Collective spin precession quantized into magnons
In a ferromagnet, all spins align. A spin wave (magnon) is a coherent precession propagating as a wave — the quantum of this excitation is the magnon.
J: exchange coupling, S: spin magnitude, k: wavevector, a: lattice spacing. Near k=0, the dispersion is quadratic:
This contrasts with acoustic phonons (linear dispersion) — magnons are "soft" at long wavelengths due to broken continuous symmetry (Goldstone mode).