Bragg’s law and Brillouin zones: a visual guide

In 1912 Max von Laue shone X-rays through a crystal and saw a pattern of sharp spots on a photographic plate. A year later, William Lawrence Bragg and his father found a remarkably simple way to explain those spots. Their equation is still how we measure the spacing of atoms in crystals — from table salt to new battery materials.

Bragg’s law

Picture a crystal as stacks of parallel planes of atoms, separated by a distance d. An X-ray beam hits the planes at a glancing angle θ. Each plane reflects a little of the beam. The reflections from neighboring planes add up strongly only when the extra path travelled by the deeper ray is a whole number of wavelengths:

2d sin θ = nλ

For any other angle the reflections cancel. So a diffraction pattern is really a list of angles at which this condition holds — and from those angles we can calculate d.

Reciprocal space

A crystal has many families of planes, each with its own spacing and orientation. Keeping track of them is much easier in reciprocal space. Every family of planes becomes a single point, placed in the direction perpendicular to the planes at a distance 2π/d from the origin. The collection of all these points is the reciprocal lattice.

In this language, diffraction happens when the change in the X-ray’s wave vector equals a reciprocal lattice vector G. This is the Laue condition, k′ − k = G. The Ewald sphere turns it into geometry: draw a sphere of radius 2π/λ; every reciprocal lattice point that lies on its surface produces a diffraction spot.

Brillouin zones

Take one point of the reciprocal lattice and draw the perpendicular bisector planes to its nearest neighbors. The smallest closed volume around the point is the first Brillouin zone. For a face-centered cubic crystal such as copper or aluminum it is a truncated octahedron; for a body-centered cubic crystal such as iron it is a rhombic dodecahedron.

Brillouin zones matter because their boundaries are exactly where the Bragg condition is met for electrons moving in the crystal. That is where energy gaps open in the band structure — the reason some materials are metals and others are insulators or semiconductors.

Why some peaks are missing

Not every reciprocal lattice point gives a spot. When a crystal has more than one atom per unit cell, waves scattered by different atoms can cancel. The structure factor F(G) = Σ fⱼ e^(iG·rⱼ) measures this. For body-centered cubic crystals, reflections with h + k + l odd vanish; for face-centered cubic, reflections with mixed odd and even indices vanish. These systematic absences are a fingerprint of the crystal structure.

Try it yourself

The 3D Brillouin zone and Bragg simulator lets you build lattices, rotate Brillouin zones in 3D, move the Ewald sphere and watch diffraction peaks appear. Switch between BCC, FCC, diamond and HCP structures to see the extinction rules in action, then explore band structures, Fermi surfaces, phonons and peak fitting.

Key points

• Bragg’s law, 2d sin θ = nλ, links diffraction angles to the spacing of atomic planes.
• In reciprocal space, each family of planes is a point; diffraction occurs when k′ − k = G.
• Brillouin zone boundaries are where electrons satisfy the Bragg condition and energy gaps appear.
• The structure factor explains which diffraction peaks disappear.