$3\alpha$ clustering in excited states of ${}^{16}\mathrm{C}$

The $\alpha$-cluster states of ${}^{16}\mathrm{C}$ are investigated by using antisymmetrized molecular dynamics. It is shown that two different types of $\alpha$-cluster states exist: triangular and linear-chain states. The former has an approximate isosceles triangular configuration of $\alpha$ particles surrounded by four valence neutrons occupying the $sd$ shell, while the latter has the linearly aligned $\alpha$ particles with ${\left(sd\right)}^2{\left(pf\right)}^2$ neutrons. It is found that the structure of the linear-chain state is qualitatively understood in terms of the $3/2_{\pi }^{-}$ and $1/2_{\sigma }^{-}$ molecular orbits as predicted by molecular-orbital model, but there exists a non-negligible ${}^{10}\mathrm{Be}+\alpha +2n$ correlation. The bandhead energies of the triangular and linear-chain rotational bands are 8.0 and 15.5 MeV, and the latter is close to the ${}^4\mathrm{He}+{}^{12}\mathrm{Be}$ and ${}^6\mathrm{He}+{}^{10}\mathrm{Be}$ threshold energies. It is also shown that the linear-chain state becomes the yrast state at $J^{\pi }={10}^{+}$ with $E_x=27.8$ MeV owing to its very large moment of inertia comparable with hyperdeformation.

Figure 1

The energy curve of the $J^{\pi }=0^{+}$ states as functions of quadrupole deformation parameter $\beta$ obtained by the angular momentum projection. Filled symbols show the energy minimum states for given values of $\beta$, while open symbols show local energy minima. There appear three different structures shown by circles, triangles, and boxes (see text). Dashed lines show the thresholds energies for $1n$, $2n$, and cluster decays.

Figure 2

The density distribution of the ground (a,b), triangular (c,d), and linear-chain (e,f) configurations at their energy minima. The contour lines show the proton density distributions and are common to the upper and lower panels. The color plots show the single-particle orbits occupied by four valence neutrons. The lower panels show the most weakly bound two neutrons, while the upper panel show the other two valence neutrons.

Figure 3

The schematic figure showing the $3/2_{\pi }^{-},1/2_{\pi }^{-}$, and $1/2_{\sigma }^{-}$ molecular orbits introduced in Ref. [19]. If the system has axial symmetry and the effect of the spin-orbit interaction is negligible, these orbits are the eigenstates of ${\widehat{j}}_z$ and ${\widehat{l}}_z$.

Figure 4

The boxes show the energy of the linear-chain configuration with $J^{\pi }=0^{+}$ as function of quadrupole deformation parameter $\gamma$. The solid line shows the overlap between the linear-chain state ($0_5^{+}$ state) and the basis wave functions.

Figure 5

The calculated and observed positive-parity energy levels of ${}^{16}\mathrm{C}$ up to $J^{\pi }={12}^{+}$ states. Open boxes show the observed states with the definite spin-parity assignments, and other symbols show the calculated result. The filled circles, triangles, and lines show the ground, triangular, and linear-chain bands, while lines show the states without cluster structure.