Dipolar and Quadrupolar Ordering inS=32Ising Systems

We have studied in the molecular-field approximation the statistical-mechanical properties of the Hamiltonian $\mathcal{H}=-h\Sigma \stackrel{}{i}{S}_i^z-\Sigma \stackrel{}{i,j}{J}_{\mathrm{ij}}{S}_i^z{S}_j^z-\Sigma \stackrel{}{i,j}{K}_{\mathrm{ij}}[{\left(S_i^z\right)}^2-\frac{1}{3}S(S+1\mathrm{}\left)\right][{\left(S_j^z\right)}^2-\frac{1}{3}S(S+1\mathrm{}\left)\right]$ for $S=\frac{3}{2}$ ions. We investigate the possibility of ordering in the independent parameters $M=〈S^z〉$ and $Q=〈{\left(S^z\right)}^2-\frac{1}{3}S(S+1\mathrm{})\mathrm{}〉$. The phase diagram is discussed for positive and negative biquadratic interactions as a function of the ratio of bilinear to biquadratic interactions and as a function of the magnetic field $h$. Successive phase transitions and triple, critical, and tricritical points are found. We also show that the model gives a qualitative understanding of the phase transitions observed in DyV${\mathrm{O}}_4$.