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

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 factorS(Q) andg(r), we first show analytically that even infinitely precise, noise-freeS(Q) data-spanning an unboundedQ-range-cannot uniquely specify bothρandg(r). This non-uniqueness arises from phase information loss, finite-dimensional projections inherent in one-dimensional pair distributions, and the mathematical insensitivity ofS(Q) to coordinated rescaling of density and radial distances. In addition, we highlight practical problems arising from mathematical methods aimed at extractingρvia Fourier transform of data. Direct calculation from integratingg(r)-1(Yarnell method) converges badly for high density because of long-range structure ing(r), and at low density because of a bias coming from the central atom ing(r). Indirect calculation from the slope off⋅[g(r)-1](Eggert method) depends sensitively on having good quality high-Qdata. To address these ambiguities, we introduce a density-sweep protocol using the empirical potential structure refinement (EPSR) within theab initioaugmented structure solving engine framework. By systematically varying trial densities around target values (±5%-50%) and evaluating both the internal EPSRR-factor and an externalR-factor based on finalF(Q), one can identify a clear minimum bracketing the trueρwithout 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 D₂O at 298 K and amorphous silica and reliably recover known densities within±5%.