How Accurate Are Modern Fundamental Parameter

THE RIGAKU JOURNAL
VOL. 21 / NO. 2 / 2004, 17–25
HOW ACCURATE ARE MODERN FUNDAMENTAL PARAMETER
METHODS?
MICHAEL MANTLER AND NAOKI KAWAHARA*
Institute of Solid State Physics, Vienna University of Technology, Vienna, Austria
* Rigaku Industrial Corporation, Osaka, Japan
Fundamental parameter methods comprise a wide variety of applications and mathematical approaches. The question of achievable accuracy requires therefore a rather detailed view,
and answers will vary with respect to the range of the analyte elements and the analyte lines,
the employed mathematical models (such as conventional fp-equations, Monte-Carlo approaches, or theoretical influence coefficients), the type of specimen (bulk material, layered
material, fused samples, etc.), and the instrumentation including the geometrical setup, excitation source, and detection system. Reference materials, their eventual dispensability (standardless methods) and—if provided—their optimum composition require careful attention. Finally,
the reliability of the tabulated values of fundamental parameters themselves is a key element
of such discussions.
1.
Basic Considerations
As with any other method, the fp-approach
can take advantage of reference materials, and
an analysis employing a standard of similar
composition as the unknown material will give
better results than by using a very different
standard. Theoretically, the best possible result
will therefore be expected if the compositions
of unknown and standard (as well as the
method of their preparation) are identical. In
this case the obtained count data are defined
only by the repeatability of the measurement,
which is the closeness of agreement between
independent results obtained with the same
method on identical test material, under the
same conditions. It is measured by the repeatability standard deviation, a purely statistical
quantity. Hence, it can be made as small as desired by repetitions and long counting times
within the general limitation of systematic deviations introduced by long-term drifts.
Therefore, in principle, fundamental parameter methods (and other methods as well) are
not inherently limited in accuracy. The achievable accuracy in an actual experiment is ultimately limited by repeatability issues, i.e., statistical errors, and the availability and quality of
reference materials, i.e., the closeness of their
assumed compositional data (“best possible estimates”) with (never observable) true values.
The discussed hypothetical case of “analyzing” an unknown by comparison with a standard of identical composition is of course kind
Vol. 21
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of extreme, particularly in view of the main advantage of fp-methods which is that similar
standards are not necessarily required. The
other extreme is to use no reference materials
at all (truly standard-less method*1). Then all
parameters in the employed mathematical
model must be known at an absolute scale
(rather than in comparison to a standard). The
obtained accuracy depends on all statistical and
systematical errors of the fundamental parameters, the systematical errors introduced by deficiencies of the mathematical models, the numerical and statistical computational errors,
and the statistical errors of the experiment.
In an attempt to explore the limitations of
such standard-less analysis by state-of-the art
instrumentation and fp-methods, experiments
have been carried out by Beckhoff et al. [1] and
Mantler et al. [2] at fully calibrated synchrotron
beam-lines of the PTB*2 laboratory at Bessy-II*3
by using certified pure element standards as
*1 In many commercial products the term Standard-less
Method is used as well, but with the understanding of employing pre-calibrated instruments, where data from reference materials are stored in the factory and remain valid
over extended periods of time due to the exceptional longterm stability of modern instruments. In the context of this
paper, [truly] standard-less is used for methods which are
based on absolute counts rather than count rate ratios, and
which use no reference materials for calibration.
*2 Physikalisch-Technische
Bundesanstalt
(National
Metrology Institute of Germany), Berlin, Germany.
*3 Berlin electron storage ring company for synchrotron
radiation, Berlin, Germany.
17
objects for testing. Measured and computed absolute photon counts (not count-rate ratios) are
in agreements of less than 4% relative for thin
layers of 5 nm–100nm of Cu and Ni. This
matches the estimated uncertainties in the complex absolute calibration of all beam-line components and proves the validity of the basic fpapproach.
Such purely standard-less methods are highly
useful for development work, for example studies of the complex indirect excitation effects in
the regime of light elements or XRF of L- and Mlines. All practical methods are, however, based
on count rate ratios and some kind of reference
materials including pure element reference
standards. This allows working with “relative
intensities”, where several factors (including
some with less well known values) in the equations of standard and unknown cancel.
2. The “Ideal” Fundamental Parameter
Approach
The simplest model for the computation of
the number of photons observed from a sample
with known type and composition and under
known experimental conditions is based upon
absorption of primary and fluorescent beam
within the sample, and the excitation of analyte
line i by absorption of a primary photon of energy E (see Fig. 1). The following equations for
the number of observed photons, ni, are valid
for a thin compound layer (see the Appendix for
a definition of the symbols):
S 1
Ω
⋅κ i ⋅ ci ⋅ i
⋅ pi ⋅ ω i
Si
4 ⋅ π ⋅ sin ψ 1
ni ∫
E0
⋅
Ei
N 0 (E ) ⋅ τ i (E )
⋅ (1e T ⋅µi*⋅ρ ) ⋅ dE
µi*
(1)
where
L
µi* ∑c
j
j 1
and c i L
 µ j (E )
µ j (i ) 
⋅
c j ⋅ µ ji*
sin ψ 2  j 1
 sin ψ 1
∑
ˆ
mi
m
i
M
Mˆ
L
and Mˆ ∑ mˆ
j
j 1
Replacing the concentrations by mass-thicknesses, m̂i , eliminates also the thickness, T :
S 1
Ω
ˆi ⋅ i
⋅κ i ⋅ m
⋅ pi ⋅ ω i
Si
4 ⋅ π ⋅ sin ψ 1
ni ∫
E0
⋅
Ei
18
⋅
N 0 (E ) ⋅ τ i (E )
⋅ (1e ∑ j mˆ j ⋅µ ji* ) ⋅ dE
ˆ
mj ⋅µ ji*
∑
j
(2)
Fig. 1. Basic geometry of fundamental parameter
approach.
For bulk materials one obtains after division of
numerator and denominator by m̂i :
S 1
Ω
⋅κ i ⋅ i
⋅ pi ⋅ ω i
Si
4 ⋅ π ⋅ sin ψ 1
ni ∫
E0
⋅
Ei
N 0 (E ) ⋅ τ i (E )
⋅dE
ˆj
m
⋅ µ ji*
j m
ˆi
∑
(3)
The count rate of a layer is a function of all
mass-thicknesses constituting the layer, while
for the bulk material it is only a function of
mass ratios and independent on the total mass,
M, (or thickness). If each of the L elements is
measured by a single analyte line, equation (2)
delivers (by an iterative algorithm) the L unknown mass-thicknesses m̂i . This is equivalent
to L1 concentrations (with the L’th concentration obtained from cL1 Í cj ) and the thickness, T. In the case of bulk materials, however,
there are only L1 unknown ratios m̂j /m̂i (or L
concentrations after complementation by
cL1 Í cj ), which makes the system of equations over-determined by 1. Unless the redundant equation is used for the analysis of an unmeasured element, the overall accuracy may be
thereby improved. An undesirable side-effect
may be that least-squares methods, as inherently used for the solution of an over-determined system of equations, have the general
tendency of minimizing the sum of errorsquares by best-matching in first place the parameters with large numerical values, i.e., the
high concentrations, at cost of low concentrations and traces.
The Rigaku Journal
The accuracy of the predicted counts, ni , depends upon
• The general validity of the model (e.g., contributions by indirect excitation, errors introduced by the simplified geometry, etc.).
• The accuracy by which the spectral distribution, N0(E ), of the primary photons is known.
• The accuracy of the fundamental parameters.
The simple model above includes only direct
excitation in a homogeneous specimen. Indirect
excitation is often seen synonymous with secondary and tertiary excitation and treated as a
correction term, but there exist a few effects
which can in specific situations by far outweigh
direct excitation. The most important contributions are:
• Secondary excitation (up to 25–30% of primary photons for bulk materials, much less
for thin layers).
• Tertiary excitation (up to 5%).
• Coster–Kronig transitions (up to 30% in L-subshells, up to 70% in M-shells, due to modifications of the electronic configuration).
• Cascading electron transfers during relaxation, whereby K-ionization initiates L-line (or
M-line) emission within the same atom. The
contribution varies considerably. It can be a
large multiplicative factor in case of efficient
[monochromatic] excitation by energies above
the K-edge.
• Excitation by photo-electrons and Auger-electrons (can be a multiplicative factor).
Special sample types such as multi-layered materials, inhomogeneous samples, non-ideal
geometries, rough surfaces, etc. must be dealt
with by further extensions to this model (or
treated by alternative methods, such as MonteCarlo computations).
3.
The “Real” Situation: Divergent Beams
In most modern instruments the X-ray tube is
situated rather close to the specimen (Fig. 2).
Therefore, the beam from the X-ray tube hits
the specimen under rather widely divergent angular directions. Three effects must be taken
into account:
• As a consequence of the lateral incidence of
the beam, the spot with the highest intensity
on the specimen is not at the center but
closer to the spot of minimum distance to the
target (following roughly the “distancesquared-law”). This can easily be made visible by placing a photographic film (wrapped
into black paper) into the specimen holder, as
shown in Fig. 2.
The uneven illumination by the primary
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2004
Fig. 2. Geometry of a crystal spectrometer. The
sample is irradiated by a divergent beam under an
angle of about 45° by the X-ray tube. The photographic image shows the intensity-distribution across
the specimen.
beam may also be an issue with imaging
techniques, where a small spot of the specimen is selected by a mask and the specimen
scanned. Only if the specimen is moved (as in
modern Rigaku Systems) rather than the
mask, a constant illumination by the primarybeam is maintained.
• The spectral distribution and intensity of the
partial beams in each direction vary due to
the varying angle of emergence from the target and transmission through the Beryllium
window. Radiation in the peripheral area is
slightly hardened up. Mainly low energies
and therefore very light elements or analyte
lines from outer shells are affected.
• Assuming a single “average” angle of incidence is only a rough approximation for a divergent polychromatic beam. While the deviation due to the geometrical divergence can be
19
Fig. 3. Computed dependence of count rates and count rate ratios of incident and observation angle. [Note: The count-rate ratios R_CrLa as a function of observation angle (at left/bottom) are practically constant at a value of 1.0. The count-rates n_NiKa , n_FeLa , n_CrLa , and
n_NiLa as a function of observation angle are all within a narrow band (at right/bottom)].
easily computed, it is difficult to estimate the
effect of spectral intensity variations, as this is
a function of tube geometry, target material,
and operating voltage.
Computed data for stainless steel samples
with varying angles are shown in Fig. 3.
Building count-rate ratios is not necessarily
helpful for all analyte lines (but certainly is in
some cases). For an incident-beam divergence of 45°10° a relative error (without accounting for variations of the primary photon
flux) in the 0.1% range for K-lines and up to
1% for low-energetic L-lines must be expected, for absolute photon counts as well as
for count rate ratios.
• While the observed fluorescent beam is quite
parallel in crystal spectrometers, this is not
the case of energy-dispersive detectors. Besides the geometrical error which can be derived from the fp-equation(s), the various angular directions can cause different detector
response properties in EDXRS.
4. The Accuracy of Fundamental Parameters
Fundamental parameters include
20
• Atomic interaction coefficients which describe
the probabilities for photo-electric effect, coherent scattering, and incoherent scattering,
• Absorption edge jump ratios, which are defined as the ratio of photo-absorption above
and below the absorption edge and are used
to compute the probability that a photon is
absorbed in a specific energy level (edge),
• The transition probability, which describes the
probability of a specific electron transfer into
the vacancy created by photo-absorption, and
• The fluorescent yield, which is the probability
that the relaxation is associated with the
emission of a photon rather than of an Augerelectron.
Additional parameters are required to describe
interaction in the regime of light elements and
low energies, such as
• Probabilities of Coster–Kronig transitions,
whereby electrons move into vacancies of a
sub-shell from an other sub-shell of an atom,
such as L1 ←L2 or L2 ←L3 or L1 ←L3, or correspondingly in the regime of the M-shell,
• Various specific Auger-probabilities (see
below), and
• Cross sections for elastic and inelastic scatterThe Rigaku Journal
ing of electrons by atoms, for the computation of indirect excitation effects by electrons.
In addition, several other parameters are required, such as the spectral distribution of tube
photons, the detector response function (“detection efficiency”), and the geometry of the instrument. For tube spectra, experimental data
[3] as well as algorithms for their computation
[4] are available. The match between theoretical
and experimental data may be impaired by the
age of a tube due to the contamination of target
and window by tungsten and copper, and by
deterioration of the target flatness. This affects
mainly absolute photon counts and cancels in
most cases effectively by building count rate ratios.
As far as the detector response function can
be treated as a purely energy dependent
“counting efficiency”, it cancels by building
count rate ratios. However, dead-time problems, and—with energy dispersive detectors—
the complex behavior of the detector response
function may require special corrections with
unpredictable errors in case of neglecting these
effects.
The accuracy by which the atomic parameters
are known depends in general on atomic number and energy (there are a few elements within
the “well known-range”, such as radioactive materials and artificial elements, for which individual data are nevertheless unavailable or unreliable). It should be kept in mind that many coefficients have never been measured and their
tabulated numerical values are derived by interpolation from neighboring atoms or energies,
or by theoretical methods.
Estimates of the accuracy of fundamental parameters by Elam [5], who has published a
freely downloadable collection of fundamental
parameters (Elam et al. [6]), are:
A general observation in the field of interac-
Cross sections,
fluorescent yields
Other parameters
Below 1–3 keV
Above 1–3 keV
10–50%
A few %
50%? Frequently
unknown
10–20%
Mass Absorption Coefficients of Zinc
1.E+04
Line: XCOM v3.1( M.J. Berger. S.M. Seltzer, 1999
Dots: McMaster et.al.
1.E+03
1.E+02
1.E+01
0
5
10
15
20
25
30
keV
m Zinc: Differences between Tables
5%
4%
3%
2%
%Difference
1%
0%
-1%
0
5
10
15
20
25
30
keV
Fig. 4. Interaction coefficients for Zinc as a function of energy. Comparison of data by McMaster et al.
and XCOM.
scale. The overall effect on count rate ratios is
typically in the range of up to 0.1%.
5. The Advantage of Building Count-rate
Ratios
The equations for directly excited photons of
element i in a compound sample, ni , and for the
pure element, Ni , show several common factors
which cancel completely by building ratios (indicated by solid lines). The same effects would
be achieved by using a compound reference
material.
In addition, the spectral distribution function,
N0(λ ), and the photo-absorption coefficient,
τ i (E ), appear in the integral of both equations
and the effect of an error will at least cancel partially. This is particularly true if the error in
these functions is closely a constant factor.
tion coefficients is that the experimental values
obtained by various authors differ mainly in the
near-absorption edge region and are often in
good agreement otherwise, as shown in Fig. 4
with data by McMaster et al. [7] and XCOM [8].
The differences are not visible in the logarithmic
Vol. 21
No. 2
2004
21
ni ,direct Ω
S 1
⋅κ i ⋅ i
⋅ pi ⋅ ω i
Si
4 ⋅ π ⋅ sin ψ 1
∫
E0
⋅
Ei
Ni N 0′ (E ) ⋅ τ i (E )
⋅ dE
µi*
Ω
S 1
⋅κ i ⋅ i
⋅ pi ⋅ ω i
4 ⋅ π ⋅ sin ψ 1
Si
∫
E0
⋅
Ei
N 0 (E ) ⋅ τ i (E )
⋅ dE
µii*
ni, sec( j →i ) ⋅c i ⋅
(a)
(b)
Ω
1
S 1
⋅
⋅κ i ⋅ i
⋅ pi ⋅ ω i
2 4 ⋅ π ⋅ sin ψ 1
Si
Si 1
⋅ pj ⋅ ω j ⋅c j ⋅
Si
∫
E0
Ej
N 0 (E ) ⋅ τ i (j ) ⋅ τ j (E )
µ j*


 sin ψ 1
µ(E )
⋅ ln 1
⋅

µ(j ) ⋅ sin ψ 1 

 µ(E )

sin ψ 2
µ(i )
⋅ ln 1
(
j
)
⋅ sin ψ 2
µ(i )
µ

 
  ⋅ dE
 
In the equations for secondary excitation the
situation is less favorable, because all factors
describing the excitation probability of the primary fluorescent photon, j, remain in effect (in
the marked box). Their error propagates directly
into the secondary excitation term and affects
the total accuracy in proportion to the contribution of secondary excitation. If fluorescent yield,
absorption edge jump, and transition probability are each affected by a statistical error of 1%,
the error propagation law gives a total error in
(relative) secondary excitation of also about 1%.
Assuming that the contribution is nsecondary /
ntotal10%, the overall result has then an error
of 0.1%.
6.
Specific Effects at Light Elements
Analysis of light elements differs from that of
medium- and high-Z elements by the fact that
• Their fluorescent yield is very low (in the
order of 103 for carbon and 104 for beryllium), and therefore the observed photon
count rates are extremely low.
• Excitation by tube photons is inefficient because there are no photons in the vicinity of
the absorption edges of light elements.
• Absorption of the fluorescent photons by the
matrix is high.
• Chemical state may affect the observed photon count rates.
22
Fig. 5. (a) Exitation of light elements by photoelectrons. (b) Contribution of Photo-electrons and
Auger-electrons to the observed carbon Ka -counts in
percents of the total photon count (computations by
Pavlinksy et al. [11]). Singular data-point: Own computations for Fe–C, for a range of different geometrical set-ups.
Excitation by Photo- and Auger-electrons
When a photon is absorbed by an atom in an
inner shell, in practically all cases a photo-electron is emitted which carries the energy-difference between the photon and the electron’s
binding energy. Because most tube-photons
have energies much higher than the binding-energy of a light element, the energy of the photoelectron may be rather high. The example in
Fig. 5 shows the situation for EPhoton20 keV,
EBinding0.28 keV, EElectronEPhotonEBinding19.72
keV. The energies of such electrons are in the
same order of magnitude as those used for example in electron probe microanalysis (EPMA),
and photo-electrons are therefore very well capable of exciting one or more other (light) elements. Other than in EPMA, electrons are emitted anywhere in the sample where photons are
absorbed rather than injected from an external
source (Fig. 6). In a similar way, Auger-electrons
contribute to indirect excitation.
The implementation of indirect excitation efThe Rigaku Journal
10
-8
10
-9
Measurement
Calculation:
without photoelectron excitation, without cascade effect
with photoelectron excitation, without cascade effect
with photoelectron excitation, with cascade effect
[a]
NB: parameters for the ionization crosssection of Bethe: b=0.209, c=1.053
[c]
[b]
10
-8
10
-9
Calculated Cr-L Intensity
Measured Cr-L Intensity
Fig. 6. Left: Electrons in electron probe microanalysis (source is outside the specimen).
Right: Photoelectrons emitted from some point within the specimen. Any point within the specimen, where tube-photons are absorbed, is such a source-point of electrons.
-10
10
0
1000
2000
3000
4000
5000
6000
7000
8000
Incident Photon Energy / eV
Fig. 7. Circles: Photon counts of CrLa , as a function of monochromatic excitation energy. [a]
Effect of excitation by photo-electrons. [b] Jump at Cr–K edge due to additional vacancies in
the K-shell, which are filled by K ←L3 transitions, freeing L3 for L3 ←M5 transitions (La 1). [c]
Contributions of Coster–Kronig transitions. After Kawahara et al. [10].
fects by electrons in fundamental parameter
software is difficult. While Monte-Carlo methods appear to be most suitable for computation
of electron interactions, Kawahara [9, 10] has
shown a successful solution for bulk materials
and thin films which is based on conventional
sets of equations (Fig. 7).
Because of the fact that the electrons can
originate from any atom in the matrix, those
will abound whose host atoms absorbs high energy photons best and are contained in high
concentration. In steel this is naturally iron.
Since the size of micro-structural entities
(grains) in iron-based materials is often in the
Vol. 21
No. 2
2004
same order of magnitude as the range of electrons and carbon-photons, the microstructure of
the material (and therefore the manufacturing
process and specimen preparation) have an essential influence. Such situations constitute still
a challenge in light element analysis and require careful selection of reference materials.
Estimates by Pavlinsky et al. [11] and own computations show that in cases with small concentrations of a light analyte element (e.g., 1% C) in
a heavy matrix up to 85% of the observed intensity may originate from indirect excitation by
electrons.
23
7.
Effects in the L- and M-Region
Coster–Kronig Transitions and Auger-effect
Coster–Kronig transitions and Auger-effect
describe essentially the same type of de-excitation of an atom. The starting point is an ionized
atom, as it exists for example after an initial absorption of a photon in an inner shell, after
emission of a photo-electron. The mechanism
of ionization is, however, of no importance.
In XRF the subsequent relaxation is favorably
a cascade of electron transitions from outer
shells (e.g., K ←[Ka 1] ←L3 followed by L3 ←
[La 1] ←M5 and possibly further transitions from
outer shells). The competitive process is an energy transfer to an electron, which is then emitted, leaving the atom in a double ionized state.
For example, when an electron transition K ←L3
takes place, the energy (equivalent to Ka 1) can
be transferred to an electron, for example in M1,
which is thereby emitted with an energy of
EKa 1EM1. The electron is identified as a KLMAuger-electron (in detail: KL3M1-Auger electron).
The Auger-process should be considered a single quantum mechanical relaxation event,
which is not bound to the quantum mechanical
selection rules for photons. In particular, there
exists no “intermediate” state with a Ka 1 photon and hence no “internal absorption”.
In this particular example, three different
shells are involved: K, L, and M. If initial ionization and subsequent electron transition refer to
the same principal shell (e.g., a LLM-type effect), it is called a Coster–Kronig transition. If
everything takes place within the same principal
shell, it is called a super-Coster–Kronig transition.
These effects have the consequence of rearranging the distribution of electrons (in effect,
of the vacancies) within the electronic shell of
the atom and thereby changing the observed
line intensities, which are proportional to the
number of pertinent vacancies. For example,
the emission of a La 1 photon (L3 ←M5) requires
a vacancy in L3. This can be created in various
ways:
• Direct ionization of the L3 shell (e.g., by a
tube-photon).
• Direct ionization of the L2-shell, followed by a
L2L3 Coster–Kronig transition.
• Direct ionization of the L1-shell, followed by a
L1L3 Coster–Kronig transition or a double transition L1L2L2L3.
• Direct ionization of the K-shell, followed by a
K ←[Ka 1] ←L3 transition.
• Direct ionization of the K-shell, followed by a
24
K ←[Ka 2] ←L2 transition, followed by a L2L3
Coster–Kronig transition.
• KL1X or KL2X Auger-effects with subsequent
Coster–Kronig-transitions. Note that Auger-effect leaves the atom in a doubly ionized state.
The change of energy levels may have a measurable effect.
• KL3X Auger-effects, where X can be any outer
shell for which EXEKEL3.
These effects can be rather large. For example
in many practical applications the ionization of
the K-edge may be much more efficient than of
the L3-edge and the cascading relaxation, amplified by Coster–Kronig transitions, may change
the observed intensity considerably. Some selected numerical values have been given by J.
Kawai [12] in detailed discussion of such effects
(also related to the influence of chemical states)
in a previous issue of the Rigaku Journal. A
rather complete set of Coster–Kronig probabilities is contained in the compilation of fundamental parameters by Elam et al. A general discussion including a variety of numerical values
of (detailed) Auger-probabilities is given by
Bambynek et al. [13].
8.
Summary
The current error-limit of absolute (truly standard-less) XRF by fundamental parameter
methods under favorable circumstances (fully
calibrated synchrotron beam-lines, medium-Z
elements) is 3–4% relative. By using relative intensities with conventional instrumentation accuracies of a few 0.1 wgt% relative can be
achieved with K-analyte lines in case of well
prepared, homogeneous specimens. Main
sources of errors are—besides experimental errors—inaccurate values of fundamental parameters including spectral distributions of tube
spectra as well as neglect of accounting for real
geometries and indirect excitation effects, particularly when light elements or L- and M-lines
are involved.
Symbols and Terminology
i, j, k,..., n : Subscripts for elements (or compounds) in any given sample/specimen. Subscript i usually designates
the analyte and j the remaining elements (or compounds) in the specimen (referred to as matrix elements
or compounds.
The subscripts are also used to indicate the (analyte or otherwise referred to) line of that element, the
energy of that line, or the absorpThe Rigaku Journal
tion edge belonging to that line.
Atomic number
Energy of a photon
Highest energy in the tube-spectrum
Ej : Energy of the absorption edge for
analyte line j (element j)
m iE : Mass absorption coefficient for element i for a given energy E
m i*: Total effective mass absorption coefficient for element i,
m i*m im i
Ni : Count-rate from analyte line in pure
element standard
ni : Count-rate from analyte line in
compound specimen
N0(E ) · dE : Incident photons within interval
(E, EdE ), per cm2 and second
(flux)
ci : Concentration (weight fraction) of
element i
t : Mass photo-absorption coefficient
(subscripts as for m )
ˆ : Indicates items “per unit area”, such
as m̂ m/A (mass/area) or
T : Thickness (of thin layers)
w : Fluorescent yield
pi : Transition probability into a vacant
shell in context of analyte line i
S : Absorption edge jump ratio
S(Photo-absorption coefficient at
high energy side of edge)/(Photoabsorption coefficient at low energy
side of edge)
Z:
E:
E0 :
P:
Vol. 21
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ω ⋅p ⋅
2004
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[5]
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Denver X-ray Conference. Steamboat Springs, USA,
2004.
T. Arai, T. Shoji and K. Omote, Advances in X-Ray
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