Uniaxial and Biaxial Quadrupolar Ordering in Magnetic Crystals: Molecular-Field Theory

The single order parameter $Q=〈{\left(S^z\right)}^2=\frac{1}{3}S(S+1\mathrm{})\mathrm{}〉$ is, in general, insufficient to describe ordering in low-symmetry magnetic crystals with quadrupolar coupling. After a brief discussion of the general situation, we study in detail the molecular-field theory of an array of quadrupoles coupled in pairs $〈12〉$ by the Hamiltonian ($J\left(12\right)>0\mathrm{}$), $\mathcal{H}=-{\Sigma }_{〈12〉}J\left(12\right)\mathrm{}{\left[\right[\frac{3}{2}\left\{\right[S^z\mathrm{}\left(1\right)]}^2-\frac{1}{3}S(S+1\mathrm{})\left\}\right\{{\left[S^z\mathrm{}\right(2\left)\right]}^2-\frac{1}{3}S(S+1\mathrm{})\}+\frac{1}{2}\eta \{{\left[S^x\mathrm{}\right(1\left)\right]}^2-{\left[S^y\mathrm{}\right(1\left)\right]}^2\left\}\right\{{\left[S^y\mathrm{}\right(2\left)\right]}^2-{\left[S^y\mathrm{}\right(2\left)\right]}^2\left\}\right]]$. For $\eta >1\mathrm{}$, both $Q$ and the biaxiality parameter $P=〈{\left(S^x\right)}^2-{\left(S^y\right)}^2〉$ become simultaneously nonzero at sufficiently low temperatures. We exhibit ($\eta , T$) phase diagrams and give the temperature dependence of the ordering for spins $S=1, \frac{3}{2}, 2, \frac{5}{2}, \mathrm{and} \infty$. The transitions in $Q$ and $P$ may be separated by introducing single-ion anisotropy into the Hamiltonian.