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In scattering situations characterized by the appearance of a rainbow, the scattering sum converges very slowly. To calculate the differential cross section in such cases the sum is transformed by the Poisson summation formula into a series of integrals, each of which is evaluated by the method of stationary phase. Previous work along these lines, reviewed here, has led to the classical, `crude semi-classical' and Airy approximations. Using a method due to Chester, Friedman and Ursell a uniform approximation is derived, valid for all angles irrespective of whether they are near or far from the rainbow angle. This new approximation involves only those quantities needed to solve the equivalent purely classical problem; it also reduces to the earlier results in appropriate limiting cases. For a particular numerical example of scattering by a Lennard-Jones potential the uniform approximation is found to agree very well with the results of an exact calculation by Hundhausen and Pauly.
Michael Berry (Tue,) studied this question.