Kinetic theory of the anistropic Heisenberg Hamiltonian

A set of kinetic equations for the correlation functions describing the simultaneous propagation of two-spin fluctuations in an anisotropic Heisenberg paramagnet [$H=-\left(\frac{1}{2}\right)\Sigma a_{\mathrm{ij}}{\overrightarrow{\mathrm{S}}}_i·{\overrightarrow{\mathrm{S}}}_i-\left(\frac{1}{2}\right)\Sigma b_{\mathrm{ij}}{S}_i^z{S}_j^z$] is obtained that reduces to earlier results in the case of vanishing anisotropy. The equations conserve the total spin and energy, and have for their equilibrium solution the spherical-model static correlation functions. A prescription for obtaining a diagrammatic expansion of the moments of the spectral density of a single-spin fluctuation mode, ${〈{\omega }^n〉}_q$, for any $n$, to lowest order in $\frac{1}{C}$, where $C$ is the number of spins in the range of the interaction is given. The kinetic equation can be used to calculate this spectral density by providing a partial summation of terms in the diagrammatic representation of the moment expansion. The spectral density obtained by solving the kinetic equation will have the correct second and fourth moments to lowest order in $\frac{1}{C}$. An approximate solution for the response of the $q=0\mathrm{}$ mode in a dipole lattice in a strong magnetic field is obtained using a constant-relaxation-time approximation, and shown to be in good agreement with the measurements in Ca${\mathrm{F}}_2$. A comparison of the theory with other attempts to calculate the free induction decay is given.