One dimensional Cherenkov processes in ferromagnetic insulator

One dimensional Cherenkov processes in ferromagnetic isolators are studied with perturbation theory under the constraint condition of conservation of energy and momentum. It is shown that the magnon-phonon interaction channels are limited and wave number dependent, which result in respectively $1/k^{2}$ and $1/k^{4}$ dependence of the lifetime and the relaxation time of long wavelength magnons. The reciprocal of relaxation time between magnons and phonons, $1/\tau_{mp}$, is found to be a linearly increasing function of the temperature as $T>$ 70 K. Based on the Sanders-Walton model, we further show that when a thermal (phonon) gradient is applied along the system, the temperature difference between the phonon bath and the magnons with wave-vector $k$ becomes more pronounced as $k$ decreasing.