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

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