A numerical method for the determination of atom-atom scattering amplitudes from the measured differential cross sections

A numerical method is given for the determination of the scattering amplitude, hence all the phase shifts, from the differential cross section at fixed energy. To obtain the phase of the scattering amplitude, the unitarity equation of scattering theory is solved, using the (experimental) cross section as input information. A straightforward iterative approach diverges for atom-atom input data, thus an appropriately modified method of solution is introduced to overcome this difficulty. The method was applied to two test cases, in both of which calculated atom-atom cross sections were used as simulated input data. Convergence to the correct phase was obtained in both examples, starting in each case with a guess phase function that differed considerably from the true solution. Convergence cannot be obtained, however, from an extremely poor starting phase. This study shows that the scattering amplitude for atomic scattering at thermal energies can be determined systematically, without use of parametrized functions, if a sufficiently accurate experimental cross section is available. A subsequent article describes a quantum mechanical procedure whereby the interaction potential can be constructed from the determined scattering amplitude.