Phenomenological Discussion of Magnetic Ordering in the Heavy Rare-Earth Metals

The rare-earth metals Gd-Tm have similar crystal structures and their magnetic properties have been partially evaluated by a number of techniques. The magnetic order is complicated, showing several phases in some cases and differing considerably in the various elements. These various orderings can be explained on a molecular field (Bragg-Williams) model if a long-range oscillatory exchange interaction whose minimum Fourier component $J(\mathrm{q})$ is at $q\ensuremath{\ne}0$, small quadrupole-quadrupole interaction, and anisotropy are included. A crystal field calculation gives axial and hexagonal anisotropies which vary along the series in a way which accounts for the observed structures. In Tb, Dy, and Ho the moment is forced into the basal plane and the order is a spiral at high $T$, becoming ferromagnetic at low $T$ because of the hexagonal anisotropy. The quadrupole-quadrupole interaction determines the change of pitch with $T$. In Er and Tm the moment is forced along the $c$ axis and the observed order, with sinusoidal variation of this moment, is found to have lowest free energy at high $T$. As $T$ is lowered, transitions to an anti-phase domain structure and then to ferromagnetism are predicted.

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