Pair Distribution Function Obtained from Electron Diffraction: An Advanced Real-Space Structural Characterization Tool

Abstract

Pair distribution function from electron diffraction (ePDF) is a remarkable technique capable of elucidating the atomic arrangement of materials with high spatial resolution, especially those that do not present sharp diffraction reflections, such as amorphous materials and nanoparticles. Higher electron-matter interaction favors the use of ePDF for materials that have weak Bragg diffraction and higher amounts of diffuse scattering. The analysis and treatment of reciprocal-space electron diffraction data are nowadays well developed, determining the ePDF g(r) results. However, efforts should be made to enable and improve quantitative analysis. Recent advances have proved that ePDF can reveal the local structure details of solid and frozen liquid materials. Interpretation of PDF real-space results requires simulations using structural references or fittings to refine the atomic structure. We believe that ePDF will become a standard structure-characterization method for the analysis of nanomaterials and amorphous materials. Atomic-scale structure determination is crucial to the understanding of nanomaterial properties and development of new technologies. Although pair distribution function (PDF) analysis by neutrons and X-ray scattering profile has been used to study materials, electron diffraction can offer advantages to characterize the atomic structure of clusters, amorphous samples, and nanomaterials. Electrons have higher scattering power than X-rays, allowing the acquirement of PDF from electron diffraction (ePDF) for small sample amounts and with time-efficient data acquisition. Compared with synchrotron X-rays and neutrons as sources for PDF, the availability of electron microscopes worldwide is advantageous. Nowadays, with the rise of methodologies and specific software for ePDF data analysis, the scientific community can benefit from advanced transmission electron microscopy (TEM) structure determination integrating commonly available TEM analyses—size, distribution, shape, and high-resolution TEM atomic visualization—with ePDF atomic structure determination, both for bulk and surface configurations. Therefore, ePDF has the potential to become a routine and advanced characterization tool for nanomaterials science. Atomic-scale structure determination is crucial to the understanding of nanomaterial properties and development of new technologies. Although pair distribution function (PDF) analysis by neutrons and X-ray scattering profile has been used to study materials, electron diffraction can offer advantages to characterize the atomic structure of clusters, amorphous samples, and nanomaterials. Electrons have higher scattering power than X-rays, allowing the acquirement of PDF from electron diffraction (ePDF) for small sample amounts and with time-efficient data acquisition. Compared with synchrotron X-rays and neutrons as sources for PDF, the availability of electron microscopes worldwide is advantageous. Nowadays, with the rise of methodologies and specific software for ePDF data analysis, the scientific community can benefit from advanced transmission electron microscopy (TEM) structure determination integrating commonly available TEM analyses—size, distribution, shape, and high-resolution TEM atomic visualization—with ePDF atomic structure determination, both for bulk and surface configurations. Therefore, ePDF has the potential to become a routine and advanced characterization tool for nanomaterials science. In materials science, the determination of the atomic structure of materials is fundamental to the understanding of their properties and development of new applications. Crystalline structures can be successfully determined using traditional techniques such as X-ray diffraction (XRD) and electron diffraction (ED), which is performed in transmission electron microscopy (TEM).1Brandon D. Kaplan W.D. Microstructural Characterization of Materials.2nd Edition. John Wiley, 2008Crossref Scopus (196) Google Scholar, 2David W. Shankland K. McCusker L. Bärlocher C. Structure Determination from Powder Diffraction Data. Oxford University Press, 2006Crossref Scopus (0) Google Scholar, 3Gemmi M. Mugnaioli E. Gorelik T.E. Kolb U. Palatinus L. Boullay P. Hovmöller S. Abrahams J.P. 3D electron diffraction: the nanocrystallography revolution.ACS Cent. Sci. 2019; 5: 1315-1329Crossref PubMed Scopus (131) Google Scholar The periodic lattice of crystals produces strong scattering signals, called Bragg diffraction peaks, allowing their phase indexation and structure determination. Compared with ED, XRD is the main technique to determine the atomic structure of crystalline materials.2David W. Shankland K. McCusker L. Bärlocher C. Structure Determination from Powder Diffraction Data. Oxford University Press, 2006Crossref Scopus (0) Google Scholar The position and intensities analysis of the Bragg peaks make it possible to determine the three-dimensional (3D) crystalline structure.1Brandon D. Kaplan W.D. Microstructural Characterization of Materials.2nd Edition. John Wiley, 2008Crossref Scopus (196) Google Scholar, 2David W. Shankland K. McCusker L. Bärlocher C. Structure Determination from Powder Diffraction Data. Oxford University Press, 2006Crossref Scopus (0) Google Scholar, 3Gemmi M. Mugnaioli E. Gorelik T.E. Kolb U. Palatinus L. Boullay P. Hovmöller S. Abrahams J.P. 3D electron diffraction: the nanocrystallography revolution.ACS Cent. Sci. 2019; 5: 1315-1329Crossref PubMed Scopus (131) Google Scholar Using single-crystal XRD (SC-XRD) and single-crystal ED (zone-axis pattern and 3D ED), crystal structures can be solved accurately. Recently, 3D ED, which is similar to SC-XRD analysis, has already been used for the analysis of single crystals ranging from micrometers to nanometers.3Gemmi M. Mugnaioli E. Gorelik T.E. Kolb U. Palatinus L. Boullay P. Hovmöller S. Abrahams J.P. 3D electron diffraction: the nanocrystallography revolution.ACS Cent. Sci. 2019; 5: 1315-1329Crossref PubMed Scopus (131) Google Scholar X-ray powder diffraction (PXRD) and selected-area ED (SAED) are the techniques frequently used for analyzing particulate crystals in these size ranges. PXRD is a reliable structure analysis technique with the capacity of accurate structure refinement. Over the years, these techniques have been refined and successfully applied to the analysis of metallic, inorganic, and organic materials, along with the determination of complex materials such as proteins.4Wilkins S.W. Celebrating 100 years of X-ray crystallography.Acta Crystallogr. A. 2013; 69: 1-4Crossref Scopus (13) Google Scholar,5Hendrickson W.A. Evolution of diffraction methods for solving crystal structures.Acta Crystallogr. A. 2013; 69: 51-59Crossref PubMed Scopus (15) Google Scholar However, new techniques are necessary to study the structure of materials that present significant diffraction peak broadening, such as amorphous and nanostructured materials. For example, nanocrystals show a highly diffuse PXRD pattern, which makes its interpretation very challenging.6Petkov V. Nanostructure by high-energy X-ray diffraction.Mater. Today. 2008; 11: 28-38Crossref Scopus (165) Google Scholar, 7Bawendi M.G. Kortan A.R. Steigerwald M.L. Brus L.E. X-ray structural characterization of larger CdSe semiconductor clusters.J. Chem. Phys. 1989; 91: 7282-7290Crossref Scopus (254) Google Scholar, 8Hall B.D. Zanchet D. Ugarte D. Estimating nanoparticle size from diffraction measurements.J. Appl. Crystallogr. 2000; 33: 1335-1341Crossref Scopus (200) Google Scholar, 9Holder C.F. Schaak R.E. Tutorial on powder X-ray diffraction for characterizing nanoscale materials.ACS Nano. 2019; 13: 7359-7365Crossref PubMed Scopus (273) Google Scholar Moreover, nanocrystals can present homogeneous and inhomogeneous strain, structural defects, morphological parameters (size and shape), and surface disorder that contribute to generating a diffuse PXRD pattern.10Gilbert B. Huang F. Zhang H. Waychunas G.A. Banfield J.F. Nanoparticles: strained and stiff.Science. 2004; 305: 651-654Crossref PubMed Scopus (394) Google Scholar For instance, to illustrate the size effect in the PXRD pattern, Figure 1A shows the PXRD patterns of ZrO2 microcrystals compared with ZrO2 nanocrystals with a mean particle size of 3 nm. It is important to point out that the high-resolution transmission electron microscopy (HRTEM) image pictured in the inset of Figure 1A confirms the crystalline nature of the ZrO2 nanocrystals. This example shows the PXRD peak broadening due to the crystal size, well described by the Scherrer equation, and how its interpretation is non-trivial for nanocrystals. In the last decade, pair distribution function (PDF) analysis based on X-ray or neutron total scattering data has become a mature and well-established method capable of providing quantitative information of the atomic structure in amorphous and nanostructured materials.11Egami T. Billinge S. Underneath the Bragg Peaks, Structural Analysis of Complex Materials.1st Edition. Pergamon Press, 2003Google Scholar,12Billinge S.J. Nanoscale structural order from the atomic pair distribution function (PDF): there’s plenty of room in the middle.J. Solid State Chem. 2008; 181: 1695-1700Crossref Scopus (109) Google Scholar In this Perspective, we analyze and review the current progress and challenges in using ED (mainly SAED) to obtain PDF, i.e., the ePDF, as an important characterization tool to obtain structural information of materials with significant degree of disorder. Instead of performing an extensive review of published articles from the literature, we provide an analysis to show the advantages of using the ED obtained in a conventional TEM to analyze nanomaterials, amorphous materials, and frozen liquids. We also show the importance of combining ePDF with theoretical calculations to obtain detailed analyses of materials' atomic arrangements. We strongly believe that the ePDF is a new functionality of TEM, especially in obtaining structural information. Materials structure can be described by their atomic ordering arrangement. At small distances, i.e., the nearest atomic neighbors, the structure is dominated by the nature of the chemical bonds with well-defined distances and angles, originating the short-range order (SRO).13Egami T. Billinge S.J.L. Underneath the Bragg Peaks: Structural Analysis of Complex Materials.2nd Edition. Pergamon Materials Series, Elsevier Science, 2012Google Scholar Crystalline materials have a periodic atomic lattice and the atomic correlation holds for long distances, i.e., they exhibit long-range order (LRO).13Egami T. Billinge S.J.L. Underneath the Bragg Peaks: Structural Analysis of Complex Materials.2nd Edition. Pergamon Materials Series, Elsevier Science, 2012Google Scholar Whereas in crystalline materials the structure (LRO) can be well determined using PXRD or ED analyses, in glasses, amorphous materials, and liquids the structure determination is a challenge. These materials frequently present intermediate atomic arrangement between SRO and LRO distances (approximately 1–10 nm), named middle-range order (MRO).12Billinge S.J. Nanoscale structural order from the atomic pair distribution function (PDF): there’s plenty of room in the middle.J. Solid State Chem. 2008; 181: 1695-1700Crossref Scopus (109) Google Scholar Likewise, nanocrystals that have sizes below 10 nm, having only a few thousand atoms and no LRO, present ordering only up to the MRO, due to size effects. The structure determination of these materials in the reciprocal space is a challenge as XRD peaks broaden, signal intensity diminishes, and peaks overlap.6Petkov V. Nanostructure by high-energy X-ray diffraction.Mater. Today. 2008; 11: 28-38Crossref Scopus (165) Google Scholar, 7Bawendi M.G. Kortan A.R. Steigerwald M.L. Brus L.E. X-ray structural characterization of larger CdSe semiconductor clusters.J. Chem. Phys. 1989; 91: 7282-7290Crossref Scopus (254) Google Scholar, 8Hall B.D. Zanchet D. Ugarte D. Estimating nanoparticle size from diffraction measurements.J. Appl. Crystallogr. 2000; 33: 1335-1341Crossref Scopus (200) Google Scholar, 9Holder C.F. Schaak R.E. Tutorial on powder X-ray diffraction for characterizing nanoscale materials.ACS Nano. 2019; 13: 7359-7365Crossref PubMed Scopus (273) Google Scholar Nanoparticles with size ranging from 20 to 200 nm can present LRO and usually show PXRD patterns with defined Bragg diffraction peaks. In this Perspective we will, therefore, use the generic term nanocrystals when broadly referring to nanoparticles with dimensions ranging from 2 nm to 10 nm. Quantitative determination of the local atomic structure (SRO and MRO) of amorphous, glasses, liquids, and nanocrystalline materials require methods that go beyond identification of the Bragg diffraction peaks. It is necessary to use methods that enable total scattering analysis, i.e., the Bragg reflections and diffuse scattering. The atomic PDF technique is a special approach to analyze the total scattering diffraction data and obtain structural information such as neighbors’ atomic distances and coordination numbers of materials.11Egami T. Billinge S. Underneath the Bragg Peaks, Structural Analysis of Complex Materials.1st Edition. Pergamon Press, 2003Google Scholar,14Klug H.P. Alexander L.E. X-Ray Diffraction Procedures: For Polycrystalline and Amorphous Materials.2nd Edition. John Wiley, New York1974Google Scholar To better understand the PDF technique, we can imagine a two-dimensional square lattice (2D crystal) and mark a central atom in this lattice, as illustrated in Figure 1B. At 0 K, the central reference atom has a first coordination sphere, where this atom is coordinated by four nearest neighbors at a distance r1=r, four next-nearest neighbors at r2=2r, another four at r3=2r, eight at r4=5r, and so on. Figure 1C shows the number of atoms (n(r)) to be found at a radial distance r away from the central atom at T=0K. At temperatures T>0K, but below the melting temperature (TM), the mean distance between the atoms increases slightly in the presence of thermal vibrations, and the amplitude and frequency of atomic displacements also increase; however, the atomic structure is still the same. Under this condition, the number of atoms around the reference atom is now better described by Figure 1D, where the atomic displacement leads to an atomic distribution around the position r (broadened signal) instead of having exactly the value of n(r) at r (discrete peak). Therefore, the simple n(r) is not a suitable parameter to represent the atomic arrangement of real solid materials. One way to describe a solid structure is to determine the correlations between all atomic positions via the interatomic distances rij within the system, i and j being the individual atoms. Thus, the distribution of interatomic distances is given by a function called atomic pair density function ρ(r), also known as the pair correlation function,11Egami T. Billinge S. Underneath the Bragg Peaks, Structural Analysis of Complex Materials.1st Edition. Pergamon Press, 2003Google Scholarρ(r)=ρ0g(r)=14πNr2∑i∑jδ(r−rij),(Equation 1) where δ(r−rij) is the Dirac delta, ρ0 is the number density of the material (number of atoms per volume, calculated by taking the number of atoms inside the unit cell divided by its volume), and N is the number of atoms. Equation 1 leads to the atomic structure determination as a function of r, similar to the n(r) graph. However, the experimental determination of all relative atomic positions is not possible. For real solid materials, the atomic structure can be described by a function that accounts for the probability of finding an atom at an interatomic distance r from another atom, frequently called atomic pair distribution function g(r).11Egami T. Billinge S. Underneath the Bragg Peaks, Structural Analysis of Complex Materials.1st Edition. Pergamon Press, 2003Google Scholar Both ρ(r) and g(r) describe the atomic structure: the former is defined directly from real space and the latter can be obtained by the Fourier transform of the structure factor (reciprocal Q-space data). Q is the magnitude of the wave vector (Q=4πsin(θ)/λ), 2θ is the angle between the incident and scattered radiation (X-rays) or particles (neutrons or electrons), and λ is the wavelength of the incident radiation or particles.11Egami T. Billinge S. Underneath the Bragg Peaks, Structural Analysis of Complex Materials.1st Edition. Pergamon Press, 2003Google Scholar Thus, it is possible to describe an atomic structure, with or without translational periodicity, using the PDF g(r), which gives a measurement in the real space of the local atomic density of a spherical shell of unit thickness at a distance r. An accurate calculation of PDF g(r) demands experimental diffraction data obtained with the highest Q possible (Qmax). This requirement can only be achieved using high-energy X-rays or using neutrons, as both sources provide diffraction data with a high wave vector (> 30 Å−1). For instance, high-energy X-rays may be obtained from synchrotron radiation sources or laboratory sources such as sealed X-ray tubes with Mo (17 keV, Qmax ≈ 17 Å−1) or Ag (22 keV, Qmax ≈ 20 Å−1) anodes.15Thomae S.L.J. Prinz N. Hartmann T. Teck M. Correll S. Zobel M. Pushing data quality for laboratory pair distribution function experiments.Rev. Scientific Instr. 2019; 90: 043905Crossref PubMed Scopus (16) Google Scholar,16Petkov V. Characterization of Materials.2nd Edition. Wiley, 2012Google Scholar It is important to point out that the energy of sealed X-ray tubes with Cu anode (8 keV, Qmax ≈ 8 Å−1) is too low and is not suitable to obtain diffraction data for PDF calculations. High Qmax can be easily achieved in an ED experiment performed in a commercial transmission electron microscope (80–300 kV).17Gorelik T.E. Neder R. Terban M.W. Lee Z. Mu X. Jung C. Jacob T. Kaiser U. Towards quantitative treatment of electron pair distribution function.Acta Crystallogr. Section B. 2019; 75: 532-549Crossref Scopus (17) Google Scholar Hence, PDF analysis can be performed using ED data in the same way as X-rays or neutron diffraction experiments. Compared with X-rays and neutrons, electrons have a much stronger interaction with matter, resulting in a stronger scattering signal, shorter-time data collection, and smaller required sample amounts. Consequently, even small amounts of materials (micrograms to nanograms) are enough to produce a signal without the use of specialized equipment, which means that a conventional transmission electron microscope can be used to obtain good-quality PDF analysis.17Gorelik T.E. Neder R. Terban M.W. Lee Z. Mu X. Jung C. Jacob T. Kaiser U. Towards quantitative treatment of electron pair distribution function.Acta Crystallogr. Section B. 2019; 75: 532-549Crossref Scopus (17) Google Scholar Another advantage of the ED is the possibility of achieving high spatial resolution. In conventional TEM, nanobeam electron diffraction (NBED) with a spot size of 3 nm is easily obtained, making it possible to analyze local order in amorphous materials.18Hirotsu Y. Ishimaru M. Ohkubo T. Hanada T. Sugiyama M. Application of nano-diffraction to local atomic distribution function analysis of amorphous materials.J. Electron Microsc. 2001; 50: 435-442Crossref PubMed Scopus (39) Google Scholar With the advent of spherical aberration-corrected (Cs-corrected) TEM, NBED with a coherent electron beam smaller than 1 nm in diameter can be achieved, thus making possible the direct observation of the local atomic structure (SRO) in amorphous materials.19Hirata A. Guan P. Fujita T. Hirotsu Y. Inoue A. Yavari A.R. Sakurai T. Chen M. Direct observation of local atomic order in a metallic glass.Nat. Mater. 2011; 10: 2833Crossref Scopus (390) Google Scholar Disadvantages come mostly from inelastic scattering and the possibility of multiple scattering (dynamical scattering regime), which affects the acquisition of reliable scattering intensity profile I(Q). Inelastic scattering contributes to the diffraction pattern primarily as background data, and its effects can be minimized by theoretical approaches or by acquiring the scattering profile using an electron energy loss spectroscopy (EELS) detector, which can filter the inelastic contribution to I(Q).17Gorelik T.E. Neder R. Terban M.W. Lee Z. Mu X. Jung C. Jacob T. Kaiser U. Towards quantitative treatment of electron pair distribution function.Acta Crystallogr. Section B. 2019; 75: 532-549Crossref Scopus (17) Google Scholar Furthermore, multiple scattering modifies diffraction intensities in such a way that experimental intensities (especially in higher angles) are much stronger than the kinematic scattering. However, multiple scattering is negligible if the electron mean free path is greater than the crystal size, which is easily achieved in nanomaterials.17Gorelik T.E. Neder R. Terban M.W. Lee Z. Mu X. Jung C. Jacob T. Kaiser U. Towards quantitative treatment of electron pair distribution function.Acta Crystallogr. Section B. 2019; 75: 532-549Crossref Scopus (17) Google Scholar Moreover, precession electron diffraction (PED) and deconvolution techniques can also be used to reduce a small contribution of dynamical scattering;20Hoque M.M. Vergara S. Das P.P. Ugarte D. Santiago U. Kumara C. Whetten R.L. Dass A. Ponce A. Structural analysis of ligand-protected smaller metallic nanocrystals by atomic pair distribution function under precession electron diffraction.J. Phys. Chem. C. 2019; 123: 19894-19902Crossref Scopus (9) Google Scholar,21Mu X. Neelamraju S. Sigle W. Koch C.T. Totò N. Schön J.C. Bach A. Fischer D. Jansen M. van Aken P.A. Evolution of order in amorphous-to-crystalline phase transformation of MgF2.J. Appl. Crystallogr. 2013; 46: 1105-1116Crossref Scopus (22) Google Scholar however, the PED technique is still under development and further studies should be done regarding ePDF applications. In any case, it is almost mandatory to work with the samples in optimal conditions to minimize damage, inelasticity, and multiple scattering. Using adequate data collection, ePDF can produce results comparable with those obtained using synchrotron or neutron sources.20Hoque M.M. Vergara S. Das P.P. Ugarte D. Santiago U. Kumara C. Whetten R.L. Dass A. Ponce A. Structural analysis of ligand-protected smaller metallic nanocrystals by atomic pair distribution function under precession electron diffraction.J. Phys. Chem. C. 2019; 123: 19894-19902Crossref Scopus (9) Google Scholar Although electrons have higher scattering power compared with X-rays, at higher scattering angles, high-Q region, the ED sensitivity is lower, i.e., the signal-to-noise ratio is poor for higher-order reflections as the amount of scattered electrons is lower compared with X-rays.22Zheng J.-C. Zhu Y. Wu L. Davenport J.W. On the sensitivity of electron and x-ray scattering factors to valence charge distributions.J. Appl. Crystallogr. 2005; 38: 648-656Crossref Scopus (25) Google Scholar As suggested by the Mott equation, the electron atomic scattering amplitude (fel(s)) is related to the X-ray atomic scattering (fx(s)) as fel(s) ∝ (1/s2)/[Z−fx(s)], where s is the scattering vector (s=2sin(θ)/λ=Q/2π) and Z is the atomic number. fel(s) is inversely proportional to the square of the scattering angle s, indicating that the sensitivity of the ED with s decreases faster compared with in X-rays.22Zheng J.-C. Zhu Y. Wu L. Davenport J.W. On the sensitivity of electron and x-ray scattering factors to valence charge distributions.J. Appl. Crystallogr. 2005; 38: 648-656Crossref Scopus (25) Google Scholar Figure 2A shows the comparison between fel and fx for carbon, showing the fast decay of fel with Q as described by the Mott equation, and Figure 2B shows the fel2, proportional to the observed diffraction intensity, I∝|f|2. By defining the relative intensity drop at high Q (Φ), for instance at 20 Å−1 (Φ20), X-rays show an intensity drop of 99.027% compared with its maximum at 0 Å−1, while electrons show Φ20 of 99.958%. In practice, this means that for high-order reflections the electron scattering amount is small, therefore higher accumulation times could be necessary to collect reliable data, especially for amorphous materials that have lower scattering probabilities compared with crystalline samples. The ePDF resolution depends upon the determination of reliable S(Q) with high Qmax; technically, reasonable ePDF results can be obtained with s higher than 3 Å−1 (Q>18 Å−1).24Skinner L.B. Huang C. Schlesinger D. Pettersson L.G. Nilsson A. Benmore C.J. Benchmark oxygen-oxygen pair-distribution function of ambient water from X-ray diffraction measurements with a wide Q-range.J. Chem. Phys. 2013; 138: 074506Crossref PubMed Scopus (281) Google Scholar The electron lower fel at higher-order structure factors do not hinder acquiring good ePDF results, as the literature has already shown the determination of trustworthy structure factors using ED even for amorphous samples.25Souza Junior J.B. Schleder G.R. Colombari F.M. de Farias M.A. Bettini J. van Heel M. Portugal R.V. Fazzio A. Leite E.R. Pair distribution function from electron diffraction in cryogenic electron microscopy: revealing glassy water structure.J. Phys. Chem. Lett. 2020; 11: 1564-1569Crossref PubMed Scopus (10) Google Scholar Figure 2C shows ED data for amorphous carbon (carbon black nanoparticles) with intensities shown as log(I) to highlight the low-intensity signal, revealing the presence of oscillations until 17 Å−1, with only 16 s of data acquisition. Carbon being one of the lightest atoms, it has low fel compared with other atoms, the amorphous carbon black material was chosen to show the possibility of acquiring reasonable ED for amorphous samples. Different techniques can also be used to collect suitable ED data. For instance, Mu and colleagues21Mu X. Neelamraju S. Sigle W. Koch C.T. Totò N. Schön J.C. Bach A. Fischer D. Jansen M. van Aken P.A. Evolution of order in amorphous-to-crystalline phase transformation of MgF2.J. Appl. Crystallogr. 2013; 46: 1105-1116Crossref Scopus (22) Google Scholar showed that acquiring the ED with different angular ranges (Q regions) can help to improve the high-order reflections, instead of using a single image. However, for this experiment the collection of high-Q regions was also restricted to a single frame due to their omega-type energy filter. Another technique that can be used is collecting the ED with different camera lengths (magnifications) so that different pixel binning will be applied in different Q regions, increasing the acquisition counting. PDF is a powerful technique that here will be referred to as the case known as particle radial distribution function, or just the radial distribution function. Detailed literature about PDF can be found elsewhere.11Egami T. Billinge S. Underneath the Bragg Peaks, Structural Analysis of Complex Materials.1st Edition. Pergamon Press, 2003Google Scholar In this section we focus on how ePDF can be successfully acquired from ED data and how this can change the way that researchers deal with nanoscale problems. To fulfill this, first we present a succinct description of how the ePDF g(r) is obtained from electron scattering data. The scheme in Figure 3 shows the sequence of processes that will be discussed hereafter to determine the local structure of materials using ePDF. A powder SAED pattern is acquired with the highest Qmax possible, typically higher than 20 Å−1, by decreasing the camera length. The resolution of the g(r) is proportional to the Qmax and besides, lower Qmax leads to peak broadening and can lead to problems in determining the peak position.24Skinner L.B. Huang C. Schlesinger D. Pettersson L.G. Nilsson A. Benmore C.J. Benchmark oxygen-oxygen pair-distribution function of ambient water from X-ray diffraction measurements with a wide Q-range.J. Chem. Phys. 2013; 138: 074506Crossref PubMed Scopus (281) Google Scholar The azimuthal integration centered precisely at the direct electron beam spot generates the scattering intensity profile I(Q). For the 2D azimuthal integration of SAED, software such as Gatan Digital Micrograph (GMS) including the Difftools26Mitchell D. Difftools: electron diffraction software tools for digital micrograph.Microsc. Res. Tech. 2008; 71: 588-593Crossref PubMed Scopus (182) Google Scholar script or DAWN27Filik J. Ashton A.W. Chang P.C.Y. Chater P.A. Day S.J. Drakopoulos M. Gerring M.W. Hart M.L. Magdysyuk O.V. Michalik S. et al.Processing two-dimensional X-ray diffraction and small-angle scattering data in DAWN 2.J. Appl. Crystallogr. 2017; 50: 959-966Crossref PubMed Scopus (218) Google Scholar can be used. If needed, digitally correcting the diffraction pattern from nonscattering imperfections such as the beam stopper, shadows, and circularity can be performed. Typically, a standard SAED pattern such as gold or aluminum is used to calibrate the reciprocal space I(Q) dimensions. The background from the thin carbon support film should also be subtracted from the scattering result.17Gorelik T.E. Neder R. Terban M.W. Lee Z. Mu X. Jung C. Jacob T. Kaiser U. Towards quantitative treatment of electron pair distribution function.Acta Crystallogr. Section B. 2019; 75: 532-549Crossref Scopus (17) Google Scholar The presence of the carbon background does not affect the sample peak positions, but the extra peaks could lead to misinterpretation of convolution and intensities, thus interfering with the final structure determination. Once the I(Q) is obtained, the structure factor S(Q) and the reduced structure factor F(Q)=Q[S(Q)−1] are calculated by adjusting a proper electron scattering factor that accounts for the material scattering profile. For some special cases, the scattering factor can be modified to account