Computational Analysis of Diffraction in Ideal Nanocrystalline Powders

Quantifying the statistical uncertainty in diffracted intensities was first investigated by Alexander, Klug and Kummer in 1948, who developed a formulation that estimated the relative uncertainty in the diffracted intensities from the relative uncertainty in the populations of diffracting particles within an irradiated powder. In this thesis, we show that this formulation becomes inapplicable for powder ensembles with particle sizes below 1 micron. In this size regime, the probability of diffraction cannot be formulated based on simple area ratios, and the classical multiplicity should be replaced by effective multiplicities. To properly relate the diffracted intensities collected by the detector to the grains participating in diffraction, we develop a modeling methodology which isolates the sampling and intensity spaces and links each diffracting particle to its own diffracted spot. The independent investigation of diffracted intensities and diffracting particle populations reveals that the uncertainties in the diffracted intensities are almost always greater than those in the diffracting particle populations. The only special case, where the two uncertainties are equal, occurs for 'large' particle sizes, where the full angular width of the particle's characteristic rocking curve is smaller than the angular resolution of the detector pixel. Our modeling results also show that the population of particles required to reach the ultimate average diffracted intensities predicted by the Debye equation depends on the size and crystallinity of the irradiated particles. Finally, the direct link between diffracting grains and diffracted intensities is not preserved in the formulation of the Debye scattering equation, and therefore the analysis and refinement of experimental diffraction data against the Debye model are shown to result in ambiguous structural parameters