Kronig-Penney Band Gaps — the 2D Companion

This is the exactly-solvable 1D counterpart to the 3D nearly-free-electron band-structure simulator: instead of numerically diagonalizing a truncated plane-wave Hamiltonian on a 2D square lattice, it solves the closed-form Kronig-Penney transcendental equation for a 1D chain of delta-function barriers, live, on a flat 2D canvas. For every crystal momentum k it finds the roots of F(E) = cos(ka) directly by root-scanning the real dispersion function — no basis truncation, no matrix at all. With the barrier strength at zero you recover the exact free-electron parabola folded into the Brillouin zone; turning it up opens a real forbidden gap at the zone boundary that grows monotonically, the same Bragg-reflection mechanism, reached by an entirely independent calculation you can watch converge and verify against the analytic free-electron limit.