= Solution
Take $(x_j,y_j,z_j)$ as fractional coordinates in the chosen <unit cell>. In the kinematic approximation the <crystal basis structure factor> is
$$
\boxed{F(hkl)=\sum_j f_j(Q)\exp[2\pi i(hx_j+ky_j+lz_j)]},\qquad Q=\frac{4\pi\sin\theta}{\lambda}.
$$
Equivalently, in Cartesian coordinates the phase is $\mathbf G_{hkl}\cdot\mathbf r_j$. Reversing every phase sign complex-conjugates the nonabsorbing amplitude and leaves its <intensity> unchanged. The <atomic scattering factor> is the <Fourier transform> of the <Chemical atom>[atom]'s <Electron number density>[electron density], in <electron> scattering units:
$$
f_j(\mathbf Q)=\int\rho_{e,j}(\mathbf r)e^{i\mathbf Q\cdot\mathbf r}\,d^3r.
$$
At zero <Bragg angle>[scattering angle] the normal form factor counts <Electron>[electrons]: $f_j(0)=Z_j$ for a neutral <Chemical atom>[atom], modified by the ionic charge for an <ion>. At increasing $Q$, different parts of the <electron> cloud scatter with different phases, reducing the coherent amplitude. Typical nonresonant <Atomic scattering factor>[atomic curves] fall from their forward value toward small values at large angle. A localized cloud falls more slowly in reciprocal space. Anomalous scattering near an absorption edge adds energy-dependent complex terms and is excluded from this simple sketch.
For a fixed reflection and fixed instrumental and geometric corrections,
$$
\boxed{I_{hkl}\propto|F(hkl)|^2.}
$$
The full amplitude also contains the lattice sum. At its <Reciprocal lattice>[reciprocal-lattice] peak, $N_c$ coherently illuminated identical cells give a peak amplitude $N_cF$; peak height and integrated peak <intensity> have different size dependences because the peak narrows with <crystal> size. Powder multiplicity, polarization, absorption, <Thermal fluctuation>[thermal motion] and measurement geometry also affect observed <Intensity>[intensities]. The basis factor determines relative <interference>, not every experimental prefactor.
\Image[/past-exam-of-the-condensed-matter-physics-course-of-the-university-of-oxford/2016-atomic-form-factor-trends.png]
{title=Forward electron counts and decreasing atomic scattering amplitudes}
{description=Original Gaussian-cloud illustrations for Cs+, Cl- and Fe, with their correct electron counts at zero transfer. Cloud widths are illustrative rather than tabulated atomic form-factor fits; the horizontal variable increases with scattering angle at fixed wavelength.}
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