Impurity effect of the $\Lambda$ particle on the structure of ${}^{18}$F and ${}_{\Lambda }^{19}$F

We perform three-body model calculations for a $sd$-shell hypernucleus ${}_{\Lambda }^{19}$F (${}_{\Lambda }^{17}\mathrm{O}+p+n$) and its core nucleus ${}^{18}$F (${}^{16}\mathrm{O}+p+n$), employing a density-dependent contact interaction between the valence proton and neutron. We find that the $B\left(E2\right)$ value from the first excited state (with spin and parity of $I^{\pi }=3^{+}$) to the ground state ($I^{\pi }=1^{+}$) is slightly changed by the addition of a $\Lambda$ particle, which exhibits the so called shrinkage effect of $\Lambda$ particle. We also show that the excitation energy of the $3^{+}$ state is reduced in ${}_{\Lambda }^{19}$F compared to ${}^{18}$F, as is observed in a $p$-shell nucleus ${}^6$Li. We discuss the mechanism of this reduction of the excitation energy, pointing out that it is caused by a different mechanism from that in ${}_{\Lambda }^7$Li.

Figure 1

The level scheme and the $B\left(E2\right)$ values for the ${}^{18}$F and ${}_{\Lambda }^{19}$F nuclei. For the ${}_{\Lambda }^{19}$F nucleus, a sum of the $B\left(E2\right)$ values for the [3${}^{+}\otimes {\Lambda }_{s_{1/2}}{]}^{J=5/2}\to {[1^{+}\otimes {\Lambda }_{s_{1/2}}]}^{J=3/2}$ and the [3${}^{+}\otimes {\Lambda }_{s_{1/2}}{]}^{J=5/2}\to {[1^{+}\otimes {\Lambda }_{s_{1/2}}]}^{J=1/2}$ transitions is shown, which corresponds to the $B\left(E2\right)$ value from the 3${}^{+}$ to the 1${}^{+}$ states in ${}^{18}$F (see the text for details). The excitation energies are shown on the top of each state. The measured $B\left(E2\right)$ value for ${}^{18}$F is $16\pm 0.6e^2{\mathrm{fm}}^4$ [26].

Figure 2

The two-particle densities of the ground state of (a) ${}^{18}$F and (b) ${}_{\Lambda }^{19}$F as a function of $r_p=r_n\equiv r$ and the opening angle between the proton and the neutron, ${\theta }_{\mathrm{pn}}$. Those densities are multiplied by a weight factor of $8{\pi }^2{r}^4\sin {\theta }_{\mathrm{pn}}$ and are given in units of fm${}^{-3}$.

Figure 3

The difference of the density distribution in units of fm${}^{-3}$ between ${}_{\Lambda }^{19}$F and ${}^{18}$F for (a) the $1^{+}$ state and (b) the $3^{+}$ state.