Monopole transitions to cluster states in ${}^{10}\mathrm{Be}$ and ${}^9\mathrm{Li}$

Isoscalar monopole transitions from the ground states to cluster states in ${}^{10}\mathrm{Be}$ and ${}^9\mathrm{Li}$ are investigated using ${}^6\mathrm{He}+\alpha$ and ${}^6\mathrm{He}+t$ cluster models, respectively. In ${}^{10}\mathrm{Be}$, significant monopole strengths are obtained for ${}^6\mathrm{He}+\alpha$ cluster resonances of ${}^{10}\mathrm{Be}\left(0_{3,4}^{+}\right)$ above the $\alpha$-decay threshold, whereas the strengths for ${}^6\mathrm{He}+t$ cluster resonances in ${}^9\mathrm{Li}$ are not enhanced because of the large fragmentation of strengths in the corresponding energy region. The monopole transition to ${}^{10}\mathrm{Be}\left(0_2^{+}\right)$ with the molecular orbital structure is relatively weak compared with those to ${}^6\mathrm{He}+\alpha$ cluster resonances. Monopole strength distributions do not directly correspond to distributions of ${}^6\mathrm{He}\left(0^{+}\right)+\alpha$ and ${}^6\mathrm{He}\left(0^{+}\right)+t$ components but reflect the component of the deformed ${}^6\mathrm{He}$ cluster with a specific orientation, which is originally embedded in the ground state.

Figure 1

Schematic configurations for (a)–(f) of the ${}^6\mathrm{He}$ cluster in the ${}^6\mathrm{He}+t$ and ${}^6\mathrm{He}+\alpha$ cluster models. Details are described in the text. The figure is taken from Ref. [52].

Figure 2

Squared overlaps between the weak-coupling cluster wave functions ${\mathrm{\Phi }}_{{}^6\mathrm{He}+\alpha \left(t\right)}^{{[I\otimes J^{\prime }]}_J}\left(D\right)$ and the strong-coupling cluster wave functions ${\mathrm{\Phi }}_{{}^6\mathrm{He}+\alpha \left(t\right)}^{(T,A,I_{z}2)}\left(D\right)$. The squared overlaps of (a) ${\mathrm{\Phi }}_{{}^6\mathrm{He}+\alpha }^{{[0\otimes 0]}_0}\left(D\right)$, (b) ${\mathrm{\Phi }}_{{}^6\mathrm{He}+\alpha }^{{[2\otimes 2]}_0}\left(D\right)$, (c) ${\mathrm{\Phi }}_{{}^6\mathrm{He}+t}^{{[0\otimes 3/2]}_{3/2}}\left(D\right)$, and (d) ${\mathrm{\Phi }}_{{}^6\mathrm{He}+t}^{{[0\otimes 3/2]}_{3/2}}\left(D\right)$ with ${\mathrm{\Phi }}_{{}^6\mathrm{He}+\alpha \left(t\right)}^{(T,A,I_{z}2)}\left(D\right)$ are shown.

Figure 3

Energy levels of ${}^{10}\mathrm{Be}\left(0^{+}\right)$ and ${}^9\mathrm{Li}(3/2^{-})$ obtained using the $D\le 8$ calculation. The experimental energy levels of ${}^{10}\mathrm{Be}\left(0^{+}\right)$ are also shown. The ${}^6\mathrm{He}+\alpha$ and ${}^6\mathrm{He}+t$ threshold energies are shown as dashed lines.

Figure 4

Distributions of the squared overlap $|\langle {\mathrm{\Psi }}_{D\le 8}^{J_k^{\pi }}|{\mathrm{\Psi }}_{D\le 15}^{J_l^{\pi }}\rangle {|}^2$ of the $J_k$ state obtained by the $D\le 8$ calculation with states obtained using the $D\le 15$ calculation. Distributions of the $J_k=0_2^{+}$, $0_3^{+}$, and $0_4^{+}$ states in ${}^{10}\mathrm{Be}$ are shown in (a)–(c), and those of $J_k=3/2_2^{-}$, $3/2_3^{-}$, $3/2_4^{-}$, and $3/2_5^{-}$ states in ${}^9\mathrm{Li}$ are shown in (d)–(g). The Breit-Wigner distributions at energies ($E_x$) with widths (${\mathrm{\Gamma }}_{\mathrm{sum}}/2$) for the resonances obtained by the $D\le 8$ calculation are shown by dashed lines. The largest value of ${\mathrm{\Gamma }}_{\mathrm{sum}}$ at $a=5$, 6, and 7 fm from Table 1 is adopted for each state.

Figure 5

ISM strength distributions of ${}^{10}\mathrm{Be}\left(0^{+}\right)$ obtained by the (c+f), $D\le 8$, and $D\le 15$ calculations. Dashed lines in the middle panel (b) for the $D\le 8$ calculation show 50-fold values of the strengths.

Figure 6

ISM strength distributions of ${}^9\mathrm{Li}(3/2^{-})$ obtained by the (c+f), $D\le 8$, and $D\le 15$ calculations. Dashed lines in the middle panel (b) for the $D\le 8$ calculation show 50-fold values of the strengths.

Figure 7

Energy-weighted ISM strengths (solid lines). Panels (a)–(c) show the distributions in ${}^{10}\mathrm{Be}\left(0^{+}\right)$ obtained by the (c+f), $D\le 8$, $D\le 15$ calculations, and panels (d)–(f) show the distributions in ${}^9\mathrm{Li}(3/2^{-})$. The Gaussian smeared distributions of the energy-weighted ISM strengths with fixed width $1/\sqrt{\pi }$ are shown by dashed lines.

Figure 8

${}^6\mathrm{He}+\alpha$ components in ${}^{10}\mathrm{Be}\left(0^{+}\right)$ obtained by the $D\le 8$ calculation. [(a)–(d)] Squared overlaps of ${}^{10}\mathrm{Be}\left(0^{+}\right)$ with ${\mathrm{\Phi }}_{{}^6\mathrm{He}+\alpha }^{{[0\otimes 0]}_0}\left(D\right)$ and ${\mathrm{\Phi }}_{{}^6\mathrm{He}+\alpha }^{{[2\otimes 2]}_0}\left(D\right)$. [(e)–(h)] Squared overlaps of ${}^{10}\mathrm{Be}\left(0^{+}\right)$ with ${\mathrm{\Phi }}_{{}^6\mathrm{He}+\alpha }^{\left(T\right)}\left(D\right)$ and ${\mathrm{\Phi }}_{{}^6\mathrm{He}+\alpha }^{\left(A\right)}\left(D\right)$.

Figure 9

${}^6\mathrm{He}+t$ components in ${}^9\mathrm{Li}(3/2^{-})$ obtained by the $D\le 8$ calculation. Panels (a)–(e) show squared overlaps of ${}^9\mathrm{Li}(3/2^{-})$ with ${\mathrm{\Phi }}_{{}^6\mathrm{He}+t}^{{[0\otimes 3/2]}_{3/2}}\left(D\right)$, ${\mathrm{\Phi }}_{{}^6\mathrm{He}+t}^{{[2\otimes 3/2]}_{3/2}}\left(D\right)$, and ${\mathrm{\Phi }}_{{}^6\mathrm{He}+t}^{{[2\otimes 1/2]}_{3/2}}\left(D\right)$. Panels (f)–(j) show squared overlaps of the ${}^9\mathrm{Li}(3/2^{-})$ with ${\mathrm{\Phi }}_{{}^6\mathrm{He}+t}^{\left(T\right)}\left(D\right)$, ${\mathrm{\Phi }}_{{}^6\mathrm{He}+t}^{\left(A\right)}\left(D\right)$, and ${\mathrm{\Phi }}_{{}^6\mathrm{He}+t}^{\left(I_{z}2\right)}\left(D\right)$.

Figure 10

ISM strengths $B(IS0;\mathrm{g}.\mathrm{s}.\to {\mathrm{\Phi }}_{{}^6\mathrm{He}+\alpha \left(t\right)}^{(T,A)}\left(D\right))$ for the transitions from the ${}^{10}\mathrm{Be}$ and ${}^9\mathrm{Li}$ ground states obtained by the $D\le 8$ calculation to ${\mathrm{\Phi }}_{{}^6\mathrm{He}+\alpha \left(t\right)}^{(T,A)}\left(D\right)$. $B(IS0;\mathrm{g}.\mathrm{s}.\to {\mathrm{\Phi }}_{{}^6\mathrm{He}+\alpha \left(t\right)}^{(T,A)}\left(D\right))$ is defined in Eq. (21).

Figure 11

Distributions of squared overlaps of ${\mathrm{\Phi }}_{{}^6\mathrm{He}+\alpha }^{\left(T\right)}\left(D\right)$ at $D=5$ fm for the ${}^{10}\mathrm{Be}\left(0^{+}\right)$ obtained by the (c+f), $D\le 8$, and $D\le 15$ calculations.

Figure 12

Distributions of squared overlaps of ${\mathrm{\Phi }}_{{}^6\mathrm{He}+t}^{\left(T\right)}\left(D\right)$ at $D=5$ fm for the ${}^9\mathrm{Li}(3/2^{-}$) obtained by the (c+f), $D\le 8$, and $D\le 15$ calculations.

Figure 13

Schematics of ISM excitations to cluster states. (a) A system of two clusters that are spinless $ls$-closed shell nuclei. (b) A system consisting of a deformed cluster and a spinless $ls$-closed cluster. (b) A system consisting of a deformed cluster and a finite-spin cluster.

Figure 14

Distributions of ${}^6\mathrm{He}\left(0^{+}\right)+\alpha$ and ${}^6\mathrm{He}\left(2^{+}\right)+\alpha$ components in ${}^{10}\mathrm{Be}\left(0^{+}\right)$. [(a) and (c)] Squared overlaps of ${}^{10}\mathrm{Be}\left(0^{+}\right)$ obtained by the $D\le 8$ calculation with ${\mathrm{\Phi }}_{{}^6\mathrm{He}+\alpha }^{{[0\otimes 0]}_0}\left(D\right)$ and ${\mathrm{\Phi }}_{{}^6\mathrm{He}+\alpha }^{{[2\otimes 2]}_0}\left(D\right)$ at $D=7$ fm. [(b) and (d)] Same but for the $D\le 15$ calculation.

Figure 15

Distributions of ${}^6\mathrm{He}\left(0^{+}\right)+t$ and ${}^6\mathrm{He}\left(2^{+}\right)+t$ components in ${}^9\mathrm{Li}(3/2^{-})$. [(a) and (c)] Squared overlaps of ${}^9\mathrm{Li}(3/2^{-})$ obtained by the $D\le 8$ calculation with ${\mathrm{\Phi }}_{{}^6\mathrm{He}+t}^{{[0\otimes 3/2]}_{3/2}}\left(D\right)$ (solid lines), ${\mathrm{\Phi }}_{{}^6\mathrm{He}+t}^{{[2\otimes 3/2]}_{3/2}}\left(D\right)$ (solid lines), and ${\mathrm{\Phi }}_{{}^6\mathrm{He}+t}^{{[2\otimes 1/2]}_{3/2}}\left(D\right)$ (dashed lines) at $D=7$ fm. [(b) and (d)] Same overlaps for the $D\le 15$ calculation.