Extension of the Ornstein-Zernike Theory of the Critical Region. II

A study is made of the terms in an expansion of the direct correlation function at the critical point. If homogeneity of long-range correlations is assumed, we find that the terms involving $m$-point correlation functions, $m>\mathrm{}2\mathrm{}$, do not dominate the terms that depend only on pair-correlation effects. For a system with a short-range pair potential, we have previously shown that this result yields, in the usual notation, $2-\eta =min[2, \frac{d(\delta -1)}{(\delta +1)}]$, where $d$ is dimensionality. It is argued that for a pair potential $V\left(\mathrm{r}\right)\sim -r^{-d-\sigma }$, for $r\to \infty$, we should expect no change in this relation for $\sigma >min[2, \tilde{s}]$, where $\tilde{s}$ is an exponent appearing in our analysis; $\tilde{s}=\frac{7}{4}$ for $d=2\mathrm{}$, and $\tilde{s}\approx 2$ for $d=3\mathrm{}$. For smaller $\sigma$, the problem is more complex, and our analysis is only suggestive; it indicates that when $\sigma <\tilde{s}$, one should be prepared to find a marked difference in the behavior of critical exponents between the $\sigma <\frac{1}{2}d$ and $\sigma >\frac{1}{2}d$ cases. For the latter, we again find $2-\eta =min[2, \frac{d(\delta -1)}{(\delta +1)}]$. We find $2-\eta =\sigma$ in both cases.