-$j$ shell and multi-shell seniority model calculations as well as by using the standard HFB approach with Skyrme force we show that the simplest three-point formula $\Delta_C^ {(3)}(N)=\frac{1}{2}\left[B(N,Z)+B(N-2,Z)-2B(N-1,Z)\right]$ can provide a good measure of the neutron pairing gap in even-$N$ nuclei. It removes to a large extent the contribution from the nuclear mean field as well as contributions from shell structure details. It is also less contaminated by the Wigner effect for nuclei around $N=Z$. We also show that the strength of $\Delta^ {(3)}_C(N)$ can serve as a good indication of the two-particle spatial correlation in the nucleus of concern and that the weakening of $\Delta^ {(3)}_C(N)$ in some neutron-rich nuclei indicates that the di-neutron correlation itself is weak in these nuclei.

Abstract:
We use shell model techniques in the complete pf shell to study pair correlations in nuclei. Particular attention is paid to the competition of isoscalar and isovector proton-neutron pairing modes which is investigated in the odd-odd N=Z nucleus 46V and in the chain of even Fe-isotopes. We confirm the dominance of isovector pairing in the ground states. An inspection of the level density and pair correlation strength in 46V, however, shows the increasing relative importance of isoscalar correlations with increasing excitation energy. In the Fe-isotopes we find the expected strong dependence of the isovector pairing strength on the neutron excess, while the dominant J=1 isoscalar pair correlations scale much more gently with neutron number. We demonstrate that the isoscalar pair correlations depend strongly on the spin-orbit splitting.

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 BCS and HFB theories which can accommodate the pairing correlations in the ground states of atomic nuclei are presented. As an application of the pairing theories, we investigate the spatial extension of weakly bound Ne and C isotopes by taking into account the pairing correlation with the Hartree-Fock-Bogoliubov (HFB) method and a 3-body model, respectively. We show that the odd-even staggering in the reaction cross sections of $^{30,31,32}$Ne and $^{14,15,16}$C are successfully reproduced, and thus the staggering can be attributed to the unique role of pairing correlations in nuclei far from the stability line. A correlation between a one-neutron separation energy and the anti-halo effect is demonstrated for $s$- and p-waves using the HFB wave functions. We also propose effective density-dependent pairing interactions which reproduce both the neutron-neutron ($nn$) scattering length at zero density and the neutron pairing gap in uniform matter. Then, we apply these interactions to study pairing gaps in semi-magic finite nuclei, such as Ca, Ni, Sn and Pb isotopic chains.

Abstract:
Simple generic aspects of nuclear pairing in homogeneous medium as well as in finite nuclei are discussed. It is argued that low-energy nuclear structure is not sensitive enough to resolve fine details of nuclear nucleon-nucleon (NN) interaction in general and pairing NN interaction in particular what allows for regularization of the ultraviolet (high-momentum) divergences and a consistent formulation of effective superfluid local theory. Some aspects of (dis)entanglement of pairing with various other effects as well as forefront ideas concerning isoscalar pairing are also briefly discussed.

Abstract:
The Relativistic Continuum Hartree-Bogoliubov (RCHB) theory, which properly takes into account the pairing correlation and the coupling to (discretized) continuum via Bogoliubov transformation in a microscopic and self-consistent way, has been reviewed together with its new interpretation of the halo phenomena observed in light nuclei as the scattering of particle pairs into the continuum, the prediction of the exotic phenomena -- giant halos in nuclei near neutron drip line, the reproduction of interaction cross sections and charge-changing cross sections in light exotic nuclei in combination with the Glauber theory, better restoration of pseudospin symmetry in exotic nuclei, predictions of exotic phenomena in hyper nuclei, and new magic numbers in superheavy nuclei, etc. Recent investigations on new effective interactions, the density dependence of the interaction strengthes, the RMF theory on the Woods-Saxon basis, the single particle resonant states, and the resonant BCS (rBCS) method for the pairing correlation, etc. are also presented in some details.

Abstract:
Several pairing schemes currently used to describe superfluid nuclei through Hartree-Fock-Bogolyubov (HFB) calculations are briefly reviewed. We put a particular emphasis on the regularization recipes used in connection with zero-range forces and on the density dependence which usually complement their definition. Regarding the chosen regularization process, the goal is not only to identify the impact it may or may not have on pairing properties of nuclei through spherical 1D HFB calculations but also to assess its tractability for systematic axial 2D and 3D mean-field and beyond-mean-field calculations.