Transition densities and form factors in the triangular $\alpha$-cluster model of $^{12}$C with application to $^{12}$C+$\alpha$ scattering

TL;DR

This paper models $^{12}$C as a triangular alpha-cluster system, calculating densities and form factors to accurately describe inelastic scattering reactions involving alpha particles.

Contribution

It introduces a simple geometric alpha-cluster model for $^{12}$C that effectively predicts scattering cross-sections and reveals intrinsic nuclear structures.

Findings

Model accurately reproduces experimental inelastic scattering data.

Transition densities reveal intrinsic vibrational modes.

Geometrical approach effectively describes alpha-cluster degrees of freedom.

Abstract

Densities and transition densities are computed in an equilateral triangular alpha-cluster model for $^{12}$C, in which each $\alpha$ particle is taken as a gaussian density distribution. The ground-state, the symmetric vibration (Hoyle state) and the asymmetric bend vibration are analyzed in a molecular approach and dissected into their components in a series of harmonic functions, revealing their intrinsic structures. The transition densities in the laboratory frame are then used to construct form-factors and to compute DWBA inelastic cross-sections for the $^{12}$C$(\alpha, \alpha')$ reaction. The comparison with experimental data indicates that the simple geometrical model with rotations and vibrations gives a reliable description of reactions where $\alpha$-cluster degrees of freedom are involved.