Proton–neutron pairing in N = Z nuclei: Quartetting versus pair condensation

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.

Isoscalar-isovector proton-neutron pairing and quartet condensation inN=Znuclei

“…2 is that the isovector and isoscalar pairing correlations coexist for any ratio of the two pairing forces. A similar conclusion was obtained recently for the isovector and isoscalar pairing forces acting on time-reversed axially deformed states [19]. It is worth stressing that the majority of HFB calculations for isovector-isoscalar pairing predict the coexistence of the two pairing phases only for particular pairing forces and/or nuclei [20,21].…”

“…Thus, when the spin-orbit is so strong that the occupancy of j = l − 1/2 can be neglected, the matrix elements of the S = 0 and S = 1 pairing forces are reduced, in the limit of large j, by a factor 2 and 6, respectively. The ratio of the matrix elements (19) and (20) is equal to 3j/(j + 1), which is not very far from the ratio between the values of the two pairing energies shown in Fig. 1 at maximum spin-orbit energy splitting.…”

“…We have also shown that in the QCM approximation the correlations in the two pairing channels coexist for any admixture of isovector and isoscalar pairing forces, which confirms the findings of Refs. [17,19].…”

“…If the isoscalar quartet is further reduced to include only the ∆ + 0 pairs, this formalism becomes formally equivalent to the one proposed in Ref. [19] for the treatement of the isovector-isoscalar pairing forces acting on axially deformed states.…”

“…An extension of the QCM approach for the axially deformed isovector-isoscalar pairing interactions was recently proposed in Ref. [19].…”