Spin waves (magnons) are collective excitations of a ferromagnet. Their dispersion ω(k) reflects the lattice geometry. At long wavelengths, ω ∝ k² (ferromagnet) vs ω ∝ k (antiferromagnet).
Heisenberg Hamiltonian: H = −J·Σ Sᵢ·Sⱼ. Spin-wave (Holstein-Primakoff): ω_k = 2JS·z·(1−γ_k) + gμ_B·B, where γ_k is the lattice structure factor. Long-wavelength: ω ∝ k² (FM) or ω ∝ |k| (AFM).