Correlating sampling and intensity statistics in nanoparticle diffraction experiments

In a previous article [Öztürk, Yan, Hill & Noyan (2014). J. Appl. Cryst. 47, 1016–1025] it was shown that the sampling statistics of diffracting particle populations within a polycrystalline ensemble depended on the size of the constituent crystallites: broad X-ray peak breadths enabled some nano-sized particles to contribute more than one diffraction spot to Debye–Scherrer rings. Here it is shown that the equations proposed by Alexander, Klug & Kummer [J. Appl. Phys. (1948), 19, 742–753] (AKK) to link diffracting particle and diffracted intensity statistics are not applicable if the constituent crystallites of the powder are below 10 nm. In this size range, (i) the one-to-one correspondence between diffracting particles and Laue spots assumed in the AKK analysis is not satisfied, and (ii) the crystallographic correlation between Laue spots originating from the same grain invalidates the assumption that all diffracting plane normals are randomly oriented and uncorrelated. Such correlation produces unexpected results in the selection of diffracting grains. For example, three or more Laue spots from a given grain for a particular reflection can only be observed at certain wavelengths. In addition, correcting the diffracted intensity values by the traditional Lorentz term, 1/cos θ, to compensate for the variation of particles sampled within a reflection band does not maintain fidelity to the number of poles contributing to the diffracted signal. A new term, cos θ_B/cos θ, corrects this problem.