X-ray Diffraction and Structure Determination

Core Idea

X-ray diffraction is the primary experimental technique for determining crystal structures. When X-rays (wavelength comparable to atomic spacings, ~1 Angstrom) scatter from a crystal, constructive interference occurs only when the Bragg condition 2d sin(theta) = n*lambda is satisfied, or equivalently when the scattering vector equals a reciprocal lattice vector (von Laue condition). The intensity of each diffraction peak is proportional to |F(G)|^2, where F(G) is the structure factor — the Fourier transform of the electron density within one unit cell. Systematic absences in the structure factor reveal the lattice type and basis arrangement.

Explainer

X-ray diffraction is the experimental backbone of crystallography and condensed matter physics. The technique works because X-ray wavelengths (~0.5-2 Angstroms) are comparable to interatomic spacings in crystals, so crystals act as natural diffraction gratings. When a beam of X-rays hits a crystal, most of it passes through, but at specific angles the scattered waves from different atoms interfere constructively and produce sharp intensity peaks — the diffraction pattern. Each peak corresponds to scattering from a family of lattice planes.

The condition for constructive interference can be expressed two equivalent ways. Bragg's law — 2d sin(theta) = n*lambda — treats the crystal as a stack of partially reflecting planes separated by distance d and requires the path difference between reflections from successive planes to be an integer number of wavelengths. The von Laue condition — Delta k = G — requires the change in wavevector to equal a reciprocal lattice vector. Both encode the same physics: the periodicity of the lattice selects discrete scattering directions. The von Laue picture is more powerful because it connects directly to the reciprocal lattice and works naturally in three dimensions.

The intensities of the diffraction peaks carry information about what sits at each lattice point. This information is encoded in the structure factor F(G) = sum_j f_j e^{iG · r_j}, where the sum runs over all atoms j in the unit cell at positions r_j, and f_j is the atomic form factor (the Fourier transform of each atom's electron density). The measured intensity at each reciprocal lattice point is I proportional to |F(G)|^2. Some reflections may have F = 0 even though G is a valid reciprocal lattice vector — these systematic absences are diagnostic. For example, FCC lattices show peaks only when h, k, l are all even or all odd, and BCC lattices show peaks only when h + k + l is even. These selection rules immediately distinguish lattice types from the diffraction pattern.

The major challenge in structure determination is the phase problem: detectors record |F(G)|^2, losing the complex phase of F(G). Since reconstructing the electron density via inverse Fourier transform requires the full complex F(G), the phase must be recovered by indirect methods. Modern techniques including direct methods, anomalous dispersion, and computational refinement have made structure determination routine for many materials, but the phase problem remains a fundamental limitation. Beyond X-rays, electron diffraction and neutron diffraction complement the technique — electrons are sensitive to electrostatic potential and work well for thin films, while neutrons scatter from nuclei and magnetic moments, providing information invisible to X-rays.