Consistency of two different approaches to determine the strength of a pairing residual interaction in the rare-earth region

Two fits of the pairing residual interaction in the rare-earth region are independently performed. One is made on the odd-even staggering of masses by comparing measured and explicitly calculated three-point binding-energy differences centered on odd-even nuclei. Another deals with the moments of inertia of the first $2^{+}$ states of well-deformed even-even nuclei upon comparing experimental data with the results of Inglis-Belyaev moments (supplemented by a crude estimate of the so-called Thouless-Valatin corrections). The sample includes 24 even-even and 31 odd-mass nuclei selected according to two criteria: They should have good rotor properties and should not correspond to low pairing-correlation regimes in their ground states. Calculations are performed in the self-consistent Hartree-Fock plus BCS framework (implementing a self-consistent blocking in the case of odd-mass nuclei). The Skyrme SIII parametrization is used in the particle-hole channel and the fitted quantities are the strengths of $|T_z|=1$ proton and neutron seniority residual interactions. As a result, the two fits yield sets of strengths in excellent agreement: about 0.1% for the neutron parameters and 0.2% for protons. In contrast, when one performs such a fit on odd-even staggering from quantities deduced from BCS gaps or minimal quasiparticle energies in even-even nuclei, as is traditional, one obtains results significantly different from those obtained in the same nuclei by a fit of moments of inertia. As a conclusion, beyond providing a phenomenological tool for microscopic calculations in this region, we have illustrated the proposition performed in the seminal paper of Bohr et al. [Phys. Rev. 110, 936 (1958)] that moments of inertia and odd-even staggering in selected nuclei were excellent measuring sticks of nuclear pairing correlations. Furthermore, we have assessed the validity of our theoretical approach which includes simple yet apparently reasonable assumptions (seniority residual interaction, parametrization of its matrix elements as functions of the nucleon numbers, and global Thouless-Valatin renormalization of Inglis-Belyaev moments of inertia).

Figure 1

The nuclear mass region of interest with a total of 24 even-even, 17 odd-neutron, and 14 odd-proton nuclei considered in this paper. The ground-state experimental quantum numbers $I^{\pi }$ are displayed (as discussed in the text, they are assumed here to correspond to the $K^{\pi }$ quantum numbers). Whenever our lowest-energy solutions $K^{\pi }$ values are inconsistent with the data, two sets of quantum numbers are displayed. In each box, the upper panel corresponds to the data where the use of parentheses means that these numbers are simply assumed, whereas the lower panel corresponds to our calculated ground-state solutions.