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BEYOND THE PAIR DISTRIBUTION FUNCTION IN X-RAY ABSORPTION SPECTRA
M. Benfatto, C. Natoli
To cite this version:
M. Benfatto, C. Natoli. BEYOND THE PAIR DISTRIBUTION FUNCTION IN X-RAY AB- SORPTION SPECTRA. Journal de Physique Colloques, 1987, 48 (C9), pp.C9-1077-C9-1084.
�10.1051/jphyscol:19879196�. �jpa-00227313�
BEYOND THE P A I R DISTRIBUTION FUNCTION I N X-RAY ABSORPTION SPECTRA M. BENFATTO and C . R . NATOLI
INFN Laboratori Nazionali di Frascati, I-00044 Frascati, Italy
INTRODUCTION:
The x-ray absorption spectroscopy (XAS) concerns the study of electronic transitions from inner shells to unoccupied states. Electronic and structural information on selected absorbing sites can be obtained in complex systems just by tuning the x-ray energy of the beam light impinging onto the sample.
In this way the atomic species under study can be selected and their environment explored ( I ) . The absorption coefficient shows a more or less.
pronounced oscillatory structure as a function of the photon energy above the rising edge. Such structure reflects the presence of neighbours around the absorbing atom. Because of the finite lifetime of the photoelectron in the final state the description of the absorbing spectra depends mainly on the geometrical structure of a finite cluster of atoms around the photoabsorber.
The size of this cluster changes with the systems under study, ranging from a single shell to severaf shells. Generally speaking a XAS spectrum can be divided in different regions (2). Going from extremely high energies, where the scattering power of the neighbours of the photoabsorber is small, to lower energies, different scattering regimes take place and the shape of the spectrum reflects this situation. By decreasing the photoelectron kinetic energy a gradual turnover occurs from a single scattering regime (SS) to the XANES full multiple scattering (FMS) regime through a intermediate region where only few low-order multiple scattering (MS) paths are relevant (IMS).
This is the " normal " situation; there might be exceptions, however, as in the case of copper K-edge, where the single scattering regime takes place in the low energy part of the spectrum. This is due to the peculiar behaviour of the relevant phase shifts at lower energy, which happen to be very small (modulo n). Up to now only the SS part (EXAFS) of the XAS spectrum. which gives
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:19879196
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information about the pair correlation function, has been the object of extended experimental and theoretical investigation. However, experimental evidence of multiple scattering effects has been found up to several hundred eV far from the edge. For this reason we think that now it is possible to exploit all the potentiality of the XAS spectroscopy for obtaining structural information. In this paper we present a unifying scheme of interpretation of the x-ray absorption spectra valid for the whole energy range based on the multiple scattering theory and at the same time we propose a method of analysis capable in principle of providing useful information about higher order correlation functions.
THEORY:
In the multiple scattering approach one calculates the absorption coefficient in the real space for a small cluster around the photoabsorber, whose size is determined by the inelastic mean free path of the photoelectron in the final state and the core-hole lifetime ( 3 ) . The advantage, in comparison with other methods, is that it can be extended to materials where there is no long range order like solutions, liquids and amorphous materials. In this formalism the polarization averaged absorption coefficient as a function of the energy &is given by ( 3 , 4 ) :
where the is the atomic absorption coefficient of the photoabsorber from a core initial state of angular momentum I (I=0 for K-edge) and the quantity ~ l c o n t a i n s all the interesting structural information concerning the environment. The expression for these quantities is given by ( 3 ) :
where I is the unit matrix, G is the matrix describing the spherical wave propagation of the photoelectron from one site to another, T, is the diagonal
shift of the photoabsorbing atom ( located at site 0 ) for angular momentum 1.
In absence of atoms around the photoabsorber (G
=
O), the absorption coefficient reduces to the atomic absorption. Normally in the XANES calculations ( 5 ) the full MS result of Eq. 2 is compared with the experiment.However it is possible to follow another approach which is more interesting for structural problems. When the condition p(TaG)<l is verified for all relevant energies of interest, where p(A) indicates the spectral radius of the matrix A , (see ref. 3 for a discussion about this important point) one can develop the matrix inverse in Eq. 2 in series, by formally writing:
thus generating the MS series. In fact insertion of this expression into Eq.1 gives
where now
These quantities represent the partial contribution of order n to the absorption coefficient of the cluster coming from all the processes where the photoelectron has been scattered n-1 times by the atoms surrounding the photoabsorber before returning to site 0 to exit into free space. Notice that X,! =1, X,I=O since there is no propagation from one site to itself and that X,I is the usual EXAFS term with spherical wave propagators (3s4). Of course the higher order terms are related to the n-particle correlation functions which are the quantities we are interested in. On mathematical and physical grounds the general functional expression for the quantities Xnlis given by:
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where
C
indicates the sum over all paths p, of order n starting from andD-
ending at'ihe central photoabsorbing atom with n - l intermediate steps on the surrounding atoms,
R P ,
indicates the corresponding total path length and the dependence of the amplitude and phase functions is on some rotationally invariant combination of scalar and vector products of vectors joining consecutive site locations, the path being simbolically denoted by Ri? The theory provides well defined expression for these quantities @ ) . Their experimental determination is clearly of the outmost importance due to the fact that they bear information on the correlation functions of order higher than 2, i.e. on the local geometry around the photoabsorber.BEYOND THE PAIR DISTRIBUTION FUNCTION:
To show that the XAS spectroscopy is capable of " structural selectivity" in Fig. 1 are compared the Mn K-edge spectra of [MnO,]- and [M~(oH,),]~+ complex in acqueous solution. The energy scales are rescaled in the ratio (R,'/R,*)~
= 0.47 where R,* and R,* are the Mn-0 distances of the two complexes, corrected for the linear term coefficient of the phase function, in order to eliminate the effect arising in the spectra from the different bond lenght and the dipendence of the phase function on the energy. The two spectra, after a further rescaling of the amplitude in order to take into account the different number of neigbouring in the two complexes ( see insert in Fig. 1 ), show a superposed behaviour in the EXAFS energy region in which the photoelectron probes the pair correlation function. Below the 160 eV ( referred to the [MnO,]- energy scale ) the two spectra are different showing that the photoelectron distinguishes between the two complexes.
The deviation from the pure EXAFS regime first begins to show up in the [MnO,]- complex and it is only due to the a, = a,
X,
term which therefore can be extracted using a subtraction procedure. The period of the signal so obtained is related to the perimeter of the 12 equivalent triangular paths Mn-0-0-Mn in the [MnO,]' complex and its Fourier trasform confirms this interpretation ( see Ref. 2 for a deeper discussion ). This is a fortunate case since the deviation from SS regime is only due to the a, term and its energy range is wide enough to extract a signal which can be used to obtain information on the three particle correlation function. Other simple tetrahedral systems have the same behaviour and the same information has been extracted from the experimeNal signal in [Cr0,12' acqeuous solution t7)0 8 0 160 2 4 0 3 2 0 4 0 0 4 8 0 5 6 0 6 4 0
ENERGY ( e V )
0 50 100 150
FIGURE 1
Comparison between normalized Mn K-edge x-ray absorption spectra of [Mn04]- and [ M ~ ( o H & ] ~ + ions from 50 mM aqueous solution of KMn0, and MnCI2. The respective energy scales are given in the upper (lower) part of the figure.
1
t
4'
I
I.
and recently in GeCI, gas by Bunker et al (8) using the same subtraction procedure. In this last case the procedure utilized was very neat and the MS signal was obtained by simply adding three experimental spectra in the following linear combination:
XANES EXAFS
- 6 -
- - - - - [ M n O l J -
-IMnOr I
[MS] = [GeCI,]
+
3 [GeH,]-
4 [GeH,CI]I I I I I I I I
In fact the first spectrum contains SS and MS contributions while the others contain just the atomic and SS signal rispectively. The Fourier trasform of the residual signal peaks at about 3.5A corresponding to the shortest (3.8A) triangular path in the tetrahedral GeCI, molecule. In this case the size of this residual MS is not very high ( it goes from 20% to 2% of the SS signal ) but it is there and can be detected.
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In large cluster of atoms it is not so simple to single out the MS paths which are responsible for the MS contributions in the experimental spectrum but in this case the signal is in general stronger than in the small cluster case due to the high multiplicity of the paths. Sometimes it can happen that just a single higher order path dominates the MS signal and in this case it is still possible to isolate a signal which bears information on the higher order correlation functions as in the previous case. This is the case of crystal silicon ('). In this system the MS contributions are very high and necessary for explaining the absorption spectra in the first 70 eV beyond the K threshold. The MS signal obtained by subtracting from the experimental spectrum the X, spherical wave contribution, due to the first seven shells around the photoabsorber, in the whole energy range above threshold is reported in Fig.2 (panel a). In the same figure the comparison among the MS, experimental and EXAFS signal is also shown. Looking at this figure it is quite evident that the main features in the spectrum in the first 70 eV above the edge arise mainly from multiple scattering effects. In the panel b of Fig.
2 is reported the X3 calculated signal coming from the all 756 scattering paths involving two neighbour atoms within the first three shells (curve a).
In the same figure it is also reported the signal which arises from 36 paths with the shortest total length.
These latter can be divided in two groups differing each other in the angles between the outgoing and incoming path vectdrs directed at the photoabsorber vertex. The first of these two groups contains 12 paths involving atoms within the first shell (curve c) while the second one comprises 24 paths involving one atom of the first and one of the second shell (curve b).
1 2 3 4 5 6 7 1 2 3 6 5 6 7
k(A-' ) k(A-1 )
FIGURE 2
Panel a: Comparison between experimental XMS signal, EXAFS contribution and difference spectrum.
Panel b: Total calculated X3 signal (curve a) very close to the experimental MS contribution , X3 signal (curve b) from the total shortest paths with lenght of 8.54 A, X3
contribution (curve c) due to the paths involving just the first shell.
described above and it is a remarkable results that the most important contribution to the MS signal comes from these 24 paths with total lenght of 8.54A involving the first two shells. Therefore the experimental MS signal so obtained does really bear information on the three particle correlation function involving the photabsorber, the silicon in first shell and another silicon in second shell. The Fourier Trasform (FT) analysis confirms this interpretation. In Fig. 3 the comparison between the FT multiple scattering experimental signal (solid line) and the theoretical X, is reported. The main peak (big arrow) due to the shortest X, paths can be easily recognized both in the FT of the experimental and theoretical signal.
, 1 1 1 I l l , I ~ 1 I
FIGURE 3
Comparison between the Fourier trasform of the calculated curve a (dot-dashed line) and the FT of the XMS spectrum of Fig.2a (solid curve). The big arrow indicates the peak at 3.6A due to the shortest double scattering pathways.
Other peaks of noticeable intensity (small arrows) at higher distances are present which can be explained taking into account other factors as: other MS pathways of longer length, some errors in the subtracting procedure or a residual EXAFS signal coming from shells farther than the seventh. In particular we think that the pronounced peak at about 5.3A is due to the focussed EXAFS contribution due to the eighth shell (collinear with the first) which was not subtracted from XeXp,. To conclude we have shown several
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examples extracted structures
where the multiple scattering signal is present and big enough to be from the experimental spectrum even in the case of open like the one discussed here.
Of course there are several open questions expecially related to the possibility of the determination of the higher order correlation functions of unknown systems but we think that the XAS spectroscopy has the potentiality to give information about bond angles and geometrical atomic arrangements in general. This potentiality should be exploited to obtain useful structural information, even if it is partial, beyond the pair distribution function.
REFERENCES :
1) For a review of the topics see the Proceeding of the "EXAFS and NEAR
EDGE STRUCTURE IV", July 7-1 1,1986 Abbaye Royale de Fontevraud (France), Ed. by P.Lagarde, D.Raoux and J.Petiau,
Journal de Physique 9, Colloque C8.
2 ) M.Benfatto, C.R.Natoli, A.Bianconi, J.Garcia, A.Marcelli, M.Fanfoni
and I.Davoli, Phys. Rev &, 5774 (1986) and references therein 3) C.R.Natoli and M.Benfatto, Journal de Physique
a,
C8-11 (1986) 4) W.L.Schaich, Phys. Rev. 829,6513 (1984)5) P.J.Durham, J.B.Pendry and C.H.Hodges, Solid State Comm.
a,
159 (1981) 6) M.Benfatto. C.R.Natoli and M.Ruiz, manuscript in preparation7 ) G.Garcia, M.Benfatto, C.R.Natoli, A.Bianconi, A.Marcelli and l . ~ i v o l i Solid State Comm.
a,
595 (1986)8) G.Bunker, C.Bouldin and D.McKeown, Poster presented at the NATO AS1 on X-Ray Spectroscopy in Atomic and Solid State Physics,
Vimeiro (Portugal) 30 Aug - 12 Sept 1987
9) A.Bianconi, A.Di Cicco, N.V.Pavel, M.Benfatto, A.Marcelli, C.R.Natoli, P.Pianetta and J.Woicik in " Multiple Scattering Effects in the K-edge X-ray Absorption Near Edge Structure of Crystal and Amorphous Silicon" Phys. Rev B in press and references therein.
