= Delta-shell Born amplitude
{title2=$f_B(\theta)=-gR\,\operatorname{sinc}[2kR\sin(\theta/2)]$}
For a real shell potential $V(r)=V_0\delta(r-R)$, define $g=2mV_0R/\hbar^2$ and $\operatorname{sinc}u=\sin u/u$, with value one at zero. Angular integration of its three-dimensional <Fourier transform> gives $4\pi V_0R^2\operatorname{sinc}(qR)$, where the <momentum transfer> has wave-number magnitude $q=2k\sin(\theta/2)$. Substitution in the <first Born approximation> proves the displayed <scattering amplitude>. It is real. Its squared <modulus> is a nonzero order-$V_0^2$ <differential scattering cross-section>, whose <optical theorem> counterpart requires the order-$V_0^2$ imaginary part of the forward amplitude from the next Born term. Thus squaring the first-order amplitude does not supply a consistently unitary second-order amplitude.
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