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Register a new userExact T=0 Eigenstates of the Isovector Pairing Hamiltonian
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We derive the exact $T=0$ seniority-zero eigenstates of the isovector pairing Hamiltonian for an even number of protons and neutrons. Nucleons are supposed to be distributed over a set of non-degenerate levels and to interact through a pairing force with constant strength. We show that these eigenstates (and among them, in particular, the ground state) are linear superpositions of products of $T=1$ collective pairs arranged into $T=0$ quartets. This grouping of protons and neutrons first into $T=1$ collective pairs and then into $T=0$ quartets represents the distinctive feature of these eigenstates. This work highlights, for the first time on the grounds of the analytic expression of its eigenstates, the key role played by the isovector pairing force in the phenomenon of nuclear quarteting.
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Neutron-proton (np-) pairing is expected to play an important role in the N Z nuclei. In general, it can have isovector and isoscalar character. The existence of isovector np-pairing is well established. On the contrary, it is still debated whether there is an isoscalar np-pairing. The review of the situation with these two types of pairing with special emphasis on the isoscalar one is presented. It is concluded that there are no substantial evidences for the existence of isoscalar np-pairing.
We address the problem of the bosonization of finite fermionic systems with two different approaches. First we work in the path integral formalism, showing how a truly bosonic effective action can be derived from a generic fermionic one with a quartic interaction. We then apply our scheme to the pairing hamiltonian in the degenerate case proving that, in this instance, several of the features characterizing the spontaneous breaking of the global gauge symmetry U(1) occurring in the infinite system persist in the finite system as well. Accordingly we interpret the excitations associated with the addition and removal of pairs of fermions as a quasi-Goldstone boson and the excitations corresponding to the breaking of a pair (seniority one states in the language of the pairing hamiltonian) as Higgs modes. Second, we face the more involved problem of a non-degenerate single particle spectrum, where one more kind of excitations arises, corresponding to the promotion of pairs to higher levels. This we do by solving directly the Richardson equations. From this analysis the existence emerges of critical values of the coupling constant, which signal the transition between two regimes, one dominated by the mean field physics, the other by the pairing interaction.
We present a new analysis of the pairing vibrations around 56Ni, with emphasis on odd-odd nuclei. This analysis of the experimental excitation energies is based on the subtraction of average properties that include the full symmetry energy together with volume, surface and Coulomb terms. The results clearly indicate a collective behavior of the isovector pairing vibrations and do not support any appreciable collectivity in the isoscalar channel.
The pairing correlation energy for two-nucleon configurations with the spin-parity and isospin of $J^pi=0^+$, $T$=1 and $J^pi=1^+$, $T$=0 are calculated with $T$=1 and $T$=0 pairing interactions, respectively. To this end, we consider the $(1f2p)$ shell model space, including single-particle angular momenta of $l=3$ and $l=1$. It is pointed out that a two-body matrix element of the spin-triplet $T$=0 pairing is weakened substantially for the $1f$ orbits, even though the pairing strength is much larger than that for the spin-singlet $T$=1 pairing interaction. In contrast, the spin-triplet pairing correlations overcome the spin-singlet pairing correlations for the $2p$ configuration, for which the spin-orbit splitting is smaller than that for the $1f$ configurations, if the strength for the T=0 pairing is larger than that for the T=1 pairing by 50% or more. Using the Hartree-Fock wave functions, it is also pointed out that the mismatch of proton and neutron radial wave functions is at most a few % level, even if the Fermi energies are largely different in the proton and neutron mean-field potentials. These results imply that the configuration with $J^pi=0^+$, $T$=1 is likely in the ground state of odd-odd $pf$ shell nuclei even under the influence of the strong spin-triplet $T$=0 pairing, except at the middle of the $pf$ shell, in which the odd proton and neutron may occupy the $2p$ orbits. These results are consistent with the observed spin-parity $J^{pi}=0^+$ for all odd-odd $pf$ shell nuclei except for $^{58}_{29}$Cu, which has $J^{pi}=1^+$.
We propose a particle number conserving formalism for the treatment of isovector-isoscalar pairing in nuclei with $N>Z$. The ground state of the pairing Hamiltonian is described by a quartet condensate to which is appended a pair condensate formed by the neutrons in excess. The quartets are built by two isovector pairs coupled to the total isospin $T=0$ and two collective isoscalar proton-neutron pairs. To probe this ansatz for the ground state we performed calculations for $N>Z$ nuclei with the valence nucleons moving above the cores $^{16}$O, $^{40}$Ca and $^{100}$Sn. The calculations are done with two pairing interactions, one state-independent and the other of zero range, which are supposed to scatter pairs in time-revered orbits. It is proven that the ground state correlation energies calculated within this approach are very close to the exact results provided by the diagonalization of the pairing Hamiltonian. Based on this formalism we have shown that moving away of N=Z line, both the isoscalar and the isovector proton-neutron pairing correlations remain significant and that they cannot be treated accurately by models based on a proton-neutron pair condensate.
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