Energy Width of Spin Waves in the Heisenberg Ferromagnet

A calculation of the energy width of spin waves whose energy is small compared to $\mathrm{kT}$ in a Heisenberg ferromagnet is presented. Full account is taken of all two-particle processes. For $\frac{2JS}{\mathrm{kT}}\gg 1$ and $\frac{2JS{a}^2{\lambda }^2}{\mathrm{kT}}\ll 1$, the energy width $\Delta {\omega }_{\lambda }$ of a spin wave of wave number $\lambda$ is $\Delta {\omega }_{\lambda }\sim {\lambda }^4{T}^2[A+B\ln (\frac{\mathrm{kT}}{2JS{a}^2{\lambda }^2})+C{\ln }^2(\frac{\mathrm{kT}}{2JS{a}^2{\lambda }^2}\left)\right]$. The disagreement with some previous treatments is explained physically by their neglect of two many-body effects.