= Solution
Use the same physical normalization for both fields. Part (b) has $E_s=E_i|f_\infty^{(1)}|^2/R^2+O(R^{-3})$, so the common <energy flux> prefactor cancels. The <differential cross-section for scalar wave scattering> is
$$
\boxed{\sigma_d^{(1)}(\widehat{\mathbf r},\widehat{\mathbf r}_0)
=|f_\infty^{(1)}(\widehat{\mathbf r},\widehat{\mathbf r}_0)|^2
=\frac{k^4}{16\pi^2}\left|\int_V(n^2-1)e^{-i\mathbf k_s\cdot\mathbf r}\,d^3r\right|^2.}
$$
The <far-field pattern> has dimensions of length in this normalization, so the <differential cross-section for scalar wave scattering> has dimensions of area per unit <solid angle>. Only the scattered field is used in this ratio; interference between the incident and scattered fields is a different contribution to the total field's <energy flux>.
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