$2\alpha +t$ cluster structure in ${}^{11}\mathrm{B}$

The $2\alpha +t$ cluster structure in ${}^{11}\mathrm{B}$ is investigated by the microscopic generator coordinate method with the Brink cluster wave functions. With a proper choice of the parameters of the effective interaction, the calculated energy spectrum shows reasonable agreement with the observed low-lying spectra of both parities. On the basis of the calculated radii, and monopole and $B\left(E2\right)$ transition strengths, several developed cluster states of ${}^{11}\mathrm{B}$ are suggested. For the negative-parity states, in addition to the well-known $3/2_3^{-}$ cluster state, the $1/2_2^{-}$ and $5/2_3^{-}$ states are also proposed as the well-developed cluster states. For the positive-parity states, it is found that many states around the $2\alpha +t$ threshold show the feature of developed clusters. In particular, the $1/2_2^{+}$ state is found to have a linear-chain-like structure, which is consistent with the previous antisymmetrized molecular dynamics calculation, but contradicts the orthogonality condition model calculation. It is also found that many of these positive-parity cluster candidates have non-negligible isoscalar dipole transition strengths, which require experimental confirmation.

Figure 1

Schematic figure of $2\alpha +t$ Brink wave function, which is used to define the squared overlap given by Eq. (16).

Figure 2

Comparison of the energy levels from the present calculation with parameter set 1, AMD [18] and OCM [21] calculations, and experiments [33]. The excitation energies are relative to the $2\alpha +t$ threshold energy. The upper and lower dashed lines represent the $2\alpha +t$ and ${}^7\mathrm{Li}+\alpha$ thresholds, respectively.

Figure 3

Comparison of the energy levels from the cluster model in Ref. [16] and the present calculations by using set 2 interactions (see Table 1 and the text). The excitation energies are relative to the ${}^7\mathrm{Li}+\alpha$ threshold energy. The upper and lower dashed lines represent the $2\alpha +t$ and ${}^7\mathrm{Li}+\alpha$ thresholds, respectively.

Figure 4

The calculated energy levels, rms matter radii (right-hand side of the energy levels), and the isoscalar monopole transition strengths which are stronger than 30 ${\mathrm{fm}}^4$ for the negative-parity states. The upper and lower dashed lines correspond to the $2\alpha +t$ and ${}^7\mathrm{Li}+t$ thresholds, respectively. Units for the energy and radius are MeV and fm, respectively.

Figure 5

Calculated $B\left(E2\right)$ transition strengths between the negative-parity states in ${}^{11}\mathrm{B}$. The transition strengths weaker than 4.80 $e^2{\mathrm{fm}}^4$ are not shown.

Figure 6

Comparison of the calculated $K^{\pi }=3/2^{-}$ bands between the present calculation ($3/2_3^{-}$, $5/2_3^{-}$, and $7/2_2^{-}$ or $7/2_3^{-}$), AMD [18] ($3/2_3^{-}$, $5/2_3^{-}$, and $7/2_3^{-}$), and the experimental band ($3/2_3^{-}$, $5/2_3^{-}$, and $7/2_3^{-}$) suggested in Ref. [22].

Figure 7

Contour plots of the squared overlap $O(R_{\alpha \alpha },{\mathit{R}}_t)$ defined by Eq. (16) for the $J^{\pi }=1/2^{-}$, $3/2^{-}$, and $5/2^{-}$ states as a function of ${\mathit{R}}_t$. The horizontal axis corresponds to the $z$ component of ${\mathit{R}}_t$, while the vertical axis corresponds to the $x$ component. The optimum distance between $2\alpha$ clusters which maximizes the squared overlap is also shown.

Figure 8

The GCM-Brink energy levels, rms radii for the mass distributions (right-hand side of the energy levels), and the isoscalar monopole transition strengths (stronger than 60 ${\mathrm{fm}}^4$) for the positive-parity states in ${}^{11}\mathrm{B}$. The upper and lower dashed lines correspond to the $2\alpha +t$ threshold energy and ${}^7\mathrm{Li}(3/2^{-})+t$ threshold energy, respectively. The units for the energy and radius are MeV and fm, respectively.

Figure 9

ISD transitions from the ground state to the positive-parity states in ${}^{11}\mathrm{B}$.

Figure 10

Positive-parity energy levels with the $B\left(E2\right)$ transition strengths (stronger than 15 $e^2{\mathrm{fm}}^4$).

Figure 11

Contour plots of the squared overlap $O(R_{\alpha \alpha },{\mathit{R}}_t)$ defined by Eq. (16) for the $J^{\pi }=1/2^{+}$, $3/2^{+}$, and $5/2^{+}$ states as a function of ${\mathit{R}}_t$. The horizontal axis corresponds to the $z$ component of ${\mathit{R}}_t$, while the vertical axis corresponds to the $x$ component. The optimum distance between $2\alpha$ clusters which maximizes the squared overlap is also shown.