Effects of $\alpha$-cluster breaking on $3\alpha$-cluster structures in ${}^{12}\mathrm{C}$

To clarify the effects of $\alpha$-cluster breaking on $3\alpha$-cluster structures in ${}^{12}\mathrm{C}$, we investigate ${}^{12}\mathrm{C}$ using a hybrid model that combines the Brink-Bloch cluster model with the $p_{3/2}$ subshell closure wave function. We have found that $\alpha$-cluster breaking caused by spin-orbit force significantly changes cluster structures of excited $0^{+}$ states through orthogonality to lower states. Spatially developed cluster components of the $0_2{}^{+}$ state are reduced. The $0_3{}^{+}$ state changes from a vibration mode in the bending motion of three $\alpha$ clusters to a chainlike $3\alpha$ structure having an open triangle configuration. As a result of these structure changes of $0^{+}$ states, the band assignment for the $2_2{}^{+}$ state is changed by the $\alpha$-cluster breaking effect. Namely, in model calculations without the $\alpha$-cluster breaking effect, the $0_2{}^{+}$ state is assigned to be the band-head of the $2_2{}^{+}$ state. However, when we incorporate $\alpha$-cluster breaking caused by the spin-orbit force, the $0_3{}^{+}$ state is regarded as the bandhead of the $2_2{}^{+}$ state.

Figure 1

Schematic of a $3\alpha$-cluster structure. The configuration is characterized by the distances between $\alpha$ clusters, $d_1$ and $d_2$, and the bending angle $\theta$.

Figure 2

Comparison of calculated energy levels of $0^{+}$ and $2^{+}$ states with experimental ones. Energies are measured from the $3\alpha$ threshold energy. Theoretical and experimental threshold energies are $-82.9$ and $-84.9$ MeV, respectively. Levels labeled as “$3\alpha$” and “$3\alpha +p_{3/2}$” show cluster model results and present results for $u_{\mathrm{ls}}=1600$ MeV, respectively.

Figure 3

Change in energy levels of $0^{+}$ states against the strength of the spin-orbit force $u_{\mathrm{ls}}$. Solid line is the energy of the $p_{3/2}$ subshell closure wave function.

Figure 4

$0^{+}$ energy surface of a single BB wave function for $3\alpha$ clusters in which equal distances between $\alpha$ clusters are assumed, i.e., $d_1=d_2\equiv d$.

Figure 5

Overlap surfaces between $0^{+}$ GCM states and single BB wave functions for $u_{\mathrm{ls}}=0,1600,3200$ MeV are shown in left, middle, and right columns, respectively. For the BB wave function, the isosceles triangle structure, $d_1=d_2\equiv d$, is assumed.

Figure 6

Overlap curves between $0^{+}$ GCM states and single BB wave functions for $u_{\mathrm{ls}}=0$ MeV $(3\alpha )$ and $u_{\mathrm{ls}}=1600$ MeV $(3\alpha +p_{3/2})$ on the $d=6.0-2.5\theta /{180}^{\circ }$ line.

Figure 7

Comparison of levels from a semimicroscopic $3\alpha$-cluster model (${\mathrm{OCM}}_{\text{K}}$ [27, 28] and ${\mathrm{OCM}}_{\text{O}}$ [29]), microscopic $3\alpha$-cluster models (THSR [15], GCM [9], RGM [10], and $3\alpha )$, and models including $\alpha$-cluster breaking component $(3\alpha +p_{3/2}$, FMD [6], and AMD [5]) with experimental data [25, 36].