By employing separable potentials for the binary, i.e., $n-n$ and n-core subsystems, we compute the three-body integral equations to solve for the $2n$-separation energies in the ground and excited states of halo nuclei like ${}^{19}\mathrm{B},$ ${}^{22}\mathrm{C},$ and ${}^{20}\mathrm{C}.$ We show through numerical analysis and also from analytical considerations that Borromean-type halo nuclei like ${}^{19}\mathrm{B}$ and ${}^{22}\mathrm{C},$ where $n-n$ and n-core are both unbound, are much less vulnerable to respond to the existence of the Efimov effect. On the contrary, those nuclei, like ${}^{20}\mathrm{C}$ in which the halo neutron is supposed to be in the intruder low lying bound state with the core, appear to be the promising candidates to search for the occurrence of Efimov states at energies below the $n-\mathrm{}\left(\mathrm{nc}\right)\mathrm{}$ breakup threshold which can be within the experimental limits to measure.

• Received 1 September 1999

DOI:https://doi.org/10.1103/PhysRevC.61.051303