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

TL;DR

This study explores the geometric configurations of alpha clusters in the second 2+ state of carbon-12, revealing that it predominantly forms an acute-angled triangle structure related to the 8Be+alpha configuration, but not as a perfect rotational band member.

Contribution

The paper introduces a detailed three-alpha cluster model analysis of the second 2+ state in 12C, clarifying its structure and relation to the Hoyle state using advanced computational methods.

Findings

Main configurations are acute-angled triangle shapes.

Approximately 2/3 of the state consists of 8Be+alpha components.

The state is not an ideal rigid rotational band member.

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 Schr\"odinger 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.