Three-$\alpha$ configurations in the $0^{+}$ states of $^{12}\mathrm{C}$

TL;DR

This paper investigates the geometric arrangements of three alpha particles in the ground and first excited states of Carbon-12 using two different cluster models, revealing dominant equilateral and isosceles triangle configurations.

Contribution

It compares two alpha-cluster models with different treatments of the Pauli principle to analyze geometric configurations in $^{12}$C states.

Findings

Equilateral triangle configurations dominate in the ground state.

Isosceles triangle configurations dominate in the first excited state.

Different models show quantitative differences in wave functions.

Abstract

Geometric configurations of three-$\alpha$ particles in the ground- and first-excited $J^\pi=0^+$ states of $^{12}$C are discussed within two types of $\alpha$-cluster models which treat the Pauli principle differently. Though there are some quantitative differences especially in the internal region of the wave functions, equilateral triangle configurations are dominant in the ground state, while in the first excited $0^+$ state isosceles triangle configurations dominate, originating from $^8{\rm Be}+\alpha$ configurations.