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The structure factor is the coherent scattering amplitude of the contents of a unit cell at a specified momentum transfer. Contributions from its atoms add with phases fixed by their positions. At a reciprocal lattice vector indexed by Miller indices, the crystal basis structure factor isChanging every phase sign conjugates the amplitude for real scattering factors and leaves the intensity unchanged. The positions here are fractions of the chosen cell edges. An atomic basis of a crystal should not be confused with a set of primitive translation vectors.

In kinematic elastic scattering the diffraction intensity of a Bragg reflection is proportional to $|S_{hkl}{|}^2$, after including geometrical, polarization, thermal-displacement and experimental factors. Powder integrated intensities also include multiplicity. The many-cell interference factor selects reciprocal-lattice vectors satisfying the diffraction geometry; the unit-cell factor determines their relative amplitudes and additional extinctions. Ring positions determine spacings through Bragg's law, while intensity patterns and systematic absences constrain lattice centering and atomic arrangements. Intensity loses the overall complex phase, so structure determination is not a direct measurement of every individual atomic phase.

X-rays scatter from electron density, giving an atomic factor that changes with momentum transfer and chemical charge distribution. Neutrons scatter coherently from nuclei with bound coherent scattering lengths, which need not track atomic number; magnetic neutron scattering is an additional channel in magnetic materials. Their different elemental/isotopic contrasts make the two probes complementary.

The X-ray wavelength is $\lambda =hc/E=1.54980\times 10^{-10}\mathrm{m}$. Write the amplitude of one conventional unit cell as the face-centred cubic selection rule times the two-atom atomic basis amplitude:The first bracket is four when the indices have equal parity and zero otherwise. Among the allowed Bragg reflections, the zinc blende structure factor therefore givesHere the atomic scattering factors are taken as real. For the requested rough estimate set $f_{\mathrm{Ga}}\simeq 31$ and $f_{\mathrm{As}}\simeq 33$. The weak difference reflections remain allowed: unequal species do not have the extra exact extinction of the equal-species diamond limit.

The first six allowed values of $h^2+k^2+l^2$ are $3,4,8,11,12,16$. Bragg's law gives the measured scattering angle $2\theta =2{\sin }^{-1}(\lambda \sqrt{h^2+k^2+l^2}/2a)$. For powder diffraction, multiply the squared amplitude by the powder diffraction multiplicity $m_{hkl}$. The last two columns use $I_{111}=100$:The first intensity column is the geometry-independent multiplicity-and-structure estimate. The second additionally assumes an unpolarized beam and a common symmetric powder-diffractometer geometry, with Lorentz-polarization factorAn overall constant cancels in either normalization. A particular measured intensity set is not fixed by the atomic numbers alone: actual atomic scattering factors decrease with momentum transfer, anomalous scattering and absorption can matter, and the instrument geometry and polarization determine angular corrections. The two columns state the approximation rather than asserting identical peak heights for all instruments.^[ref]