Phase shifts in the collision of massive particles

In this paper phase shifts for inverse power potentials and their superpositions are considered. These potentials govern a large number of collision processes involving massive particles. But they are problematic as they do not conform to the norms obeyed by standard potentials: The usual rules for estimating the error of the phase shifts break down and an abnormally large number of phase shifts have to be computed, even if they are small. A sufficiency condition is deduced for the applicability of the Born approximation which shows that, for a given potential strength and l, surprisingly, the Born approximation is good at low energies and bad at high energies. The magnitude and direction of the error committed are also estimated. These conclusions are then verified in a number of special cases, for instance, the inverse fourth power potential used in the scattering of electrons by atoms and the Lennard--Jones potential used in the scattering of beams of molecular hydrogen with mercury atoms