Spin Waves — Magnon Dispersion & Ferromagnetic Resonance

Collective spin precession in a ferromagnet: long-wavelength spin waves (magnons) have quadratic dispersion ω∝k², excited by microwave fields at FMR frequency

1.00
1.00
0.50
0.40
Heisenberg ferromagnet (1D chain): Ĥ = −J Σ Sᵢ·Sᵢ₊₁ − gμ_B B₀ Σ Sᵢᶻ. Spin wave dispersion: ℏω(k) = 2JS(1−cos(ka)) + gμ_B B₀ ≈ JSa²k² + ω_Larmor for small k. Quantized spin waves are magnons with bosonic statistics. Ferromagnetic resonance (FMR) occurs at k=0: ω_FMR = gμ_B B₀/ℏ (Kittel formula). The gap at k=0 is set purely by the applied field; exchange governs the k²-dependence. Magnon dispersion is measured by inelastic neutron scattering and Brillouin light scattering.