Cluster-shell competition in light nuclei

We demonstrate whether the cluster structure dissolves or remains when the shell-model-like model space is introduced in addition to the cluster model space in light nuclei. Although the binding energies of ${}^8\mathrm{Be}$, ${}^{10}\mathrm{Be}$, and ${}^{10}\mathrm{B}$ become larger by about 1–2 MeV by adding shell-model-like basis states to the $\alpha +\alpha +N+N+\cdots$ basis states, the $\alpha ‐\alpha$ structure is a dominant configuration of the ground states. However, $\alpha$-breaking wave functions strongly mix in ${}^{12}\mathrm{C}$, and the decrease of the energy from the $3\alpha$ configuration by about 6 MeV is a clue to resolving a long-standing problem of the binding energies of ${}^{12}\mathrm{C}$ and ${}^{16}\mathrm{O}$. The improved version of antisymmetrized molecular dynamics (AMD), AMD superposition of selected snapshots (AMD triple-S), is used to show the cluster-shell competition of these nuclei.

Figure 1

The energy curve of the $\alpha ‐\alpha$ system $\left(0^{+}\right)$ with respect to the distance between the centers of the two $\alpha$ clusters. The energy does not coincide with twice the energy of the $\alpha$ cluster $(2\times -27.57\mathrm{MeV})$ even at a large $\alpha ‐\alpha$ distance, since the relative distance is fixed. The corresponding kinetic energy of the zero-point oscillation is $\hslash \omega ∕4=4.86\mathrm{MeV}$.

Figure 2

The energy convergence of ${}^8\mathrm{Be}\left(0^{+}\right)$ with respect to the number of trial AMD basis states. The first three basis states have the $\alpha +\alpha$ structure with the relative distances of 2, 3, and 4 fm. The energy calculated using these three is shown as the dotted line, and the basis states of the $\alpha +p+p+n+n$ model space are included in the calculations represented by the lower-lying dots.

Figure 3

The energy convergence of ${}^{10}\mathrm{Be}\left(0^{+}\right)$ with respect to the number of trial AMD wave functions. The basis states from 1 to 300 are those of the $\alpha +\alpha +n+n$ model space with the relative $\alpha ‐\alpha$ distance of 2, 3, and 4 fm, and after 300, shell-model-like wave functions of the $\alpha +2p+4n$ model space are added.

Figure 4

The energy convergence of ${}^{10}\mathrm{B}\left(3^{+}\right)$ with respect to the number of trial AMD wave functions. The basis states from 1 to 200 are those of the $\alpha +\alpha +p+n$ model space with the relative $\alpha ‐\alpha$ distance of 2, 3, and 4 fm, and after 200, shell-model-like wave functions of the $\alpha +3p+3n$ model space are added.

Figure 5

The energy curve of ${}^{12}\mathrm{C}$ with respect to the expectation value of the principal quantum number $({\vec{a}}^{\mathrm{†}}\bullet \vec{a})$. The solid line and dotted line represent the results with the spin-orbit interaction and without it, respectively.

Figure 6

The energy convergence of ${}^{12}\mathrm{C}\left(0^{+}\right)$ with respect to the number of trial AMD basis states. The basis states from 1 to 100 are those of the $3\alpha$ model space, and those from 101 to 600 are $\alpha +\alpha +2p+2n$ model space with relative $\alpha ‐\alpha$ distances of 2, 3, and 4 fm. After 601, the wave functions with the shell-model-like $\alpha +4p+4n$ model space are added.

Figure 7

The nuclear chart of the cluster-shell competition. $\mathrm{\Delta }$ represents the increase of the binding energy when the breaking of one of the $\alpha$ clusters is taken into account. For ${}^{12}\mathrm{C}$, $\mathrm{\Delta }\left(2\right)$ is also shown, which is the increase of the binding energy when the breaking of two of the $\alpha$ clusters is taken into account.