International
Tables for Crystallography Volume I Xray absorption spectroscopy and related techniques Edited by C. T. Chantler, F. Boscherini and B. Bunker © International Union of Crystallography 2024 
International Tables for Crystallography (2024). Vol. I. ch. 8.9, pp. 979987
https://doi.org/10.1107/S1574870720004723 Nanoclusters^{a}Department of Materials Science and Chemical Engineering, Stony Brook University, Stony Brook, NY 11794, USA,^{b}Department of Physics, University of Washington, Seattle, WA 98195, USA,^{c}Department of Chemistry, University of Illinois, Urbana, IL 61801, USA, and ^{d}Division of Chemistry, Brookhaven National Laboratory, Upton, NY 11973, USA Metal clusters comprise an important class of functional nanomaterials due to their tuneable size, structure, shape and support, which are key factors that affect their physicochemical properties and functions. In cluster research, the main objectives are the design of new materials with the desired properties and the development of new methods for characterizing these materials. In this chapter, methods of extended Xray absorption finestructure (EXAFS) analysis developed in the last two decades for the structural characterization of monometallic and bimetallic nanoclusters are reviewed. Keywords: Xray absorption; monometallic and bimetallic nanoclusters; size, shape and composition determination. 
Clusters with sizes ranging from subnanometre to 10 nm in diameter have attracted broad interest due to their nonbulklike properties. For example, the continuous band structure of metals breaks up into discrete electronic states when the particle size approaches the Fermi wavelength of an electron (Johnston, 2002). For small clusters, multiple absorption peaks are assigned to singleelectron intraband resonances, and due to this some clusters exhibit strong fluorescence emission upon UV photoexcitation, which makes them good candidates for biolabels and lightemitting sources (Lin et al., 2009). These properties of small clusters cannot be simply described by scaling laws, and more complicated sizedependent relationships should be considered. Other important physical parameters that do not have analogues in bulk crystalline counterparts are geometry, electronic/atomic structure, shape and composition, all of which have a significant influence on cluster properties.
Additionally, the properties of clusters can be affected by their surroundings, such as ligands, organic solvents, supports and gases, and this environment is often sensitive to, and thus can change as a result of, the operating conditions. Such a sensitivity to the surroundings provides opportunities for producing materials with novel properties (Gates, 1995) and, simultaneously, provides challenges for both experimental and theoretical methods in linking the properties of clusters to their size, shape, structure and composition. For example, electronmicroscopy (EM) methods are commonly utilized to image clusters, but their resolving power greatly diminishes when timeresolved measurements (Zheng et al., 2009) or/and realistic catalytic environments (Hansen et al., 2002) are required. These requirements pose the same challenges to Xray photoelectron spectroscopy (XPS), although the availability of highflux Xrays based on synchrotron radiation and recent advances in instrumentation have enabled XPS measurements at ambient pressure (Tao et al., 2008) and at the liquid–solid interface (Brown et al., 2013).
Other techniques such as Xray diffraction and smallangle Xray scattering can also provide information about particle size, but those methods work best for particles with sizes larger than several nanometres. Xray absorption spectroscopy (XAS) can easily adapt to in situ/operando modes (Frenkel et al., 2014). Compared with EMbased techniques, which probe a few hundreds to thousands of particles, XAS provides excellent statistical information owing to the large fluxes (10^{10}–10^{13} photons per second interrogating the sample) available at today's synchrotrons. XAS detects local structure, which makes it ideal for the investigation of small (less than 2–5 nm in dimeter) clusters where large disorder is expected.
In the rest of this chapter, we will review XAS methods (owing to space limitation, we focus on the extended Xray absorption fine structure only) developed in recent decades for characterizing the structure of monometallic and bimetallic clusters.
Fouriertransformed (FT) EXAFS spectra exhibit one or more peaks corresponding to the contributions of different photoelectron paths that connect the absorbing atom and its neighbouring atoms. In the most commonly used form of EXAFS analysis, these paths are characterized by three structurerelated parameters that are unique for each path and can be quantified by fitting the theoretical EXAFS spectrum to experimental data. They are the coordination number (CN), the bond distance (R) and the meansquared bondlength disorder, also known as the EXAFS Debye–Waller factor (σ^{2}). Among these the CN is a key parameter that is used for the determination of the size and shape of well defined clusters. It is defined asHere, N_{A} is the total number of Atype atoms in the cluster, N_{AA(i)} is the total number of A–A nearestneighbour pairs within the same coordination shell and n_{i} is the coordination number of the ith shell at radius R_{i} around the absorbing atom in a monometallic cluster.
For small monometallic clusters, EXAFS spectra usually show one prominent peak in rspace located between 2 and 3 Å. This peak, which depending on the type of absorbing atom can be split into two or more peaks owing to Ramsauer–Townsend resonance (McKale et al., 1988; Rehr et al., 1994), corresponds to the first nearestneighbour (1NN) metal–metal bond. Basing on the CN of this bond, several methods are available for size determination. One such method was proposed by MontejanoCarrizales et al. (1997), who found that N_{AA}, the total number of atoms N_{A} and the coordination number n_{1} of regular polyhedra can be analytically expressed as a function of cluster order (L), which is defined as the number of spacings between adjacent atoms along the edge of the cluster. By comparing n_{1} obtained from EXAFS analysis against model structures with known geometrical characteristics (Table 1 and Fig. 1), the size/geometry of the clusters under investigation can be estimated.

Calvin and coworkers developed another useful method to approximate the size of spherical clusters (Calvin et al., 2003). In this method, the average CN of the ith shell (n_{i}) for the cluster can be written asIn equation (2), N_{i} is the ithshell CN of the bulk structure, r_{i} is the scattering path length for the ith shell and R is the radius of the cluster. The advantages of this approach are (i) the cluster radius can be directly extracted from the fit and (ii) it also takes into account single scattering paths at higher distances. Indeed, no matter how narrow the cluster size distribution is, the size/geometry of the clusters cannot be obtained only with knowledge of the first nearest CN because of the correlation between the cluster size and shape in terms of their effects on the 1NN coordination numbers. To address this issue, Frenkel and coworkers proposed the utilization of the different functional behaviours of the 1NN, 2NN, 3NN, 4NN and 5NN coordination numbers as a function of cluster size for studying ideal polyhedral clusters (Frenkel, 1999; Frenkel et al., 2001; Nashner et al., 1997). For instance, knowledge of the 1NN CN is not sufficient to distinguish between a 55atom icosahedron and a 79atom truncated octahedron. The degeneracy of the 1NN CNs will be lifted when comparing n_{2} or even n_{3} owing to their uniqueness for each of these two geometries, as shown in Fig. 1(b).
For irregular geometries of relaxed clusters, analytical calculations cannot be performed. Frenkel and coworkers proposed the use of a histogrambased method in which a radial distribution function (RDF) of nearestneighbouring shells is calculated for any given set of atomic coordinates and the CNs are then obtained by integrating the RDF within the shells of interest (Frenkel et al., 2005; Glasner & Frenkel, 2007). This method is not limited by the shapes and symmetries of a very small number of regular polyhedral clusters, making it a very robust strategy for size/geometry determination.
When the analysis of EXAFS data extends beyond the 1NN contribution, multiplescattering (as demonstrated in Fig. 2) effects should be included in the model. Using multiplescattering analysis, Frenkel and coworkers were able to identify the icosahedral geometry of a monolayerprotected Au_{13} cluster (Frenkel et al., 2007; Menard et al., 2006) and determine the shapes of well defined supported nanoclusters (Frenkel, 1999; Frenkel et al., 2001; Roldan Cuenya et al., 2010).
Jentys also proposed the estimation of the mean size and shape of clusters by comparing a set of CNs obtained by EXAFS analysis with those of models with different geometries (Jentys, 1999). This method uses a hyperbolic function to correlate the relationship between the average CN of the ith (1 ≤ i ≤ 5) shell and the total atom number in an f.c.c. cluster,Using the known information on 13atom to 7500atom clusters, a, b, c and d were quantified by applying a nonlinear leastsquares fitting. The particle shape was found to have a minor influence on the CNs of the first and second nearest shells but a significant influence on those of the higher shells. Therefore, one can first estimate the cluster size by using n_{1} and further determine the cluster shape by comparing the ratio of n_{3} to n_{1}. A similar idea has been extended by Beale and Weckhuysen to a larger number of atom packings and shapes with the use of the Hill function (Beale & Weckhuysen, 2010).
Heterometallic clusters composed of two or more metals are of great interest in a broad range of fields (Ganguly et al., 2013; Sasaki et al., 2011; Sun et al., 2000; Tao et al., 2008). The most important factors affecting their properties are the size, morphology and mixing patterns of the different elements in the cluster. Several analytical methods developed based on EXAFS data for characterizing heterometallic nanoclusters will be described here.
In nanoclusters, just as in bulk alloys, one should discriminate between homogeneity and randomness. Both characteristics can be carefully characterized by EXAFS (Frenkel, 2012; Frenkel et al., 2013). Different types of bimetallic configurations are demonstrated in Fig. 3. Compared with the cluster in Fig. 3(a), which has perfect shortrange (and longrange) order, the atomic distribution is random for the cluster in Fig. 3(b). For such random alloys (in which A and B are mixed statistically), n_{AA} and n_{AB} have the same ratio as the bulk concentrations of A and B atoms in the cluster,In equation (4), analogously to the definition of the coordination number for a homometallic pair, the coordination number for heterometallic bonds is defined asIn addition, it is also important to compare the average CN of the A–metal (AM) pairs n_{AM} with that of the B–metal (B) pairs n_{BM}. These indices are defined as n_{AM} = n_{AA} + n_{AB}, n_{BM} = n_{BA} + n_{BB}.
By analysing EXAFS data, the CNs of AM and BM pairs can be obtained, and information about the patterns of mixing or segregation of alloying elements can be analysed by modelling. If n_{AM} < n_{BM}, and if the cluster size and composition distribution is narrow, then this inequality points to the preferential location of A atoms near the surface, with smaller numbers of nearest neighbours, while B atoms are preferentially located in the cluster core, where the CN of the nearest neighbours is larger.
For a more formal characterization of segregation or mixing tendencies, including characterization of the randomness of well mixed alloys, we introduce a shortrange order parameter α (Frenkel, 2012; Frenkel et al., 2013), similar to its definition by Cowley for bulk alloys (Cowley, 1950), As shown in Fig. 3, α (−1 ≤ α ≤ 1) can be applied to clusters with different degrees of homogeneity and randomness. For alloys that favour or disfavour clustering of like atoms, α will be positive or negative, respectively. In two dimensions, it is −1 for systems with perfect order, zero for random alloys and 1 for systems without the formation of a heterometallic bond. This parameter is therefore essential for the characterization of nanoalloys, such as coreshell, random or clusteroncluster types.
In addition to the composition pattern, the cluster size can be determined by methods similar to those described above for monometallic particles with the knowledge of the average number of metal–metal neighbours per metal atom,Another similar method to estimate the atomic distribution in a bimetallic cluster has been proposed by Hwang et al. (2005). In this method, several parameters were defined:
The extents of alloying of element A (J_{A}) and B (J_{B}) were given by
Several cases were discussed and here we list five of them.
(i) J_{A} = 0 and J_{B} = 0 → separated homometallic clusters.
(ii) J_{A} = J_{B} = 100% → perfectly alloyed clusters.
(iii) J_{A} < 100% and J_{B} < 100% → partial A and B atoms tend to segregate; in this case, if J_{A} < J_{B} it indicates an Arich core and a Brich shell.
(iv) J_{A} > 100% and J_{B} < 100% → both A and B atoms prefer B rather than A.
(v) J_{A} > 100% and J_{B} > 100% → higher ratio of heterometallic bonds than homometallic bonds.
To reliably obtain the CNs of metal–metal bonds, the EXAFS fitting should be performed concurrently for both absorption edges with obvious constraints imposed on the heterometallic bonds,
To analyze highershell data, it is necessary to add the multiplescattering path contributions, which may be comparable to the amplitude of the singlescattering paths.
Some bimetallic systems contain two elements, for example Ir and Pt, that are close to each other in the periodic table. In this case the EXAFS region of the Ir edge spectrum intrudes into the spectrum of Pt. This phenomenon could be extended to any materials with adjacent absorption edges, for instance, the Ti K edge and Ba L_{3} edge of perovskite, BaTiO_{3}. With overlapping XAFS signals, accurately analysing the XAFS data of these systems could be problematic. Separating these EXAFS signals can be performed experimentally in selected cases. As an example, Ravel and coworkers used diffraction anomalous finestructure (DAFS) measurements to separate the finestructure signals from barium and titanium (Ravel et al., 1999). In another approach, Glatzel and coworkers used highenergy resolution fluorescence detection, which enables the separation of emission lines from different elements (Glatzel et al., 2005). Here, we introduce an analytical method that can be broadly applied to any combination of elements that have overlapping absorption edges in order to deconvolute their EXAFS signals analytically.
Menard and coworkers reported a new method and successfully deconvoluted the XAFS signals of overlapped Ir and Pt L_{3} edges (Menard et al., 2009). The overlapped XAFS signals could be split into three parts: (i) the Ir EXAFS in the Pt L_{3} edge before the Pt L_{3} edge, (ii) the Ir EXAFS in the Pt L_{3} edge and (iii) the Pt EXAFS in the Pt L_{3} edge. These three contributions could be described by the EXAFS equationsand
In the process of fitting, appropriate constraints and strategies should be applied.
(i) The nonlinear leastsquares fitting of equations (11) and (12) to experimental data should be performed concurrently.
(ii) The factor A = Δμ_{0,Ir}/Δμ_{0,Pt}, where Δμ_{0,Ir} and Δμ_{0,Pt} are the changes in the absorption at the edge steps, is necessary in fitting because the extraction of χ(k) includes a normalization to these edge steps.
(iii) The correction to the threshold energy (in eV) for the Ir EXAFS at the Pt L_{3} edge should be defined as ΔE_{0,Ir} − (349 + ΔE_{0,Pt}), where 349 eV is the difference between the empirical threshold energies. Such a large energy origin shift is necessary in this method since it accounts for a unique k = 0 reference point for the Ir EXAFS extending beyond the Pt edge when the Pt edge EXAFS is transformed to kspace.
(iv) Constraints should be made to Ir–M paths because the Ir EXAFS in the Ir L_{3} and Pt L_{3} edges describe the same coordination environments of Ir atoms.
The artefacts in EXAFS analysis arise owing to the broad range of sizes and compositions of cluster ensembles, and the corrective strategies that should be undertaken are described below. Heterogeneity of bondlength and compositional distributions is a common situation that occurs not only in each individual cluster but can be present due to the changes between multiple clusters. In addition, theoretical calculations reveal the importance of dynamic structural disorder (a contribution of the lowfrequency component to the bond dynamics of clusters that may cause shape changes and even mobility over a several picosecond time scale) for the catalytic activity of metal clusters (Rehr & Vila, 2014).
For small clusters, the effects of capping ligands (Carter et al., 1997), steric effects (Shiang et al., 1995), crystalline defects (de la Rubia & Gilmer, 2002) and interaction with adsorbates (LópezCartes et al., 2005) could result in imperfection in the crystalline lattice of clusters. The enhanced surface tension causes a decrease in the lattice parameter (Mays et al., 1968). The bonds near the surface of nanoclusters are more strained than those inside the core, which accordingly results in a strong variation in the interatomic distances between outside and inside bonds (Huang et al., 2008). Such nonGaussian or interparticle disorder would cause a problem in EXAFS analysis if it is not considered. Yevick & Frenkel (2010) examined the effects of surface disorder on EXAFS modelling of metallic clusters with 147 and 923 atoms. They applied the empirical distortion function f(r) to each atom within the clusters to simulate surfacetension effects:This distortion function satisfies the conditions f(0) = 1 and f(R) = B ≤ 1 for the atoms at the centre and the periphery, respectively. Multiplying the radial distortion function (equation 13) to all atomic coordinates yields the new positions of atoms in the distorted cluster. The parameter A (1.00001 ≤ A ≤ 1.05) corresponds to the curvature of the distortion curve to simulate the uniformly (Woltersdorf et al., 1981) and nonuniformly distorted structures (Huang et al., 2008). The B parameter lies in the range 0.95–1.0, which agrees well with physically reasonable bondlength truncation effects in small clusters. Fig. 4(a) shows the distortion function f(r) with different curvatures A as a function of r/R. Different distortion functions represent different bondlength distributions of the first nearestneighbour bonds or different radial relaxation of the surface tension within a cluster. The results exhibited the enhanced surface disorder in metal clusters (less than 5 nm). If such disorder is unaccounted for in analysis, it may result in a significant underestimation of the particle size (Fig. 4b) and an overestimation of the nearestneighbour distances. To minimize the errors in the analysis because of surface relaxation, one can passivate the cluster surface with H_{2}, which increases the bulklike order in the clusters (Kang et al., 2006; Sanchez et al., 2009), or use complementary techniques, such as the pair distribution function (PDF; Chapman & Chupas, 2013), when the asymmetric disorder causes artefacts in the conventional EXAFS analysis. In 2018, a new method of mapping the EXAFS spectra onto the pair radial distribution function g(r) of neighbours was developed based on the artificial neural network approach (Timoshenko et al., 2018). Utilizing this method, Timoshenko, Frenkel and coworkers obtained the characteristics of monometallic and bimetallic nanoparticles directly from g(r) and observed more detailed information about placement of dilute Pd atoms in the goldrich nanoparticles than is available through coordinationnumber analysis (Timoshenko et al., 2019).
Other than the intraparticle disorder, the interparticle disorder also affects the EXAFS results. The general understanding is that the CNs derived from EXAFS analysis overestimate the average cluster size if the size distribution of clusters is broad. To interpret it, Frenkel et al. (2011) consider a symmetric arbitrary distribution ρ of cluster order (L) approximated as the Gaussian function,where is the average cluster order and σ_{s} is the standard deviation in L. The average coordination numbers are calculated over all clusters with the size distribution ρ(L):Equation (15) contains a weighting factor, N(L), which indicates the number of atoms in a cluster of order L. This correction is required because ρ(L) is commonly obtained by electronmicroscopy measurements as a frequency distribution that depends on cluster size, not volume, whereas volume averaging is required for CN measurements by EXAFS. To estimate the effect of ρ(L) on the average CNs, clusters with cuboctahedral geometry are assumed. The calculated average CNs for various values of as a function of σ_{s} are plotted in Fig. 5. If σ_{s} = 0 (all clusters are identical), = n_{0}, the CN in each cluster. When σ_{s} increases > n_{0} because the larger clusters contribute more to the average than the smaller clusters. For systems with a relatively narrow size distribution (σ_{s} < 0.5 for small clusters and σ_{s} < 1 for larger clusters) the EXAFS predictions are not significantly affected (Fig. 5). Frenkel et al. (2011) also examined the effects of σ_{s} on r and σ^{2}, and observed that even for very poorly defined cluster sizes, with σ_{s}/L ≃ 2/3 and typical choices of r(L), the resulting corrections do not exceed 0.015 Å for r and 0.00015 Å^{2} for σ^{2}.
In summary, the following guidance can be offered for the interpretation of the CNs in clusters. (i) If the distribution of particles sizes is known, the theoretical average CN can be obtained by applying equation (15). (ii) For a quasiGaussian distribution, the average CN can be determined from Fig. 5 for different values of σ_{s}. (iii) If σ_{s} is incorrectly assumed to be too narrow, the EXAFS CNs overestimate the mean cluster size. (iv) A combination of EXAFS and microscopic analyses is necessary to find out whether the models proposed based on each method agree with each other (Agostini et al., 2014). If the EXAFSderived cluster size is found to be outside the measured size distribution, such a discrepancy suggests that there must be other factors that have not yet been considered, for example the presence of ultrasmall clusters that cannot be directly measured by electron microscopy, or that the wrong model was chosen to describe the EXAFS results. (v) The effects of σ_{s} on r and σ^{2} are relatively weak compared with those on the CNs.
Next, we discuss the effects of compositional disorder on the CNs of heterometallic nanoclusters determined by EXAFS analysis. For demonstration, a simple system which contains clusters of the same size but of different compositions is considered. We will assume the distribution of compositions across the ensemble of clusters to be a Gaussian,where x = N_{A}/N is the fraction of A atoms in each cluster, is the average composition over all clusters and σ_{c} represents the standard deviation in the distribution of ρ(x). Furthermore, assuming a random ordering of the atoms within each cluster, the partial CN n_{AA} in each cluster is defined aswhere N_{A} is the number of A atoms in the given cluster and N is the total number of atoms in each cluster. Compared with the equations used by Hwang et al. (2005) and Frenkel (2007), which require that n_{AA} = xn_{MM} for randomly distributed atoms within the clusters, equation (17) is more general, is accurate for all clusters and correctly calculates the CNs over the entire compositional range. The exact formulas for the partial CNs, averaged over the ensemble of clusters, areFor clusters with 100 atoms, calculated partial CNs which have been normalized by n_{MM} are shown in Fig. 6. The results indicate that the ensemble average can be smaller for narrow compositional distributions, or larger for broad distributions, than the CNs predicted by the equation n_{AA} = xn_{MM}. The two sets of values, and n_{AA}, agree for , for which the normal distribution coincides with the binomial distribution. To summarize, for bimetallic systems with broad composition distributions, corrections can be applied to characterize the equivalent cluster of the samples.
An important limitation in the analysis and modelling of EXAFS data is the ensembleaveraging nature of the technique. A possible remedy is to employ singlenanoparticle spectroscopy (Nie & Emory, 1997; Xu et al., 2008). Xu and coworkers demonstrated the benefit of singlenanoparticle spectroscopy over ensembleaveraging techniques by studying catalysis by a single gold nanoparticle at the singlemolecule level. They discovered two distinct nanoparticle groups (about 6 nm) which show different catalytic activities (Xu et al., 2009). To develop Xray spectroscopy at the singleparticle level, one of the key challenges is the design of the probe. Hitchcock and Toney have published an overview of spectromicroscopy methods that include nanoprobebased systems (Hitchcock & Toney, 2014). Challenges to the existing nanoprobe methods have been summarized by Frenkel & van Bokhoven (2014).
Characterization of nanoclusters in situ or in operando conditions is now one of the most important requirements for modern applications of synchrotronbased spectroscopic methods. Accordingly, there are growing demands in developing timeresolved and operando techniques. For instance, to bridge the `pressure gap' between spectroscopic and electromicroscopic techniques, Frenkel, Stach and coworkers demonstrated the advantage of using a microreactor for nanocatalysis studies at ambient temperature and pressure (Li et al., 2015; Zhao et al., 2015). The catalytically active site is often only a minority of the atoms in the catalysts, and sensitivity is therefore an issue. Ferri and coworkers employed modulation excitation and filtering of the corresponding spectra with the excitation frequency to increase the sensitivity to the differences in the spectra, which provides a better chance of capturing the structure of the active site selectively (Ferri et al., 2011). Intermediate species can be detected by a phase delay of the structural difference. Pump–probe measurements are able to capture structural variation on microsecond, nanosecond, picosecond and even femtosecond time scales (Bressler & Chergui, 2004). In the literature related to this technique, most such measurements involve a photocatalytic process during which a flash of light induces electron excitation, after which a structural change occurs and is detected in an ultrafast time domain (Bressler et al., 2009; Smolentsev et al., 2014).
To summarize, nanoclusters have been studied for several decades with an increasing accuracy and level of detail that parallel the development of analytical methods. Of these methods, the Xray absorption spectroscopy technique has proved to be uniquely capable of capturing multiple attributes of structure and measuring their dynamic changes in real time owing to externally controlled conditions. The combination of the ongoing developments of clustersynthesis methods, new synchrotron nanoprobe methodologies and the increased use of in situ/operando characterization techniques are promising new directions for future progress in this field in the next decade.
Funding information
AIF and RGN gratefully acknowledge support for this work from the US Department of Energy, Office of Basic Energy Sciences under grant No. DESC0022199. YL and JJR gratefully acknowledge support for this work from the US Department of Energy, Office of Basic Energy Sciences under grant No. DEFG0203ER15476.
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