Wave function of ${}_9\mathrm{Be}$ in the three-body $\left(\alpha \alpha n\right)$ model

A simple analytic expression of the three-body wave function describing the system $\left(\alpha \alpha n\right)$ in the ground state ${\frac{3}{2}}^{-}$ of ${}_9\mathrm{Be}$ is obtained. In doing this, it is assumed that the $\alpha$ particles interact with each other via the $S$-wave Ali-Bodmer potential including the Coulomb term, and the neutron-$\alpha$ forces act only in the $P$-wave state. This wave function is constructed by trial and error method via solving in this way a kind of inverse problem when the two-body $\alpha \alpha$ potential is recovered from a postulated three-body wave function. As a result, the wave function is an exact solution of the corresponding three-body Schrödinger equation for experimentally known binding energy and for the $\alpha \alpha$ potential whose difference from the Ali-Bodmer one is minimized by varying the adjustable parameters on which the postulated wave function depends.