Extracting short-ranged interactions from structure factors

TL;DR

This paper investigates the challenges of extracting short-range interparticle interactions from structure factors in scattering experiments, highlighting conditions where inversion is difficult and proposing new methods to gain insights from invariant points.

Contribution

It demonstrates that at low concentrations, different short-range attractive potentials produce similar structure factors, and introduces the concept of isosbestic points for better potential characterization.

Findings

Structure factors at low concentration are insensitive to potential shape.

Inversion from g(r) is well conditioned in energetic fluid regimes.

Isosbestic points can reveal effective potential range.

Abstract

Inverting scattering experiments to obtain effective interparticle interactions is generally a poorly conditioned problem. L. Reatto (Phil. Mag. A 58, 37 (1986)) showed that for atomic liquids close to the triple point, inversions are hard because the structure closely resembles that of an equivalent hard-sphere fluid. Here I demonstrate that at low concentrations and for particles with short-ranged attractive potentials, S(k) also exhibits a very weak dependence on potential shape. Instead, different potentials can generate an S(k) that closely resembles that of the Baxter model with a similar second-virial coefficient. By contrast, in this energetic fluid regime, the inversion of an attractive interaction from real-space correlations such as the radial distribution function g(r) is well conditioned. Nevertheless, one may extract further information from S(k) by measuring isosbestic points, values of k where the scattering intensity I(k) or the structure factor S(k) is invariant to changes in interaction-potential well-depth. These points suggest a new extended corresponding states principle for particles in solution based on the packing fraction, the second osmotic virial coefficient, and a new measure of effective potential range.