Origin of Anisotropy in Gadolinium Crystal Using a New Spin Hamiltonian

Single crystal rare-earth magnets, such as hexagonal-close-packed gadolinium, usually have a large second order anisotropy K_2 and a negative first order anisotropy K_1 at low temperatures, which are difficult to explain using microscopic theories. An atomic scale effective spin Hamiltonian \mathcal F\\boldsymbol S_i\ is proposed, which, apart from the usual isotropic nearest neighbor coupling J, consists of two new terms that are different for in-plane and out-of-plane neighbors and which are characterized by two new couplings C_1 and C_2, respectively. The hybrid Monte–Carlo method is utilized to sample this system to the desired Boltzmann-like distribution \exp(-\mathcal F/k__\rm BT). It is found that K_2 and K_1 are compatible with the experimental values and arise naturally from the exchange anisotropy C_1 and C_2, which are less than 0.01\% in magnitude of the isotropic exchange energy J. This new model spin Hamiltonian can also be applied to study other magnetic properties.