Comparison between X-ray Absorption Spectroscopy, Anomalous

11040

J. Phys. Chem. B 1997, 101, 11040-11050

Comparison between X-ray Absorption Spectroscopy, Anomalous Wide Angle X-ray Scattering, Anomalous Small Angle X-ray Scattering, and Diffraction Anomalous Fine Structure Techniques Applied to Nanometer-Scale Metallic Clusters D. C. Bazin,*,† D. A. Sayers,‡ and J. J. Rehr§ LURE, UniVersite´ Paris XI, Baˆ t 209D, Orsay, 91405, France, Department of Physics, North Carolina State UniVersity, Raleigh, North Carolina 27695-8202, and Department of Physics, Washington UniVersity, Seattle, Washington, 98195 ReceiVed: July 1, 1997; In Final Form: October 9, 1997X

The emergence of the synchrotron radiation center in the past decade has resulted in a tremendous development of various techniques. XAS is an efficient tool for describing local organization in materials such as nanometerscale metallic clusters but is insensitive to polydispersity. Ab initio calculations of the diffraction diagram show clearly that the crystallographic network as well as the distribution of the two metals inside the species can also be found. Near the diffraction peak (0,0,0), information regarding the size distribution and the morphology as well as the distribution of the metals inside the cluster can be obtained through a so-called Debye function analysis. For the DAFS technique, a wedding between diffraction and absorption, ab initio calculations will allow us to underline its advantages and its limitations. Finally, a synthesis of all the information given by these techniques is made in order to define a methodology in the building of a structural model.

Introduction X-ray methods can be applied to a wide range of scientific disciplines from basic physics to material science. One can think of two fundamental channels for the photon-matter interaction: the absorption (the energy of the photon is lost within the target) and the scattering processes, either elastic (Thomson for free electron) or inelastic (Compton). The interaction of a photon with nuclei is out of the X-ray range as considered here, namely, between 3 and 25 keV. The emergence of a synchrotron radiation center in the past decade has resulted in a tremendous development of various techniques, i.e., the X-ray absorption spectroscopy (XAS), the anomalous wide angle X-ray scattering (AWAXS), the anomalous small angle X-ray scattering (ASAXS), and the diffraction anomalous fine structure (DAFS), which use some of the particular properties of the beam such as the wide spectral range. Up to now, only experimental studies have been reported that take into account the development of the different experimental setups at several synchrotron radiation centers. In this report, through ab initio calculations, we discuss the information given by each of the techniques and indicate their advantages as well as their disadvantages when nanometer-scale mono- or multimetallic clusters are considered. Finally, a synthesis is made in order to define a methodology in the building of a structural model concerning the metallic part of the material. I. X-ray Absorption Spectroscopy (XAS) For the absorption process, when the X-ray photon energy (E) is tuned to the binding energy of some core level of an atom, an abrupt increase in the absorption coefficient µ(E), known as the absorption edge, occurs. For atoms, either in an molecule or embedded in a condensed phase, the absorption coefficient displays oscillatory variation as a function of photon †

Universite´ Paris XI. North Carolina State University. § Washington University. X Abstract published in AdVance ACS Abstracts, December 1, 1997. ‡

S1089-5647(97)02131-7 CCC: $14.00

energy. X-ray absorption spectra are commonly (and roughly) separated into several parts according to the spectral region. The so-called “near-edge X-ray absorption fine structure” (NEXAFS), also named the “X-ray absorption near edge structure” (XANES), is the structure beginning before the absorption threshold E0 up to about 40 eV (E0 - 100 eV to E0 + 40 eV) and the other one is called “extended X-ray absorption fine structure” (EXAFS: E0 + 40 eV to E0 + 1500 eV). If the first absorption spectra were recorded at the beginning of the century,1 the understanding of the physics process of the EXAFS part begins only two decades ago.2,3 The basic model is the following. The absorption of an X-ray photon by an atom ejects a photoelectron, which is scattered by neighbors. An interference process builds up between the wave function of the outgoing electron and its scattered parts, leading to a modulation of the absorption coefficient. Far beyond the absorption edge, the electron is assumed to be free. The scattering is weak and involves only one neighbor atom (single scattering approximation). I.1. L Edge of 5d Transition Metals. It is well-known that one can obtain information on the electronic state of the absorbing atom by studying white lines or threshold spikes that appear at the LII,III edges of the transition metals. These spikes have been explained qualitatively by Y. Cauchois et al.4 in terms of the high density of final states to which the transitions are made. From theoretical considerations, W, the probability per unit time for a transition from an initial state |i〉 to a final state 〈f| by the action of a perturbation (here, it is an X-ray photon) is given by the golden rule:5

W ) 2π|h|〈i|H|f〉/2F(Ef)

(1)

where F(Ef) is the density of final state. In the dipole approximation, the transition is restricted by the following selection rules: ∆L ) (1, ∆J ) 0, (1. Therefore, the LII X-ray absorption edge probes those final states that are characterized by a total angular momentum quantum number J ) 3/2 while the LIII edge probes those states with total angular momentum quantum number J ) 3/2 and J ) 5/2. In the case © 1997 American Chemical Society

Comparison of Techniques Applied to Clusters

J. Phys. Chem. B, Vol. 101, No. 51, 1997 11041

Figure 3. Size effect on the XANES spectrum for 13 Pt (line), 19 Pt (dots), and 43 Pt (dashes).

Figure 1. Intensity of the peak at the LIII edge (proportional to the d electron vacancies) in the case of two model compounds: the platinum and the gold metallic foils.

Figure 4. Rough comparison between the LIII edge of reference compounds such as the platinum foil and the alloy platinum-palladium. Figure 2. Comparison between the white line associated with the platinum foil: calculated with FeFF6 (dots); experimental values (line).

of platinum, the spin-orbit interaction has to be taken into account. Under some assumption, the magnitude of the splitting ESO is simply giving by ESO ) (5/2)ζd where ζd is the spinorbit parameter for d states. Basically, the 5d5/2 states are shifted toward higher energies while the 5d3/2 states are shifted toward lower energies. F. W. Lytle et al.6,7 have reported experimentally that the intensity of the peak at the LIII edge is proportional to the d electron vacancies, and Figure 1 illustrates this dependence in the case of two compounds: the platinum and the gold foils. Quantitative techniques for the determination of the number of unoccupied d electron states using the LII,III edges have been also developed.8,9 As pointed by S. Zabinsky et al.,10 the LIII platinum edge is generally reproduced in the framework of a muffin-tin multiple scattering approach by using large clusters of few hundreds of atoms in which the absorbing atom is always at the center of the cluster. The result is compared to the experimental spectrum for a 147 atom cluster (Pt147) in Figure 2, showing an overall agreement concerning both the white line and the other structures. In the case of a nanometer-scale metallic cluster, we have shown recently that a strong correlation exists between the intensity of the white line and the size of the cluster through ab initio calculation.11 The intensity of the white line varies slowly with the size of the cluster (Figure 3). Thus, at the opposite of numerous studies that measure electronic transfer through a comparison with the white line of the platinum foil, we show that assuming charge transfer is not necessary to explain the variations of the white line intensity. Indeed, the density of

state of the nanometer-scale cluster being far from the bulk one, the intensity of the white line has to be different anyway. When heterobonds exist inside the metallic cluster, a rough comparison between the LIII edges of reference compounds such the platinum foil and the alloy platinum-palladium, for example (Figure 4), shows that in fact all the absorption spectrum is perturbed by the substitution platinum-palladium around platinum atom. More precisely and as pointed out by E. K. Hlil et al.12 in the case of the platinum-based alloys, the modifications on the platinum LII,III edges can be interpreted as resulting from a variation of the number of holes in the 5d band, corresponding to an electron transfer to platinum from the 3d metal. Size effect (as shown in the case of the monometallic) as well as the composition of the coordination sphere can thus change dramatically the L edge. From an experimental point of view, the LIII white line is at the center of the electronic charge transfer either between the nanometer-scale metallic particle and the support or between the two metals that are present inside the cluster.6,7 Numerous studies have been done using the shape of the edge as a chemical probe to determine the chemical state of the metal.13 Either supported14 or embedded in polymer matrixes,15 metallic particles have been the object of many studies of either the K or L edge depending of the metal of interest. They were done after16 or during the reaction,17 the studies being done in situ or ex situ. For example, one of the first studies done by F. W. Lytle et al.18 has shown the effect of chemical environment on the magnitude of the LIII platinum white line. Finally, for the L edge, ab initio simulations have to go through a fine calculation of the density of states which cannot be performed yet by usual code. These calculations of the density of states are performed in a tight-binding model

11042 J. Phys. Chem. B, Vol. 101, No. 51, 1997

Bazin et al.

Figure 5. Near-edge part of the absorption spectrum of the copper foil that exhibits three main features A, transition 1s to 4p in the 3d10 configuration that belongs to the NEXAFS part, B, and C.

including spd hybridization, which is detailed elsewhere.19 Even if some tendencies are expected regarding the size of the cluster or the position of the atom inside the cluster, it is not possible yet to proceed to a significant quantitative analysis. I.2. K Edge of 3d Transition Metals. The XANES part of the K absorption spectrum of most elements of the periodic table can be simulated. Full multiple scattering calculations can be performed by using the CONTINUUM code developed by M. Natoli20 or the FEFF code written by J. Rehr,21 both codes having the same physics at the root with two different mathematical expressions. Since the information of interest is contained in the oscillations superimposed onto the otherwise smooth atomic absorption coefficient µ0, one can define χ(k) as

χ(k) ) [µ(k) - µ0(k)]/µ0(k) k ) [(2m/h2)Ec]1/2

(2)

where the atomic absorption coefficients are expressed as function of k, the module of the wave vector and Ec the difference between the energy of the photons and the electron binding energy. In the FeFF code, the oscillations are expressed as a sum of different multiple scattering contributions. Each contribution can be expressed in the following form

χn(k) ) χn0(k) exp(-Ln/λn - 2k2σ2n) χn0(k) ) Fn(k) sin(kLn + θn(k)) (3) Here, n represents different single or multiple scattering paths and Ln is the total path length. F and θn are the amplitude and phase that depend on k, on the specifics of the scattering path involved, and on the atomic potential parameters. Here, what we want is to discuss the nature of the information relative to this part of the spectrum. Let us consider the near-edge part of the absorption spectrum of the copper foil that exhibits three main features A, transition 1s to 4p in the 3d10 configuration, which belongs thus to the NEXAFS part, B, and C (Figure 5). This face-centered cubic (fcc) lattice is generally reproduced in the multiple scattering framework in the muffin-tin approximation by using large clusters in which an absorbing atom is always at the center of the cluster, the surface atom contribution to the total cross section being negligible. In the case of a 13 atom cluster, it is essential to consider each kind of atom inside the cluster as a signal coming from the surface, and the central atoms are definitely not the same. Clearly, a 13 atom environment is not enough to produce the resonance C (Figure 6). A similar approach was made for a 55

Figure 6. Near-edge part of the absorption spectrum as calculated with the FeFF program for clusters of 13 and 55 atoms.

copper atom cluster (Figure 6). The major point that has to be made is that the presence of the features B and C of the copper foil K edge are already present.22,23 Since the surface atoms have a non isotropic environment, features are present but with lower intensities than for the copper foil spectrum. Finally, regarding the structure A, as we have mentioned before for the L edge, this structure normally has to be band structure dependent. Thus, for the same reason we have given for the L edge, this feature is size dependent. These simple calculations on nanometer-scale copper clusters have been confirmed by different experimental results. For example, in the case of a copper cluster in a solid argon matrix24 similar XANES results have been obtained. More recently, the absence of features B and C have been measured by EXAFS in a study of copper clusters implanted in a AlN matrix.25 In conclusion, for nanometer-scale material, it is definitely not possible to simulate their XANES part for the K edge with a linear combination of the XANES of well-crystallized reference compounds. Such an analysis, called P.C.A. (principal component factor analysis), has been recently developed26 to follow the temperature-programmed reduction of copperpalladium bimetallic catalysts. It assumes that the absorbance in a set of spectra can be mathematically modeled as a linear sum of individual (uncorrelated) components known as factors. Such an approach can lead thus to a misunderstanding of the physical process. In the case of a copper cluster, ab initio calculations clearly indicate that a 13 atom environment is not enough to produce all the features present at the edge, and thus, the XANES part can be used as a fingerprint of the cluster size. I.3. Extended X-ray Absorption Fine Structure (EXAFS). The relation between the modulated part χ(k) and the structural parameters contained in the absorption spectrum, has been established in numerous theoretical studies2,3 and can be written as

χ(k) )

∑j Nj/kRj2fj(k) exp(-ΓRj/k) exp(-2σj2k2) × sin(2kRj + φj(k)) (4)

where k is the wave vector of the photoelectron, j refers to the different coordination shells around the absorbing atom, each shell containing Nj equivalent atoms. A Debye-Waller factor σj takes into account the fact that a spread in distance exists in material, and we assume that this distribution is Gaussian. Generally, the electron mean free path Γ(k) is introduced in order to reflect the probability that the electron is inelastically scattered



Figure 7. Modules of the Fourier transform of a metallic platinum foil taken at different temperatures.

by its environment. The backscattering amplitude Fj(k) is essentially the magnitude of the transform of the scattering potential, and its shape is a measure of the type of scattering atom. A Fourier transform (FT) of the EXAFS with respect to the photoelectron wavenumber peaks at distances corresponding to the nearest-neighbor coordination shells of atoms. Thus, between χ(k) and the radial distribution function around the central atom, there is just a simple FT relationship on which is based the whole analysis procedure. Conventional analysis of the EXAFS modulations contains several well-known steps,27 namely the subtraction of the background absorption and normalization, the building of the χ(k) function, the FT of χ(k) in order to obtain a pseudoradial distribution function, and finally the curve-fitting procedure in k space to obtain the structural parameters. One of the key points of the analysis is linked to the transferability hypothesis, i.e., the fact that phase shift and amplitude can be extracted from reference compounds that have well-separated Fourier transformed peaks. Nevertheless, special attention has to be paid in the case where the radial distribution function cannot be expressed by a Gaussian function. Finally, a large k range increases the amount of information contained in the EXAFS curve. Let us just recall that the number of independent points N is

N ) 2∆k∆R/π

(5)

with ∆k the k range for which data are available and ∆R the range in R space used for the fitting. In the case of the platinum foil, such anharmonic disorder is illustrated by plotting the modules of the FT and observing a displacement to the short distance of the maximum versus the temperature instead of the normal thermal expansion (Figure 7). This implies the utilization of other formalisms for the data analysis procedure.28 I.3.a. Monometallic Clusters. In the case of platinum clusters, it is easy to build the EXAFS modulations as well as the module of the FT (Figure 8). These plots show that the EXAFS spectroscopy is suitable for very small clusters, the number of first neighbors varying rapidly for them but not for bigger ones, i.e., containing more than a thousand atoms, clusters for which the modules of the FT is very (too) similar to the module of the platinum metallic foil. EXAFS spectroscopy is thus a structural probe, sensitive only to the local order (because of the mean free path term λ) around one given type of atom in the medium (well defined by its X-ray absorption edge). This technique is now an invaluable tool for catalyst science, which establishes a link between the structural characteristics of the metallic part of the catalyst and its chemical activity.

Figure 8. Modules of the Fourier transform associated with clusters of 13 and 55 platinum atoms compared to the modulus of platinum metallic foil.

Figure 9. Fourier transform associated with a metallic copper foil calculated with and without multiple scattering contributions.

I.3.b. Multiple Scattering Considerations. One of the major improvements of the X-ray absorption spectroscopy has been done through accurate calculations of multiple scattering processes present just after the edge. In the case of the FeFF code, since the first comparison with experiments done for the copper, the platinum and the titanium metallic foils up to the fourth shell, numerous compounds have been tested. All these experimental “acid” tests show the validity of one of the basic assumptions relative to the code: a convergence of the calculation based on the Rehr-Albers29 scattering matrix formulation can be made with a small number of paths. As an example, in the case of copper, only 56 paths among 60 billion in a cluster of radius 12.5 Å have a significant contribution. In the case of nanometer-scale metallic cluster, in order to illustrate the importance of multiple scattering process, we have compared in Figure 9 two FT calculated between 2.4 and 15 Å-1 where multiple scattering processes are or are not taken into account for the copper foil. As we can see, multiple scattering is essential beyond the first shell, and in fact as proved before by S. Zabinski,10 the forward and near-forward scattering are the most important. For a nanometer-scale metallic cluster of copper, this kind of calculation is essential for the determination of the morphology of the cluster. As shown by R. B. Greegor et al.,30 there are simple relations among the different average coordination, the shape, and the size of the cluster. Thus, an accurate calculation including single and multiple scattering process of the different shells is necessary to determine the morphology of the metallic cluster. In the case of platinum,


Bazin et al.

Figure 11. Evolution of the different coordination numbers versus the number of monometallic clusters. Figure 10. Fourier transform associated with a metallic platinum foil calculated with and without multiple scattering contributions.

TABLE 1: Average Number (N1, N2, N3, N4) of Neighbors Nt

N1

N2

N3

N4

diameter (Å)

13 147 1415 13, 1415

5.5 8.9 10.5 8.9

1.8 4.0 5.0 4.0

3.7 13.0 18.5 13.8

0.9 6.1 9.1 6.5

8.3 13.8 24.9

owing to the white line, the FT has been made between 3.3 and 15 Å-1 (Figure 10). It seems that under these conditions, the effect of the multiple scattering effect is rather weak.31 I.3.c. Effect of Polydispersity. Nevertheless, this technique has, like some others, some limitations. One of these is the fact that EXAFS is insensitive to polydispersity.32 It is very easy to underscore that point. In Table 1, we have calculated N1, N2, N3, and N4, the first, second, ... average number of neighbors given by this technique. It is clear that the results given by mixing clusters that have 13 atoms and 1415 atoms are similar to the coordination number associated with a cluster of 147 atoms. Another case is given by bimetallic systems. A lot of studies have been made on such compounds because basically EXAFS seems to be one of the few techniques that give the distribution of the two metals inside the cluster. In the case of homogeneous system for which the core of the cluster is composed of N atoms of A (NA) and the surface is made of N atoms of B (NB), the total coordination (NAA + NAB) is equal to 12 for the A atom and less than 12 for the B atom.33 More generally, we have in the case of the bimetallic cluster, the following relationship:

NAA + NAB ) 12 > NBA + NBB, (NA)NAB ) (NB)NBA, σAB ) σBA, RAB ) RBA (6) Nevertheless, this propriety is no more valid when a monometallic cluster coexists with bimetallic ones. In this case, as shown in Figure 11, the number of heterobonds decreases significantly as the content of monometallic cluster increases. The first point of the curve (the number of atoms of A is equal to 13) corresponds to the different coordination labeled NAA, NAB, NBA, and NBB. Then a cluster of 13 A atoms are added, and we see perfectly the decreases of the coordination NAB, the others being constant. If most of the A atoms are involved in monometallic clusters, we have then NAA + NAB < NBA + NBB, and thus, the distribution of the two metals inside the cluster given by the different coordination, if only a bimetallic cluster exists, is simply false. This point has never been clearly

discussed in the different experimental studies done on bimetallic systems. I.3.d. Examples of in Situ Studies. The fact that in situ studies can be performed has transformed this scientific curiosity of the 70’s into a major characterization technique used by academic research groups as well as by industrial companies. One of the first studies dedicated to catalyst materials was published by F. W. Lytle et al.34 where they compared EXAFS and hydrogen chemisorption data for osmium, iridium, and platinum catalysts. Many types of cells have then been used for the study of catalysts under reactive gases at high temperatures under atmospheric pressures.35 One of the major successes of EXAFS spectroscopy has been obtained with the characterization of the preparation steps of the catalysts. Three steps constitute generally the usual preparation modes, namely, impregnation, calcination, and reduction. Each one has a major impact on the final structural characteristics of the material. For example, several studies have shown the importance of the choice of the precursor as well as the chlorine content during the impregnation steps.36 Moreover, the value of the temperature of calcination is a factor in the formation of heteroatomic bonds in the case of several bimetallic catalysts.37 Finally, the temperature and the time of reduction control the size of the metallic cluster finally generated.38,39 In fact, numerous studies have adopted this approach and several general reviews exist in this research field.40 Most of the bimetallic systems are easy to study owing to their great difference from a backscattering function point of view, but it is not the case for a system like the platinum-rhenium bimetallic catalysts. For this catalyst, one can use the proximity of the L edges of the two metals to follow “simultaneously” the evolution of their electronic states during reduction to obtain information on the formation or lack of formation of a bimetallic catalyst.41 I.4. Conclusion. As we have seen, the totality of the absorption spectrum (NEXAFS, XANES, and EXAFS) allows for a correlation between the electronic information given by the edge and the structural information given by the edge and EXAFS modulations. In fact, this chemical tool has been coupled to more classical techniques such infrared spectroscopy,42 temperature-programmed reduction (TPR),43 phase chromatography,44,45 and in the last case a correlation has been done between the structural parameter and the catalytic activity. Nevertheless, the lack of information regarding the local order after the first shell as well as on the support has pushed the community to the diffraction technique. II. Anomalous Wide Angle X-ray Scattering (AWAXS) For the past few years, AWAXS experiments have been performed in several synchrotron radiation centers, dealing with



Figure 13. Radial distribution function (RDF) calculated for a cluster of 13 atoms (line) and 55 atoms (dots) of platinum. Figure 12. Diffraction diagram calculated for fcc nanometer-scale platinum cluster of N atoms.

various kinds of systems.46,47 On supported metallic catalysts, P. Ratnasamy et al.48,49 have shown that the radial distribution method and the method of direct intensity calculation are rather complementary to each other. For the method of direct comparison of calculated and experimental intensities, the first step is to set up a plausible model of the material and to calculate the interatomic distance between all pairs of atoms. The corresponding intensity is then calculated from the Debye formulas as a function of q and compared with the coherently scattered intensity observed experimentally. The model is then refined until a satisfactory agreement is reached between the two curves. II.1. Associated Formalism. We recall here just some elements associated with this technique.50,51 In AWAXS, the energy dependence of the atomic scattering factor f(q,E) near an absorption edge is used, and this parameter is expressed in electron units as

f(q,E) ) f0(q) + f ′(E) + if ′′(E)

(7)

where q bisects the angle between the incident and scattered directions, which in turn, defines the scattering plane. f(q,E) gives the amplitude of the radiation coherently scattered by a single atom and is composed of an energy independent part f0(q) and a real and imaginary energy dependent corrections f ′(E) and f ′′(E). The energy independent part f0(q) is in fact the usual form factor related to the electron density of the atom.51

∫0∞ F(r) [sin(qr)]/qrr2 dr

f0(q) ) 4π

(8)

It is well-known that for forward scattering, this parameter tends toward Z plus a relativistic correction in the case of medium and heavy elements.52 For the real and imaginary energy dependent corrections, which originate mostly from tightly bound inner electrons, these terms are related through the Kramers-Kronig dispersion relationship. Also, absorption spectroscopy and anomalous diffraction are intimately related, the imaginary part being linked to the photoeffect cross section σ(E) by the relation

∫0∞f ′′()/( 2 - E2) d

f ′(E) ) (2/π)P

f ′′(E) ) Eσ(E)/(2hcre) (9)

Finally, the diffraction intensity I(q) is given by the Debye scattering equation.53

I(q) )

∑i ∑j fi(q)fj(q) [sin(qRij)]/qRij

(10)

In this equation, I(q) is the angle dependent intensity from coherent scattering, the sums over i and j are over all the atoms, Rij is the distance between the atoms i and j, and fi and fj are the angle dependent atomic scattering factors. II.2. Analysis Procedure. Details on the data analysis can be found in several studies.46 What we recall here is the fact that to perform a Fourier analysis of the diffraction diagram, it is convenient to introduce the partial structure factor Smn(s), s being the wave vector that is related to the pair distribution function by the following relation:

∫0∞(Fmn(r) - Fn0) r sin(sr) dr

Smn(s) ) (4π/s)

with s ) 4π(sin θ)/λ (11) where Fmn(r) is the pair distribution function, i.e., the density of atoms of type n at a distance r from an atom of type m, and Fn0 is the average atomic density of atoms of type n in the sample. We have then

I(s) )

xmfm(s)fn(s)Smn(s) ∑n ∑ m

(12)

where x is the atomic fraction. II.3. Application to Monometallic Clusters. We now discuss the possibility of using AWAXS to study very small clusters. In Figure 16, we see the results using calculated atomic scattering factors and neglecting the thermal disorder and the Compton scattering in a first approach. This last approximation is valid. Since platinum is a heavy element, the Compton scattering is in a small diffuse background that increases slowly when q increases. As pointed out by P. Gallezot et al.,54,55 the different diffraction diagrams (Figure 12) and their FT (Figure 13) show clearly that even for extremely small clusters, the Bragg diffraction peaks of a fcc network are clearly visible, and thus, the nature of the network can be directly determined even for very small clusters. In the case of supported materials, as is the case generally, we have to discuss the differential intensities, i.e., the difference between two diffraction diagrams taken at two energies before the edge. In the case of a monometallic cluster, the differential is similar to the diffraction intensity as shown in eq 10:


∆I(q) )

Bazin et al.

∑i ∑j ∆fi(q)∆fj(q) [sin(qRij)]/qRij ) (∆f(q))2 ∑ ∑ [sin(qRij)]/qRij i j

(13)

Moreover, the splitting between the Bragg peaks is extremely sensitive to the size of the clusters and thus allows for a direct measure of the cluster size.56 Finally, we point out that when the signal coincides with the observed one, the particle can be considered as a perfect single crystal; i.e., the structures of the surface atoms and inner atoms are identical.57 At the opposite, in the case of a zinc ferrite crystallite, a precise analysis of the broadening of the different diffraction peaks suggests that a monolayer surface atom form an amorphous-like atomic structure rather than the zinc ferrite crystalline structure found inside the particle.58 II.4. Effect of Polydispersity. In the case of size distribution, the Debye function analysis method (simulation done with a linear combination of ab initio calculations of a diffraction diagram) as developed by W. Vogel et al.59 can be used, and thus, information regarding the size distribution is available. On this point, the absorption spectroscopy and the diffraction method can be considered as complementary methods. II.5. Applications to Bimetallic Clusters. Bimetallic systems are more and more the subject of AWAXS studies. In a previous study of platinum-molybdenum catalysts,60,61 the similarity between the WPSFs (weighted sum of partial structure factor) for the platinum pairs (platinum-molybdenum) and those for the molybdenum pairs (molybdenum-platinum) suggests that molybdenum does not form pure clusters but does form bimetallic ones having the fcc structure. Nevertheless, no conclusion was made about the distribution of molybdenum inside the cluster. A new analysis of the same data with the method of direct intensity gives information on the distribution of the metals inside the cluster.62 Now we would like to know if a simple measurement of the fwhm (full width at half maximum in Å-1) of the 111 peak is enough to get information on the distribution of the two metals inside the cluster. For that purpose, we calculated the diffracted intensity for a fcc cuboctahedron of 2057 atoms. For the first cluster, we considered a statistical distribution of platinum and cobalt atoms inside the cluster (containing thus 1029 atoms of platinum and 1028 atoms of cobalt). For the second cluster, we considered that one kind of metal is at the surface and the other one at the core (1415 atoms of platinum define the core and 642 atoms of cobalt define the surface). For f0(q), we used the Cromer and Mann parameters,63 and Table 2 gives the values of f ′ and f ′′ for each element according to the Sasaki tables.64 For the first cluster (Figure 14), the WPSF’s (weighted sum of partial structure factor) at the platinum and cobalt edges are the same. At the opposite and as previously reported, the WPSF’s at the platinum and cobalt edges are different (Figure 15). In Table 3, we have reported the different values measured for the fwhm for the two clusters. As we can see in the Table 3, the fwhm are the same in the case of a statistical distribution and quite different if a preferential distribution between the two metals exists inside the metallic cluster. These simple calculations show that simple TABLE 2: f ′ and f ′′ for Platinum and Cobalt Atoms at Different Energies E (eV)

f ′ (Pt)

f ′′ (Pt)

f ′ (Co)

f ′′ (Co)

7650 7700 11450 11540

-5.03 -5.05 -12.25 -15.35

7.49 7.41 3.93 3.88

-4.60 -6.73 0.03 0.04

0.48 0.47 2.03 2.00

Figure 14. WPSF’s calculated at the Pt LIII edge (dots) and at the Co K edge (line) for a statistical distribution.

Figure 15. WPSF’s calculated at the Pt LIII edge (dots) and at the Co K edge (line) for a preferential distribution.

TABLE 3: Full Width at Half-Maximum As Measured for the Two Clusters Considered statistical distribution ∆k (Å

-1)

Pt atoms at the core, Co atoms at the surface

Pt LIII

Co K

Pt LIII

Co K

0.183

0.17

0.19

0.14

measurements of the fwhm of the (1,1,1) peak is enough to get information on the distribution of the two metals inside the cluster. II.6. Conclusions. Simple calculations show that the equivalent of a Scherrer relationship can be applied to differential functions. For monometallic catalysts, it is possible to extract the size and the network of the cluster and on bimetallic clusters to get the distribution of the metals inside the crystallite. Moreover, regarding an experimental point of view, calculation of the structure of the WPSFs shows that only a small part of the spectra need to be measured, and thus, this approach can significantly decrease the acquisition time devoted to such study. III. Anomalous Small Angle X-ray Scattering (ASAXS) III.1. Case of Monometallic Clusters. In the case of small metallic clusters, Figure 16 shows the diffraction diagram near the (0,0,0) peak calculated using the Debye equation.10 For fcc (as well as for bcc) clusters, we observe different modulations after the (0,0,0) peak, and thus, information regarding the crystallographic network as well as on the size on the particle65,66



Figure 16. Central diffusion as calculated using the Debye equation near the (0,0,0) peak for cuboctahedra fcc platinum cluster of N atoms. Figure 18. Differential obtained near the (0,0,0) peak as calculated at the LIII Pt edge (line) and at the cobalt K edge (dots) for a statistical distribution.

Figure 17. Central diffusion as calculated using the Debye equation near the (0,0,0) peak for hemicuboctahedra fcc platinum cluster of N atoms.

are available. Nevertheless, we have to point out that physical parameters such as the interatomic distance modify the position of these modulations. Also, if other kinds of morphology such as a hemicuboctahedron (Figure 17) are considered, no modulations exit.59,65-66 III.2. Case of Bimetallic Clusters. In the case of bimetallic cluster, this technique is able to give information on the distribution of the two metals inside the cluster. For that purpose, we calculated the diffracted intensity near the (0,0,0) peak for a fcc cuboctahedron of 561 atoms. For the first cluster, we considered a statistical distribution of platinum and cobalt atoms inside the cluster containing 280 atoms of platinum and 281 atoms of cobalt. For the second one, we considered a cluster with one kind of metal at the surface, the other at the core (309 atoms of platinum define the core, and 252 atoms of cobalt define the surface). For f0(q), we used the Cromer and Mann parameters,63 and Table 2 gives the values of f ′ and f ′′ for each element according to the Sasaki tables.64 If we consider now the differential obtained for the two distributions of metals inside the metallic cluster (statistical and preferential), it is clear that anomalous small angle scattering can give information on the distribution of the two metals inside the particle. In the first case (Figure 18), the two differentials are similar. At the opposite, Figure 19 exhibits very different differentials for the metal at the core and the metal at the surface. III.3. Conclusion. These simple calculations show that the anomalous small angle scattering technique is extremely sensitive to the size distribution of the monometallic cluster, much more than the X-ray absorption spectroscopy or the diffraction technique. The anomalous small angle scattering is thus a remarkable complementary tool. Regarding the distribution of

Figure 19. Differential obtained near the (0,0,0) peak as calculated at the LIII Pt edge (line) and at the cobalt K edge (dots) for a preferential distribution.

the metals inside the cluster, we have shown through ab initio calculations that anomalous small angle scattering can give such information. As shown above, in the case of a statistical distribution, the two differentials are similar. At the opposite, for a preferential distribution, the calculation gives very different differentials. IV. Some Considerations of Diffraction Anomalous Fine Structure (DAFS) Reported a long time ago,67 DAFS can be considered as a wedding between absorption and diffraction, combining the sensitivities of the two techniques: the long-range structural information and the chemical and local structure selectivity.68 With the development of the synchrotron radiation center, several groups69,70 have applied this high beam consuming method to different kinds of samples (thin films, multilayers, powder and single crystal). In the case of a nanometer-scale metallic particle, one easy way to calculate the DAFS modulations is to build the EXAFS modulations associated with each kind of atom present inside the cluster using the FeFF code, for example, in order to get the anomalous term f ′ and f ′′. f ′′ is related to the photoabsorption section by the optical theorem, and f ′ and f ′′ are related to each other by the Kramers-Kroning relationship.9 Such approach has been used for a cluster of 13 atoms of platinum.


Bazin et al. same authors have shown and measured in the case of copper that the DAFS and XAS signals have, respectively, the forms -cos (2kRj + δj) and sin(2kRj + δj). V. Discussion

Figure 20. f ′ and f ′′ associated with the central atom (0,0,0).

Figure 21. DAFS spectrum as calculated for a 13-platinum (line) and a 55-platinum (dots) cluster.

Figure 22. Comparison between EXAFS and DAFS modulations for a cluster of 13 atoms.

To illustrate the calculation procedure, we first calculate the EXAFS modulations of the central atom, then obtain the anomalous term f ′′ through the optical theorem, and then calculate f ′ with the Kramers-Kroning relationship (Figure 20). The Debye equation is then used to obtain the diffraction diagram. Finally, a diffraction peak is selected, and when its intensity versus the photon energy is plotted, it is easy to build the DAFS spectrum. The DAFS spectra have been calculated for clusters of platinum containing 13 and 55 atoms (Figure 21). Finally, we have plotted the EXAFS and DAFS modulations obtained for a 13 atoms cluster of platinum according to the normalization procedure developed by C. E. Bouldin et al.71 As they have previously pointed out, the DAFS and EXAFS oscillations are both about 10% peak to peak when normalized to their corresponding edge step. Finally, regarding the frequency, the

Let us begin by a simple comparison between the Fourier transform of the EXAFS modulations (Figure 8) and the radial distribution function obtained from diffraction data (Figure 13), the two Fourier transforms being done on a similar k range. It shows clearly that the diffraction technique is better than absorption spectroscopy for the determination of the local order after the first shell. It is due obviously to the fact that the low k range of the absorption spectrum is dominated by multiple scattering processes that lead to a loss of information in the high R range. This advantage nevertheless has to be taken with care. If X-ray scattering factors have a similar k dependence whatever the atomic number Z, and this is true for wide as well as for small angle scattering, it is definitely not the case with X-ray absorption spectroscopy. Amplitude and phase scattering change significantly with the atomic number for the last one, and thus, the chemical sensitivity of the absorption spectroscopy technique is really higher. Moreover, as we have recalled previously, information on the electronic state of the absorber atom as well as on the local structure is available through ab initio calculations of the “edge” part. Finally, regarding the structural information obtained by AWAXS, it is in many case ambiguous especially in the case of an overlap of the shells in the same range of distances. That restriction comes from the fact that the technique provides n independent structural factors and thus, it is not possible to decorrelate the n (n + 1)/2 individual partial structure factors. In fact, there is an evident disadvantage with all the techniques because we need to assume the nature of the network (given by the AWAXS technique for example) and the morphology (which can be confirmed by the ASAXS) to get the size of the particle from the different coordination numbers given by the absorption spectroscopy. For bimetallic as well as for multimetallic particle, information regarding the distribution of the two metals inside the cluster can be given by the different techniques as well, and thus, a coherent structural model position has to be in line with the different results. Let us consider now the analysis procedure. One of the major limitations of the anomalous diffraction comes from the fact that only a small k range of data is available. It is especially the case in heterogeneous catalysis for which the elements of interest are 3d (and 5d) transition metals, which have a K (L) edge around 10 keV. This implies that the maximum value for k is thus close to 10 Å-1 and means that the classical analysis procedure based on the Fourier transform fails at the step of data normalization. Thus, we have to move to ab initio calculations using the Debye equation.10 As shown in this report, we can applied a pseudo-Scherrer formula on the differential or compare directly with ab initio calculation, and this direct analysis procedure allows to obtain the size and the network of the cluster as well as the distribution of the metals inside the particle. Regarding the X-ray absorption spectroscopy, we see that ab initio calculations have to be done for the XANES part using the FeFF code, for example, instead of a linear combination of the XANES of well-crystallized reference compounds and that for the modulations after the edge, multiple scattering process have to be taken into account for an accurate determination of



TABLE 4: Information Obtained by the Different Techniques XAS

AWAXS

ASAXS

Nanometer-Scale Monometallic Cluster network difficult easy difficult size distribution difficult difficult easy size easy possible easy morphology difficult difficult easy Nanometer-Scale Bimetallic Cluster distribution of the easy difficult difficult metals inside the cluster

References and Notes DAFS

difficult easy easy easy not evaluated here

the network and the morphology of the cluster. The last one is intimately dependent on the different average coordination. From an experimental point of view, it is clear that for supported low-loading metallic clusters, fewer photons are needed to get an absorption spectrum than to build a differential from the diffraction diagrams. The anomalous phenomena for wide as well as for small angle is based on a rather weak change in the scattering factor. Obviously, it is even more critical for the DAFS technique. This explains why EXAFS is still widely used while the AWAXS experiments just begin to contribute to the understanding of the physicochemical phenomena linked to catalyst science. From an experimental point of view, we can thus consider absorption spectroscopy X as a starting point for the structural determination of the material. Finally, we gathered all this discussion in Table 4, where the different techniques and all the information we can obtain from them are listed. VI. Conclusion and Perspective Numerous X-rays techniques related to synchrotron radiation have been used for structural characterization of nanometerscale materials, and each one has its own advantages and disadvantages. Through ab initio calculations, we have tried to point out their advantages and disadvantages and, thus, the fact that to build an accurate structural model, scientists need information from all these techniques, whatever the composition of the crystallite, namely, metallic or oxide metallic. The simultaneous use of the different techniques that have been considered is necessary owing to the complexity of the material for which we have to take into account the size distribution as well as the fact that different families of clusters can exist at the same time inside the material. From a data analysis point of view, we have shown that for the edge part of the absorption spectra, linear combination is not an accurate method and ab initio calculations have to be done. For the EXAFS part, special attention has to be paid to relate the coordination number to the size of the cluster or to the distribution of the different components inside the cluster. In the case of anomalous diffraction, the Scherrer law can be extended to the differential in the case of monometallic and bimetallic systems. It is clear, however, that only a correlation with their specific properties, catalytic activities, for example, can lead to a significant breakthrough for various physicochemical process. The fact that the different techniques considered in this report deals with X-ray photons allows for the conception and the realization of in situ reaction cells. Numerous techniques have been developed for absorption spectroscopy and anomalous diffraction. This paper shows that a new family of reaction cells will appear with the emergence of third generation synchrotron radiation being compatible with the different X-ray techniques and the measurements of their chemical properties.

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Comparison between X-ray Absorption Spectroscopy, Anomalous

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