Simplified modeling of cluster-shell competition in carbon isotopes

We study the cluster-shell competition in carbon isotopes. Two parameters are introduced for the ${}^{12}\mathrm{C}$ core to characterize the transition from a cluster state to a shell-model state as a measure of the cluster-shell competition: one expresses the relative distance between α clusters and the other does the dissolution of one of the α clusters. We show the energy curves as functions of these parameters. The calculation is performed by combining the simplified modeling of cluster-shell competition for the ${}^{12}\mathrm{C}$ core and AMD triple-S. As the neutron number increases, the $\alpha \text{−}\alpha$ distance becomes smaller; on the contrary, the dissolution of α clusters is enhanced. The origins of the anomalously small $B(E2)$ value of ${}^{16}\mathrm{C}$ are discussed from cluster-shell competition point of view, and the appearance of pure cluster state around the 10 MeV region is suggested.

Figure 1

Energy convergence of ${}^{14}\mathrm{C}$ and ${}^{16}\mathrm{C}$ calculated with SMSO and AMD triple-S. The horizontal axis is the number of introduced basis states ($N$), where distribution of the valence neutrons is randomly generated. In the upper-right panel, the “Normal” and “$S_z=1$ mixed” show the calculations with $S_z=0$ only and $S_z=1$ mixed basis set, respectively.

Figure 2

Calculated energies of ${}^{12}\mathrm{C}$ (upper panel), ${}^{14}\mathrm{C}$ (middle panel), and ${}^{16}\mathrm{C}$ (lower panel) as functions of Λ. Solid and open circles, solid and open squares, solid and open triangles correspond to $R_1=3.0$, 2.5, 2.0, 1.5, 1.0, and 0.5 fm, respectively.

Figure 3

Energy levels of ${}^{16}\mathrm{C}$.

Figure 4

The $0^{+}$ energy levels of ${}^{16}\mathrm{C}$. Here “Base 1” shows the basis states with $R_1=0.5$ (fm) and $\Lambda =0.8$, and “Base 2” shows $R_1=2.5$ (fm) and $\Lambda =0.0$.