General Theory of Spin-Wave Interactions

An ideal model of a ferromagnet is studied, consisting of a lattice of identical spins with cubic symmetry and with isotropic exchange coupling between nearest neighbors. The aim is to obtain a complete description of the thermodynamic properties of the system at low temperatures, far below the Curie point. In this temperature region the natural description of the states of the system is in terms of Bloch spin waves. The nonorthogonality of spin-wave states raises basic difficulties which are examined and overcome. The following new results are obtained: a practical method for calculating thermodynamic quantities in terms of a nonorthogonal set of basic states; a proof that in 3 dimensions there do not exist states (shown by Bethe to exist in a one-dimensional chain of spins) in which two spins are bound together into a stable complex and travel together through the lattice; a calculation of the scattering cross section of two spin waves, giving a mean free path for spin-spin collisions proportional to T^-7{2} at low temperatures; and an exact formula for the free energy of the system, showing explicitly the effects of spin-wave interactions. Quantitative results based on this theory will be published in a second paper.