Kinetic equation for spin waves in a weak alternating magnetic field

From the equation for the density matrix, a kinetic equation is obtained for the magnon population numbers $n_k$ in the first nonvanishing approximation in the small parameter $hA_k/\gamma_k$, where $h$ is the amplitude of the alternating field, $A_k$ is the constant of the coupling between the alternating field and the magnons, and $\gamma_k$ is the damping frequency of magnons with wave vector $k$. A diagram method is used to show that only in this approximation and, in addition, only for small frequency difference $\omega-2\omega_k\ll\omega_k$, does the expression for the rate of change of $n_k$ have the simple Lorentzian form postulated earlier by White and Sparks.