In a crystal, atoms are connected by interatomic bonds (modeled as springs). When atoms vibrate collectively, the normal modes of vibration form phonons — quantized quasiparticles of lattice vibration.
This gives a sinusoidal dispersion curve, bounded by the Brillouin zone edge k = π/a. At small k (long wavelength), ω ≈ vₛk (acoustic, linear — behaves like sound). The optical branch in diatomic lattices has ω(k=0) = √(2κ/μ), where adjacent atoms move out of phase.
The quantization of energy: E = ℏω(n + ½). Phonons are bosons — Bose-Einstein statistics determine thermal occupancy: ⟨n⟩ = 1/(e^(ℏω/kT) − 1). This underlies the Debye model of specific heat and thermal conductivity in solids.