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

We use a microscopic cluster model to investigate the ${}^{15}\mathrm{C}$ nucleus. Two variants are considered: a two-cluster ${}^{14}\mathrm{C}+n$ description and a three-cluster ${}^{13}\mathrm{C}+n+n$ approach. We show that ${}^{15}\mathrm{C}$ bound-state energies converge to similar values, provided that the number of ${}^{14}\mathrm{C}$ 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 ${}^{15}\mathrm{F}$ spectrum, where all known states are unbound.

Figure 1

${}^{14}\mathrm{C}$ states included in the ${}^{14}\mathrm{C}+n$ two-cluster model. Only the lowest experimental states are shown.

Figure 2

Convergence of the ${}^{15}\mathrm{C}$ binding energies for $J=1/2^{+}$ (top) and $J=5/2^{+}$ (bottom). For the ${}^{13}\mathrm{C}+n+n$ model, ${\ell }_x$ corresponds to the maximum value of the $n+n$ orbital momentum. The dotted lines represent the ${}^{14}\mathrm{C}$ energy.

Figure 3

Energy curves in the three-cluster (a) and two-cluster (b) models for $J=1/2^{+}$. The dashed lines correspond to a limited basis (see text).

Figure 4

As in Fig. 3 but for $J=1/2^{-}$.

Figure 5

${}^{15}\mathrm{C}$ spectrum in the three- and two-cluster models.

Figure 6

${}^{15}\mathrm{F}$ spectrum in the three- and two-cluster models.