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

TLDR

Rare‑earth metals Gd–Tm share similar crystal structures, yet their magnetic ordering is complex, exhibiting multiple phases that vary markedly among the elements. A Bragg‑Williams molecular‑field model incorporating a long‑range oscillatory exchange interaction (J(q) minimum at nonzero q), weak quadrupole‑quadrupole coupling, and crystal‑field‑derived axial and hexagonal anisotropies explains the observed magnetic structures across the series. The model predicts that in Tb, Dy, and Ho the moments lie in the basal plane forming a spiral at high temperature that becomes ferromagnetic at low temperature due to hexagonal anisotropy, while in Er and Tm the moments align along the c‑axis with sinusoidal order at high temperature, transitioning to anti‑phase domains and then ferromagnetism as temperature decreases.

Abstract

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.