Rietveld Refinement of Electron Diffraction Patterns of Nanocrystalline Materials Using MAUD: Two-Beam Dynamical Correction Implementation and Applications

Abstract

Nanocrystalline (NC) materials have widespread industrial usage. X-ray and neutron diffraction techniques are primary tools for studying the structural and microstructural features of NC materials. Selected area electron diffraction (SAED) patterns collected using a transmission electron microscope (TEM) on polycrystalline nanostructured materials, featuring nested rings, that are analogous to Debye-Scherrer patterns, possess similar potentials to aid materials characterisation. The utility of SAED patterns is further enhanced by the possibility of applying crystallographic approaches, like full pattern fitting procedures, based on Rietveld refinement algorithms, enabling the evaluation of material features, such as crystallite size, lattice distortions, defect structures, and the presence of secondary phases even from very small volume scale. In this paper, we have discussed the possibilities afforded by a Rietveld code applied to SAED patterns of NC materials, including the mathematical implementation of the two-beam dynamical correction model in MAUD software (version 2.9995), and a critical discussion of the results obtained on different NC materials.

Keywords: Rietveld refinement; electron diffraction; nanocrystalline materials.

Figure 1

Low magnification BF image of NC CeO₂ showing small agglomerates of particles (a). A relatively higher magnification BF image (b) with the corresponding SAED pattern from the field of view in (c).

Figure 2

Low magnification BF image of NC Si showing large agglomerates of particles (a). A relatively higher magnification BF image (b) with the corresponding SAED pattern from the field of view in (c).

Figure 3

Results of the Rietveld refinements of the NC-CeO₂ sample under (a) the two-beam dynamical correction (${\mathrm{R}\mathrm{w}\mathrm{p}}_{\mathrm{n}\mathrm{o}\_\mathrm{b}\mathrm{k}\mathrm{g}}=$ 12.79%) and (b) the kinematical approximation (${\mathrm{R}\mathrm{w}\mathrm{p}}_{\mathrm{n}\mathrm{o}\_\mathrm{b}\mathrm{k}\mathrm{g}}=$ 12.82%). The dotted line is the experimental intensity profile, and the red continuous line is the calculated profile.

Figure 4

Results of Rietveld refinements of the NC-Si sample under (a) the two-beam dynamical correction (${\mathrm{R}\mathrm{w}\mathrm{p}}_{\mathrm{n}\mathrm{o}\_\mathrm{b}\mathrm{k}\mathrm{g}}=$ 6.61%) and (b) the kinematical approximation (${\mathrm{R}\mathrm{w}\mathrm{p}}_{\mathrm{n}\mathrm{o}\_\mathrm{b}\mathrm{k}\mathrm{g}}=$ 7.01%). The dotted line is the experimental intensity profile, and the red line continuous is the calculated profile.

Figure 5

BF image showing the presence of thick NC CoFe₂O₄ agglomerates (a) with the SAED pattern from the field of view in (b). A high magnification image with overlapping grains (c).

Figure 6

Results of Rietveld refinements of the NC CoFe₂O₄ sample under (a) the two-beam dynamical correction (${\mathrm{R}\mathrm{w}\mathrm{p}}_{\mathrm{n}\mathrm{o}\_\mathrm{b}\mathrm{k}\mathrm{g}}=$ 13.42%) and (b) the kinematical approximation (${\mathrm{R}\mathrm{w}\mathrm{p}}_{\mathrm{n}\mathrm{o}\_\mathrm{b}\mathrm{k}\mathrm{g}}=$ 14.75%). The dotted line is the experimental intensity profile, and the red line is the calculated profile. Blue and green lines correspond to Co₃O₄ and CoFe₂O₄ phases, respectively.

Figure 7

BF image showing the presence of thin Au-Ag agglomerates (a) with the SAED pattern from the field of view in (b). A high-magnification image showing the orientation of selected overlapping grains (c).

Figure 8

Results of Rietveld refinements of the Au-Ag sample under (a) the two-beam dynamical correction (${\mathrm{R}\mathrm{w}\mathrm{p}}_{\mathrm{n}\mathrm{o}\_\mathrm{b}\mathrm{k}\mathrm{g}}=$ 16.06%), (b) the kinematical approximation (${\mathrm{R}\mathrm{w}\mathrm{p}}_{\mathrm{n}\mathrm{o}\_\mathrm{b}\mathrm{k}\mathrm{g}}=$ 16.08%), and (c) the two-beam dynamical correction but without accounting for planar defects (${\mathrm{R}\mathrm{w}\mathrm{p}}_{\mathrm{n}\mathrm{o}\_\mathrm{b}\mathrm{k}\mathrm{g}}=$ 17.33%). The dotted line is the experimental intensity profile, and the red continuous line is the calculated profile.

Figure 9

BF image showing the presence of thin NC NaYF₄ agglomerates (a) with the SAED pattern from the field of view in (b). A high magnification image showing the orientation of some individual grains with the presence of a film shown using red arrows (c).

Figure 10

Results of Rietveld refinements of the NC NaYF₄ sample under (a) the two-beam dynamical correction (${\mathrm{R}\mathrm{w}\mathrm{p}}_{\mathrm{n}\mathrm{o}\_\mathrm{b}\mathrm{k}\mathrm{g}}=$ 23.45%) and (b) the kinematical approximation (${\mathrm{R}\mathrm{w}\mathrm{p}}_{\mathrm{n}\mathrm{o}\_\mathrm{b}\mathrm{k}\mathrm{g}}=$ 23.48%). The dotted line is the experimental intensity profile, and the red line is the calculated profile. The red arrows in the figure show the effect of the amorphous thin film (see Figure 9c) on the Rietveld fitting.