Magnitude of Exchange Interactions in Gadolinium

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 Tc one obtains J=2.9°K, while from the low-temperature magnetization curve and simple spin-wave theory one finds J=1.8°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 T72 and important corrections arising from zone boundary effects were included. The detailed analysis yields a slightly improved low-temperature value, J=2.1°K. It also demonstrates that the magnetization follows a T32 law very closely from low temperatures up to about 0.8 Tc 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.