Copyright 2004 by Marcel Dekker, Inc. All Rights Reserved.

IðqÞ ¼ Ið0Þexpð q 2 R 2 g =3Þ; where Ið0Þ is the scattering intensity at zero angle[30]. For spherical p ffiffiffiffiffiffiffi paticles, the particle radius R g relates to the particleradius as R ¼ R g 5=3 . Rg is the volume-weighted average in a polydispersedispersion. As a general guide, the Guinier approximation applies whenRq V 1, although this may vary depending on particle geometry and polydispersity.For values of Rq > 1, I( q) oscillates from spherical scatterers due toconstructive and destructive interference of x-rays scattered from individualparticles. A representation of the scattering data as a Porod plot [I( q)q 4 versusq] visually enhances the oscillations for more accurate curve fitting of theexperimental and modeled intensities. Fitted curves are usually generatedfrom analytic expressions based on the shape and polydispersity of a collectionof independent scatterers, although atomistic models have also beenreported in the literature [31]. For example, P( q) for a monodisperse sphere ofconstant core density is [13,30]" # 2sinðqRÞ qRcosðqRÞPðqÞ ¼ 3ðqRÞ 3The sample polydispersity strongly affects P( q) and must be accounted for tofit scattering curves of real samples. The scattering intensity can be calculatedwhen the shape of the size distribution is known usingIðqÞ ¼Z l0NðRÞPðqÞR 6 dRð1Þwhere N(R) is the normalized number distribution as a function of particleradius, typically taken as a Gaussian or log-normal distribution. The shape ofthe size distribution must be assumed to determine the nanocrystal size andsize distribution from the scattering data; however, this can be checked bycomparison with histograms generated from TEM images of nanocrystals. Afit of Eq. (1) to the experimental data provides the average radius andstandard deviation of the particles. The polydispersity damps the scatteringoscillations in the Porod region and standard deviations in particle sizeabove f20% nearly eliminate the oscillations, decreasing the usefulness ofSAXS for sizing nanocrystal samples with broad size distributions.For nanocrystal dispersions in which interparticle interactions are importantor for concentrated solutions where the molecular motion isinhibited, S( q) p 1 and I( q) deviates from P( q) at low q values [10,30].Positive deviations from the shape factor indicate interparticle attraction andmay be characterized by the presence of dimers or the flocculated particles (ora negative second virial coefficient) [10]. Conversely, a negative departure ofthe scattering curve from the shape factor indicates strong interparticlerepulsion.Copyright 2004 by Marcel Dekker, Inc. All Rights Reserved.