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1-
The
parameterless
correction
method
in
X-ray
microanalysis
Eric
Van
Cappellen
University
of
Antwerp,
RUCA,
Groenenborgerlaan
171,
B-2020
Antwerp,
Belgium
(Received
December
15, 1989;
accepted
January
7,1990)
Résumé. 2014
Une
méthode
permettant
de
corriger
les
effets
d’absorption
et
de
fluorescence
en
mi-
croanalyse
X
d’échantillons
transparents
pour
la
microscopie
en
transmission
(S.)
TE.M.
est
présen-
tée.
Cette
procédure
n’exigeant
la
connaissance
d’aucun
paramètre
comme
l’épaisseur
ou
la
densité
de
l’échantillon
ni
d’aucun
coefficient
comme
les
coefficients
d’absorption
de
masse
ou
encore
des
co-
efficients
de
rendement
de
fluorescence,
est
extrêmement
utile
et
pratique
pour
effectuer
des
analyses
de
routine
d’échantillons
minces
et
a
été
appelée :
"
parameterless
correction
method".
De
surcroit
tout
effet
dépendant
de
l’épaisseur
de
l’échantillon,
comme
la
présence
de
couches
de
surface
ou
l’effet
d’orientation
crystallographique
(l’effet
Borrmann),
peuvent-être
détectés
par
cette
méthode.
Abstract.
2014
The
parameterless
correction
method
to
perform
absorption
and
fluorescence
correc-
tions
in
X-ray
microanalysis
of
tranparent
(S.)
TE.M.
specimens
is
presented
and
supported.
This
correction
procedure
requires
no
external
parameters
such
as
foil
thickness
or
density,
and
no
coeffi-
cients
such
as
mass
absorption
coefficients
or
fluorescence
yield
coefficents
and
is
therefore
suitable
for
fast
routine
thin
film
microanalysis.
Furthermore,
thickness
dependent
artefacts
such
as
surface
and
contamination
layers
and
crystallographic
orientation
effects
(the
Borrmann
effect)
can
be
de-
tected
by
the
method.
Microsc.
MicroanaL
MicroshUct. 1
( 1990)
FEBRUARY
1990,
PAGE
1
Classification
PhysicsAbstracts
06.50--07.80-78.70
1.
Introduction.
It
is
now well
established
that
even
in
thin
film
X-ray
microanalysis
absorption
and
sometimes
fluorescence
corrections
are
needed
to
acquire
accurate
quantitative
data.
In
the
plane
parallel
foil
model
an
absorption
correction
already
requires
three
external
parameters :
the
foil
thickness,
the
density
and
the
take-off
angle
of
the
X-rays.
The
classical
fluorescence
corrections
of
Tixier
[1,
2]
and
Nockolds
[3]
all
derived
for
a
plane
parallel
foil,
require
even
more
parameters
and
are
therefore
generally
omitted.
Moreover
the
models
strongly
disagree
on
the
correction
amplitudes
and
the
hypothetical
plane
parallel
shape
is
seldomly
encountered
in
experimental
situations.
The
secondary
or
fluorescence
emission
of a
spectral
line
can
always
be
expanded
in
terms
of
T,
the
mass
thickness
of
the
specimen
defined
along
the
optical
axis
of
the
microscope.
Computer
simulations
of
this
secondary
radiation
in
wedge
shaped
specimens
revealed
that
this
polynomial
in
T
without
constant
term,
is
smooth
and
that
the
first
or
linear
term
is
directly
proportional
Article available at http://mmm.edpsciences.org or http://dx.doi.org/10.1051/mmm:01990001010100
2
with
the
wedge
angle
a
[4,
5].
To
evaluate
this
influence
of
the
specimen
shape
on
the
secondary
emission,
simulations
for
a
Cu-50
at%
Cr
alloy
are
shown
in
figure
1
for
diflerent
wedge
angles
(a
=
0°,
10°,
20°,
30°
and
40°).
This
graph
shows
the
dramatic
increase
in
fluorescence
emission
with
increasing
wedge
angle.
For
a
thickness
of
50
nm,
a
thickness
very
common
in
transmission
or
scanning
transmission
electron
microscopy
the
secondary
emission
is
twice
as
important
for
a
wedge
of
10°
as
for
a
plane
parallel
foil
whereas
for
a
wedge
of
40°
this
intensity
is
already
six
times
as
large.
Fig.
1.
-
Secondary
or
fluorescence
radiation
in
wedge
shaped
targets
(Cu-50at%
Cr)
[4].
The
slope
of
the
curves
in
the
origin
(0,0)
is
directly
proportional
with
the
wedge
angle
a.
In
other
words,
a
very
modest
change
in
the
shape
of
the
specimen
can
have
a
large
influence
on
the
secondary
emission.
This
confirms
our
conviction
that
no
fluorescence
correction
can
be
entirely
satisfactory
unless
the
exact
shape
of
the
specimen
is
introduced
in
the
correction
procedure.
Since
X-rays
can
propagate
considerable
distances
in
the
specimen
when
compared
with
the
typical
dimensions
of
the
primary
excitation
volume,
the
secondary
excitation
volume
will
be
some
orders
of
magnitude
larger
than
the
primary
one.
As
a
consequence
the
specimen
shape
should
not
only
be
known
in
the
neighbourhood
of the
electron
irradiated
area
but
in
a
much
larger
area,
typically
some
tenths
of a
millimeter
or
even
more
in
diameter.
Even
if
a
fluorescence
correction
procedure
based
on
an
on-line
computer
simulation
for
an
arbitrary
target
geometry
would
be
developed
in
the
future
it
remains
extremely
doubtful
whether
it
will
ever
be
feasible
to
determine
the
exact
specimen
shape
over
such
a
large
area
for
a
routine
analysis.
Tb
overcome
the
practical
problems
enumerated
above
an
alternative
correction
procedure
has
been
developed
capable
of
correcting
for
absorption
and
for
fluorescence
to
a
large
extend
and
was
called
the
parameterless
correction
method
[6-8].
3
2.
Principle
of the
parameterless
correction
method.
2.1
METHODOLOGY. -
The
name
"parameterless
correction
method"
is
proposed
because
no
external
parameters
are
needed
during
execution.
This
implies
other
concessions.
The
major
one
is
that
not
one
but
several
spectra
are
needed
for
one
analysis.
The
spectra
are
taken
at
different
sites
on
the
specimen
with
different
thickness.
It
is
obvious
that
the
specimen
must
have
a
homogeneous
composition
in
the
analysed
area
which
in
practice
must
be
about
1
x
0.1
03BCm
in
size.
A
characteristic
line
of
an
energy
dispersive
X-ray
spectrum
can
always
be
considered
as
being
composed
of
primary
and
secondary
characteristic
radiation
both
attenuated
due
to
absorption
in
the
specimen.
Both
components
can
be
expressed
as
polynomials
in
T
the
thickness
or
mass
thickness
of
the
foil.
The
intensity
of
a
characteristic
line
vanishes
monotonically
at
zero
thickness,
consequently
it’s
polynomial
in
T
will
have
no
constant
term.
Hence
the
ratio
of
two
net
peak
integrals,
as
encountered
in
the
Cliff
Lorimer
[9]
equation
is
a
polynomial
in
the
thickness
with
a
constant
term.
This
constant
term
times
the
proper
k-factor
is
the
mass
concentration
ratio
of
the
considered
elements.
From
the
different
spectra
taken
at
different
thicknesses
uncorrected
mass
concentration
ratios
are
calculated
with
the
Cliff
Lorimer
technique.
These
results
are
plotted
versus
the
foil
thickness
and
a
least
square
fit
is
used
to
extrapolate
the
mass
concentration
ratio
to
zero
thick-
ness.
As
will
be
demonstrated
lateron
this
extrapolated
value
is
free
from
absorption
and
major
fluorescence
effects.
In
order
to
make
the
procedure
"parameterless"
the
foil
thickness
or
mass
thickness
used
for
the
extrapolation
must
be
substituted
by
an
internal
measure.
This
substitute
henceforth
termed
"internal
thickness
measure"
must
monotonically
tend
to
zero
for
vanishing
specimen
thickness
so
as
to
obtain
the
same
extrapolation
value
as
with
the
true
thickness.
We
shall
now
discuss
two
possible
candidates
for
this
internal
measure.
2.2
INTERNAL THICKNESS
MEASURE.
2.2.1
Bremsstrahlung.
-
In
the
quantitative
method
for
biological
tissues
proposed
by
Hall
[10]
the
X-ray
intensity
in
a
background
window
is
used
as
indirect
measure
for
the
specimen
thickness.
At
first
sight
a
Bremsstrahlung
band,
free
from
characteristic
radiation,
is
indeed
a
good
substitute
measure
for
the
foil
thickness,
but
as
will
be
experimentally
demonstrated,
not
all
white
radiation
originates
in
the
specimen.
As
a
consequence
Bremsstrahlung
does
not
totally
vanish
for
a
hypothetical
zero
thickness
specimen
and
must
therefore
be
rejected
as
internal
thickness
measure.
2.2.2
Characteristic
radiation.
-
As
already
mentioned,
characteristic
radiation
can
be
ex-
pressed
as
a
smooth
monotone
polynomial
in
the
thickness
without
a
constant
term.
Consequently
the
foil
thickness
can
be
expanded
as
the
inverse
smooth
monotone
polynomial
in
the
intensity
of
the
peak
of
the
considered
element
without
a
constant
term,
on
condition
that
this
element
is
only
present
in
the
specimen.
A
net
peak
integral
or
a
sum
of
net
peak
integrals
will
prove
to
be
an
excellent
internal
thickness
measure.
3.
Mathematical
formulation.
Two
slightly
different
but
equivalent
formulations
of the
parameterless
correction
method
will
be
presented.
The
first
one
is
straightforward
and
starts
from
the
assumption
that
a
characteristic
line
of
an
element
X
consists
of
primary
and
secondary
radiation
both
corrected
for
absorption.
4
The
second
approach
starts
from
the
Cliff-Lorimer
equation
extended
with
an
absorption
and
a
fluorescence
correction.
Both
approaches
lead
to
the
same
conclusions
since
the
absorption
and
fluorescence
corrections
of
the
Cliff-Lorimer
equation
are
transformed
into
expressions
contain-
ing
the
primary
and
secondary
emission
of
the
considered
elements.
3.1
FIRST APPROACH.
3. 1. 1
The
intensity
Ix
of a
characteristic peak
of an
element X -
The
primary
emission
of
an
el-
ement
X,
corrected
for
absorption
in
the
direction
of the
detector
can
be
approximated
as
follows
(e.g.
[1,
2]) :
All
distances
are
expressed
in
mass-length
units
and
p( t)
is
the
mass-depth
ionization
distribution
function
along
the
optical
axis.
The
total
mass
thickness
of the
foil
along
this
optical
axis
is
de-
noted
T.
Cx
and
Mx
are
respectively
the
mass
concentration
and
the
mass
absorption
coefficient
of
element
X
whereas
s(t)
is
the
distance
the
primary
X-ray
photons
must
travel
in
the
specimen
in
the
direction
of
the
detector
before
leaving
the
surface.
This
function
strongly
depends
on
the
shape
of
the
specimen
and
the
detection
direction.
The
constant
C
takes
account
of
the
fluores-
cence
yield
of
the
excited
shell
of
element
X,
the
weight
factor
of
the
line
under
consideration,
the
fraction
of
the
isotropic
primary
emission
emitted
towards
the
detector
and
last
but
not
least
the
detector
efficiency
for
the
considered
line.
Equation
(1)
can
always
be
expanded
in
a
power
series
in
T :
It
should
be
noted
that
there
is
no
constant
term
and
that
for
an
infinitesimally
thin
film
the
first
(linear)
term
will
be
dominant.
The
secondary
emission
IX
of
an
element
X,
corrected
for
absorption
in
the
direction
of
the
detector
can
be
approximated
as
follows
[4,
5] :
This
equation
is
a
four
dimensional
integral
(three
dimensional
over
the
specimen
volume
(dV)
and
one
dimensional
over
the
optical
axis
(dt)).
The
vector
r connects
the
site
where
the
primary
emission
of
an
element
Y
is
emitted
with
the
site
where
secondary
emission
of
element
X
is
generated
by
the
primary
radiation
of
element
Y.
q(r,
t)
is
analogously
the
distance
to
be
bridged
by
the
fluorescence
radiation
before
leaving
the
surface
in
the
direction
of
the
detector.
The
constant
C’
now
mainly
contains
the
fluorescence
yields
of
the
shells
of
elements
X
and
Y
and
their
respective
weight
factors,
the
ionisation
yield
of
element
X,
the
mass
absorption
coefficient
of
element
Y
in
pure
X,
the
mass
concentrations
of
the
elements
X
and
Y.
Equation
(3)
can
also
be
expanded
in
a
power
series
in
T
and
in
the
case
of wedge
shaped
speci-
mens,
computer
simulations
showed
that
the
linear
term
is
proportional
to
the
wedge
angle
[5].
The
total
intensity
Ix
of
element
X
is
given
by
the
sum
of
(2)
and
(4) :
5
and :
3.1.2
The
mass
thickness
T. -
Equation
(5)
can
be
inversed
so
as
to
yield
the
mass
thickness
T :
Note
that
all
the
coefficients
p-y,
fit,
at
and
bXi
strongly
depend
on
the
shape
of
the
specimen
and
the
detection
geometry.
3.1.3
The
intensity
ratio
of
two
elementsA
and
B.
-
As
usual
for
a
thin
film
ratio
technique
the
ratio
of
two
peak
intensities
of
elements A
and
B
is
considered :
The
mass
thickness
T
can
be
replaced
by
(7) :
The
constant
terms
AAB0
and
BtB
are
equal
to
the
ratio
IAIIB
for
mass
thickness
tending
to
zero
and
read :
The
intensity
ratio
IA IIB
can
be
converted
into
a
mass
concentration
ratio
when
divided
by
the
Cliff-Lorimer
kAB -factor.
This
mass
concentration
ratio
depends
on
T
(or
Ix)
and
is
therefore
called
the
"uncorrected
mass
concentration
ratio"
and
is
denoted
(CA/CB)’ .
This
ratio
tends
to
a
fixed
value
as
a
limit
when
T or
Ix
approaches
zero :
On
the
other
hand,
the
exact
mass
concentration
ratio
CA/CB
can
be
derived
from
the
Cliff-
Lorimer
equation
on
condition
that
no
absorption
and,
as
a
consequence,
no
fluorescence
of
X-rays
occur
in
the
specimen.
This
condition,
fulfilled
in
an
infinitesimal
thin
plane
parallel
foil,
reads :
-
From
equations
(11)
and
(12)
it
is
clear
that
strictly
speaking
the
extrapolated
value
of
the
uncor-
rected
mass
concentration
ratio
(11)
is
not
equal
to
the
exact
ratio
(12).
However
the
coefficients
pi
and
If
represent
respectively
the
primary
and
the
secondary
emission
of
an
element
X
when
T
the
specimen
mass
thickness
(or
IX)
approaches
zero.
It
is
therefore
beyond
doubt
that
fi
is
much
smaller
than
pt
or :
....
T-..
T-..
6
This
assumption
will
be
explicitly
demonstrated
in
paragraph
4.
Consequently
in
the
limit,
for
T
or
Ix
approaching
zero,
one
has :
This
last
statement
proves
the
parameterless
correction
method.
In
practice,
uncorrected
mass
concentration
ratios
are
measured
at
different
sites
with
différent
mass
thickness
and
are
plotted
as
a
function
of
a
net
peak
integral
Ix.
By
means
of
a
least
square
fit
through
these
experimental
points
the
extrapolation
to
Ix
=
0
is
obtained.
Equation
(14)
states
that
this
extrapolated
value
is
equal
to
the
exact
mass
concentration
ratio
CA/CB.
-
Remarks.
The
net
peak
integral
Ix
used
for
the
extrapolation
can
be
replaced
by
a
sum
of
net
peak
integrals.
The
only
condition
is
that
it
should
be
characteristic
radiation.
Obviously
the
peaks
IA
and
IB
can
also
be
used.
It
has
been
shown
that
for
a
perfect
plane
parallel
foil
the
linear
term
of fluorescence
emission
equals
zero
(11
=
0)
[5].
Hence
If
=
fB
=
0
so
that
from
(11)
and
(12)
it
is
clear
that
for
T
or
Ix
approaching
zero
(CA/CB)’ ==
(CA/CB)
without
any
assumption.
The
equations
for
primary
(1)
and
secondary
(3)
emission
are
derived
by
assuming
a
one
dimensional
excitation
volume
along
the
optical
axis.
Actually
the
one
dimensional
integration
along
the
optical
axis
from
0
to
T
should
be
replaced
by
a
three
dimensional
integration
over
the
real
excitation
volume.
However,
this
would
not
affect
the
conclusions
of
the
parameterless
correction
method
since
T
is
then
simply
replaced
by
V
the
real
excitation
volume.
Equation
(5)
then
becomes :
and
consequently :
With
equations
(15)
and
(16)
a
completely
analogous
derivation
is
possible
as
with
equations
(3)
and
(7).
The
intensity
of
two
elements A
and
B
now
reads :
and :
All
the
conclusions
drawn
in
the
previous
paragraph
remain
valid.
The
reason
why
the
initial
reasoning
was
carried
out
with
the
mass
thickness
T
instead
of
the
excitation
volume
V
is
that
in
transmission
or
scanning
transmission
electron
microscopy
the
notion
mass
thickness
is
more
common
than
the
notion
excitation
volume.
It
is
indeed
easier
to
claim
that
two
spectra
were
taken
at
différent
(mass)
thicknesses
than
with
différent
excitation
volumes.
The
mass
thickness
of
an
area
can
be
directly
associated
with
the
transparancy
of
it
while
by
no
means
the
excitation
volume
(size
and
intensity)
can
be
visualized.
7
3.2
SECOND
APPROACH. -
The
second
more
classical
approach
starts
from
the
Cliff
Lorimer
equation
but
is
entirely
equivalent
with
the
first
one
since
the
same
equations
for
primary
and
secondary
emission
will
be
used.
The
corrected
Cliff
Lorimer
equation
reads
[9] :
IA
and
lE
are
the
measured
intensities
and
A
and
F are
respectively
the
absorption
and
fluores-
cence
correction
factors.
The
classical
notation
I(x)
(x
=
J.LeCosec
0)
for
the
absorption
correction
of a
peak
will
not
be
used
here
since
strictly
speaking
it
is
only
valid
for
a
plane
parallel
foil,
and
it
could
lead
to
confusion
with
fluorescence
correction
terms.
"ax"
and
"fX"
respectively
denote
the
absorption
and
fluorescence
corrections
of
the
peak
of
element
X
and
are
defined
as
follows :
and
I0X
is
the
total
primary
emission
of
a
specific
line
of
element
X
generated
in
the
specimen.
The
definitions
(20)
and
(21)
can
be
transformed
with
equations
(1)
and
(3) :
Equations
(22)
and
(23)
can
be
expanded
in
power
series
in
T :
The
respective
ratios
for
two
elements A
and
B read :
Using
(26)
and
(27)
the
general
correction
factor
of
the
Cliff
Lorimer
equation
becomes :
8
where :
Note
that
if
=
flx lpx
when
referring
to
the
polynomials
(2)
and
(4).
(CA/CB)’
denotes
again
the
uncorrected
mass
concentration
ratio
and
is
equal
to
the
uncorrected
Cliff
Lorimer
equation :
This
can
be
transformed
to :
This
polynomial
can
again
be
transformed
into
a
polynomial
in
Ix
with
coefficients
G’ABi,
but
obviously Gô B
=gaz.
The
limit,
as
T
or
Ix
approaches
zero,
of
(CA/CB)’
equals :
If
approximation
(13)
is
valid,
equation
(32)
again
states
that
the
extrapolation
value
of
the
un-
corrected
mass
concentration
ratio
approaches
the
exact
ratio
and
since :
equation
(32)
equals
(11).
The
parameterless
correction
method
depends
on
the
validity
of
con-
dition
(13)
which
is
also
equivalent
with :
This
condition
is
examined
in
the
next
paragraph.
4.
Systematic
error
estimation :
lim
(IF
/Ik) .
The
systematic
extrapolation
error
governed
by
condition
(34)
will
now
be
examined.
Expres-
sions
for
the
primary
IPX
and
secondary
IX
emissions
of
an
element
X
are
obtained
for
a
small
mass
thickness
T,
and
are
evaluated
using
computer
simulations
of
secondary
emission
in
wedge
shaped
targets.
9
4.1
THE
PRIMARY
EMISSION
IPX
FOR
SMALL T.
-
The
number
of
ionizations
of
the
j-shell
of
element
X
in
an
infinitesimal
thin
film
is
given
by
the
formula
of
Castaing
[2] :
where
Cx
and
Ax
are
respectively
the
mass
concentration
and
the
atomic
weight
of
element
X.
dt
is
the
mass
thickness
of
the
foil
and
el x (E)
is
the
ionization
cross
section
for
the j-shell
of
element
X.
The
primary
X-ray
intensity
of
line
6,
dis
(line 6
is
a
characteristic
line
of
the
j-shell
of
element
X)
is
equal
to
the
number
of
ionizations
(35)
multiplied
by
the
fluorescence
yield
wu
of
the
j-shell
and
the
weight
factor
WX
of
the
considered
line
6 :
Since
an
infinitesimal
thin
foil
is
considered, E
the
energy
of the
exciting
electrons
can
be
replaced
by
Eo,
the
accelerating
potential.
Furthermore
no
absorption
effects
occur,
in
other
words
the
detected
primary
emission
is
equal
to
the
generated
intensity
multiplied
by
the
fraction
emitted
towards
the
detector.
If
the
solid
angle
spanned
by
the
detector
is
denoted
dÇ2,
IX
equals :
4.2
THE
SECONDARY
EMISSION
IX
FOR
SMALL T.
-
’Ib
make
an
evaluation
possible
If
will
be
derived
for
a
wedge
shaped
specimen.
In
the
final
equation
numerical
data
acquired
from
computer
simulations
will
be
used
[5].
Consider
an
infinite
wedge
with
a
wedge
angle
a.
Primary
radiation
of
an
element
Y
can
excite
element X
and
generate
secondary
radiation
of
element
X.
If
only
the
tip
of
the
wedge
is
irradiated
by
the
electron
beam
(Fig.
2).
The
primary
radiation
of
element
Y
generated
in
the
tip
of
the
wedge
reads
(see
(37)) :
The
fraction
of
this
radiation
reaching
the
element
of
volume
dV
in
the
wedge
equals :
where
dÇ2
is
the
solid
angle
spanned
by
the
element
of
volume
dV
andyy
the
mass
absorption
coefficient
of
the
¡-line
of
element
Y.
In
dV
the
X-atoms
will
absorb
a
certain
primary
intensity
of
element
Y
equal
to :
u
u
x
is
the
mass
absorption
coefficient
of
the
y-line
in
a
target
consisting
of
pure
X
atoms
and
dlrl
is
the
distance
travelled
by
the
primary
Y-radiation
in
dV.
The
secondary
radiation
of
element
X
generated
in
dV
and
emitted
towards
the
detector
reads :
10
Fig.
2.
-
Model
used
to
calculate
the
secondary
emission
when
only
the
tip
of
the
wedge
is
irradiated.
(rx - 1)
/rx
is
the
ionization
yield
of
the
j-shell
of
element
X
and
wu
is
the
fluorescence
yield
of
the
same
shell.
vvi
is
the
weight
factor
of
the
8-line
in
the
set
of
all j-shell
ionizations
and
exp
(-pxq(r))
takes
account
of
absorption
of
the
secondary
8-line
emission
in
the
direction
of
the
detector.
The
solid
angle
spanned
by
the
detector
is
denoted
dÇ2.
The
total
fluorescence
contribution
is
obtained
by
an
integration
of
dV
over
the
whole
specimen
volume :
or:
4.3
THE
RATIO
"IFX
/ IJ:"
FOR
SMALL
T.
-
The
ratio
IJ:
/ II
for
small
T
reads
(Eq.
(43)
divided
by Eq. (37)) :
Since
this
ratio
is
independent
of
T
it
equals
the
limit
as
T
approaches
zero.
To
evaluate
the
magnitude
of
equation
(44)
numerical
data
acquired
with
computer
simulations
of
fluorescence
radiation
in
wedge
shaped
targets
will
be
used
[5].
For
small
T
it
can
be
shown
that
the
four
di-
mensional
integral
used
for
the
simulations
is
equivalent
to
the
three
dimensional
integral
(44)
11
multiplied
by
the
mass
thickness
T.
In
other
words
the
three
dimensional
integral
(44)
is
the
lin-
ear
coefficient
of
the
fluorescence
polynomial
in
T.
It
was
demonstrated
that
this
coefficient
is
proportional
to
the
wedge
angle
a
and
the
proportionality
factor
will
be
denoted
KXY :
!{ xy
depends
on
the
mass
absorption
coefficients
03BCX
and
My
and
of
the
geometry
of
the
specimen.
It
was
also
shown
that
!{Xy
is
quite
insensitive
to
the
specimen
orientation
in
the
electron
beam
and
to
the
X-ray
detection
direction.
Equation
(44)
is
the
fluorescence
to
primary
intensity
ratio
of
the
b-line
of
element
X,
in
which
only
the
secondary
emission
generated
by
the
y-line
belonging
to
the
i-shell
ionization
of
element
Y
is
considered.
1
fact
all
the
lines
belonging
to
the
i-shell
ionizations
have
to
be
consid-
ered
since
they
all
contribute
to
the
fluorescence
emission
of
element
X.
All
these
lines
are
very
close
in
energy,
consequently
they
all
have
approximately
the
same
mass
absorption
coefficient
and
the
same
ionization
cross
section.
Furthermore
the
sum
of
ail
the
weight
factors
of
the
lines
of
a
set
equals
one
(2:
Wf
1
The
secondary
to
primary
intensity
ratio
of
element
X
taking
into
account
all
the
lines
of
the
i-
shell
ionizations
of
element
Y
as
possible
primary
X-ray
excitation
sources
of
the
X
atoms
then
reads :
.
T"’ -
.:
The
ionization
cross
section
ratio
03C8iY
(Eo)
/03C8jX
(Eo)
can
be
approximated
with
the
BETHE
for-
mula
fl 11 :
Eo
and
EX
(EiY)
are
respectively
the
accelerating
potential
of
the
electrons
and
the
critical
ioniza-
tion
energy
of
the
j-shell
(i-shell)
of
atom
X(Y).
The
number
of
electrons
in
the
j-shell
(i-shell)
of
element
X
(Y)
is
denoted
ZX
(ZiY).
4.4
EVALUATION
OF
THE
SYSTEMATIC
EXTRAPOLATION
ERROR. -
All
the
terms
in
equation
(46)
will
now
be
discussed
separately.
Let
us
first
consider
the
ionization
cross
section
ratio
03C8iY
(Eo)
/03C8jX
(E0).
Whenever
two
K-
or
L-
lines
are
considered,
the
ratio
ZiY/ZjX
equals
1
since
for
elements
beyond
Ne
the
K-
and
L-lines
always
contain
2
and
8
electrons.
Another
frequent
occuring
situ-
ation
is
that
L-lines
of
an
element
(X)
are
excited
by
the
K-lines
of
another
element
(Y).
In
this
case
the
ratio
ZiY/ZjX
equals
1/4.
The
critical
excitation
energy
ratio
EjX/EiY
is
certainly
smaller
than
one
since
the
energy
of
the
exciting
line
must
be
higher
than
the
energy
of
the
excited
line.
The
same
conclusion
is true
for
the
ra tio ln
(EolEi )
/~n
EOIEJX
In
other
words
the
ionization
cross
section
ratio
03C8iY
(E0)/03C8jX
(Eo)
is
always
smaller
than
unity.
Usually
the
atom
mass
ratio
Ax /Ay
is
also
less
than
1
because
the
heavier
element
excites
the
other.
The
fluorescence
yield
Wjy
of
element
Y,
the
ionization
yield
¡ix - 1/ ¡ix
of
élément
X
and
the
mass
concentration
Cy
of
element
Y
are
also
parameters
smaller
than
1.
The
mass
absorption
12
coefficient
J-l9
is
typically
of
the
order
of
100
cm2/g
and
the
wedge
angle a
does
not
exceed
90°.
A
rough
estimate
for
the
constant
KXY
is
1
x
10-4
g/
(cm2
deg.)
(see
next
paragraph).
Generally
speaking
lim
(IFX/IPX)
will
always
be
less
than
1
% ,
a
value
only
trespassed
in
extreme
situations.
This
statement
will
now
be
checked
for
a
Cu-50
at%
Cr
alloy.
From
the
fluorescence
simulations
in
Cu-50
at%
Cr
wedges
we
know
that
[5].
The
mass
absorption
coefficient
of
the
Cu-Ka
line
in
a
pure
chromium
target
is
extremely
high
and
equals :
03BCCrCu=
250
cm 2/g.
The
fluorescence
yield
of
the
Cu-K line,
03C9KCu
is
equal
to
0.39
and
the
mass
ratio
Acr/Acu
is
0.817
[12].
The
ionization
cross
section
ratio
03C8KCu
(100
kV)/03C8KCr
(100
kV)
is
calculated
with
the
critical
ionization
potentials
EKCu
and
Ec,
which
are
respectively
equal
to :
EKCu =
8.943
keV
and
EKCu
=
5.957
keV
[13] :
The
only
remaining
unknown
is
the
ionization
yield
(rKCr-1)
/rKCr.
The
absorption
jump
ratio
rcr
is
calculated
from
mass
absorption
coefficients
values
in
a
pure
chromium
target.
The
ratio
of
the
curve
fit
value
above
and
below
EKCr
yields
the
absorption
jump
ratio
rcr
[5] :
Consequently
the
ionization
yield
of
the
chromium
K-shell
equals :
Finally
the
secondary
to
primary
emission
ratio
reads :
where
a
the
wedge
angle
is
expressed
in
degrees.
Note
that
this
ratio
is
equal
to
1
%
for
a
wedge
angle
of
38°.
From
this
residual
ratio
one
can
estimate
the
error
on
the
final
extrapolated
mass
concentra-
tion
ratio
since :
or:
13
where :
IFB
=
0
is
the
worst
case
situation
encountered
in
binary
systems
such
as
a
Cu-50
at%
Cr
alloy.
From
(53)
it
is
clear
that :
or
approximately :
The
relative
error
on
C’A
is
the
highest
when
CB
approaches
1
or
in
other
words
when
there
is
only a
small
fraction
of fluorescing
material
and
a
large
fraction
of the
exciting
element
present
in
the
specimen.
However
the
error
on
the
extrapolated
mass
concentration
C’A
will
never
exceed
a
which
in
its
turn
never
exceeds
the
residual
ratio
lim
(IFA/IPA).
For
the
Cu-50
at%
Cr
example
the
extrapolated
concentrations
would
be :
The
error
is
0.25
wt%
and
as
will
be
demonstrated
with
experimental
results
(see
Sect.
6)
the
errors
on
the
extrapolated
values
due
to
statistical
fluctuations
are
typically
of the
order
of 0.5
to
1.0
wt%.
Consequently
the
systematic
error
due
to
the
linear
term
of
the
fluorescence
polynomial
in
T
is
smaller
than
the
error
introduced
by
statistical
fluctuations
and
can
therefore
be
neglected.
An
experimental
confirmation
of
this
conclusion
will
be
given
in
paragraph
6
where
the
experimental
results
on
a
highly
fluorescing
system
(Fe-Ni)
will
be
discussed.
5.
Practical
implementation
of the
method.
5.1
INTERNAL
THICKNESS
MEASURE. -
In
section
2.2
it
was
already
mentioned
that
Bremsstrahlung
is
not
a
suitable
internal
thickness
measure
especially
in
the
limit
for
zero
thick-
ness.
This
statement
will
now
be
verified
by
comparing
the
results
obtained
on
a
Cr
85
at%
-Al
alloy
with
a
Bremsstrahlung
window
and
with
a
net
peak
integral
(Cr-Ka).
The
specimen
was
examined
in
a
low
background
(carbon)
holder
at
100
kV
Ten
spectra
were
taken
at
different
thicknesses
but
all
in
very
transparant
areas
and
the
dead
time
of
the
multi
channel
analyser
never
exceeded
35
%.
The
lifetime
of
the
spectra
is
60
s.
The
uncorrected
mass
concentration
ratios
(CAl/CCr)’
are
calculated
with
a
standardless
metallurgical
thin
film
program
provided
by ltacor
Northern,
which
uses
calculated
k-factors.
No
reliable
standard
specimen
was
available
and
as
will
be
clear
from
the
results
the
calculated
k-factor
is
accurate
enough
especially
for
comparative
experiments.
The
parameterless
correction
procedure
was
carried
out
with
10
Bremsstrahlung
windows
at
different
energies
and
with
the
net
peak
integral
of
the
Cr -
Ka
line.
All
the
results
with
the
Bremsstrahlung
windows
agree
within
0.15
at%
and
the
average
extrapolated
concentrations
are :
14
The
Cr-Ka
peak
analysis
yields
less
aluminium :
The
results
of
the
Cr-K,
peak
analysis
and
the
19.5-20.0
keV
Bremsstrahlung
window
analysis
are
plotted
in
figure
3.
From
this
figure
can
be
deduced
that
about
40
counts
in
the
19.5-20.0
keV
window
are
not
origi-
nating
from
the
specimen
although
a
low
background
holder
was
used
for
the
measurements.
Fig.
3.
-
In
spite
of
the
use
of
a
low
background
specimen
holder,
the
analysis
with
a
Bremsstrahlung
win-
dow
(19.5-20.0
keV)
exhibits
the
presence
of
spurious
X-rays
(about
40
counts).
When
no
low
background
holder
can
be
used
or
when
for
some
other
reason
the
amount
of
spurious
X-rays
increases
the
difference
between
the
two
extrapolation
values
becomes
more
important.
It
is
therefore
imperative
that
the
parameterless
correction
procedure
should
make
use
of
characteristic
radiation
as
internal
thickness
measure.
This
characteristic
radiation
can
be
a
net
peak
integral
or
the
sum
of
net
peak
integrals.
15
5.2
POLYNOMIAL
FIT.
-
A
polynomial
has
to
be
fitted
through
the
experimental
points
so
as
to
match
equation
(9).
However,
when
the
degree
of
the
fitted
polynomial
is
too
high
the
curve
fit
will lose
its
physical
significance,
because
of
the
statistical
fluctuations
of
the
experimental
points.
The
exirapolation
to
net
peak
zero
then
becomes
hazardous.
On
the
other
hand
a
systematic
error
can
also
be
induced
when
the
degree
of
the
fit
is
too
low.
As
an
example
the
spectra
for
figure
3
were
taken
in
a
very
thin
area
so
that
a
linear
curve
fit
seems
sufficient.
Although
a
linear
regression
is
sufficient
in
most
of
the
cases,
especially
when
the
spectra
are
acquired
in
transparant
areas
of
the
specimen,
it
can
be
inadequate
for
systems
with
a
strongly
absorped
element.
Then
BAB1
is
not
negligible
(Eq.
(9))
and
a
parabolic
curve
fit
is
needed.
Fig-
ure
4
shows
an
example
were
the
linear
fit
clearly
fails.
It
is
again
a
Cr-Al
alloy
and
the
9
spectra
were
taken
till
just
over
the
transparancy
limit
at
100
kV
Figure
4
shows
a
linear
fit
with
the
6
first
points
and
a
parabolic
fit
with
all
the
points
(9).
Moreover
a
situation
where
a
parabolic
fit
is
inadequate
was
never
encountered.
Fig.
4.
-
An
example
where
a
linear
fit
fails
and
a
parabolic
fit
is
needed
to
extrapolate
the
aluminium
concentration
of
a
Cr-Al
alloy.
Finally
it
is
to
the
operators
common
sense
to
judge
whether
a
linear
or
parabolic
fit
is
to
be
used
for
a
specific
analysis.
When
only
transparant
areas
are
analysed
one
of
both
will
always
be
adequate
to
extrapolate
the
corrected
mass
concentration
ratio.
16
5.3
EXTERNAL
INFLUENCES.
-
In
this
paragraph
some
phenomena
which
could
introduce
er-
rors
in
the
parameterless
correction
method
will
be
discussed.
Some
of
these
phenomena
can
be
traced
by
the
method
and
subsequently
avoided
or
at
least
reduced.
5.3.1
Surface
layers.
-
Surface
layers
can
be
classified
in
two
categories ;
extrinsic
and
intrinsic
layers.
5.3.1.1
Extrinsic
surface
layers.
-
Extrinsic
surface
layers
are
layers
which
are
deposited
onto
the
specimen
by
preparation
techniques
(jet
polishing,
cleaning,
ion
thinning...)
or
by
contamina-
tion
build
up
during
the
microscopical
investigation.
These
layers
usually
contain
elements
which
are
not
present
in
the
specimen
and
are
therefore
easily
detectable.
Although
they
increase
the
primary
excitation
volume
they
seem
to
have
no
significant
influence
on
the
quantitative
results
obtained
with
the
parameterless
correction
method
on
condition
that
characteristic
radiation
is
used
as
external
thickness
measure.
This
experimentally
verified
statement
is
equivalent
to
say
that
the
absorption
correction
for
the
analysed
elements
within
the
extrinsic
layer
is
negligible.
5.3.1.2
Intrinsic layers. -
Intrinsic
layers
are
layers
which
originally
belong
to
the
specimen.
Caused
or
not
by
preparation
techniques
the
composition
of the
outermost
layer
of
the
specimen
can
differ
from
the
bulk
composition,
e.g. :
selective
desorption
of
certain
elements
at
the
surface
[14].
The
structure
and
composition
of
the
surface
can
also
be
modified
by
the
electron
beam
(radiation
damage).
Since
these
layers
contain
the
same
elements
as
the
bulk
their
influence
on
the
X-ray
spectra
is
intrinsic
and
cannot
be
eliminated
easily.
However
the
parameterless
cor-
rection
method
often
enables
to
trace
them
since
these
intrinsic
surface
layers
are
approximately
uniformly
thick
and
their
relative
influence
increases
with
decreasing
specimen
thickness.
As
a
consequence
the
extrapolation
curve
for
an
element
X
will
exhibit
a
sudden
bend
for
decreasing
specimen
thickness.
In
other
words,
a
deviation
from
linearity
on
the
left
hand
side
of
the
experimental
curve
is
an
indication
for
a
surface
layer.
If
this
layer
cannot
be
avoided,
fairly
good
quantitative
results
can
still
be
obtained
by
eliminating
the
obviously
deviating
experimental
points.
Only
the
mea-
surements
in
the
thicker
regions
where
the
relative
influence
of
the
surface
layers
is
negligible
are
then
considered
for
the
extrapolations.
Consequently
the
final
results
will
be
less
accurate
but
still
more
acceptable
than
a
quantitative
analysis
obtained
with
only
one
spectrum
precisely
taken
in
the
thinnest
region.
5.3.2
The
Borrmann
effect.
-
E.D.
spectra
of
crystalline
specimens
can
be
affected
by
the
ori-
entation
of
the
specimen
in
the
electron
beam.
The
main
idea
is
that
a
plane
wave
interacting
with
a
periodic
potenial
can
generate
a
standing
wave
in
the
crystal
depending
on
the
exact
im-
pact
direction.
The
ionization
probability
of
an
atom
is
proportional
with
the
electron
density
11/; 12,
which
for
a
periodical
wave
function 03C8,
might
differ
from
one
atom
site
to
another.
If
such
a
situation
occurs
the
emission
of
characteristic
radiation
from
certain
elements
will
be
favoured
when
compared
to
others
and
consequently
the
spectrum
will
be
distorted.
CHERNS
and
co-
workers
[15],
demonstrated
that
this
orientation
effect
can
be
very
crucial
for
X-ray
microanalysis
when
a
number
of
Bragg
reflections
are
strongly
excited
and
especially
for
crystal
thicknesses
of
the
order
of
10
nm.
For
thicker
targets
multiple
scattering
will
become
dominant
and
|03C8|2
will
gradually
lose
its
periodicity
until
all
atom
sites
are
equally
excited.
This
phenomenon
is
traced
by
the
parameterless
correction
method
in
the
same
way
as
are
intrinsic
surface
layers.
The
extrapolation
curve
will
again
exhibit
a
strange
behaviour
for
decreas-
ing
specimen
thickness.
The
bormann
effect
disturbs
the
spectra
up
to
much
larger
thicknesses
than
intrinsic
surface
layers,
however
the
remedy
to
this
problem
is
much
easier
since
a
small
tilt
of
the
specimen
away
from
the
strongly
excited
Bragg
reflections
destroyes
the
responsible
standing
wave.
17
6.
Some
examples
of
application
of
the
method.
Two
distinct
systems
will
be
investigated.
The
first
testing
material
is
a
Cr
70
at%
-Al
alloy
and
it
was
chosen
because
of
the
important
absorption
correction
due
to
the
presence
of
aluminium.
The
second
set
of
results
concerns
Fe-Ni
alloys
which
are
known
to
be
very
strongly
fluorescing
systems.
6.1
Cr
70
at%
-Al.
-
The
specimens
were
electropolished
with
a
mixture
of
90
%
methanol
(CH30H)
and
10
%
perchloric
acid
(HC104).
The
spectra
were
acquired
at
100
kV,
a
low
back-
ground
holder
was
used
and
strong
Bragg
reflections
were
carefully
avoided.
A
first
multi
spectra
analysis
makes
use
of
10
measuring
sites
more
or
less
aligned
perpendic-
ular
to
the
specimen
edge
and
parallel
to
the
tilt
axis.
After
the
spectra
acquisitions
the
specimen
holder
was
tilted
from
+
35°
to -
35°
so
as
to
visualize
the
wedge
shape
geometry
of
the
specimen.
The
SEM
image
in
figure
5
shows
the
9
first
contamination
spots
in
profile
from
which
could
be
deduced
that
the
wedge
angle
is
approximately
11.5°.
Fig.
5.
-
SEM
image
showing
the
contamination
spots
due
to
parameterless
correction
analysis.
The
spec-
imen
was
tilted
over
an
angle
of
70°
and
is
clearly
wedge
shaped
in
the
neighbourhood
of
the
hole.
18
The
reason
for
using
SEM
is
that
it was
impossible
in
transmission
to
obtain
an
adequate
contrast
and
brightness
of
the
image
so
as
to
visualize
the
whole
thickncss
range.
The
spectra
of
the
thinnest
points
show
small
Si
and
Cl
peaks
due
to
contamination.
When
only
the
9
analysis
points
of
figure
5
are
considered
for
the
curve
fit,
a
linear
least
square
fit
is
more
than
adequate.
When
the
10th
point
is
added
it
is
preferable
to
use
a
parabolic
fit.
’IWo
internal
measures
for
the
mass
thickness
are
used :
the
Cr-Ka
net
peak and
the
sum
of the
Cr-Ka
+
Al-Ka
net
peaks.
Table
1
gives
an
overview
of
the
results.
Thble
I.
-
Results
of a
10
sites
parameterless
correction
procedure
on
a
Cr
70
at% -
Al
alloy.
Both
the
Cr -
Ka
net
peak
and
the
sum
of
the
Cr -
Ka
+
Al -
Ka
net
peaks
are
used
as
an
internal
parameter for
the
riiass
thickness.
The
results
with
the
linear fit
were
obtained
by
orriitting
the
thickest
analysis point.
Tb
confirm
the
previous
results
a
second
analysis
was
carried
out
in
an
other
area
of
the
same
specimen.
It
concerns
a
12
point
analysis
and
in
each
measure
site
spectrum
of
80
000
counts
was
acquired.
Subsequently
the
spectra
were
renormalized
to
a
lifetime
of
100
s
in
order
to
be
used
for
a
parameterless
correction.
This
slightly
modified
procedure
ensures
that
all
the
spectra
are
statistically
equivalent.
The
disadvantage
compared
to
the
normal
procedure
is
the
relatively
long
real
acquisition
time
in
the
very
thin
areas.
The
region
where
the
12-point
analysis
was
carried
out
is
thin
enough
so
that
a
linear
curve
fit
is
sufficient.
The
results
are
surveyed
in
table
II.
The
results
of
tables
1
and
II
are
in
good
agreement
with
each
other
and
with
the
presumed
composition
Cr
70
at%
-Al.
Depending
on
the
accuracy
of
this
presumed
composition
an
exper-
imental
k-factor
could
now
be
derived,
but
this
would
only
be
a
little
larger
than
the
calculated
and
used
k-factor.
The
results
quite
obviously
demonstrate
that
the
parameterless
correction
procedure
is
a
powerful
way
of
correcting
for
absorption
even
in
systems
with
strongly
absorbed
peaks.
6.2
THE
Fe-Ni
SYSTEM.
-
Iron-nickel
alloys
are
strongly
fluorescing
materials.
The
energy
of
the
Ni-Ka
peak
(7.47
keV)
is
sufficient
to
excite
the
K-shell
of
iron
(Efe=
7.083
keV)
[13].
Ni-K
radiation
therefore
generates
secondary
Fe-K
radiation
with
a
very
high
yield.
Consequently
Fe-Ni
alloys
are
extremely
well
suited
to
investigate
whether
the
parameterless
correction
proce-
dure
sufficiently
corrects
for
fluorescence.
19
Table
II.
-
Results
of a
12
sites parameterless
correction
procedure
on
a
Cr
70
at% -
AI
alloy.
Both
the
Cr -
Ka
and
the
sum
of the
Cr -
Ka
and
the
Al-
Ka
net peaks
are
used
as
an
internal parameter
for
the
mass
thickness.
6.2.1
A
composition
without absorption
correction.
-
Table
III
is
a
survey
of
the
mass
absorption
coefficients
of
the
Fe-Ka
and
Ni-Ka
lines
in
iron
and
nickel
targets.
Thanks
to
the
presence
of
an
absorption
edge
in
between
the
energies
ufFe-Ka
(6.40
keV)
and
Ni-Ka
(7.47
keV)
in
iron,
a
specific
composition
exists
for
which
the
Fe-Ka
and
Ni-Ka
lines
have
an
identical
mass
absorption
coefficient.
In
other
words
no
absorption
correction
is
needed
for
this
specific
alloy,
and
the
differences
between
uncorrected
concentrations
or
concentration
ratios
and
the
real
values
are
only
due
to
fluorescence
in
the
specimen.
This
specific
composition
is
obtained
straightforwardly
(Thb.
IV)
and
for
the
sake of
simplicity
this
alloy
will
be
referred
to
as :
Fe
10-Ni.
Thble
III.
-
The
mass
absorption
coefficients
of the
Fe -
Ka
and
Ni -
Ka
lines
in
pure
Fe
and pure
Ni
Thble
IV. 2013
The
composition
of
a
Fe-Ni
alloyfor
which
no
absorption
correction
is
needed.
Specimens
of
this
alloy
and
of
two
other
compositions,
Fe
50-Ni
and
Fe
90-Ni
were
thinned
elec-
trochemically.
However
some
Fe
10-Ni
specimens
were
prepared
by
ion
thinning
so
as
to
obtain
different
geometries.
This
is
achieved
by
placing
the
ion
guns
at
different
angles
and
specimens
20
with
wedge
angles
of
10, 40
and
60
degrees
and
referred
to
as
03B110,
a40
and
a60
were
obtained.
If
the
extrapolation
value
of
the
correction
procedure
is
noticably
influenced
by
the
real
geometry
of
the
specimen
this
will
certainly
be
observed.
Two
supplementary
electrochemically
Fe
10-Ni
specimens
denoted
pol 1
and
pol
2
were
also
examined.
Two
Fe-Ni
alloys
are
used
to
derive
an
experimental
kNiFe-factor :
Fe
(50.00 ±
0.07)
wt%
-Ni
and
Fe
(90.01 ±
0.01)
wt%
-Ni.
The
result
reads :
The
experiments
with
the
Fe
10-Ni
specimens
are
carried
out
with
this
k-factor.
6.2.2
Experimental
results for
the
Fe
10-Ni
alloy.
-
The
five
Fe
10-Ni
specimens
are
examined
in
a
single
tilt
low
background
holder
at
100
kV
and
all
the
spectra
are
taken
in
electron
transparant
areas.
Since
the
specimens
have
approximately
the
composition
given
in
table
IV,
no
absorption
correction
is
needed
even
for
the
thickest
regions.
On
each
specimen
3
independent
parameterless
analysis
are
carried
out
and
the
results
are
sur-
veyed
in
table
V
Thble
V -
Compositions
measured
with
the
parameterless
correction
procedure
(in
at% )
on
differ-
ent
Fe
10 -
Ni
specimens.
The
relatively
large
errors
are
due
to
the
fact
that
here
fluorescence
is
the
only
phenomenon
which
influences
the
uncorrected
concentrations
(or
concentration
ratios).
In
other
words
the
uncorrected
values
will
be
somewhat
influenced
by
the
local
changes
in
the
specimen
geometry.
Consequently
the
errors
on
the
extrapolation
values
will
be
larger
than
for
specimens
where
ab-
sorption
dominates.
No
distinction
can
be
made
between
the
différent
specimens
and
all
the
results
are
consistent
with
the
presumed
composition
(Thb.
IV).
The
weighted
average
of
the
results
of
table
V
can
be
considered
as
the
most
accurate
estimation
of
the
final
composition
and
is
listed
in
table
VI.
21
Table
VI.
-
Thefinal
composition
of the
Fe
10-Ni
alloy
measured
with
the parameterless
correction
procedure.
This
final
result
(Tab.
VI)
clearly
demonstrates
that
the
parameterless
correction
procedure
corrects
succesfully
for
fluorescence,
although
generally
with
less
accuracy
than
for
absorption.
However,
the
accuracy
can
be
improved
by
increasing
the
number
of
spectra.
Apart
from
mentioning
that
the
uncorrected
concentrations
undergo
rather
serious
fluctua-
tions
due
to
local
geometry
changes
it
has
not yet
been
demonstrated
that
fluorescence
corrections
are
actually
needed.
Therefore
in
figure
6
the
uncorrected
iron
concentrations
of
the
first
analysis
of
specimen
al0
are
plotted
versus
the
sum
of
the
net
peak
integrals
of
Fe-Ka
and
Ni-Ka.
It
is
seen
from
figure
6
that
within
the
transparancy
limit
at
100
kV
the
uncorrected
Fe
concentration
undergoes
a
relative
increase
of
17%
(1.57
at% ).
Since
this
increase
cannot
be
ascribed
to
an
absorption
phenomenon,
it
is
entirely
caused
by
the
generation
of
secondary
Fe-Ka
radiation.
Fig.
6.
-
Uncorrected
Fe
concentration
(at% )
versus
the
sum
of the
net
peak
integrals
of Fe-Ka
and
Ni-Ka.
The
apparant
increase
of the
iron
concentration
with
specimen
thickness
is
a
pure
fluorescence
phenomenon.
All
spectra
were
taken
in
electron
transparent
area.
7.
Conclusion.
Experimental
results
reported
elsewhere
[16]
as
well
as
the
applications
mentionned
in
sec-
tion
6
clearly
demonstrate
the
power
of
this
correction
procedure.
Irrespective
of
the
shape
and
composition
of
a
specimen
the
absorption
and
fluorescence
phenomena
are
succesfully
corrected
22
for.
Beside
this
unquestionable
advantage
and
also
in
contrast
with
the
classical
approaches,
the
described
correction
procedure
provides
the
final
results
with
an
accuracy
figure.
This
statisti-
cal
error,
which
typically
lies
between
0.2
and
1.0
at-wt%
easily
permits
to
distinguish
between
well
and
badly
performed
experiments,
although
it
does
not
account
for
the
systematic
error
in-
troduced
by
the
use
of
a
wrong
k-factor.
About
this
k-factor
problem
it
should
be
mentioned
that
Carpenter et
aL
(1988)
[17]
used
the
method
precisely
to
determine
k-factors
provided
with
accuracy
estimates.
Furthermore,
mass
thickness
dependent
artefacts
such
as
surface
layers
or
a
crystallographic
ori-
entation
effect
(the
Bormann
effect
which
can
become
very
important
in
the
very
thin
sections
of
the
specimen
are
detected
by
the
method.
Finally
the
main
achievement
of
this
correction
procedure
probably
remains
the
fact
that
no
external
parameters
such
as
thickness
and
density
of
the
specimen,
mass
absorption
coefficients,
fluorescence
yields
etc....
are
needed.
This
guarantees
a
fast
and
non
speculative
processing
of
the
acquired
spectral
data.
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