Magnon lifetime of one dimensional insulating ferromagnetic chain under finite temperature

A magnon-phonon interaction model is built in one dimensional insulating ferromagnetic chain.By using Matsubara Green function theory,we have studied the magnon damping?Im Σ*(1)(k),and calculated the magnon damping ?Im Σ*(1)(k) curve in the Brillouin zone for different parameters,and different temperatures in the system.It is concluded that at the boundary of Brillouin zone there is a strong magnon damping.But the magnon damping is very small on zone of small wave vectors,and magnon damping is maximal at the boundary of Brillouin Zone.According to expression ?I m Σ*(1)(k) =(2τ),the magnon lifetime can be discussed.