Approximate treatment of coupled-channels effects in sub-barrier fusion

The authors show how an 'adiabatic' approximation to the usual coupled-channels equations gives a simple way of calculating and understanding the sub-barrier fusion cross sections in rotational nuclei. The effect of coupling of a 0+ ground state to a low-lying 2+ excited state may be calculated by solving two elastic scattering problems with central potentials differing by an amount related to the coupling strength. The important feature of this method is that the full angular momentum algebra is included rigorously, allowing all S-matrix elements of physical interest to be generated from the two elastic S-matrices obtained from a standard optical-model code. In particular, the inelastic cross section may be obtained in a manner which is numerically simpler than the distorted-wave Born approximation and which is also more accurate for strongly deformed systems. The observed enhancement in sub-barrier fusion cross sections for deformed heavy ions can be understood simply in this approximation and the coupling to excited states leads to a simple phenomenological expression which is an extension of the formula of Wong (1973). For a given deformation the enhancement is shown to be more important for systems with a large Coulomb barrier.