DETERMINATION OF RADIONUCLIDE CONCENTRATIONS
OF U AND Th IN UNPROCESSED SOIL SAMPLES
By
EDWARD NICHOLAS LAZO
A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN
PARTIAL FULFILLMENT OF THE REQUIREMENTS
FOR THE DEGREE OF DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
1988
"~~z'lF `~
So as it turned out, just as I suspected from the start, this dissertation was a lot of
work. It took a lot of time, pulling me around its ins and outs for over five years. It took
lots of long days and lots of weeks without weekends. It took me from the valley of the
shadow of death, so to speak, to the heights of exhilaration and joy. It was among the most
important things in my life, and certainly was the primary thing for which I strove for all
that time. And throughout that time I had a fairly vague but very warm feeling as to why
I was doing this, and that feeling kept me going during this work.
So putting that vague feeling into words, this dissertation is dedicated to my parents, Dr.
Robert Martin Lazo and Rosemarie Lazo, who taught me by their example that learning is a
large part of what life is all about. And that the other biggest part of life is the satisfaction
that comes with trusting yourself enough to follow through on your dreams.
During this time I met the woman who is now my wife, I acquired two nephews, one
sisterinlaw, and one brotherinlaw, rounding out a very eventful time for the Lazo clan.
This dissertation is also dedicated to my family; my wife, my two brothers, my sister, their
families, and my two cats Max and Milli, who helped me all along the way. It is especially
dedicated to my wife, Corinne Ann Coughanowr, who has supported me, encouraged me,
helped me, put up with me, and continued to love me through the worst of times. To you
all, I love you.
ACKNOWLEDGEMENTS
This publication is based on work performed in the Laboratory Graduate Participation
Program under contract #DEAC05760R00033 between the U.S. Department of Energy
and Oak Ridge Associated Universities.
In that the production of this dissertation has been a very difficult process which I
could not have finished without the help of numerous others, I would like to acknowledge
those who have given me so much invaluable assistance.
I would like to thank Corinne Ann Coughanowr, my wife, who provided me with excel
lent advice, guidance, and support throughout the project.
I would like to thank Dr. Genevieve S. Roessler, committee chair, University of Florida,
who provided technical and procedural guidance.
I would like to thank Dr. David Hintenlang, committee member, University of Florida,
who provided guidance which helped assure a quality final product.
I would like to thank Dr. Edward E. Carroll, committee member, University of Florida,
who taught me enough instrumental expertise to properly perform experiments.
I would like to thank Dr. Emmett Bolch, committee member, University of Florida,
who provided support in soil sample analysis.
1 would like to thank Dr. A. G. Hornsby, committee member, University of Florida,
who provided support for work in soil moisture content determination.
I would like to thank Dr. II. Van Rinsvelt, committee member, University of Florida,
who provided support in EDXRF analysis.
I would like to thank Dr. Barry Berven, committee member, Oak Ridge National
Laboratory (ORNL), who provided me with technical guidance and the managerial backing
necessary to ensure the purchase of the equipment necessary for this work.
I would like to thank Dr. Guven Yalcintas, committee member, ORNL, who provided
me with technical assistance throughout the work at ORNL.
I would like to thank Dr. Joel Davis, University of Tennessee at Chattanooga, who
provided invaluable assistance, guidance and technical expertise in every aspect of the work
performed at ORNL. Without Dr. Davis' help and friendship this dissertation would have
required much more time and would not have been half as fun as it was.
I would like to thank Dr. Keith Eckerman, ORNL, who provided expert assistance in
development of the mathematical model, which is the core of this dissertation, and in just
about any other areas where I needed help. Again, without the assistance and friendship
of Dr. Eckerman this work would have been very much more tedious.
I would like to thank Dr. Jeff Ryman, ORNL, who provided assistance in the develop
ment of the transport mathematics used in the mathematical model.
I would like to thank Dr. George Keogh, ORNL, who provided assistance in developing
the analytical mathematics used in the computer model.
I would like to thank Debbie Roberts, ORNL, who performed several invaluable exper
iments for me after I had left ORNL, and who performed the soil assay against which I am
gaging my technique.
I would like to thank Dr. Rowena Chester, ORNL, who provided managerial backing
for the project and its purchases.
I would like to thank Dr. Mark Mercier, Nuclear Data Incorporated, who helped
introduce me to peak shaping and provided invaluable assistance in the development of the
peak shaping programs used in this work.
I would like to thank John Hubble, National Bureau of Standards, who as the "God of
all Cross Sections" provided me with the latest cross sectional data and plenty of friendly
encouragement and expert advice.
I would like to thank Dr. Raymond Gunnink, Lawrence Livermore National Laboratory,
who provided me with the peak shaping program GRPANL and helped me to understand
the theory behind the program.
I would like to thank Dr. Wayne Ruhter, Lawrence Livermore National Laboratory,
who nursed me through the intricacies of GRPANL and analyzed several of my peaks to
verify my program.
I would like to thank Isabell Harrity of Brookhaven National Laboratory who provided
invaluable assistance, all the way up to the last minute, in getting this document prepared
using TEX.
I would like to thank Dr. Eric Myers who also provided last minute advice as to how
to get TgXto do its thing.
Finally, I would like to thank Oak Ridge Associated Universities who provided me with
a Laboratory Graduate Research Fellowship so that I could work at Oak Ridge National
Laboratory for two years and complete this project.
TABLE OF CONTENTS
ACKNOWLEDGMENTS ..........
LIST OF TABLES .............
LIST OF FIGURES ............
ABSTRACT .. .. . .. .... ..
CHAPTERS
I INTRODUCTION ............
Soil Sample Assay for Radionuclide Content
Standards Method for Gamma Spectroscopic
Radionuclides of Interest . . . . .
Process Sensitivity ............
Statement of Problem ..........
XRay Fluorescent Analysis . . . .
Assay Technique .............
Literature Search . . . . . .
II MATERIALS AND METHODS .....
Peak Shaping ..............
A Fitting Peak .. .........
A Fitting Background ........
Page
. ili
Sxi
xiv
. xv
...........
...........
...........
. . . . . .
...........
Assay of Soil Samples
...........
. . . . . .
...........
...........
...........
...........
...........
...........
Soil Moisture Content and Attenuation Coefficients . . .
Soil Attenuation Coefficient . . . . . . . .
Soil Moisture Content . . . . . . . . .
System M odel .. .. .. .. .. .. .. .. .. . ..
Introduction . . . . . . . . . . . .
Technique Description ..................
Mathematical Model ...................
Compton Scatter Gamma Production of Fluorescent X Rays
Compton scatter gamnna model . . . . . .
Mathematical model ..................
Electron Density . .. .. .. .. .. .. .. .. ..
Natural Production of Fluorescent X Rays . . . .
S32
S33
. 37
. . . 37
. . . 37
. . . 38
. . . 41
. . . . 51
. . . . 51
. . . 52
. . . 60
. . . . 61
Isotopic Identification . . . . . . . . . . . . .
Error Analysis . . . . . . . . . . . . . . .
Introduction . . . . . . . . . . . . . . .
Least Squares Peak Fitting .......................
Covariance Matrix and Functional Error . . . . . . . .
Error Propagation .. .. .. . .. .. .. . .. . .. . .
Linear Function Fitting .. .... ........ ...... ...
Experimental Procedure . . . . . . . . . . . . .
Sample Counting . . . . . . . . . . . . . .
Data Analysis . . . . . . . . . . . . . . .
II RESULTS AND CONCLUSIONS . . . . . . . . . .
Experimental Results .......
Assay Results ..........
Peak Fitting Results .......
Conclusions . . . . . .
. . . . . . . . . . . 88
. . . . . . . . . . . 88
. . . . . . . . . . 112
Recommended Future Work .......................
. 63
S 65
S .65
S .66
S 69
S 71
S 72
S74
S74
. 86
S88
APPENDICES
A EQUIPMENT AND SETUP ........................ 124
System Hardware ............. ... .......... 124
The ND9900 MCA ......... .. .... ........ 124
The ADC ................... .............. 125
The IIPGe Detector ........................... 125
XRF Excitation Source and Transmission Sources . . . . . . 125
The XRF Excitation Source Holder and Detector Shield . . . .. 126
System Calibration ................ ......... 127
Mass Attenuation Coefficients . . . . . . . .. .. . 134
Pulse Pileup . . . . . . . . . . . . . . . . 142
Compton to Total Scatter Ratio in Soil . . . . . . . . . 142
B UNSUCCESSFUL ANALYSIS TECHNIQUES . . . . . . . 146
Sample Inhomogeneity Analysis . . . . . . . . .. . 146
Reasons for Inhomogeneity Analysis Failure . . . . . . . . 149
Soil Moisture Content Analysis . . . . . . . . . . . 158
Reason for Soil Moisture Content Analysis Failure . . . . . .. 161
C COMPUTER PROGRAMS ....................... 163
Peak Shaping Programs .......................... 163
POLYBK.BAS ............................ 164
BKG.BAS . . . . . . . . . . . . . . . . 172
PEAKFIT.BAS ............................ 175
Geometry Factor Programs ......................... 184
DIST.FOR .. . . . . . . . . . . . ... 185
IMAGE.FOR .................... ........ 190
COMPTON.FOR .................. ......... 196
ASSAY.FOR .. .. . .. .. .. .. .. . .. .. .. .. .. 205
. . . . . . . . . . . . . . 211
REV6.FOR . . . . . .. . . . . . . . . 212
COMDTA.FOR ............................ 215
XRFDTA.FOR ............................ 218
GEOM5A.FOR .............................. 221
GEOM 5C.FOR ........... .............. ... 223
GEOM 5E.FOR ............................ 225
GEOM5G.FOR ............................. 227
GEOM SI.FOR ........................ ..... 229
GEOM5K.FOR .............................. 231
GOEM5M.FOR ............................ 233
GEOM50.FOR ............................ 235
Sample Description Programs . . . . . . . .. . . 237
SAMPLE2.FOR ............................ 238
SAMPLE3.FOR ............................ 240
SAMPLE4.FOR ............................ 242
SAMPLEU1.FOR ........................... 244
SAMPLEU1A.FOR ........................... 246
SAMPLETH1.FOR ........................... 248
SAMPLETH1A.FOR .... ..................... 250
SAMPLENJAU.FOR .......................... 252
SAMPLENJATH.FOR . . ................... ...... 254
SAMPLENJBU.FOR .......................... 256
SAMPLENJBTI.FOR .......................... 258
SAMPLEUSA.FOR ........................... 260
SAMPLEUSB.FOR ........................... 262
SAMPLEUSC.FOR ........................... 264
SAMPLEUSD.FOR ........................... 266
S2XRF.FOR .. .. .. .. .. .. .. .. . .. . .. .. 268
S3XRF.FOR ... .. .. .. .. .. .. .. .. .. . .. .. 272
S4XRF.FOR. . . . . . . . . . . . . . . . 276
U1XRF.FOR . . . . . . . . . . . . . . . . .. 280
U1AXRF.FOR ............................ ..... 284
TH1XRF.FOR ..................... ....... 288
TIIH AXRF.FOR ............................ 292
NJAUXRF.FOR ............................ .... 296
NJATHXRF.FOR ........................... 300
NJBUXRF.FOR ................... ......... 304
NJBTHXRF.FOR ........................... 308
Date File Programs
USAXRF.FOR
USBXRF.FOR
USCXRF.FOR
USDXRF.FOR
S312
S 316
S 320
324
LIST OF REFERENCES .......................... .... 328
BIOGRAPHICAL SKETCII ......................... 330
............................
............................
............................
............................
LIST OF TABLES
Table
1. Uranium 238 Decay Chain ..............
2. Thorium 232 Decay Chain ..............
3. Summary of DOE Residual Contamination Guidelines
4. U and Th KShell Absorption and Emission . . .
5. Co57 and Eu155 Emission Energies and Yields . .
6. Co57 and Eu155 Physical Characteristics . . .
7. Typical Soil Linear Attenuation Coefficients . . .
8. Isotopic Concentrations: ppm vs. pCi/gm . . . .
9. Soil Assay Results for U and Th Contaminated Soil .
10. Assay Sensitivity to the Number of Fitting Points Used
11. Measured vs. Fitted Detector Response for U1 . .
Measured vs.
Measured vs.
Measured vs.
Measured vs.
Measured vs.
Measured vs.
Measured vs.
Measured vs.
Fitted Detector Response for Ula . .
Fitted Detector Response for NJAU .
Fitted Detector Response for NJBU .
Fitted Detector Response for USC . .
Fitted Detector Response for USD . .
Fitted Detector Response for Sample 2
Fitted Detector Response for Sample 3
Fitted Detector Response for Sample 4
Page
. . . . . . 7
.......... . 32
. . . . . . 11
. . . . . . 32
. . . . . 32
. . . . . . 36
. . . . . . 63
. . . . . . 93
....... .. 94
. . . . . . 95
. . . . . . 96
. . . . . . 97
. . . . . . 98
. . . . . . 99
. . . . . . 100
. . . . . . 101
. . . . . . 102
. . . . . . 103
20. Measured vs. Fitted Detector Response for Thl
. . . . . . 104
21. Measured vs. Fitted Detector Res
22. Measured vs. Fitted Detector Res
23. Measured vs. Fitted Detector Res
24. Measured vs. Fitted Detector Res
25. Measured vs. Fitted Detector Res
26. Sample Physical Characteristics
27. Measured Sample Linear Attenuat
28. Comparison of KaI Peak Areas as
29. Peak Fit Results for Sample U1
30. Peak Fit Results for Sample Ula
31. Peak Fit Results for Sample NJA
32. Peak Fit Results for Sample NJB
33. Peak Fit Results for Sample USC
34. Peak Fit Results for Sample USD
35. Peak Fit Results for Sample 2
36. Peak Fit Results for Sample 3
37. Peak Fit Results for Sample 4
38. Peak Fit Results for Sample Thl
39. Peak Fit Results for Sample Thla
40. Peak Fit Results for Sample NJA
41. Peak Fit Results for Sample NJB
42. Peak Fit Results for Sample USA
43. Peak Fit Results for Sample USB
A1. Shield Material XRay Emission
ponse for Thla . . . . . . 105
ponse for NJATh . . . . . .. 106
ponse for NJBTh . . . . . .. 107
ponse for USA . . . . . . . 108
ponse for USB . . . . . . .. 109
. . . . . . . . . . . 110
ion Characteristics . . . . . . 111
Determined by PEAKFIT and GRPANL 112
. . . . . . . . . . . 113
. . . . . . . . . . 113
U . . . . . . . . . . 114
U . . . . . . . . . . 114
. . . . . . . . . . 115
. . . . . . . . . . 115
. . . . . . . . . . . 116
. . . . . . . . . . . 117
. . . . . . . . . . . 118
. . . . . . . . . . 119
. . . . . . . . . . 119
Th . . . . . . . . . 120
Th . . . . . . . . . 120
. . . . . . . . . . 121
. . . . . . . . . . 121
Energies . . . . . . . . 127
A2. NBS Source, SRM 4275B7, Emission Rates
. . . . . . 129
A3. NBS Source, SRM 4275B7, Physical Characteristics . . . . . 129
A4. System Calibration Parameters . . . . . . . . . . 133
A5. Water Attenuation Coefficients, p (E)f2,, Actual and Calculated Values . 140
A6. Water Attenuation Coefficients, p (E)oo
Calculated Values vs. Target Distance from the Detector . . . . 141
A7. Representative Soil Elemental Concentrations . . . . . . . 144
A8. Compton to Total Scatter Coefficients for Soil at 150 keV and 100 keV . 144
A9. Average Compton to Total Scatter Ratio for Soil . . . . . . 145
B1. Relative Sample Separation vs. Solution Matrix Condition . . . ... 151
B2. TargetDetector Distance vs. Measured Peak Area . . . . . .. 154
LIST OF FIGURES
Figure Page
1. Typical Gamma Ray Spectral Peak and Background . . . . .... . 16
2. Lorentzian X Ray as Seen Through the Gaussian Response of a Detector . 22
3. Typical Th K,~ XRay Spectral Peak . . . . . . . .... ... .25
4. Polynomial and Step Function XRay Peak Background . . . .... ... 29
5. Source Target Detector Physical Geometry . . . . . .. ... .39
6. Source Target Detector Spatial Geometry . . . . . ... . . 49
7. Compton Scatter Spatial Geometry . . . . . . . . . . 53
8. Exploded View of Target Holder Assembly . . . . . . .... ... .76
9. Target in Place above Detector ..................... .78
10. Target in Place above Detector Showing Laser Alignment System . . .80
11. ND9900 Multichannel Analyzer, ADC, Amplifier, and Detector Power Supply . 82
12. Typical XRF K,1 Peak on MCA ................... ... 84
B1. Relative Sample Separation vs. Solution Matrix Condition . . . . 152
B2. TargetDetector Distance vs. Measured Peak Area . . . . . .. 155
Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy
DETERMINATION OF RADIONUCLIDE CONCENTRATIONS
OF U AND Th IN UNPROCESSED SOIL SAMPLES
By
Edward Nicholas Lazo
December, 1988
Chairman: Genevieve S. Roessler
Major Department: Nuclear Engineering Sciences
Work with systems used to assay soil samples for U238 and Th232 indicated that the
need existed to more directly measure the concentrations of these radionuclides. An xray
fluorescent analysis system was developed here to directly measure the concentrations of
these radionuclides in bulk (125 gm), unprocessed (not dried and not ground to uniform
particle size), soil samples. This technique improves on gamma spectroscopic analysis be
cause progeny equilibrium is not required, improves on neutron activation analysis because
bulk samples are assayed, and improves on both methods because standard soil samples are
not needed for system calibration.
The assay system developed equates a measured K1a xray peak area to a calculated
"Geometry Factor" (GF) times the unknown soil sample radionuclide concentration. From
this equation the radionuclide concentration is determined. Spectral data are generated
by irradiating the soil sample with Co57 gammas to induce fluorescent x rays which are
measured using an intrinsic Ge detector. The Co57 sources, the sample, and the detector
are oriented to optimize the production of fluorescent x rays. Transmission gamma rays
are then used to determine the sample linear attenuation coefficient at the Ka, energy of
interest.
Peak areas are determined by shaping spectral data to a Voigt Profile using an algorithm
from the peak shaping program GRPANL. The steeply sloping nature of the Compton
backscatter hump on which the IKa xray peaks rest necessitated the development of a
unique polynomial/erfc background function which is subtracted prior to peak shaping. The
GF of a sample is the calculated number of K x rays which would be counted in the full
energy spectral peak if the contamination concentration in the sample were one picoCurie
per gram. This calculated GF includes considerations of the sample linear attenuation
coefficient, fluorescence induced by unscattered source gammas, fluorescence induced by
singly scattered Compton gammas which account for approximately 15% of all production,
and natural fluorescence production.
Experimentally, thirteen test samples were analyzed using this method, gamma spectro
scopic analysis, and neutron activation analysis. Results compared very well with gamma
spectroscopic analysis. Neutron activation analysis of small portions of each sample did not
match well with the results of either of the other methods due to sample inhomogeneities.
CHAPTER I
INTRODUCTION
In returning to school to pursue a Ph.D. in health physics, I knew that I would have to
complete an original research project and I knew that I wanted my research to be practical in
nature. I wanted to pick some existing process or procedure and inject it with "SCIENCE"
to facilitate its operation and improve its accuracy and precision. I stumbled upon such a
process in need of science during a summer working experience in 1983. After obtaining
a fellowship to go to Oak Ridge National Laboratory to develop an improved process, I
discovered that such a process would have a much more general application than I had
originally thought.
Soil Sample Assay for Radionuclide Content
The summer position that sparked this dissertation involved health physics work for
the Formerly Utilized Sites Remedial Action Program, known as FUSRAP. Begun in 1974,
FUSRAP is a Department of Energy (DOE) project to clean up 26 contaminated sites
within the United States. Twentytwo of the sites were formerly used during the Manhattan
Engineer's District (MED) project in World War II. The other four sites are civilian and
were added by Congress in 1976.
Sites range from a contaminated floor drain at Lawrence Berkeley Laboratory to a
contaminated Th ore processing plant and several surrounding residential properties in
northern New Jersey, containing approximately 100,000 cubic yards of contaminated soil.
In general, all sites are contaminated with varying levels of U, Th, their progeny, or some
mixture thereof. Sites typically include at least one building and the surrounding lands.
Clean up typically includes scrubbing and/or vacuuming of contaminated building surfaces
to remove contamination, destruction of facilities too contaminated or too uneconomical
to clean up, and digging up of contaminated soils. All contaminated wastes are deposited
in a controlled and monitored temporary storage area to await their ultimate disposition.
Wastes are generally low level and thus their ultimate disposition will be in the low level
waste repository of the state or compact area from which the wastes came.
One portion of this process that is of scientific interest and could stand some improve
ment is the assay of soils to determine whether or not they are contaminated. During
the course of site decontamination, many soil samples are taken. Preliminary soil samples
are taken to determine the approximate extent and concentration of radionuclides present.
Periodic soil samples are taken during soil excavation to determine whether preliminary
estimates were correct and to locate previously unidentified radionuclide deposits. Final
soil samples are taken to confirm that all contaminated soil has been removed. With so
many samples being collected, quick and accurate assay becomes important.
All soil samples are assayed twice: once when they arrive at the lab as wet, inhomo
geneous soil, and once after they have been processed. The standards comparison method
for ganina spectroscopic analysis is used to assay the soil samples. This method will be
discussed in detail later. Soil sample processing involves drying the soil in an oven, grinding
the soil into a powder that will fit through a standard 200 mesh per square inch sieve, and
stirring the powdered soil into a relatively homogeneous mixture. Soil sample processing
adds approximately 2 days to sample analysis time. Samples are analyzed twice because,
while the first analysis is fast enough to meet stingy construction schedules, it is not ac
curate enough to meet quality control guidelines. When a "fudge factor" is applied, the
fudge factor being the average ratio of sample analysis results for processed vs. unprocessed
samples, the results of the analysis of unprocessed soil samples are accurate enough to use
and to guide further work. Even with the fudge factor, analysis of unprocessed soil samples
is not sufficiently accurate to prove, for example, that an area is free of contamination and
needs no further work. The second analysis, of the processed soil, is accurate but takes too
long to meet construction schedules.
The process in need of development, then, was an assay technique that was accurate but
could be performed on unprocessed soil samples. This would eliminate the timeconsuming
step of sample drying and grinding.
Upon selecting this topic, I received a Laboratory Graduate Participation Fellowship
from Oak Ridge Associated Universities (ORAU) to pursue the research at Oak Ridge Na
tional Laboratory (ORNL). The Radiological Survey Activities section (RASA), currently
called the Measurement Applications and Development (MAD) section, of the Health and
Safety Research Division (HASRD) sponsored this work because it has been in the business
of performing radiological assessment surveys of various contaminated government sites
around the country. At the MAD lab at ORNL soil samples are processed in the same
method as used by the FUSRAP analysis lab, and the standards comparison method of
gamma spectroscopic analysis is used to determine radionuclide concentrations. As with
the FUSRAP project, soil sample processing is a timeconsuming endeavor.
Further research indicated that the processing of soil samples prior to analysis was
standard procedure at most soil assay labs. Thus a procedure that eliminated the processing
step would be universally useful.
My research also indicated that there was a second drawback to standard gamma spec
troscopic techniques. A description of the standards comparison method of gamma spec
troscopic analysis will help provide a better understanding of this problem.
Standards Method for Gamma Spectroscopic Assay of Soil Samples
The standards method for ganmma spectroscopic assay of soil or any other sort of ra
dioactive sample is a simple process. A sample of unknown radionuclide content is placed
in a fixed geometry, relative to a detector, and a spectrum is collected for a fixed length
of time. A sample containing a known amount of radionuclide is then placed in the same
geometry as that used to count the unknown sample, and a second spectrum is collected.
By comparing these two spectra, the identity and amount of radionuclide in the unknown
sample can be determined.
Qualitatively, the presence of a radionuclide in a sample is determined by the presence
of spectral peaks at energies characteristic of that radionuclide. For example, Co57 emits
gannma rays at 136 keV and 122 keV in a known ratio. If spectral peaks of these energies
and of proper relative intensity are present in a spectrum, then Co57 is probably present
in the unknown sample.
Quantitatively, the concentration of a radionuclide in a sample is determined by com
paring the area of a spectral peak generated by an unknown sample to the area of a spectral
peak generated by a sample of known concentration. For example, a sample known to con
tain 100 pCi of Co57 is counted in a standard geometry relative to a detector. After one
hour of counting this known sample, the area beneath the 136 keV peak is 10,600 counts
and the area beneath the 122 keV peak is 85,500 counts. Next, an unknown sample is
counted, in the same geometry in which the known sample was counted, and after one
hour of counting the areas beneath the 136 keV and 122 keV peaks are 21,200 counts and
171,000 counts, respectively. The unknown sample resulted in peak areas twice those of the
known sample and thus the unknown contains 200 pCi of Co57. Further details of gamma
spectroscopy can be found in Knoll.1
The drawback to this technique is that it relies upon the known sample, referred to as
the standard, being physically similar to each unknown sample counted. This is because the
density, moisture content, consistency, and elemental makeup of a sample will determine
that sample's radiation attenuation properties. A homogeneous sample of given properties
containing 100 pCi of a radionuclide will attenuate a given fraction of the gammas emitted by
that radionuclide. A second homogeneous sample, of different properties but also containing
100 pCi of the same radionuclide, will attenuate a different fraction of the gammas emitted
by that radionuclide. Therefore two homogeneous samples containing the same amount of
a radionuclide can yield spectra with characteristic gamma energy peaks of different areas.
This makes it very important that the standards chosen match the unknowns as closely as
possible.
Unfortunately, the standards used for analysis are often significantly different in atten
uation properties from the unknown samples. Dry, ground, and homogeneous standards are
obviously different from unprocessed, wet, inhomogeneous unknown samples. And although
standards can be fairly similar to processed unknown samples, mineral content differences
do result in differences in attenuation properties. Thus the process to be developed should
take the attenuation properties of each unknown sample into account in order to properly
determine radionuclide content.
Radionuclides of Interest
Two elements that are of particular interest to both FUSRAP and MAD are U and
Th. These are common contaminants at sites around the country. Many of the MED sites
of FUSRAP became contaminated while receiving, processing, or shipping U to be used
in the fabrication of the first atomic bomb. The Grand Junction, Colorado, site that the
MAD program is surveying is contaminated with mill tailings from U mining operations.
Two of the civilian FUSRAP sites were chemical plants involved in processing Th from ore.
Although less prevalent, Th is also found in mill tailings and, thus, is of interest to the
MAD program.
The main reason to develop an assay technique specifically designed to detect U and
Th is that U238 and Th232, the most common radioisotopes of U and Th respectively,
emit only low energy and low yield gamma rays. In order to perform gamma spectroscopic
analysis of these radioisotopes, gamma rays emitted by progeny must be used, equilibrium
of the parent with the progeny must be assumed, and concentrations must be inferred from
the presence of the progeny. Table 1 lists the U238 decay chain and the radiations emitted
by each member. Table 2 shows equivalent information for the Th232 decay chain.
Unfortunately, U and Th require long periods to reach equilibrium, and equilibrium
may not have been reached in the soil samples to be analyzed. Also, each decay product
has its own rate of dissolution in ground water. Thus as contamination waits in the soil to
be sampled, varying amounts of U, Th, and their progeny dissolve and diffuse. This also
confuses the equilibrium situation. Since equilibrium can not always be correctly assumed,
the assay technique to be developed must also directly measure U and Th and should not
rely on measurements of progeny.
Process Sensitivity
Since the process to be developed is to be practical in nature, some guidelines as to
sensitivity and accuracy should be followed. Since process application is soil assay for
contaminated sites undergoing decontamination, it is sensible to use guidelines established
by DOE for releasing sites for unrestricted public use.
TABLE 12
U238 Decay Chain
S Major Radiation Energies
MeV) and Intensities
Radionuclide IIalfLife Alpha Beta Gamma
4.15 (25%)
4.20 (75%)
Th234
Pa234
(Branches)
Pa234
(.13%)
U234
(99.8%)
Th230
Ra226
Rn222
Po218
(Branches)
Pb214
(99.98%)
At218
(.02%)
Bi214
(Branches)
U238
4.59E9 a
24.1 d
1.17 min
6.75 h
2.47E5 a
8.0E4 a
1.602E3 a
3.823 d
3.05 min
26.8 min
2.0 s
19.7 min
0.103 (21%)
2.29 (98%)
1.75 (12%)
0.53 (66%)
1.13 (13%)
0.33 (0.019%)
0.65 (50%)
0.71 (40%)
0.98 (6%)
1.0 (23%)
1.51(40%)
3.26(91%)
4.72 (28%)
4.77 (72%)
4.62 (24%)
4.68 (76%)
4.60 (6%)
4.78 (95%)
5.49 (100%)
6.00 (100%)
6.65 (6%)
6.70 (94%)
5.45 (.012%)
5.51 (.008%)
0.063 (3.5%)+
0.765 (0.30%)
1.001 (0.60%)+
0.100 (50%)
0.70 (24%)
0.90 (70%)
0.053 (0.2%)
0.068 (0.6%)
0.142 (0.07%)
0.186 (4%)
0.510 (0.07%)
0.295 (19%)
0.352 (36%)
0.609(47%)
1.120 (17%)
1.764 (17%)
TABLE 1 (continued)
Major Radiation Energies
(MeV) and Intensities
Radionuclide HalfLife Alpha Beta Ganuna
Po214 164.0 us 7.69 (100%) 0.799 (0.014%)
(99.98%)
TI210 1.3 min 1.3 (25%) 0.296 (80%)
(.02%) 1.9 (56%) 0.795 (100%)
2.3 (19%) 1.31 (21%)
Pb210 21.0 a 3.72 (2E6%) 0.016(85%) 0.047 (4%)
0.061(15%)
Bi210 5.10 d 4.65 (7E5%) 1.161 (100%)
(Branches) 4.69 (5E5%)
Po210 138.4 d 5.305 (100%) 0.803(0.0011%)
(100%)
TI206 4.19 min 1.571 (100%)
(.00013%)
Pb206 Stable
NOTES + Indicates those gamma rays that are commonly used to identify U238. Equilibrium
must be assumed.
TABLE 22
Th323 Decay Chain
S Major Radiation Energies
(MeV) and Intensities
Radionuclide HalfLife Alpha Beta Gamma
3.95 (24%)
4.01 (76%)
Ra228
Ac228
Th228
Ra224
Rn220
Po216
Pb212
Bi212
(Branches)
Po212
(64%)
TI208
(36%)
6.7 a
6.13 h
1.91 a
3.64 d
55.0 s
.15 s
10.65 h
60.6 min
304.0 ns
3.10 min
Stable
0.055 (100%)
1.18 (35%)
1.75 (12%)
2.09 (12%)
0.346 (81%)
0.586 (14%)
1.55 (5%)
2.26 (55%)
1.28 (25%)
1.52 (21%)
1.80 (50%)
NOTES: + Indicates those gamma rays that are commonly used to identify Th232. Equilibrium
must be assumed.
Th323
5.34 (28%)
5.43 (71%)
5.45 (6%)
5.68 (94%)
6.29 (100%)
6.78 (100%)
6.05 (25%)
6.09 (10%)
8.78 (100%)
Pb210
1.41E10 a
0.34 (15%)+
0.908 (25%)+
0.96 (20%)+
0.084 (1.6%)
0.214 (0.3%)
0.241 (3.7%)
0.55 (0.07%)
0.239 (47%)
0.300 (3.3%)
0.040 (2%)
0.727 (7%)+
1.620 (1.8%)
0.511 (23%)
0.583 (86%)+
0.860 (12%)
2.614 (100%)+
The decontamination criteria established by DOE are based on the "Homestead Farmer"
scenario. This scenario assumes that a farmer will homestead on contaminated lands, will
grow all his/her own food on the land, will raise and graze his/her own livestock on the
land, will drink water from wells on the land, and eat fish from a stream running through
the land. Limiting radionuclide concentrations were calculated such that the homestead
farmer would not build up radionuclide body burdens greater than those suggested by the
National Council on Radiation Protection and Measurements (NCRP). Table 3 lists relevant
guidelines. Soil radionuclide content limitations are in units of pCi/gm of dry soil.
Statement of Problem
The objective of this research is to develop a fast and economical technique for lab
oratory assay of U and Th in an inhomogeneous sample consisting of moist, chunky soil
compressed into a plastic, cylindrical jar. The product of this assay should be the isotopic
concentrations of U238, U235, Th232, and Th230 in pCi per gram of dry soil averaged
over the entire sample. To accomplish this assay the technique must determine the dry
soil weight, must be sensitive to U and Th isotopic concentrations from approximately 100
pCi/gm to 2000 pCi/gm, and must account for the effects of sample inhomogeneity.
Current teclmiques for the nondestructive assay of U in soil samples include neutron
activation analysis and gamma spectroscopy. Gamma spectroscopy and its limitations have
been discussed previously. Neutron activation depends upon the availability of a large
neutron source. Since it is advantageous to develop a process that is as simple, portable,
and as inexpensive as possible, neutron activation can be ruled out. The technique chosen for
this application is xray fluorescent analysis (XRF). Details of the technique are described
in the following sections.
TABLE 3 (a, b, c) 3
Summary of DOE Residual Contamination Guidelines
Soil Guidelines (Maximum limits for Unrestricted Use)
Radionuclide Soil Concentration (pCi/g) above Background
Ra226 5 pCi/g averaged over the first 15 cm of
Ra228 soil below the surface.
Th232 15 pCi/g when averaged over any 15 cm
Th230 thick soil layer below the surface layer.
Other Soil guidelines will be calculated on a
radionuclides site specific basis using the DOE manual
developed for this use.
a: These guidelines take into account ingrowth of Ra226 from Th230, and
Ra228 from Th232 and assume secular equilibrium. If either Th230 and
Ra226, or Th232 and Ra228 are both present, not in secular equilibrium,
the guidelines apply to the higher concentration. If other mixtures of ra
dionuclides occur, the concentrations of individual radionuclides shall be
reduced so that the dose for the mixtures will not exceed the basic dose
limit.
b: These guidelines represent unrestricteduse residual concentrations above
background, averaged across any 15 cm thick layer to any depth and over
any contiguous 100m2 surface area.
c: If the average concentration in any surface or below surface area less than
or equal to 25m2 exceeds the authorized limit or guideline by a factor of
V/lO/A where A is the area of the elevated region in square meters, limits
for "HOT SPOTS" shall be applicable. These hot spot limits depend on the
extent of the elevated local concentrations and are given in the supplement.
In addition, every reasonable effort shall be made to remove any source of
radionuclide that exceeds 30 times the appropriate soil limit irrespective of
the average concentration in the soil.
XRay Fluorescent Analysis (XRF)
Atoms can be ionized, i.e., have one or more electrons removed, via several processes. In
the case of xray fluorescence, gamma or x rays incident on an atom undergo photoelectric
reactions resulting in the ionization of the atom. The ionized atom then deexcites via the
emission of x rays. An upper shell electron falls into the hole vacated by the ionized electron
and x rays, equal in energy to the difference in shell energies, are emitted. Since elements
have characteristic atomic energy levels, the emitted x rays are characteristic of the element
and can be used to identify the element. The intensity of the emitted x rays is proportional
to the concentration of the element in the xray emitting material. This technique can thus
be said to directly determine U and Th concentrations in soil samples. For a more detailed
description of xray fluorescent spectrometry see Woldseth.4
Assay Technique
In overview, the assay process is simple. Each sample is irradiated to induce fluorescence
and the emitted fluorescent x rays are detected by a hyperpure intrinsic Ge planar detector.
The areas under the Ka& xray peaks from U and Th are determined by a spectral analysis
system. The Ka peak was chosen because the Kshell lines are highest in energy, thus
minimizing attenuation effects, and the Kai line is the most predominant Kshell line. The
areas of these peaks are used to determine U and Th concentrations. This determination
involves two steps; the determination of sample moisture content and attenuation properties,
and the handling of sample inhomogeneity. The isotopic fractions are determined by looking
at the relative intensities of gamma rays from U and Th daughters. The details of these
processes are discussed in subsequent sections.
This research is divided into two broad sections; development of a mathematical model
of the assay system, and experimental verification of that model. The model is divided into
three sections. The first section involves the development of a peak shaping program to
accurately determine the areas of the Kai xray peaks of U and Th. The second section
involves the determination of the sample moisture content and attenuation properties. This
is done by measuring how gamma rays are transmitted through the sample. The third
section uses the peak areas, determined in section one, and the soil moisture content and
attenuation coefficients, determined in section two, to mathematically model the sample so
that an accurate assay can be performed. In the third section the final result of the analysis,
the radionuclide concentrations of U and Th in the soil sample, is calculated.
Literature Search
In order to learn more about existing techniques of gamma spectroscopic analysis, x
ray fluorescent analysis, peak shaping tecluiques, soil moisture determination, and assay of
inhomogeneous samples, a computer literature search was performed. The central research
library at ORNL performed the search, looking through Chemical Abstracts, Physical Ab
stracts, and the DOE Energy Data Base. Many references which discussed these topics
were located, however no references were found which discussed data analysis techniques
similar to that presented in this work were located.
CHAPTER II
METHODS AND MATERIALS
This chapter describes the theoretical basis for the U and Th assay technique. The
theory is divided into three sections: peak shaping, soil moisture content and attenuation
properties, and system modeling. Following this theory are descriptions of the error analysis
and the experimental procedure used in this work.
Peak Shaping
In order to determine the physical properties physical measurements must be made. In
the case of this soil sample assay technique, the induced fluorescent x rays emitted by the
target are the physical quality measured. The measurement takes the form of an energy
spectrum. The number of x rays emitted by the sample is proportional to the concentration
of U and/or Th in the sample. The areas of the Kai xray peaks are the number of x
rays that hit the detector and are counted in the full energy peak. These areas, then, are
proportional to the concentrations of U and/or Th in the soil sample. The details of the
proportionality are discussed in subsequent sections. This section describes the method
used to determine peak areas.
The fitting of spectral data to mathematical functions is known as peak shaping or
peak fitting. As stated above, the motivation for fitting peaks is to accurately determine
the peak area which is proportional, in this case, to the concentration of U and/or Th in a
soil sample. The "art" of peak fitting has been steadily perfected over the years, particularly
with the advent of high resolution semiconductor detectors. A good overview of current
theory and of the variety of functions available to fit peaks and backgrounds is provided by
Prussin.5 As an introduction to the theory of peak shaping he states that:
As is well known, the shape of a photopeak from monoenergetic photons
in spectra taken with semiconductor detectors is closely approximated by
a Gaussian with more or less severe tailing below the centroid. The peak
is joined smoothly to a lowerenergy continuum of small curvature until it
meets the relatively sharp Compton edge. This continuum, which is pro
duced mainly by the loss of some of the energy of photoelectrons from the
sensitive volume of the detector, leads to the appearance of a steplike dis
tribution upon which the main intensity is superimposed. Under conditions
of low input rate, short counting times with stable electronics and negligi
ble background at higher energies, the highenergy edge is indeed found to
be nearly Gaussian. The low energy edge begins to deviate from Gaussian
form at fractions of the peak maximum in the range of .5 to .01 depending
upon the detector type, its quality and its history. While Gaussian shape
results from statistical spread due to fluctuations in electronhole pairs
produced in the stopping process and random noise from the amplifying
electronics, the low energy tailing represents pulseheight degradation from
a number of phenomena including charge trapping and recombination, en
ergy loss of primary and secondary electrons in the insensitive volume of
the detector or by bremsstrahlung.
Figure 1 shows a typical spectral gamma peak and its component parts; the Gaussian
peak and step function background. Typically, peak fitting programs will fit the background
to some function, subtract the background from beneath the peak, and fit the remaining
data to some peak function. The following sections will discuss the peak and background
fitting functions used in this work.
A Fitting Peak
Peak shaping and peak area determination are commonly done by gamma spectroscopy
systems in an efficient manner. Such systems usually contain long computer programs to
locate all spectral peaks, to determine and subtract the baselines from beneath those peaks,
and to determine peak areas by summing of channel counts or by least squares fitting to a
Gaussian shape.
FIGURE 1
Typical Gamma Ray Spectral Peak and Background
1000000
100000
V)

a 10000
0
0
1000=
* Spectral Data
Gaussian Fit
ERFC Background
0**
100
1200
1250
1300
Channel Number
I
1350
I I
The xray fluorescent analysis system described in this paper uses its own peak shaping
program for the following reasons. First, since only the Ka& peaks from U and Th will
be used, only two peaks at known energies, need to be determined. It is not necessary
to search the entire spectrum to shape each peak and to calculate the area of all possible
peaks since only the Kai peaks are of interest. This eliminates much of the computational
software necessary for large spectral analysis programs, thus decreasing processing time.
Since most processing programs are quite fast, this is not the most important reason to
have a separate peak shaping program. The second and more important reason is that
most spectral analysis programs perform Gaussian peak shaping, which is inappropriate for
x rays. This results in inaccurate area determination.
The spectral response of a detector system can be mathematically described as a con
volution of the detector system's inherent response function and the energy distribution of
the "monoenergetic" incident radiation (Knoll1 pp 732739).
N(H)= R( ,E) x S(E)dE
where
N (H) = the differential pulse height spectrum,
R (H, E) = the differential probability that a pulse of
amplitude II originates from a photon of
energy within dE of E,
S (E) = the photon energy distribution.
Detector system response functions are typically Gaussian (Knoll1 pp 434440). Mo
noenergetic gamma rays emitted by the deexcitation of a nucleus in an excited state are
actually not monoenergetic but are distributed in energy about a central value. This distri
bution is described by the function S(E). The width of this energy distribution is inversely
proportional to the mean lifetime of the excited nuclear state (Evans6 pp 397403). This is
directly attributable to the Heisenberg uncertainty principle such that (Evans6 pp 397403)
r (eV) =.66E 15 (eV s) /tm (8)
where
r = energy distribution width (eV),
.66E 15 (eV s) = Plank's Constant/27r,
t, = mean lifetime of excited state.
NOTE: half life (t/z2) = t,/ln(2)
Therefore, for a gamma ray to have an energy distribution width greater than 1 eV,
its mean life would have to be less than 1E15 s. Since most gamma rays are emitted
from radionuclides with half lives much longer than that, the width of ganuna ray energy
distributions is zero for practical purposes. Since the width of the energy distribution for
gamma rays is so small, S(E) is effectively a delta function. The convolution of a delta
function energy distribution and a Gaussian distribution detector response results in a
Gaussian shape spectral peak for gamma rays (Knoll1 pp 434440).
X rays, however, are generated by electrons falling from upper to lower orbitals, as
described in a previous section. These transitions take place very rapidly, and therefore the
emitted x rays have fairly large widths which increase with increasing energy.7,8 Experimen
tal measurements have shown K, x rays to have widths of from 1 eV for Ca to 103 eV for
U.9 Xray energy distributions must therefore be described by a Lorentzian distribution10
and an xray spectral peak must therefore be described by the convolution of a Gaussian
detector response function and a Lorentzian xray energy distribution.11 Mathematically,
this convolution is written as
C(E) = G (E') xL (E E') dE',
J 0
where
G (E') = Gaussian distribution function,
= Aexp (.5 ((E' E.) /)2)
E' = convolution dummy variable,
E. = peak centroid,
a = Gaussian peak standard deviation,
A = Gaussian peak height constant, and
L (E E') = Lorentzian distribution function,
= A'/ ((E E' E)2 + .25r) ,
E = energy,
E' = convolution dummy variable,
Eo = peak centroid,
r = Lorentzian peak full peak width at half
the maximum peak height,
A = Lorentzian peak height constant.
The resulting convolution, C (E), can be solved numerically in the following manner.12
C (E) = A" (exp (X2) x (C1 + C2 x X2 + C3 x (1 2X2)))
+A" x C4 x P (X),
where
X" = (1/2)((E E.)/,)2,
C1 = 1 (i/v) (F/a),
C2 = (1/2v7) (r/lV2),
C3 = (1/8)(r/a) ,
C4 = (2/7rV)(/o),
B(X) = (exp (X)) (f (X)),
S() ((exp(n/4)) ) x (1 cosh(nX)) and
n=l
A" = new peak height constant.
This is a numerical equation in four unknowns; E., r, o, and A". This equation
lends itself to weighted least squares fitting to the spectral data. The result of this fitting
will be values for the above four unknowns and their associated errors. Figure 2 shows
the Lorentzian distribution of the incident x rays, the Gaussian response function of the
detector system, and the resulting convoluted distribution that is the spectral xray peak.
It should be remembered that the objective of this exercise is to determine number of x
rays that hit the detector. As will be explained further in the section describing data analysis
and the section describing detector system calibration, the measured number of x rays will
be compared to the calculated number of x rays to determine concentrations of U and Th
in the soil. By properly calibrating the detector system, the area under the convoluted peak
will be proportional to the number of x rays that hit the detector. Determination of the
area of the convoluted peak is thus the desired end result of this peak analysis. Therefore it
should be noted that the Lorentzian xray distribution and the Gaussian detector response
function, shown in Figure 2, are for reference only and will not actually be seen in the
spectrum or have their areas calculated.
With the peak parameters determined, the peak area can be determined. Since the
convolution function is rather complex, the peak area is determined by numerical integra
tion. The peak shaping program defines the spectral peak as having a beginning channel
and an ending channel and performs the integration between these limits. Only a small
FIGURE 2
Lorentzian X Ray as Seen Through
the Gaussian Response of a Detector
10000000 
1000000
100000
0
10000
1000
920
Gaussian
 Lorentzian
Convolution
940
940
960 980
Channel Number
1000
1000
1020
portion of the peak area lies beyond those limits and is accounted for by use of an equation
from Wilkinson.12 Wilkinson's equation determines the fractional area beyond a specified
distance from a peak centroid. It should be noted that the numerical integration is per
formed on the fitted peak function and not on the actual spectral data. Figure 3 shows a
typical xray spectrum in the vicinity of the Ka xray peak of Th.
Thus once the spectral data is fit to the proper peak shape and the four fitting parame
ters are known, the peak area can be determined. Since least squares fitting techniques also
lend themselves to convenient error analysis, the errors associated with the above fitting
parameters can be found and propagated to determine the error in the peak area.
A Fitting Background
As was previously mentioned, before a proper peak shape can be determined, the back
ground must be subtracted from the peak. In the case under consideration in this work both
the U and Th Ka~ peaks lie on top of a large, steeply sloping background (see Figure 3).
This background is the sum of the Compton continuum step function background, described
by Prussin5 and others13, 14, and gammas from the excitation source that backscatter in
the target and hit the detector.
The step function portion of the background is described in several well known peak
fitting programs as a complementary error function, erfc.13, 15, 16 While details of the
functions used vary slightly from program to program, most use an equation of the form
SB (X) = A X erfc (( Y) /r) ,
where
SB (X) = step background value at channel X,
A = amplitude,
FIGURE 3
Typical Th K1a Spectral Peak
1500
0 500 1000
Channel Number
X = peak centroid, and
a = detector response function width for
peak centered at X.
The numeric approximation to this function used in this work in HYPERMET16 and
in GRPANL15 is
SB(X,) = BL + (BH BL)x Y (X) / Y(X) ,
j=1 j=l
where
SB (Xi) = step background value at channel Xi,
BL = average background value on the low energy
side of the peak,
BH = average background value on the high energy
side of the peak,
i
Y (Xi) = the sum of the gross channel counts from the
first peak channel to channel Xi, and
N
Y (Xi) = the sum of the gross channel counts from the
first peak channel to the last peak channel.
The above algorithm assumes that the background to either side of the peak is relatively
flat. This will not necessarily be the case for all spectra and is certainly not the case for
this work.
GRPANL,15 in addition to using the above step function, allows the use of two different
background slopes, one for each side of the peak. The average slope of the background under
the peak is then the average of the background slopes from either side of the peak. The
change in background attributable to this slope is then equal to the vertical change of a
line, having the average background slope, over a horizontal change equal to the number
of channels in the peak. The actual vertical change in the background is equal to the
difference between the number of counts in the last low energy side background channel
and the number of counts in the first high energy side background channel. Then the
vertical background change due to the step function is equal to the actual vertical change
minus the vertical change due to the slope. The use of this rationale results in a step
function background whose slope at either side of the peak fits smoothly with the actual
background slopes.
Unfortunately, the background slope of the spectrum under consideration in this work
does not change uniformly from the low energy side of the peak to the high energy side.
Rather, the slope on the low energy side is very steep, changes very quickly, and then
approaches the slope on the high energy side. The average background slope, as described
above, will therefore be too steep and the vertical change attributable to the slope will exceed
the actual vertical change. Under these circumstances, GRPANL will fit the background to
a smoothly changing slope without a step change.
While this might seem contrary to theory, Baba et al.14 state, for large peaks in a
multiple group or even for smaller single peaks, that the peak areas and centroids are
determined with sufficient accuracy by using a properly fit straight line, curved, or step
function background. As mentioned earlier, peak fitting is as much an art as a defined
science.
Bearing all of the above in mind, the approach used in this work was to use both the
step function and the sloping background. The background is easily fit to a third or fourth
order polynomial. In this work then, the vertical change in background beneath the peak is
attributed half to the polynomial and half to the step function. The resulting background is
shown in Figure 4. Source listings of POLYBK.FIT and BKG, the codes used to accomplish
the background determination and subtraction, are supplied in Appendix C.
FIGURE 4
Polynomial and Step Function XRay Peak Background
1000000
S_ \ D~c .grouLl.iU.
0 100000
0
10000 
920 940 960 980 1000 1020
Channel Number
Ill that the steeply sloping nature of this background is somewhat unusual in spec
troscopy, some explanation as to its origin is warranted. The excitation source, Co57,
which emits gammas at 122 keV and 136 keV, was chosen because of the proximity of its
gamma energies to the Kshell absorption energy. At these energies, U and Th have high
cross sections for photoelectric reactions with Kshell electrons. Table 4 shows the absorp
tion and emission energies for U and Th. Table 5 shows the emission energies and yields
for Co57. Table 5 also lists emission energies and gamma yields for Eu155. The latter
radionuclide is used for transmission measurements which are explained in a subsequent
section. Table 6 shows relevant source physical properties.
Unfortunately, when 122 keV ganunas Compton scatter at approximately 180 degrees,
the resulting gamma is 83 keV. This is called a backscatter gamma. The backscatter gamma
from an incident 136 keV gamma is 89 keV. These backscatter gammas are at inconvenient
energies because they form the majority of the background beneath the U and Th Ka, x
ray, thus somewhat obscuring the peaks. And the shape of the background depends upon
the geometry of the scattering soil sample. That is, the size and relative position of the
source, soil sample, and detector determine what scatter angles, and thus what energies,
will be seen as backscatter gammas. Thus the shape of the background does not lend itself
to simple theoretical treatment and a third or fourth order polynomial fit, as mentioned
earlier, is necessary. The peak areas calculated by this technique are used in subsequent
analyses to determine the soil sample concentrations of U and Th.
TABLE 4
U and Th KShell Absorption and Emissionl7
KShell K.i Ka2
Element Absorption Emission Emission
U 115.591 keV 98.434 keV 94.654 keV
Th 109.63 keV 93.350 keV 89.957 keV
TABLE 5
Co57 and Eu155 Emission Energies and Yields18
Backscatter
Element Emission Energy Gamma Yield Energy
Co 57 122.063 keV .8559 82.6 keV
136.476 keV .1061 89.0 keV
Eu 155 105.308 keV .207 74.6 keV
86.545 keV .309 64.6 keV
*: The gamma yields for Eu 155 are not known to the same precision as
those of Co57. Europium155 sources, therefore, are described by gamma
emission rates, Activity (Ci) x Yield (gammas/s). The listed yields are for
estimation purposes only.
TABLE 6
Co57 and Eu155 Physical
Characteristics
Soil Moisture Content and Attenuation Coefficients
In order to properly analyze spectral data, the soil attenuation coefficient as a function
of energy must be known. The details of their use are described in a subsequent section.
Since the goal of this assay is to determine the soil U and Th concentrations in units of pCi
per gm of dry soil, the water weight fraction is needed to determine the soil dry weight.
Fortunately, both of these parameters are easily measured.
Co57 Eu155
Activity 5 mCi 15% 2 mCi 15%
(1 October 1985) (1 April 1986)
Half Life 271.7 d 1741 d
Soil Attenuation Coefficient
It is well known that as monoenergetic gamma rays pass through any medium, the
fraction of uncollided gammas, as a function of thickness of the medium, is given by
Transmission Fraction = exp (p (E) po),
where
p (E) = mass attenuation coefficient at the
energy E, (cm/gm2),
Po = density of the attenuating medium,
(gm/cc), and
x = thickness of the attenuating medium (cm).
For a monoenergetic point source, with emission rate Ao, the number of gammas which
strike and are detected by a detector of area AD located at distance r from the source is
AA (E) x AD x 7 (E) x CT
A (E)42 (1)
where
Ao (E) = source gamma emission rate at energy E
(Gammas/s),
AD = detector surface area (cm2),
q (E) = detector intrinsic energy efficiency at
energy E, gammass counted in the full energy.
peak per ganuna hitting the detector),
CT = pulse pileup corrected live time (s),
r = distance from source to detector (cm).
Thus for a monoenergetic gamma passing through an attenuating medium, the number of
gammuas counted in the full energy peak can be described by the product of the above two
attenuations:
(E A (E) x AD x {!(E) x CT
A (E)= xexp( i(E)pox),
where A (E) = full energy peak area at energy E.
Next, once the above measurement is made and A (E) is determined, the attenuating
object can be removed from between the source and detector and the measurement of
A (E) repeated. This time, however, the new measurement, A' (E), is described by Eq. 1
alone since no attenuating object is between the source and the detector. The ratio of
A (E) /A' (E) is then proportional to the objects transmission fraction at energy E:
A(E) A, CT
x x exp (p (E) pz),
A' (E) A(E') CT' 
where all terms are as defined previously.
The terms that differ from one measurement to the next are A, (E) and CT. The source
emission rate, A, (E), changes from measurement to measurement because of source decay.
If the measurements are made sequentially, this change is small, but it is always finite. The
count live time also varies from measurement to measurement due to pulse pileup. Pulse
pileup corrections are discussed in detail in a subsequent section. Since A. (E) and CT vary
from measurement to measurement, they remain to be accounted for in the above ratio. All
other terms divide out.
Since all the terms in the above equation are measured except the transmission fraction
term, the transmission fraction can be calculated. If the thickness of the attenuating object
is known, then the attenuation coefficient can be determined.
TF(E)= exp (p (E)poP),
where
TF (E) = transmission fraction for gammas at
energy E, gammass transmitted through
the object uncollided per gamma incident
on the object), and
other terms are as previously defined.
Therefore
t (E) x p = (1/x) x In(TF(E)),
where
pt(E)x po = object linear attenuation coefficient, (cm).
In the case where the attenuating object is a cylindrical jar of soil, this equation results
in the soil's linear attenuation coefficient at energy E. This information is used in the
next phase of this assay process, dealing with data analysis, to eventually determine U and
Th concentrations. As will be described in the next section, the soil's linear attenuation
coefficient is necessary at four energies; 136 keV and 122 keV, which are the energies of the
Co57 gamma rays used to induce xray fluorescence in U and Th, 98 keV, the energy of
the K,, x ray from U, and 93 keV, the energy of the Ka1 x ray from Th.
It should be noted here that the "soil" in the jar is actually a mixture of dry soil and
water. For the purposes of simplicity the term "soil" will be used to refer to this soil water
mixture.
Since Co57 is used to induce xray fluorescence, the same source can be used to measure
transmission gamma rays and thus determine the soil's linear attenuation coefficients the
energies of the Co57 ganmmas. Unfortunately, no clean and calibrated source of U or Th x
rays is available. In this case, clean refers to a source that emits x rays only at the energy
of interest. Additional x rays or garm as will complicate the transmission spectra, add
background, and generally complicate the results such that true peak areas at the energies
of interest are hard to determine.
Fortunately, over a small energy range attenuation coefficients can be described as a
simple function of energy (personal conversation with John Hubble):
In (p (E)) = A + B x hi (E) + C x (In (E))2,
or
p (E) = exp (A + B x In(E) C x (In(E))2),
where A, B, and C are constants.
Therefore, the gamma rays from Eu155, at 105 keV and 86 keV, are also used and the
soil's linear attenuation coefficients at these energies are measured. The four data points,
two from Co57 gammas and two from Eu155 gammas, are then fit to the above equation,
using a least squares fit technique, and the linear attenuation coefficients at the U and
Th Ka, xray energies can be calculated from the resulting curve fit. Table 7 shows typical
soil linear attenuation coefficients.
TABLE 7
Typical Soil Linear Attenuation Coefficients
Measured Curve Fit
Energy (keV) p (E) (1/cm) p (E) (1/cm)
136.476 0.20505 0.20517
122.063 0.21505 0.21479
105.308 0.23114 0.23132
98.428 0.24056
93.334 0.24866
86.545 0.26159 0.26155
Soil Moisture Content
The above analysis of soil linear attenuation coefficients assumes that the "soil" in the
jar consists of everything in the jar, water and soil. As will be seen, this is the appropriate
linear attenuation coefficient to be determined here. The moisture fraction of this soil
is also needed for the data analysis for final U and Th concentration determination. As
such, after the transmission and XRF measurements have been made, each sample jar is
placed in a microwave oven and dried in the jar. Jar weights before and after drying are
used to determine soil moisture weight fraction. Soil sample densities, before drying, were
determined by dividing the known wet soil weight by the know bottle volume.
Initially, soil moisture content was to be determined via use of the same transmission
gamma rays described in the last section. Unfortunately, the set of four simultaneous
equations that were to be used resulted in a nonunique solution set instead of one unique
answer. Although this approach could not be used, the details of this approach and the
reasons for its failure are included in Appendix B.
System Model
Introduction
In general, XRF determination of elemental concentrations is done by comparing the
area of a peak from an unknown sample to the area of a peak from a sample of known ele
mental concentration, called a standard. The concentration of that element in the unknown
sample is simply the ratio of unknown sample peak area to standard peak area times the
elemental concentration in the standard.
This technique assumes that the measurement geometry and attenuation properties of
the unknown sample are identical to those of the known standard. Practically speaking,
this means that both the standard and unknown must be as close to physically identical as
possible and must be measured using the same detector and in the same position. To achieve
this for soil samples, standards and unknowns usually are dried soil that has been crushed
into powder form, thoroughly mixed into a homogeneous mass, and put into containers.
These containers can then be exposed to an xray excitation source and the fluorescent x
rays can be counted. Experiments of this type are easily reproducible.
Unfortunately, standards and unknowns do not always match. The attenuation prop
erties of a soil sample vary with elemental concentrations and soil makeup. Clays, for
example, have different attenuation properties than black dirt. Comparison of unknowns
to standards of different attenuation properties may lead to erroneous results. This may
be compensated for by using very thin samples such that attenuation is not a factor. With
large samples, however, attenuation variations will cause problems.
To eliminate this problem and to eliminate the need to dry and crush soil samples,
and thus significantly decrease sample processing time, the technique described in this
paper is an absolute technique. That is, this technique does not compare unknown samples
to known standards to determine elemental concentrations. This technique can be used
on unprocessed samples which may be inhomogeneous. The teclmique also provides a
quantitative measure, in the form of a X2 value of statistical significance, of whether the
sample is too inhomogeneous to be analyzed without prior processing.
Technique Description
Figure 5 shows the sourcetargetdetector geometry used for this XRF analysis tech
nique. In this configuration, the target is exposed to excitation gammas from the source and
emits fluorescent x rays which are seen at the detector. The detector is shielded from direct
exposure to the sources by the Pb and W source holder. The spectrum seen by the detector
FIGURE 5
Source Target Detector Physical Geometry
TARGET
POINT
SOURCE
POINT
SOURCE
DETECTOR
is composed of gammna rays from the source which have backscattered in the target and hit
the detector, and of fluorescent x rays from the U and/or Th in the soil. The number of
fluorescent x rays counted by the detector is proportional to the U or Th concentration in
the target. In overview, the assay technique is quite simple.
The target can be thought of as many small point sources. The fluorescent x rays
produced at each point source contribute separately to the the full energy photopeak of
Ka, x rays seen by the detector. A mathematical model of each point source is used to
calculate the xray contribution from each point source. These calculated individual point
source contributions can be summed to yield a calculated total detector response. The
calculated response is then compared to the actual measured response, in a least squares
sense, to determine the contamination concentration in the target sample.
Mathematical Model
All of the equations in this section stem from well known first principles. To begin
with, it is well known that the excitation gamma ray flux (FL) that reaches a point in the
target can be described by
ER(E)
FL(E) = 4ER x exp(/p(E)poR2), (2)
where
FL (E) = excitation gamma flux at a point in the
target, (gammas/cm2s),
ER (E) = source emission rate at energy E,
(gammas/s),
RI = distance from the source to the point, (cm),
pI (E) po = sample mass attenuation coefficient at
energy E, p (E) (gm/cm') times sample
density, Po (gm/cm3), and
R2 = that portion of the total distance that
lies within the attenuating sample, (cm).
The photoelectric reaction rate (RX) at the point, due to the above excitation gamma
flux, can be described by
RX (E)= FL(E)x PE (E) x x AD, (3)
where
RX (E) = photoelectric reaction rate at the point,
(reactions/s) / (pCi/gm of dry soil),
FL (E) = excitation gamma flux at a point in the
target, (gammas/cm2s) ,
PE (E) = photoelectric cross section for U or Th at
energy E, (cm2/atom),
V = volume of the point source, (cm3),
AD = atom density of U or Th,
(atoms/cm3of soil) / (pCi/gm of dry soil),
= .037 (dis/s) / (pCi) x x p,,
and:
.037 = the number of disintegrations per second
per pCi of activity,
A = disintegration constant for U or Th,
(s1),
Note : the units of .037 A are
(atoms/pCi), and
p, = soil bulk density,
(gm of dry soil)/ (cm3 of soil).
The fluorescent yield (FY) at the point, due to the above photoelectric reaction rate,
can be described by
FY (E',E) = RX (E) x KS (E) x KY (E'),
where
FY (E', E) =
the flux of fluorescent x rays of energyE' at
the point, that are caused by excitation gammas
of energy E,
((Ka, x rays) /s) / (pCi/gm of dry soil),
RX (E) = photoelectric reaction rate at the point,
(reactions/s) / (pCi/gm of dry soil),
KS (E) = fraction of photoelectric reactions that result
in K shell vacancies,
(K shell vacancies) / (photoelectric reaction),
= (Rk 1)/Rk
Rk = K shell Jump Ratio, and
KY (E') = fraction of K shell x rays that are K&I
x rays, (K~1 x rays) / (K shell x ray).
The flux at the detector (FD), of the Ka~ x rays that hit the detector, due to the above
xray fluorescent yield, can be described by
FDE' FY (E',E)x DA
FD (E) = DA x exp(p (E')por2),
4irr?
where
FD (E') = the flux of fluorescent x rays of energy E' that
hit the detector,
((Ka1 x rays) /s) / (pCi/gm of dry soil),
FY (E', E) = the flux of fluorescent x rays of energy E' at
the point, that are caused by excitation gammas
of energy E,
((Kai T rays) /s) / (pCi/gm of dry soil),
DA = detector area, (cm2) ,
rl = distance from the point to the detector, (cm),
p (E') x p = sample mass attenuation coefficient at energy
E', (gm/cm2), times sample density, (gm/cm3),
and,
r2 = that portion of the total distance that
lies within the attenuating sample, (cm).
It should be noted that this equation does not include any terms to account for small
angle scatter, and correspondingly small energy change, x rays which are mathematically
removed from the x ray beam but would actually still hit the detector and be counted in
the full energy peak. Looking at the geometry of the situation, the largest scatter angle
which would leave an x ray still traveling toward the detector is ten degrees. Integrating
the KleinNishina differential scattering cross section (Evans6 pp 677689) over 27r, for do,
and over ten degrees, for dO, the ratio of this to the total scattering cross section is .029.
Considering that ten degrees is the upper bound for scatters which will still hit the detector,
this 3% error can be ignored.
As evidence of this, if small angle scatter were a significant contributor to the total peak
area, peaks would be broadened on the low energy side of their centroids and would not be
well described by the peak fitting equations shown earlier. Data in Chapter III shows that
the measured peaks are fit very well by the previously described peak fitting equations. As
such it is concluded that small angle scatter of x rays is not a significant problem and need
not be accounted for here.
Finally, the fluorescent signal (FS), the number of K,1 x rays that are counted in the
full energy peak at energy E', due to the above flux at the detector, can be described by
FS (E') = FD (E') x DE (E') x CT,
where
FS (E') = the number of counts in the full energy peak at
energy E', ie. peak area,
(Kai rays) / (pCi/gm of dry soil),
FD (E') = the flux of fluorescent x rays of energy E' that
hit the detector,
((K.a z rays) /s) / (pCi/gm of dry soil),
DE (E') = the detector intrinsic energy efficiency at
energy E',
(x rays counted) / (x ray hitting the detector),
and,
CT = total counting time, (s), corrected for pulse
pileup as described in a subsequent section.
This equation can also be written in the following more useful form
DR, (E') = GF (E'),
where
DR, (E') = detector response at energy E' to point
node i,
(counts/s) / (pCi/gm of dry soil),
GF, (E') = FD (E') x DE (E') x CT,
= geometry factor at energy E' for point
node i,
(counts/s) / (pCi/gm of dry soil).
The above equations are a mathematical description of the fluorescent x ray flux,
counted by a detector, due to a single point target irradiated by a point excitation source.
Looking at equation 3 more closely, the term AD, atom density of U or Th at the point
node, is in units of (atoms/cm3 of soil)/(pCi/gm of soil). The geometry factor, GFT, is
thus normalized to a contamination concentration at the point node of 1 pCi/gm. And the
detector response to a point node contaminated to any concentration, C pCi/gm, is a linear
function of the geometry factor.
Since both DR, (E') and GFC (E') are "per pCi/gm of dry soil", the detector response
to a point node contaminated to a concentration of "C" pCi/gm, would be
DR, (E') = C x GF (E'),
where
DR, (E') = FS (E')
= detector response at energy E' to point
node i,
(counts/s) / (C pCi/gm of dry soil),
C = contamination concentration at point
node i, pCi/gm of dry soil,
GFj (E') = FD (E')x DE (E') x CT,
= geometry factor at energy E' for point
node i,
(counts/s) / (pCi/gm of dry soil).
If a large target of uniformly distributed contamination, with a concentration of C
pCi/gm, were broken into nodes small enough to approximate point nodes then the fluo
rescent x rays from each point node could be calculated by the above equations assuming
that the source target detector geometry was sufficiently well described. The total signal
from the large target and seen at the detector would then be the sum of the signals from
each of the target point nodes. The detector response could then be modeled as the sum of
all the point node geometry factors.
DR = C x GFI,
where
DR = the photopeak area as measured by a
detector, (counts/s),
C = the uniformly distributed concentration
of contamination in the target,
(pCi/gm of dry soil), and
GFi = the calculated geometry factor for
point node i,
(counts/s) / (pCi/gm of dry soil).
These equations thus make up a mathematical model of a physical situation. The
model can be experimentally verified by calculating all the nodal Geometry Factors, GFj,
for a particular geometry and then making an actual measurement of the signal, DR, from
a target in that geometry. The sum of the geometry factors, referred to hereafter as the
"target geometry factor", times the contamination concentration in the source should equate
to the signal seen at the detector. This model can then be used to assay unknown target
samples. The unknown contamination concentration of a sample is given by
DR
C=
SGFi'
where all terms are as previously defined.
This equation is thus the basis for target sample assay. Figure 6 shows the spatial
relationships of the source, target, and detector that were used to experimentally verify the
mathematical model.
To further verify the model and to provide a better assay of the target, each target is
measured in more than one geometry. This is accomplished easily by varying the target to
detector distance between measurements. A graph of detector response vs target geometry
factor is closely approximated by a straight line
DR(P) =C x GF(P),
where
DR (P) = fluorescent signal seen at the detector
from a target at position P,
(counts/s),
C = contamination concentration in the
target, pCi/gm,
FIGURE 6
Source Target Detector Spatial Geometry
Target Cylinder
Center
Y Z
Detector Point Source
Center
@ Origin 
Point Source
GF (P) = target geometry factor, or, the sum of
all point node geometry factors for a
target located at position P,
(counts/s) / (pCi/gm).
Using the contamination concentration, C, as the fit parameter, the data collected from
measurements made at several different geometries is then least squares fit to the above
equation. This yields a value for C, the contamination concentration in the target sample,
which is the desired result of the assay.
Compton Scatter Gamma Production of Fluorescent X Rays
The fluorescent xray production described by the previous equations is due to unscat
tered source gammas undergoing photoelectric interactions in the target. Since the target is
thick, gammas will also Compton scatter. If the scatter angle is small enough, the scattered
gamma will still be of sufficient energy to undergo a photoelectric interaction in the sam
ple and produce more fluorescent x rays. As will be discussed in the experimental results
section, approximately 15% of the fluorescent x rays produced are due to singly Compton
scattered gammas. It is therefore important to calculate this production term and include
it in the model.
Compton scatter gamma model
Qualitatively, Compton scatter gamma production of fluorescent x rays is due to ex
citation source gammas which undergo a single Compton scatter interaction at point A
in a target, change direction and energy, and then undergo a photoelectric interaction at
point B in a target. Quantitatively, this is slightly more complicated to describe. Consider
two target points, A and B, as shown in Figure 7. Using equations similar to those used
previously, the flux at scatter point A is calculated, the scatter flux and gamma energy of
gammas which are scattered toward point B are calculated, the photoelectric reaction rate
and fluorescent xray production at point B are calculated, and the number of fluorescent
x rays which are counted by the detector is calculated. In a fashion similar to that used
previously, the microscopic calculations for each point in a large target are summed into
the macroscopic total fluorescent xray production due to Compton scatter gammas.
Mathematical model
Looking at Figure 7, the excitation gamma flux at point A is given, again, by Eq. 2
ER (E)
FLI (E) = ER(E x exp (MP(E)poR2),
where
FL1 (E) = excitation gamma flux at a point in the
target, (gammas/cm2s) ,
ER(E) = source emission rate at energy E,
(gammas/s),
R, = distance from the source to the point, (cm),
p (E) po = sample mass attenuation coefficient at
energy E, p (E) (gm/cm2), times sample
density, Po (gm/cm3),
R2 = that portion of the total distance that
lies within the attenuating sample, (cm), and
E = energy of the incident gamma.
The flux that arrives at point B is dependent upon the flux that arrives at point A,
and on the differential Compton scatter cross section for scatter through an angle 0. The
FIGURE 7
Compton Scatter Spatial Geometry
R sin
in 0 6d
X AXIS
d4
INCIDENT
KleinNishina differential scatter cross section, in units of (cm2/electron) / (dl), is given
by (Evans6 pp 677689)
doa = rx X do x [ ],
where
do = differential cross section,
(cm2/electron) ,
ro = classical electron radius, (cm),
d2 = sin (0) dOdo
and
0 = gamma ray scatter angle with respect
to the original direction of motion,
S= rotational angel about the original
direction of motion,
[. ] = terms from equation, see reference 6 (pp 677 689).
Using these two equations, the reaction rate for gammas scattering at A into the solid
angle dil about 0, in other words towards B, is given by
RX = FL1 (E) x do x EDens x Vol,
where
RX = scatter reaction rate, (scatters/s),
FLi (E) = flux of excitation gammas at point A,
(gammas/cm2s) ,
dr = Klein Nishina differential scatter cross
section, (cm2/electron) ,
= r, x dfx x [...]
EDens = electron density at point A, (electrons/cm3) ,
Vol = volume of point A, (cm3).
The energy of the scattered gamma is given by (Evans6 pp 677689)
,! mo c2
1 cos(0) + (/a)'
where
E' = energy of the scattered gamma, (keV),
0 = scatter angle,
mo c2 = electron rest mass,
= 511keV,
E
m0 c2
E = energy of the incident gamma, (keV).
The flux at point B due to Compton scatter at point A is then described by
RX
FL2 (E') = si( exp (P Po X),
X2 sin (0) d~do
where
FL2 (E') = flux at point B due to Compton scatter
at point A, (gammas/cm2s),
E' = energy of scattered gamma, (keV),
RX = scatter reaction rate, (scatters/s),
exp (p Po X) = attenuation factor for
gammas passing through soil,
and
pt = soil attenuation coefficient
at energy E',
p = soil density, (gm/cm3),
X = distance from point A to
point B, (cm),
X2 sin (0) dOdO = surface area through which
garmnas, scattered at point
A into d2 about 0,
pass upon reaching point B.
But since the reaction rate, RX, contains the term do which contains the term sin(0)
dO do, this will cancel out of the numerator and denominator leaving
FL1 (E) x r2 x [* ] x EDens x Vol
FL2 (E') = X exp (p X),
where all terms are as previously defined.
Knowing the flux at point B and the energy of the incident gammas, allows the calcu
lation, using the equations described in the previous section, of the photoelectric reaction
rate at point B due to scatter in point A, the fluorescent xray production rate at point B
due to scatter in point A, the x ray attenuation from point B to the detector due to scatter
in point A, and the number of fluorescent xrays from point B due to scatter in point A
counted by the detector. To determine the entire production of fluorescent x rays at point
B, scatter from every other point node in the target that results in gammas reaching point
B must be calculated. A summation of all these contributions yields a Compton Geometry
Factor for point B, analogous to the Geometry Factor calculated in the previous section for
unscattered gammas. The result of these equations is a summation equation completely
analogous to that derived in the previous section
CDR, (E') = C x CGFi (E'),
where
CDRi (E') = detector response at energy E'
to Compton scatter production at point
node i,
(counts/s) / (CpCi/gm of dry soil),
C = contamination concentration at point
node i, (pCi/gm of dry soil),
CGF, (E') = Compton geometry factor at energy E'
for point node i,
(counts/s) / (pCi/gm of dry soil).
As before, the contribution from each point node of a large target could be summed to
yield the total calculated detector response due to Compton scatter gamma production. The
Compton scatter production portion of the detector response cannot, however, be measured.
Actually the measured detector response is the sum of fluorescent xray production due to
unscattered excitation gammas and Compton scattered excitation gammas, and these two
contributions cannot be physically separated. The solution is to alter the model such that
the measured detector response is modeled as being the sum of the unscattered gamma
Geometry Factors and the Compton scattered Geometry Factors
DR. (E') = C x (GFi (E') + CGF, (E')),
where all terms are as previously defined.
Then, for a large target of uniform contamination concentration C pCi/gm of dry soil,
the detector response is modeled as
DR = C x (GF, (E') + CGF, (E')),
where
DR = the photopeak area as measured by a
detector, (counts/s),
C = the uniformly distributed concentration
of contamination in the target,
(pCi/gm dry soil),
GFi = the calculated geometry factor for
point node i,
(counts/s) / (pCi/gm of dry soil),
CGF, = the calculated Compton geometry factor
for point node i,
(counts/s) / (pCi/gm of dry soil).
This set of equations then constitutes a mathematical model of the fluorescent xray
production due to unscattered gammas and Compton scatter gammas. As stated before,
the desired result of the assay, the value of C, could theoretically be calculated using only
one measurement made in one geometry. In this work, multiple measurements are used to
achieve a more statistically significant answer. As discussed earlier, this model is verified
by actual measurements of targets in several known geometries. As will be discussed in the
results section, the model is in very good agreement with actual measurements.
Electron density
The electron density used above is a parameter which will vary from sample to sample
due to changes in density, elemental makeup, and water content. Electron density must
therefore be calculated for each sample. This is accomplished in a simple fashion using the
transmission measurement described earlier.
The total linear attenuation coefficient is made up of an absorption coefficient, a pair
production coefficient, a Compton scatter coefficient, and a coherent scatter coefficient.
A conversation with John Hubble of the National Bureau of Standards and a followup
computer study indicated that for various soils, the ratio of Compton scatter coefficient
to total linear attenuation coefficient is approximately constant for a given energy gamma.
Appendix A details the computer study done to verify this and to arrive at an appropriate
average Compton to total ratio. Knowing this ratio, the measured total linear attenuation
coefficient can be used to calculate the electron density as follows
cale = CTR x Iea"'
where
aelc = calculated Compton linear attenuation
coefficient as ratioed from the total
linear attenuation coefficient, (cm'),
pnea = measured total linear attenuation
coefficient, (cm), measured as
described in a previous section,
CTR = ratio of Compton linear attenuation
coefficient to total linear attenuation
coefficient,
but
Fi"le = EDens x a KN
where
EDens = soil electron density, (electrons/cm2),
,N" = Klein Nishina Compton scatter cross
section, (cnm2/electron).
therefore
eale
Cate
EDens = 
oKN
where all terms are as previously defined.
Natural Production of Fluorescent X Rays
Since progeny of both U238 and Th232 emit U and Th x rays (see tables 1 and 2) the
natural xray production rate was also calculated. As will be seen in the results section,
this term contributed less than 1% to the total fluorescent xray production.
The decay chains of U238 and Th232 are very similar. In both cases, the parent alpha
decays to the first progeny, the first progeny beta decays to the second progeny, and the
second progeny beta decays to the parent element with an atomic weight four less than that
of the parent. As the second progeny decays, the beta is emitted leaving the metastable
decayed atom looking very much like a parent atom with one extra electron. During the
complicated events that follow beta decay, the decayed atom emits x rays that are of energy
characteristic of the parent atom.
In the case of U, U238 alpha decays to Th234, which beta decays to Pa234, which
beta decays to U234. As the Pa234 decays to U234, U x rays are emitted. ICRP report
#3819 gives the emission rate of these x rays as 0.00232 Ka,/decay.
In the case of Th, Th232 alpha decays to Ra228, which beta decays to Ac228, which
beta decays to Th228. As the Ac228 decays to Th228, Th x rays are emitted. ICRP
report #3819 gives the emission rate of these x rays as 0.0428 K,1/decay.
Since natural decay production of x rays contributes so little to the total production,
the assumptions that Th232 is in equilibrium with Ac228, and that U238 is in equilibrium
with Pa234 will introduce little error. As such, one pCi of Th232 will be in equilibrium
with one pCi of Ac228, and one pCi of U238 will be in equilibrium with one pCi of Pa234.
The production rate of U and Th x rays can now be calculated.
For thorium
K,,yield= (0.0428 Ka x (0.037 decay/s
Sdecay) pCi Th 232 '
= 0.001584 K s
pCi Th 232
For uranium
Klyield = 0.00232 Kdeci 0 .037 decay/s
decay) pCi U 238 /
= 0.00008584 K, i/
pCi U 238
These terms are in the correct units to be added directly into the previously described
mathematical model at the point where fluorescent xray production in each point node is
calculated. With this small correction added, the mathematical model is complete.
Isotopic Identification
As mentioned earlier, the two isotopes which are of principle interest for this assay
technique are U238 and Th232. In fact, these are the only two isotopes of U and Th
which can be seen at small concentrations using XRF. This is because XRF is dependent
upon the number of atoms present. In the cases of other U or Th isotopes, tremendously
high numbers of curies would have to be present before there would be enough atoms of
these isotopes to be seen by XRF. This is due to the very long half lives of U238 and Th232
with respect to their other isotopes, since the number of curies is equal to the number of
atoms times the decay constant. Table 8 illustrates this point.
The sensitivity of the experimental setup tested in this work is approximately 50 pCi/gm
of Th232, which corresponds to 500 ppm. The system sensitivity then, in terms of number
of atoms required, is approximately 500 ppm. The table clearly shows that huge quantities,
in terms of pCi/gm, of all the isotopes except U238 and Th232 would be required to reach
concentrations 500 ppm. Since these huge concentrations are rarely seen, and since one of
the objectives of this assay system is to achieve a low sensitivity in terms of pCi/gm, it can
be concluded that this XRF assay technique cannot be used for U and Th isotopes other
than 238 and 232 respectively.
TABLE 8
Isotopic Concentrations: PPM vs. pCi/gm
Concentration Concentration
Isotope (ppm) (pCi/gm)
U238 500 168.1
U235 500 1.081E3
U234 500 3.125E6
Th232 500 54.65
Th234 500 1.158E13
Th230 500 1.009E7
Th228 500 4.098E11
It is also reasonable to conclude that all U and Th seen by XRF is U238 and Th
232 respectively. Based on the above table, such large quantities of other isotopes would
be required before these isotopes could be seen by XRF, the radiation levels of the sources
would be too large for analysis using sensitive Ge detectors. It is also extremely unlikely that
such large quantities of other isotopes would be found. Even if U238 were in equilibrium
with Th234, U234, and Th230, or if Th232 were in equilibrium with Th228, while the
curie contents of the sample would be high in these other isotopes, the ppm concentrations
of these other isotopes would be much to low to register using XRF. It would thus be safe
to assume that all of the signal seen at the detector was from U238 or Th232.
Prior knowledge of the nature of the process which lead to the contamination might
point toward high concentrations of other isotopes. For example, excavation near a plant
which previously processed Th230 to remove Ra226 would be expected to show high Th
230 levels and natural Th232 levels. Again though, unless the Th230 levels were extremely
high, XRF would not be of any use.
Unfortunately then, this XRF technique is not useful in determining the concentrations
of isotopes other than U238 and Th232. If it were known that the contamination in
question was primarily U238 and/or Th232, progeny equilibrium with U238 and Th232
could be assumed thus establishing an upper limit of other isotopic concentrations. This
would not, however, constitute an adequate assay unless the upper limit were below some
lower bound of regulatory concern.
However, the analysis technique used, that is measuring the attenuation properties of
the sample at the energies of the gamma rays of interest, and the calculation of geometry
factors, could be applied to gamma spectroscopic techniques or to neutron activation anal
ysis techniques to improve their accuracy. This work is, however, beyond the scope of this
project.
The value of this technique is that it measures U238 and Th232 directly and without
relying on equilibrium with progeny. Neutron activation and its associated neutron source
was previously required to achieve this independence.
Error Analysis
Introduction
A soil contamination assay must be accompanied by an estimate of the error associated
with the measured contamination concentration. The assay system presented in this work
is based on fitting measured data to mathematical models using the least squares technique.
Least squares analyses lend themselves well to propagation of error from one curve fit to
the next, as well as to calculation of the chisquared test statistic as a measure of goodness
of fit.
This work begins with measured spectral data and the error associated with each data
point. This data is fit to a mathematical model of an xray peak as viewed through a Ge
detector, and the peak parameters are determined. These parameters are used to determine
the area of the peak and the error associated with the peak area. This operation is repeated
for several different target configurations yielding several peak areas and their associated
errors. These areas are then fit to a mathematical model which predicts the detector
response as a function of target geometry and contamination concentration. The only
fitting parameter of this model is the contamination concentration in the target, which
is calculated by least squares fitting the previously calculated peak areas to the modeled
function. The error in the fitting parameter is extracted from the least squares fitting
process and the desired result, determination of the soil contamination concentration and
its associated error, is achieved.
Least Squares Peak Fitting
The least squares fitting technique is fairly simple and is described in numerous books
and articles. The mathematics used in this work was taken from Forsythe et al.20 This
technique was used in this work to properly determine the spectral peak areas and their
associated errors.
Least squares fitting is an iterative technique based on minimizing the square of the
difference between a measured value and a value calculated based on a mathematical model.
The minimization is performed by properly choosing the parameters of the mathematical
model. The model may be linear in these parameters or nonlinear. Each successive iteration
refines the fitting parameters such that the sum of the squares is minimized. To begin, the
mathematical model is chosen. An initial guess as to the fitting parameters is also required.
F (XY : P1, P2, P3,..., Pn), = Yi,
where
Xi = independent variable,
Pn = fit parameters of the mathematical model,
1i = dependent variable.
Note: the "1" indicates that the fitting parameters are the current guess. The previous guess,
or for the first iteration, the initial guess, will be used to determine the new, or current,
guess.
Using current guess as to the fitting parameters, the sum of the squares of the difference
between the measured values and the calculated values is calculated.
S = (F (X),  y)2
where
S = sum of squares,
F (X), = ,
= calculated dependent variable based
on current fittingparameters,
yi = measured dependent variables.
To mininiize this equation
dS dS dS dS
dPI dP2 dP3 dPn
This creates a set of "n" independent equations each looking
like this
dS dF (X)
dP1 = 2 x (F (X)i yi x dP1 0,
where
F (Y,), = F(X,: PL1,P21, P31,...,Pn).
This equation is mathematically correct, however only the initial guess parameters are
known at this point. Fortunately F (X,), can be approximated by a Taylor expansion,
truncated after the first order terms, knowing 1. the values of F (Xi)o which are based on
the previous best guess of the fitting parameters, and 2. the function partial derivatives at
each X,
dF (X,)o
F(X,)l = F (X1)o + (P11 Plo) x
dF(X,)o dF(X,)o
+ (P2, P2o) x d + (P31 P3o) x dP
dP2 dP3
dF (Xi)o
+ ... (Pni Pno) x d
dPn
where all terms are previously defined.
Substituting this into the least squares minimization equation yields
dS dF(X)o dF (X,)o x (F(X) y,)= ,
dP1i dp1 x ["] + dP xF() =0
where
[* ] = DP1 x d ) + DP2
dP1
dF (X,)o dF (X )o dF (X,).
x + DP3 x o+ + + DPn x
dP2 dP3 dPn '
DPn = Pnl Pno.
This can be rewritten as
dF(X,)"o dF (X,)
dF x [**) ] = Z (XP x (y, F(X,)o).
As previously stated, similar equations are generated for each differential equation
dS
dPI
dS dS
=0, 0 ..." = 0.
dP2 dPn
This system of equations lends itself to the matrix form
DFt (n, m) x DF (m, n) x A (n, 1) = DFt (n, m) x DY (m, 1),
where
DF(m,n) =
/ dF(X,)
dP1
dPI
dPF(X)o
SdP1
dF(XI)o
dP2
dF(X2 )
dP2
dF(X )o
dP2
dF(X1)o
dP3
dF(X2)0
dP3
dF(XPm)
dP3
. dPn
S. dPn
dF(X), P
dPn
DFt (n, m) = the transpose of DF (m, n),
n = the number of parameters in the fitting function,
m = the number of data points used in the fit,
/ (Pl, P10o) \
(P21 P2o)
A(n,1)= (P31 P30)
\(Pn Pno)
/(y, F(X,)o)
(y2 F (X2)0)
DY(m, 1)= (Y F (X)o)
\(ym F (X)0)
This matrix equation is solved by Gaussian elimination to yield the values of the A (n, 1)
matrix. These values are used to update the parameter guesses from the initial guesses of
Plo, P20, ..., Pno, to P11, P21, ..., Pni, since Pil = Pio + A (i, 1). New values of F (X,)
are calculated based on these new parameters, these new parameters become the "initial
guesses" for the next iteration, and the matrix solution is repeated. This iteration process
is continued until the sum of squares is "minimized" based on some predetermined cutoff
criteria. The result of the minimization is the calculation of the "best" fitting parameters
for the mathematical model.
Covariance Matrix and Functional Error
As a byproduct of this method, the variances and covariances of the fitting parameters
are calculated. The inverse of the matrix product, DFE x DF, is defined as the covariance
matrix 20
(DF' (n, m) x DF (m, n))1 = Covar (n, n).
The diagonal values of this matrix, Covar(i,i), are the variances of the model fitting
parameters.
Covar (1,1) = 02 (P),
Covar (2, 2) = o2 (P2),
Covar (3,3) = '2 (P3),
Covar (n, n) = 2 (Pn).
The covariance matrix is diagonally symmetrical, with the off diagonal elements being
the covariances of the various parameters, for example
Covar (1, 2) = 2 (P1, P2).
These values are used to calculate the errors associated with fitting parameters at
various stages of determining the final solution, the soil contaminant concentration and its
associated error. But in addition to knowing the errors associated with fitting parameters,
the errors associated with functions of those fitting parameters are necessary. For this
work, only linear functions of fitting parameters are used. The error associated with a
linear function of fitting parameters is given by the following equation20
if: F(X : P1,P2,P3,...,Pn),
where : PI, P2, P3, ..., Pn and their associated errors
are known,
then for: Q (X : P1, P2, P3,..., Pn),
n nL
'2 (Q (X,)) = Pi2 x (Pi)+ Pi x Pj x Covar (Pi, Pj). (4)
i=l ,i=1
This equation and the covariance matrix as determined via the least squares process
described here are all that is needed to properly propagate error through the peak fitting
portion of this work.
Error Propagation
The various stages of the assay analysis have all been thoroughly described. This section
will describe how error is propagated through these calculations.
To begin with, a spectral peak is collected. The error associated with each spectral
data point is
The first manipulation performed on this data is the calculation of the background
beneath the xray peak being analyzed. This calculation is performed in two steps. First, a
polynomial background (POLYBK) is calculated using data points to the right and left of
the peak. These points are least squares fit to a fourth order polynomial yielding the best fit
and a covariance matrix. Using the covariance matrix and Eq. 4, the error associated with
each calculated data point is determined, Var(POLYBKi). Second, the "complimentary
error function (erfc)" background (ERFBK) is determined. This is a numerical process, the
error associated with each of these points is estimated to be
0.2 (ERFBKS) = ERFBKj.
The two backgrounds, polynomial and erfc, are assumed to each contribute equally to
the complete background, thus the complete background is equal to
POLYBK, + ERFBKj
BKi =
2
and,
o 2 (BKi) = C2 (POLYBKI) + o2 (ERFBKi).
The next step in the analysis is to subtract the background from the spectral data
(SPEC) to yield an estimated peak. The variance associated with each of the peak points
is then
0.2 (PKi) = 02 (SPECi) + .2 (BKi).
The peak is then least squares fit to a Voigt peak shape, yielding the covariance matrix
for this process. Equation 4 is then used to calculate the error associated with the numeri
cally calculated peak area. These steps are repeated for several geometric configurations of
the soil sample target yielding several peak areas and their associated errors.
Linear Function Fitting
Once the peak areas and their associated errors have been determined, the last step
of the assay is to determine the concentration of contamination in the sample jar and its
associated error. As described earlier, the measured detector response is a linear function
of calculated GFs, the slope of this line being the concentration of contamination in the
sample jar. The measured detector response data, the error associated with this data, and
calculated GFs need only to be fit to a linear function to determine the desired assay result.
Easier still, the fitted line must pass through the origin since if the calculated GF for a
sample were zero, the measured detector response would also have to be zero. This sort of
statistical data fitting is common to most introductory statistic books. The treatment used
here is from handouts prepared for a statistics class at Cornell University 21
For any linear function of X,
Y = x F (X),
a=1
where
aa = the ath of m fitting parameters,
F. (X) = the a'h of m linear functions of X,
then the values of the fitting parameters a, are given
by,
Em E= I" F.F (XY)a
ai = )H (a, i),
a. 0"2
where
H1 (i, i) = the covariance matrix,
ab = the standard deviation of
detector response b,
and,
1i,j)= E= F, (X,) Fj(X)
a .
For a simple function such as DR = C x GF, where C is the unknown fitting parameter
al, = DR, X = GF, and Fi (X) = X = GF, these equations are extremely easy to solve
and yield the desired result of this analysis, the value of C. The error in the value of C is
also quite simple.
a' = H1 (ii).
This then yields the desired result of this analysis, the soil contamination concentration,
C, and its associated error.
Experimental Procedure
Sample Counting
Once the experimental apparatus is constructed and properly calibrated, as described
in Appendix A, the processing of samples is straight forward and requires only eight steps.
1. Samples are placed in 100 ml plastic jars and weighed. The soil may be wet and not
completely uniformly mixed. The sample jars are of radius 2.32 cm with an active
height of 6.50 cm.
2. The sample is weighed, wet or dry in the jar, and the average jar weight, 20.7 0.1
gm, is subtracted to obtain the sample weight.
3. The sample jar is placed in a known geometry above a Ge detector, described in detail
in Appendix A, such that the center of the sample is directly in line with the center of
the detector. In this work, the jar center is 11.6 0.1 cm from the detector window,
and the detector window to detector distance is 0.5 cm.
4. A Co57 source is placed directly in line with the center of the detector such that the
sample jar is between the source and the detector. The source to detector window
distance is 89.4 0.2 cm. The transmitted fraction of 136 keV and 122 keV gammas
from Co57 is then measured. Count duration is dependent upon source strength. In
this work, count times of one hour are used. Shorter times could be used by placing
the source closer to the detector such that the detector system dead time remained
below 20% or so, and the total counts under the peaks in question were statistically
significant. In this work, peak areas are on the order of 100,000 counts.
5. The Co57 source is replaced with an Eu155 source and step three is repeated, mea
suring the transmitted fraction of 105 keV and 86 keV gammas from Eu155. The
transmission fractions are then used to calculate the sample linear attenuation coeffi
cients as described in a previous section.
6. The next step is sample XRF analysis. The sample is placed such that the center of the
sample is again directly above the center of the detector. The source holder and sample
holder assemblies are shown in exploded view in Figure 8. Figure 9 shows the sample in
its holder positioned above the detector and above the source holder shield. Using the
center of the detector as the origin of an XYZ coordinate system, the sample center
is located at coordinate (10.5 0.1 cm, 0.0 cm, 0.0 cm). Figure 10 shows the sample
in position and the laser alignment beam used to ensure that it is properly centered.
The source holder, described in detail in Appendix A, holds two sources located in the
XZ plane formed by the long axis of the sample bottle and the detector center. The
sources are located at coordinates (4.42 0.01 cm, 0.0 cm, 4.42 0.01 cm) and (4.42
0.01 cm, 0.0 cm, 4.42 1 0.01 cm).
Once the sample is properly positioned, the sources are placed in their holders and the
sample is irradiated for one hour. In this work, fluorescent xray peaks ranged in area
from 20,000 counts to 1,500,000 counts depending upon the U or Th concentration in
the samples. Figure 11 shows the multichannel analyzer system used to collect spectral
information. Figure 12 shows a typical spectrum on the MCA.
7. For the next seven counts, the sample is next raised 0.3 0.01 cm and step six is
repeated. The yields a total of eight counts with the sample center being at 10.5, 10.8,
11.1, 11.4, 11.7, 12.0, 12.3, and 12.6 cm from the detector. The detector and XRF
excitation sources remain in fixed positions for all eight counts.
8. Finally the sample is dried, if necessary, in an oven or a microwave and weighed again.
The dry soil weight and soil moisture fraction are thus determined.
FIGURE 8
Exploded View of Target Holder Assembly
l I I
C~1 :~ :a
"~
'?1
? .. l
i l
FIGURE 9
Target in Place Above Detector
79
f
l i
i JW
FIGURE 10
Target in Place Above Detector
Showing Laser Alignment System
I
FIGURE 11
ND9900 Multichannel Analyzer, ADC,
Amplifier, and Detector Power Supply
Ai a
FIGURE 12
Typical XRF Ka Peak on MCA
