$2\alpha +t$ cluster feature of the $3/2_3^{-}$ state in ${}^{11}\mathrm{B}$

We reanalyze $2\alpha +t$ cluster features of $3/2^{-}$ states in ${}^{11}\mathrm{B}$ by investigating the $t$-cluster distribution around a $2\alpha$ core in ${}^{11}\mathrm{B}$, calculated with the method of antisymmetrized molecular dynamics. In the $3/2_3^{-}$ state, a $t$ cluster is distributed in a wide region around $2\alpha$, indicating that the $t$ cluster moves rather freely in angular as well as radial motion. From the weak angular correlation and radial extent of the $t$-cluster distribution, we propose an interpretation of a $2\alpha +t$ cluster gas for the $3/2_3^{-}$ state. In this study, we compare the $2\alpha +t$ cluster feature in ${}^{11}\mathrm{B}(3/2_3^{-}$) with the $3\alpha$ cluster feature in ${}^{12}\mathrm{C}(0_2^{+}$) and discuss their similarities and differences.

Figure 1

Schematic figure of $2\alpha +t$ cluster structure.

Figure 2

Energies of $3/2^{-}$ states of ${}^{11}\mathrm{B}$ measured from the $2\alpha +t$ threshold and those of $0^{+}$ states of ${}^{12}\mathrm{C}$ measured from the $3\alpha$ threshold. ${}^7\text{Li}+\alpha$ and ${}^8\text{Be}+\alpha$ threshold energies are also shown. Theoretical values calculated using $m=0.62$ and those for $m=0.60$ are shown in comparison with the experimental energy levels.

Figure 3

(Top) Energy curve of the $\alpha +t$ system for ${}^7\mathrm{Li}$ and that of the $2\alpha$ system for ${}^8\mathrm{Be}$. The $J^{\pi }=3/2^{-}$ and $J^{\pi }=0^{+}$ projected energies are plotted as functions of the intercluster distance $D_{\alpha -\alpha \left(t\right)}$ of the BB cluster wave function. (Bottom) GCM amplitudes for the ${}^7\mathrm{Li}(3/2^{-}$) state were obtained by the $\alpha +t$ cluster GCM calculation, and those for the ${}^8\mathrm{Be}(0^{+}$) state were obtained by the $2\alpha$ cluster GCM calculation.

Figure 4

Energy of the $(J^{\pi },K)=(3/2^{-},+3/2)$ states projected from BB wave functions for the $2\alpha +t$ cluster and that of the $0^{+}$ states for $3\alpha$ cluster calculated with m62. This figure shows the energy surface on the $(R_z,R_x)$ plane for a fixed $2\alpha$ core with the $2\alpha$ distance $D=3$, 4, and 5 fm.

Figure 5

Overlap $U(D,\mathbf{R})$ of the BB wave function for $2\alpha +t$ cluster with the $\mathrm{AMD}+\mathrm{VAP}$ wave functions for ${}^{11}\mathrm{B}\left(3/2_1^{-}\right),{}^{11}\mathrm{B}(3/2_2^{-}$), and ${}^{11}\mathrm{B}(3/2_3^{-}$) calculated with the m62 interaction. The overlap is plotted on the $(R_z,R_x)$ plane for $D=3$, 4, and 5 fm. The $\alpha$ positions $(R_z,R_x)=(\pm D/2,0)$ in the $2\alpha$ core are shown by black circles.

Figure 6

Overlap $U(D,\mathbf{R})$ of the BB wave function for a $3\alpha$ cluster with the $\mathrm{AMD}+\mathrm{VAP}$ wave functions for ${}^{12}\mathrm{C}\left(0_1^{+}\right),{}^{12}\mathrm{C}(0_2^{+}$), and ${}^{12}\mathrm{C}(0_3^{+}$), calculated with the m62 interaction. The overlap is plotted on the $(R_z,R_x)$ plane for $D=3$, 4, and 5 fm. The $\alpha$ positions $(R_z,R_x)=(\pm D/2,0)$ in the $2\alpha$ core are shown by black circles.

Figure 7

$\theta$ dependence of overlap $U(D,\mathbf{R})$ for ${}^{11}\mathrm{B}$ and ${}^{12}\mathrm{C}$ calculated with the m62 interaction in the $R=D$ case.

Figure 8

Same as Fig. 7, but for the m60 result.

Figure 9

$\theta$ dependence of overlap $U(D,\mathbf{R})$ for ${}^{11}\mathrm{B}(3/2_3^{-}$) and ${}^{12}\mathrm{C}(0_2^{+}$) calculated with the m62 interaction.

Figure 10

Same as Fig. 9, but for the m60 result.

Figure 11

$R$ dependence of $U(D,\mathbf{R})$ for ${}^{11}\mathrm{B}\left(3/2_1^{-}\right),{}^{11}\mathrm{B}\left(3/2_3^{-}\right),{}^{12}\mathrm{C}(0_1^{+}$), and ${}^{12}\mathrm{C}(0_2^{+}$) calculated with the m62 interaction. $\theta$ is fixed at ${90}^{\circ }$.

Figure 12

Schematic figure of the 1D toy model for the $t(\alpha$)-cluster motion in the $\theta$ direction around the $2\alpha$ core with $D=2R$. (a) The angular $\left(\theta \right)$ motion 1D in the intrinsic frame. (b) The $\theta$ dependence of the $t$ wave function ($U$) around the $2\alpha$ and the absolute value ($\left|U\right|$) and (c) that of the $\alpha$ wave function around the $2\alpha$.