# Nonlocalized motion in two-dimensional container of $\alpha$ particles in $3^-$ and $4^-$ states of $^{12}$C

## TL;DR

This paper investigates the negative-parity states of $^{12}$C using a container model, revealing nonlocalized clustering structures that differ from traditional rigid triangle configurations, with high similarity to GCM wave functions.

## Contribution

It introduces a shift parameter in the container model to study negative-parity states and demonstrates the nonlocalized clustering structure in $^{12}$C's $3^-$ and $4^-$ states.

## Key findings

- High similarity (96% and 92%) to GCM wave functions for $3^-$ and $4^-$ states.
- Negative-parity states exhibit nonlocalized clustering, contrasting traditional rigid structures.
- The container model effectively describes these states with variational calculations.

## Abstract

The first $3^-$ and $4^-$ states of $^{12}$C are studied in the present container model, in which the shift parameter is introduced to break the parity symmetry for projecting out the negative-parity states. Taking the limit as the shift parameter approaches zero and by variational calculations for one-deformed size parameter, the local energy minima are obtained for the $3^-$ and $4^-$ states. It is found that the obtained single THSR (Tohsaki-Horiuchi-Schuck-R\"{o}pke) wave functions for $3^-$ and $4^-$ states are 96% and 92% equivalent to the corresponding GCM wave functions, respectively. The calculated intrinsic densities further show that these negative-parity states of three clusters, different with the traditional understanding of rigid triangle structure, are found to have nonlocalized clustering structure in the two-dimensional container picture.