Some Effects of Anisotropy on Spiral Spin-Configurations with Application to Rare-Earth Metals

This is a theoretical study of effects of anisotropy on spiral spin-configurations, emphasizing the problem of the magnetic ordering in rare-earth metals. The principal results are as follows: It is shown that the ferromagnetic spiral observed at low temperatures by Wilkinson, Koehler, Wollan, and Cable in erbium can be the classical ground state of a spin Hamiltonian containing exchange and anisotropy terms, provided the latter includes terms of at least the fourth power in the spin variables. Furthermore, the observed cone angles (which imply large deviations from configurations possible with exchange forces alone) can be obtained with anisotropy forces much smaller than the exchange forces. The spin-wave spectrum and susceptibility for ferromagnetic spirals were also considered. The former has the interesting properties that $\omega \left(\mathrm{k}\right)$ is linear in k for small k (even though the net moment is not zero), and there are two distinct branches, as contrasted with the case of simple antiferromagnetic spirals. For high temperatures, calculations are made on the basis of the molecular field approximation. It is shown that a small easy-axis anisotropy implies that at the highest transition temperature, $T_c$, the ordered spin configuration is a static longitudinal spin wave; the average spins are collinear, their lengths varying sinusoidally through the crystal. As $T$ decreases below $T_c$, the amplitude of the wave grows in order ${(T_c-T)}^{\frac{1}{2}}$, other Fourier components entering in higher order. The perpendicular components remain zero until a second transition temperature is reached, below which they begin to order. Since this complex type of behavior, which has been observed in erbium, can also occur with the same exchange and anisotropy constants needed to give the observed ground state, the possibility exists of describing erbium through the whole temperature range by this type of theory.