Tensor Coupling and the Vector Shell Model

Abstract

Because of the large tensor force contribution to the nuclear binding energy, which results in a situation of near degeneracy for the low-lying states, the tensor force may produce a large mixing of the low states even in the absence of tensor matrix elements between them. The type of coupling among the low states resulting from this near degeneracy is investigated by perturbation theory; the high-lying states, which are considered largely responsible for the effects of the tensor force, are eliminated by applying closure. An intermediate-coupling model for the low states emerges which is very similar to the customary one based on a vector force in that the effective nuclear potential for the low states is shown to consist of a central two-body force (with a spin-dependence different from that of the elementary two-body central force), plus a strong vector force, plus a tensor force which is probably weaker than the elementary two-body tensor force. The effective vector force is principally a three-body force, and hence may be expected to show a quite different "hole"-particle relationship than the one- or two-body vector forces usually assumed in the shell model. Because of the neglect in the wave function of the high-lying states which are mixed in directly by the tensor force, the model is expected to be valid for light nuclei only. The $\ensuremath{\beta}$ decay of ${\mathrm{B}}^{12}$ is discussed briefly, and is shown to be compatible with the tensor force as the sole spin-orbit force in the elementary two-body interaction.