Bragg Gaps in the Brillouin Zone

This simulator diagonalizes the real nearly-free-electron "central equation" for a 2D square lattice, live, in your browser. A finite plane-wave basis built from the crystal's reciprocal lattice is projected onto the Schrödinger equation to give an N×N symmetric matrix; a Jacobi eigensolver finds its eigenvalues at every point along the Γ→X→M→Γ high-symmetry path, exactly as a real electronic-structure code would. With the periodic potential off you see the empty-lattice approximation — free-electron parabolas simply folded into the first Brillouin zone. Turn the potential on and Bragg reflection at the zone boundary splits the degenerate plane waves apart, opening real energy gaps at X and M that you can watch grow, while the reciprocal lattice and the scanned k-point are drawn live on the square Brillouin zone below the bands.