Critical Scattering and Integral Equations for Fluids

Critical scattering predictions of the Percus-Yevick, hypernetted-chain, and Yvon-Born-Green integral equations for fluids are described for general dimensions $d$. The first two equations are unsatisfactory. The Yvon-Born-Green equation predicts critical scattering of Ornstein-Zernike form for $d>\mathrm{}4\mathrm{}$, but for $d<~4$ the compressibility, $K_T$, remains bounded unless the net correlation function, $g\left(r\right)-1\mathrm{}$, becomes negative near criticality for intermediate and large $r$: In that case scaling behavior occurs with $\eta =4-d$ and $K_T\to +\infty$ for $d>\mathrm{}{d}_0\simeq 2.2$.