Semiclassical approximation for the scattering volume in cold-atom collisions

A semiclassical result is derived for the scattering volume for near-threshold p-wave scattering by a potential falling off as 1/R(n), n >= 6, with R being the interatomic separation. This result depends only on the s-wave zero-energy classical action and properties of the long-range potential and hence can be related to the semiclassical result for the scattering length. The exact result for the scattering volume for the Lennard-Jones (2n - 2, n), n >= 6, is derived and it is shown that the corresponding semiclassical result is within 1% for potentials supporting at least two rotationless bound levels. Comparisons are made for the scattering volume with a variety of results in the literature for a range of systems supporting between 0 and 83 rotationless bound levels. The semiclassical result is generally within 5% and was never in error by more than 15%, which occurred for a case supporting no bound states.