Solid State Physics & Crystallography

Brillouin Zones & Wigner-Seitz

Construct Wigner-Seitz primitive cells in real space and 1st/2nd Brillouin Zones in reciprocal lattice space $\vec{k}$. Visualize Bragg reflection planes $\vec{k} \cdot \frac{\vec{G}}{2} = \left(\frac{G}{2}\right)^2$ and electron energy band gaps.

Lattice Type Square 2D

Wavevector |k| 1.25 Å⁻¹

Zone State 1st Brillouin Zone

Crystal Lattice Setup

2D Lattice Geometry

Lattice Constant a 3.0 Å

Visual Overlay

Show 1st Brillouin Zone (Central Polygon)

Show 2nd Brillouin Zone

Show Reciprocal Vector Badges

Controls

Reciprocal Lattice & Brillouin Zone Equations

Reciprocal Vector Basis: b₁ = 2π (a₂ × ẑ) / |a₁ × a₂|
Bragg Condition: k · (G / 2) = (G / 2)² (Perpendicular bisectors of reciprocal lattice)
1st Brillouin Zone: Wigner-Seitz cell surrounding origin in reciprocal k-space!

Developer Reference

Core Algorithm & Standalone Script

// Wigner-Seitz cell and Brillouin zone geometric solver
const canvas = document.getElementById('canvas');
const ctx = canvas.getContext('2d');
// k * (G/2) = (G/2)^2