Transcript No Slide Title

UofO- Geology 619
Electron Beam MicroAnalysisTheory and Application
Electron Probe MicroAnalysis (EPMA)
Quantitative Analysis:
Intensities to Concentrations
Modified from Fournelle, 2006
K ratios
Recall Castaing’s approach to quantitative
analysis, where specimen intensities are ratioed
to standard intensities:
where K is the “K ratio” for element i, I is the
X-ray intensity of the phase and subscript i is
the element.
With the counts acquired on BOTH unknowns
and standards on the same instrument, under the
same operating conditions, we assume that
many physical parameters of the machine that
would be needed in a rigorous physical model
cancel each other out (same in numerator and
denominator).
unk
i
std
i
I
Ki 
I
Castaing’s First Approximation
unk
i
C
unk
i
st d
i
I

I
st d
i
C
st d
i
 KiC
Castaing’s “first approximation” follows this
approach. The composition C of element i of the
unknown is the K ratio times the composition of the
standard. In the simple case where the standard is the
pure element, then, the fraction K is roughly equal to
the fraction of the element in the unknown.
...close but not exact
unk
i
C
unk
i
st d
i
I

I
st d
i
C
st d
i
 KiC
However, it was immediately obvious to Castaing
that the raw data had to be corrected in order to
achieve the full potential of this new approach to
quantitative microanalysis.
The next two slides give a graphic demonstration of
the need for development of a correction procedure.
Raw data needs correction
This plot of Fe Ka Xray intensity data
demonstrates why we
must correct for matrix
effects. Here 3 Fe alloys
show distinct variations.
Consider the 3 alloys at
40% Fe. X-ray intensity
of the Fe-Ni alloy is
~5% higher than for the
Fe-Mn, and the Fe-Cr is
~5% lower than the FeMn. Thus, we cannot
use the raw X-ray
intensity to determine
the compositions of the
Fe-Ni and Fe-Cr alloys.
(Note the hyperbolic functionality of the upper and lower curves)
Absorption and Fluorescence
• Note that the Fe-Mn alloys
plot along a 1:1 line, and so is a
good reference.
• The Fe-Ni alloys plot above
the 1:1 line (have apparently
higher Fe than they really do),
because the Ni atoms present
produce X-rays of 7.478 keV,
which is greater than the Fe K
edge of 7.111 keV.Thus,
additional Fe Ka are produced
by this secondary fluorescence.
Z
24
25
26
28
el
Cr
Mn
Fe
Ni
Ka keV K edge keV
5.415
5.989
5.899
6.538
6.404
7.111
7.478
8.332
• The Fe-Cr alloys plot below the 1:1 line (have apparently lower Fe than they
really do), because the Fe atoms present produce X-rays of 6.404 keV, which
is greater than the Cr K edge of 5.989 keV. Thus, Cr Ka is increased, with Fe
Ka are “used up” in this secondary fluorescence process.
Two approaches to corrections
In his 1951 Ph.D. thesis, Castaing laid out the two approaches that
could be used to apply matrix corrections to the data:
• an empirical ‘alpha factor’ correction for binary compounds, where
each pair of elements has a pair of constant a-factors representing the
effect that each element has upon the other for measured X-ray
intensity, and
• a more rigorous physical model taking into account absorption and
fluorescence in the specimen. This later approach also includes atomic
number effects and became known as ZAF correction.
Z A F
unk
i
C
Iiunk ZAFi unk st d
 st d
st d Ci
Ii ZAFi
In addition to absorption (A) and fluorescence (F), there are two other
matrix corrections based upon the atomic number (Z) of the material:
one dealing with electron backscattering, the other with electron
penetration (or stopping). These deal with corrections to the generation
of X-rays.C is composition as wt% element (or elemental fraction).
We will now go through all these corrections in some detail, starting
with the Z correction, which has two parts, the stopping power
correction, and the backscatter correction.
Note that all these corrections require close attention to exactly what
feature’s value is being input: the target (matrix), or the X-ray in
question.
Stopping Power Correction
Incident electrons lose energy in
inelastic interactions with the inner
shell electrons of the target. The
“stopping power” (energy lost by HV
electrons per unit mass penetrated) is
not constant but drops with increasing
Z. A higher number of X-rays will be
produced in higher Z targets. Thus, if
the mean Z of the unknown is higher
than that of the standard, a downward
correction in the composition must be
applied. The stopping power
correction factor is “S”, and can be
approximated by:
Reed, 1996, Fig. 8.6, p. 135
Stopping power of pure elements for 20 keV electrons
Z 1.166  Emean 
S  ln


A 
J
where J=11.5+ Z and Emean= (E0+Ec)/2
(J is the mean ionization energy; J, Z and A are of
the target, Emean is of the X-ray)
Backscatter Correction
As we discussed earlier, the
fraction of high energy incident
elections that are backscattered
(h) increases with atomic number.
There then will be relatively less
incident electrons penetrating into
higher Z specimens, resulting in a
smaller number of X-rays. Thus,
if the mean Z of the unknown is
higher than that of the standard, a
upward correction in the
composition must be applied. The
backscatter correction factor is
“R”.
Reed, 1996, Fig. 2.11, p. 17
R can be approximated by
1
R
1 [0.008  (1  W)  Z]
where W = Ec/E0 (the inverse of overvoltage),
and Z is of the target, and W is of the X-ray
Z correction
The total atomic number correction is formed
by multiplication of the R and S of the
unknown and standard thusly:
Z = Rstd/Runk * Sunk/Sstd
Overall the backscatter and the stopping power corrections
tend to cancel each other out.But if there is a (small)
correction, it is usually in the direction of the backscatter
correction.
Beers Law
The intensity I of X-rays that pass
through a substance are subject to
attenuation of their initial intensity
I0 by the material over the distance
they travel within the material.
The attenuation follows an
exponential decay with a
characteristic linear attenuation
length 1/m, where m is the (linear)
absorption coefficient. Beers Law
can also be expressed in terms of
mass, using density terms:
I = I0 exp -(m/r)(r Z)
where (m/r) is the mass absorption coefficient (cm2/g), r is
the material density (g/cm3), and Z is the distance (cm)
Als-Nielsen and McMorrow, 2001, Fig 1.10, p. 19
Mass Absorption Coefficients
Mass absorption coefficients
(MACs) have been tabulated* for
many X-rays through many
substances (though some are
extrapolations). They exist as a
matrix of numbers: absorption of a
particular X-ray line (emitter, e.g.
Ga ka) by a absorber or target (e.g.
As) will have one value (51.5).
Note that the absorption of As Ka
by Ga is a totally different
phenomenon with a distinct MAC
(221.4) .
Emitter = X-ray
(here, Ka)
Absorber =
matrix material
*See following discussion
Goldstein et al, 1992, p. 750.
Mass Absorption Coefficients
Emitter = X-ray
(here, Ka)
Terminology:
“the mass absorption of Ga Ka
by As”
Question for you: Is the mass
absorption of As Ka by Ga the
same as the mass absorption of
Ga Ka by As?
Why or why not?
Absorber =
matrix material
Goldstein et al, 1992, p. 750.
Absorption
X-rays produced within the material will
be propagated in all directions, and will
suffer attenuation in the process. Note
that the path length of travel of the X-ray
to the spectrometer is z cosecy, where y
(psi)is the takeoff angle (cosec = 1/sin).
Castaing’s approach was to integrate the
Beer’s Law equation over the depth at
the given y, producing the absorption
correction factor f(c) where c is defined
as m cosec ywhere m is the MAC.
f ( c) 
emergent i ntensi ty
generated intensi ty
The absorption (A) correction is then
defined as
A= f(c)std / f(c)sample
Reed, 1993,, p. 219
Absorption
To be able to correct for this absorption
of the measured X-rays, we need to know
how the production of X-rays varies
with depth (Z) in the material.
The distribution of X-rays as a function
of depth is known as the f(rz) [phi-rhoz] function, where a “mass depth”
parameter is used instead of simple z
(bottom right).
The f(rz) function is defined as the
intensity generated in a thin layer at some
depth z, relative to that generated in an
isolated layer of the same thickness.
Reed, 1993, p. 219
Absorption
One commonly used simplified form
(Philibert 1963) is
f ( c) 

(1 h)



1  (s )  1 h  1 (s )
c
c
where c = m cosec y , s is a measure
of electron absorption and depends on
effective electron energy, where
A
4.5  105
h  1.2  2
s  1.65
1.65
E0  Ec
Z
The Philibert approximation breaks
down, however, at the near surface,
creating errors when dealing with low
energy light elements, and we need to
go to more complicated and accurate
forms of the f(rz) function.
Reed, 1993, p. 219
Phi-Rho-Z
(combining the absorption correction with
stopping power and backscatter loss)
f(rz) [phi-rho-z] Curves
To be able to correct
properly for absorption -particularly for light
elements, the exact shape
of the f(rz) [phi-rho-z]
curve must be known.
Each X-ray has its own
curve. There are 3 main
parameters that affect the
shape of the curve:
•E0 (accelerating voltage)
• Ec (critical excitation energy of a particular
element line
• mean Z of the material
Reed, 1993, p. 220
Tracer Method
The f(rz) [phi-rho-z] curves are
usually determined by the “tracer
method”, where successive layers
are deposited by vacuum
evaporation. The tracer layer B is
deposited atop substrate A, with
successive layers of A deposited
on top.
Characteristic X-rays from the
tracer element are measured
(“emitted”) and then a generation
curve is calculated by correcting
each step for absorption and
fluorescence effects
“Quadralateral” Phi-Rho-Z
model!
What constitutes a
model?
Is the following
case, the x-ray
intensity as a
function of depth
can be modeled
using the geometry
of a rectangle!
And it works!
Fluorescence Correction
The X-rays produced within a
specimen have the potential for
producing a second generation of
X-rays: this is secondary
fluorescence, generally shortened
to fluorescence. This occurs when
the characteristic X-ray has an
energy greater than the absorption
edge energy of another element
present in the specimen.
As we saw earlier, Ni Ka (7.48
keV) is able to fluoresce Fe Ka
(Ec 7.11 keV). This effect is
maximized when there is a small
amount of the fluoresced element
present, e.g. Fe in a Ni-Fe alloy.
Reed gives an example where the Fe
intensity is 142% of what it “should be”.
Also, the continuum above an absorption
edge also causes fluorescence, although
this is generally weak.
Reed, 1996, Fig. 8.10, p. 139
Fluorescence Correction
The form of the correction F is
F
1
If
1
Ip
where If/Ip is the ratio of emitted X-rays from fluorescence, compared to the
X-ray intensity from inner shell ionization. In a compound, this term is
summed overall all the elements that could fluorescence the element of
interest.
Fluorescence Problems
Secondary fluorescence is an
important issue that must be
appreciated. Generated X-rays are
not scattered nearly as much as
incident electrons, and thus the
generated X-rays can travel
relatively long distances (50 um in
Fig 3.49) within the specimen and
produce a second generation of Xrays. If the specimen (and
standards) are relatively large
(=homogeneous), this is not a
problem. However, if minor or
trace elements are being analyzed
in small grains (Phase 1 in Fig
16.10) and the host phase (2) has
high abundance, an error may be
made in the EPMA analysis.
Goldstein et al. p. 142; Reed 1993, p. 258
Fluorescence across boundaries
Secondary fluorescence is a potential
source of analytical error across linear
boundaries, either horizontal (e.g. thin
films) or vertical (e.g., diffusion
couples).
In the example here of a vertical
interface between untreated Cu and Co,
there is NO diffusion. However, the
resulting EPMA profiles clearly imply
there is diffusion. There is NO diffusion
– there is only secondary fluorescence
across the boundary. Cu Ka X-rays can
excite Co, to the extent that there is
apparently 1 wt% Co about 15 um away
from the boundary within the Cu. But Co
Ka cannot excite Cu, so only the
continuum X-rays can create secondary
fluorescence, which is less – but
certainly distinguishable, an apparent 0.5
wt% Cu at 10 um from the boundary in
the Co.
Reed 1993, p. 259-260
Secondary Fluorescence Correction
A recent article (below left) reports an innovative approach
to correcting the secondary fluorescence (SF) in diffusion
couples and from adjacent phases. This utilizes a complex
Monte Carlo program called PENELOPE (Penetration and
Energy Loss of Positrons and Electrons) that permits
complicated geometric models of electron and X-ray behavior
in materials. SF can be simulated in a model that represents the
actual specimen (e.g. Fig 1 below), and then subtracted from
the observed data (right figure).
Matrix Correction Programs
The raw X-ray intensities are first corrected for:
• background contribution (MAN or off-peak)
Beam drift correction:
BN
• beam drift (i.e. counts are normalized)
• deadtime
• interferences* (if appropriate)
IC  I U
Where :I C
TU
TN
BU
is the beam drift corrected unknown intensity
is the uncorrected unknown intensity
is the nominal beam current
is the count time for the unknown intensity
is the count time for the nominal beam current measurement
is the beam current for the unknown intensity
IU
BN
TU
TN
BU
and then the K-ratios are input into an automated matrix correction program.
To run, the correction calculations must assume an initial composition for
the unknown -- because the magnitude of each factor is proportional to the
abundance of the element times its correction in a pure end member. The
assumed composition is a normalized (to 100%) value of the K-ratio. Based
upon the first iteration with this assumed composition, the result gives a more
truer composition, which then is the input for the second iteration. The
process is iterated until convergence, usually 3-5 times.
* Probe for EPMA does the interference correction within the matrix correction, a far better approach compared to the
normal (antiquainted) procedure of correcting the data after the matrix correction is completed.
Quantitative Interference Corrections:
u
CA
Element A (Interfered)
Element B (Interfering)

Element A + Element B
[ ZAF]s A
where :
Intensity
[ ZAF]s A I sA (  A )
[ ZAF]u A
u
IA
( A )
CsA
I sA (  A )
[ ZAF ]u A
and :
u
IA
( A )
CsA
u
IA
( A )
s
IA ( A )
CsA
[ ZAF ]s
is the unknown k-ratio and
is the ZAF correction factor of the unknown
is the unknown intensity for element A at 
is the standard intensity for element A at 
is the concentration of the element in the standard
A
is the ZAF correction for the element in the standard
Wavelength
I ( A ) 
u
C
u
A
C As

[ ZAF ]u A
s
[ ZAF ] A
[ ZAF ]s A
CBs
CBu
I Bs (  A )
u
[ ZAF ] A
I As (  A )
Where the following notation has been adopted :
Cij
is the concentration of element i in matrix j
[ ZAF ]j
i
is the ZAF (atomic #, absorption and fluorescence) correction term for matrix j
j
sI i (  i )
i
(Z and A are for wavelength
element i)
 i at
and F is for the characteristic line
for

.
i
is the measured x-ray intensity excited by element i in matrix j at wavelength
refers to an interference standard which contains a known quantity of the
interfering element B, but none of the interfered with element A.
ZAF options
One currently widely used matrix correction program is CITZAF, developed
by John Armstrong (then CIT, now NIST) and implemented in our Probe for
EPMA software. There are several options, which we elucidate here, but that
generally we do not modify them from the default values. Probably the only
parameter you would ever modify would be mass absorption coefficients
(there are different ones for the light elements).
Alpha correction
In the early decades of probing when computer power was
negligible, the alpha correction technique was widely used, as
it required less number crunching and relied mainly on
empirical data and less on complex physical models and
physics. Today, however, there may be a rekindled interest in
this approach, as it may “work better” in many cases.
Ziebold and Ogilvie binary a-factors
In 1963-4, Ziebold and Ogilvie* developed a corrections for some binary
metal alloys, with an equation in the form
(1  K1 )
(1  C1 )
 a 12
K1
C1
where a12 is the a-factor for element 1 in the binary with element 2, K is
the K-ratio, and composition (fractional) is C. This equation can be
rearranged in the form
C1
 a12  (1  a12 )C1
K1
If experimental data exist for binary alloys, then a plot of C1/K1 versus C1
is a straight line with a slope of (1- a 12), leading to determination of a 12.
Such a hyperbolic relationship between C1 and K1 was shown to be correct
for several alloy and oxide systems, but it was difficult to find appropriate
intermediate compositions for many binary systems.
*Quantitative Analysis with the Electron Microanalyzer, Analytical Chemistry, Vol 35, May 1963, p. 621-627;
An Empirical Method for Electron Microanalysis, Analytical Chemistry, Vol 36, Feb. 1964, p. 322-327.
Ziebold and Ogilvie ternary a-factors
Ziebold and Ogilvie showed that a corrections could be developed for
some ternary metal alloys, with an equation in the form
(1  K1 )
(1 C1 )
 a 123
K1
C1
where a123 is the a-factor for element 1 in the ternary with elements 2 and
3, and is defined as
a C  a 13 C3
a123  12 2
C2  C3
This equation can be rearranged
C1
 a 123  (1  a 123 )C1
K1
Similar relationships can be written for elements 2 and 3, and used to
calculate a-factors for the 3 binary systems of the ternary.These a-factors
were limited to a particular E0 and takeoff angle.
Bence-Albee multicomponent systems
Bence and Albee* in 1968 showed that this approach could be extended to
silicates and other minerals, i.e. a system of n components, where for the
Cn
nth component a b-factor could be found
 bn
kn
where
k1a n1  k2 a n2  k3a n3    k na nn
bn 
k1  k2  k3     kn
where an1 is the a-factor for the n1 binary.
These factors were determined for a limited set of conditions, i.e. 15 and
20 keV, and take off angles of 52.5° and 38.5°.
The 1968 Bence and Albee paper is one of the most highly cited papers in
the geological literature (over ~20,000 citations).
* Empirical correction factors for the electron microanalysis of silicates and oxides, J. Geology, Vol. 76, p.
382-403; also see Albee and Ray, Correction Factors for Electron Probe Microanalysis of Silicates, Oxides,
Carbonates, Phosphates, and Sulfates, Analytical Chemistry, Vol 42, Oct 1970, p. 1408-1414.
Calibration Curves
One can also use a multistandard calibration curve
method.
Generally this is only for
certain unique situations
such as trace carbon in
steel.
Evaluating matrix corrections
In 1988, John Armstrong* reviewed the Bence-Albee (a-factor) correction
scheme for EPMA of oxide and silicate minerals. He evaluated the old
factors, and revised some, using a -factors calculated from newer ZAF and
f(rz) algorithms, and showed “that with some modifications the a -factor
corrections can be as accurate as any other correction procedure currently
available and much easier and quicker to process.”
*Bence-Albee after 20 years: review of the accuracy of a-factor correction procedures for oxide and silicate
minerals, in Microbeam Analysis-1988, p. 469-76.
Armstrong also reviewed† ZAF and f(rz) corrections and suggested that
some of these correction algorithms “produce poorer results in the analysis
of silicate and oxide minerals than some of the earlier corrections”. He
specifically was referring to various corrections that were optimized for
metal alloys
† Quantitative analysis of silicate and oxide materials: comparison of Monte Carlo, ZAF and f(rz)
procedures, in Microbeam Analysis-1988, p. 239+
Before we forget....
Unanalyzed elements
The matrix corrections assume that all elements present (and interacting
with the X-rays) will be included. There are situations, however, where
either an element cannot be measured, or not easily, and thus the analyst
must make explicit in the quantitative setup the presence of unanalyzed
element/s -- and how they are to be input into the correction.
Typically oxygen (in
silicates) is calculated
“by stoichometry”.
Elements can also be
defined in set
amounts, or relative
proportions, or “by
difference” – although
this later method is
somewhat dangerous
as it assumes that
there are no other
elements present.
Unanalyzed oxygen
One complication for oxygen is variable valence states of elements
such as Fe. Robust software will allow you to enter case by case
different valence states.
In some cases, if oxygen is not included, there can be errors in the
matrix corrections of some elements, as the presence of O, OH, and
H2O can affect the actually measured elements, as there may be
significant absorption of those x-rays by the oxygen present*.
* Tingle, T.N., Neuhoff, P., Ostgergren, P., Jones, R.E. and Donovan, J.J. (1996)
The effect of “missing” unanalyzed oxygen on quantitative electron probe
microanalysis of hydrous silicate and oxide minerals. GSA Abstracts, 28, 212.
Impact of unaccounted for oxygen
Consider: Apophyllite -- KCa4Si8O20(F,OH)•8 H2O
Which has LOTS of oxygen which typically is “unanalyzed” and
therefore not involved in the matrix correction
Solution: Iterate a
fixed amount of H2O
(16 atoms of H =
1.76 wt% H plus
stoichometric O) per
formula to achieve
good results.
As shown in the
bottom analysis
where the H2O is
missing, there is up
to 3% relative error
for cations.
Physical Parameters Needed
The ZAF corrections require accurate and precise knowledge
about many physical parameters, such as
• Electron stopping power
• Mean ionization potentials
• Backscatter coefficients
• X-ray Ionization cross sections
• Mass absorption coefficients
• Surface ionization potentials
• Fluorescent yields
Additional Corrections:
•Area Peak Factor (APF) corrections:
•Volatile Element corrections:
•Standard Drift corrections:
“State of EPMA parameters”
As David Joy points out in his 2001 article “Constants for
Microanalysis”, there are problems in our knowledge of many
parameters:
• there are experimental stopping power profiles for 12 elements and
12 compounds, which raise questions about the traditional Bethe
equation
• only half of the elements whose K lines are used for EPMA have
measured K shell ionization cross-sections ; only 6 elements have
measured L shell cross-sections; there are zero M shell cross-sections
• K shell fluorescent yields are the best documented parameters; there
are gaps in the data for L shell yields; there are only 5 measured M
shell yields
• despite the fact that backscatter coefficients have been measured for
100 years, the data has many gaps and is of poor precision (i.e. 30%)
At the Eugene EPMA workshop in September 2003, John
Armstrong reviewed the state of EPMA matrix corrections
• Big problem with software/manufacturers, not documenting which
corrections used. Some have picked "improved" parameters which do not fit
with the other parameters, e.g. in some, where no continuum fluorescence
correction, the absorption correction was tweaked to take fluor into account,
and then when later fluorescence corrections developed, to use this in addition
to absorption correction, has an overcorrection for fluorescence.
• Problem with researchers not stating in their publications which correction
they used; NIST is trying to develop some protocols which people can
reference (brief notation with pointer to NIST for full description).
• There are a few errors/typos in the long accepted X-ray tables (i.e., Bearden)
– 3 are major errors.
• Actually measured mass absorption factors are rare! Measurements exist for
Na Ka by Al; Si Ka by Al and Ni; Mg Ka by O, Al, Ti and Ni; and Al Ka by
O, Na .....
• There is over 30% variation in published values of some macs for
geologically relevant elements; they can’t all be correct!
“So what do we do?”
We have discussed various ways to correct the raw data, the goal being
to come up with the most accurate and precise analytical procedures to
give us the most trustworthy data.
We have just mentioned that everything is not as rosy as one would
hope.
So, can we trust the numbers we get out of the probe? In many/most
cases, given care, yes. But we cannot blindly look at the electron
probe and computer as a black box!
Stay tuned for an upcoming installment, where we discuss standards,
accuracy and precision in EPMA.