= Solution
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> is
$$
\boxed{S_{hkl}=\sum_{j=1}^N f_j(\mathbf Q)\exp[2\pi i(hx_j+ky_j+lz_j)]}.
$$
Changing 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.
Back to article page
