Important role of the spin-orbit interaction in forming the1/2+orbital structure in Be isotopes

The structure of the second $0^{+}$ state of ${}^{10}\mathrm{Be}$ is investigated using a microscopic $\alpha +\alpha +n+n$ model based on the molecular-orbit (MO) model. The second $0^{+}$ state, which has dominantly the ${(1/2\mathrm{}}^{+}{)}^2$ configuration, is shown to have a particularly enlarged $\alpha$-$\alpha$ structure. The kinetic energy of the two valence neutrons occupying along the $\alpha$-$\alpha$ axis is reduced remarkably due to the strong $\alpha$ clustering and, simultaneously, the spin-orbit interaction unexpectedly plays an important role in making the energy of this state much lower. The mixing of states with different spin structure is shown to be important in negative-parity states. The experimentally observed small-level spacing between $1^{-}$ and $2^{-}$ $(\sim 300$ keV) is found to be evidence of this spin-mixing effect. ${}^{12}\mathrm{Be}$ is also investigated using the $\alpha +\alpha +4n$ model, in which four valence neutrons are considered to occupy the ${(3/2\mathrm{}}^{-}{)}^2{(1/2\mathrm{}}^{+}{)}^2$ configuration. The energy surface of ${}^{12}\mathrm{Be}$ is shown to exhibit similar characteristics, that the remarkable $\alpha$ clustering and the contribution of the spin-orbit interaction make the binding of the state with the ${(3/2\mathrm{}}^{-}{)}^2{(1/2\mathrm{}}^{+}{)}^2$ configuration properly stronger in comparison with the closed p-shell ${(3/2\mathrm{}}^{-}{)}^2{(1/2\mathrm{}}^{-}{)}^2$ configuration.