Inelastic $^{16}$O +$^{12}$C rainbow scattering to the $2^+$ (4.44 MeV) state of $^{12}$C was measured at the incident energies, $E_L$ = 170, 181, 200, 260 and 281 MeV. A systematic analysis of the experimental angular distributions was performed using the coupled channels method with an extended double folding potential derived from realistic wave functions for $^{12}$C and $^{16}$O calculated with a microscopic $alpha$ cluster model and a finite-range density-dependent nucleon-nucleon force.The coupled channels analysis of the measured inelastic scattering data shows consistently some Airy-like structure in the inelastic scattering cross sections for the first $2^+$ state of $^{12}$C, which is somewhat obscured and still not clearly visible in the measured data. The Airy minimum was identified from the analysis and the systematic energy evolution of the Airy structure was studied. The Airy minimum in inelastic scattering is found to be shifted backward compared with that in elastic scattering.
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