The nature of x-ray scattering from geo-nanoparticles: Practical considerations of the use of the Debye equation and the pair distribution function for structure analysis

Chemical Geology 329 (2012) 3–9

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Chemical Geology j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / c h e m g e o

The nature of x-ray scattering from geo-nanoparticles: Practical considerations of the use of the Debye equation and the pair distribution function for structure analysis Richard Harrington a, b,⁎, Reinhard B. Neder c, John B. Parise a, b, d a

Department of Geosciences, Stony Brook University, Stony Brook, NY 11794, USA Department of Chemistry, Stony Brook University, Stony Brook, NY 11794, USA Kristallographie und Strukturphysik, Universität Erlangen, Staudtstrasse 3, D-91058 Erlangen, Germany d Photon Sciences Division, Brookhaven National Laboratory, Upton, NY, 11973, USA b c

a r t i c l e

i n f o

Article history: Accepted 22 June 2011 Available online 30 June 2011 Keywords: X-ray diffraction Debye equation Nanoparticles Pair distribution function PDF Ferrihydrite

a b s t r a c t Interpretation of elastic scattering from nanocrystals is critical to building and testing models of the short and medium range atomic arrangements in environmentally relevant nanoparticles. Along with information about atomic arrangements, the diffraction pattern arising from a collection of nanoparticles contains information on the grain size, shape and defect structure and cannot simply be treated as a broadened collection of Bragg peaks. The Debye equation, a sum over all pairs of atoms in the particle, calculates the scattering at each value of Q, rather than at discrete values of hkl. Especially when atomic arrangements are well known, modeling using the Debye equation allows the refinement of particle shape, allowing the investigation of dominant growth axes. We show the example of 6-line ferrihydrite, fitting the diffraction pattern using a disc of 2.3(1) nm in the ab plane and 6(2) nm along the c axis. A powerful and intuitive way to help examine the diffraction pattern is to take the Fourier transform of the normalized total scattering (Bragg and diffuse), giving the pair distribution function (PDF). The PDF represents a bond length distribution of the material weighted by the respective scattering powers of the contributing atoms. Using examples from the literature, we show that by analyzing the medium range correlations (~ 5–15 Å), the structure of a nanoparticle can be distinguished using PDFs generated from model structures. © 2011 Elsevier B.V. All rights reserved.

1. Introduction Nanomaterials are ubiquitous in the natural environment (Banfield and Zhang, 2001). Recent interest in nanomaterials is driven by size dependent (Waychunas and Zhang, 2008) catalytic (Bell, 2003), optical (Fukumi et al., 1994), magnetic (Ghosh et al., 2006), electrical (Gilbert et al., 2004), thermal (Burda et al., 2005) and other physical properties. Another size dependent property of nanomaterials is their x-ray diffraction pattern. Crystalline materials with large grain sizes produce Bragg peaks at discrete scattering angles; a direct consequence of their crystallinity and the effectively infinite periodicity of their atomic arrangements. From these diffraction patterns, it is now almost routine, especially for single crystal data, to determine the long range structure (unit cell contents) ab initio. This is slightly more involved for powder data, but with modern instrumentation and software, structures can be refined by the Rietveld method, if a suitable starting model is available (Evans and Radosavljevic Evans, 2004). For nanoparticles, long range order is diminished and, ⁎ Corresponding author at: Department of Geosciences, Stony Brook University, Stony Brook, NY 11794, USA. E-mail address: [email protected] (R. Harrington). 0009-2541/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.chemgeo.2011.06.010

depending on the size of the nano-grain, the diffraction pattern appears as a series of broadened maxima, rather than sharp peaks associated with a crystalline powder pattern. As crystallite size decreases, Bragg peaks broaden due to the incomplete destructive interference close to the Bragg condition (Cullity and Stock, 2001). Analysis of this peak broadening for samples with crystallite size below ~ 200 nm can garner information on the size of the crystallites in a sample; Scherrer developed a method for performing this crystallite size analysis based on the full width at half maximum (FWHM) of peaks (Scherrer, 1918). This technique has been successfully used in many studies since (Klug and Alexander, 1954; Langford and Wilson, 1978; Calvin et al., 2005; Harrington et al., 2006; Smilgies, 2009). As crystallite size decreases further, the Bragg peaks continue to broaden and may become indistinguishable from background. If the nanoparticle remains structurally perfect and only the size is decreased and the number of defects, etcetera, remains the same, the diffraction pattern can be modeled using Bragg peaks with diminished height but the same integrated intensity. Generally, this is not the case; nanoparticles have increased strain, defects and different shapes. Scattering events due to these non-long range effects manifest as part of ‘diffuse scattering’, which can be 10 6 less intense than the Bragg scattering from a crystalline material. On instrumentation with

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optimized peak-to-background discrimination, such as is found at modern synchrotron x-ray storage rings, the diffuse scattering can be distinguished from the background. High signal to noise discrimination in the diffraction pattern of a nanoparticle is critical for interpretation of the real space structures giving rise to the diffraction pattern. The brilliance afforded by a synchrotron can be around ten orders of magnitude greater than that from a laboratory source. This means that a one second exposure on a synchrotron source is equivalent to around 300 years exposure on a lab source. So the low intensity of the diffuse scattering is compensated by the high flux of the synchrotron source and the full diffraction pattern can be analyzed. Diffuse scattering, present in any diffraction pattern, does not appear at discrete 2θ positions. Because of the dominance of the Bragg component in a powder diffraction pattern from a crystalline material, the diffuse scattering component is usually ignored and incorporated into a “background correction” in a Rietveld refinement. In the majority of cases where the average structure of a highly crystalline material is the object of the study, Rietveld refinement provides excellent results. However diffraction from an ensemble of nanocrystals have a diffuse scattering component of comparable intensity with the Bragg peaks; information on the atomic arrangements and any defects such as stacking faults, can be obtained from interpretation of the total scattering (Neder et al., 2007). 2. The Debye equation For a crystalline material, the diffraction pattern can be calculated if the unit cell size, symmetry and contents (the atom types and their fractional coordinates) are known using the structure factor equation: N

2πiðhun + kvn + lwn Þ

Fhkl = ∑ fn e 1

Where fn is the atomic form factor of atom n; h, k and l are the Miller indices, u, v and w are the fractional coordinates of atom n and the sum extends over all atoms in the unit cell. This simply produces a set of delta functions at each value of 2θ where the Bragg condition is satisfied. The result can be multiplied by the Debye–Waller factor and convoluted with instrumental effects to produce a more realistic model of the diffraction pattern. When dealing with scattering from objects with little crystallinity such as nanoparticles, it is not enough to calculate the Bragg peaks alone. Rather than calculating the structure factor of each hkl peak at discrete scattering angles, the total scattering must be calculated at any arbitrary scattering angle. This is most conveniently expressed in terms of momentum transfer, Q, an angular quantity normalized by wavelength. Q is defined as: Q =

Debye equation is over all atoms in the grain, the computational overhead for a calculation of a crystalline sample with crystallites of N500 nm and N10 9 atoms, even with modern computers, is prohibitive (Grover and McKenzie, 2001). For nanocrystals of a few hundred to a few thousand atoms the calculation is more tractable. Nanoparticles of different shapes and sizes can be defined as a limited collection of unit cells and, using atomic positions from known bulk structures, their diffraction pattern calculated by the Debye equation using a program such as DISCUS (Proffen and Neder, 1997, 1999; Neder and Proffen, 2008). Even if the underlying structure is the same, the diffraction pattern of a nanoparticle can be markedly different depending on the size and shape of the particle. Indeed, Hall (2000) showed that for very small particles, not just the relative intensity but also the position of a Bragg peak can change with different growth axes. Fig. 1 illustrates this shape dependence with three calculated x-ray diffraction patterns of hematite (α-Fe2O3) particles with different shapes: a plate with 30 × 30 × 1 unit cells, a rod with 2 × 2 × 18 unit cells and a block with 4 × 4 × 2 unit cells. The patterns are so different that one would be hard pressed to identify that they have the same underlying structure. In many cases, peaks that appear in one pattern are not evident in the others. Peaks with 0kl indices, such as the (011) peak at ~ 2.5 Å − 1 are very sharp in the diffractogram of the plate, which contains a significant number of unit cells in the ab plane, but much less so in the other two patterns. It should be noted that this is not preferred orientation, although the effects are somewhat similar; the difference here is due to the predominant growth axis. For materials stable under an electron beam, information taken from TEM images provides an excellent starting point for analysis of nano-grain shape using the Debye equation. For electron sensitive materials or for in situ and time resolved diffraction of particle growth, Debye modeling may still provide an excellent complement to small angle x-ray and/or neutron scattering. It should be noted, however, that if the structure is not well known, adding the particle shape and size to the refinement against a data set which has precious little peaks can cause over refinement and the answer returned may not be unique. 3. Example — Debye modeling of 6-line ferrihydrite Ferrihydrite is a ubiquitous iron oxyhydroxide; an important phase in the iron cycle, it is often found where iron has rapidly precipitated out of solution (Jambor and Dutrizac, 1998). Ferrihydrite is an exclusively nanocrystalline phase, the two forms, named 2-line and 6-line for the number of broad Bragg peaks in their diffraction pattern, have the same underlying structure with different sized

4π sinθ λ

Where θ is the scattering angle and λ is the wavelength of the radiation used. A more robust method that allows calculation of intensity as a continuous function of Q, and not just at specific hkl points is the Debye equation (Debye, 1915; Warren, 1990): 2

IðQ Þ = ∑ fm + ∑ ∑ fm fn m

n m;m≠n

sinðQ rmn Þ Qrmn

Here I(Q) is the intensity at a given value of Q, fm is the atomic form factor for atom m, Q is the momentum transfer as previously defined and rmn is the interatomic distance between atoms m and n. The Debye equation is completely general and assumes no periodicity, it is simply a sum over all atoms in the particle; therefore, the scattering intensity at each point, Q, whether arising from Bragg or diffuse scattering, is calculated using this method (Pinna, 2005). Since the sum in the

Fig. 1. Normalized diffractograms of hematite (α-Fe2O3) with different shapes: plate with 30× 30× 1 unit cells (black), rod with 2 × 2 × 18 unit cells (blue) and block with 4 × 4 × 2 unit cells (red) calculated using the Debye equation.

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coherent scattering domains, coincidently 2 nm and 6 nm, respectively (Michel et al., 2007b). Because of the lack of a crystalline analogue, the structure of this important phase has eluded research for decades. Recently, Michel et al.(2007a), using pair distribution function (PDF) analysis of high energy x-ray total scattering data (see later discussion), published a single phase model for ferrihydrite based on the structure of Akdalaite. This model was criticized for not fitting the powder diffraction pattern. Manceau(2009) showed that powder pattern calculated assuming full crystallinity does not provide an adequate fit to the diffraction pattern. Fig. 2 shows the diffraction pattern for 6-line ferrihydrite, along with the powder pattern calculated from the model proposed by Michel assuming large particles in the micrometer size range. It can be clearly seen that the two do not match in peak intensity and in some cases peak position. A clear example of this is the (012) peak (labeled). In the calculated powder pattern, this is one of the most intense peaks. In this instance, comparing the diffraction pattern of a nanoparticle with that of a crystalline sample, as is routinely done using the powder diffraction file, is not comparing apples to apples. The correct structure could easily be missed if size and shape effects are not taken into account. In the experimental data, the peak is not only around half as intense, but shifted to higher Q. Fig. 3 shows the diffraction pattern of rotationally symmetrical ellipsoidal particles with diameters 2.3(1) nm and 6(2) nm in the ab plane and along the c axis, respectively calculated using the Debye equation and refined using the program DIFFEV (Proffen and Neder, 1997). The initial structure used was that proposed by Michel et al.(2007a); the atom positions were fixed and the unit cell constants allowed to refine, returning values of a = 5.99(3) Å and c = 8.56(4) Å. The final R-value returned was 8.8%, using a weighting scheme with weights = 1/intensity. No defects or surface effects were included in the refinement. This is compared with the same experimental pattern as in Fig. 2. The diffraction pattern calculated using the Debye equation provides a much better fit to the data and is, again, rather different to the diffraction pattern calculated assuming large particle size. 4. The pair distribution function (PDF) A powerful and conceptually simple way to examine the underlying atomic arrangement giving rise to the diffraction data is to take the Fourier transform of the total scattering (Bragg and diffuse contributions) to obtain the pair distribution function (PDF) (Billinge and Kanatzidis, 2004; Proffen and Page, 2004). The PDF represents a bond length distribution of the material weighted by the respective

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Fig. 3. Experimental diffractogram of 6-line ferrihydrite (blue) compared with a diffractogram calculated using the Debye equation (red) of nanoparticles of with dimensions of 2.3 nm in the ab plane and 6.3 nm along the c axis. The structure was modeled as an uniaxial ellipsoid with the unique axis parallel to the c axis. The calculated powder pattern was convoluted with a Pseudovoigt function. The parameters for this function were kept fixed at values that had been refined for a CeO2 standard.

scattering powers of the contributing atoms. The PDF is similar to a Patterson map, the difference being that the former takes into account the total scattering (Bragg and Diffuse) from a sample, the latter just the Bragg scattering and so a Patterson map only yields interatomic vectors within a unit cell. To obtain the PDF in practice, several corrections are required before the Fourier transform is carried out. It is necessary to take a diffraction pattern of the empty sample holder under exactly the same conditions as the sample is exposed to and subtract this from the pattern of the sample, leaving only the scattering from the sample. This is necessary as there is no way to separate the scattering from the sample and the scattering from the sample holder, air scattering, detector dark current and so on once the Fourier transform has been carried out (Chupas et al., 2003) — in a Rietveld refinement these contributions are treated as background. To isolate the elastic scattering from the sample the effects of Compton scattering, which is dominant at high Q when a non-energy-discriminating detector is used, multiple scattering, absorption, Laue diffuse scattering and so on are removed Finally, the pattern is normalized against the scattering power of the constituent atoms, producing the structure function, S(Q) (Proffen and Kim, 2009). It is this quantity which is Fourier transformed to give the PDF, G(r): Gðr Þ = 4πr ½ρðr Þ−ρ0  =

Fig. 2. Experimental diffractogram of 6-line ferrihydrite (blue), collected at beamline 11-ID-B at the APS, λ = 0.2128 Å, compared with the a diffractogram calculated assuming large particles in the micrometer size range (red) from the ferrihydrite model proposed by Michel et al. (2007a).

2 ∞ ∫ Q ½SðQ Þ−1 sinðQ r ÞdQ π 0

Where ρ(r) and ρ0 are the microscopic pair density and the average number density, respectively, and r is the radial distance. Computer programs such as PDFgetX2 are available to carry out this process (Qui et al., 2004). The Fourier transform is over all Q space, from zero to infinity. An infinite Qmax is, in practice, impossible. The termination of the diffraction pattern at a finite value of Qmax produces Fourier ripples, which manifest as oscillations in the PDF, increasing the error. The higher the Qmax used in the Fourier transform, the less the noticeable the Fourier ripples; it has been shown that termination with Qmax greater than 30 Å − 1 produces minimal errors due to termination ripples (Toby and Egami, 1992). Diffuse scattering contributes over the entire Q space of a diffraction pattern. If the deviation from local structure occurs only at one site, a delta function in real space, the Fourier transform will cover the entirety of diffraction space (Egami and Billinge, 2003). If the defects are a little

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more extended, say with a relaxed immediate neighborhood, or if there is short range order between defects, then the diffuse scattering will become modulated and will carry information on the structure of the defect. For this reason, it is important to collect the diffraction pattern of a nanoparticle to a high value of Q for PDF studies. The real space resolution is related to the Qmax used in the Fourier transform, δr ≈ π/Qmax (Billinge and Kanatzidis, 2004). In this way, the PDF provides a probe to study the local structure of materials. For a laboratory diffractometer using Cu Kα1 radiation (λ = 1.54 Å), the largest Qmax achievable is ~ 8 Å -1 (at 2θ = 180°). To obtain a high Qmax, low wavelength (high energy) x-rays are necessary. This can be achieved in the laboratory using Mo (Q max = 15 Å − 1 ) or Ag (Qmax = 21 Å −1) radiation, if angular coverage, 2θ, of 180° is possible. Most PDF experiments make use of synchrotron x-radiation. In addition to the much greater flux offered by these sources over lab sources, most beamlines offer tunable wavelength. This allows the user to select the wavelength to avoid fluorescence and obtain a high value of Qmax. X-rays of greater than 100 keV are routinely used in PDF experiments. Initially, the PDF technique was used to study liquid systems (Debye and Menke, 1930), but with the advent of synchrotron sources, it is being applied to a wider range of materials. For example, recent studies have used this technique to examine crystalline materials in which the local structure has been shown to depart from the long range structure (Petkov et al., 2000; Jeong et al., 2001; Peterson et al., 2001; Kim et al., 2007). High energy, high brilliance sources coupled with fast read out area detectors (Chupas et al., 2007b) allow diffraction patterns suitable for PDF analysis to be collected in seconds, making extreme conditions (Ehm et al., 2007; Chapman et al., 2010) and time resolved studies possible (Xu et al., 2011). Recently, the PDF analysis of x-ray total scattering data has been successfully applied to nanoparticles (Neder and Korsunskiy, 2005; Korsunskiy et al., 2007; Billinge, 2008; Petkov, 2008; Billinge et al., 2010). A PDF can be divided up, somewhat arbitrarily, into three parts: the local, medium and long range structures. The local structure, below ~5 Å, gives information on the first few coordination spheres. The medium range structure ranges from ~ 5 Å to ~ 15 Å and the long range structure involves correlations greater than this. In the PDF of a crystalline material, the long range structure should agree with the average structure determined by a single crystal or Rietveld refinement. The PDF of a crystalline material tails off at high r due to the finite resolution of the instrument (Proffen et al., 2003). For nanoparticles, the long range structure is either absent of minimized due to the small size of the particle under investigation; the longest observed correlation corresponds to the diameter of the nanoparticle

Fig. 4. PDFs calculated from the three diffraction patterns in Fig. 1 using a Qmax of 35 Å-1. The difference in the height above the zero line indicates the number of neighbors at a given distance.

(Page et al., 2011). It is here that the shape of the particle manifests. Fig. 4 shows the Fourier transform of the three hematite phases from Fig. 1, calculated using a Qmax of 35 Å − 1. There is very little difference between the three PDFs in the local structure and not much difference below 10 Å. Above this the intensities of the peaks differ from one PDF to another due to the shape of the nanoparticle, although the position of each peak remains at the same value of r. Contrary to the x-ray pattern in diffraction space, Fig. 1, it is trivial to see that these PDFs are from particles with the same underlying structure. Petkov et al. (Petkov et al., 2009a) investigated crystalline, spherical nanoparticulate and tetrapod nanoparticulate forms of maghemite (γ-Fe2O3). They found that the shape of the nanoparticle affected some peaks more than others and postulated that this may be a way to investigate nanoparticles of unknown shape. 5. The importance of medium range structure X-ray absorption spectroscopy (XAS) and especially extended xray absorption fine structure (EXAFS), a technique widely used in the study of nanoparticles (Bargar et al., 2008; Sunil et al., 2009; Jeong et al., 2010), provides element specific information on the local structure (b~5 Å). The local environment of an ion, around the first and second coordination shells, can be very similar in a number of closely related structures. For example in iron oxides and iron oxide–hydroxides, if the coordinating species, oxidation states and coordination number are the same, the bond length will not deviate much from structure to structure, as stated by Pauling's second rule (Pauling, 1929) and elaborated on by bond valence theory (Brown and Altermatt, 1985). Thus, from the limited information provided by EXAFS it can be difficult to distinguish between structural models. Fig. 5 shows an experimental PDF of ferrihydrite, collected at beamline 11-ID-B at the Advanced Photon Source (APS) with x-rays of λ = 0.2128 Å and a Qmax of 24 Å − 1 used in the Fourier transform. The calculated model is that of goethite, α-FeOOH, fit to the data with a least squares algorithm using the program PDFgui (Farrow et al., 2007). The fit to the local structure, below 5 Å, is reasonable (Rw = 25%); the intensities are not a perfect match, but the peak positions match the data well. If this was the only data available, it may be enough to give encouragement that the structure model for goethite would provide an excellent starting model for further refinement (Toner et al., 2009). Inspection of the fit at medium range interatomic distances (5 b r b 15 Å) shows that this initial impression that goethite is a reasonable starting point is not justified: the fit beyond 5 Å is extremely poor (Rw = 97%). It was

Fig. 5. PDF of ferrihydrite, split into short (blue) and medium range (dark blue) structure. The calculated phase is that of goethite; the short range fit (red) is reasonable, while the medium range fit (grey) is poor.

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emphasis on this medium and long range order, revealed by high energy x-ray PDF that enabled the solution of the structure of ferrihydrite (Michel et al., 2007a, 2010a). It should be noted that, unlike PDF, XAS is element specific and sensitive to low concentrations of dopants. In this way, XAS and PDF are complementary, rather than competitive techniques. 6. Examples Recent success stories in the use of total scattering data include the work of Michel et al. who showed that ~ 2 nm FeS, both freshly precipitated and aged under hydrothermal conditions, exhibit the mackinawite structure (Michel et al., 2005). Further, this study showed that the PDF can be used to place limits on the likelihood of a multiphase model (not likely) and used the damping of the PDF to allow the in situ determination of particle size in this extremely air sensitive material. With the current excitement over mackinawiterelated superconductors (McQueen et al., 2009), this system is likely to receive much attention in the future, especially if the composition FeS turns out to be superconducting. The PDF technique has been used to discriminate between phases of TiO2 nanoparticles. The stable phase of TiO2 at ambient conditions is rutile. This changes as the crystallite size decreases; below about 10 nm anatase is the stable phase. Zhang et al. (2008) investigated the structure of nano amorphous TiO2 using PDF and EXAFS spectroscopy. They found that the particles contained a small, strained anatase core below a highly distorted shell. The average Ti–O bond length was 1.940 Å and the average Ti coordination number was 5.3, rationalized by the truncation of the Ti–O octahedra at the nanoparticle surface. Petkov et al. applied this technique to bacterial and fungal produced nanoparticulate manganese oxide (Petkov et al., 2009b). Nanomaterials produced in such a manner are of potential commercial interest due to the microbes' ability to produce phases which are difficult to prepare using conventional routes. It was found that bacterial and fungal routes produce nanoparticles of different phases, birnessite-type layers and todorokite-type tunnels, respectively, determined by comparing the PDFs of the nanoparticles to PDFs of standard materials and seeing if the peaks matched the position of the peaks in the unknown PDF. This is similar to the standard procedure of matching powder diffraction patterns of unknown phases to patterns from a database such as the powder diffraction file. Due to the considerations discussed earlier, the matching of diffraction patterns of nanoparticles to a crystalline standard is not trivial using patterns in diffraction space, but is much simpler when the pattern is viewed in real space as a PDF. Using high energy x-ray diffraction with pair distribution function analysis, the structure of the nanoparticle grown inside protein cages can be determined in situ. Michel et al. (2010b) used ferritin to template the growth of iron oxyhydroxide nanoparticles. They found that as the number of Fe atoms per cage increased, the peaks in the diffraction pattern became sharper. The PDFs for these samples showed the same local and intermediate structure for each loading level, the difference showing up in the longer range structure as a change in grain size. From the PDFs, the authors concluded that the iron oxyhydroxide phase grown in ferritin has the same structure as inorganically derived ferrihydrite (Michel et al., 2007a). Jolley et al., (2010, 2011) performed analyses on a range of biologically produced particles, including environmentally significant phases ferrihydrite, γ-Fe2O3, FeS, Mn3O4 and TiO2. It was found that although the use of larger protein templates produced nanoparticles of greater size, often the size of the nanoparticles did not match the size of the cage. This is explained by there being several nucleation sites at the walls of the protein with several grains growing simultaneously and independently. Rather than a single crystal per cage, several grains grow inside each cage. Recently, use of differential pair distribution function (d-PDF) analysis (Kramer, 2007) has allowed study of very small changes

Fig. 6. From Harrington et al.(2010), showing the d-PDF of arsenate sorbed on ferrihydrite. All distances are in Å. Figure reprinted with permission.

within a sample. For example, Chapman et al. used this technique to determine that hydrogen, when stored in Prussian Blue, is associated with the pore space rather than with an manganese ion (Chapman et al., 2005, 2006). The d-PDF involves subtracting the PDF of a reference material, such as unloaded framework, from that of a similar material, or the same material under slightly different conditions, such as a gas loaded framework. The difference represents vectors between the framework atoms and the sorbed molecules. It should be noted that this technique differs from experiments taking advantage of anomalous scattering or neutron isotope effects. Chupas et al. investigated the kinetics of platinum nanoparticles growing on titania support using the d-PDF technique. This allowed only the Pt–Pt correlations to be recovered (Chupas et al., 2007a, 2009). In this way, they were able to follow the reaction in situ and ex situ from the first deposition of Pt on the substrate, to the growth of face-centered-cubic Pt nanoparticles. This study suggests that the initial formation is of small (b1 nm) particles, followed by agglomeration of these nanoparticles, followed by annealing to form larger, well ordered particles. The d-PDF method can also be applied to surface species. Waychunas et al. (1996b), using a laboratory Mo source, studied arsenate sorption on 2 nm ferrihydrite. As–O and Fe–As correlations were observed; the Fe–As distance matched that observed in previous EXAFS studies (Waychunas et al., 1993, 1996a; Sherman and Randall, 2003), confirming bidentate binuclear coordination. More recently, Harrington et al. (2010) used synchrotron radiation to investigate the binding mechanism and observed not only nearest neighbor As–O and Fe–As correlations, but also As–O correlations from further in the particle and the second nearest neighbor Fe–As peak, Fig. 6. 7. Summary With the growing interest in natural nanoparticles, an appreciation of their diffraction properties is critical. The main points of this article can be summarized as follows: • The broad diffraction features arising from a nanoparticulate sample are not simply the Bragg peaks broadened. Diffuse scattering, scattering not arising as a consequence of long range order, is present in every diffraction pattern but is of similar intensity to Bragg scattering in the diffraction pattern of nanoparticles below ~ 10 nm. Much information is contained within this scattering and any analysis which disregards it is incomplete. • The Debye equation, a sum over all scatters in the sample, can be used to calculate all scattering, Bragg and diffuse, and is a much

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more rigorous way to analyze the diffraction pattern of nanoparticulate samples. • The pair distribution function (PDF), the Fourier transform of the total scattering — Bragg and diffuse, provides a convenient and conceptually simple method of viewing the diffraction pattern. As short and medium range peaks are not highly affected by particle shape and size, it is much easier to compare the diffraction pattern of a nanoparticle with a reference diffraction pattern (arising from a real sample or calculated) using this real space technique. • When carrying out a diffraction experiment for PDF analysis, it is imperative that diffraction patterns of nanoparticles be collected to high values of diffraction vector, Q, as the diffuse scattering has very little Q dependence.

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