United States
Environmental Protection
Agency
Environmental Sciences Research
Laboratory
Research Triangle Park NC 27711
EPA GOO ? 80 070
Apni 1980
Research and Development
Mathematical
Techniques for
X-ray Analyzers
-------
RESEARCH REPORTING SERIES
Research reports of the Office of Research and Development, U.S. Environmental
Protection Agency, have been grouped into nine series. These nine broad cate-
gories were established to facilitate further development and application of en-
vironmental technology. Elimination of traditional grouping was consciously
planned to foster technology transfer and a maximum interface in related fields.
The nine series are:
1. Environmental Health Effects Research
2. Environmental Protection Technology
3. Ecological Research
4. Environmental Monitoring
5. Socioeconomic Environmental Studies
6. Scientific and Technical Assessment Reports (STAR)
7. Interagency Energy-Environment Research and Development
8. "Special" Reports
9. Miscellaneous Reports
This report has been assigned to the ENVIRONMENTAL PROTECTION TECH-
NOLOGY series. This series describes research performed to develop and dem-
onstrate instrumentation, equipment, and methodology to repair or prevent en-
vironmental degradation from point and non-point sources of pollution. This work
provides the new or improved technology required for the control and treatment
of pollution sources to meet environmental quality standards.
This document is available to the public through the National Technical Informa-
tion Service, Springfield, Virginia 22161.
-------
EPA-600/2-80-070
April 1980
MATHEMATICAL TECHNIQUES FOR X-RAY ANALYZERS
by
Robin P. Gardner and Kuruvilla Verghese
Center for Engineering Applications of Radioisotopes
North Carolina State University
Raleigh, North Carolina 27650
Grant No. R-802759
Project Officer
T. G. Dzubay
Atmospheric Chemistry and Physics Division
Environmental Sciences Research Laboratory
Research Triangle Park, N. C. 27711
ENVIRONMENTAL SCIENCES RESEARCH LABORATORY
OFFICE OF RESEARCH AND DEVELOPMENT
U. S. ENVIRONMENTAL PROTECTION AGENCY
RESEARCH TRIANGLE PARK, N. C. 27711
-------
DISCLAIMER
This report has been reviewed by the Environmental Sciences Research
Laboratory, U. S. Environmental Protection Agency, and approved for publica-
tion* Approval does not signify that the contents necessarily reflect the
views and policies of the U. S. Environmental Protection Agency, nor does
mention of trade names or commercial products constitute endorsement or
recommendation for use.
ii
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ABSTRACT
A research program was initiated with the overall objective of develop-
ing mathematical techniques and subsequent computer software to process
energy-dispersive X-ray fluorescence spectra for elemental analyses of air-
borne particulate matter collected on filters. The research concerned two
areas: (1) determination of characteristic X-ray intensities and (2)
determination of elemental amounts from the known characteristic X-ray intens-
ities. In the first area, efforts primarily concentrated on developing and
implementing the library, linear least-squares method and included the two
common non-linear aspects of XRF pulse-height spectra: excitation source
background and pulse pile up. A detector response function model was also
developed for Si(Li) detectors to alleviate the necessity for obtaining and
storing the extensive complete library spectra for every element of interest.
This approach gave improved accuracy, greatly reduced the experimental effort
required, and was capable of accounting for variations in detector calibra-
tion and resolution without requiring extensive additional experimental
effort.
In the second research area, the fundamental parameters method was
developed by Monte Carlo simulation. Data were collected for several shapes
of particles deposited on filters. Empirical correction factors for various
practical cases of interest based on these simulations are reported. In
addition some more general Monte Carlo simulations were performed for a
number of cases. The conditions treated included monoenergetic photon (radio-
isotope) exciting sources; continuous (X-ray machine) photon exciting sources;
homogeneous samples of arbitrary thickness; primary, secondary, and tertiary
fluorescence in multi-component samples; and the inclusion of source and
characteristic radiation scattering effects.
This report was submitted in fulfillment of Grant No. R-802759 by North
Carolina State University under the partial sponsorship of the U.S. Environ-
mental Protection Agency. This report covers a period from May 15, 1974, to
May 14, 1979, and work was completed as of May 14, 1979.
iii
-------
CONTENTS
Abstract iii
Figures vi
Tables viii
Acknowledgements ix
1. Introduction 1
2. Conclusions 2
3. Recommendations 3
4. Determination of X-Ray Intensities 4
5. Determination of Elemental Amounts 6
References 7
Appendix: List of Reports, Theses, and Publications
Either Entirely or Partially Supported by
this Grant 7
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FIGURES
No.
1 X-ray spectrum containing eight elements excited with a
titanium secondary fluorescer XRF system
Residuals obtained from the X-ray "y spectrum shown in
Figure 1 for various simulated errors after least-
squares analysis 6
Case 1. Good statistics. X-ray spectra and residuals for
relative amounts of backscatter, Cr, Mn, and Fe in the
ratios 1.00, 1.00, 0.05, and 1.00 analyzed without the
Mn library spectrum 7
Case 1. Good statistics. X-ray spectra and residuals for
relative amounts of backscatter, Cr, Mn, and Fe in the
ratios 1.00, 1.00, 0.05, and 1.00 analyzed with the
Mn library spectrum
Case 2. Poor statistics. X-ray spectra and residuals for
relative amounts of backscatter, Cr, Mn, and Fe in the
ratios 1.00, 1.00, 0.05, and 1.00 analyzed without the
Mn library spectrum 8
Case 2. Poor statistics. X-ray spectra and residuals for
relative amounts of backscatter, Cr, Mn, and Fe in the
ratios 1.00, 1.00, 0.05, and 1.00 analyzed with the
Mn library spectrum 8
Comparison of an experimental high counting rate spectrum of
55Fe with that predicted by parabolic and polynomial
approximated pulse shapes 10
Monte Carol-predicted photon spectra from the Ag K-ot x-ray
(22.163 keV) scattered from a 5 mm thick Al target 11
Comparison of the observed backscatter regions of the X-ray
spectrum from a 1(*9Cd source from a 5 mm thick Al sample
with that predicted from Monte Carlo-generated shape
standards
VI
-------
FIGURES (continued)
10 Illustration of the four major spectral features of the two
K-0! X rays of gallium in a Si (Li) spectrum 13
11 The response function model compared to the experimental
pulse-height spectrum for manganese 14
12 The response function model compared to the experimental
pulse-height spectrum for nickel 14
13 The response function model compared to the experimental
pulse-height spectrum for gallium 15
14 The response function model compared to the experimental
pulse-height spectrum for arsenic 15
15 Variation of sulfur enhancement with layer thickness
and sphere diameter for various calcium compounds 17
16 Schematic diagram of the arrangement and nomenclative of the
rectangular secondary source target and circular sample in
relation to the circular detector and collimator for the
Monte Carlo simulation of a secondary fluorescer EDXRF
system 21
17 A typical experimental secondary source intensity
distribution 21
18 Monte Carlo simulation and experimental results for the
relative intensities from a pure nickel sample at various
angles for the Case l(a) and Case 2(b) system
parameters given in Table 1 22
19 Monte Carlo simulation and experimental results for
the relative intensities from pure nickel, pure iron,
and pure chromium samples at various angles for the
Case 3 system parameters given in Table 1 22
20 Monte Carol simulation and experimental results for the
relative intensities from a 42 percent nickel - 58 per
cent iorn binary sample at various angles for the Case
3 system parameters given in Table 1 23
VI1
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TABLES
No. Page
1 Results of a library least-squares analysis on a sample 5
2 Results of a library least-squares analysis for two
simulated particulate spectra containing chromium,
manganese, and iron 6
3 Comparison of intensities predicted by the Monte Carlo
simulation and the explicit analytical relationships 19
4 Comparison of relative intensities for Honte Carlo and
numerical calculations 20
Vlll
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ACKNOWLEDGEMENTS
R.K. Stevens, T.G. Dzubay of the Environmental Sciences Research
Laboratory, U.S. Environmental Protection Agency and D. Rickel of Northrop
Services, Inc. are gratefully acknowledged for the guidance and help that they
provided during the conduct of this research.
The following graduate students contributed significantly to this
research: Alan R. Hawthorne, Faruk Arinc, Lucian Wielopolski, and Joseph M.
Doster.
IX
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SECTION 1
INTRODUCTION
Energy-dispersive X-ray fluorescence (EDXRF) has only been recently
developed as a tool for elemental analysis. The use of this principle has
probably become of interest largely due to the advent of the high resolution
Si(Li) detectors. Combining these detectors with modern multichannel analy-
zers allows one to accumulate characteristic X-ray pulse-height spectra of
many elements simultaneously that are resolved from one another^ Therefore,
the X-ray analyst interested in EDXRF analysis must be concerned with computational
procedures for determining characteristic X-ray intensities for pulse-height
energy spectra.
A technique that was developed some time ago for the determination of
radioisotope amounts in mixtures from their gamma-ray pulse-height spectra is
the library least-squares method (1,2). The primary difference between the
radioisotope gamma-ray application and the photon-excited, energy-dispersive
X-ray fluorescence intensity application is that the presence of the exciting
source in the X-ray fluorescence case gives rise to spectral features that do
not obey the basic library linear least-squares assumption. This basic assumption
Is that the unknown spectrum is a linear combination of all the individual or
pure library spectra for the elements involved. This difference was first
addressed by Trombka and Schmadebeck (3) for the X-ray fluorescence analysis
of thick, homogeneous samples that might be encountered in space exploration
studies. In the present work (4,5) it is addressed for the X-ray fluorescence
analysis of airborne particulates collected on filters and the library least-
squares method is developed and demonstrated for this case.
One of the major problems associated with any type of X-ray fluorescence
analysis is that one must often account for sample matrix effects. The
intensity of the characteristic X-ray from a given element is a function of
all the other elements in the sample matrix through the combined phenomena of
enhancement and absorption. The present research addressed this problem
specifically for airborne particulates collected on filters (6,7) and for
more general cases such as for homogeneous and heterogeneous samples.
Since most of the work described here has been published in the open
literature, the present report has been shortened by eliminating the details
that can be found there. All of the published work that has been either
entirely or partially supported by this grant is listed in the Appendix.
-------
SECTION 2
CONCLUSIONS
Two major mathematical methods have been developed and tested for the
EDXRF analysis of airborne participates collected on filters: (1) the library
least-squares analysis of characteristic X-ray pulse-height spectra for
determining pure element intensities and (2) the Monte Carlo simulation of
characteristic X-ray intensities from particulates of various sizes, shape
factors, and compositions for determining elemental amounts from characteristic
X-ray intensities. As a result of this work the library least-squares analysis
method is now being routinely used by the Environmental Sciences Research
Laboratory in the analysis of about 10,000 samples per year.
In addition to these two major mathematical methods, other models have
been developed that are useful for certain analysis conditions. In the
determination of characteristic. X-ray intensities these include: (1) Monte
Carlo simulation of the sample scattered exciting photon radiation, (2) a
model and method for accounting for pulse pile up, and (3) a model for simu-
lating pure element characteristic X-ray pulse-height spectra based on a
model for the Si(Li) response function. In the simulation of characteristic
X-ray intensities from known elemental compositions these include: (1) Monte
Carlo simulation of discrete photon energy radioisotope exciting sources for
homogeneous samples, (2) Monte Carlo simulation of continous and discrete
energy X-ray machine exciting sources for homogeneous samples, and (3) the
development of a simple semi-empirical model for the prediction of X-ray
intensities from particulate and heterogeneous samples that are simply
characterized.
-------
SECTION 3
RECOMMENDATIONS
It is recommended that the library least-squares method based on the
simple model of the pure element library spectra obtained via the model of
the Si(Li) response function be implemented for the determination of charac-
teristic X-ray intensities for the EDXRF analysis of airborne participates
collected on filters.
It is also recommended that matrix correction factors be generated by
Monte Carlo simulation for cases of general interest that can be identified
for the same application. The methods and subsequent computer programs have
been developed and are available to accomplish these tasks.
Additional work is required to delineate the fundamental processes that
govern the Si(Li) response function and obtain a more fundamental model for
it.
-------
SECTION 4
DETERMINATION OF X-RAY INTENSITIES
The library least-squares method for the determination of characteristic X-ray
pulse-height intensities has been developed for photon excitation of airborne
participates collected on filters (4,5). In this case the exciting source
radiation is scattered almost entirely by the filter and can be considered to
have a constant shape. Different filter thicknesses can be assumed to only
alter the intensity of the exciting source scattered spectrum. Therefore, the
exciting source scattered spectrum can be obtained from a clean filter and
treated as a separate pure library spectrum. This is shown to be a practical
and accurate method of treating the exciting source scattered radiation in the
case of airborne particulates on filters. For the more general case where the
shape of the exciting source scattered radiation varies with sample composi-
tion, Monte Carlo simulation of the scattered portion of the spectrum has been
developed and demonstrated for a wide variety of homogeneous samples (8,9).
There are a number of minor problems associated with applying the library
least-squares method. These include: (1) statistical counting rate fluctu-
ations, (2) gain shift, (3) zero shift, and (4) missing library spectra.
These problems have been studied by simulation techniques. In the first study
the first three problems were examined for eight elements excited by a titan-
ium secondary fluorescer XRF system. The results of this study are given in
Table 1 and are shown in Figures 1 and 2. It is clear from these results that
the effect of even small gain and zero shifts are very dramatic and one must
minimize electronic drifts of this type if the method is to yield accurate
results.
In a second study the problem of detecting missing library spectra was ex-
amined by simulation methods. Spectra for the three elements chromium, mangan-
ese, and iron in various relative amounts and for various levels of counting
statistics were simulated by appropriate use of the existing library least-
squares method with spectra. The results are given in Table 2 and in Figures
3 through 6. From these results one may conclude that: (1) the reduced chi-
square value is an accurate indicator that elements have been missed if those
elements represent about their equal share of the total spectrum, (2) the
predicted standard deviations for each elemental amount are good estimates
of the errors Involved if the controlling source of error is the statistical
fluctuations of the counting rates and the chi-square value is close to unity,
and (3) the residuals of the least-squares analysis are a very sensitive vis-
ual indicator of missed elements. It would probably be quite easy to develop
a quantitative mathematical treatment of the visual indication given by the
residuals based on the counting rate differences in adjacent channels. How-
ever, a suggested alternative (5) is to first use all available library
-------
spectra in an Initial "screening" least-squares analysis. Those library
spectra that give negative amounts or amounts smaller than some prescribed
value are omitted and the least-squares analysis is performed again.
TABLE 1. RESULTS OF A LIBRARY LEAST-SQUARES ANALYSIS OF A SAMPLE
Element
Amount
Relative standard deviations (%)
) Simulation A*
Simulation B*
Reduced chisquare values: 0.964
75.63
Simulation C*
Argon
Aluminum
Silicon
Phosphorus
Sulfur
Chlorine
Potassium
Calcium
Lead
0.434
3.098
6.653
0.028
6.659
0.332
1.001
1.460
11.721
0.508
6.135
1.291
106. 781
1.055
4.989
1.070
0.657
10. 903
0.569
4.476
1.239
11.789
1.525
5.033
1.027
0.732
2. 648
0.512
6.554
1.282
194,391
0.981
4. 284
1.089
0.669
253.989
4.75
*Simulations A, B, and C are when errors are due only to statistical counting
rate fluctuations, a 1 percent gain shift, and a 0.2 channel zero shift,
respectively.
» te 500 -
u n ooo -
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CHANNEL NUMBER.
130
Figure 1. X-ray spectrum containing eight elements excited with a titanium
secondary fluorescer XRF system.
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21 42 63 84 105
CHANNEL NUMBER, i
126
Figure 2. Residuals obtained from the X-ray y spectrum shown in Figure 1
for various simulated errors after least-squares analysis.
TABLE 2. RESULTS OF A LIBRARY LEAST-SQUARES ANALYSIS FOR TWO
SIMULATED PARTICULATE SPECTRA CONTAINING CHROMIUM, MANGANESE, AND IRON
2
Amounts (yig/cm ) Relative standard deviations (%)
Element Case 1 Case 2 Without Mn Library With Mn Library
(Good statistics) (Poor statistics) Case 1 Case 2 Case 1Case 2
Chromium 13.7
Manganese 0.685
Iron 13.7
0.274
0.0274
0.274
0.34 2.68 0.36 2.80
3.46 18.93
0.27 2.15 0.28 2.17
Reduced chi-square values:
3.20
1.37
1.10
1.30
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Figure 3. Case 1. Good statistics. X-ray spectra
and residuals for relative amounts of
backscatter, Cr, Mn, and Fe in the ratios
1.00, 1.00, 0.05, and 1.00 analyzed without
the Mn library spectrum.
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Figure 4. Case 1. Good statistics. X-ray spectra
and residuals for relative amounts of
backscatter, Cr, Mn, and Fe in the ratios
1.00, 1.00, 0.05, and 1.00 analyzed with
the Mn Library included.
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Figure 5. Case 2. Poor statistics. X-ray spectra
and residuals for relative amounts of
backscatter, Cr, Mn, and Fe in the ratios
1.00, 0.02, 0.002, and 0.02 analyzed
without the Mn library spectrum.
Figure 6. Case 2. Poor statistics. X-ray spectra
and residuals for relative amounts of
backscatter, Cr, Mn, and Fe in the ratios
1.00, 0.02, 0.002, and 0.02 analyzed with
the Mn library included.
-------
A problem that Is encountered in some cases in the application of the
library least-squares method is that of the pulse pile-up phenomenon. At
high counting rates two (or more) pulses may be analyzed simultaneously giving
rise to sum pulses of the wrong magnitude and correspondingly fewer correct
individual pulses. This phenomenon can be partially accounted for by the use
of electronic pile-up rejection methods. However, it is impossible to
eliminate all sum pulses by this method and, therefore, a mathematical model
and correction technique has been developed and demonstrated (10,12). It is
found that the model and correction technique work quite well and are parti-
cularly suited to use with detection systems that include electronic pulse
pile-up rejectors.
The model for pulse pile up (10-12) is based on assuming that only two
pulses are analyzed simultaneously and that the interval probability distribu-
tion applies. To obtain explicit analytical expressions the pulses are as-
sumed to be either parabolic in shape or to be capable of approximation by a
low-order polynomial. Results of model predictions based on the parabolic
shape and the polynomial approximated pulses are shown in Figure 7. It is
obvious that the model is capable of good accuracy in either case, but the
polynomial approximated pulses give significnatly better results.
The previously mentioned problem of obtaining the shape of the source
backscattered spectrum for the more general problem of homogeneous samples has
been studied by using Monte Carlo simulation (8,9). Monte Carlo models were
developed that considered the five cases: (1) single Rayleigh (coherent)
scattering, (2) single Compton (incoherent) scattering, (3) double Compton
scattering, (4) Compton-Rayleigh scattering, and (5) Rayleigh-Compton scatter-
ing. A typical set of predicted results are shown in Figure 8 for the last
four cases. The single Rayleigh case (and multiple Rayleigh scattering) is
trivial since no change in energy occurs and the normal Gaussian-shaped pulse
with the resolution imposed by the detector is observed. The spectra from
these five cases were then used as librarXqSpectra in a least-squares program
to fit the experimental spectrum from a Cd source. The results of the
predicted least-squares fit of the individual library spectra are compared to
experimental results in Figure 9. It is clear that this technique is quite
accurate and should prove useful for homogeneous samples of known thickness.
Another problem in the application of the library least-squares method is
the necessity for preparing samples and obtaining the response from all the
pure elements of interest to the analyst to obtain library spectra. This is a
tedious and frustrating procedure and has the disadvantages that: (1) a large
amount of experimental work is required, (2) a large amount of computer
storage is necessary, (3) inaccuracies arise due to impurities in the samples
and the analyzer system, and (4) any change in the analyzer system (particu-
larly the detector) requires that the library spectra be accumulated again.
To alleviate this problem a model of pure element library spectra has been
constructed based on first obtaining a model for the Si(Li) detector response
function (13,14). This approach has been demonstrated for two Si(Li) detec-
tion systems of varying complexity and for a range of elements from manganese
to arsenic for excitation with molybdenum K X rays. All of the disadvantages
previously listed are alleviated and, in addition, one has the advantage with
this method that ratios of the K-a and K-g X rays need not be fixed. One
-------
IP 60
CHANNEL NUMBER
80
100
60
CHANNEL NUMBER
Figure 7. Comparison of an experimental high
counting rate spectrum of 55Fe with
that predicted by parabolic and
polynomial approximated pulse shapes.
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22.5
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22.5
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22.5
Energy (keV )
Figure 8. Monte Carlo-predicted photon spectra from the
Ag K-a X ray (22.163 keV) scattered from a
5 mm thick Al target.
11
-------
Experiment
++ LSQ fit using
Monte Carlo shape standards
315
330
345
360
375
390 «05
Channel number
Figure 9. Comparison of the observed backscatter region of the X-ray
spectrum from a Cd source from a 5mm thick Al sample wit
that predicted from Monte Carlo-generated shape standards.
only has to have available a few standards containing known amounts of a
mixture of elements to utilize the Si(Li) response function model for
producing pure element standard libraries*
The Si(Li) response function model for photons up to about 30 keV
exhibits five major spectral features* The lowest energy feature results from
Compton scattering within the detector. In most cases this occurs at pulse-
height energies lower than the lower level discriminator of the spectrometer
and. therefore, has no practical importance. The second feature is a flat
continuum from the lowest energy up to the center of the main photopeak. The
third feature is a truncated exponential tail to the low-energy side of each
photopeak. The fourth feature is a Gaussian-shaped silicon X-ray escape peak.
The fifth and final feature is the main photopeak which is also Gaussian-
Shaped. The last four of these features are illustrated for the two K-ex
X rays of gallium in Figure 10.
The four parameters required in the response function were obtained for
13 pure element spectra for a double guard-ring Si(Li) detector system. The
experimental and model pulse-height spectra for four representative elements
are shown in Figures 11 through 14. This data had to be manipulated some-
what since contaminants were present in almost all of the original spectra.
In many cases unwanted photopeaks were stripped from the original data. As a
result of this some of the model spectra do not correspond to experimental
results as well as others. Of the four spectra given two are representative
12
-------
..
OftLLIUM
-------
o
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10'
MflNGRNESE
10'
10'
10'
10*
I
80 131 182
CHflNNEL NUMBER
233
335
386
Figure 11. The response function model compared to the experimental
pulse-height spectrum for manganese.
NICKEL
v>
o
o
10
131 182
CHflNNEL NUMBER
233
335
386
Figure 12. The response function model compared to the experimental
pulse-height spectrum with nickel.
14
-------
GflLLIUM
o
o
131 182
CHflNNEL NUMBER
233
28*
335
386
Figure 13. The response function model compared to the experimental
pulse-height spectrum for gallium.
flRSENIC
I
80
131 182
CHflNNEL NUMBER
233
335
386
Figure 14. The response function model compared to the experimental
pulse-height spectrum for arsenic.
15
-------
SECTION 5
DETERMINATION OF ELEMENTAL AMOUNTS
For the determination of elemental amounts from characteristic X-ray
intensities a Monte Carlo simulation for various sample matrices has been
developed (6,7). Results were obtained for participates deposited on filters
excited by photons. Idealized particle shapes such as spheres and right
circular cylinders of various lengths were considered. Empirical correction
factors for typical cases of practical interest were reported. Secondary
fluorescence effects were included in the simulation. This appears to be a
useful approach for correcting for sample matrix effects when characteristics
of the particulates in the sample are fairly well known.
An example result is shown in Figure 15 for the case of sulfur enhance-
ment by calcium in the form of either CaO, CaCO_, or CaSO,.2(H20) as calcula-
ted for the idealized cases of uniform spheres or uniform layers. In the
case of the CaO and the CaCO the sulfur is assumed to be in trace quantities
that are uniformly distributed. It is obvious that a significant amount of
enhancement occurs in these practical cases of interest and that it must be
taken into account if good accuracy is to be obtained.
In addition to this application of primary interest other cases of
practical interest have also been treated by Monte Carlo simulation. These
include: monoenergetic wide-angle photon excitation of homogeneous
samples (15,16), monoenergetic and continuous narrow-beam photon excitation of
homogeneous samples (17), and monoenergetic, secondary-fluorescer photon
excitation of homogeneous samples (18). Recently a simple model based on
Monte Carlo simulated results has been developed for particulate powder
samples (19).
The fundamental parameters method consists of using the fundamental
parameters of X-ray fluorescence such as photoelectric cross-sections and
fluorescence yields in rigorous mathematical relationships for predicting the
characteristic X-ray intensities from the elemental amounts in samples. These
rigorous mathematical relationships can only be obtained in explicit analyti-
cal form for the simple case of a thick homogeneous sample with a constant
angle of entrance of the monoenergetic photon exciting source radiation and a
constant angle of exit of the characteristic X rays. However, one may obtain
primary, secondary, and tertiary fluorescence in these rigorous relationships
for this specific case. Primary, secondary, and tertiary fluorescence are
defined as the characteristic X rays produced from the element of interest,
either directly from the exciting radiation (primary), indirectly through the
characteristic X rays from a second element (secondary) or indirectly through
the characteristic X rays from a third element, which in turn was produced by
the characteristic X rays of a second element (tertiary).
16
-------
X
I
O
LU
LU
O
LU
x
UJ
z
It
I
x
a
I
w
u_
o
g
cc
1.0
1.5
5 10 15 20
SPHERE DIAMETER (urn)
t 1.4
1.3
1.2
'0
a - CoO
a - CoCO.
5 10 15
LAYER THICKNESS (j/m)
20
Figure 15. Variation of sulfur enhancement with layer thickness and sphere
diameter for various calcium compounds. For CaO and CaCO It is
assumed that sulfur as a trace element is uniformly dispersed
throughout the sphere or layer.
In a series of
Gardner and Hawthorne (15-18) have extended the
of infinite thickness ^J * '^^f^^ that symmetry exists about the
cular problem is fortunately two a circular detector. This makes
17
-------
It is possible in treating this problem to accomplish complete variance reduction
on the basis of physical principles. This means that the resulting computer
program designed to make the Monte Carlo calculations is as efficient as possible
and that this is attained without the fear of introducing bias. Typical calculation
times for the computer program on an IBM 360/165 computer are 15 seconds to
obtain relative standard deviations of 2% or better for three components in
one sample when primary, secondary and tertiary fluorescence are considered.
The FORTRAN statements for the computer program XRAY designed to accomplish
this calculation are available from the authors.
The Monte Carlo simulation was verified by making calculations for an
extreme analyzer geometry, which was chosen to approximate the conditions of
fixed angle of exciting source entry and fixed angle of characteristic X-ray
exit so that the explicit analytical solution derived by Sherman (20,21) is
also valid. A comparison of the results obtained by the two methods is given
in Table 3. Note that there is no statistical difference between the Monte
Carlo results and those from the analytical relationships. Note also that
verification of the Monte Carlo simulation with the explicit analytical solution
has the advantage over the use of experimental results in that while only the
total intensities could be checked by experiment, the explicit analytical solutions
provide verification of the primary, secondary and tertiary fluorescence contributions
as well.
Reference 17 described the adaptation of this Monte Carlo simulation to
the continuous and discrete exciting source photon spectra from conventional
X-ray tubes. This is accomplished by obtaining a probability distribution of
the X-ray tube exciting source spectrum and sampling from the cumulative dis-
tribution constructed from it to begin each Monte Carlo history. The Monte
Carlo simulation was verified in this case with the results given by Shiraiwa
and Fujino (22,23). They used the analytical relationships for primary, secondary
and tertiary fluorescence in homogeneous thick samples for the fixed entrance
and exit angles and discrete photon spectra with numerical integration to obtain
results for a continuous spectra of exciting photons as would be obtained with
a conventional X-ray tube. By using the identical fundamental parameters and
form of the X-ray tube spectrum that they used in the Monte Carlo simulation
the results are directly comparable. These results are given in Table 4. Note
that the average relative error for all results was only 0.95%, while the average
Monte Carlo predicted relative standard deviation was 1.47%. This indicates
that the Monte Carlo simulation is well within the experimental accuracy attain-
able with XRF analyses. The computer times required for these calculations were
about 45 seconds per case.
Reference 18 describes the extension of the Monte Carlo method to secondary
fluorescer X-ray machines. This was accomplished by assuming that a distribu-
tion of X-ray intensities from all positions on the secondary fluorescer could
be obtained. A simple experimental method for accomplishing this was developed
and demonstrated. It consisted of placing a small piece of the fluorescer of
interest at known locations on the target and measuring the resulting intensity
with a detector located at a fixed distance above the target. The general
secondary fluorescer geometry assumed is shown in Figure*16.,
18
-------
TABLE 3. COMPARISON OF INTENSITIES PREDICTED BY THE MONTE CARLO SIMULATION AND THE EXPLICIT ANALYTICAL
RELATIONSHIPS
Component
Primary Ni
Primary Fe
Primary Cr
Secondary Ni-Fe
Secondary Ni-Cr
Secondary Fe-Cr
Tertiary Ni-Fe-Cr
Intensity
0.086829
0.33410
0.19167
0.023921
0.014772
0.062348
0.0052
Sam pi e 1
Error in
Monte Carlo
prediction
(%)
+2.27
+0.03
-1.91
+0.75
-1.16
-0.08
-1.92
Monte Carlo
predicted
standard
deviation
«)
2.37
1. 87
1.65
1.91
1.68
1.79
1.95
Intensity
0.59657
0.042168
0.096741
0.022259
0. 052694
0.0039275
0.0026
Sample 2
Error in
Monte Carlo
prediction
(%)
-0.31
-0. 64
-1.80
-0.71
-1.70
+1.84
-7. 69
Monte Carlo
predicted
standard
deviation
a)
1.47
1,66
1.70
1.96
1.98
1.92
2.49
-------
TABLE 4. COMPARISON OF RELATIVE INTENSITIES3 FOR MONTE CARLO AND NUMERICAL CALCULATIONS
Composition HI X-rays Fe X-rays Cr X-rays
<%>
XI Fe Cr Primary Primary Secondary Total Primary Secondary-Nl Secondary-Fe Tertiary Total
15 70 15
(Monte Carlo) 0.0651(2.04%) 0.5277(1.34%) 0.0220(1.51%) 0.5497(1.28%) 0.1525(1.57%) 0.0042(1.40%) 0.0473(1.43%) 0.0023(1.60%) 0.2063(1.19%)
(::u.-erieaa)c 0.064 0.532 0.022 0.558 0.155 0.004 0.047 0.0021 0.2C3
15 35 50
(Monte Carlo) 0.0694 (2.04%) 0.1753 (1.74%) 0.0068 (1.98%) 0.1821 (1.67%) 0.4788 (1.18%) 0.0164 (1.39%) 0.0527 (1.28%) 0.0027 (1.48%) 0.5'3r>5 (l.C3%)
(Xa-aerical) O.C6S 0.175 0.007 0.182 0.489 0.016 0.052 0.0025 0.559
20 55 25
Olor.ta Carlo) 0.0914 (1.99%) 0.3614 (1.47%) 0.0201 (1.66%) 0.3815 (1.39%) 0.2458 (1.41%) 0.0101 (1.41%) 0.0530 (1.38%) 0.0036 (1.56%) 0.3124 (1.122)
(Numerical) 0.090 0.363 0.020 0.383 0.250 0.010 0.052 0.0033 0.315
fo
° 25 65 10
(Monte Carlo) 0.1152 (1.94%) 0.5279 (1.30%) 0.0396 (1.46%) 0.5675 (1.21%) 0.1018 (1.69%) 0.0048 (1.42%) 0.0312 (1.49%) 0.0027 (1.65%) C. 14C4 (1.25;;)
(N-Jserical) 0.114 0.533 0.040 0.573 0.103 0.005. 0.031 0.0025 0.141
25 25 50
(Monte Carlo} 0.1246 (1.94%) 0.1253 (1.77%) 0.0086 (2.00%) 0.1339 (1.65%) 0.4733 (1.19%) 0.0288 (1.41%) 0.0369 (1.31%) 0.0033 (1.50%) 0.5422 (1.04%)
(Xuserical) 0.123 0.125 0.009 0.134 0.483 0.029 0.037 0.0031 0.552
40 50 10
(Monte Carlo) 0.2064 (1.80%) 0.4061 (1.35%) 0.0539 (1.48%) 0.4600 (1.20%) 0.0097 (1.69%) 0.0085 (1.45%) 0.0035 (1.54%) 0.0035 (1.69%) 0.1350 (1.26%)
0.205 0.410 0.053 0.463 0.101 0.009 0.023 0.0033 0.135
|i
40 30 30
(Mante Carlo) 0.2155(1.80%) 0.1859(1.60%) 0.0234(1.77%) 0.2093(1.42%) 0.2853(1.37%) 0.0278(1.45%) 0.0316(1.43%) 0.0048(1.59%) 0.3495(1.11%)
(Nuaerlcal) 0.213 0.187 0.024 0.210 0.290 0.028 0.031 0.0046 0.354
70 20 10
(Monte Carlo) 0.4766 (1.53%) 0.1659 (1.57%) 0.0497 (1.59%) 0.2156 (1.25%) 0.0972 (1.73%) 0.0191 (1.58%) 0.0091 (1.78%) 0.0031 (1.82%) 0.1235 (1.31%)
(Munerlcal) 0.478 0.167 0.043- 0.210 0.098 0.019 0.009 0.0029 0.129
80 1C 10
(Xonte Carlo) 0.6116 (1.43%) 0.0842 (1.72%) 0.0324 (1.66%) 0.1166 (1.31%) 0.0969 (1.77%) 0.0246 (1.66%) 0.0046 (1.98%) 0.0019 (1.90%) 0.1280 (1.36X)
(Xunerical) 0.615 0.085 0.033 0.118 0.097 0.025 0.005 0.0019 0.128
aAverage predicted relative standard deviation for all the total Intensities » 1.47%. Average most probable relative error for
.all the total Intensities 0.95%.
Predicted relative standard deviations given in parentheses.
c:."ucerical results taken fron Ref. [6].
-------
RECTANGULAR
SECONDARY SOURCE
TARGET
(0.0,0)
Figure 16. Schematic diagram of the arrangement and nomenclative of the
rectangular secondary source target and circular sample in
relation to the circular detector and collimator for the
Monte Carlo simulation of a secondary fluorescer EDXRF system.
A typical experimental secondary source intensity distribution is shown
in Figure 17. Typical Monte Carlo results as compared to experimental data
are shown in Figures 18, 19, and 20. It is clear from these results that
the method is quite accurate.
-0 5 10 '5 03
DISTANCE ALONG THE SECONDARY FLUORESCER TARGET FROM ONE END (mm)
Figure 17. A typical experimental secondary source intensity distribution.
21
-------
1.3 T-
NJ
04
o CALCULATION MONTE CARLO
I EXPERIMENTAL
O PURE Ni MONTE CARLO
a PURE Fe MONTE CARLO
6 PURE Cr MONTE CARLO
EXPERIMENTAL Ni
04
,0. 20° 30° 40' 50° 60° 70°
ANGLE OF SAMPLE WITH HORIZONTAL PLANE, e, (degrees)
10° 20° 30° 40° 50° 60° 70°
ANGLE OF SAMPLE WITH HORIZONTAL PLANE ,es,(degrees)
Figure 18. Monte Carlo simulation and experimental
results for the relative intensities
from a pure nickel sample at various
angles for the Case l(a) and Case 2(b)
system parameters given in Table I.
Figure 19. Monte Carlo simulation and experi-
mental results for the relative in-
tensities from pure nickel, pure
iron, and pure chromium samples at
various angles for the Case 3 system
parameters given in Table I.
-------
t.s-r
1.2 -
O Ni 42 % MONTE CARLO
D Ft 58 % MONTE CARLO
EXPERIMENTAL NI
EXPERIMENTAL Fe
Kf 30° 30° 40* 50° 60°
WWLE Of SAMPLE WITH HORIZONTAL PL ANE , 6S ,
70°
Figure 20. Monte Carlo simulation and experimental results for the
relative intensities from a 42 percent nickel - 58 per-
cent iron binary sample at various angles for the Case
3 system parameters given in Table 1.
The primary anticipated use of the Monte Carlo simulation is in the
calibration of EDXRF analyzer systems. Although the calibration could be
performed in a number of ways, the primary use of the Monte Carlo simulation
would almost always be in providing "bench mark" responses for samples of
known composition and other pertinent properties. If one adopts the usual
experimental technique of obtaining responses as the response from the sample
of interest divided by the response from a pure sample of the appropriate
element, then Monte Carlo simulations could be used to provide accurate
"synthetic" standards of this type. -These could be used instead of actual
standards with experimental responses for preparing empirical calibration
relationships. The advantages of this procedure include: (1) being able to
obtain better accuracy since alternate elemental analyses are not required,
(2) eliminating the expense and time-consuming process of preparing actual
standards and obtaining experimental responses, and (3) being able to select
the range of standard compositions and other properties pertinent to the
analyses to be performed without physical limitations.
Another second order use of the Monte Carlo simulation is in the optimum
geometrical design of secondary fluorescer EDXRF analyzer systems. In order to
use the Monte Carlo simulation in this way it is desirable to modify it to
include the scatter of the exciting radiation (Compton and Rayleigh) so that
23
-------
one will be able to evaluate what is usually taken as either the signal-to-
noise or signal-to-background ratios. This modification should be relatively
easy to make in principle, but in practice some of the pertinent interaction
cross sections may be inaccurate or unavailable.
This brings up the additional point of interest to the authors which is
to provide predictions of complete spectral responses from EDXRF analyzer
systems so that more of the available information can be used for analysis
purposes. While it is true that much more elemental analysis information is
contained in the characteristic X-ray responses than in the scattered response,
nevertheless additional useful information is available from this latter
source. The approach of utilizing all of the available spectral information
is presently being investigated.
While the present Monte Carlo simulations are very efficient since essen-
tially complete variance reduction techniques have been employed, the calcula-
tions are still quite time consuming and expensive. Therefore, the Monte Carlo
simulation will probably not often be used in routine calculation such as in
the small computers association with EDXRF analyzer systems or even in off-line
computations on large computers for individual sample data processing. There
is a need for the development of simple approximate models to replace the Monte
Carlo simulation for routine use. It appears that one very promising approach
is to use either the average angle approach (6,7) directly or a suitable modi-
fication of it. This approach essentially consists of obtaining the average
entrance and exit angles for a given EDXRF analyzer system by Monte Carlo
simulation. Then with these average angles one may use either the explicit
analytical equations originally derived by Sherman (20,21) for thick homogen-
eous samples or a suitable equation like that derived by Dzubay and Nelson (24)
for other types of samples to simulate sample responses. One modification of
this would be to use two or more different radiation paths at fixed entrance
and exit angles within the range of these angles that exist in the EDXRF
analyzer system. This approximation is presently under study.
24
-------
REFERENCES
1. Salmon, L. Analysis of Gamma-Ray Scintillation Spectra by the Method of
Least-Squares. Nuclear Instruments and Methods, 14:193-198, 1961.
2. Schonfeld, E., A. H. Kibbey, and W. Davis, Jr. Determination of Nuclide
Concentrations in Solutions Containing Low Levels of Radioactivity by
Least-Squares Resolution of the Gamma-Ray Spectra* Nuclear Instruments
and Methods, 45:1-21, 1966.
3. Trombka, J. I., and R. L. Schmadebeck. A Numerical Least-Square Method
for Resolving Complex Pulse-Height Spectra. NASA SP-3044, National
Aeronautics and Space Administration, Gbddard Space Center, Washington,
D. C., 1968. 170 pp.
4. Arinc, F*, R. P. Gardner, L. Wielopolski, and A. R. Stiles. Application
of the Least-Squares Method to the Analysis of XRF Spectral Intensities
from Atmospheric Participates Collected on Filters. Advances in X-Ray
Analysis, 19:367-380, 1976.
5. Arinc, F., L. Wielopolski, and R. P. Gardner. The Linear Least-Squares
Analysis of X-Ray Fluorescence Spectra of Aerosol Samples Using Pure
Element Library Standards and Photon Excitation. In: X-Ray Fluorescence
Analysis of Environmental Samples, T. G. Dzubay, ed. Ann Arbor Science
Publishers, Inc., Ann Arbor, Michigan, 1977. pp. 227-240.
6. Hawthofne, A. R., R. P* Gardner, and T. G. Dzubay. Monte Carlo Simulation
of Self-Absorption Effects in Elemental XRF Analysis of Atmospheric
Participates Collected on Filters. Advances in X-Ray Analysis, 19:323-337,
1976.
7. Hawthorne, A. R., and R. P. Gardner. Monte Carlo Applications to the
X-Ray Fluorescence Analysis of Aerosol Samples. In: X-Ray Fluorescence
Analysis of Environmental Samples, T. G. Dzubay, ed. Ann Arbor Science
Publishers, Inc., Ann Arbor, Michigan, 1977. pp. 209-220.
N
8. Arinc, S. Faruk. Mathematical Methods for Energy Dispersive X-Ray
Fluorescence Analysis. Ph.D. Thesis, North Carolina State University,
Raleigh, North Carolina, 1976. 164 pp.
9. Arinc F., and R. P. Gardner. Models for Correcting Backscatter Non-
linearities in XRF Pulse-Height Spectra. Transactions of the American
Nuclear Society, Supplement No. 3, 21:37-38, 1975.
10. Wielopolski, Lucian, and Robin P. Gardner. Prediction of the Pulse-
Height Spectral Distortion Caused by the Peak Pile-Up Effect. Nuclear
25
-------
Instruments and Methods, 133:303-309, 1976.
11. Gardner, Robin P., and Lucian Wielopolski. A Generalized Method for
Correcting Pulse-Height Spectra for the Peak Pile-Up Effect Due to
Double Sum Pulses. Part I. Predicting Spectral Distortion for Arbitrary
Pulse Shapes. Nuclear Instruments and Methods, 140:289-296, 1977.
12. Wielopolski, Lucian, and Robin P. Gardner. A Generalized Method for
Correcting Pulse-Height Spectra for the Peak Pile-Up Effect Due to
Double Sum Pulses. Part II. The Inverse Calculation for Obtaining
True from Observed Spectra. Nuclear Instruments and Methods, 140:297-
303, 1977.
13. Wielopolski, L., and R. P. Gardner* Development of the Detector Response
Function Approach for the Library Least-Squares Analysis of Energy-
Dispersive X-Ray Fluorescence Spectra. Advances in X-Ray Analysis,
22:317-323, 1979.
14. Wielopolski, L., and R. P. Gardner, Development of the Detector Response
Function Approach in the Least-Squares Analysis of X-Ray Fluorescence
Spectra. Nuclear Instruments and Methods, 165:297-306, 1979.
15. Gardner, Robin P., and Alan R. Hawthorne. Monte Carlo Simulation of the
X-Ray Fluorescence Excited by Discrete Energy Photons in Homogeneous
Samples Including Tertiary Inter-Element Effects. X-Ray Spectrometry,
4:138-148, 1975.
16. Hawthorne, Alan R., and Robin P. Gardner. Fundamental Parameters
Solution to the X-Ray Fluorescence Analysis of Nickel-Iron-Chromium
Alloys Including Tertiary Corrections. Analytical Chemistry, 48:2130-
2135, 1976.
17. Hawthorne, Alan R. , and Robin P. Gardner. Monte Carlo Simulation of
X-Ray Fluorescence from Homogeneous Multielement Samples Excited by
Continuous and Discrete Energy Photons from X-Ray Tubes. Analytical
Chemistry, 47:2220-2225, 1975.
18. Gardner, R. P., L. Wielopolski, and J. M. Doster. Adaptation of the
Fundamental Parameters Monte Carlo Simulation to EDXRF Analysis with
Secondary Fluorescer X-Ray Machines. Advances in X-Ray Analysis,
21:129-142,, 1978.
19. Hawthorne, Alan R., and Robin P. Gardner. A Proposed Model for
Particle-Size Effects in the X-Ray Fluorescence Analysis of Heterogeneous
Powders that Includes Incidence Angle and Non-Random Packing Effects.
X-Ray Spectrometry, 7(4):198-205, 1978.
20. Sherman, J. The Theoretical Derivation of Fluorescent X-ray Intensities
from Mixtures. Spectrochimica Acta, 7:283-306, 1955.
21. Sherman, J. Simplication of a Formula in the Correlation of Fluorescent
X-ray Intensities from Mixtures. Spectrochimica Acta, 15:466-470, 1959.
26
-------
22. Shiraiwa, T., and N. Fujino. Theoretical Calculation of Fluorescent
X-ray Intensities in Fluorescent X-ray Spectrochemical Analysis.
Japanese Journal of Applied Physics, 5(10), 886-899, 1966.
23. Shiraiwa, T., and N. Fujino. Theoretical Calculation of Fluorescent
X-ray Intensities of Nickel-Iron-Chromium Ternary Alloys. Bulletin of
the Chemical Society of Japan, 40:2289-2296, 1967.
24. Dzubay, T. G., and R. 0. Nelson. Self Absorption Corrections for X-Say
Fluorescence Analysis of Aerosols. Advances in X-Ray Analysis, 18:619-631,
1974.
27
-------
APPENDIX
List of Reports, Theses, and Publications Either
Entirely or Partially Supported by this Grant
Reports
Gardner, R. P., F. Arinc, E. Efird, L. Wielopolski, A. R. Hawthorne, and
K. Verghese. Mathematical Techniques for X-Ray Analyzers, Technical Progress
Report, U.S.E.P.A. Grant No. R-802759, for period May 15, 1974, to May 14,
1975.
Gardner, R. P., F. Arinc, A. R. Hawthorne, L. Wielopolski, G. R. Beam, and
K. Verghese. Mathematical Techniques for X-Ray Analyzers, Technical Progress
Report, U.S.E.P.A. Grant No. R-802759, for period May 15, 1975, to May 14,
1976.
Theses
Arinc, S. Faruk. Mathematical Methods for Energy Dispersive X-Ray Fluores-
cence Analysis. Ph.D. Thesis, North Carolina State University, Raleigh,
North Carolina, 1976. 164 pp.
Beam, George R. Gamma-Ray Transport and X-Ray Fluorescence Calculations by
Invariant Imbedding. M.S. Thesis, North Carolina State University, Raleigh,
North Carolina, 1977. 136 pp.
Hawthorne, Alan R. Mathematical Models for Interelement and Matrix Effects
in X-Ray Fluorescence Analysis. Ph.D. Thesis, North Carolina State
University, Raleigh, North Carolina, 1977. 185 pp.
Wielopolski, Lucian. Utilization of Si(Li) Detector Response Function and
Pile-Up Model in the Analysis of X-Ray Spectra. Ph.D. Thesis, North Carolina
State University, Raleigh, North Carolina, 1979. 85 pp.
Journal Articles
Arinc, F., and R. P. Gardner. Models for Correcting Backscatter Nonlinear-
ities in XRF Pulse-Height Spectra. Transactions of the American Nuclear
Society, Supplement No. 3, 21:37-38, 1975.
Arinc, F., R. P. Gardner, L. Wielopolski, and A. R. Stiles. Application of
the Least-Squares Method to the Analysis of XRF Spectral Intensities from
Atmospheric Particulates Collected on Filters. Advances in X-Ray Analysis,
19:367-380, 1976.
28
-------
Arinc, F. L. Wielopolski, and R. P. Gardner. The Linear Least-Squares
Analysis of X-Ray Fluorescence Spectra of Aerosol Samples Using Pure Element
Library Stanrdards and Photon Excitation. In: X-Ray Fluorescence Analysis of
Environmental Samples T. G. Dzubay, ed. Ann Arbor Science Publishers, Inc.,
Ann Arbor, Michigan, 1977. pp. 227-240.
Gardner. Robin P., and Alan R. Hawthorne. Monte Carlo Simulation of the
X-Ray Fluorescence Excited by Discrete Energy Photons in Homogeneous Samples
Including Tertiary Inter-Element Effects. X-Ray Spectrometry, 4:138-148,
1975.
Gardner, R. P., L. Wielopolski, and K. Verghese. Application of,Selected
Mathematical Techniques to Energy-Dispersive X-Ray Fluorescence Analysis.
Atomic Energy Review, 15(4):701-754, 1977.
Gardner, R. P., L. Wielopolski, and J. M. Doster. Adaptation of the Funda-
mental Parameters Monte Carlo Simulation to EDXRF Analysis with Secondary
Fluorescer X-Ray Machines. Advances in X-Ray Analysis, 21:129-142, 1978.
Gardner, R. P., L. Wielopolski, and K. Verghese. Mathematical Techniques
for Quantitative Elemental Analysis by Energy Dispersive X-Ray Fluorescence.
Journal of Radioanalytical Chemistry, 43:611-643, 1978.
Gardner, R. P., and J. M. Doster. The Reduction of Matrix Effects in X-Ray
Fluorescence Analysis by the Monte Carlo, Fundamental Parameters Method.
Advances in X-Ray Analysis, 22:343-356, 1979.
Hawthorne, Alan R., and Robin P. Gardner. Monte Carlo Models for the Inverse
Calculation of Multielement Amounts in XRF Analysis. Transactions of the
American Nuclear Society, Supplement No. 3, 21:38-39, 1975.
Hawthorne, Alan R., and Robin P. Gardner. Monte Carloa Simulation of X-Ray
Fluorescence from Homogeneous Multielement Samples Excited by Continuous
and Discrete Energy Photons from X-Ray Tubes. Analytical Chemistry,
47:2220-2225, 1975.
Hawthorne, A. R., R. P. Gardner, and T. G. Dzubay. Monte Carlo Simulation of
Self-Absorption Effects in Elemental XRF Analysis of Atmospheric Particulates
Collected on Filters. Advances in X-Ray Analysis, 19:323-337, 1976.
Hawthorne, Alan R., and Robin P. Gardner. Fundamental Parameters Solution
to the X-Ray Fluorescence Analysis of Nickel-Iron-Chromium Alloys Including
Tertiary Corrections. Analytical Chemistry, 48:2130-2135, 1976.
Hawthorne, A. R., and R. P. Gardner. Monte Carlo Applications to the X-Ray
Fluorescence Analysis of Aerosol Samples. In: X-Ray Fluorescence Analysis
of Environmental Samples, T. G. Dzubay, ed. Ann Arbor Science Publishers,
Inc., Ann Arbor, Michigan, 1977. pp. 209-220.
Hawthorne, Alan R., and Robin P. Gardner. A Proposed Model for Particle-
Size Effectsin the X-Ray Fluorescence Analysis of Heterogeneous Powders that
Includef Incidence A^le'and Non-Random Racking Effects. X-Ray Spectrometry,
29
-------
7(4):198-205, 1978.
Wielopolski, Lucian, and Robin P. Gardner. A Simple Accurate Model for
Correcting XRF Pulse-Height Spectra for Pulse Pileup. Transactions of the
American Nuclear Society, Supplement No. 3, 21: 39-41, 1975.
Wielopolski, Lucian, and Robin P. Gardner. Prediction of the Pulse-Height
Spectral Distortion Caused by the Peak Pile-Up Effect. Nuclear Instruments
and Methods, 133:303-309, 1976.
Wielopolski, Lucian, and Robin P. Gardner. A Generalized Method for Correct-
ing Pulse-Height Spectra for the Peak Pile-Up Effect Due to Double Sum Pulses.
Part II. The Inverse Calculation for Obtaining True from Observed Spectra.
Nuclear Instruments and Methods, 140:297-303, 1977.
Wielopolski, L., and R. P. Gardner* Development of the Detector Response
Function Approach for the Library Least-Squares Analysis of Energy-Dispersive
X-Ray Fluorescence Spectra. Advances in X-Ray Analysis, 22:317-323, 1979.
Wielopolski, L., and R. P. Gardner. Development of the Detector Response
Function Approach in the Least-Squares Analysis of X-Ray Fluorescence Spectra.
Nuclear Instruments and Methods, 165:297-306, 1979.
30
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TECHNICAL REPORT DATA
(Please read Instructions on the reverse before completing)
REPORT NO.
EPA-600/2-80-070
RECIPIENT'S ACCESSION-NO.
TITLE AND SUBTITLE
MATHEMATICAL TECHNIQUES FOR X-RAY ANALYZERS
REPORT DATE
April 1980
PERFORMING ORGANIZATION CODE
AUTHOR(S) ~~~ ~ ' ~
Robin P. Gardner and Kuruvilla Verghese
PERFORMING ORGANIZATION REPORT NO.
. PERFORMING ORGANIZATION NAME AND ADDRESS
Center for Engineering Applications of Radioisotopes
North Carolina State University
Raleigh, N.C. 27650
0. PROGRAM ELEMENT NO.
1AD712B BB-041 FY-79
1. CONTRACT/GRANT NO.
R-802759
12. SPONSORING AGENCY NAME AND ADDRESS
Environmental Sciences Research Laboratory RTF, NC
Office of Research and Development
U.S. Environmental Protection Agency
Research Triangle Park, N.C. 27711
13. TYPE OF REPORT AND PERIOD COVERED
Final. S-7A S-7Q
14. SPONSORING AGENCY CODE
EPA/600/09
15. SUPPLEMENTARY NOTES
Mathematical techniques and subsequent computer software were developed to
process energy-dispersive x-ray fluorescence spectra for elemental analysis of
airborne particulate matter collected on filters.
The research concerned two areas: (1) determination of characteristic x-ray
intensities and (2) determination of elemental amounts from the known characteristic
x-ray intensities. In the first area, efforts primarily concentrated on developing
and implementating of the library, linear least-squares method and included the two
common non-linear aspects of XRF pulse-height spectra: excitation source background
and pulse pile up. A detector response function model was also developed for Si(Li)
detectors to alleviate the necessity for obtaining and storing extensive complete
library spectra for every element of interest. This approach gives improved accurac
greatly reduces the experimental effort required, and is capable of accounting for
variations in detector calibration and resolution without requiring extensive
additional experimental effort.
In the second research area the fundamental parameters method was developed by
by Monte Carol simulation. Data were collected for several shapes of particles
deposited on filters. Empirical correction factors for various practical cases of
interest based on these simulations are reported.
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KEY WORDS AND DOCUMENT ANALYSIS
DESCRIPTORS
b.lDENTIFIERS/OPEN ENDED TERMS.
COS/
Air pollution
*Particles
*x-ray fluorescence
*Chemical elements
*Applications of mathematics
*Compter systems programs
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