$\alpha$-cluster structures and monopole excitations in ${}^{13}\mathrm{C}$

The structure of $1/2^{\pm }$ states in ${}^{13}\mathrm{C}$ up to around the $3\alpha +n$ threshold ($E_x=12.3$ MeV) is investigated with a full four-body $3\alpha +n$ orthogonality condition model (OCM) calculation, where the $3\alpha$ OCM, the model space of which is the subspace of the $3\alpha +n$ model, describes well the structure of the low-lying states of ${}^{12}\mathrm{C}$, including the $2_2^{+},0_3^{+}$, and $0_4^{+}$ states, which have been recently observed above the Hoyle state ($0_2^{+}$). A full spectrum up to the $1/2_5^{-}$ ($1/2_3^{+}$) state is reproduced consistently with the lowest five $1/2^{-}$ (three $1/2^{+}$) states of experimental spectrum. It is shown that the $1/2_2^{-}$ and $1/2_3^{-}$ states are characterized by the dominant cluster configurations of ${}^9\mathrm{Be}(3/2^{-},1/2^{-})+\alpha$, while the ground state $1/2_1^{-}$ has a shell-model-like structure. The observed monopole transition strengths to the $1/2_{2,3}^{-}$ states are consistently reproduced for the first time. These results indicate that the excited $1/2^{-}$ states have cluster structures. They are compared to those by the previous work with the shell model. However, the $1/2_1^{+}$ state is found to have a loosely bound neutron structure in which the extra neutron moves around ${}^{12}\mathrm{C}\left(\text{g.s.}\right)$ core with $1S$ orbit, reflecting the fact that this state appears by $1.9$ MeV just below the ${}^{12}\mathrm{C}(\mathrm{g}.\mathrm{s}.)+n$ threshold, while the $1/2_2^{+}$ and $1/2_3^{+}$ states are characterized by ${}^9\mathrm{Be}+\alpha$ structures. We found that the $1/2_5^{+}$ state located above the $3\alpha +n$ threshold is the Hoyle analog state in ${}^{13}\mathrm{C}$, the wave function of which is described by product states of constituent clusters, ${\left(0S\right)}_{\alpha }^3{\left(S\right)}_n$, with the probability of 52%.

Figure 1

Jacobi-coordinate systems in the $3\alpha +n$ model. Panels (a)–(d) correspond to the respective ($p\mathrm{th}$) Jacobi-coordinate systems ($p=1\sim 4$). See the text.

Figure 2

Energy spectra of ${}^{12}\mathrm{C}$ obtained by the $3\alpha$ OCM calculation with the three-body force $V_{3\alpha }$ in Eq. (7), compared with the experimental data (see the text for the details).

Figure 3

Low-lying energy levels of ${}^9\mathrm{Be}$ obtained by the $2\alpha +n$ OCM calculation together with the experimental data (see the text for the details).

Figure 4

Energy levels of the $1/2^{-}$ and $1/2^{+}$ states of ${}^{13}\mathrm{C}$ obtained by the $3\alpha +n$ OCM, compared with the experimental data. The experimental levels of the $1/2_{1,2,3,5}^{-}$ and $1/2_{1,2,3}^{+}$ states are taken from Ref. [58] and from Ref. [37] for the $1/2_4^{-}$ state, respectively. The threshold of the ${}^9\mathrm{Be}(5/2^{-})+\alpha$ channel at $E_x=13.1$ MeV, located between the ${}^9\mathrm{Be}(1/2^{+})+\alpha$ and ${}^9\mathrm{Be}(1/2^{-})+\alpha$ channels, is presented by the dashed arrow in the left-hand side of the panel.

Figure 5

Spectroscopic factors $S_L^2$ of the ${}^{12}\mathrm{C}\left(J_C^{\pi }\right)+n$ channels ($J_C^{\pi }=0_1^{+},1_1^{-},2_1^{+},3_1^{-},0_2^{+}$) and ${}^9\mathrm{Be}\left(J_9^{\pi }\right)+\alpha$ ($J_9^{\pi }=3/2_1^{-},1/2_1^{-},5/2_1^{-},1/2_1^{+}$) in the five $1/2^{-}$ states of ${}^{13}\mathrm{C}$ defined in Eq. (18).

Figure 6

Overlap amplitudes of the ${}^{12}\mathrm{C}+n$ channels and ${}^9\mathrm{Be}+\alpha$ channels for the five $1/2^{-}$ states of ${}^{13}\mathrm{C}$ defined in Eqs. (16) and (17). In the panels we present only the overlap amplitudes with the $S^2$ factor larger than $0.2$ in Fig. 5.

Figure 7

Spectroscopic factors $S_L^2$ of the ${}^{12}\mathrm{C}\left(J_C^{\pi }\right)+n$ channels ($J_C^{\pi }=0_1^{+},1_1^{-},2_1^{+},3_1^{-},0_2^{+}$) and ${}^9\mathrm{Be}\left(J_9^{\pi }\right)+\alpha$ ($J_9^{\pi }=3/2_1^{-},1/2_1^{-},5/2_1^{-},1/2_1^{+}$) in the five $1/2^{+}$ states of ${}^{13}\mathrm{C}$ defined in Eq. (18).

Figure 8

Overlap amplitudes of the ${}^{12}\mathrm{C}+n$ channels and ${}^9\mathrm{Be}+\alpha$ channels for the five $1/2^{+}$ states of ${}^{13}\mathrm{C}$ defined in Eqs. (16) and (17). In the panels we present only the overlap amplitudes with the $S^2$ factor larger than $0.2$ (see Fig. 7).

Figure 9

Radial behaviors of the dominant $S$-wave single-$\alpha$ orbit (real line) with the occupation probability of $48\%$ and the dominant $S$-wave extra neutron orbit, ${\left(S\right)}_n$, (dashed line) with the occupation probability of $65\%$ in the $1/2_5^{+}$ state, which are obtained by diagonalizing the single-$\alpha$-cluster density matrix and single-nucleon density matrix of the extra neutron defined in Eqs. (13) and (14), respectively. See the text for the details.