An introduction is given to the methods and results of some recent researches into statistical thermodynamics bearing upon the correlation functions of magnetic moments in Heisenberg-coupled spin-only magnets, and their intimate connection with neutron-scattering theory and practice is brought out. T h e interrelationships between the correlation function, the relaxation function, the generalized susceptibility, the power spectrum of the fluctuations and the neutron scattering are explained, and it is shown that insights into any one of these aspects can serve to illuminate the others. Different forms of approximate theory are seen to be suitable in approaching the topic through these different avenues ; the method of moments for analysing the power spectrum and the connection with Green function theory are described in particular. Some examples are calculated in molecular-field theory and with various other simple approximations ; expressions for the frequency spectrum of the susceptibility are obtained in various temperature ranges. T h e extreme forms of the frequency spectrum in appropriate conditions as a set of delta functions, as a pseudo-Gaussian and as a pseudo-Lorentzian are derived, and the transitions between these cases are considered. The approximation of spin diffusion and its limitations are analysed; the problem of the dynamical slowing-down of magnetic fluctuations near the phase-transition point is examined and the current position in this enquiry is set out.

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By comparing measured and calculated versions of the generalized magnetic response function of nickel over a wide range of temperatures it is shown that the much-discussed simple model of an itinerant-electron ferromagnet based on a random-phase-approximation treatment of the Hubbard Hamiltonian accounts, in its essentials, for the magnetism of that substance. Thermal neutron inelastic scattering techniques were used to explore the whole Brillouin zone of wave vectors K in nickel up to energies ~0 of some 2kBTC over the range of temperatures up to 2Tc ; the computations were based on direct evaluation of the Lindhard expression for the generalized susceptibility, using a tight-binding band structure, followed by exchange enhancement of the spin part as per Izuyama, Kim and Kubo. The contribution to the neutron scattering from orbital motion of the electrons was estimated to be small, and in a special experimental investigation a satisfactorily low upper limit on this quar~tity for our present purposes was observed.In the course of the article we attempt to illuminate with our computational results the theoretical content of the random-phase approximation, and to show how it accounts for spin waves, paramagnons and a variety of other phenomena. For instance at low temperatures, where the collective or spinwave mode is the most prominent feature, the Stoner-mode states are so strongly mixed with the spln-wave states by the electron-electron interaction that little trace is left of the ' band of Stoner modes ' often referred to in elementary accounts of this phenomenon. With increase of temperature, the severe broadening of the spin wave and the characteristic style of evolution of the magnetic response function through the critical temperature are correctly predicted, outside a narrow range of conditions about the critical point. The dramatic divergence of the susceptibility characteristic of the phase transition is confined much more closely towards the origin of K,co space than could be accounted for by free-electron gas theory, and this is explained. On the other hand what appears to be a case of breakdown of the random-phase approximation is detected in the ferromagnetic state, where in a certain set of conditions at large K and small co an excess of longitudinal spin correlation was measured over that expected theoretically.Aside from the aforementioned aberrant region, however, absolute agreement between experiment and theory essentially within the statistical error is obtainable by assuming that the effective Coulomb interaction parameter Ieft(K) falls off gently betweer~ K=0 and Kmax in a manner consistent with the predictions of Singwi et al. ; with increase of temperature, Ierf(0) declines gently and the amount of fall-off with K increases. It is shown that the theory of the Heisenberg ferromagnet fails entirely to account for the observations--lndeed, fails to correlate any of the data on nickel--and that at least three-quarters of the known magnetic stiffness of nickel must be due to band-theory effects...

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