DensEst: an automated empirical potential-based means of determining the densities of disordered materials from total scattering data

Abstract We investigate the fundamental limits of using total-scattering measurements to simultaneously determine the atomic number density ( ρ ) and pair distribution function ( g ( r )) of disordered materials. Building on rigorous Fourier-transform relationships between the structure factor S ( Q ) and g ( r ), we first show analytically that even infinitely precise, noise-free S ( Q ) data-spanning an unbounded Q- range-cannot uniquely specify both ρ and g ( r ). This non-uniqueness arises from phase information loss, finite-dimensional projections inherent in one-dimensional pair distributions, and the mathematical insensitivity of S ( Q ) to coordinated rescaling of density and radial distances. In addition, we highlight practical problems arising from mathematical methods aimed at extracting ρ via Fourier transform of data. Direct calculation from integrating g ( r ) − 1 (Yarnell method) converges badly for high density because of long-range structure in g ( r ), and at low density because of a bias coming from the central atom in g ( r ). Indirect calculation from the slope of f ⋅ [ g ( r ) − 1 ] (Eggert method) depends sensitively on having good quality high- Q data. To address these ambiguities, we introduce a density-sweep protocol using the empirical potential structure refinement (EPSR) within the ab initio augmented structure solving engine framework. By systematically varying trial densities around target values ( ± 5 % – 50 % ) and evaluating both the internal EPSR R -factor and an external R -factor based on final F ( Q ), one can identify a clear minimum bracketing the true ρ without reliance on external equations of state or arbitrary fitting ranges. We showcase the effectiveness of the method by application to supercritical krypton at multiple pressures, liquid D 2 O at 298 K and amorphous silica and reliably recover known densities within ± 5 % .