Periodic δ-comb model of a 1-D crystal: cos(ka) = cos(qa) + (P/qa) sin(qa). Find allowed energy bands and forbidden gaps from the |·|≤1 corridor, then watch each band fold into the first Brillouin zone k ∈ ±π/a. Free-electron parabola overlaid for reference.
About this model
The Kronig–Penney model treats a one-dimensional crystal as a periodic δ-function comb. Allowed Bloch wave numbers satisfy cos(ka) = cos(qa) + (P/qa) sin(qa), where a is the lattice spacing, q relates to energy, and P measures barrier strength. Energies with |right-hand side| ≤ 1 form bands; elsewhere gaps open. Each band is then folded into the first Brillouin zone k ∈ [−π/a, π/a], with a free-electron parabola overlaid for reference. The model is a single-particle, one-dimensional, zero-range periodic potential — no phonons, no disorder, and no three-dimensional Brillouin-zone geometry. You vary P and a to widen or narrow gaps and watch how free-electron states rearrange into Bloch bands.
Who it's for: Solid-state physics and physical chemistry courses introducing nearly-free electrons, Bloch’s theorem, and band gaps.
Key terms
Kronig–Penney model
Brillouin zone
Energy bands
Band gap
Bloch wave
Periodic potential
How it works
Kronig–Penney δ-comb model of 1-D band structure: solve cos(ka) = cos(qa) + (P/qa) sin(qa) graphically, see allowed bands and forbidden gaps, then watch the dispersion E(k) fold into the first Brillouin zone — the canonical introduction to crystal energy bands.
Frequently asked questions
Why are some energies forbidden?
Bragg reflection from the periodic lattice hybridizes right- and left-moving waves near zone boundaries. The transcendental Kronig–Penney condition then has no real k when |cos(qa)+(P/qa)sin(qa)| > 1, opening gaps. Gaps are not caused by ‘missing’ free-electron states but by lattice scattering.
What does folding into the first Brillouin zone mean?
Any Bloch wave can be labeled with a reduced wave vector in [−π/a, π/a] plus a band index. Extended-zone free-electron segments are mapped back by reciprocal-lattice translations G = 2πn/a. The overlay parabola helps you see which free-electron pieces become which bands.
Does larger P always mean larger gaps?
Stronger δ barriers (larger |P|) generally widen gaps and flatten bands, but gap sizes also depend on energy through the (P/qa)sin(qa) term. At special energies the sine factor can suppress the perturbation; the simulator shows this energy dependence explicitly.