Analysis ofBe9ΛandC13Λ

The hypernuclei ${}_{\Lambda }{}^{}{}_{}{}^{}\mathrm{Be}_{}^9{}_{}{}^{}$ and ${}_{\Lambda }{}^{}{}_{}{}^{}\mathrm{C}_{}^{13}{}_{}{}^{}$ are analyzed with the core nuclei treated in the alpha-particle model. Using a 2-body $\alpha -\alpha$ potential that fits the low-energy ${}_{}{}^1{}_{}{}^{}S_0^{}{}_{}{}^{}$ phase shifts and an effective 3-body interaction which is introduced to simulate the effect of nucleon exchanges where all three alpha clusters are involved, the binding energy and charge form factors of ${\mathrm{C}}^{12}$ can be correctly accounted for. The $\alpha -\Lambda$ interaction is obtained by folding a purely attractive $\Lambda$-nucleon potential into the nucleon density distribution of the alpha particle, with the depth determined from the binding energy $B_{\Lambda }$ of ${}_{\Lambda }{}^{}{}_{}{}^{}\mathrm{He}_{}^5{}_{}{}^{}$. Two $\alpha -\Lambda$ potentials have been considered. These correspond to purely attractive $\Lambda$-nucleon interactions with intrinsic ranges appropriate to 2-pion and $K$-meson exchanges, respectively. The calculation is done with a Monte Carlo method, with both the upper and lower bounds of the eigenvalue determined. The results show that to obtain agreement with the experimentally determined values of ${\mathrm{B}}_{\Lambda }$ for ${}_{\Lambda }{}^{}{}_{}{}^{}\mathrm{Be}_{}^9{}_{}{}^{}$ and ${}_{\Lambda }{}^{}{}_{}{}^{}\mathrm{C}_{}^{13}{}_{}{}^{}$, it is necessary to use a $\Lambda$-nucleon potential with an intrinsic range less than about 1 F. Together with the conclusion reached by Bodmer, from a comparison of the hypernuclear results with the low-energy $\Lambda -p$ scattering data, that purely attractive $\Lambda$-nucleon interactions with intrinsic ranges $b\lesssim 1$ F can be ruled out, our analysis gives a strong indication that the $\Lambda$-nucleon interaction is not purely attractive, but has a repulsive core. Also, it is found from our calculation that the values for $B_{\Lambda }$ are very sensitive to the structure of the core nuclei, which indicates that $\Lambda$ particles may be used as a probe to study the structure of complex nuclei.