= Solution
{c}
Separate the entire-system atom index into a cell translation $\mathbf R$ and an index $p$ within a single <primitive unit cell>. The potential becomes
$$
V(\mathbf x)=\sum_{\mathbf R\ \mathrm{all\ cells}}\sum_{p\ \mathrm{one\ cell}}u_p(\mathbf x-\mathbf R-\mathbf r_p).
$$
The same $u_p$ is used in every translated cell because the <atomic basis> repeats. Normalize <plane waves> in volume $\Omega$ by $\langle\mathbf x|\mathbf k\rangle=\Omega^{-1/2}e^{i\mathbf k\cdot\mathbf x}$. For $\mathbf G=\mathbf k'-\mathbf k$, substitution and $\mathbf s=\mathbf x-\mathbf R-\mathbf r_p$ give
$$
\langle\mathbf k'|V|\mathbf k\rangle=\frac1\Omega\left[\sum_{\mathbf R\ \mathrm{all\ cells}}e^{-i\mathbf G\cdot\mathbf R}\right]\left[\sum_{p\ \mathrm{one\ cell}}e^{-i\mathbf G\cdot\mathbf r_p}\int d^3s\,e^{-i\mathbf G\cdot\mathbf s}u_p(\mathbf s)\right].
$$
For a real static scalar potential, define the <atomic scattering factor> in the positive-exponent convention of the question by
$$
\boxed{b_p(\mathbf G)=\int d^3s\,e^{i\mathbf G\cdot\mathbf s}u_p(\mathbf s),\qquad S(\mathbf G)=\sum_{p\ \mathrm{one\ cell}}b_p(\mathbf G)e^{i\mathbf G\cdot\mathbf r_p}.}
$$
The second bracket in the matrix element is $S(\mathbf G)^*$. At a <reciprocal lattice vector> every term of the first bracket is one, giving the number of cells $N_c$. Therefore the stated transition-rate rule, equivalently the first <Born approximation> and <Fermi's golden rule> up to the final-state kinematic factors, yields
$$
\boxed{I(\mathbf G)\propto|\langle\mathbf k'|V|\mathbf k\rangle|^2=\frac{N_c^2}{\Omega^2}|S(\mathbf G)|^2\propto|S(\mathbf G)|^2.}
$$
The harmless complex conjugation records the distinction between the source's positive-exponent <Fourier transform> and the usual positive-exponent incoming <plane wave>. Reversing every Fourier sign gives the equivalent amplitude convention. Only the cell-translation sum ranges over the full crystal; the <crystal basis structure factor> contains the atoms of one cell.
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