Magnetic and Thermal Properties of Gd ( OH ) 3

Measurements of the low-temperature specific heat have shown that Gd${(\mathrm{OH})}_{3}$ undergoes a co-operative transition at 0.94 \ifmmode\pm\else\textpm\fi{} 0.02 K. To determine the interactions giving rise to this transition, a series of accurate susceptibility and magnetic-specific-heat measurements were made at temperatures high compared to the transition, and analyzed using asymptotically exact series expansions. The susceptibility measurements were made using an audiofrequency mutual-inductance method at temperatures between 1.4 and 4.2 K and 14.7 and 20.0 K. The magnetic specific heat was determined using two different techniques. One was the conventional calorimetric method, and measurements were made at temperatures between 0.4 and 15 K. An estimate of the lattice specific heat and a comparison with calorimetric measurements of the diamagnetic isomorph La${(\mathrm{OH})}_{3}$ are given. In the other method, the magnetic specific heat was determined from the field dependence of the adiabatic differential susceptibility, using the method of Casimir and du Pr\'e (CdP). For this, a special 4.5-MHz tunnel-diode oscillator was used, together with a sensitive temperature controller and a superconducting solenoid. Measurements were made in fields up to 15 kOe at temperatures between 5 and 68 K. Using an iterative procedure, the leading terms in the susceptibility expansion were found to be ${\ensuremath{\chi}}_{T}^{\ensuremath{\parallel}}=\frac{\ensuremath{\lambda}}{(T\ensuremath{-}{\ensuremath{\theta}}_{\ensuremath{\parallel}}^{\ensuremath{\infty}}+\frac{{B}_{2\ensuremath{\parallel}}}{T}+\frac{{B}_{3\ensuremath{\parallel}}}{{T}^{2}}+\ensuremath{\cdots})}$ with $\ensuremath{\lambda}=7.815\ifmmode\pm\else\textpm\fi{}0.008$ emu K/mole, ${\ensuremath{\theta}}_{\ensuremath{\parallel}}^{\ensuremath{\infty}}=0.02\ifmmode\pm\else\textpm\fi{}0.10$ K, ${B}_{2\ensuremath{\parallel}}=2.05\ifmmode\pm\else\textpm\fi{}0.10$ ${\mathrm{K}}^{2}$, and ${B}_{3\ensuremath{\parallel}}=\ensuremath{-}0.59\ifmmode\pm\else\textpm\fi{}0.10$ ${\mathrm{K}}^{3}$, where $\ensuremath{\parallel}$ denotes measurements parallel to the crystal $c$ axis and the superscript $\ensuremath{\infty}$ denotes correction to an infinitely long sample shape. For the magnetic specific heat, the leading terms were found to be $\frac{{C}_{M}}{R}=\frac{{C}_{2}}{{T}^{2}}+\frac{{C}_{3}}{{T}^{3}}+\ensuremath{\cdots}$, with ${C}_{2}=4.09\ifmmode\pm\else\textpm\fi{}0.05$ ${\mathrm{K}}^{2}$ and ${C}_{3}=\ensuremath{-}4.2\ifmmode\pm\else\textpm\fi{}0.7$ ${\mathrm{K}}^{3}$. The unusually small error in ${C}_{2}$ reflects the fact that the CdP method determines $\frac{{C}_{M}{T}^{2}}{R}$ directly, without the necessity of first correcting for the much larger lattice specific heat. To analyze for the interactions, the dominant contributions were assumed to be of the Heisenberg form ${J}_{n}{\stackrel{\ensuremath{\rightarrow}}{\mathrm{S}}}_{0}\ifmmode\cdot\else\textperiodcentered\fi{}{\stackrel{\ensuremath{\rightarrow}}{\mathrm{S}}}_{n}$ with ${J}_{n}$ restricted to nearest and next-nearest neighbors plus the calculated magnetic dipole-dipole coupling summed over all neighbors. These assumptions were tested as part of the analysis and upper limits were obtained for ${J}_{3}$ and for possible anisotropic nondipolar interactions. The analysis was also checked against additional susceptibility measurements perpendicular to the $c$ axis at temperatures between 1.4 and 4.2 K and magnetization measurements in fields up to 14 kOe at temperatures between 1.1 and 4.2 K. The final results gave ${J}_{1}=0.180\ifmmode\pm\else\textpm\fi{}0.005$ K and ${J}_{2}=\ensuremath{-}0.017\ifmmode\pm\else\textpm\fi{}0.005$ K, indicating that the system should approximate to an assembly of loosely coupled antiferromagnetic chains, but the situation is complicated by the magnetic dipole interactions which are comparable in strength. To investigate the nature of the ordering, accurate susceptibility measurements were made at temperatures between 0.6 and 1.4 K again using an inductance method. The results were interpreted as characteristic of an antiferromagnet with predominantly but not exclusively nearest-neighbor interactions. The cooperative properties of such a system are difficult to calculate, and even though all the important terms in the microscopic Hamiltonian have been determined, it has not proved possible to predict the precise nature of the ordered state. A number of possible states are discussed which probably approximate to the actual ground state, but further theoretical work on this problem is called for.