  The elastic scattering cross section measured at energies $E\lesssim 10$ MeV/nucleon for some light heavy-ion systems having two identical cores like $^{16}$O+$^{12}$C exhibits an enhanced oscillatory pattern at the backward angles. Such a pattern is known to be due to the transfer of the valence nucleon or cluster between the two identical cores. In particular, the elastic $\alpha$ transfer has been shown to originate directly from the core-exchange symmetry in the elastic $^{16}$O+$^{12}$C scattering. Given the strong transition strength of the $2^+_1$ state of $^{12}$C and its large overlap with the $^{16}$O ground state, it is natural to expect a similar $\alpha$ transfer process (or inelastic $\alpha$ transfer) to take place in the inelastic $^{16}$O+$^{12}$C scattering. The present work provides a realistic coupled channel description of the $\alpha$ transfer in the inelastic $^{16}$O+$^{12}$C scattering at low energies. Based on the results of the 4 coupled reaction-channels calculation, we show a significant contribution of the $\alpha$ transfer to the inelastic $^{16}$O+$^{12}$C scattering cross section at the backward angles. These results suggest that the explicit coupling to the $\alpha$ transfer channels is crucial in the studies of the elastic and inelastic scattering of a nucleus-nucleus system with the core-exchange symmetry. 
Elastic and inelastic alpha transfer in the $^{16}$O+$^{12}$C scattering