Relation between shrinkage effect and compression of rotational spectrum in $^7_\Lambda$Li hypernucleus

It has been shown experimentally that the $^6$Li nucleus shrinks by adding a $\Lambda$ particle while its spectrum is also compressed. We discuss the compatibility of these two effects, contrasting also with a relation between the shrinkage effect and the stability of the spectrum in the $^9_\Lambda$Be hypernucleus. To this end, we employ two-body $d$-$^5_\Lambda$He and $\alpha$-$^5_\Lambda$He cluster models for the $^7_\Lambda$Li and $^9_\Lambda$Be hypernuclei, respectively. We first argue that a Gaussian-like interaction between two clusters leads to a stabilization of the spectrum against an addition of a $\Lambda$ particle, even though the intercluster distance is reduced. In the case of $^7_\Lambda$Li, the spin-orbit interaction between the intercluster motion and the deuteron spin has to be considered also. We show that the shrinkage effect makes the expectation value of the spin-orbit potential larger, lowering the excitation energy for the $^6$Li($3^+)\otimes \Lambda(1/2^+)$ level.