$_{\Lambda}^9 $Be $(\frac{3}{2}^+,\frac{5}{2}^+)$ spin-orbit splitting within three-body cluster model

The configuration-space Faddeev equations have been applied to study spin-flip spacing for the first excited states of $_{\Lambda}^9 $Be hypernucleus considered as a three-body cluster system $\alpha \alpha \Lambda $[1]. In this model the phenomenological potentials for inter-cluster interactions are used [2]. Calculated are the binding energy for the ground and first excited states and $(\frac{3}{2}^+,\frac{5}{2}^+)$ spin-orbit splitting for different types of $\alpha \Lambda$ and $\alpha \alpha$ potentials. The overbinding of the system when Ali-Bodmer type $\alpha \alpha$ potential is used and dependence of the interaction on the orbital quantum number $\alpha \Lambda$ are discussed. The spin-orbit component of $\alpha \Lambda$ interaction is taken by the potential [3] having one range Gaussian form. Our results are in a qualitative agreement with those obtained with the combined RGM and Faddeev cluster calculations [3]. Cluster model that includes a repulsive three cluster $\alpha \alpha \Lambda$ potential [4] was also studied. 1. O. Hashimoto, H. Tamura, Prog.Part. Nucl. Phys. 57, 564 (2006); 2. E. Cravo, A.C. Fonseca, Y. Koike, PRC 66 (2002) 014001; 3. http://xxx.lanl.gov/find/nucl-th/1/au:+Fujiwara\_Y/0/1/0/all/0/1 Y. Fujiwara, at al. PRC70, 047002 (2004); 4. M. Shoeb, PRC74, 064316 (2006).