International
Tables for
Crystallography
Volume H
Powder diffraction
Edited by C. J. Gilmore, J. A. Kaduk and H. Schenk

International Tables for Crystallography (2018). Vol. H, ch. 2.4, pp. 111-113

Section 2.4.6. Rietveld refinement with electron diffraction data

T. E. Gorelika and U. Kolba

2.4.6. Rietveld refinement with electron diffraction data

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The Rietveld refinement method was initially developed for neutron diffraction data (Rietveld, 1967[link], 1969[link]). It has now become a standard technique which is extensively used with neutron, laboratory X-ray and synchrotron diffraction data. A detailed description of the method can be found in Chapter 4.7[link] .

Compared with the popularity of Rietveld refinement in X-ray and neutron powder diffraction, its application to powder electron diffraction data is very limited. So far, Rietveld refinement with electron diffraction data has only been done for nanocrystalline Al, α-MnS (Gemmi, Fischer et al., 2011[link]), hydroxyapatite (Song et al., 2012[link]), intermetallic AuFe (Luo et al., 2011[link]), TiO2 (Weirich et al., 2000[link]; Tonejc et al., 2002[link]; Djerdj & Tonejc, 2005[link], 2006[link]) and MnFe2O4 (Kim et al., 2009[link]). An example of a fit with powder electron diffraction data obtained by Rietveld refinement for hydroxyapatite is shown in Fig. 2.4.7[link].

[Figure 2.4.7]

Figure 2.4.7 | top | pdf |

Rietveld analysis result with powder electron diffraction data of hydroxyapatite. Reproduced from Song et al. (2012[link]) with permission from Oxford University Press.

Two major factors limit the application of Rietveld refinement to electron powder diffraction. First, electron powder diffraction data are collected from a sample volume far smaller than that used in an X-ray experiment. Therefore, the average statistics are poor compared with those of X-ray data. Nevertheless, electron powder diffraction data from a small sample area or thin films can give specific information which is difficult to obtain using other methods. Second, the presence of dynamical effects in the electron diffraction data hinders quantitative assessment of reflection intensities. Dynamical effects are strongest in zone-axis electron diffraction geometry, when many beams belonging to the same systematic rows are excited simultaneously. In powder electron diffraction crystals are randomly oriented towards the electron beam, thus making the fraction of zonal patterns low, thereby reducing the dynamical scattering in the data (see Section 2.4.2[link] for a more detailed discussion).

Within the limit of kinematical diffraction, the principle of Rietveld refinement is the same for electrons and X-rays, except the electron atomic scattering factors are different. The refinement procedure can thus be performed using existing programs if it is possible to input the scattering factors for electrons. Most of the reported Rietveld refinements on electron powder diffraction data have been performed using FullProf (Rodríguez-Carvajal, 1993[link]); a refinement in MAUD (Lutterotti et al., 1999[link]) has also been reported (Gemmi, Voltolini et al., 2011[link]).

Electron powder diffraction patterns are recorded on an area detector. For a Rietveld refinement the two-dimensional diffraction patterns have to be integrated into one-dimensional profiles. The zero shift is treated as for the X-ray data integrated from a two-dimensional position-sensitive detector. Details about electron diffraction data processing and calibration are given in Section 2.4.3.4[link].

The background in electron powder patterns is a complex combination of inelastic scattering, scattering from the supporting film (when it is present) and other factors. For the Rietveld refinement procedure the background of a one-dimensional integrated profile is fitted by a polynomial function. If a supporting thin amorphous carbon film is used, the background can include broad rings, which after the one-dimensional integration can produce pronounced broad peaks. These peaks are difficult to subtract using a model based on a polynomial function; therefore, these intensities may hamper the powder diffraction profile matching (Kim et al., 2009[link]). In some cases, the background can even include radially non-symmetric features originating from the shape of the tip within the electron source (see Fig. 2.4.8[link]); it can have blooming due to oversaturated CCD pixels, or streak shadows due to the fast transmission electron microscope beam-shutter movement. In these cases, a diffraction pattern from the adjacent `empty' area of the sample can be acquired and subtracted from the diffraction pattern of the material prior to the integration into one dimension. This procedure allows elimination of some of the artifacts discussed above, which otherwise after the one-dimensional integration may be falsely interpreted as diffraction peaks, and are generally more difficult to fit.

[Figure 2.4.8]

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Powder electron diffraction pattern of nanocrystalline gold demonstrating non-symmetrical background features.

Unit-cell parameters are mostly subject to the error due to the accuracy of the electron diffraction camera-length calibration. Although examples have been published showing 0.3% accuracy of the camera-length calibration, in most cases accuracy of about 2% can be achieved (Williams & Carter, 2009[link]). The effective camera length depends on many instrumental parameters such as the convergence of the electron beam, the diffraction lens focus, the mechanical position of the sample within the objective lens, or the hysteresis of the electromagnetic lenses. Thus, while the ratio of the lattice parameters within one aligned diffraction pattern can be very precise, the absolute values might not be.

Atomic displacement parameters can be refined from electron powder diffraction data; however, the interpretation of the results can be manifold. For nanocrystalline materials, which have a relatively high surface-to-volume ratio, the surface effect can be enhanced compared with that of the bulk. Thus, the average atomic displacement factors can increase because of the high fraction of near-surface relaxed atoms. Consequently, the isotropic displacement parameter B resulting from the Rietveld refinement can be relatively high. Local heating (Reimer, 1984[link]) during the electron illumination may also contribute to higher average displacement parameters. Finally, if the electron beam exceeds a material-dependent threshold acceleration voltage, it can cause knock-on damage (Williams & Carter, 2009[link]) in both organic and inorganic materials. This is a dynamical process which can cause both material loss and rearrangement of atoms. The presence of defects resulting from the rearrangement of atoms may lead to an increase in the average displacement factors. Nevertheless, the refinement using polycrystalline anatase data showed the expected displacement parameters of 1.4 (1) Å2 for Ti and 1.9 (2) Å2 for oxygen (Weirich et al., 2000[link]). Of all the parameters used during Rietveld refinement, the displacement parameters and atomic coordinates are probably the most sensitive to a possible dynamical-scattering contribution in the data. It is noticeable that after the refinement of the anatase structure the atomic coordinates converged to reasonable positions: [0, ¼, 0.1656 (5)] for oxygen (Weirich et al., 2000[link]) compared with the previous range obtained in neutron diffraction studies of [0, ¼, 0.16686 (5)] (Burdett et al., 1987[link]) to [0, ¼, 0.20806 (5)] (Howard et al., 1991[link]).

The relative ratio of two components in a mixture can be determined using the Hill–Howard approach (Hill & Howard, 1987[link]): the relative weight of a phase in a mixture of phases is proportional to the scaling factor of the phase given by the Rietveld refinement combined with the mass and the volume of the unit cell of the component. The relative content of a mixture of anatase and brookite was successfully determined from electron powder diffraction data (Djerdj & Tonejc, 2005[link], 2006[link]).

For the modelling of the Bragg reflection shape the Pearson VII function can be used (Weirich et al., 2000[link]; Kim et al., 2009[link]), although recently the more popular pseudo-Voigt peak shape function has been used (Tonejc et al., 2002[link]; Djerdj & Tonejc, 2005[link], 2006[link]) and provides a satisfactory fit between the experimental and calculated data.

The average crystalline domain size can be determined using line-broadening analysis. The measured intensity profile is a convolution of the physical line profile given by the sample with the instrumental profile broadening. When expressed in terms of the scattering angle θ, the width of the electron diffraction peaks is much smaller than that for X-rays. On the other hand, electrons generally have a smaller coherence length than X-rays. As a result, for the same material, the effective peak width for electron diffraction is larger than that for powder X-ray data (Song et al., 2012[link]). Because of this, it is sometimes difficult to separate the domain size and the instrumental contributions to the peak broadening. Therefore, the average domain size obtained after the refinement procedure should be cross-checked with the domain size determined from TEM images obtained, for instance, using the dark-field technique (Williams & Carter, 2009[link]).

In electron diffraction various instrumental parameters can affect the peak width. The energy spread of the electrons causes additional broadening of diffracted spots. This effect can be partially reduced by energy filtering of the diffraction patterns (Kim et al., 2009[link]; Egerton, 2011[link]). Finally, the electron diffraction camera length must be large enough that the detector broadening is much smaller than the peak width, as demonstrated in Fig. 2.4.9[link]: large values of the camera length (`zoomed in' diffraction patterns) result in thinner, better separated peaks.

[Figure 2.4.9]

Figure 2.4.9 | top | pdf |

Electron powder diffraction profiles of gold nanoparticles (range 2–6 nm−1) recorded at different electron diffraction camera lengths.

Preferred orientation can be an issue for electron powder diffraction: when the powder material is supported on a thin film, the crystals tend to orient themselves with their most developed facet facing the support. As a result, the relative intensities of the diffracted peaks are modified (Kim et al., 2009[link]). Texture within nanocrystalline powders introduced by the sample preparation on a support for TEM can be analysed using electron powder diffraction patterns recorded at different tilt positions of the sample. Refinement of the preferred orientation of two different materials – nanocrystalline aluminium and α-MnS powders – showed that the aluminium particles tend to have strong preferred orientation due to their facet morphology, while α-MnS particles are randomly oriented (Gemmi, Fischer et al., 2011[link]).

Although dynamical effects are believed to be reduced for nanocrystalline materials and additionally reduced by data collection from non-oriented crystals, the dynamical component of the scattering cannot be neglected. For the dynamical correction using the two-beam approximation formalism of equation (2.4.12)[link], the reader is referred to Section 2.4.2[link]. For a range of electron-beam energies from 20 to 50 kV it has been shown that polycrystalline electron diffraction patterns of aluminium crystals smaller than 9 nm have a dynamical scattering component below 10% (Horstmann & Meyer, 1962[link]). For polycrystalline MnFe2O4 with an average crystal size of 11 nm measured using a 120 kV electron beam, the ratio of the kinematical to dynamical contributions in the structure factor was about 1:1.5 (Kim et al., 2009[link]). The application of the small (less than 3%) correction for the dynamical component during Rietveld refinement of nanocrystalline intermetallic Au3Fe1−x improved the refined long-range order parameter of the alloy (Luo et al., 2011[link]).

In summary, the Rietveld refinement technique applied to electron powder diffraction data is a new area of research. It can be successfully carried out for small volumes of nanocrystalline materials, for which the small electron beam is an advantage. Results obtained from Rietveld analysis of electron powder diffraction data of nanocrystalline materials are encouraging. The refinement for powders containing large crystal grains is problematic because of dynamical scattering present in the data. There are also uncertainties caused by instrumental effects. The dynamical effects can be accounted for using the Blackman formalism, while the influence of diverse instrumental parameters needs further systematic study.

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