= Solution
The radial <S wave> vanishes at the hard surface, so outside the sphere $u_0(r)=A\sin[k(r-R)]$. Comparison with $A\sin(kr+\delta_0)$ gives
$$
\boxed{\delta_0(k)=-kR\pmod\pi,\qquad a_0=-\lim_{k\to0}\frac{\tan\delta_0}k=R.}
$$
The <hard-sphere S-wave scattering> amplitude and <cross section> are therefore
$$
\boxed{f_0(k)=-e^{-ikR}\frac{\sin(kR)}k,\qquad
\frac{\sigma_0}{4\pi R^2}=\left(\frac{\sin x}{x}\right)^2,\quad x=kR.}
$$
The three prominent features are the finite low-energy limit $\sigma_0\to4\pi R^2$, the oscillations with exact zeros at $x=n\pi$ for positive <integers> $n$, and the decreasing envelope $\sigma_0\le4\pi/k^2$. The zero-energy value is four times the geometric area $\pi R^2$. Near zero the normalized <cross section> is $1-x^2/3+O(x^4)$. The large-$k$ fall here concerns the <S wave> alone; summing all <partial waves> gives a different high-energy total.
\Image[/past-exam-of-the-theoretical-physics-ii-course-of-the-university-of-cambridge/2019-q2-scattering.png]
{title=Hard-sphere S-wave cross-section and delta-shell Born angular scattering}
{description=Dimensionless curves calculated directly from the derived scattering amplitudes. The left panel concerns only the S wave, while the right panel shows different incident wave numbers for the weak delta-shell model.}
Back to article page
