Statistical Mechanical Theory of Ferromagnetism. High Density Behavior

The partition function of the Ising model of ferromagnetism is examined in the limit of high density in the anticipation that in the limit of infinite density one recovers the Weiss molecular field. The formal parameter of expansion is $\frac{1}{z}$ where $z$ is the number of spins in the range of the exchange potential (not restricted to nearest neighbor interactions). In the absence of long-range order, only ring diagrams in the cluster expansion contribute. These give a divergence in the specific heat at $k{T}_c=\Sigma {j\ne i}^{}{v}_{\mathrm{ij}}$ where $v_{\mathrm{ij}}$ is the exchange potential. This is the molecular field value for the Curie point $T_c$. In the presence of a magnetic field the partition function is evaluated for fixed magnetic moment $M$ in the same approximation, $M$ being determined by minimization. This results in a susceptibility differing from the molecular field theory and hence an inconsistency in the theory.