One can show that in general δ δ s ( k ) ≈ ≈ − − k ⋅ ⋅ a s + O ( k 2 ) for small k (i.e. low energy scattering). The parameter a s of dimension length is defined as the scattering length. For our potential we have therefore a = r 0 , in other words the scattering length for a hard sphere is just the radius. (Alternatively one could say that an arbitrary potential with s-wave scattering length a s has the same low energy scattering properties as a hard sphere of radius a s .) To relate the scattering length to physical observables that can be measured in a scattering experiment we need to compute the cross section σ σ . In scattering theory one writes the asymptotic wavefunction as (we assume there is a finite ranged scatterer at the origin and there is an incoming plane wave along the z -axis):