Application of random-phase approximation to vibrational excitations of double-$\Lambda$ hypernuclei

Using the Hartree-Fock plus random phase approximation ($\mathrm{HF}+\mathrm{RPA}$), we study the impurity effect of $\Lambda$ hyperon on the collective vibrational excitations of double-$\Lambda$ hypernuclei. To this end, we employ Skyrme-type $\Lambda N$ and $\Lambda \Lambda$ interactions for the HF calculations, and the residual interactions for the RPA are derived with the same interactions. We find that the inclusion of two $\Lambda$ hyperons in ${}^{16}$O shifts the collective states toward higher energies. In particular, the energy of the giant monopole resonance of ${}_{\Lambda \Lambda }^{18}$O, as well as that of ${}_{\Lambda \Lambda }^{210}$Pb, becomes higher. This implies that the effective incompressibility modulus increases owing to the impurity effect of $\Lambda$ particles, if the $\beta$-stability condition is not imposed.

Figure 1

Neutron and proton single-particle levels of ${}^{16}$O (solid lines) and ${}_{\Lambda \Lambda }^{18}$O (dashed lines) obtained with the Skyrme-Hartree-Fock method.

Figure 2

Strength distributions for the electric dipole [E1; (a)], the electric quadrupole [E2; (b)], and the electric octupole [E3; (c)] excitations of the ${}^{16}$O nucleus (dashed lines) and of the double-$\Lambda$ hypernucleus ${}_{\Lambda \Lambda }^{18}$O (solid and dotted lines). Solid lines were obtained by including the residual $NN$, $\Lambda N$, and $\Lambda \Lambda$ interactions, while dotted lines were obtained by including only the residual $NN$ interactions. Strength distributions are weighted by the Lorentzian function with a width of $1.0$ MeV. For peaks indicated by arrows, the transition densities are shown in Fig. 3.

Figure 3

Transition densities for the giant dipole (left panel) and the giant quadrupole (middle panel) resonances and for the high-lying octupole state (right panel) in ${}_{\Lambda \Lambda }^{18}$O (solid lines) and ${}^{16}$O (dashed lines). The corresponding states are denoted by arrows in Fig. 2. Top (a), middle (b), and bottom (c) panels display the transition densities for neutrons, protons, and $\Lambda$ hyperons, respectively.

Figure 4

Transition density for the low-lying E1 state at $E=12.8$ MeV in the ${}_{\Lambda \Lambda }^{18}$O hypernucleus. Dashed, dotted, and solid lines show the contributions of neutrons, protons, and $\Lambda$ hyperons, respectively.

Figure 5

Strength distributions for the isoscalar monopole mode for ${}^{16}$O and ${}_{\Lambda \Lambda }^{18}$O nuclei (a) and for ${}^{208}$Pb and ${}_{\Lambda \Lambda }^{210}$Pb nuclei (b). The meaning of each line is the same as in Fig. 2.

Figure 6

Transition densities for the isoscalar giant monopole resonances of ${}_{\Lambda \Lambda }^{18}$O (left panels) and ${}_{\Lambda \Lambda }^{210}$Pb (right panels), with comparisons to the transition densities for ${}^{16}$O and ${}^{208}$Pb. The meaning of each line is the same as in Fig. 3.

Figure 7

Binding energy per particle in infinite nuclear matter for a fraction of the $\Lambda$ particle of $x_{\Lambda }=0$ (dashed line) and $x_{\Lambda }=0.11$ (solid line). Neutron and proton densities are set to be equal, that is, ${\rho }_n={\rho }_p=(1-x_{\Lambda })\rho /2$.