International
Tables for Crystallography Volume H Powder diffraction Edited by C. J. Gilmore, J. A. Kaduk and H. Schenk © International Union of Crystallography 2018 
International Tables for Crystallography (2018). Vol. H, ch. 2.4, pp. 113114

An extensive description of pair distribution function (PDF) analysis covering data acquisition, reduction and interpretation can be found in Chapter 5.7 . Here, only a short outline is presented, concentrating on aspects that are specific to PDFs obtained by electron diffraction.
Poorly crystalline and amorphous materials exhibit no longrange order and therefore show no pronounced Bragg peaks in diffraction patterns. Nevertheless, owing to defined bonding geometry, these materials do have a specific local arrangement of atoms, denoted as shortrange order. The shortrange order can be analysed using the PDF obtained from the total scattering profile. The PDF can provide general information about the degree of order, the character of local atomic packing and the size of the correlation domains. The total scattering function is collected over a wide range of reciprocal space and includes not only the Bragg reflections (if present), but also the diffuse scattering information between them (Egami & Billinge, 2003).
The PDF G(r) represents the probability of finding a pair of atoms with an interatomic distance r, weighted by the scattering power of the individual atoms. After normalization and suitable corrections, the reduced scattering function F(Q) is derived. [In the PDF analysis, the scattering vector Q, which is related to the scattering angle θ as is used, instead of S = sin θ/λ.] The PDF can be calculated by the Fourier transformation of F(Q) into direct space (Warren, 1990; Egami & Billinge, 2003; Farrow & Billinge, 2009).
Powder diffraction data for PDF analysis should be measured over a sufficiently large range of the scattering angle θ; therefore, neutron or synchrotron sources or laboratory Xray data with a shortwavelength source (Mo or Ag anode) are used. Powder electron diffraction data, with their flexibility in electron diffraction camera length, short wavelength and nuclear scattering at large scattering angles, can also cover the desired large range of scattering angles and are therefore highly suitable for PDF analysis. In addition, atoms have a much larger scattering cross section for electrons than for Xrays or neutrons, allowing sufficient signal collection from very small volumes. Finally, electrons can be focused with lenses down to a few nanometres. All these reasons make electron diffraction analysis attractive for the study of the structure of nanovolumes. The electron PDF is therefore a powerful tool for the investigation of the structures of amorphous or poorly crystalline thin films, or for small sample volumes of inhomogeneous samples.
There are several practical issues to consider when collecting electron diffraction data for PDF analysis:
Energy filtering. Traditionally, electron diffraction data for PDF analysis are collected using energy filtering in order to exclude the inelastic scattering contribution. However, quantitative or semiquantitative electron PDFs can be obtained without filtering (Abeykoon et al., 2012).
Multiple scattering/dynamical effects. In order to keep the contribution of nonkinematic scattering low, the sample thickness and the nanoparticle size should be as small as possible. Generally, particles 10 nm and smaller should scatter kinematically, and this is the size range that benefits most from PDF analysis (Abeykoon et al., 2012).
Powder average. Proper statistics are important for PDF analysis. In order to decrease measurement errors one can increase the illumination area on the sample (or the selectedarea aperture in the case of SAED), collect several diffraction patterns from different areas and average them.
Scattering angle range. A large θ range is essential for PDF analysis. An electron diffraction experiment offers significant flexibility in selecting the scattering range through the adjustment of the electron diffraction camera length and illumination wavelengths. Additionally, in order to enhance the data quality, merging of different scattering ranges recorded in a set of diffraction patterns is possible (Petersen et al., 2005).
An electron diffraction pattern is a combination of signals produced by elastically and inelastically scattered electrons. The inelastic component is a result of electron energy loss due to plasmon or innershell excitation, electron Compton or thermal diffuse scattering (Egerton, 2011). For crystalline materials with distinct Bragg peaks the inelastic scattering is not particularly critical, as it mainly contributes to the background in diffraction patterns and can be neglected when only the intensities of the Bragg peaks are analysed. For PDF analysis the total scattering profile is used; thus, the inelastic scattering, which can significantly modify the scattering profile, needs to be considered (Ishimaru, 2006). Two strategies are followed in this respect: (i) energy filtering of diffraction patterns, which is the more accurate approach but demands specific instrumentation, and (ii) subtraction of the background scattering taken from an area adjacent to the sample (i.e. from the supporting film), which assumes that the main inelastic scattering component originates from the support, and the contribution from the sample can be neglected (Cockayne, 2007). The validity of this approximation depends on the level of quantification intended in the particular study.
The PDF formalism presented above is based on the singlescattering approximation. Multiple scattering, which is much stronger in electron diffraction than for Xrays and neutrons, significantly affects the total scattering profile and therefore the PDF. The multiplescattering effects can modify the peak positions in the PDF as well as the relative intensities of the peaks, the latter being more sensitive to multiple scattering (Anstis et al., 1988). It has been shown that for amorphous materials, owing to the contribution of the multiple scattering, the total scattering profile depends on the thickness of the foil (Childs & Misell, 1972; Rez, 1983). Knowledge of the film thickness allows extraction of the singlescattering distribution. An improved agreement with the expected PDF was shown for hydrogenated amorphous silicon (Anstis et al., 1988) and amorphous germanium (Ankele et al., 2005) using the singlescattering profile.
Experimentally, it is difficult to determine the sample thickness along the incidentelectronbeam direction. In this case, the thickness parameter employed in calculations can be varied, adjusting the amplitudes of the PDF. An estimate for the sample thickness is found when the optimal fit is obtained. Different input values of the thickness result in different principal gradients of the oscillations. Once a reasonable fit is found, the correct thickness is determined and the contribution of multiple scattering can be eliminated (Ankele et al., 2005). This method was applied to amorphous NiNb alloy, allowing an estimate of the foil thickness, and thereafter improved the fit to the PDF obtained from Aganode Xray scattering experiments (Ankele et al., 2005).
Alternatively, the wavelength dependence of the multiplescattering term can be used. A set of diffraction patterns of a glassy carbon film was collected from the same sample (apparently having the same thickness) using different wavelengths (Petersen et al., 2005). These patterns were then processed in order to retrieve the singlescattering profile of tetrahedral amorphous carbon, which showed an improved fit to the reduced scattering function obtained with neutrons (Petersen et al., 2005). This method can be applied to materials for which significant multiple scattering is expected and the thickness of the foil cannot be determined a priori. For very thin films the contribution of the multiple scattering is very low and, therefore, often neglected.
The PDF of elemental materials arising from only one contributing atomic scattering function can be directly interpreted in terms of coordination numbers and allows conclusions to be drawn about the local structure. PDF analysis of amorphous silicon prepared by deposition showed the existence of voids in the structure (Moss & Graczyk, 1969) which anneal on progressive heating. PDF investigation of amorphous carbon films prepared by arc plasma deposition showed that the material mainly consists of tetrahedrally coordinated carbon rather than having a graphitic structure (McKenzie et al., 1991).
For ZrNi and ZrCu metallic glasses, partial PDFs were obtained by reverse Monte Carlo simulation (McGreevy & Pusztai, 1988) and fitted to the experimentally obtained electron scattering data. The analysis of the polyhedral statistics showed that the average coordination number of Cu was 11, while for Ni it was less than 10 (Hirata et al., 2007). Study of amorphous FeB alloys (Hirata et al., 2006) and Fe_{90}Zr_{7}B_{3} (Hirotsu et al., 2003) by PDF analysis allowed detection of nanoscale phase separation resulting in the formation of a mixture of different clusters.
Nanocrystals can be efficiently analysed by electron PDF analysis, giving information complementary to TEM imaging. The electron PDF of detonation nanodiamonds (DND) was used to estimate the average domain size (Zhang, 2011). Studies of phase separation in AgCu alloys showed the complex behaviour of the material with variation of temperature (Chen & Zuo, 2007). In the first stage, the nanodomains of the two terminal phases (Ag and Curich) are built; in the second stage, dewetting of the thin film and formation of large Ag and Cu grains occur. A comparison of electron PDFs from nanocrystalline, partially ordered and amorphous parts of silica glasses (Kovács Kis et al., 2006) allowed the estimation of the degree of order developed by changing the connectivity and orientation of the undistorted SiO_{4} tetrahedra. Indirect detection of hydrogen atoms was performed from a modified distribution of atomic distances in soot samples using electron PDF analysis (Kis et al., 2006).
With an increase in the particle size the deviations from the kinematical scattering become severe. Nevertheless, the electron PDF calculated for 100 nm Au crystals reproduced the simulated data quite well: the peak positions and relative amplitudes were not significantly modified (Abeykoon et al., 2012).
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