Spin Motions in a Classical Ferromagnet

The static and dynamic behavior of a simple cubic lattice of classical spins with Heisenberg interactions has been examined by computer on arrays of up to 8192 spins with periodic boundary conditions. Equilibrium values of the energy and magnetization at various temperatures are obtained by Monte Carlo calculations. The results indicate that the magnetization is well approximated by the formula ${(1-\frac{T}{T_c})}^{\beta }$, with $0.32\lesssim \beta \lesssim 0.36$ for the full range of temperature from zero to the Curie point $T_c$. Spin arrays whose energy and magnetization agree with the ensemble averages at a given temperature are taken as characteristic of that temperature. These are then employed to obtain instantaneous and time-displaced spin-correlation functions, the latter involving the numerical solution of the equations of motion of the spin system. The time-displaced correlations show considerable structure. For the most part, spin-wave theory agrees with the low-temperature results. Raising the temperature slows down and smears the structure of the time-displaced correlations. The slowing is much less pronounced than the drop in magnetization. The pulse emanating from a single misaligned spin in a lattice at zero temperature is also calculated.