Proper usage of Scherrer's and Guinier's formulas in X-ray analysis of size distribution in systems of monocrystalline CeO2 nanoparticles

Small-angle X-ray scattering (SAXS) and X-ray diffraction (XRD) techniques are widely used as analytical tools in the optimization and control of nanomaterial synthesis processes. In crystalline nanoparticle systems with size distribution, the discrepant size values determined by using SAXS and XRD still lacks a well-established description in quantitative terms. To address fundamental questions, the isolated effect of size distribution is investigated by SAXS and XRD simulation in polydisperse systems of virtual nanoparticles. It quantitatively answered a few questions, among which the most accessible and reliable size values and what they stand for regarding the size distribution parameters. When a finite size distribution is introduced, the two techniques produce differing results even in perfectly crystalline nanoparticles. Once understood, the deviation in resulting size values can, in principle, resolve two parameters size distributions of crystalline nanoparticles. To demonstrate data analysis procedures in light of this understanding, XRD and SAXS experiments were carried out on a series of powder samples of cubic ceria nanoparticles. Besides changes in the size distribution related to the synthesis parameters, proper comparison of XRD and SAXS results revealed particle-particle interaction effects underneath the SAXS intensity curves. It paves the way for accurate and reliable methodologies to assess size, size dispersion, and degree of crystallinity in synthesized nanoparticles.