Consistent description for cluster dynamics and single-particle correlation

Cluster dynamics and single-particle correlation are simultaneously treated for the description of the ground state of ${}^{12}\mathrm{C}$. The recent development of the antisymmetrized quasicluster model (AQCM) makes it possible to generate $jj$-coupling shell-model wave functions from $\alpha$ cluster models. The cluster dynamics and the competition with the $jj$-coupling shell-model structure can be estimated rather easily. In the present study, we further include the effect of single-particle excitation; the mixing of the two-particle–two-hole excited states is considered. The single-particle excitation is not always taken into account in the standard cluster model analyses, and the two-particle–two-hole states are found to strongly contribute to the lowering of the ground state owing to the pairing-like correlations. By extending AQCM, all of the basis states are prepared on the same footing, and they are superposed based on the framework of the generator coordinate method (GCM).

Figure 1

$0^{+}$ energy curves of ${}^{12}\mathrm{C}$ calculated with AQCM. The horizontal axis shows the parameter $R$ representing the distance between the quasiclusters with the equilateral triangular configuration. The dotted, solid, and dashed curves are the cases of $\mathrm{\Lambda }$ equal to 0.0, 0.2, and 0.4.

Figure 2

Energy convergence for the lowest $0^{+}$ state of ${}^{12}\mathrm{C}$ as a function of Hilbert space dimension. On top of 18 AQCM basis states, 50 proton two-particle–two-hole states (numbers 19 to 68) and 50 neutron two-particle–two-hole states (numbers 69 to 118) are added successively.

Figure 3

$0^{+}-2^{+}$ energy spacing of ${}^{12}\mathrm{C}$. The column “AQCM” shows the result obtained after diagonalizing the Hamiltonian consisting of the 18 AQCM basis states with $K=0$. The column “$p$-2p2h” shows the result after adding 50 two-particle–two-hole states for the protons within $K=0$. In the column “$n$-2p2h”, 50 two-particle–two-hole states for the neutrons are mixed, where $K$ quantum number is still fixed to $K=0$. The last column “$K$-mixing” shows the energy of the $2^{+}$ state when $K$ mixing is invoked.

Figure 4

Energy convergence for the $0^{+}$ state of ${}^{12}\mathrm{C}$; 120 two-particle–two-hole basis states are coupled to the 18 AQCM basis states. The basis states from 19 to 58 on the horizontal axis are excited states of the two protons, from 59 to 98 are excited states of the two neutrons, and from 99 to 138 are excited states of a proton and a neutron.