Magnon Spin Waves: Heisenberg Ferromagnet

Spin wave dispersion ω(k) = 4JS(1−cos k) + D·k² (anisotropy gap). Real-space spin precession, group velocity vg = dω/dk, density of states, and magnon gap all respond to exchange J and anisotropy D.

ω = — | vg = — | Gap = —