Long-Range Exchange Interactions from Spin-Wave Resonance

The presence of the quartic term, in addition to the usual quadratic term, in the spin-wave dispersion relation has been established to within 3% accuracy by spin-wave resonance experiments in a number of Permalloy films. High-precision measurements at 70 Gc/sec permitted observation of spin wavelengths as short as 225 Å. The coefficient of the quartic term, which involves the fourth moment of the exchange interaction, leads to a long-range interaction $〈{\rho }^2〉=120\mathrm{}r_1^{}{}_{}{}^2$, where $r_1$ is the nearest-neighbor distance; this is in good agreement with the range determined previously from spin-wave interaction measurements on a similar sample. Since the films are predominantly polycrystalline, anisotropy in the exchange could not be distinguished. $〈{\rho }^2〉$ determined from the coefficient of the $T^{\frac{5}{2}}$ term in the magnetization, measured by ferromagnetic resonance in the same type of sample, is smaller by a factor of about five. However, with such a long range, the validity of the usual expansion of the spin-wave energy to describe the magnetization is doubtful.