Biquadratic exchange and first-order ferromagnetic phase transitions

We have calculated some properties of a ferromagnet using a Heisenberg Hamiltonian with a biquadratic exchange term added. The constant-coupling approximation was employed. If the strength of the biquadratic exchange relative to the bilinear is given by a dimensionless parameter $\alpha$, there is a critical value ${\alpha }_c$ at which the phase transition changes character from second order for $\alpha <{\alpha }_c$ to first order for $\alpha >{\alpha }_c$. The spontaneous magnetization, the exchange energy, and the spin-correlation function show discontinuous jumps at $\alpha ={\alpha }_c$ and unstable behavior for $\alpha >{\alpha }_c$.