${}^8\mathrm{He}$ and ${}^9\mathrm{Li}$ cluster structures in light nuclei

The possibility of the ${}^8\mathrm{He}$ and ${}^9\mathrm{Li}$ clusters in atomic nuclei is discussed. Until now most of the clusters in the conventional models have been limited to the closures of the three-dimensional harmonic oscillators, such as ${}^4\mathrm{He}$, ${}^{16}\mathrm{O}$, and ${}^{40}\mathrm{Ca}$. In the neutron-rich nuclei, however, the neutron-to-proton ratio ($N/Z$) is not unity, and it is worthwhile to think about more neutron-rich objects with $N>Z$ as the building blocks of cluster structures. Here the nuclei with the neutron number six, which is the subclosure of the $p_{3/2}$ subshell of the $jj$-coupling shell model, are assumed to be clusters, and thus we study the ${}^8\mathrm{He}$ and ${}^9\mathrm{Li}$ cluster structures in ${}^{16}\mathrm{Be}$ (${}^8\mathrm{He}+{}^8\mathrm{He}$), ${}^{17}\mathrm{B}$ (${}^8\mathrm{He}+{}^9\mathrm{Li}$), ${}^{18}\mathrm{C}$ (${}^9\mathrm{Li}+{}^9\mathrm{Li}$), and ${}^{24}\mathrm{C}$ (${}^8\mathrm{He}+{}^8\mathrm{He}+{}^8\mathrm{He}$). Recent progress of the antisymmetrized quasi-cluster model (AQCM) enables us to utilize $jj$-coupling shell-model wave functions as the clusters rather easily. It is shown that the ${}^8\mathrm{He}+{}^9\mathrm{Li}$ and ${}^9\mathrm{Li}+{}^9\mathrm{Li}$ cluster configurations cover the lowest shell-model states of ${}^{17}\mathrm{B}$ and ${}^{18}\mathrm{C}$, respectively. To predict cluster states with large relative distances, we increase the expectation value of the principal quantum numbers by adding the nodes to the lowest states under the condition that the total angular momentum is unchanged (equal to $J^{\pi }=0^{+}$). As a result, developed cluster states are obtained around the corresponding threshold energies. The rotational band structure of ${}^{24}\mathrm{C}$, which reflects the symmetry of equilateral triangular configuration ($D_{3h}$ symmetry) of three ${}^8\mathrm{He}$ clusters, also appears around the threshold energy. We suggest a novel mechanism whereby the spin-orbit interaction induces the clustering, which is distinctive of neutron-rich nuclei.

Figure 1

Expectation value for the principal quantum number of the harmonic oscillator ($n$) for the $0^{+}$ state of ${}^{16}\mathrm{Be}$ as a function of the distance between two ${}^8\mathrm{He}$ clusters. Dotted and dashed lines are results for the protons and neutrons, respectively.

Figure 2

Energy curves for ${}^{16}\mathrm{Be}$ measured from the two-${}^8\mathrm{He}$ threshold as a function of the distance between two ${}^8\mathrm{He}$ clusters. Solid, dotted, and dashed lines correspond to the $0^{+}$, $2^{+}$, and $4^{+}$ states, respectively.

Figure 3

Energy for the $3/2^{-}$ state of ${}^{17}\mathrm{B}$ measured from the ${}^9\mathrm{Li}\text{−}{}^8\mathrm{He}$ threshold as a function of the distance between ${}^9\mathrm{Li}$ and ${}^8\mathrm{He}$ (dashed line). Energy curve for the $3/2^{-}$ state orthogonal to the state having the optimal distance of 3 fm (solid line).

Figure 4

Energy for the $0^{+}$ state of ${}^{18}\mathrm{C}$ measured from the two-${}^9\mathrm{Li}$ threshold as a function of the distance between two ${}^9\mathrm{Li}$ clusters (dashed line). Energy curve for the $0^{+}$ state orthogonal to the state having the optimal distance of 2.5 fm (solid line).

Figure 5

Intrinsic density of ${}^9\mathrm{Li}\text{−}{}^9\mathrm{Li}$ with the relative distance of 4 fm on the $xz$ plane (${\mathrm{fm}}^{-3}$): (a) protons, (b) neutrons.

Figure 6

Expectation value for the principal quantum number of the harmonic oscillator for the $0^{+}$ state of ${}^{24}\mathrm{C}$ with the equilateral triangular configuration of three ${}^8\mathrm{He}$ clusters as a function of the distance between two ${}^8\mathrm{He}$ clusters. Dotted and dashed lines correspond to the result for the protons and neutrons, respectively.

Figure 7

Energy curves of ${}^{24}\mathrm{C}$ with the equilateral triangular configuration of three ${}^8\mathrm{He}$ clusters measured from the three-${}^8\mathrm{He}$ threshold as a function of the distance between two ${}^8\mathrm{He}$ clusters. (a) Positive-parity $K=0$ band, $0^{+}$, $2^{+}$, and $4^{+}$, and (b) negative-parity $K=3$ band, $3^{-}$, $4^{-}$, and $5^{-}$.

Figure 8

The spin-orbit energy for the $0^{+}$ state of ${}^{24}\mathrm{C}$ with the equilateral triangular configuration of three ${}^8\mathrm{He}$ clusters as a function of the ${}^8\mathrm{He}\text{−}{}^8\mathrm{He}$ distance. The dotted line at $-32.0$ MeV shows the spin-orbit energy three times the contribution in a free ${}^8\mathrm{He}$ cluster.

Figure 9

Rotational band structure of three-${}^8\mathrm{He}$ configuration. Solid and dashed lines are the results for the $K=0$ (positive parity, $0^{+},2^{+},4^{+},\cdots$) and $K=3$ (negative parity, $3^{-},4^{-},5^{-},\cdots$) bands, respectively.