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

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$(\alpha ,{\alpha }^{\mathrm{\prime }})$ 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.