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A self-consistent analysis is reported of traditional diffraction-based particle size determination techniques applied to synthetic diffraction profiles generated with the Patterson approach. The results show that dimensions obtained from traditional techniques utilizing peak fitting or Fourier analysis for single-crystal nanoparticles have best-case error bounds of around 5%. For arbitrarily shaped particles, lower error magnitudes are possible only if the zeroes of the thickness fringes are used. The errors for sizes obtained by integral-breadth- and Fourier-decomposition-based techniques depend on the shape of the diffracting domains. In the case of integral-breadth analysis, crystal shapes which scatter more intensity into the central peak of the rocking curve have lower size errors. For Fourier-decomposition analysis, crystals which have non-uniform distributions of chord lengths exhibit nonlinearities in the initial ranges of the normalized Fourier cosine coefficient versus column length (|A L | versus L) plots, even when the entire rocking curve is used in the decomposition. It is recommended that, in routine analysis, all domain size determination techniques should be applied to all reflections in a diffraction pattern. If there is significant divergence among these results, the `average particle size(s)' obtained might not be reliable.

The results of a systematic rigorous study on the accuracy of lattice parameters computed from X-ray diffraction patterns of ideally perfect nanocrystalline powder and thin-film samples are presented. It is shown that, if the dimensions of such samples are below 20 nm, the lattice parameters obtained from diffraction analysis will deviate from their true values. The relative deviation depends on the relevant size parameter through an inverse power law and, for particular reflections, depends on the angular peak positions. This size-dependent error, Δa/a, is larger than the precision of typical X-ray diffraction measurements for ∼20 nm-thick diffracting domains, and it can be several orders of magnitude larger for particles smaller than 5 nm.

A rigorous study of sampling and intensity statistics applicable for a powder diffraction experiment as a function of crystallite size is presented. This analysis yields approximate equations for the expected value, variance and standard deviations for both the number of diffracting grains and the corresponding diffracted intensity for a given Bragg peak. The classical formalism published in 1948 by Alexander, Klug & Kummer [J. Appl. Phys.(1948),19, 742–753] appears as a special case, limited to large crystallite sizes, in the present analysis. It is observed that both the Lorentz probability expression and the statistics equations used in the classical formalism are inapplicable for nanocrystalline powder samples.

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 θB/cos θ, corrects this problem.

In this study, the sampling statistics of X-ray diffraction data obtained from polycrystalline nanopowders are studied through analytical formulations and numerical modelling. It is shown that the very large acceptance angles of crystalline nanoparticles can cause issues in computing the number of diffracting grains scattering into a given Bragg reflection. These results intimate that formulations previously tested and verified for polycrystalline aggregates with grains larger than 500 nm should be revalidated for particles with coherent scattering lengths below 10 nm.