Magnon Dispersion — Spin Waves

Collective Spin Excitations

Magnons are quantized spin waves — the collective precessional modes of a ferromagnet or antiferromagnet. They are the magnetic analogs of phonons and carry angular momentum ℏ.

In a ferromagnet, the dispersion is quadratic: ω ~ Jk² (gapless unless anisotropy). In an antiferromagnet, it is linear: ω ~ c|k| (relativistic). The gap is set by anisotropy Δ = gμ_B H_A.

At finite T, thermally excited magnons reduce magnetization: M(T) = M₀(1 − (T/T_C)^{3/2}) — the Bloch T^{3/2} law from spin-wave theory.

FM: ω(k) = 2JS(1−cos ka) + gμ_B B
≈ JSa²k² + gap (small k)
AFM: ω(k) = 2JSa k √(1+...) (linear)
Magnon #: n_k = 1/(e^{ℏω/kT}−1)

Exchange J 1.0 Zeeman field B 0.0 Anisotropy D 0.0 Temperature T/T_C 0.3 Mode: FM

Magnon Dispersion — Spin Waves in Magnets

Collective Spin Excitations