New benchmarks in the modelling of X-ray atomic form factors

Abstract

Analytical representations of X-ray atomic form factor data have been determined. The original data, f 0(s;Z), are reproduced to a high degree of accuracy. The mean absolute errors calculated for all s = sin θ/λ and Z values in question are primarily determined by the precision of the published data. The inverse Mott–Bethe formula is the underlying basis with the electron scattering factor expressed by an expansion in Gaussian basis functions. The number of Gaussians depends upon the element and the data and is in the range 6–20. The refinement procedure, conducted to obtain the parameters of the models, is carried out for seven different form factor tables published in the span Cromer & Mann [(1968), Acta Cryst. A24, 321–324] to Olukayode et al. [(2023), Acta Cryst. A79, 59–79]. The s ranges are finite, the most common span being [0.0, 6.0] Å−1. Only one function for each element is needed to model the full range. This presentation to a large extent makes use of a detailed graphical account of the results.