The Phase Diagram of Simple Metamagnets Using a Cluster Variation Method.

The Ising metamagnet is a system of Ising spins that is coupled by an antiferromagnetic interaction between the nearest neighbours and by a ferromagnetic interaction between the next nearest neighbours, or vice-versa. This model was reviewed by Kincaid and Cohen, who determined the phase diagram using mean field theory (MF). It was also treated by means of the Bethe or pair approximation by Meijer and Stamm. Kincaid and Cohen found in the MF approximation that the phase diagram changes in character when the reduced coupling constant, e lies above or below some critical value, e*. At values above e*, the second order transition line ends in a TCP point. At values below e*, the second order transition line ends in a CE point. On the other hand, the results of Meijer and Stamm show that, in the pair approximation, T* is always a critcal end point. Since the MF and pair approximations are generally regarded as particular cases of the more general cluster variation method of Kikuchi, it was important to see whether the use of clusters larger than one (MF) or two (pair approximation) sites would yield more detailed information on the nature of the end point. The method used is the natural iteration technique and the critical points are located by means of the finding the zeroes of an appropriately chosen subdeterminant of the Hessian of the free energy with respect to the cluster variables. Using the appropriate conditions for the critical points, the following results are obtained:. In the case of 2d square, when e lies in the interval, 0.25 (LESSTHEQ) (VBAR)er- (LESSTHEQ) 0.3 T* is TCP. Outside of this range, T* is a CE point. For the case of simple cubic (lattice) for which the 2d square was chosen as a basic cluster, T* was always CE point. For the first rank superstructure bcc (= 2sc) case, a tetrahedron was chosen as a basic cluster, and the result obtained was always of the TCP point type for the various values of e. For the case of the second rank superstructure bcc (= 2 diamond), it was possible to obtain a Neel temperature, but when the field H exceeded one unit the state became a ferromagnetic state. There are two different fcc structures. One is called the "B-structure" and it corresponds to the space group Fd3m. In zero field for the ferrogmagnetic case, critical temperatures are obtained for different values of e. These results are in good agreement with the results for the exact series expansion of Dalton and Wood. For non-zero field, BCE points and CE points are obtained for the value of e = -0.7 and -0.5. If e less than -0.5, the value of T* is contrary to expectation. The other fcc structure "A-structure" corresponds to the space group R(,3m). The treatment of this problem is different since the 2 sets of clusters are related through the two point clusters, Y(,i), consequently we have to introduce a minor iteration. For the value of e = -0.35 again BCE and CE points merge into each other, and above this value T* was CE point.