Influence of neglected small-angle scattering in radial distribution function analysis

Materials containing inhomogeneities (density-fluctuations) of much greater than atomic size produce scattering at very small angles, which may go unobserved in many X-ray, electron, and neutron scattering experiments. For liquids and for amorphous and polycrystalline solids composed of one atomic species, an approximate expression for the reduced radial distribution function obtained from intensity measurements which neglect the small-angle scattering is shown to be Gexp(r) = 4πr{ρ(r) − ρ₀[1 + (η²(ω)/ρ₀²)γ(ω, r)]} where ρ(r) is the atomic distribution function, ρ₀ is the average atomic density, (ω) is the average square of atomic density fluctuations, γ(ω,r) is the density fluctuation correlation function, and ω is a volume element larger than the average atomic volume but smaller than the scale of long-range density fluctuations. This expression is also valid for systems composed of more than one type of atom where ρ(r) is a weighted average of pair distribution functions and [(ω)/ρ₀²]γ(ω,r) for X-ray scattering describes electron density fluctuations The neglect of small-angle scattering gives rise to a G_exp(r) which appears, from its slope at small r, to correspond to a material of greater average atomic density ρ_0,exp than that of the sample being studied. These results are illustrated by application to fluid argon (ρ_0,exp/ρ₀ = 1.17 near the critical point), to amorphous silicon (ρ_0,exp/ρ₀ = 1.13), and to phase separated PbO–B₂O₃ glasses (ρ_0,exp/ρ₀ = 1.07 for 24 wt. % PbO).