Abstract:
We describe the ground state of the isovector pairing Hamiltonian in self-conjugate nuclei by a product of collective quartets of different structure built from two neutrons and two protons coupled to total isospin T=0. The structure of the collective quartets is determined by an iterative variational procedure based on a sequence of diagonalizations of the pairing Hamiltonian in spaces of reduced size. The accuracy of the quartet model is tested for N=Z nuclei carrying valence nucleons outside the $^{16}$O, $^{40}$Ca, and $^{100}$Sn cores. The comparison with the exact solutions of the pairing Hamiltonian, obtained by shell model diagonalization, shows that the quartet model is able to describe the isovector pairing energy with very high precision. The predictions of the quartet model are also compared to those of the simpler quartet condensation model in which all the collective quartets are assumed to be identical.

Abstract:
We discuss the treatment of isovector pairing by an alpha-like quartet condensate which conserves exactly the particle number, the spin and the isospin. The results show that the quartet condensate describes accurately the isovector pairing correlations in the ground state of systems with an equal number of protons and neutrons

Abstract:
The theory of quartet condensation is further developed. The onset of quartetting in homgeneous fermionic matter is studied with the help of an in-medium modified four fermion equation. It is found that at very low density quartetting wins over pairing. At zero temperature, in analogy to pairing, a set of equations for the quartet order parameter is given. Contrary to pairing, quartetting only exists for strong coupling and breaks down for weak coupling. Reasons for this finding are detailed. In an application to nuclear matter, the critical temperature for alpha particle condensation can reach values up to around 8 MeV. The disappearance of alpha particles with increasing density, i.e. the Mott transition, is investigated. In finite nuclei the Hoyle state, that is the second 0+ state of 12C is identified as an 'alpha-particle condensate' state. It is conjectured that such states also exist in heavier n-alpha nuclei, like 16O, 20Ne, etc. The sixth 0+ state in 16O is proposed as an analogue to the Hoyle state. The Gross-Pitaevski equation is employed to make an estimate of the maximum number of alpha particles a condensate state can contain. Possible quartet condensation in other systems is discussed briefly.

Abstract:
Isoscalar (T=0,J=1) and isovector (T=1,J=0) pairing correlations in the ground state of self-conjugate nuclei are treated in terms of alpha-like quartets built by two protons and two neutrons coupled to total isospin T=0 and total angular momentum J=0. Quartets are constructed dynamically via an iterative variational procedure and the ground state is represented as a product of such quartets. It is shown that the quartet formalism describes accurately the ground state energies of realistic isovector plus isoscalar pairing Hamiltonians in nuclei with valence particles outside the 16O, 40Ca and 100Sn cores. Within the quartet formalism we analyse the competition between isovector and isoscalar pairing correlations and find that for nuclei with the valence nucleons above the cores 40Ca and 100Sn the isovector correlations account for the largest fraction of the total pairing correlations. This is not the case for sd-shell nuclei for which isoscalar correlations prevail. Contrary to many mean-field studies, isovector and isoscalar pairing correlations mix significantly in the quartet approach.

Abstract:
We study the isoscalar (T=0) and isovector (T=1) pairing correlations in N=Z nuclei. They are estimated from the double difference of binding energies for odd-odd N=Z nuclei and the odd-even mass difference for the neighboring odd-mass nuclei, respectively. The empirical and BCS calculations based on a T=0 and T=1 pairing model reproduce well the almost degeneracy of the lowest T=0 and T=1 states over a wide range of even-even and odd-odd N=Z nuclei. It is shown that this degeneracy is attributed to competition between the isoscalar and isovector pairing correlations in N=Z nuclei. The calculations give an interesting prediction that the odd-odd N=Z nucleus 82Nb has possibly the ground state with T=0.

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 systematic investigation of the rotating $N=Z$ even-even nuclei in the mass $A=58-80$ region has been performed within the frameworks of the Cranked Relativistic Mean field, Cranked Relativistic Hartree Bogoliubov theories and cranked Nilsson-Strutinsky approach. Most of the experimental data is well accounted for in the calculations. The present study suggests that there is strong isovector $np$-pair field at low spin, the strength of which is defined by the isospin symmetry. At high spin, the isovector pair field is destroyed and the data are well described by the calculations assuming zero pairing. No clear evidence for the existence of the isoscalar $t=0$ $np$-pairing has been obtained in the present investigation.

Abstract:
A simple model based on the group SO(5) suggests that both the like-particle and neutron-proton components of isovector pairing correlations in odd-A nuclei are Pauli blocked. The same effect emerges from Monte Carlo Shell-model calculations of proton-rich nuclei in the full fp shell. There are small differences between the two models in their representation of the effects of an odd nucleon on the competition between like-particle and neutron-proton pairing, but they can be understood and reduced by using a two-level version of the SO(5) model. On the other hand, in odd-odd nuclei with N not equal to Z SO(5) disagrees more severely with the shell model because it incorrectly predicts ground-state isospins. The shell model calculations for any fp-shell nuclei can be extended to finite temperature, where they show a decrease in blocking.

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.