Effect of the Tensor Force on the Level Structure ofLi6andLi7

Abstract

The splittings of the ${}_{}{}^3{}_{}{}^{}S_1^{}{}_{}{}^{}$ and ${}_{}{}^1{}_{}{}^{}S_0^{}{}_{}{}^{}$ states of ${\mathrm{Li}}^6$ and the $P$ doublet states of ${\mathrm{Li}}^7$ by the tensor force are calculated by a variational method which includes the effect of configuration interaction. Other forces which would contribute to the splitting, such as spin-dependent central forces or vector spin-orbit forces, are not considered. The method of calculation is based on the use of a variational function of the form $\psi ={\psi }_0+\lambda t^{\prime }{\psi }_0$, where $t^{\prime }$ is essentially the tensor force, treated as a perturbation on a central force oscillator wave function, ${\psi }_0$. The effect of the tensor force is shown to be equivalent to a mixture of ordinary and spin-exchange central forces plus a vector type spin-orbit force of rather complicated structure. Using a Hu-Massey Gaussian shape tensor potential, an $S$-state splitting of 1.4 Mev is found for ${\mathrm{Li}}^6$ and an inverted $P$-doublet splitting of 380 kev is found for ${\mathrm{Li}}^7$. A Yukawa shape potential would give similar results. In view of the approximations made in the analysis, these results are in reasonable agreement with the experimental splittings of 3.5 Mev and 480 kev for ${\mathrm{Li}}^6$ and ${\mathrm{Li}}^7$ respectively. The tensor force is found to contribute about 12 Mev to the binding energy of these nuclei and to introduce a 6 percent admixture of excited states into the ground state. The importance of configuration interaction is shown by a second-order perturbation calculation neglecting configuration interaction which gives entirely different results—a negligible $S$-state splitting for ${\mathrm{Li}}^6$ and a normal $P$ doublet structure for ${\mathrm{Li}}^7$. The effect of the tensor force on the $P$-doublet separation in ${\mathrm{Be}}^7$ and the $F$ doublet separation in ${\mathrm{Li}}^7$ is discussed briefly.