Pair distribution function and structure factor of spherical particles

TL;DR

This paper develops an analytical method to account for finite size effects in pair distribution functions and structure factors of spherical particles, enabling better interpretation of neutron diffraction data for nanoparticle analysis.

Contribution

It introduces an envelope function that transforms the pair distribution function of an infinite solid into that of a finite spherical particle, and applies this to predict particle size distributions from diffraction data.

Findings

Envelope function accurately models finite size effects.

Method successfully predicts gold nanoparticle size distribution.

Exact calculation of peak broadening for spherical particles.

Abstract

The availability of neutron spallation-source instruments that provide total scattering powder diffraction has led to an increased application of real-space structure analysis using the pair distribution function. Currently, the analytical treatment of finite size effects within pair distribution refinement procedures is limited. To that end, an envelope function is derived which transforms the pair distribution function of an infinite solid into that of a spherical particle with the same crystal structure. Distributions of particle sizes are then considered, and the associated envelope function is used to predict the particle size distribution of an experimental sample of gold nanoparticles from its pair distribution function alone. Finally, complementing the wealth of existing diffraction analysis, the peak broadening for the structure factor of spherical particles, expressed as a convolution derived from the envelope functions, is calculated exactly for all particle size distributions considered, and peak maxima, offsets, and asymmetries are discussed.