1D diatomic chain: acoustic and optical branches, band gap at Brillouin zone boundary
Chain Parameters
Phonon dispersion: A diatomic chain with masses m₁ and m₂ and spring constant C has two branches: acoustic ω_A(k) and optical ω_O(k). A bandgap opens at the Brillouin zone boundary k = π/a, with width Δω = √(2C/m₁) − √(2C/m₂). Waves with frequency in the gap cannot propagate (imaginary k). This is the origin of phononic bandgaps and acoustic metamaterials.