n-body expansion for EXAFS analysis
Fabio Iesari
University of Toyama
iesari@sci.u-toyama.ac.jp

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1 / 11XAS cross section

The polarization averaged XAS cross-section for transitions to a dipole selected final state ofangular momentum l0 can be written asX
1
1L,L−100σ(ω) = σ0 =[T (I − G T ) ]0,0 =(t l0 ) 2l0 + 1 m00

where σ0 is the atomic cross-section, T and G are the atomic scattering (phase-shift) andphotoelectron propagator matrices in a local basis, indexed by i, j running over the different
atoms.
Electron interaction (T ) and structure (G ) are decoupled
Non-linear relationship between geometry and XAS signal: σ ∼ (I − GT )−1
mathematical consequence of the strong coupling of the photoelectron with the surrounding
atoms (more difficulties in analyzing XAS data)[A. Filipponi, A. Di Cicco, C.R. Natoli, Phys. Rev. B 52, 15122 (1995)]

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2 / 11Multiple Scattering

A first approach to this problem is to use the so-called MS expansion. Where the norm ofthe GT matrix (maximum modulus of its eigenvalues) satisfies ||GT || < 1 then the formalmatrix expansion:
T (I − GT )−1 = T (I + GT + GTGT + GTGTGT + .....)
is convergent and gives rise to the MS series. The above condition will certainly
hold above√
a given energy since not only the elements of the G matrix decrease like 1/ E but also||T || = max |ti | tends to zero much more rapidly with energy.
Writing down the series we obtain:

XX 0ij0Xσ(ω) = σ0 χ20i0 +
χ3 +
χ0ijk0+ ···4
1 +i6=0Fabio Iesari (Toyama)i6=ji6=0,j6=0

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i6=j6=ki6=0,k6=0

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The χn signals are oscillating functions of the type:χn (k) = A(k, R) sin(kRp + φ(k, R))
where A and φ are smooth functions of k and of the geometrical parameters R. The relevant
frequency of the signal is the path length Rp .
An important inconvenience of the MS expansion is that it has no simple relationship with thestructure (geometry): χ2 terms probe the relative position of atoms 0 and i, χ3 terms probe thepositions of the atoms 0, i and j. They are sensitive to the two-particle and three-particle
distribution respectively.

Note: single scattering approximation (retain only the first χ2 term) may be sufficient in somecases, but information limited to pair distributions

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For χ4 terms the situation becomes more
complicated: a generic 0ijk0 path probes
four-particle correlations, but special paths
like 0i0k0 or 0iji0 and 0i0i0 probe lower
order correlations. In general at order n, inχn there are paths involving all particle
distributions from 2 to n if n is even orfrom 3 to n if n is odd.
A different and potentially more powerfulapproach to the solution of the XAS
structural problem is based on a n-bodydecomposition of the cross-section. The
main idea is to reduce the dimension of theproblem to the actual local physicalquantities of interest, related to n-atomproperties where n  N (N being thenumber of atoms of the system).

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n-body expansion

The total x-ray absorption cross-section for n atoms σ(0, i, j...n) can be expanded in terms of theirreducible n-body cross sections of lower order:XXX
σ(0, i, j..., n) = σ0 +σ (2) (0, i) +σ (3) (0, i, j) +σ (4) (0, i, j, k) + · · · + σ (n) (0, i, j, ..., n)i(i,j)(i,j,k)The dimensionless quantities γ (n) = σ (n) /σ0 , represent the irreducible n-body contributions to theσ(E )−σ0 (E )structural XAS term χ(E ) =. In this way we arrive to an equivalent expansion for theσ0 (E )experimentally measurable structural signal χ(E ) that differs substantially from the MS series:
XXXχ(0, i, j..., n) =γ (2) (0, i) +γ (3) (0, i, j) +γ (4) (0, i, j, k) + · · · + γ (n) (0, i, j, ..., n)iFabio Iesari (Toyama)(i,j)(i,j,k)n-body in EXAFS

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The irreducible n-body γ (n) signals are the central quantities in the GNXAS approach(http://gnxas.unicam.it) since they are associated with well defined n-body arrangements ofthe atoms.
simple linear relationship among structure (expressed in terms of 2, 3, 4, ... n-bodydistributions) and signal is obtained;
due to mean free path effects, the higher order n-body terms are generally smaller than thelower order ones, so that convergence with few terms is expected;

hierarchical relationship between different n-body configurations: a n-body configuration
contains several n − m-body sub-configurations that are not independent. This allows toreduce the number of n-body coordinates just to the independent ones.

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Real XAS data: configurational average
Having defined the irreducible γ (n) MS signals, the general expression of the XAS structural term
is given in terms of the n-body distribution functions gn (r ) (r is a generic set of n-bodycoordinates):
Z ∞hχ(k)i =0dr 4πr 2 ρ g2 (r ) γ (2) (r , k) +Z

dr1 dr2 dφ 8π 2 r12 r22 sin(φ)
Z× ρ2 g3 (r1 , r2 , φ)γ (3) (r1 , r2 , φ, k) + dr1 dr2 dφdr3 dΩ 8π 2 r12 r22 r32× sin(φ)ρ3 g4 (r1 , r2 , φ, r3 , Ω)γ (4) (r1 , r2 , φ, r3 , Ω, k) · · ·The integrals, because of the short range nature of the kernels γ (n) , are actually limited to aregion of linear dimensions of the order of few Å . This is due to the strong electron interaction(mean free path).
This equation should be compared with the well known expression for the static structure factor(disordered system) which can be measured by using x-ray or neutron diffraction:
Z4πρ ∞(g2 (r ) − 1)r sin(kr )dr .
S(k) = 1 +k0
The different nature of the kernels makes the structural information on the g2 (r ) obtainable in
the two cases largely complementary (short vs. medium range information). Moreover, XAS cangive information beyond the pair distribution. For crystals, where diffraction gives obviously aricher information about atomic positions, XAS can provide unique data about local correlatedvibrations.
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Example: fcc Ni

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