Effect of Exchange Interaction on the Low-Temperature Ordering of GdCl3, Dysprosium Ethyl Sulfate, and Some Rare-Earth Hydroxides

The low-temperature ordered states of ${\mathrm{Gd}}^{+++}$ ions in gadolinium trichloride and hydroxide lattices, ${\mathrm{Dy}}^{+++}$ ions in dysprosium ethyl sulfate and hydroxide lattices, and ${\mathrm{Tb}}^{+++}$, ${\mathrm{Ho}}^{+++}$, ${\mathrm{Nd}}^{+++}$, and ${\mathrm{Er}}^{+++}$ ions in their respective hydroxide lattices are investigated taking into account both the dipole-dipole and exchange interactions. The method used is essentially that of Luttinger and Tisza for dipole-dipole interaction, later generalized by Niemeyer to include exchange interaction. A linear dependence of the energy eigenvalues upon the exchange-interaction constants is found. For Gd${\mathrm{Cl}}_3$ and Gd${\mathrm{}\left(\mathrm{OH}\right)\mathrm{}}_3$, which are the only cases where the values of the exchange-interaction constants are experimentally known, ferromagnetic and antiferromagnetic orderings, respectively, corresponding to the lowest energy are predicted in agreement with the experimental observations. For Nd${\mathrm{}\left(\mathrm{OH}\right)\mathrm{}}_3$, where a ferromagnetic low-temperature ordering is predicted by the dipole-dipole interaction, an antiferromagnetic ordering may prevail if antiferromagnetic exchange interactions of suitable values are considered. This is in accordance with the observation of Wolf, Meissner, and Catanese. For the remaining cases, a limit for the next-nearest-neighbor exchange constant is obtained which will yield the ferromagnetic ordered states as observed experimentally.