Investigation of single- and double-$\mathrm{\Lambda }$ hypernuclei using a beyond-mean-field approach

A beyond-mean-field approach consisting of angular momentum projection techniques and generator coordinate method based on Skyrme–Hartree-Fock calculations is employed to investigate single- and double-$\mathrm{\Lambda }$ hypernuclear systems. The density-dependent $N\mathrm{\Lambda }$ interactions derived from the Nijmegen soft-core potentials are used. Rotational energy spectra and electric-quadrupole transition strengths $B\left(E2\right)$ of the hypernuclei ${}_{\mathrm{\Lambda }}{}^{13}\mathrm{C},{}_{\mathrm{\Lambda }\mathrm{\Lambda }}{}^{14}\mathrm{C},{}_{\mathrm{\Lambda }}{}^{21}\mathrm{Ne}$, and ${}_{\mathrm{\Lambda }\mathrm{\Lambda }}{}^{22}\mathrm{Ne}$ are presented and compared with those of the corresponding core nuclei ${}^{12}\mathrm{C}$ and ${}^{20}\mathrm{Ne}$. The shrinkage effect of the $\mathrm{\Lambda }\mathrm{s}$ is demonstrated by the $B\left(E2\right)$ values, the charge radii, and the shape deformation $\beta$ of the nuclear core. It is found that the reduction of the $B\left(E2\right)$ values in ${}_{\mathrm{\Lambda }}{}^{13}\mathrm{C}$ and ${}_{\mathrm{\Lambda }\mathrm{\Lambda }}{}^{14}\mathrm{C}$ is mainly caused by the shrinkage of the charge radii of the nuclear cores, while the reduced shape deformations also play important roles; but the contrary is the case in ${}_{\mathrm{\Lambda }}{}^{21}\mathrm{Ne}$ and ${}_{\mathrm{\Lambda }\mathrm{\Lambda }}{}^{22}\mathrm{Ne}$. Comparison between this and other theoretical models are made, and the differences between them are illuminated.

Figure 1

Potential-energy surfaces as functions of deformation parameter $\beta$ from mean-field calculation and AMP for (a) ${}^{12}\mathrm{C}$ and (b), (c) ${}_{\mathrm{\Lambda }}{}^{13}\mathrm{C}$ and ${}_{\mathrm{\Lambda }\mathrm{\Lambda }}{}^{14}\mathrm{C}$ using the NSC89 or (d), (e) NSC97f force. The symbols represent the GCM states with given $J^{\pi }$, located at the average deformation $\overline{\beta }$ of Eq. (16). Offsets are used for hypernuclei according to the (double-)$\mathrm{\Lambda }$ binding energies to keep the ground states of all the (hyper)nuclei at the same level. The spin doublets are represented by one single curve or symbol due to their negligible splitting.

Figure 2

Collective wave functions $g_{\alpha }^J\left(\beta \right)$ for the GCM states of the ground band with $J^{\pi }=0^{+},2^{+},4^{+}$ (a) of ${}^{12}\mathrm{C}$, and the change ${\mathrm{\Delta }}_J$, Eq. (17), due to the addition of double-$\mathrm{\Lambda }$ for (b)–(d) ${}_{\mathrm{\Lambda }\mathrm{\Lambda }}{}^{14}\mathrm{C}$ using the NSC97f force. The arrows indicate $\overline{\beta }$ for each state of ${}^{12}\mathrm{C}$ and ${}_{\mathrm{\Lambda }\mathrm{\Lambda }}{}^{14}\mathrm{C}$.

Figure 3

Energy spectra of ${}^{12}\mathrm{C},{}_{\mathrm{\Lambda }}{}^{13}\mathrm{C}$, and ${}_{\mathrm{\Lambda }\mathrm{\Lambda }}{}^{14}\mathrm{C}$, using the $N\mathrm{\Lambda }$ interaction parameters NSC89 and NSC97f. Electric-quadrupole transition strengths $B\left(E2\right)$ are shown in ${\mathrm{e}}^2$ ${\mathrm{fm}}^4$ along the arrows. The experimental data are taken from Refs. [56, 57, 58, 59] for ${}^{12}\mathrm{C}$ and Ref. [6] for ${}_{\mathrm{\Lambda }}{}^{13}\mathrm{C}$.

Figure 4

Same as in Fig. 1, but for ${}^{20}\mathrm{Ne},{}_{\mathrm{\Lambda }}{}^{21}\mathrm{Ne}$, and ${}_{\mathrm{\Lambda }\mathrm{\Lambda }}{}^{22}\mathrm{Ne}$.

Figure 5

Same as in Fig. 3, but for ${}^{20}\mathrm{Ne},{}_{\mathrm{\Lambda }}{}^{21}\mathrm{Ne}$, and ${}_{\mathrm{\Lambda }\mathrm{\Lambda }}{}^{22}\mathrm{Ne}$. The experimental data are taken from Ref. [56].

Figure 6

Nuclear density distribution of the ground states for ${}^{12}\mathrm{C}$ and ${}^{20}\mathrm{Ne}$. An identical color bar is used for both nuclei.