The effects of the spin-orbit and tensor interactions in nuclei

Abstract

We devise a nucleon-nucleon interaction V c + xV so + yV t to study the effects of the spin-orbit and tensor interactions by varying x and y. For negative parity 1 p−1 h states in 16O, we find that in a TDA calculation, the weighted sum of the eigenenergies Σ J (2 J + 1) Σ i E i ( J, T) is zero for a pure two-body spin-orbit interaction or a pure tensor interaction. If a minor closed shell nucleus like 12C is taken as the core, the tensor interaction lowers the p 1 2 single-particle energy relative to p 3 2 . This is precisely opposite to what the two-body spin-orbit interaction does. For the 1 + states in 12C, a 1 p−1 h diagonalization leads to a near collapse with E ∗(T = 0) ∼ 0.9 MeV and E ∗(T = 1) ∼ 3.7 MeV as compared with experiment (12.7 MeV for T = 0 and 15.1 MeV for T = 1). Only a full scalar shell model calculation brings these states up to a respectable energy. We find that in a full shell model calculation, the excitation energies of these 1 + states are surprisingly insensitive to the spin-orbit interaction over a rather wide range of the parameter x and there does not appear to be any phase transition as we vary x. We reconsider the old problem of the effect of the spin-orbit and tensor interactions on the nearly vanishing Gamow-Teller matrix element 14C( J = 0, T = 1) → 14N( J = 1, T = 0). We calculate the single-particle energies with the same interaction that is used for the residual particle-particle or particle-hole matrix elements.