An electron in a crystal feels a periodic potential U(r) = U(r+R) with the lattice's periodicity. Bloch's theorem says its states can be expanded in plane waves ei(k+G)·r built only from k plus reciprocal-lattice vectors G. Projecting the Schrödinger equation onto this basis gives the central equation, a real symmetric matrix diagonalized live here:
H(k)_{G,G'} = (ħ²/2m)|k+G|² δ_{G,G'} + U_{G-G'}
diagonalize H(k) → eigenvalues E_n(k), n = 1,2,3,...
With U = 0 this is the "empty lattice approximation": the free-electron parabola ħ²k²/2m simply folds into the first Brillouin zone, band by band, with no gaps. Turning on UG (only the shortest reciprocal vectors carry a nonzero Fourier component here — a common, physically legitimate simplification) couples plane waves that are nearly degenerate, i.e. where |k+G| ≈ |k+G'|. That is exactly the Bragg condition, and near-degenerate perturbation theory gives a two-level splitting
E_± = (E_G + E_G')/2 ± √[ ((E_G − E_G')/2)² + U_G² ]
so a gap of size 2|UG| opens wherever a Bragg plane crosses the path — most visibly at the Brillouin-zone boundary points X and M plotted here.
- UG slider — the periodic potential's Fourier component; watch the gaps at X and M grow linearly with it.
- Lattice constant a — rescales the free-electron energy ħ²/2m·(π/a)², so a smaller a stretches the whole band structure upward (tighter confinement, more curvature).
- Basis size — how many reciprocal-lattice shells (plane waves) are kept; more shells reveal higher bands and sharpen where they fold.
- k-Point Scan — sweeps k continuously along the standard high-symmetry path Γ(0,0) → X(π/a,0) → M(π/a,π/a) → Γ, the same path shown in every real DFT/tight-binding band-structure plot, with a live marker on the square first Brillouin zone below.
This is the textbook nearly-free-electron model — the same mechanism (Bragg reflection at a Brillouin-zone boundary) that opens the fundamental gap in real semiconductors and insulators.