Movitated by the on-going $\gamma$-ray experiments at the Alternating Gradient Synchrotron at Brookhaven National Laboratory, we discuss the energy splittings of the ${5/2}_1^{+}-{3/2}_1^{+}$ doublet in ${}_{\Lambda }^9\mathrm{Be}$ and the ${3/2}_1^{-}-{1/2}_1^{-}$ doublet in ${}_{\Lambda }^{13}\mathrm{C}$ for which the $\Lambda N$ spin-orbit ( $\mathrm{LS}$) and antisymmetric spin-orbit ( $\mathrm{ALS}$) forces are relevant. In the microscopic $2\alpha +\Lambda$ ( $3\alpha +\Lambda$) model for ${}_{\Lambda }^9\mathrm{Be}$ ( ${}_{\Lambda }^{13}\mathrm{C}$), all the available Nijmegen one-boson-exchange (OBE) model $\Lambda N$ interactions lead to a wide range of splittings of $0.08–0.20\mathrm{MeV}$ in ${}_{\Lambda }^9\mathrm{Be}$ and $0.39–0.96\mathrm{MeV}$ in ${}_{\Lambda }^{13}\mathrm{C}$. On the other hand, if we use information from quark-model $\Lambda N$ interactions which have generally large $\mathrm{ALS}$ force, the splittings become about half of the smallest OBE model prediction.

• Received 14 July 1999

DOI:https://doi.org/10.1103/PhysRevLett.85.270