Reciprocal Lattice: The Hidden Map of Crystal Diffraction

The reciprocal lattice is a second lattice built from the real crystal lattice, and it is the natural language of X-ray diffraction. While the ordinary (direct) lattice lists where atoms sit in space, the reciprocal lattice lists the possible diffraction directions: every point in it corresponds to a set of crystal planes that can reflect X-rays. It is a hidden map: every diffraction pattern ever recorded is a photograph of it.

This matters because crystals are identified and their atomic arrangements solved by reading diffraction patterns, and those patterns only make sense in reciprocal space. The famous Bragg law of X-ray crystallography is a statement about reciprocal-lattice vectors. Without this map, the double helix of DNA and the structure of every drug molecule could not have been decoded.

Key Concepts

• Direct lattice vectors (a₁, a₂, a₃): the three edge vectors of the real unit cell, measured in metres.
• Reciprocal lattice vectors (b₁, b₂, b₃): defined by b₁ = 2π(a₂×a₃) / (a₁·(a₂×a₃)), with cyclic permutations. Unit: inverse metres (m⁻¹).
• Orthogonality rule: bᵢ·aⱼ = 2π if i = j, and 0 otherwise. Each reciprocal vector is perpendicular to a plane of the direct lattice.
• Reciprocal lattice vector G: G = h·b₁ + k·b₂ + l·b₃, where h, k, l are the Miller indices of a set of planes. Its magnitude |G| = 2π/d, where d is the spacing of those planes.
• Laue condition: diffraction happens when the change in the X-ray wavevector equals a reciprocal lattice vector: Δk = G.

Formulas

• Cubic crystals: the reciprocal of a simple cubic lattice of constant a is simple cubic of constant 2π/a. The reciprocal of FCC is BCC and vice versa.
• Plane spacing: d = a / √(h² + k² + l²) for a cubic crystal, so |G| = 2π√(h²+k²+l²) / a.
• Bragg's law: 2d·sinθ = nλ, with n = 1, 2, 3, …, λ the X-ray wavelength and θ the angle between the ray and the planes. This is exactly the Laue condition written in ordinary geometry.

Worked Example

A cubic crystal has a = 2.0 Å. X-rays of λ = 1.54 Å diffract from the (110) planes. Find the plane spacing and the Bragg angle for n = 1.

1. Plane spacing: d = a / √(1² + 1² + 0²) = 2.0 / 1.414 = 1.414 Å.
2. Bragg's law: sinθ = λ / (2d) = 1.54 / (2 × 1.414) = 1.54 / 2.828 = 0.5446.
3. θ = arcsin(0.5446) ≈ 33.0°. So the detector must sit at 2θ ≈ 66° from the incident beam.

Common Misconceptions

• "The reciprocal lattice is just 1/a." It has the right idea for a 1D chain, but in 3D the directions matter: each reciprocal vector is perpendicular to direct-lattice planes, and the 2π factor is part of the standard physics definition.
• "Every (hkl) gives a diffraction peak." Selection rules cancel some: in FCC, h, k, l must be all odd or all even, so (100) is missing.
• "Bragg's law and the Laue condition are different rules." They are the same physics: Bragg's law is the Laue condition Δk = G expressed with the scattering angle.

Real-World Applications

• X-ray crystallography: protein and drug structures are solved by indexing diffraction spots to reciprocal-lattice points.
• Electron diffraction in microscopes: transmission electron microscopes image reciprocal space directly to identify nanoparticles.
• Semiconductor quality control: X-ray rocking curves measure how perfectly silicon wafers match their ideal reciprocal lattice.

The reciprocal lattice turns crystal planes into a lattice of their own, with diffraction exactly when Δk = G. Reading it through Bragg's law measures atomic spacings with angstrom precision.