Bragg diffraction
X-rays scattering from a crystal lattice. When waves reflected from successive lattice planes arrive in phase, they constructively interfere — producing the sharp diffraction peaks that reveal atomic structure.
2d sin θ = nλ
Reading atomic structure with waves
When X-rays strike a crystal, they scatter from the regularly-spaced atoms arranged in lattice planes. William Lawrence Bragg and his father William Henry Bragg showed in 1913 that the scattered waves constructively interfere only at specific angles, producing sharp intensity peaks. This discovery — earning them the Nobel Prize in Physics in 1915 — launched the field of X-ray crystallography, which has since determined the structures of millions of molecules from table salt to DNA.
The key insight is geometric: waves reflecting from the second lattice plane travel an extra distance of 2d sinθ compared to waves from the first plane, where d is the spacing between planes and θ is the angle of incidence. When this extra path equals a whole number of wavelengths (nλ), the reflected waves arrive in phase and reinforce each other. At all other angles, the waves from different planes partially or fully cancel.
From path difference to intensity
Each lattice plane contributes a reflected wave with the same amplitude but a phase shift proportional to the extra path length. For N planes with spacing d, the total reflected amplitude is the sum of a geometric series: A ∝ Σ exp(i·k·2d·sinθ·n) for n = 0 to N−1. The resulting intensity is I ∝ sin²(Nδ/2) / sin²(δ/2), where δ = (2π/λ)·2d·sinθ is the phase difference between adjacent planes.
This function has sharp peaks (Bragg peaks) whenever δ = 2πn, which is exactly Bragg’s law: 2d sinθ = nλ. The peaks become sharper as more planes contribute (larger N), which is why real crystals with millions of planes produce extremely narrow diffraction lines. Between the main peaks, there are N−2 subsidiary maxima — visible in the plot below when you increase the number of planes.
A century of discovery
X-ray crystallography has been behind more Nobel Prizes than any other experimental technique. The Braggs themselves determined the structure of diamond and sodium chloride. In 1953, Rosalind Franklin’s X-ray diffraction photograph of DNA (the famous “Photo 51”) provided the crucial evidence for the double helix structure. Dorothy Hodgkin solved the structures of penicillin and vitamin B12, earning her the 1964 Nobel Prize in Chemistry.
Today, synchrotron light sources produce X-ray beams billions of times brighter than laboratory tubes, enabling protein crystallographers to determine structures of enormous biological machines — ribosomes, viral capsids, membrane channels — at atomic resolution. The diffraction patterns are far more complex than the simple 1D model here, but the underlying physics is identical: Bragg’s law, applied to a three-dimensional lattice.
Electrons, neutrons, and atoms
Bragg diffraction is not limited to X-rays. Any wave with a wavelength comparable to the lattice spacing will diffract: electron diffraction (used in transmission electron microscopy), neutron diffraction (sensitive to light atoms and magnetic ordering), and even atom diffraction from crystal surfaces. The de Broglie relation λ = h/p connects particle momentum to wavelength, meaning that matter itself behaves as a wave and satisfies Bragg’s law when scattered from periodic structures.
Modern variants include powder diffraction (randomly oriented microcrystals producing ring patterns), Laue diffraction (white-beam, fixed crystal), and serial femtosecond crystallography using X-ray free-electron lasers, which can capture diffraction from single molecules before they are destroyed by the intense beam.