defined by: where dσ cal L d Ω ( θ$_{cm}$, E$_{x}$ ) is the calculated distorted-wave Born approximation (DWBA) cross section corresponding to 100% energy-weighted sum rule (EWSR) for the L-th multipole. The fractions of the EWSR, a$_{L}$ ( E$_{x}$ ), for various multipole components were determined by minimizing χ $^{2}$. This procedure is justified since the angular distributions are well charac- terized by the transferred angular momentum L , according to the DWBA calculations for α scattering. It was confirmed that the MDA fits were not affected by including L > 7. The DWBA calculations were performed following the method of Satchler and Khoa [45], using the density-dependent single-folding model for the real part, obtained with a Gaussian α -nucleon potential, and a phenomenological Woods-Saxon potential for the imaginary term. Therefore, the α -nucleus interaction is given by: where V$_{F}$ ( R ) is the real single-folding potential obtained by folding the ground-state density with the density-dependent α -nucleon interaction: where v$_{DDG}$ ( r, r $^{′}$, ρ ) is the density-dependent α -nucleon interaction, | r − r $^{′}$| is the distance between the center of mass of the α particle and a target nucleon, ρ ( r $^{′}$) is the ground-state density of the target nucleus at the position r ′ of the target nucleon, β =1.9 fm $^{2}$, and t =1.88 fm. W is the depth of the Woods-Saxon type imaginary part of the potential, with the reduced radius R$_{I}$ and diffuseness a$_{I}$ . These calculations were performed with the computer code PTOLEMY [46, 47], with the input values modified [48] to take into account the correct relativistic kinematics. The shape of the real part of the potential and the form factor for PTOLEMY were obtained using the codes SDOLFIN and DOLFIN [49]. We use the transition densities and sum rules for various multipolarities described in Refs. [13, 50, 51]. The radial moments were obtained by numerical integration of the Fermi mass distribution with the parameter values from Ref. [52] (listed in Table I). 8 U ( r ) = V$_{F}$ ( r ) + iW/ (1 + exp (( r − R$_{I}$ ) /a$_{I}$ )) (2) v$_{DDG}$ ( r, r $^{′}$, ρ ) = − v (1 − βρ ( r $^{′}$) 2 / $^{3}$) exp ( −| r − r $^{′}$| $^{2}$/t$^{2}$)) (3) dσ exp d Ω ( θ$_{cm}$, E$_{x}$ ) = 7 ∑ L =0 a$_{L}$ ( E$_{x}$ ) × dσ cal L d Ω ( θ$_{cm}$, E$_{x}$ ) (1)