Two alternating masses M and m connected by springs of constant κ. The dispersion relation has two branches: acoustic (in-phase, ω→0 at k=0) and optical (out-of-phase, finite ω at k=0). Click the dispersion curve to select a k-point and see the atomic motion.
The gap between acoustic and optical branches at the zone boundary (k=π/a) is Δω = √(2κ/m) − √(2κ/M). Larger mass ratio → wider gap. Infrared-active lattice vibrations correspond to the optical branch at k≈0.