= Hard-sphere S-wave scattering
{title2=$\sigma_0=4\pi R^2[\sin(kR)/(kR)]^2$}
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>.
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