Spatial Structure of the $^{12}$C Nucleus in a 3$\alpha$ Model with Deep Potentials Containing Forbidden States

TL;DR

This paper investigates the spatial structure of the $^{12}$C nucleus's low-lying states using a 3-alpha model with deep potentials, revealing different configurations for bound states and resonances, and highlighting phase transitions in the system.

Contribution

It introduces an exact orthogonalization method to treat Pauli forbidden states in a 3-alpha model, providing new insights into the structure and phase transitions of $^{12}$C states.

Findings

Bound states mainly involve (2,2) and (4,4) partial waves.

Hoyle resonance has a different structure, resembling $^8$Be + $ extalpha$.

Spatial configurations vary significantly between bound and resonant states.

Abstract

The spatial structure of the lowest 0$_1^+$, 0$_2^+$, 2$_1^+$ and 2$_2^+$ states of the $^{12}$C nucleus is studied within the 3$\alpha$ model with the Buck, Friedrich, and Wheatley $\alpha \alpha$ potential with Pauli forbidden states in the $S$ and $D$ waves. The Pauli forbidden states in the three-body system are treated by the exact orthogonalization method. The largest contributions to the ground and excited 2$_1^+$ bound states energies come from the partial waves $(\lambda, \ell)=(2,2)$ and $(\lambda, \ell)=(4,4)$. As was found earlier, these bound states are created by the critical eigenstates of the three-body Pauli projector in the 0$^+$ and 2$^+$ functional spaces, respectively. These special eigenstates of the Pauli projector are responsible for the quantum phase transitions from a weakly bound "gas-like" phase to a deep "quantum liquid" phase. In contrast to the bound states, for the Hoyle resonance 0$_2^+$ and its analog state 2$_2^+$, dominant contributions come from the $(\lambda, \ell)=(0,0)$ and $(\lambda, \ell)=(2,2)$ configurations, respectively. The estimated probability density functions for the $^{12}$C(0$_1^+$) ground and 2$_1^+$ excited bound states show mostly a triangular structure, where the $\alpha$ particles move at a distance of about 2.5 fm from each other. However, the spatial structure of the Hoyle resonance and its analog state have a strongly different structure, like $^8$Be + $\alpha$. In the Hoyle state, the last $\alpha$ particle moves far from the doublet at the distance between $R=3.0$ fm and $R=5.0$ fm. In the Hoyle analog 2$_2^+$ state the two alpha particles move at a distance of about 15 fm, but the last $\alpha$ particle can move far from the doublet at the distance up to $R=30.0$ fm.