Faddeev Treatment of ${}_{}{}^6{}_{}{}^{}\mathrm{Li}$ with a Separable Potential

The nucleus ${}_{}{}^6{}_{}{}^{}\mathrm{Li}$, together with its first excited state ${}_{}{}^6{}_{}{}^{}\mathrm{Li}$*, is treated as a three-body problem consisting of an $\alpha$ particle, a proton, and a neutron, interacting through separable potentials in pairs. The $N-N$ potential includes a tensor force in addition to a central ($S$-wave) term of the Yamaguchi type. The $p$-wave part of the $\alpha -N$ potential is parametrized in terms of the sum of a central and a small spin-orbit term which fits the experimental $P_{\frac{3}{2}}$ and $P_{\frac{1}{2}}\alpha -N$ phase shifts quite accurately. The smallness of the spin-orbit part justifies its omission from the full three-body equations. The binding energies and the wave functions for the ground ($J^P=1^{+}$) and excited ($J^P=0^{+}$) states of ${}_{}{}^6{}_{}{}^{}\mathrm{Li}$ are calculated by solving the coupled Faddeev equations numerically. The calculated binding energy for the $1^{+}$ state is a little higher than, and that for the $0^{+}$ state a little lower than, the corresponding experimental values, corrected for Coulomb effects.