Spin Fluctuations in Heisenberg Paramagnets. II. Kinetic Theory and Approximate High-Temperature Solutions

A set of kinetic equations describing the time evolution of the two-spin correlation functions $〈S^{-}(q_1,t)S^{+}(q_2,t)〉$ and $〈S^z(q_1,t)S^z(q_2,t)〉$ is derived. The derivation is based upon a cluster expansion of the equations of motion and an intuitively plausible renormalization. The resultant equations may be interpreted graphically, and the derivation of the kinetic equations taken to be a partial justification of the summation of a particular set of graphs in the moment expansion for the spectral density. The kinetic equations have the property that they conserve the total spin $\Sigma 〈\overrightarrow{\mathrm{S}}(\overrightarrow{\mathrm{q}},t)·\overrightarrow{\mathrm{S}}(-\overrightarrow{\mathrm{q}},t)〉$ and the total energy $\Sigma q^{}V\left(q\right)\mathrm{}〈\overrightarrow{\mathrm{S}}(\overrightarrow{\mathrm{q}},t)·\overrightarrow{\mathrm{S}}(-\overrightarrow{\mathrm{q}},t)〉$. The second and fourth moments of the spectral density for fluctuations in the magnetization predicted by these equations are correct to lowest order in $\frac{1}{c}$, where $c$ is the number of spins in the range of the interaction. Numerical calculations of the spectral density, based upon an approximate solution of the kinetic equations that is valid outside of the critical region, are compared with the neutron-scattering data for a wide range of temperatures and wave vectors and found to be in good agreement. The temperature dependence of the diffusion coefficient in the high-temperature region is calculated and compared with experiment.