Structure of ${}_{\mathrm{\Lambda }}{}^9\mathrm{Be}$ and ${}_{\mathrm{\Lambda }\mathrm{\Lambda }}{}^{10}\mathrm{Be}$ using the beyond-mean-field Skyrme-Hartree-Fock approach

Based on the beyond-mean-field Skyrme-Hartree-Fock model, the up-to-date Skyrme-type $N\mathrm{\Lambda }$ interaction, SLL4, is used to investigate the properties of ${}_{\mathrm{\Lambda }}{}^9\mathrm{Be}$ comprehensively. Energies of different configurations, such as ${}^8\mathrm{Be}\otimes \mathrm{\Lambda }\left[000\right]1/2^{+}, {}^8\mathrm{Be}\otimes \mathrm{\Lambda }\left[110\right]1/2^{-}, {}^8\mathrm{Be}\otimes \mathrm{\Lambda }\left[101\right]3/2^{-}$, and ${}^8\mathrm{Be}\otimes \mathrm{\Lambda }\left[101\right]1/2^{-}$ are given and used to study the effects of $\mathrm{\Lambda }$ occupying different orbitals. The calculated energy spectra, including both positive- and negative-parity levels, are given and compared to the experimental data. The observed positive-parity spin doublets ($3/2^{+},5/2^{+}$) are successfully reproduced, but the energy difference needs further investigation. The two well-known band structures corresponding to the genuine hypernuclear states and the ${}^9\mathrm{Be}$-analog states are also obtained and compared with the observed ones. The shrinkage effect of $\mathrm{\Lambda }$ occupying $\mathrm{\Lambda }\left[000\right]1/2^{+}$ is investigated through the density distributions of nuclear core. And finally, the $E2$ transition rates are given and compared with the observed data and with the results of the hypernuclear particle-rotor model. Properties of ${}_{\mathrm{\Lambda }\mathrm{\Lambda }}{}^{10}\mathrm{Be}$ are also studied to show the completeness of this current model.

Figure 1

PESs obtained from the mean-field calculations with different configurations. The inserted graph shows the spin-orbit splitting near the spherical shape, and ${\mathrm{\Lambda }}_1, {\mathrm{\Lambda }}_2, {\mathrm{\Lambda }}_3$, and ${\mathrm{\Lambda }}_4$ represent the single-$\mathrm{\Lambda }$ orbitals $\mathrm{\Lambda }\left[000\right]1/2^{+}, \mathrm{\Lambda }\left[110\right]1/2^{-}, \mathrm{\Lambda }\left[101\right]3/2^{-}$, and $\mathrm{\Lambda }\left[101\right]1/2^{-}$, respectively.

Figure 2

Angular momentum projected energy curves for ${}^8\mathrm{Be}$ and ${}_{\mathrm{\Lambda }}{}^9\mathrm{Be}$. Panel (a) is for ${}^8\mathrm{Be}$ while panels (b), (c), (d), and (e) are for ${}_{\mathrm{\Lambda }}{}^9\mathrm{Be}$ corresponding to the configurations ${}^8\mathrm{Be}\otimes \mathrm{\Lambda }\left[000\right]1/2^{+}, {}^8\mathrm{Be}\otimes \mathrm{\Lambda }\left[110\right]1/2^{-}, {}^8\mathrm{Be}\otimes \mathrm{\Lambda }\left[101\right]3/2^{-}$, and ${}^8\mathrm{Be}\otimes \mathrm{\Lambda }\left[101\right]1/2^{-}$, respectively. Some low-lying GCM levels are shown at average deformations $\overline{\beta }$. For the convenience of comparison, panels (b), (c), (d), and (e) are offset upward by 7, 0, −6, and −5 MeV, respectively.

Figure 3

The calculated positive-parity energy levels for ${}^8\mathrm{Be}, {}_{\mathrm{\Lambda }}{}^9\mathrm{Be}$, and ${}_{\mathrm{\Lambda }\mathrm{\Lambda }}{}^{10}\mathrm{Be}$ with respect to the ${}^8\mathrm{Be}$+$\mathrm{\Lambda }$(+$\mathrm{\Lambda }$) threshold. The experimental data of ${}^8\mathrm{Be}$ are from Ref. [66] and the observed (double-)$\mathrm{\Lambda }$ binding energies are from Refs. [17, 67].

Figure 4

Density distribution of nuclear matter for the $0_1^{+}$ and $2_1^{+}$ states of ${}^8\mathrm{Be}$ [panels (a) and (d)] and ${}_{\mathrm{\Lambda }\mathrm{\Lambda }}{}^{10}\mathrm{Be}$ [panels (c) and (f)], and for the $1/2_1^{+}$ and $5/2_1^{+}$ states of ${}_{\mathrm{\Lambda }}{}^9\mathrm{Be}$ [panels (b) and (e)] with configuration ${}^8\mathrm{Be}\otimes \mathrm{\Lambda }\left[000\right]1/2^{+}$. The six figures share the same color bar. The unit is set as ${\mathrm{fm}}^{-3}$.

Figure 5

Energy spectra of ${}^8\mathrm{Be}$, and of ${}_{\mathrm{\Lambda }}{}^9\mathrm{Be}$ with configurations ${}^8\mathrm{Be}\otimes \mathrm{\Lambda }\left[000\right]1/2^{+}, {}^8\mathrm{Be}\otimes \mathrm{\Lambda }\left[110\right]1/2^{-}, {}^8\mathrm{Be}\otimes \mathrm{\Lambda }\left[101\right]3/2^{-}$, and ${}^8\mathrm{Be}\otimes \mathrm{\Lambda }\left[101\right]1/2^{-}$, respectively. The observed data of ${}^8\mathrm{Be}$ are from Ref. [66] and those of ${}_{\mathrm{\Lambda }}{}^9\mathrm{Be}$ are from Ref. [2, 5, 14, 68, 69].

Figure 6

Density distributions of $\mathrm{\Lambda }$ hyperon for the band heads of ${}_{\mathrm{\Lambda }}{}^9\mathrm{Be}$ with configurations ${}^8\mathrm{Be}\otimes \mathrm{\Lambda }\left[000\right]1/2^{+}$ [panel (a)], ${}^8\mathrm{Be}\otimes \mathrm{\Lambda }\left[110\right]1/2^{-}$ [panel (b)], ${}^8\mathrm{Be}\otimes \mathrm{\Lambda }\left[101\right]3/2^{-}$ [panel (c)], and ${}^8\mathrm{Be}\otimes \mathrm{\Lambda }\left[101\right]1/2^{-}$ [panel (d)], respectively. Panels (c) and (d) share the same color bar. The unit is set as ${10}^{-2}{\mathrm{fm}}^{-3}$ uniformly.