Biquadratic Exchange and Quadrupolar Ordering

We have studied in the molecular field approximation the statistical mechanical properties of the Hamiltonian $$\mathcal{H}={\displaystyle \sum _{\mathrm{mn}}}{\mathrm{J(m-n)S}}_m{\mathrm{·S}}_n-{\displaystyle \sum _{\mathrm{mn}}}{\mathrm{j(m-n)(S}}_m{\mathrm{·S}}_n{)}^2$$ , for spin one ions. Other authors have already shown that the biquadratic term can cause the magnetic transition to become first order.¹ Closer examination shows that one must consider, in addition to the usual long‐range order parameter ${\mathrm{〈S}}_z\mathrm{〉=M}$ , the possibility of ``quadrupolar'' ordering in the independent parameter ${\mathrm{Q=〈S}}_z^2\mathrm{〉-}\frac{2}{3}$ . If the biquadratic exchange is sufficiently large the quadrupolar ordering will appear at a temperature T_Q which is different from that at which dipolar ordering M appears, so that two separate phase transitions are found. The phase diagram is discussed as a function of the ratio of bilinear to biquadratic exchange, and the interplay between quadrupolar and dipolar ordering is considered.