On Small-Angle Critical Scattering

Abstract

Recent theoretical studies of simplified models of cooperative assemblies of atoms or molecules suggest that the pair correlation function G(r) at the critical is not given by the Ornstein—Zernike (O.Z.) expression but that this relation becomes asymptotically valid for large distances r away from the critical. Based on some further heuristic considerations we suggest that the low-angle scattering of radiation or particles from real three-dimensional cooperative assemblies at the critical composition and temperatures, above but near the critical is essentially given by the Fourier transform of a G(r) which is an interpolation between the O.Z. expression and the critical G(r) suggested by the simplified models. We derive thus the form of the relative, reduced small angle scattering intensity and indicate in what ways departures from the O.Z. predictions are expected and comment on their magnitude. These calculations may possibly explain certain small but systematic anomalies observed in small angle x-ray scattering from critical binary fluid mixtures. In a particular instance we can thus show that the apparent linearity of an O.Z. plot (reciprocal intensity versus essentially the square of the scattering angle) is not sufficient evidence for the validity of the O.Z. theory at the critical point. We point out the possibility of two types of clustering (``ferromagnetic'' and ``anti-ferromagnetic'') in binary mixtures and indicate the usefulness of the Zimm cluster integral as a measure of the first type in clarifying the interpretation of scattering data. We speculate briefly on the possibility of ``critical cluster'' formation.