Hard-sphere limit

The hard-sphere limit of a repulsive spherical potential makes penetration into a sphere of radius $a$ vanish. Its low-energy S wave scattering length tends to $a$, and its quantum total cross-section tends to $4\pi a^2$.

For a hard sphere of radius $R$, the exterior radial S wave is proportional to $\sin [k(r-R)]$. Matching it to $\sin (kr+{\delta }_0)$ gives scattering phase shift ${\delta }_0=-kR$ modulo $\pi$ and scattering length $R$. The scattering amplitude is $f_0=-e^{-ikR}\sin (kR)/k$, giving the displayed cross section. The zero-energy limit is $4\pi R^2$, its zeros occur at positive integer multiples $kR=n\pi$, and its high-energy envelope decreases as $k^{-2}$. This is one partial wave, not the total high-energy hard-sphere result obtained by summing all partial waves.