Three-$\alpha$ configurations of the second $J^{\pi }=2^{+}$ state in ${}^{12}$C

Abstract

We investigate geometric configurations of $\alpha$ (${}^4$He nucleus) clusters in the second $J^{\pi }=2^{+}$ state of ${}^{12}$C, which has been discussed as a rotational band member of the second $0^{+}$ state, the Hoyle state. The ground and excited $0^{+}$ and $2^{+}$ states are described by a three-$\alpha$ cluster model. The three-body Schrödinger equation with orthogonality conditions is accurately solved by the stochastic variational method with correlated Gaussian basis functions. To analyse the structure of these resonant states in a convenient form, we introduce a confining potential. The two-body density distributions together with the spectroscopic information clarify the structure of these states. We find that main configurations of both the second $0^{+}$ and $2^{+}$ states are acute-angled triangle shapes originating from the ${}^8$Be($0^{+}$)$+\alpha$ configuration. However, the ${}^8$Be$+\alpha$ components in the second $2^{+}$ state become approximately 2/3 because the 8Be subsystem is hard to excite, indicating that the state is not an ideal rigid rotational band member of the Hoyle state.

Comments: 7 pages, 3 figures