Equivalence between two- and three-cluster models: Application to the ${}_{15}C$ and ${}_{15}F$ nuclei

We use a microscopic cluster model to investigate the ${}_{15}C$ nucleus. Two variants are considered: a two-cluster ${}_{14}C+n$ description and a three-cluster ${}_{13}C+n+n$ approach. We show that ${}_{15}C$ bound-state energies converge to similar values, provided that the number of ${}_{14}C$ states is sufficient. We also study negative-parity states, and show that the three-cluster model is more appropriate. With the same nucleon-nucleon interaction, we analyze the ${}_{15}F$ spectrum, where all known states are unbound.