We analyze the elastic scattering of the $\alpha $ + $^{16}$O system at 240 and 400 MeV within the framework of the Coulomb-modified Glauber model by using an effective nucleon-nucleon ($NN$) amplitude considering a $q^{4}$ component. The calculated results show a fairly good agreement with the experimental data. The near-side and the far-side decompositions of the elastic cross section have also been performed by following Fuller's formalism. The presence of a nuclear rainbow is evidenced by a classical deflection function. We also investigate the effect of a $q^{4}$ component of the effective $NN$ amplitude on the elastic and the reaction cross sections. The effective $NN$ amplitude is found to be helpful in reproducing the $\alpha $ + $^{16}$O elastic scattering data.

Keywords: Coulomb-modified Glauber model, Effective $NN$ amplitude, In-medium $NN$ total cross section, Elastic scattering, $\alpha $ + $^{16}$O