Spin-orbit splitting in nuclei near closed shells

Abstract

We have studied the spin-orbit (s.o.) splitting arising from the effective two-body s.o. interaction in the Brueckner-Hartree-Fock (BHF) approximation, considering all of the most commonly used realistic nucleon-nucleon (NN) interactions and three different nuclear models. We conclude that the s.o. interaction alone cannot account for the observed s.o. splitting in nuclei. We derive the one-body s.o. potential arising from the effective NN s.o. interaction in the BHF approximation. The derivation applies to the interaction of a nucleon with a spin-saturated (s.s.) core of nucleons. The expression we obtain is more than a factor of two stronger than some similar expressions obtained by previous investigators. The enhancement results from the exchange term, which had previously been ignored. We discuss, and obtain expressions for, the effective s.o. interaction arising, in the local-density approximation, from the most commonly used realistic NN potentials, namely the Reid soft core (RSC), Reid hard core (RHC), Hamada-Johnston (HJ), Yale, and Gammel-Thaler (GT) potentials. We find that they yield two distinct sets of values for the strength, $\text{−}\text{1}\text{2}\text{πS}{}^{30}$, of the one-body potential: the RSC, RHC, and HJ potentials give $\text{−}\text{1}\text{2}\text{πS}{}^{30}\text{≈ 59}\text{MeV}\text{·}\text{fm}{}^5$ while the Yale and GT potentials give $\text{−}\text{1}\text{2}\text{πS}{}^{30}\text{≈ 73}\text{MeV}\text{·}\text{fm}{}^5$. We have calculated the s.o. splitting for three different nuclear models: (a) the spherical harmonic oscillator (h.o.) shell model, (b) a hybrid model which uses a Fermi distribution to describe the nucleons in the s.s. core and h.o. orbitals to describe the single-particle levels, and (c) the self-consistent Brueckner model of Negele. For each of these, quantitative results for the splitting are obtained for the five modern NN potentials listed above. Significant differences between the three models are found, models (b) and (c) giving better agreement with experiment than (a) for the heavier nuclei. We have also used the h.o. model to verify the numerical accuracy of our approach. We find that the effective s.o. interaction gives splittings which are roughly compatible with experiment for the normally unoccupied levels of the entirely s.s. nuclei ¹⁶O and ⁴⁰Ca. For all the other levels of the nuclei studied, it appears that the s.o. interaction alone lacks sufficient strength to account for the entire s.o. splitting.