  Recent measurements for fusion cross section at energies around the Coulomb
barrier have systematically indicated a significant deviation of fusion cross
sections from a prediction of double-folding model. It has been argued that the
deviation can be accounted for if one uses a large value of surface diffuseness
parameter for a Woods-Saxon internuclear potential. We investigate this problem
using large-angle quasi-elastic scattering, that is a good counterpart of
fusion reaction. Towards a reconciliation of the apparent anomaly in the
diffuseness parameter for fusion reactions, we also discuss possible
ingredients which are missing in present nuclear reaction models. Those include
a deviation from the frozen density approximation, the effect of
antisymmetrization and the role of coordinate dependent moment of inertia for
the relative motion between the colliding nuclei.
