Magnitude of Exchange Interactions in Gadolinium

Abstract

The effective nearest-neighbor exchange interaction $J$ in gadolinium can be deduced from experiment in two ways. From a high-temperature expansion of the susceptibility and the measured value of the Curie temperature $T_c$ one obtains $J=2.9\mathrm{}°$K, while from the low-temperature magnetization curve and simple spin-wave theory one finds $J=1.8\mathrm{}°$K. In order to clarify the reason for this discrepancy the low-temperature spin-wave problem has been analyzed in detail. Contributions to the magnetization from two different spin-wave modes were evaluated up to order $T^{\frac{7}{2}}$ and important corrections arising from zone boundary effects were included. The detailed analysis yields a slightly improved low-temperature value, $J=2.1\mathrm{}°$K. It also demonstrates that the magnetization follows a $T^{\frac{3}{2}}$ law very closely from low temperatures up to about $0.8T_c$ because of a complicated cancellation of the higher order effects. Finally it is shown qualitatively that the remaining difference between the high- and low-temperature $J$ values can probably be explained by the effect of interactions beyond nearest neighbors and it is suggested that these are on the average more antiferromagnetic than ferromagnetic.