Droplet Model for Tricritical Points: Metamagnetic Transition

Abstract

The phase transition of a two-sublattice spin-½ Ising antiferromagnet in an external magnetic field is studied on the basis of a modified form of Fisher's droplet model for critical phenomena. It is assumed that the interaction also contains an intrasublattice ferromagnetic coupling so that the model is appropriate for metamagnetic substances. By taking into account the contribution to the free energy, both of antiferromagnetic (AF) and of ferromagnetic (F) clusters, it is found that the phase boundary of the AF phase consists of a line of critical points at small fields when the AF clusters first become critical, but this line merges into a line of a first-order phase transition for magnetic fields above a threshold value $H_t$ when the F clusters first become energetically favorable. The end point at ($H_t$, $T_t$) of the first-order phase transition is a tricritical point, in agreement with the results of the Landau and of the molecular-field theories. Two kinds of critical behavior at the tricritical point are found. In what is called a tricritical point of the first kind, only the AF clusters become critical at ($H_t$, $T_t$). In this case, the difference $M_t-M_{-}$ of the magnetizations of the two coexisting phases on the first-order transition goes to zero linearly at the tricritical point. The free energy does not have a homogeneous form near this point. At a tricritical point of the second kind, both AF and F clusters become critical. In this case it is found that $(M_{+}-M_{-})\sim {\left|\frac{(T-T_t)}{T_t}\right|}^{\zeta }$, where, in general, $\zeta$ is different and less than unity. Under certain conditions it is found that the free energy has a homogeneous form near ($H_t$, $T_t$). The critical behavior near ($H_t$, $T_t$) is discussed in detail for both kinds of tricritical points. Better agreement with experiments is found than in the case of the molecular-field theory. The tricritical point of dysprosium aluminum garnet seems to be of the second kind, but more detailed measurements are needed. The two-fluid critical mixing point in ${\mathrm{He}}^3$-${\mathrm{He}}^4$ is briefly discussed.