Inelastic mean free path and phase-shift determinations in NiO, using EXELFS

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Inelastic mean free path and phase-shift determinations in NiO, using EXELFS

Mohammad Tafreshi, Stefan Csillag, Zou Yuan, Christian Bohm, Elisabeth Lefèvre, Christian Colliex

To cite this version:

Mohammad Tafreshi, Stefan Csillag, Zou Yuan, Christian Bohm, Elisabeth Lefèvre, et al.. Inelastic

mean free path and phase-shift determinations in NiO, using EXELFS. Journal de Physique I, EDP

Sciences, 1993, 3 (7), pp.1649-1659. �10.1051/jp1:1993207�. �jpa-00246823�

Classification

Physic-s

Abstracts

71.20 78.70D 61.14

Inelastic

_mean

free path and phase-shift determinations in NiO, using EXELFS

Mohammad A. Tafreshi

(I),

Stefan

Csillag ('),

Zou Wei Yuan

('),

Christian Bohm

('),

Elisabeth Lefbvre

(2)

and Christian Colliex

j2)

(')

Institute of

Physics,

University of Stockholm,

Vanadisvhgen

9, S- ii 3 46 Stockholm, Sweden (2) Laboratoire de Physique des Solides, Bit. 510, Universitd Paris Sud, 91405 Orsay, France

(Receit'ed 22 September1992, accepted in final form 2 March 1993)

Abstract. Structural and chemical information about the local atomic environment in

a

specimen

can be obtained

by analysing

the EXELFS-modulations

occurring beyond

_energy loss ionization edges.

Using

spectra ^{recorded from} _a thin NiO sample, ^this _paper reviews the problems and

possibilities

associated with distance and

intensity analysis

in EXELFS.

Knowing

_a priori the

structure of the sample, _{one can} determine unknown parameters involved in the

analysis

of the data

such _{as an} average value of the inelastic _mean free

path

for the

ejected

electrons, a linear approximation for the k-dependence of the oxygen d~~, oxygen-oxygen pair d~~_~, ^{and nickel}

backscattering

d~

~~, phase shifts.

1. Introduction.

The energy distribution of electrons

passing through

a thin foil of material is _{a measure} of the electronic excitations

produced

in the

sample by

the beam.

During

inelastic

scattering

events, the incident electrons excite inner-shell electrons. Interference between the

outgoing

excited

inner-shell electron _waves and electron _waves backscattered from the

surrounding

atoms

affects the

energy-loss

spectrum. ^The

probability

for _a

given

_energy loss increases _or decreases

depending

on whether constructive _or destructive interference _occurs

[Il.

Approximating

the

ejected

electron _wave function at the

backscattering

atom

by

_a

plane-

wave while

assuming single scattering

formalism with _a small

scattering angle,

the

interference

amplitude

_can be described

[2] by

:

where

N~

^{is the number of atoms in shell}

j,

_i~ is the

radius,

_W~

(k)

is the total

phase change

of the electron _wave after

t?avelling through

the field of the

emitting

and

backscattering

atoms,

f~(k)

is the

backscattering amplitude

which

depends

on the type ^of

backscattering

atoms,

A is the mean free

path

^{for inelastic}

scattering

of the

ejected

^{electron, and} _«~ ^is a parameter

including

thermal vibrations and static disorders of atoms.

The total

phase shift, W~(k)

^{for K-shell ionisation}

edges

is described

[3] by

:

mj(k>

_~

ma(k>

₊

m~(k>

_ar

(2>

where

W~(k)

and

W~(k)

are

phase

^{shifts due}

respectively

to the central and

backscattering

atoms.

Changing f~(k)

to

f~(k,

r) ^and

W~(k)

to

4l~(k, r),

ⁱⁿ

expression (21,

recasts this

expression

from the

plane-wave approximation

to full curved-wave formalism

[4].

Depending

on the _a

priori

known

parameters, equation (I)

can

provide

means for

deriving

information about the local atomic environment,

[I.e.,

_q,

N~, ^A,, 4l~(k), W~(k), f~(k)

and

«jl 15].

2.

Experimental

data

acquisition.

EELS spectra ^{have been recorded under} a

primary

beam

voltage

of 100 kV with _a Vacuum Generators dedicated STEM

equipped

with _a Gatan

parallel

EELS spectrometer

(see

Bouchet et al.

[6]

for _a

complete description

of the instrument

operated

in

Orsay). During

_spectrum

acquisition,

the incident

probe

is

permanently

scanned _over

specimen

_areas of

typically

10 _to 30 _x 10 _to 30

nm~.

The

angle

of acceptance ^{of the inelastic electrons is determined}

by

the size of the inner hole in the annular dark field

detector,

which

corresponds

to about 25 mrad at the exit surface of the

specimen.

NiO

specimens

were

specially prepared by

C-E-

Rojas

as part ^of a

study

on model

catalysts.

After

impregnation

and calcination of _a mixture of

amorphous

silica and nickel nitrate, the reduction to metallic nickel is carried _out in _a

thermoprogrammed

reduction

(TPR)

system. A

large peak,

observed

during

this

procedure

at 392

°C,

is

normally

associated with the presence

of bulk NiO. In order _to be

observed,

the

samples

were crushed in _an agata mortar and

suspended

ⁱⁿ ethanol before

deposition

and

drying

_{on a} thin

holey

carbon foil. The

particles

observed showed _an average size of 90 _nm but looked

generally

like clusters of smaller size.

Selected _area diffraction pattems

correspond

to

crystalline

NiO. EELS spectra are recorded

from _areas close to the

edge

of such thin

microcrystals protruding

over the holes of the

supporting

films. The average thickness _was estimated _to be less than 0.5

A,~,

where A,~ ^{is the inelastic} mean free

path

of the incident electrons. This _was done _to avoid the

superposition

of

plasmon

_type satellites, due to

multiple

inelastic events, over the first 50 _ev

following _{_the edge.}

Typical

spectra are shown in

figure

I. When

focusing

_on

edge

fine structures, the energy

dispersion

is _{set to cover a}

typical

100 eV window.

Energy

resolution better than I eV can then be achieved with _a sufficient

signal-to-noise

ratio. The

analysis

of the

origin

of the fine structures observed within the first 50 eV above threshold is discussed in another paper

by

Kurata et al.

[7].

^For

EXAFS-type studies,

the whole energy range from the O-K

edge

to the next

edge (I.e., Ni-L~~

at 855

eV)

must be recorded

(see Fig. lb)

with _a sufficient

signal-to-

noise ratio, in order to detect the

periodic

oscillations _over this extended energy loss domain.

Since the

Ni-L~

and

Ni-L~ edges

are

relatively

close in energy and thus

overlap,

structure

obtained

beyond

these two

edges

cannot be used for EXELFS

analysis,

otherwise such _an

analysis

would have

hopefully strengthened

and confirmed the results obtained from the O-K

edge.

30 General comments about the

analysiso

The structure

parameters

^{of NiO}

(I,e.,

of Nacl

type

^with a ₌ 4,1767

1) _previously

_determined

by

other

techniques [8] together

with the

energy-loss

spectra ^{recorded from the} oxygen

K-edge

in _a thin NiO

sample (Fig. lb),

have been used to :

Perform and

investigate

the

EXELFS-intensity analysis.

Determine the average value of the inelastic _mean free

path

of electrons

ejected

from oxygen ^atoms in NiO.

Obtain

experimental

values for

4lo_o (I,e., phase

shifts due to the O-O

pair),

and _a mixture of the

experimental

and theoretical values for

4lo_

and 4l_N,

(I.e., phase

shifts due to oxygen central and nickel

backscattering

atom

respectively),

as well _as the linear fit _to these

phases

_over the

experimental

data range.

Provided

optimum sample

thickness is

achieved,

the _errors introduced in the determination of the _non

phase-shift

corrected interatomic distances and

corresponding intensities, originate mainly

from the

following

_{sources :}

Low resolution effects and truncation artefacts introduced _{as a} consequence of the short data interval and limited number of data

points.

These effects _can however be estimated and corrected for

by analysing

simulated spectra

corresponding

_to the _same conditions _as the

experimental

one.

Inaccuracy

in determination of the K~~~

K~,~ value,

caused

by

_an inaccurate choice of the

Eo position (I.e.,

_energy threshold

position)

and inaccurate calibration of the energy axis.

The main effect of this

inaccuracy

is _a shift of all measured distances with

equal

percentage in the _same direction

[9].

§~ 27

O 7

U fi

]O ⁵

1 f3

520 540 560 580 600

Energy Lou (cV) Ch&""tl ~~.

~~ b>

Fig.

I. al Energy ^loss spectrum

displaying

the _near edge structure of the O-K edge from _a thin NiO sample b) _energy loss spectrum ^{from the} same NiO _area exhibiting the O-K and Ni-L~~ edges. Energy loss interval is from 460 _to 948 eV.

The _near

edge

structure of the O-K

edge

in NiO has been

theoretically

calculated

[10].

Comparing

the

experimental

_spectrum with the

theoretically

calculated spectrum ^allows an

accurate determination of the

Eo position.

The value obtained for

Eo using

this

procedure

is

found _to be 530 eV, I-e- 10 eV below the

position

of the most intense

peak (observe

that the

same

value,

I-e. 530 eV, has also been used

by Leapman

et al.

[I II).

Since calibration of the energy axis has been

accurately

determined

by using

the energy difference between oxygen K-

edge

and nickel L~

~

edges,

the

inaccuracy

introduced

by

this step ^{should be}

negligible.

Inaccurate

background

subtraction _can

give

rise to false _structures in the RDF

(I.e.,

Radial Distribution Function _or

magnitude

of the

FFT)

at small I-values.

Background

subtraction has been carried out

by fitting

two third order

polynomial

functions to different

parts ^{of the data interval of interest. In order} to minimize the

inaccuracy,

the

fitting regions

have been chosen _as

long

as

possible

(I,e., one from channel 157 to 691 and the other from channel 435 _to

803,

_see

Fig, lb).

The first

peak,

obtained in the RDF

(Fig.

2b,

peak Po),

is _more

likely

due _to this

background

subtraction effect and _to the truncation artefacts introduced

by

^the second

peak. Analysis

of

simulated data shows that this

peak

has _a few percent ^influence on the

intensity

of

Pi (I.e., Ip,,

see Tab. I and

Fig. 2b),

and _{none on} its

position.

Since intensities in _our

analysis

are determined

by fitting

the RDF of simulated data _to the

experimental

data, the effect of this

peak

can be

compensated

for

by including

the

corresponding frequency

in the simulation

model.

Q _~

W fi

§

_I

E ]

« s s a 1 4 o la u u 13 1«

A'~ AngsUom

a> b>

Fig. 2. al Spectrum from

figure

16 after background subtraction, conversion from energy to k-space, multiplication by k~ and filter function _; b) Magnitude of the FFT of (al, P4 is not observable due to its low intensity (less than 0.5 times Ip~ and 0.4 times Ip~), however its position ^{should be} at about 3.88 A.

4. Data

analysis.

The data

analysis procedure

used in this

work,

which has been

developed by

the Stockholm group and discussed

extensively by

^Tafreshi et al.

[9, 12],

includes the

following

_steps

selection of

Eo Position

at 530 eV,

subtracting

the main

background,

conversion from energy space ^to

k-space, multiplying

the data

by

^k~ in order _to compensate ^{for the}

damping

effect of the

k-dependent

factors

[I,e., Ilk, f~

^{(k) and} «-term, see

Eq. II II

and

multiplying by

_a filter

function which varies _as cosine square ^at the ends of the data range but is flat _over 50 fib of the central

region (Fig. 2a).

As a consequence of the short data interval

corresponding

to the

wavelengths

of the

principal

modulations, it is useful to zero-extend the data outside the

original

data interval

prior

to the Fourier transform in order _to

improve

the apparent resolution.

However this

improvement

is achieved _at the _cost of increased transformation time and the appearance of smaller satellite

peaks

due to the truncation effect.

Special

care must be taken when

interpreting

these

peaks.

In order to minimize

inaccuracy,

fifteen different data intervals have been chosen and Fourier transformed. The _mean and standard deviations of the FFT results obtained from these

intervals

(Fig.

^2a and

Fig. 2b), (Tab. I),

is

being

used _as the

experimental

^values.

Table I. The lst column is the central and backscatterer atoms in the

first

_seven shells

expected

_to

give

most contribution _to the EXELFS

modulations,

the 2nd column is the

expected

values

of

the

corresponding

interatomic distances in

~ngstrom

;

3rd, expe;imental

non

phase-shift

cot.i"ected distances in

~ngstrom _4th, dijfiet.ences

bemleen the known interatomic distances

(I,e.,

column

2)

and the

experimental

_non

phase-shift

corrected

(I,e.,

column 3

)

_; sth,

expe;imental

intensities

(I.e., height ofthe peaks

) _; 6th,

expected

coo;dination

numbers

;

7th, expe;imental

_q~o (w>ill ^be discussed in the

following)

in ;adian.

2 3 4 5 6 7

O-Ni 2.09 1.81 ± 0.02 0.28 _± 0.02

lpi

= 100 _± 0 6 0.30 _± 0,14

O-O 2.95 2.65 _± 0.02 0.30 _± 0.02

Ip2

₌ ⁸⁰ ± 2 12 4. 84 _± 0,19

O-Ni 3.62 3.36 _± 0.04 0.26 _± 0.04

Ip3

₌ 41 _± 2 8 3, I1 _± 0.23

O-O 4,18 6

O-Ni 4.67 4.32 _± 0.04 0.35 _± 0.04

Ips

=

36 _± 3 24 0.92 _± 0.31

O-O 5.12 4.79 _± 0.06 0.33 _± 0.06

Ip~

=

34 _± 2 24 4.39 _± 0.40

O-O 5.91 5.53 _± 0.02 0.38 _± 0.02

Ip~

=

32 _± 2 12 3.09 _± 0.19

4.I ANALYSIS OF THE PHASE SHIFTS. For many elements the

phase

shift

[W~(k)],

^is

approximately

linear in k _over the

analysing

interval and _can be written

[13]

_as

~°j(k>

+ " ~

~°a(k>

₊

ibb(k>

# §~o + §~I k.

(3>

While _q~o is _{a constant}

phase

term which _cannot affect the distances obtained from the

RDF,

q~j introduces _a shift of these distances towards lower values. For those elements for which

W~

(k)

cannot

accurately

be described

by equation (3),

this

expression

_can still be considered _as

a linear

approximation.

A

comparison

between

expressions

I and 3

implies

that the

experimental

distance obtained from the RDF is

equal

to r +

q~j/2.

Thus

subtracting

the

expected

^{value of} ; from the

experimentally

measured value

(I,e.,

_{; +}

~j/2), yields q~j/2.

In

general

the

imaginary

and real parts,

corresponding

to a

particular peak

in the

magnitude

spectrum ^of a Fourier

transform,

can be used to calculate the

phase

value _«

= arctan

(Im/Re ).

This is the

phase

of the _wave

corresponding

to that

particular peak,

at the

beginning

of the transformed data interval. Since the

frequency corresponding

to this _wave is

already

known

(I,e.,

the

position

of the

peak

in the

magnitude spectrum),

it is

possible

to determine the

phase

value of this _wave in

regions

outside the transformed data interval.

In _{our case} the

imaginary

and real parts

corresponding

to a

particular peak

in the

magnitude

spectrum

(see Fig. 2b), yield

the

phase

value _at k~~~

(in

the

k-space). Using

this

phase

and the

corresponding frequency

allows _us to determine the value of _q~o

w

(I,e., phase

^shift at

ko or

Eo).

Column 7 in table I shows the obtained

phase

shifts with correction for

w

(I.e., ~o)

for different atomic shells.

Analysis

of the simulated data indicates that _{as an} effect of

overlapping neighbouring

shells

(or

low

resolution),

with the

exception

of the

peaks Pi

and

P~

^the

position

of other

peaks

ⁱⁿ the RDF

(particularily P~),

cannot

accurately

^describe the

corresponding

interatomic distances.

Since the obtained value of _q~o

depends

_on the

position (I.e.,

distance _or

frequency)

of the

peaks,

the

relatively

_poor agreement ^{between the} _q~o ^{values obtained}

using

these

peaks (particularily

for P~, see

Fig. 2b)

_can

mainly

be

explained by

^{this effect.} Curved-wave effects

[4]

also introduce _a

slight r-dependence

in the

phase

shifts.

The

experimental

value for the

non-phase

shift corrected first O-O distance is 2.65 _± 0.02

1

(see

^Tab.

I),

while the

expected

distance for this shell is

2.951 _(see

_{Tab. Il.}

Subtracting

2.95

from 2.65 _± 0.02 and

multiplying by

two

gives

_q~

j as 0.60 _±

0.041.

_{Since the}

experimental

q~o for this shell is 4.84 ± 0.19

radians,

the

phase

shift for the first O-O shell in the

analysing

interval

(I,e.,

2.3 _to 8.5

l~ ')

_can be written _{as :}

4lo_o

=

(0.60

_± 0.04

)

k

(4.84

_± 0.19

(4)

where k is in

l~

' and 4l in radians. The

phase

shift calculated

by

Teo & Lee

[14], using

the

plane-wave approximation

in the k-interval 3.8 to 8.5

l~ ',

leads to

4lo_o

=

0.67 k 3.49.

Using 4lo_ (I,e.,

_oxygen central atom

phase shift)

calculated

by

Teo & Lee and

4l_o

calculated

by

McKale

[4], considering

curved-wave formalism, _over a shorter k-interval

(I.e.,

from 3.8

l~

_{to 8.5}

l~') yields 4lo_o

= 0.59 k 4.90 (see

Fig. 3a).

-6 00 5 50

, ,

, ,.:'~--.

, :

, ". ',

,

,

".

, 5 00 :~ "..., ,_'

', _{i ".,} "

-8.00

.. , I [ I, ".. '

". ',

~, ".., ',

'" ',

450 I "., ',

, c

' ,

~,

,

~

£ _".

,

".. ',

i

". ^"

... , ~ _".

".., '

400 ". _,

".., ',

",

"".. ',

".._ '

".._

....

~

"

3 50

3 oo 16

A"~ A"1

a> _i~~

Fig. 3. k-dependent phase shifts

: a) ^{the dashed} curve is d~~_~ using d~o ^and d~

o calculated

by

Teo and Lee [14], where d~

o is calculated

using

the plane-wave

approximation.

The dotted _curve is the 4~o-o, ^where d~o- ^{is the} same as in the previous _case but d~_~ is calculated

using

the curved-wave

formalism [4]. The solid line is the

experimental phase

shift extracted from this work, I.e. _a linear fit _to the 4~o-o values _over the

corresponding

k-interval (from 2.3 _to 8.5 A~ ^' b) 4~

~~

values _: the dashed _curve is calculated using the plane-wave

approximation

[14], while the dotted _{curve uses} the curved-wave

formalism [4]. The solid line is the linear fit _to d~

~, obtained in the present study.

The

good

agreement ^between

theoretically

calculated and

experimentally

determined values for

4lo_o

and the accuracy

[15]

of the calculated

4lo_,

support ^{the claim} that the calculated

values for 4l

o,

using

curved-wave formalism, _{are accurate.} The first order

polynomial

fitted to this

phase

shift _over the

experimental

_range

(I,e.,

2.3 to 8.5

l~ yields 4l_o

=

-0.25k+

0.54,

subtraction of this from the

experimentally

obtained

4lo_o (I.e., expression (4))

allows _{us to} determine the

phase

shift of the central _{atom over} the

experimental

data range ^{as :}

4~o_ =

(0.35

± 0.04 k

(5.38

_± 0,19

). (5)

This _can be

compared

to

4lo_

=

0.40 k 5.10, the value obtained

using

the

phase

shift calculated

by

Teo & Lee _{over a} shorter data range

(I.e.,

3.7 to 8.5

l~

_~).

The

experimentally

obtained value for the first

non-phase

shift corrected O-Ni distance is 1.81 _± 0.02

1 _(see

_Tab.

_Ii.

_{This value}

together

with the

expected

value for this distance

(I.e.,

2.091)

_{leads to}

4lo_~~ =

(0.56

_± 0.04 ) k

(0.30

_± 0.14

(6)

Subtracting

the

previously

determined

4lo_ (I.e., expression (5))

from this

yields

the nickel

backscattering phase

shift

(see Fig. 3b)

4l

~, ⁼

(0.21

_± 0.06

)

k ₊

(5.08

_± 0.24

(7)

While the calculated

[14] phase

shift

using

the

plane-wave approximation

over a shorter k- interval

(I.e.,

3.8 to 8.5

l~ ')

resulted in 4l

~, ⁼ + 0.10 k ₊

4.60,

the calculated

[4]

values

using

curved-wave formalism _{over a} k-interval

equal

to the

experimental

interval leads to 4l ~, ⁼ + 0.00 k ₊ 5.00

(see Fig. 3b).

The

disagreement

between

experimental

and theoretical values of

4l_~,

can be

explained partly by

inaccuracies in the calculated

backscattering phase

shift values

(a

_common

problem

for

relatively heavy

atoms

[15]),

and

partly by

the fact that the data interval used for EXELFS

analysis

includes the lower part ^{of the k-interval}

(see Fig. 3),

where the

reliability

of the

calculated

phase

shift values is limited and the

validity

of the linear

approximation

questionable.

The

phase

^shifts values for

4l_o

^and

4l_~,

^{have been calculated}

by

McKale

[4]

for two distances

(I.e.,

2,10 and

3.751

_{in the}

case of 4~_o ^{and 2.75 and}

41

_{in the}

case of

4l_~,),

therefore linear

interpolations

have been used _to obtain the calculated

phase

shifts in

figure

3

corresponding

to 2.95 and

2.091distances respectively.

Since the 4l

N, has _not been calculated

(by

Teo & Lee

[14]),

the 4l_N, ⁱⁿ

figure

3b has been obtained

by

a linear

interpolation

between the calculated 4l

~~ and 4l

c~.

4.2 INTENSITY _ANALYSIS. To the _extent that the inelastic _mean free

path

of the

ejected

electron

(I.e., A,)

_over the actual k-interval _can be considered to be

independent

of

k,

the

intensity

of the

peaks

in the RDF _can be described

by

the

following relationship

:

>~

fit 2

Ip~ ^cc S~

fl

_e ^~'

(8)

r~

where Ip~ ^{is the}

experimentally

measured

intensity (I,e., height

of the

peak)

due to the shell

j

(see ^Tab.

I),

^and

S~ is _a parameter

describing

the combined effect of the

product

of

backscattering amplitude

and disorder term

[I.e., f~(k) exp(-

2

ml k~)]

_on the

intensity.

Since both the first and fifth

peaks

in the RDF

(see Fig. 2b)

contain the _same backscatterer

atoms

(I,e., nickel), Sj

should

approximately

^be

equal

to S~. ^{On this account}

using

equation (8),

the ratio of

lpi /Ip~ yields

2(1"~

rj)

A, =

(9)

In

(lpi N~ ;()

In

(Ip~

^N

j

r))

Since all parameters ⁱⁿ

equation (9)

_are

known,

it is therefore

possible

_to determine the value of

A,

_as 6.5

1.

_{It should be mentioned however that the}

same calculation could also be carried out

using intensity

of the third

peak (I,e., Ip~)

which also

corresponds

to a nickel shell.

However the low

intensity

and poor resolution of this

peak

_suggests the _use of the fifth

peak

in the calculations.

Using equation (8),

the ratio of

Ipj/Ip~

leads _to the

following expression.

S~~

lpi N~ r(

_{i-j r~}

= exp 2

(10)

So

_I

p~ N

i~

^A

This

expression

describes

S~,/So

_{as a} function of, _among

others,

the measured intensities

corresponding

_to shell I and 2.

Using

^this

expression together

^{with the}

already

calculated A, ^value (I.e.,

6.51), _lpi

_and

_Ip~,

the value of

S~,/So

ratio _can be calculated _as 0.96.

In order to prove the accuracy of the results, _a NiO

EXELFS-spectrum including

the first ten shells around the oxygen as central _atom has been simulated _over the _same k-interval _as the

experimental

data (see

Fig. 4a).

The known values of the interatomic

distances,

coordination numbers, and the

previously

determined values of 4~o_,

4l_o

and 4l_~~ have been used. In order

to better

approximate

the real

physical

conditions, _a

background equal

to the

remaining

background

in the

experimental

RDF

(Fig. 2b)

has also been included in the simulated data.

Since the resolution is _not sufficient to

distinguish

the difference between different disorder parameters due to the difl'erent

shells,

and since before

attempting

to Fourier transform, _a

compensation

for the

damping

effect of the

k-dependent

factors has been

applied,

the disorder

parameters ^{has been taken} as zero, while the obtained values for A,

(I.e.,

A, ₌

6.51)

_and

S~,/So (I.e., SN,/So

₌

0.96)

have been used.

Analysis

of the simulated spectrum ^indicates that, due to the effects of low resolution and trucation artefacts, the ratios between intensities in the

RDF,

are not

equal

to the

expected

ratios

originating

from the EXELFS modulations. This in _{turn means} that

using

the

experimental

intensities in table I _{can cause} a

relatively large

_error in the determination of A, ^{values and}

S~~/So

ratios. In other words we cannot use the measured intensities

(I,e., lpi, Ip~

^and

Ip~)

without

considering

the effect of low resolution and truncation artefacts.

To solve this

problem

we simulate data with different _A~ and

S~,/So

values _to find those values which

give

the best fit between the intensities of

Pi,

P2 and P5 (see

Fig. 2b)

in the RDFS of the simulated and

experimental

spectrum. ^{The best fit}

(Figs. 4a, b)

^{is achieved} for A, =

6.51

and

SN,/So

₌ 1,10.

The standard deviation of the measured intensities

(Tab.

I, column

5),

causes error

margins

in determined values of the

A,

^and SN~/So ^ratio.

Considering

these deviations the determined values become

A,

₌ ^6.5 ± 0.7

1

_and

SN,/So

=

I.10 _± 0.04

respectively.

Assuming approximately

the _same disorder parameters «~, ^{for all atomic shells around the} oxygen ^{centre atom,} the

S~,/So

becomes _a parameter

describing

the average ratio between the

backscattering amplitudes

of the nickel and oxygen atoms

(I,e., f~,/fo). Using

calculated

backscattering amplitudes

based _on curved-wave formalism

[4],

the average value of the

fNi/fo

_over the

analysed

interval

(I.e.,

2.3 to 8.5

l~

~)

yields

I.

II,

I-e- a very

good

agreement with _our

experimental

results.

Using

calculated values based _on the

plane-wave approximation [13],

_a shorter interval

(I.e.,

3.8 to 8.5

l~'), _yields

1.57.

4.3 THE _EFFECTS OF MULTIPLE SCATTERING.

Equation

(I takes into account

only

_a

single

backscattering

from the

neighbouring

atoms, while the

ejected

electron in

principle

_can be backscattered after

multiple scattering [16]

from different atomic shells. Due to the

longer

effective

path (I,e., longer distance), multiple scattering

of the

ejected

electron does _not affect either the

position

or the

intensity

of the

peaks corresponding

to the first interatomic distance.

Theoretical and

experimental

work about the

multiple scattering

effect has been

performed by

^{different authors}

[10, 16-20]. Study

of the MS effect _{on near}

edge

structure of the NiO shows

[10]

that

beyond approximately

20 eV above the O-K

edge

the so-called type I -MS is indeed dominant, so that _an

expansion

ⁱⁿ order to take other types ^{of MS into} account is not

necessary. The type ^{I-MS is when}

scattering

occurs from _atoms which _are

arranged

z 3 « s 7

h-1

a)

3 5 6 la 14 15 is

Angs~om

b)

Fig.

4. a) Simulated EXELFS data including the first _ten atomic shells around _an oxygen atom, over

the _same k-interval _as in figure 2a, ^with A, =

6.5 A and SNI/S~ ₌ ^I, lo b) magnitude of the FFT of (al.

approximately colinearly

with the central _{atom so} that the

intervening

atom focuses the

ejected

electron _wave onto the

backscattering

_atom and enhances

scattering

from it. This _can, in _turn, enhance the

intensity

of _some

pealcs

in the

experimental

RDF

(see Fig. 2b).

In NiO, the atoms in the first shell have coordinates

(1/2,

0,

0)

and lie

colinearly

^with atoms in the fourth shell with coordinates

(1,

0,

0)

and hence the

multiple scattering

can affect the

intensity

of the P4 in the

experimental

^RDF. The second shell has coordinates

(1/2,

1/2,

0)

and

can enhance the

intensity

of the seventh

shell,

which has coordinates

(I,

1,

0).

The

intensity

of the ninth shell, with coordination

(I,

I,

I)

_can also be

amplified by

atoms in the third shell,

which have the coordinates

(1/2, 1/2, 1/2).

Since the

positions

of the nearest

peaks,

Pi and P2, have been used in the

phase

shift

analysis, multiple scattering

cannot have any

significant

effect _on the results of this

analysis.

In the

intensity analysis lpi, Ip~

and

Ip~

have been used which indicates that _even here

multiple scattering

effects _can

only

have a very limited effect on the results.

The

relatively high intensity

of the 7th and 9th

peak

in the

experimental

RDF

compared

with the RDF of the simulated data, _can be

partly explained by

the noise effects and

partly by

the

multiple scattering

contribution.

5. Discussion.

EXELFS

analysis

_over several short data intervals indicates that the

phase

shift values obtained in this work _are valid _even for shorter

intervals,

which in _turn

implies

that

they

_can be used for

typical

EXELFS

analysis.

The curved-wave formalism suggests ^{that the obtained}

4lo_o

^{is valid for} r =

2.951

_and

4l_~,

^for _;

=

2.091.

_{However calculated}

phase

shifts

using

curved-wave formalism show that the

phase

shift

corresponding

to a

particular

distance is also valid within _a certain interval

around it without any

significant changes [4].

Comparing

the values for

4l~,

obtained in this work

[I.e., (-0.21±0.06)k

₊

(5.08

± 0.24

)]

with

corresponding

values obtained

using

curved-wave formalism

[4] (I.e.,

+ 0.00 k ₊

5.00),

indicates _a

discrepancy

between the

slopes

which _cannot be

explained by

the

error

margins.

It should be mentioned here that the

slope

of the calculated

phase

shift in the

higher

_{energy part} is about

0.181,

_{which is close to the}

experimentally

determined value.

The value obtained for

4l_~,

^{in this} work is _a linear fit _to this

phase

and cannot therefore be

regarded

as the real

phase.

The

discrepancy

between calculated and

experimentally

obtained values

emphasizes

the

necessity

of

performing

EXELFS

analysis

_on

specimens

with _a known _structure in order to determine

experimental

values for the

backscattering phase

^shift and

amplitude, particularly

for relative

heavy backscattering

atoms.

As discussed

here,

the

intensity

of _a

peak

in the RDF _can be influenced

by

^{several factors}

by

noise _as well _as

by

low resolution and truncation artefacts due _to the

neighbouring peaks.

For _an accurate

intensity analysis,

these effects should therefore be considered and accounted

for. In other words when resolution in the RDF is

relatively

low

(which

is the

typical

_case in the

EXELFS),

the effect of

overlapping peaks

makes the _use of back Fourier

filtering procedure

rather inaccurate and therefore _we have

adapted

the method described in this work for data

analysis.

6.

Summary.

For _an accurate EXELFS

intensity analysis,

the effect of low resolution and truncation

artefacts must also be considered. The method used in this

work, analysing

simulated spectra

corresponding

to the same conditions _as the

experimental

_{ones, can} be used to test the

reliability

of the obtained results.

In

spite

of the limitations above, an accurate

intensity analysis

is still

possible by fitting

the RDF from _a simulated

spectra

to the RDF of the

experimental

data.

Since

relying entirely

on calculated values of the

backscattering phase

^{shift of}

relatively heavy

backscatterer atoms can lead to

large

errors, EXELFS

analysis

of known

specimens

seems to be necessary ^to determine

experimental

values of these effects.

The average value of the _mean free

path

of

inelastically

scattered electrons in the

typical

EXELFS

region (I,e.,

2.3-8.5

l~')

in NiO has been estimated _to be A, ₌ ^6.5 ± 0.7

ji.

The linear

approximation

for the nickel

backscattering phase

shift in the

analysed

k-interval

(I.e.,

2.3-8.5

l~ ')

_{at a} distance of about

2.091,

_can be described

by

4l ~, ⁼

(0.21

_± 0.06 k ₊

(5.08

_± 0.24

).

The

phase

shift

corresponding

to the O-O shell in the _same k-interval, where the backscatter oxygen ^atom is _at about

2.951,

as well as a linear fit to it _can be described

by

4lo_o

₌

(0.60

_± 0.04 k (4.84 _± 0.19 )

The linear fit _to the oxygen central _atom

phase

shift in the

analysed

k-interval _can be described

by

4lo_

=

(0.35

_± 0.04 k

(5.38

_± 0.19 ).

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