Uniform approximation in scattering by spheres

A new approximation to the scattering amplitude from an impenetrable sphere at short wavelengths is developed. In contrast with Fock's theory, it remains valid at large scattering angles, where it can be matched with the usual semiclassical approximations. The accuracy, both for the magnitude and for the phase of the scattering amplitude, is improved by one or more orders of magnitude over previously known approximations. The approximation remains accurate even at size parameters below unity, bridging the gap between short and long wavelengths. Large-angle diffraction can be interpreted as a tunnelling effect.