Qualitative Aspects of Neutron-Nuclear Interactions and the Optical Model

The appearance of broad maxima and minima in the energy dependence of neutron-nuclear total and scattering cross sections below 100 MeV is given a simple explanation. Within the framework of the Glauber diffraction approximation, the general conditions for the occurrence of these maxima and minima are obtained in terms of an integral, over impact parameters, involving a phase-shift function. A modification of the approximation is employed so that the formalism may be applied below 100 MeV. The assumption that the average phase difference between that portion of the neutron wave which traverses the nucleus and that which does not is equal to an integral multiple of $\pi$ at cross-section maxima and minima is shown to be incorrect. The approximations are illustrated with some optical-model calculations which indicate that the positions of the maxima and minima shift to higher energies with increasing target mass number. Some numerical calculations using the Schrödinger equation with square-well and Woods-Saxon potentials are compared with cross sections calculated from the approximation. Near the maxima, appreciable contributions come from a number of partial cross sections. The appearance of these maxima implies neither a resonance in any single partial wave nor a maximum in any single partial cross section. A brief consideration of the energy dependence in the total, absorption, and scattering cross sections separately, reveals that the principal qualitative features of the total cross section are manifest in the scattering cross section.