A low-wave-vector expansion for reflectivity

We expand an accurate, closed-form equation for the reflectivity in a power series around q2=0, where q2 is the z component of the wave-vector transfer in a film of constant composition. We examine, in particular, the case where q2=0 at the critical edge. Such an example might be realized through a neutron-reflectivity experiment on a microemulsion that exhibits surface layering. We find the coefficients of this low-wave-vector expansion to be the moments of the scattering-length density profile. We demonstrate how the first moment of the profile can be obtained from an accurate reflectivity measurement