Small Angle X-Ray Scattering Technique for the Particle Size Distribution of Nonporous Nanoparticles

Table 1

Summary of the small angle X-ray data analysis methods.

Methods	Underlying assumptions	Advantages	Disadvantages
Average size determination method (ADM); [27–45]	All nanoparticles in a system are assumed to be equally sized; determine the average size of nanoparticles	Straightforward and easy to implement	Typically nanoparticles have various sizes and shapes
Parametric distribution model (PDM); [46–55]	A parametric form of particle size distribution is assumed; a particle shape is typically fixed; a particle is sparsely distributed in a sample	Straight forward and easy to implement	For many cases, we do not have whether a particle size distribution follows a simple parametric form
Integral transform method (ITM); [56–70]	The shape of particle is assumed; a particle is sparsely distributed in a sample	Various forms of the particle size distribution are considered	It often involves complex numerical integrations which may become divergent or unstable
Numerical method (NM); [71–94]	The shape of particle is assumed; a particle is sparsely distributed in a sample	A finite approximation of the ITM method; more computationally feasible	The approximation is often solved by an optimization procedure, which is nonlinear, so it is hard to obtain the global optimal approximation