()
[]
)(/2
2/)1(
2
2
2
2
0
2
0
0
)2(12exp
)()(
)(
kR
m
St
j
j
jj
j
ekk
kR
kSkfN
k
λ
σσχ
−
+−
+−=
∑
)2(actan)1()(2)(2sin
0 SCj
j
kmkkkR
. (5)
Here R
0
is the distance of central atom to the closely packed atom, σ
S
and σ
t
are the structural (or
statistic) disorder and thermal disorder, respectively. The difference between Eqs. (3) and (5) is that the
22
2
j
k
e
σ
−
term in Eq. (3) is converted to
)
]
2/)1(
2
2
2
)2(12exp
m
St
kk
+−
+−
σσ
in amplitude and
(m+1)actan(2kσ
S
) is added in the phase shift.
2.2 Multiple-scattering theory
The EXAFS equation (3) is derived based on the single-scattering approximation, which assumed that
the final state of the photoelectron is modified by a single backscattering process of neighboring atoms.
Clearly there are other cases if the photoelectron is scattered by more than one neighboring atoms
before returning to the absorbing atom. The single-scattering approximation is valid as long as the
scattering is small. The addition of further scatterings such as scattering from one neighbor to the next
before interfering with the outgoing wave is sometimes important if the scattering is strong, especially
in the low-k region for a low-Z nearest neighbor. Practically, for extracting the structural information
beyond the first shell by EXAFS, the consideration of multiple-scattering processes occurring in
different shell atoms is often necessary. For collinear configuration in which three or atoms are aligned
or almost aligned, the multiple-scattering XAFS contributions can even be larger than the high shell
single-scattering contribution.
Historically, the theoretical treatment of multiple-scattering EXAFS was first proposed by Lee and
Pendry who classified the multiple-scattering contribution in terms of effective path length.[6] This
approach takes the advantage of the Fourier transform in the EXAFS data analysis: since we are
ultimately interested in Fourier transforming the absorption coefficient to position space, at a given
radial distance we can only consider those multiple-scattering effects which contribute to the Fourier
transform at this particular radius and neglect all the others. Its main advantages are its computational
efficiency and accuracy; moreover, it provides a geometrical parametrization and interpretation of
EXAFS, which is ideal for analyzing interatomic distances and other structural quantities. For example,
the double-scattering shown in Fig. 3 is characterized by the effective path length R
eff
= (R
1
+R
2
+R
3
)/2.
In general, these effective path lengths are long and involve successive large-angle scattering events. In
k-space they give rise to high frequency oscillations which tend to cancel out. The multiple-scattering
becomes important only in the low-k region where the electron free path becomes very long.
A major advance in the path-by path XAFS theory, which overcomes all the major computational
difficulties of the multiple-scattering expansion, is based on a separable representation of the
Green’s-function propagators developed by Rehr and Albers.[7][8] For an N-leg path Γ with scatterers
at
R
1
, R
2
, . . . ,R
N
= R
0
, the result for the XAFS amplitude is:
[]
⎭
⎬
⎫
⎩
⎨
⎧
⋅⋅⋅
=
Γ
−
⋅+⋅⋅+++
Γ
12
2/2
21
)2(
2
0
22
121
Im)( FFFMTre
e
Sp
N
l
p
N
i
N
L
σ
δρρρ
ρρρ
χ
. (6)
Here
ρ
i
= p(R
i
−R
i−1
), p=(E−V
mt
)
1/2
is the photoelectron momentum measured with respect to the
muffin-tin zero, F
i
is the scattering matrix at site i, M
l
is the termination matrix for the final state of
angular momentum l. Finally, when one substitutes k=(p
2
−k
F
2
)
1/2
, it is clear that the result for χ
Γ
can be
recast exactly in terms of the standard XAFS equation (3) , but with an effective scattering amplitude
f
eff
,
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