Transition from vibrational to rotational character in low-lying states of hypernuclei

In order to clarify the nature of hypernuclear low-lying states, we carry out a comprehensive study of the structure of ${}_{\phantom{00000.}\mathrm{\Lambda }}^{145\text{–}155}\mathrm{Sm}$ hypernuclei, which exhibit a transition from vibrational to rotational character as the neutron number increases. To this end, we employ a microscopic particle-core coupling scheme based on a covariant density functional theory. We find that the positive-parity ground-state band in the hypernuclei shares a similar structure to that of the corresponding core nucleus. That is, regardless of whether the core nucleus is spherical or deformed, each hypernuclear state is dominated by the single configuration of the $\mathrm{\Lambda }$ particle in the $s_{1/2}$ state ($\mathrm{\Lambda }{s}_{1/2}$) coupled to one core state of the ground band. In contrast, the low-lying negative-parity states mainly consist of $\mathrm{\Lambda }{p}_{1/2}$ and $\mathrm{\Lambda }{p}_{3/2}$ configurations coupled to plural nuclear core states. We show that, while the mixing amplitude between these configurations is negligibly small in spherical and weakly deformed nuclei, it strongly increases as the core nucleus undergoes a transition to a well deformed shape, which is consistent with the Nilsson wave functions. We demonstrate that the structure of these negative-parity states with spin $I$ can be well understood based on a naive $LS$ coupling scheme, with total orbital angular momentum $L=[I\otimes 1]$ and spin angular momentum $S=1/2$.

Figure 1

The total energy in the mean-field approximation for Sm isotopes as a function of the the intrinsic quadrupole deformation $\beta$ (the solid lines). The energy curves for ${}^{146}\mathrm{Sm}$ and ${}^{152}\mathrm{Sm}$ are shifted upward by 12 MeV, while those for ${}^{148}\mathrm{Sm}$ and ${}^{154}\mathrm{Sm}$ are shifted by 26 MeV. The squares of the collective wave functions, $|g_{nI}{\left(\beta \right)|}^2$, for the GCM ground states ($0_1^{+}$) are also shown with the dashed curves.

Figure 2

The yrast levels of the Sm isotopes calculated with the MR-CDFT (the red lines) in comparison to the experiment data (the black lines) taken from Ref. [24]. The figure also shows the scaled levels (the blue lines) with a multiplicative factor of $f=E_{2^{+}}^{\exp .}/E_{2^{+}}^{\mathrm{MR}\text{−}\mathrm{CDFT}}$, that is $E_{I^{+}}^{\mathrm{Scaled}}=f\times E_{I^{+}}^{\mathrm{MR}\text{−}\mathrm{CDFT}}$.

Figure 3

(a) The ratio $R_{4/2}$ of the excitation energy for the first $4^{+}$ state to that for the first $2^{+}$ state for the Sm isotopes as a function of the mass number. (b) The electromagnetic transition strength, in units $e^2{\mathrm{fm}}^4$, from the first $2^{+}$ state to the ground state, $B(E2:2_1^{+}\to 0_1^{+})$, as a function of the mass number for the Sm isotopes. The experiment data are taken from Ref. [24].

Figure 4

The spectrum for the first positive parity states in the ${}_{\mathrm{\Lambda }}\mathrm{Sm}$ isotopes. The locations of the spin doublet states are arranged based on the yrast levels of the even Sm core nuclei.

Figure 5

The low-lying negative-parity states in the ${}_{\mathrm{\Lambda }}\mathrm{Sm}$ isotopes obtained with the microscopic particle-core coupling scheme based on the covariant density functional theory. The arrows indicate the $E2$ transition strengths, given in units of $e^2{\mathrm{fm}}^4$.

Figure 6

A schematic picture for hypernuclear states based on a naive $LS$ coupling scheme. It is assumed that a $\mathrm{\Lambda }$ particle is in a $p$ orbital and coupled to spherical core states.

Figure 7

(a) The energy levels of the $1/2_1^{-}$ and $3/2_1^{-}$ states in the ${}_{\mathrm{\Lambda }}\mathrm{Sm}$ hypernuclei as a function of the mass number. The solid line and the dashed line indicate the results of the coupled-channels and the single-channel calculations, respectively. (b) The energy splitting between the $1/2_1^{-}$ and $3/2_1^{-}$ states shown in the upper panel.

Figure 8

The probability $P_k$ of the dominant components in the wave function of (a) the $1/2_1^{-}$ state and (b) the $3/2_1^{-}$ state as a function of the mass number of the ${}_{\mathrm{\Lambda }}\mathrm{Sm}$ isotopes.

Figure 9

(a) The energy of the $J^{\pi }=1/2_1^{-}$, $1/2_2^{-}$, and $3/2_1^{-}$ states of the ${}_{\mathrm{\Lambda }}{}^{155}\mathrm{Sm}$ hypernucleus as a function of the quadrupole deformation $\beta$ of the core nucleus. These curves are obtained with the microscopic particle-rotor model calculation by setting the excitation energies of the core states to be zero. (b) The probability $P_k$ of the dominant components in the $1/2_1^{-}$ state as a function of the deformation parameter, $\beta$. (c) Same as (b), but for the $1/2_2^{-}$ state.

Figure 10

The relation among the $E2$ transition strengths for transitions between low-lying negative parity states in a well deformed hypernucleus. The solid line indicates strong transitions, for which the $B\left(E2\right)$ value is larger than the $B\left(E2\right)$ value for the transition from the first excited state with $(I,L,J)=(2,3,5/2)$ to the ground state with $(I,L,J)=(0,1,1/2)$, where $I$, $L$, and $J$ are the spin of the core nucleus, the total orbital angular momentum of the hypernuclei and the total spin of the hypernuclei, respectively. The dashed lines denote weaker transitions. See Table 3 for the actual values of the transition strengths. We note that the band with $m_L=0$ should be lower in energy than the other two with $m_L=\pm 1$ for a well deformed hypernucleus, as we have discussed in Figs. 5 and 6.