Size and Concentration Determination of Colloidal Nanocrystals by Small-Angle X-Ray Scattering

The accurate determination of the dimensions of a nano-object is paramount to the de- velopment of nanoscience and technology. Here, we provide procedures for sizing quasi- spherical colloidal nanocrystals (NCs) by means of small-angle x-ray scattering (SAXS). Using PbS NCs as a model system, the protocols outline the extraction of the net NC SAXS pattern by background correction and address the calibration of scattered x-ray intensity to an absolute scale. Different data analysis methods are compared, and we show that they yield nearly identical estimates of the NC diameter in the case of a NC ensemble with a monodisperse and monomodal size distribution. Extending the analysis to PbSe, CdSe and CdS NCs, we provide SAXS calibrated sizing curves, which relate the NC diameter and the NC band-gap energy as determined using absorbance spectroscopy. In compari- son with sizing curves calibrated by means of transmission electron microscopy (TEM), we systematically find that SAXS calibration assigns a larger diameter than TEM calibration to NCs with a given band gap. We attribute this difference to the difficulty of accurately sizing small objects in TEM images. To close, we demonstrate that NC concentrations can be directly extracted from SAXS patterns normalized to an absolute scale, and we show that SAXS-based concentrations agree with concentration estimates based on absorption spectroscopy.