High-Temperature Expansions for the Classical Heisenberg Model. I. Spin Correlation Function

Abstract

The diagrammatic representation of the first nine coefficients for loose-packed lattices and the first eight coefficients for close-packed lattices in a high-temperature series expansion of the zero-field spin correlation function ${〈{\mathbf{S}}_{f}\mathbf{\ifmmode\cdot\else\textperiodcentered\fi{}}{\mathbf{S}}_{g}〉}_{\ensuremath{\beta}}$ is presented. This calculation exploits the order-of-magnitude simplifications which occur in treating the quantum-mechanical spin operators in the Heisenberg model as isotropically interacting classical vectors of length ${[S(S+1)]}^{\frac{1}{2}}$. This semiclassical approximation---the "classical" Heisenberg model---appears to be excellent for some critical properties of interest if $S>\frac{1}{2}$. A recursion relation is seen to obviate the need to consider the sizeable classes of disconnected diagrams and diagrams containing articulation points. The utility of the high-temperature series for ${〈{\mathbf{S}}_{f}\mathbf{\ifmmode\cdot\else\textperiodcentered\fi{}}{\mathbf{S}}_{g}〉}_{\ensuremath{\beta}}$ is discussed. It contains information which is relevant to current experiments and is not contained in the high-temperature expansions for the thermodynamic functions (e.g., susceptibility, specific heat), as well as providing an efficient method of extending the series for all the thermodynamic functions together. As an example of the applicability of the series expansion of ${〈{\mathbf{S}}_{f}\mathbf{\ifmmode\cdot\else\textperiodcentered\fi{}}{\mathbf{S}}_{g}〉}_{\ensuremath{\beta}}$ to obtain information concerning the short-range magnetic order to be expected for $T>{T}_{c}$, a calculation of the elastic paramagnetic neutron-scattering cross section for normal cubic spinels with nearest-neighbor $A\ensuremath{-}B$ and $B\ensuremath{-}B$ exchange interactions is given, and contact is made with experiments on Mn${\mathrm{Cr}}_{2}$${\mathrm{O}}_{4}$.