Part 1 — The Monatomic Chain
where the phonon is born

Strip the crystal to its skeleton: identical masses m, identical springs K, one atom per cell, spacing a. Displace one and the whole line rings. Every possible ringing is a sum of normal modes, each a travelling wave with its own ω and k — and each mode, quantised, comes in lumps of ℏω. That lump is the phonon. Pick a mode below and watch the atoms actually move it.