Alpha cluster structures and monopole excitations in $^{13}$C

The structure of $1/2^{\pm}$ states in $^{13}$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 $^{12}$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 $^{9}$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. On the other hand, the $1/2^{+}_{1}$ state is found to have a loosely bound neutron structure in which the extra neutron moves around $^{12}$C(g.s) core with $1S$ orbit, reflecting the fact that this state appears by $1.9$ MeV just below the $^{12}$C(g.s)+$n$ threshold, while the $1/2^{+}_{2}$ and $1/2^{+}_{3}$ states are characterized by $^{9}$Be+$\alpha$ structures. We found that the $1/2^+_5$ state located above the $3\alpha+n$ threshold is the Hoyle analogue state in $^{13}$C, the wave function of which is described by product states of constituent clusters, $(0S)^3_{\alpha}(S)_n$, with the probability of $52$ %.