Determination of the Phase of the Scattering Amplitude from the Differential Cross Section

The unitarity relation of quantum scattering theory is studied to show how it can be utilized to obtain the phase of the scattering amplitude that corresponds to a given differential cross section. The main results are the following: (i) Provided the cross section is sufficiently small, repeated iterations of the unitarity equation yield a series which converges uniformly to the correct phase of the scattering amplitude. (ii) A modified iteration procedure can be introduced which provides a sequence of alternating upper and lower bounds on the phase of the scattering amplitude. Subject to a certain restriction on the magnitude of the cross section, the upper and the lower bounds so generated converge, respectively, from above and from below to the true value of the phase. (iii) A high-energy approximation to the unitarity relation, referred to as the small-phase approximation (SPA), is introduced. It is shown to be solvable by means of an iteration sequence, and upper and lower bounds to its solutions are constructed. (iv) Using some very crude approximations, we solve the SPA equation for the phase of the scattering amplitude explicitly in terms of the cross section. Since extremely oversimplifying assumptions are employed, this can be regarded only as a primitive model for a possible behavior of the exact solutions. The outline of these results is followed by a discussion of the practical applicability of the above methods for obtaining the phase of the scattering amplitude from the experimental cross section, in which their potential significance for atom-atom collision theory is stressed.