Effect of random anisotropy on magnetic properties of amorphous systems

We consider the random anisotropy model for amorphous magnetism by making a local-mean-field approximation (LMFA) on arrays of spin-one particles. Hysteresis loops and the temperature ($T$) dependence of several thermodynamic quantities are presented for various values of the ratio of the strength of the exchange ($J$) to the strength of the uniaxial anisotropy ($D$). Using the LMFA limits us to systems with a small number ($N$) of spins, of which we explicitly consider $N=64, 216, \mathrm{and} 1000$. We assume periodic boundary conditions on a system with $N^{\frac{1}{3}}$ spins along an edge, nearest-neighbor coupling of constant strength, and six nearest neighbors (as for a simple cubic lattice). For $J>\mathrm{}0\mathrm{}$ the free energy of spin-glass-like states is higher than that of corresponding states with remanent magnetization. The dependence of the coercive field ($B_c$) on $J$ and $D$ is discussed and the apparent discrepancy of Chi and Alben vis à vis Callen, Liu, and Cullen concerning the behavior of $B_c$ for large $D$ is clarified. A calculation of the temperature dependence of $B_c$ is presented which is reminiscent of experimental results. This random anisotropy is found to give rise to a second peak in the specific heat for suitable values of $\frac{D}{J}$. The magnetic susceptibility (${\chi }_T$) is calculated for both positive and negative $J$ and shows positive and negative paramagnetic Curie-Weiss temperatures, respectively. The slopes of the ${\chi }_T^{-1\mathrm{}}\mathrm{}\left(T\right)\mathrm{}$ curves for $T$ well above the critical temperature ($T_c$) have values that are roughly equal to $\frac{3}{2}$, the value appropriate to $D=0\mathrm{}$ and $S=1\mathrm{}$. The local order parameter $q$ is used to identify $T_c$, which correlates well with the critical temperature identified from other thermodynamic quantities. The presence of the random anisotropy is found to reduce $T_c$ by up to about 25%. The results of several temperature-dependent calculations are summarized in a phase diagram and regions of paramagnetic, random ferromagnetic, and random antiferromagnetic (or spin-glass-like) behavior are identified.