Relation between shrinkage effect and compression of the rotational spectrum in the ${}_{\Lambda }^7$Li hypernucleus

It has been shown experimentally that the ${}^6$Li nucleus shrinks by adding a $\Lambda$ particle while its spectrum is also compressed. We discuss the compatibility of these two effects, contrasting also with a relation between the shrinkage effect and the stability of the spectrum in the ${}_{\Lambda }^9$Be hypernucleus. To this end, we employ two-body $d$-${}_{\Lambda }^5$He and $\alpha$-${}_{\Lambda }^5$He cluster models for the ${}_{\Lambda }^7$Li and ${}_{\Lambda }^9$Be hypernuclei, respectively. We first argue that a Gaussian-like interaction between two clusters leads to a stabilization of the spectrum against an addition of a $\Lambda$ particle, even though the intercluster distance is reduced. In the case of ${}_{\Lambda }^7$Li, the spin-orbit interaction between the intercluster motion and the deuteron spin has to be considered also. We show that the shrinkage effect makes the expectation value of the spin-orbit potential larger, lowering the excitation energy for the ${}^6$Li($3^{+})\otimes \Lambda (1/2^{+})$ level.

Figure 1

The experimental low-lying spectra for ${}^6$Li, ${}_{\Lambda }^7$Li, ${}^8$Be, ${}_{\Lambda }^9$Be, ${}^{12}$C, and ${}_{\Lambda }^{13}$C nuclei. The energies are denoted in units of MeV. The energies for the 1${}^{+}\otimes$1/2${}^{+}$ and 2${}^{+}\otimes 1/2^{+}$ levels of the ${}_{\Lambda }^7$Li nucleus are obtained by spin-averaging the observed level energies for the 1/2${}^{+}$ (0 MeV) and the 3/2${}^{+}$ (0.069 MeV) states and for the 5/2${}^{+}$ (2.05 MeV) and the 7/2${}^{+}$ (2.52 MeV) states [19], respectively. The energy for the 2${}^{+}\otimes 1/2^{+}$ level of ${}_{\Lambda }^9$Be is the spin-averaged energy between the 5/2${}^{+}$ (3.024 MeV) and 3/2${}^{+}$ (3.067 MeV) states [21]. In the ${}_{\Lambda }^{13}$C nucleus, only the energy for the 3/2${}^{+}$ state has been measured [17], while the energy for the 5/2${}^{+}$ state is expected to be lower than the energy of the 3/2${}^{+}$ state by at most 0.36 MeV [18].

Figure 2

Low-lying spectra for ${}^8$Be and ${}_{\Lambda }^9$Be obtained with the two-body cluster model of $\alpha +\alpha$ and $\alpha {+}_{\Lambda }^5$He, respectively. The energies are denoted in units of MeV. The energies measured from the two-body thresholds are also shown in parentheses. The arrows indicate the reduced $E2$ transition probabilities, $B\left(E2\right)$, from the first excited state to the ground state.

Figure 3

The radial wave functions $u_L\left(R\right)$ defined as ${\psi }_L\left(\mathit{R}\right)=u_L\left(R\right)/R·Y_{LM}\left(\widehat{\mathit{R}}\right)$ obtained with the two-body cluster model. The dashed and the solid lines are for the ${}^8$Be and ${}_{\Lambda }^9$Be nuclei, respectively. The upper panel shows the wave functions for the ground state with $L=0$, while the lower panel shows the wave functions for the first excited state with $L=2$.

Figure 4

The intercluster potential for the $\alpha +\alpha$ system (upper panel) and for the $\alpha {+}_{\Lambda }^5$He system (lower panel). The dashed lines show the result of the fitting with a single Gaussian function.

Figure 5

The central part (upper panel) and the spin-orbit part (lower panel) of the intercluster potential. The dashed lines show the potentials for the ${}^6$Li nucleus, while the solid lines show those for the ${}_{\Lambda }^7$Li hypernucleus.

Figure 6

Low-lying spectra for ${}^6$Li and ${}_{\Lambda }^7$Li obtained with the two-body cluster model of $\alpha +d$ and ${}_{\Lambda }^5$He+$d$, respectively. The results obtained with and without the spin-obit intercluster potential are compared. The energies are denoted in units of MeV. The energies measured from the two-body thresholds are also shown in parentheses. The arrows indicate the reduced $E2$ transition probabilities, $B\left(E2\right)$, from the first excited state to the ground state, where the calculated values are denoted in units of $e^2$$$fm${}^4$.

Figure 7

(a) The radial wave functions for the ground state with $L=0$ for ${}^6$Li (dashed line) and ${}_{\Lambda }^7$Li (solid line). (b) Those for the first excited state with $L=2$ obtained without including the spin-orbit intercluster potential. (c) The overlap between the spin-orbit potential and the unperturbed wave functions for $L=2$. (d) The same as Fig. 7b, but obtained by including the spin-orbit potential.

Figure 8

(a) The energy difference between the $L=2$ ($1d$) and the $L=0$ ($2s$) states for a Gaussian potential $V\left(R\right)=V_0{e}^{-\beta R^2}$ for an $\alpha +\alpha$ system as a function of the depth parameter $-V_0$. The solid line is obtained by numerically solving the Schrödinger equation, while the dashed line is the result of the approximate formula given in Eq. (12) based on the Bohr-Sommerfeld quantization rule. The dotted line denotes the energy difference for a classical rotor, evaluated with the root-mean-square (rms) intercluster distance for the $2s$ state shown in Fig. 8c. (b) The energy of the $L=0$ ($2s$) state as a function of the depth parameter $-V_0$ for the Gaussian potential. (c) The rms intercluster distance for the $2s$ state.