  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.
