Structure of low-lying states of ${}^{12}\mathrm{C}$ and ${}_{\mathrm{\Lambda }}{}^{13}\mathrm{C}$ in a beyond-mean-field approach

Based on the beyond-mean-field Skyrme-Hartree-Fock model, we investigate the energy spectra and possible cluster structure in ${}^{12}\mathrm{C}$ and its $\mathrm{\Lambda }$ hypernucleus ${}_{\mathrm{\Lambda }}{}^{13}\mathrm{C}$. The up-to-date Skyrme-type $N\mathrm{\Lambda }$ interaction SLL4 and the reduced $NN$ interaction SGII are employed. Low-lying energy spectra of ${}^{12}\mathrm{C}$, including the lowest five $0^{+}$ states, are predicted and discussed in detail, and found in good agreement with experimental results. The electric quadrupole transition rates and the monopole matrix elements between the $0^{+}$ states are also examined. Although typical cluster structures for the $0_2^{+}$ and $0_3^{+}$ states are absent, a linear-chain-like $3\alpha$ structure is found for the $0_4^{+}$ state. An energy spectrum including both the positive- and negative-parity levels is also given and structures of the corresponding hypernuclear states are discussed in detail. Impurity effects of a ${\mathrm{\Lambda }}_s$ or ${\mathrm{\Lambda }}_p$ hyperon on the nuclear core states of ${}_{\mathrm{\Lambda }}{}^{13}\mathrm{C}$ are demonstrated by the change of collective wave functions.

Figure 1

The potential energy surfaces (PESs) of MF (thin curves) and AMP states (thick curves): $0^{+}$ states for ${}^{12}\mathrm{C}$ (solid black curves), $1/2^{+}$ for ${}_{\mathrm{\Lambda }}{}^{13}\mathrm{C}$ (dashed red curves). The markers indicate the energy, Eq. (6), and average deformation, Eq. (13), of the excited states $0_{\alpha }^{+}$ and $1/2_{\alpha }^{+}$, $\alpha =1,...,5$.

Figure 2

Energy levels of ${}^{12}\mathrm{C}$($0^{+}$) (left columns) and ${}_{\mathrm{\Lambda }}{}^{13}\mathrm{C}$($1/2^{+}$) (right columns). The broad resonance state $0_4^{+}$ at $10.3\pm 3\text{MeV}$ [46] in the first column is in the second column decomposed into $0_3^{+}$ and $0_4^{+}$ states at $9.04\pm 1.45$ and $10.56\pm 1.42\text{MeV}$, respectively [8]. Experimental ${}_{\mathrm{\Lambda }}{}^{13}\mathrm{C}$($1/2_1^{+}$) results are from Refs. [52, 53, 55]. The thresholds of various decay channels are also indicated.

Figure 3

Two-dimensional nuclear-matter density distributions of the first five $0^{+}$ states in ${}^{12}\mathrm{C}$ (top row) and of the corresponding $1/2^{+}$ states in ${}_{\mathrm{\Lambda }}{}^{13}\mathrm{C}$ (bottom row). The average quadrupole deformation $\overline{\beta }$ and the charge radius $R_c$ (in fm) are also indicated in the lower left and right corners, respectively.

Figure 4

Weights of the natural states in the collective subspace, $g_{\alpha }^{J=0}$, Eq. (11), for the first five GCM states of ${}^{12}\mathrm{C}\left(0^{+}\right)$ and ${}_{\mathrm{\Lambda }}{}^{13}\mathrm{C}(1/2^{+})$.

Figure 5

The energy levels of ${}^{12}\mathrm{C}$ (left side) and ${}_{\mathrm{\Lambda }}{}^{13}\mathrm{C}$ (right side), comparing experimental and theoretical values. Colors indicate the predominant core configurations.

Figure 6

Collective wave functions of the (hyper)nuclear states shown in Fig. 5.

Figure 7

Same as Fig. 3, but for the $0_1^{+}$, $2_1^{+}$, $4_1^{+}$ states of ${}^{12}\mathrm{C}$ (top row), and for the band-head states of the three hypernuclear ${\mathrm{\Lambda }}_p$ configurations in Fig. 5 (bottom row).