Tetrahedral $4\alpha$ and ${}^{12}\mathrm{C}+\alpha$ cluster structures in ${}^{16}\mathrm{O}$

I have investigated structures of the ground and excited states of ${}^{16}\mathrm{O}$ with the method of variation after spin-parity projection in the antisymmetrized molecular dynamics model combined with the generator coordinate method of ${}^{12}\mathrm{C}+\alpha$ cluster. The calculation reasonably reproduces the experimental energy spectra; $E2, E3, E4$, and $IS1$ transitions; and $\alpha$-decay properties. The formation of $4\alpha$ clusters has been confirmed from nucleon degrees of freedom in the AMD model without assuming the existence of any clusters. They form “tetrahedral” $4\alpha$- and ${}^{12}\mathrm{C}+\alpha$ cluster structures. The ${}^{12}\mathrm{C}+\alpha$ structure constructs the $K^{\pi }=0^{+}$ band consisting of the $0_2^{+}, 2_1^{+}$, and $4_1^{+}$ states and the $K^{\pi }=0^{-}$ band of the $1_2^{-}, 3_2^{-}$, and $5_1^{-}$ states. The $0_1^{+}, 3_1^{-}$, and $4_2^{+}$ states are assigned to the ground band constructed from the tetrahedral $4\alpha$ structure. The $0_1^{+}$ and $3_1^{-}$ are approximately interpreted as $T_d$ band members with the ideal tetrahedral configuration. The ground-state $4\alpha$ correlation plays an important role in the enhancement of the $E3$ transition strength to the $3_1^{-}$. The $4_2^{+}$ state is not the ideal $T_d$ member but constructed from a distorted tetrahedral $4\alpha$ structure. Moreover, significant state mixing of the tetrahedral $4\alpha$ and ${}^{12}\mathrm{C}+\alpha$ cluster structures occurs between $4_1^{+}$ and $4_2^{+}$ states, indicating that the $T_d$ configuration of $4\alpha$ is rather fragile at $J^{\pi }=4^{+}$.

Figure 1

Energy levels of ${}^{16}\mathrm{O}$ calculated by (a) the VAP and (b) $\mathrm{VAP}+\alpha \mathrm{GCM}$ and (c) those of the experimental data. The positive- and negative-parity levels are shown by circles and triangles, respectively. The ground and ${}^{12}\mathrm{C}+\alpha$ bands are connected by dashed and solid lines, respectively.

Figure 2

(Color on-line) Density distributions in the intrinsic states obtained by the VAP for the $0_1^{+}, 4_1^{+}, 3_1^{-}, 1_1^{-}$, and $2_1^{-}$ states. The densities integrated along the $Y, X$, and $Z$ axes are plotted on the (left) $X-Z$, (middle) $Y-Z$, and (right) $X-Y$ planes, respectively.

Figure 3

Same as Fig. 2 but for the $0_2^{+}, 2_1^{+}, 4_2^{+}$, and $5_1^{-}$ states.

Figure 4

${}^{12}\mathrm{C}\left(0_1^{+}\right)+\alpha$ component in the positive-parity states obtained by the $\mathrm{VAP}+\alpha \mathrm{GCM}$.

Figure 5

Same as Fig. 4 but for the negative-parity states.

Figure 6

Occupation probability of $N$ shells in the harmonic oscillator expansion in the states obtained by the VAP.

Figure 7

Occupation probability of $N$ shells in the harmonic oscillator expansion in the states obtained by the $\mathrm{VAP}+\alpha \mathrm{GCM}$.