Energy average of the scattering matrix in picket-fence models

The optical-model potential at low energy is defined by the requirement that it yields a suitably defined energy average of the scattering function. It is argued that the quantities which appear in the expression for this energy average derived by Lane and Thomas in the framework of R-matrix theory can be identified with those which appear in a convenient parametrization of the fine structure cross section. The accuracy of this identification is illustrated with the help of various picket-fence models. We discuss the independence of the results with respect to the size of the energy domain covered by the experimental data and to the nature of the R-matrix boundary parameters. Also we discuss terms that can be included if one wishes to discuss fluctuations about a smooth average