Structures of $p$-shell double-$\mathrm{\Lambda }$ hypernuclei studied with microscopic cluster models

$0s$-orbit $\mathrm{\Lambda }$ states in $p$-shell double-$\mathrm{\Lambda }$ hypernuclei (${}_{\mathrm{\Lambda }\mathrm{\Lambda }}{}^{A}Z$), ${}_{\mathrm{\Lambda }\mathrm{\Lambda }}{}^8\mathrm{Li}$, ${}_{\mathrm{\Lambda }\mathrm{\Lambda }}{}^9\mathrm{Li}$, ${}_{\mathrm{\Lambda }\mathrm{\Lambda }}{}^{10,11,12}\mathrm{Be}$, ${}_{\mathrm{\Lambda }\mathrm{\Lambda }}{}^{12,13}\mathrm{B}$, and ${}_{\mathrm{\Lambda }\mathrm{\Lambda }}{}^{14}\mathrm{C}$ are investigated. Microscopic cluster models are applied to core nuclear part and a potential model is adopted for $\mathrm{\Lambda }$ particles. The $\mathrm{\Lambda }$-core potential is a folding potential obtained with effective $G$-matrix $\mathrm{\Lambda }\text{−}N$ interactions, which reasonably reproduce energy spectra of ${}_{\mathrm{\Lambda }}{}^{A-1}Z$. System dependence of the $\mathrm{\Lambda }\text{−}\mathrm{\Lambda }$ binding energies is understood by the core polarization energy from nuclear size reduction. Reductions of nuclear sizes and $E2$ transition strengths by $\mathrm{\Lambda }$ particles are also discussed.

Figure 1

(Top) Excitation energy shift ${\delta }_{\mathrm{\Lambda }}\left(E_x\right)$, (middle) nuclear size difference $(R_N-R_{N,\mathrm{gs}})$, (bottom) excitation energy shift plotted against the nuclear size difference. The results obtained with ESC08a(Hyb), ESC08a(DI), and ESC08a(DD) are shown. The experimental values of ${\delta }_{\mathrm{\Lambda }}\left(E_x\right)$ are shown in the top panel, and those plotted against the size difference calculated with ESC08a(Hyb) are shown in the bottom panel.

Figure 2

$\epsilon$ dependence of energies for the ${}^6\mathrm{Li}\left(1_1^{+}\right)$ core plotted as functions of the nuclear size $R_N\left(\epsilon \right)$. Left: single-$\mathrm{\Lambda }$ energy $\left[E_{\mathrm{\Lambda }}\left(\epsilon \right)\right]$ and two-$\mathrm{\Lambda }$ energy $\left[E_{\mathrm{\Lambda }\mathrm{\Lambda }}\left(\epsilon \right)\right]$. $2E_{\mathrm{\Lambda }}\left(\epsilon \right)$, which correspond to the two-$\mathrm{\Lambda }$ energy for the $V_{\mathrm{\Lambda }\mathrm{\Lambda }}=0$ case is also shown. Right: nuclear energy $\left[E_N\left(\epsilon \right)\right]$, total energy $E(\epsilon ;{}_{\mathrm{\Lambda }}{}^{A-1}Z)=E_N\left(\epsilon \right)+E_{\mathrm{\Lambda }}\left(\epsilon \right)$ of ${}_{\mathrm{\Lambda }}{}^{A-1}Z$, and total energy $E(\epsilon ;{}_{\mathrm{\Lambda }\mathrm{\Lambda }}{}^{A}Z)=E_N\left(\epsilon \right)+E_{\mathrm{\Lambda }\mathrm{\Lambda }}\left(\epsilon \right)$ of ${}_{\mathrm{\Lambda }\mathrm{\Lambda }}{}^{A}Z$. The values of $E_N\left(\epsilon \right)$ and $E\left(\epsilon {;}_{\mathrm{\Lambda }}^{A}Z\right)$ are shifted by $-5$ MeV and $+5$ MeV, respectively.

Figure 3

Energy counting in ${}_{\mathrm{\Lambda }}{}^7\mathrm{Li}\left(1_1^{+}\right)$ and ${}_{\mathrm{\Lambda }\mathrm{\Lambda }}{}^8\mathrm{Li}\left(1_1^{+}\right)$ for the ${}^6\mathrm{Li}\left(1_1^{+}\right)$ core. $R_N\left(\epsilon \right)$ dependences of $E_N$, $E_N+E_{\mathrm{\Lambda }}$, $E_N+E_{\mathrm{\Lambda }\mathrm{\Lambda }}$, and $E_N+2E_{\mathrm{\Lambda }}$ calculated with ESC08a(Hyb) are shown by dash-dotted, solid, dashed, and dotted lines, respectively.

Figure 4

Top: the values of $\mathrm{\Delta }{B}_{\mathrm{\Lambda }\mathrm{\Lambda }}$, ${\mathcal{V}}_{\mathrm{\Lambda }\mathrm{\Lambda }}^{\mathrm{bond}}$, and ${\mathcal{V}}_{\mathrm{\Lambda }\mathrm{\Lambda }}^{\mathrm{bond}}+{\delta }_{\mathrm{\Lambda }}\left(E_N\right)$ for ${}_{\mathrm{\Lambda }\mathrm{\Lambda }}{}^{A}Z$ calculated with ESC08a(Hyb). $[{\delta }_{\mathrm{\Lambda }}\left(E_N\right)$ is the nuclear energy change ${\delta }_{\mathrm{\Lambda }}\left(E_N\right)$ in ${}_{\mathrm{\Lambda }}{}^{A-1}Z$.] For comparison, the theoretical $\mathrm{\Delta }{\overline{B}}_{\mathrm{\Lambda }\mathrm{\Lambda }}$ (spin-averaged values) of the OCM cluster model calculation in Ref. [11] (${\mathrm{OCM}}^{2002}$) and those in Ref. [44] (${\mathrm{OCM}}^{2010}$) are also shown. Bottom: $\mathrm{\Delta }{B}_{\mathrm{\Lambda }\mathrm{\Lambda }}$, $\mathrm{\Delta }{B}^{\mathrm{w}}/{\mathrm{ocp}}_{\mathrm{\Lambda }\mathrm{\Lambda }}$, and $\mathrm{\Delta }{B}^{\mathrm{w}}/{\mathrm{ocp}}_{\mathrm{\Lambda }\mathrm{\Lambda }}+2{\delta }_{\mathrm{\Lambda }}\left(E_N\right)$ for ${}_{\mathrm{\Lambda }\mathrm{\Lambda }}{}^{A}Z$.

Figure 5

Energy spectra calculated with ESC08a(Hyb) and the experimental spectra. The experimental data for ${}^{A-2}Z$ are taken from Refs. [62, 63], and those for ${}_{\mathrm{\Lambda }}{}^{A-1}Z$ are from Refs. [2, 64, 65, 66, 67]. The excitation energy of ${}^8\mathrm{Be}\left(2^{+}\right)$ is calculated with the resonance energy obtained by the resonating group method (RGM). The experimental data for ${}_{\mathrm{\Lambda }}{}^{A-1}Z$ are the spin-averaged values reduced from the excitation energies of spin doublet states.

Figure 6

Energy spectra calculated with ESC08a(Hyb) and experimental spectra. The experimental data for ${}^{A-2}Z$ are taken from Refs. [63, 68, 69], and those for ${}_{\mathrm{\Lambda }}{}^{A-1}Z$ are from Refs. [2, 67, 70, 71, 72, 73, 74]. For the experimental spectra of ${}_{\mathrm{\Lambda }}{}^{12}\mathrm{B}(1/2_1^{-})$ and ${}_{\mathrm{\Lambda }}{}^{13}\mathrm{C}\left(2^{+}\right)$, non-spin-averaged values $E_x\left(1^{-}\right)$ and $E_x(3/2^{+})$ are used, respectively. For ${}_{\mathrm{\Lambda }}{}^{12}\mathrm{B}(3/2_2^{-})$, the experimental values of $E_x\left(1^{-}\right)$ in ${}_{\mathrm{\Lambda }}{}^{12}\mathrm{B}$ and $E_x\left(2^{-}\right)$ of the mirror state in ${}_{\mathrm{\Lambda }}{}^{12}\mathrm{C}$ are averaged by assuming the same Coulomb shift in ${}_{\mathrm{\Lambda }}{}^{12}\mathrm{B}(3/2_2^{-})\text{−}{}_{\mathrm{\Lambda }}{}^{12}\mathrm{C}(3/2_2^{-})$ as that in ${}^{11}\mathrm{B}(3/2_2^{-})\text{−}{}^{11}\mathrm{C}(3/2_2^{-})$.

Figure 7

Excitation energy shift ${\delta }_{\mathrm{\Lambda }}\left(E_x\right)$ in ${}_{\mathrm{\Lambda }}{}^{A-1}Z$ and ${\delta }_{\mathrm{\Lambda }\mathrm{\Lambda }}\left(E_x\right)$ in ${}_{\mathrm{\Lambda }\mathrm{\Lambda }}{}^{A}Z$ calculated with ESC08a(Hyb). The experimental values of ${\delta }_{\mathrm{\Lambda }}\left(E_x\right)$ in ${}_{\mathrm{\Lambda }}{}^{A-1}Z$ are also shown.