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A Combined Fit of Total Scattering and Extended X-ray Absorption Fine Structure Data for Local-Structure Determination in Crystalline Materials

01 Oct 2009-Journal of Applied Crystallography (International Union of Crystallography)-Vol. 42, Iss: 5, pp 867-877

Abstract: Reverse Monte Carlo (RMC) refinements of local structure using a simultaneous fit of X-ray/neutron total scattering and extended X-ray absorption fine structure (EXAFS) data were developed to incorporate an explicit treatment of both single- and multiple-scattering contributions to EXAFS. The refinement algorithm, implemented as an extension to the public domain computer software RMCProfile, enables accurate modeling of EXAFS over distances encompassing several coordination shells around the absorbing species. The approach was first tested on Ni, which exhibits extensive multiple scattering in EXAFS, and then applied to perovskite-like SrAl1/2Nb1/2O3. This compound crystallizes with a cubic double-perovskite structure but presents a challenge for local-structure determination using a total pair-distribution function (PDF) alone because of overlapping peaks of the constituent partial PDFs (e.g. Al—O and Nb—O or Sr—O and O—O). The results obtained here suggest that the combined use of the total scattering and EXAFS data provides sufficient constraints for RMC refinements to recover fine details of local structure in complex perovskites. Among other results, it was found that the probability density distribution for Sr in SrAl1/2Nb1/2O3 adopts Td point-group symmetry for the Sr sites, determined by the ordered arrangement of Al and Nb, as opposed to a spherical distribution commonly assumed in traditional Rietveld refinements.

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Title: A combined fit of total scattering and extended X-ray
absorption fine structure data for
local-structure
determination
in
crystalline materials
Author(s):
V.
Krayzman
I.
Levin
J.
C. Woicik
Th. Proffen
T.
A.
Vanderah
M. G. Tucker
Intended for: Journal of Applied Crystallography
Los Alamos National Laboratory, an affirmative actionlequal opportunity employer, is operated by the Los Alamos National Security, LLC
for the
National Nuclear Security Administration
of
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rm of this contribution, or
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publisher identify this article as work performed under the auspices of the U.S. Department of Energy. Los Alamos National
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s right
to
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Form 836
(7106)

lournal of
Applied
Crystallography
ISSN
0021-8898
Received 20 April 2009
Accepted
19
lune 2009
© 2009 International Union
of
Crystallography
Prim
ed
in Singapore - all rights
reser
ve
d
1. Introduction
research papers
A combined fit of total scattering and extended
X-ray absorption fine structure data for
local-
structure determination in crystalline materials
V.
Krayzman/,b I.
levin/*
J.
C.
Woicik/
Th.
Proffen/
T.
A.
Vanderah
a
and
M.
G. Tucker
d
"Ceramics Division, National Institute
of
Standards and Technology, Gaithersburg,
MD
20899,
USA
, bDepartment
of
Materials Science and Engineering, University
of
Maryland, College
Park,
MD
20742,
USA,
CLujan
Neutron Center,
Los
Alamos National Laboratory,
Los
Alamo
s,
NM,
USA, and
diS
IS,
Rutherford Appleton Laboratory, Didcot, UK. Correspondence e-mail: igor.levin@nisl.gov
Reverse
Monte
Carlo
(RMC)
refinements
of
local
structure
using . a
simultaneous
fit
of
X-ray/neutron
total
scattering
and
extended
X-ray
absorption
fine
structure
(EXAFS)
data
were
developed
to
incorporate
an
explicit
treatment
of
both singJe- and multiple-scattering
contributions
to
EXAFS.
The
refinement algorithm,
implemented
as
an
extension
to the public
domain
computer
software RMCProfile,
enables
accurate
modeling
of
EXAFS
over
distances encompassing several
coordination
sheUs
around
the absorbing
species.
The
approach
was first tested
on
Ni, which exhibits extensive multiple
scattering
in
EXAFS,
and then applied to perovskite-like SrAltI2Nb,1203' This
compound
crystal1izes with a cubic double-perovskite
structure
but
presents
a
chaUenge for local-structure
determination
using a
total
pair-distribution
function
(PDF)
alone because
of
overlapping
peaks
of
the
constituent
partial
PDFs
(e.g.
AI-O
and
Nb-O
or
Sr-O
and
0-0).
The
results
obtained
here
suggest
that
the combined use
of
the total
scattering
and
EXAFS
data
provides
sufficient constraints for
RMC
refinements to
recover
fine details
of
local
structure
in complex perovskites.
Among
other
results, it was found
that
the
probability density distribution for
Sr in SrAl1l2Nb1l203
adopts
Td
point-group
symmetry for
the
Sr sites,
determined
by
the
ordered
arrangement
of
Al and Nb,
as
opposed
to a spherical distribution
commonly
assumed in traditional Rietveld
refinements.
Local atomic
arrangements
in complex crystalline materials
often deviate from the
average
configurations described by the
space-group symmetry.
Common
examples include solid
solutions
that
are
10caUy
heterogeneous
and
systems with
displacive disorder.
In
many
cases, subtle local-structure
details critically affect
the
functional
properties
of
materials.
Several experimental techniques
can
probe
various aspects
of
local structure,
but
comprehensive
quantitative
determination
of
local atomic
order
remains a chaUenge (Billinge & Levin,
2007).
mation.
However,
a typical distance
range
(~5
ft.)
probed
by
EXAFS
encompasses only
the
first few
coordination
sheUs
around
the
absorber,
which
is
insufficient for analyses of
nanoscale
structural
heterogeneities.
Another
technique
that
emerged
recently as a powerful
probe
of
local/nanoscale
atomic
order
relies
on
X-ray/neutron
elastic total
scattering
(i.e. a
sum
of
the
Bragg and diffuse
diffracted intensities)
to
extract
the
instantaneous
atomic
PDF
(Egami & Billinge, 2003). This
PDF
is
obtained
via a Fourier
transform
of
the
properly
normalized total scattering intensity.
The
total scattering
PDF
is
chemically unresolved
but
extends
over
distances up to
several
tens
of
nanometres. Thus,
EXAFS
and
X-ray/neutron
total
scattering
provide
highly comple-
mentary
information. Unfortunately,
quantitative
refinements
of
local
structure
using a
simultaneous
fit
of
both
types
of
data
are
rare (Bins ted et
aI.
, 2001;
Winterer
et at., 2002; Krayzman et
at., 2008), in
part
because
of
the lack
of
appropriate
software
and established
methodology
for combined refinements.
Extended
X-ray
absorption
fine
structure
(EXAFS)
has
been
for decades the
primary
tool for
quantitative
local-
structure
determination. A
Fourier
transform
of
EXAFS
data
yields a version
of
the
instantaneous
pair-distribution function
(PDF)
around
the
absorbing
species which
is
modified by
several nonstructural
parameters
that
describe scattering
of
a
photoelectron
on
neighboring atoms. AdditionaUy, multiple-
scattering contributions to
EXAFS
render
it sensitive to
bond
angles, thereby providing
three-dimensional
structural infor-
/.
Appl.
Crysl.
(2
009).
42,
867-877
Traditional
EXAFS
refinements
are
performed
by fitting a
signal, calculated for a small atomic cluster
around
the
absorber
species, to
the
experimental
data. Nonstructural
doi
:
10.1107/S0021889809023541
867

research papers
parameters that affect
EXAFS
are calculated prior to refine-
ments using
ab
initio theory
(Ankudinov
et
al.,
1998). Local
structure
is
described using a small number of independent
variables that include coordination numbers, interatomic
distances and their associated
Debye-Waller
(D-W)
factors,
and, if multiple scattering
is
included,
bond
angles (Ravel &
Newville, 2(05). A
PDF
underlying the
EXAFS
signal
is
commonly assumed to
be
a sum
of
the Gaussian peaks. Non-
Gaussian distance distributions can be modeled using higher-
order
cumulants,
but
reliable determination
of
these para-
meters from
EXAFS alone
is
rarely feasible.
The
total scattering
PDF
can yield model-independent
local-structure information by fitting Gaussian profiles to the
first
few
PDF
peaks (Bozin et at., 2007; Qiu et al., 2005);
however, the applicability
of
these analyses
is
limited by the
peak overlap. Typically, a structural model
is
assumed and
refined against the experimental
data
.
One
approach to
PDF
analyses relies on a small unit cell, frequently
under
symmetry
constraints, to describe the structure (Farrow
et
at.,
2007).
Information on local, medium-range
and
long-range atomic
order
can be deduced by selecting an appropriate distance
range for a
PDF
fit:
that
is
,
the
structural model and inter-
pretation of the refined parameters may vary with the distance
range used .. These Rietveld-style real-space refinements are
gaining popularity because
of
the small number
of
structural
variables, which facilitates interpretation
of
the results.
However, despite evident strengths
and
notable successes,
especially as applied to nanomaterials (Masadeh
et al., 2007;
Pradhan
et
at., 2(07), this
method
suffers from several
limitations. For example, it cannot provide accurate treat-
ments of solid solutions, non-Gaussian distributions
of
interatomic distances and local chemical/displacement corre-
lations.
In contrast, reverse
Monte
Carlo
(RMC)
refinements that
employ large atomic ensembles to
fit
experimental
data
using
a variation
of
the Metropolis algorithm enable explicit treat-
ment
of
atomic disorder without a priori assumptions about
the distribution
of
interatomic distances (McGreevy, 2001).
For most systems, the
PDF
alone
is
insufficient to recover a
complete set
of
essential structural characteristics. Simulta-
neous fitting
of
the total scattering
data
in real
and
reciprocal
spaces partly alleviates this problem, while additional inclu-
sion
of
Bragg intensities
as
a
separate
data
set further
constrains average-structure characteristics (Tucker
et at.,
2007). However, in many cases, even these measures cannot
prevent unphysical distances/configurations and, therefore,
various geometric restraints must be imposed
on
interatomic
distances and bond angles
in
order
to obtain chemically sound
structural models (Tucker
et at., 2007).
The
situation
is
parti-
cularly difficult for structures that exhibit a substantial overlap
of
the peaks
of
the constituent partial
PDFs
.
Recently, we
demonstrated
that
simultaneous fitting
of
X-ray/neutron total scattering and
EXAFS
data
diminishes
and sometimes completely obviates the
need
for geometric
restraints (Krayzman
et
at
., 2008). However, these studies
considered only single-scattering contributions to
EXAFS,
which significantly limited the
extent
of
EXAFS
data
included
868
v.
Krayzman el
a/.
Combining
total scattering and
EXAFS
data
in
the refinements. In the
present
work, we developed an
algorithm for a
joint
RMC
fit of the total scattering and
EXAFS
data
that
enables accurate
treatment
of both single-
and
multiple-scattering
EXAFS
contributions; like the
previous version, this algorithm was implemented as an
extension to
the
public domain RMCProjite computer code
(Tucker
et
at.
, 2007).
The
software enables accurate modeling
of
EXAFS
data
over
a distance range up to
~5
A,
which
includes several coordination shells
around
the absorber.
We first tested this
approach
on
Ni
metal because
it
exhibits
extensive multiple-scattering effects
in
EXAFS
data.
Our
combined refinements reproduced the experimental data and
yielded
Ni-Ni
displacement correlation parameters that
matched those obtained from
the
direct analyses of peak
widths in the neutron
PDF. Subsequently, we applied the
method to Sr(Al
lIz
NbJl2)03
and
its solid solutions with
SrTi0
3
.
Sr(AIJI2NbJl2)03 crystallizes with a relatively simple cubic
double-perovskite
structure
but presents a challenge for local-
structure refinements using
either
total scattering
data
or
EXAFS alone because
of
the overlap
of
the
Al-O
and
Nb-
o peaks
in
the total PDF, and the inability to measure Al
EXAFS
owing to its overlap with an
Sr
L-edge.
2. Experimental
The
SAN
(SrAI
lI2
Nb
Jl2
0
3
) and SA NT (SrAlo.
4325Nbo4325-
Ti
o
.
135
0
3
) samples were
prepared
using solid-state methods
from
SrC0
3
(99.99%),
Al
2
0
3
(99.99%, 0.
3/lm),
NbzOs
(99.9985%) and
Ti0
2
« 10 p.p.m. Pl. Prior to heating, the
samples
(~20
g each) were mixed by grinding with an agate
mortar
and pestle for 30 min, pelletized, and placed on beds
of
same-composition sacrificial powder
supported
on alumina
ceramic.
After
an initial overnight calcination at 1223
K,
the
samples were
heated
at 1473 K (18 h), then at 1833 K (112 h)
and then again
at
1833 K (110 h); after this final heating the
samples were cooled to 1723 K
and
annealed at that
temperature
for
144
h.
Samples were furnace-cooled to
973
K
and then air-quenched on the bench-top. X-ray diffraction
patterns were collected using a powder diffractometer
equipped with a
Ge
incident
beam
monochromator (Cu
Kcx]
radiation) and a position-sensitive detector. Both the SAN
and the
SANT
samples contained just-detectable « 1 %)
amounts of
Sr
s
Nb
4
0
15
in
addition to the major perovskite-like
phase.
The
SAN sample exhibited a highly
ordered
arrange-
ment of Al and Nb as confirmed by Rietveld refinements using
X-ray diffraction
data
.
The
SANT
composition with 13.5% Ti
was selected from
our
studies
of
B-cation order/disorder
behavior
in
the
Sr
Ail/
z
Nbl/
z
O
r
SrTi0
3
system, to be described
elsewhere. This
SANT
composition exhibited weak
but
still
discernible long-range AllNb order, making it a suitable
candidate for comparison with
SAN
.
Neutron total scattering
data
were collected at room
temperature on
the
NPDF
instrument at the Lujan Neutron
Scattering
Center
(Los Alamos National Laboratory).
The
samples were loaded in vanadium containers. The
PDFGetN
software (Peterson
el
at., 2000) was used for data correction
and normalization.
The
PDF
,
G(r),
was obtained as a sine
I. Appl.
Cr
y
si.
(2009).
42,
867-877

Fourier
transform
of
the
normalized
scattering
intensity S(Q)
according to
Qm
a:<
G(r) = (2/rr) J Q[S(Q) -
1]
sin(Qr)
dQ
. (1)
o
A value of
Qmax
~
35
A was used in
the
Fourier
transform.
EXAFS
data
for
the
K-edges
of
Ni,
Nb
and
Sr
were
measured
at
the NIST X23A2
beamline
of
the
National
Synchrotron
Light
Source
.
The
data
were
collected in transmission using a
metal foil for Ni
and
powder
samples
for
SAN/SANT.
The
Athena
software
package
(Ravel
& Newville, 2005) was used
to extract the
EXAFS
oscillations.
Preliminary
EXAFS
fitting
was accomplished with
the
Artemis
software
(Ravel
&
Newville,
2005).
Conventional
Rietveld
refinements to
determine
the
average
structures
were
conducted
using
the
GSAS software
package
and
the
neutron
diffraction
data.
The
refined lattice
parameters
and
atomic
positional
parameters
were used to
generate
starting
atomic
configurations for
the
RMC
refine-
ments.
These
refinements
were
performed
using
the
RMCProjile software (Tucker el al., 2007), which was modified
to
enable
a
simultaneous
fit
of
the
total
scattering
and
EXAFS
data
. RMCProjile
adopts
the
PDF
representation
T(r), which
is
related
to G(r) as
T(r)
=
(~c;
b)
2 [G(r) + 4rrr
Po].
(2)
where
Po
is
the
average
atom
number
density, c;
is
the fraction
of
atoms
of
type i
and
b;
is
the
corresponding
neutron
scat-
tering length.
3.
EXAFS
fit
in
combined RMC refinements
The
RMC
method
relies
on
a
Monte
Carlo
'random
walk'
algorithm to identify a
configuration
that
yields the best match
to experimental
data
.
Unlike
downhill algorithms used
in
Rietveld-style refinements,
atomic
moves
that
worsen
the
discrepancy
between
the
calculated
and
experimental
data
are
accepted with a probability
that
decays
exponentially
as
the
misfit increases.
The
target
function,
R?op
minimized
during
the
fit
,
is
the
total residual
R~
o
t
= L
R7
=
L(1/an
L
(Yj.IC
-
yt
P
)2
,
(3)
i i j
where
R;
is
the
agreement
factor for
the
ith
data
set
(or
restraint),
yr
1c
and
Y/,P
refer
to
the
calculated
and
experi-
mental signals/restraints, respectively,
j specifies individual
data
points in each
data
set,
and
lid;
is
the weight
that
controls
the
contribution
of
the
ith
data
set
to
the
target function.
Calculations
of
the
target
function
components
associated
with the
PDF
, S(Q),
Bragg
intensities
and
various polyhedral
constraints
have
been
described
previously (Tucker et al.,
2007).
Therefore,
we limit
the
discussion
here
to
the
details
specific to
EXAFS
.
In
the
present
RMC
refinements,
the
contributions
of
instantaneous local
atomic
configurations to
the
EXAFS
/. Appl.
Cr
Y
5!.
(
2009)
.
42,
867-877
research papers
signal are included explicitly
without
any
assumptions
about
the
type
of
atomic
distributions
or
atomic
motions
.
The
single-
scattering
contribution
to
EXAFS
for the absorbing
atom
i
can
be
calculated as
(Mustre
de
Leon
el
al., 1991)
x
jSi
nglc)(k)
= L S6;ffi/k)IF/rr,
k)1
j
krt;
x sin[2krij +
1/I;(k)
+ rp/rr,
k)]exp[-2r;j/A(k)],
(4)
where
S6;
is
the
amplitude
reduction
factor, ffij(k)
is
the total
central
atom
loss factor, j specifies
the
neighboring atoms
around
the
absorber,
r;j
is
the
interatomic
separation,
k
is
the
photoelectron
wavenumber,
Fj(rr,
k)
= Ifj(rr,
k)1
exp[rp/rr. k)]
is
the
complex
back
scattering
amplitude,
1/Ij(k)
is
the total
scattering
phase
shift for the
absorbing
atom
and
A(k)
is
the
photoelectron
mean
free
path.
The
summation
is
carried
out
over
all the
neighboring
atoms
within a
sphere
of
radius R
max
,
and
the
resulting
EXAFS
signal
is
averaged
over
all the
absorber
atoms
in
the
configuration.
Common
multiple-scat-
tering
processes
that
contribute
significantly to
EXAFS
include
double
and
triple
scattering
(Appendix
A)
.
Double-
scattering
contributions
can
be
calculated
according to
Xjd
o
UbIC)(k)
= L S6;ffi/k) I
r;:f(2)
(!?
,
k)1
j,n
kr;/jn r
nj
X sin[2kr
c
rr
+ 1/I/k) +
rp~:1(2)(!?,
k)]exp[-2r
err
/A(k)):
(5)
where
F1:
1
(2
)(!?
,
k)
is
the
effective
amplitude
for
the
scattering
angle!? (see
Appendix
A for
details)
and
refr
= (rjj +
rin
+
rn;)/2.
Triple-scattering
contributions
are
described
using analogous
equations.
The
only
structural
parameter
that
enters
equation
(4) for
single
scattering
is
the
interatomic
distance
r;j:
that
is
, the
single-scattering
EXAFS
is
uniquely
determined
by
the
absorber-scatterer
partial
PDF.
For
multiple scattering,
EXAFS
depends
on
three-body
distribution
functions
that
can
be
readily
computed
for
any
given
atomic
configuration.
According
to
our
computational
procedure
, al\ nonstruc-
tural
parameters
(i.e.
scattering
amplitudes,
phase
shifts etc.)
are
calculated
prior
to
refinements
using
the
average-structure
model. Variations
of
scattering
amplitudes
due
to small
changes in
interatomic
distances
during
refinements can
be
neglected, as
commonly
assumed
in traditional
EXAFS
analyses
(Ravel
& Newville, 2005). Values
of
F1:
1
(2\{)
,
k)
and
F1:
I
(J
)({),
k) for
double
and
triple
scattering
, respectively,
are
calculated on a fixed
{)
grid
and
then
linearly
interpolated
for
an
arbitrary
angle
during
the
RMC
fit.
EXAFS
data
can
be
fitted in
either
k
or
r space;
however
, an r-space
fit
is
typically
preferred.
In
the
first
step
of
the
refinements,
the
EXAFS
signal X(k)
is
calculated
for all
the
absorber
atoms
in
the
starting atomic
configuration
according to
equations
(3)
and
(4)
and
then
Fourier
transformed
into
real
space as
k,
X(n)(r)
=
[l/(2rr)I
/
2]
J exp(2ikr)kn X(k) dk, (6)
k,
where
n
is
the
k-weighting factor.
After
each
atomic move,
EXAFS
contributions
for all
the
absorber
atoms within a
V. Krayzman el al
.•
Combining total scattering and
EXAFS
dala 869

research papers
Table 1
Displacement
c
orrelation
parameters
for stl'ccessive
nearest
neighbors
(N
N)
in
Ni calculated for
the
RMC-refined
configuration.
NN
Longitudinal
Transverse
1st
(011)
0.36
0.09
2nd
(002)
0.02
0.04
3rd
(012)
0.08
4rd
(022)
0.16
5rd
(013)
0.02
sphere
of
radius
Rm
ax
are
recalculated. Typically,
Rmax
is
limited to
.::::
5 A to maintain reasonable computing times.
Traditional
EXAFS
analyses often treat an amplitude
reduction factor
(~)
and a threshold energy (Eo) as adjustable
parameters. Refinements
of
S
ij
compensate for systematic
errors due to sample/instrument problems
(i.e. sample inho-
mogeneity), whereas adjustments of
Eo
correct deficiencies
in
the muffin tin interatomic potentials and scattering ampli-
tudes. Moreover, separate
Eo
shifts (flEo) often must be
introduced for distinct
absorber-scatterer
paths. In
our
RMC
refinements, both ~
and
Eo
were
kept
fixed. Values
of
these
parameters are determined using traditional
EXAFS
fits
prior
to the
RMC
refinements. According to
our
results, self-
consistent potential calculations implemented in version 8.20
of
FEFF
(Ankudinov et al., 2002) are sufficiently accurate to
obviate the need for multiple values
of
flE
o
.
Several
EXAFS
data
sets can
be
included
in
the refine-
ments. For a real-space fit, the contribution
of
each data set to
the total residual
is
calculated
as
where
1/o1xAFS
is
the weight assigned to a particular EXAFS
data set and the subscripts calc and exp refer to calculated and
experimental values, respectively.
In most cases, both total scattering
and
EXAFS
are domi-
nated by systematic errors and, therefore, a rigorous assign-
ment of weights in the total agreement function [equation (3)]
is
unfeasible.
We
implemented a weighting scheme that adjusts
values of
if;
[equation (3)] during the refinements to maintain
the relative contributions
of
different
data
sets
[i.e.
It;
in
equation (3)] to the total-agreement function fixed. The
reliability of the refined structural characteristics was analyzed
by repeating the analyses using different distributions
of
the
weights.
4. Results
and
discussion
4.1. Nickel metal
Metallic nickel was chosen as a test system because a
Fourier transform
of
the
Ni
K-edge
EXAFS
data features a
well separated first-nearest-neighbor peak and strong
multiple-scattering effects within the collinear chains formed
by the first and fourth nearest neighbors. This type of
EXAFS
data
is
typical for all face
centered
cubic metals.
870
v.
Kray
zman
e!
a/ .
Combining
total scattering and
EXAFS
data
Rietveld refinements using
neutron
diffraction data yielded
an
Ni
atomic displacement
parameter
(ADP)
Vi
so =
0.00573 (3) A
2.
The
first-nearest-neighbor peak
in
the neutron
PDF
was satisfactorily fitted using a Gaussian profile with
'Ni-Ni
= 2.487 (1)1 and
~i-Ni
= 0.0064 (1)
A2.
A single-shell
fit
of
the first peak
in
EXAFS
generated
'Ni-Ni = 2.485 (2) A
and
O'~i-Ni
= 0.0064 (2) A
2,
which agree with the PDF-derived
values. Refinement
of
sij
and flEo during the EXAFS
fit
produced
sij
= 0.84 (2) and flEo = +3.3
eV
(hereafter, flEo
is
specified with respect to the edge inflection point). The refined
value
of
sij
was smaller than
sij
= 0.953 predicted by FEFFB.20;
however, such relatively small differences
are
not unusual as a
result
of
experimental errors. Refinements using a fixed value
of
sij
= 0.953 yielded
~
= 0.0073 (2)
A2
.
Good
agreement
between the 'Ni-Ni
and
~ values
obtained
by fitting the first-
nearest-neighbor peaks in the
PDF
and
EXAFS
suggests that
the two
data
sets
are
consistent with each
other
and, therefore,
amenable to simultaneous refinements.
RMC
refinements were
performed
using
an
atomic config-
uration
of
4000 atoms arranged in the cubic lOa x lOa x lOa
box
(a
= 3.5323 A
is
the lattice
parameter
of Ni). The total
PDF, EXAFS
data
and Bragg profile were fitted simulta-
neously.
The
fitting ranges were 2.1-17.5 A for the
PDF
and
1.5-5.2
A for EXAFS.
The
profile shape and background
parameters for the Bragg profile were obtained using Rietveld
refinements and kept fixed during the RMC
fit.
Satisfactory
agreement between the calculated
and
experimental profiles
for all the
data
sets was
obtained
(Fig. 1).
The
D-W
factor
O'~i-Ni
~
0.0064
A2
for the first
Ni-Ni
coordination
in
both the
PDF
and
EXAFS, as estimated from
the first peak
fit
,
is
significantly smaller than the value
of
2V
iso
~
0.0114 A
2
expected for uncorrela ted atomic motion.
The
correlation
parameter
<p
= 1 -
~
Ni-N;!(2V
iso
)
is
equal to
0.4 (1), which corresponds to a preference for the in-phase
displacements
of
the nearest-neighbor
Ni
atoms.
RMC
refinements enable calculations
of
any type
of
interatomic correia tions. For example, displacement pair
correlations can
be
evaluated as
(8)
where
Uj
is
the displacement
of
an atom from the lattice site i
having a radius vector
rio
Dt
and
D2 are the unit vectors along
the directions
of
interest, and angular brackets indicate an
average over the atomic sites
i
and
j.
Table 1 summarizes displacement correlation parameters
calculated for several successive coordination shells using the
refined
RMC
configuration.
Both
longitudinal and transverse
correlations were considered.
The
correlation parameters
calculated from
our
RMC-refined configuration up to the
fourth coordination shell agree reasonably well with those
obtained from the peak widths
in
the
neutron
PDF.
Our
values
for the correlation
parameters
also agree with those reported
previously by Jeong
et
al
. (2003).
1 Hereafter, the numbers
in
parentheses refer to a single estimated standard
deviation.
/.
Appl.
Crys!. (2009).
42,
867-877

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