Low-temperature phase transitions in terbium(x)dysprosium(1-x) competing anisotropy alloys

Tb$\sb{\rm x}$Dy$\sb{1-\rm x}$ is probably the simplest example of a magnetic system with competing anisotropy. Tb and Dy are both easy-plane ferromagnets at low temperature, with well characterized anisotropy energies, such that Tb moments prefer to lie along the $\rm\vec b$ axis in the hcp structure, while Dy moments prefer the $\rm\vec a$ axis. It has been predicted, using a two-subnetwork "average crystal" model, that the low-temperature phase of the alloys with x between 0.86 and 0.76 exhibits a non-symmetry magnetization direction, a non-collinear spin structure, and twelfth-order anisotropy. However, the approximation is suspect when used to describe magnetic structure in alloys near their phase boundaries, such as those between the ferromagnetic and the non-collinear phase. Using a more realistic probability distribution for site occupation, the fluctuations of the spin orientation and the spin-spin correlation function have been calculated as a function of composition in the non-collinear phase. It was found that the mean square fluctuation in spin orientation is proportional to $\sqrt{\rm x\sb{b}-x}$ where x$\sb{\rm b}$ is the critical concentration which separates the $\rm\vec b$ axis ferromagnet from the non-collinear magnet. The average orientation as calculated from the average crystal model also varies as $\sqrt{\rm x\sb{b}-x}$. Therefore, the simple two-subnetwork model is not valid, at least near the phase boundaries. Upon including the fluctuations in the calculation of the phase boundaries, it has been found that the fluctuations from the average site occupation always increase the range of stability of the non-collinear phase. These characteristics, (1) a non-collinear phase space and (2) a widening of this phase space with the inclusion of fluctuations, should be valid for other similar alloy systems, and therefore the model should be useful in predicting the low-temperature phase characteristics of a variety of binary alloys, and in modifying previous calculations based only on the average crystal model.