Condensed matter physics

Crystal structure and the reciprocal lattice

A crystal repeats a motif on a Bravais lattice; its reciprocal lattice describes periodicity in wave-vector space.

\mathbf{R}=\sum_i n_i\mathbf{a}_i,\qquad \mathbf{a}_i\cdot\mathbf{b}_j=2\pi\delta_{ij}

Definition: Structure and diffraction

Choose primitive vectors and an atomic motif at each lattice point. Different unit cells can describe the same crystal.

Rotate the graphene sheet to inspect sublattices A and B; the Ising lattice below shows spin order weakening as temperature rises.

Physical model

Choose primitive vectors and an atomic motif at each lattice point. Different unit cells can describe the same crystal.

\mathbf R=\sum_i n_i\mathbf a_i,\quad \mathbf a_i\cdot\mathbf b_j=2\pi\delta_{ij}

The reciprocal basis obeys aᵢ·bⱼ=2πδᵢⱼ. Elastic diffraction transfers a reciprocal-lattice vector; Bragg’s law is the equivalent description using plane spacing.

Example: Worked example

For a simple cubic lattice of constant a, d₁₀₀=a; first-order Bragg diffraction requires 2a sinθ=λ.

Solution

Substitute d₁₀₀=a into 2d sinθ=λ for m=1: 2a sinθ=λ.

Quick check

Fourier transformation turns spatial periodicity into discrete peaks in reciprocal space. For scattering vector q, diffraction requires q=G, a reciprocal-lattice vector; intensity also depends on the structure factor of the atomic motif. Thus crystals with the same Bravais lattice but different bases can have different diffraction patterns. This provides a route to infer plane spacings, symmetry, and atomic positions from X-ray or neutron diffraction.

For an orthogonal lattice with constants a, b, c, planes (hkl) have spacing dₕₖₗ=1/√[(h/a)²+(k/b)²+(l/c)²]. In a cubic crystal this reduces to dₕₖₗ=a/√(h²+k²+l²). Combining d with Bragg’s law λ=2d sinθ determines lattice constants from diffraction angles. Some reflections are nevertheless extinguished by the structure factor, so not every index triplet produces an observed peak.

Miller indices (hkl) are integer coordinates of a plane normal in the reciprocal basis, not point coordinates in real space. Crystal symmetry groups equivalent planes and is classified by space groups. Together with the structure factor, systematic absences can identify lattice type.

In crystal diffraction, which set contains the elastic momentum transfer?

How many atomic sublattices does graphene’s honeycomb unit cell contain?

References

Charles Kittel (2004). Introduction to Solid State Physics