Transition densities and form factors in the triangular α -cluster model of C12 with application to C12+α scattering

Abstract

Densities and transition densities are computed in an equilateral triangular $\ensuremath{\alpha}$-cluster model for $^{12}\mathrm{C}$, in which each $\ensuremath{\alpha}$ 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 distorted-wave Born approximation inelastic cross sections for the $^{12}\mathrm{C}(\ensuremath{\alpha},{\ensuremath{\alpha}}^{\ensuremath{'}})$ reaction. The comparison with experimental data indicates that the simple geometrical model with rotations and vibrations gives a reliable description of reactions where $\ensuremath{\alpha}$-cluster degrees of freedom are involved.