Determination of particle concentration by small angle X-ray scattering (SAXS)

Small-angle X-ray scattering (SAXS) is a well-established method to obtain structural information on inhomogeneities in materials at the nanoscale, typically between 1 nm and 100 nm, and is thus perfectly suited for nanoparticulate systems. For sufficiently monodisperse spherical particles, the observed oscillations of the scattered intensity as a function of the momentum transfer, which is directly related to the scattering angle and the wavelength of the incident X-rays, enable the size determination of nanoparticles. In order to determine their concentration in a suspending medium, the absolute differential scattering cross section has to be determined, thus the ratio of the scattered intensity to the incident intensity. Assumptions on the particle shape are required, which can be based on microscopic techniques like electron microscopy. Furthermore, the electron density difference between the particles and the suspending medium needs to be known.In this standard, only the concentration of sufficiently monodisperse spherical nanoparticles is treated. Here, the differential scattering cross section can be calculated based on the form factor, which depends only on the momentum transfer q and the particle radius r. Furthermore, this International Standard is limited to dilute systems. A dilute system in the sense of SAXS means that particle interactions are absent. In case of long-range interactions (Coulomb forces between the particles), special care has to be taken and a reduction of the concentration might be necessary.