Title Page
 Table of Contents
 List of Tables
 List of Figures
 Materials and methods
 Results and conclusions
 Biographical sketch

Title: Determination of radionuclide concentratins of U and Th in unprocessed soil samples
Full Citation
Permanent Link: http://ufdc.ufl.edu/UF00089970/00001
 Material Information
Title: Determination of radionuclide concentratins of U and Th in unprocessed soil samples
Physical Description: Book
Language: English
Creator: Lazo, Edward Nicholas
Publisher: Edward Nicholas Lazo
Publication Date: 1988
 Record Information
Bibliographic ID: UF00089970
Volume ID: VID00001
Source Institution: University of Florida
Holding Location: University of Florida
Rights Management: All rights reserved by the source institution and holding location.
Resource Identifier: alephbibnum - 001130251
oclc - 20139548

Table of Contents
    Title Page
        Page i
        Page ii
        Page iii
        Page iv
        Page v
    Table of Contents
        Page vi
        Page vii
        Page viii
        Page ix
        Page x
    List of Tables
        Page xi
        Page xii
        Page xiii
    List of Figures
        Page xiv
        Page xv
        Page xvi
        Page 1
        Page 2
        Page 3
        Page 4
        Page 5
        Page 6
        Page 7
        Page 8
        Page 9
        Page 10
        Page 11
        Page 12
        Page 13
    Materials and methods
        Page 14
        Page 15
        Page 16
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    Results and conclusions
        Page 88
        Page 89
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    Biographical sketch
        Page 330
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Full Text






"~~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

sister-in-law, and one brother-in-law, 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.


This publication is based on work performed in the Laboratory Graduate Participation

Program under contract #DE-AC05760R00033 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.



LIST OF TABLES .............

LIST OF FIGURES ............

ABSTRACT .. .. . .. .... ..


I INTRODUCTION ............

Soil Sample Assay for Radionuclide Content

Standards Method for Gamma Spectroscopic

Radionuclides of Interest . . . . .

Process Sensitivity ............

Statement of Problem ..........

X-Ray Fluorescent Analysis . . . .

Assay Technique .............

Literature Search . . . . . .


Peak Shaping ..............

A Fitting Peak .. .........

A Fitting Background ........


. ili



. 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 . . . .

. 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

. 86



A EQUIPMENT AND SETUP ........................ 124

System Hardware ............. ... .......... 124
The ND-9900 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


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






S 316

S 320


LIST OF REFERENCES .......................... .... 328

BIOGRAPHICAL SKETCII ......................... 330







1. Uranium 238 Decay Chain ..............

2. Thorium 232 Decay Chain ..............

3. Summary of DOE Residual Contamination Guidelines

4. U and Th K-Shell Absorption and Emission . . .

5. Co-57 and Eu-155 Emission Energies and Yields . .

6. Co-57 and Eu-155 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 NJA-U .

Fitted Detector Response for NJB-U .

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


. . . . . . 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

A-1. Shield Material X-Ray Emission

ponse for Th-la . . . . . . 105

ponse for NJA-Th . . . . . .. 106

ponse for NJB-Th . . . . . .. 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

A-2. NBS Source, SRM 4275-B-7, Emission Rates

. . . . . . 129

A-3. NBS Source, SRM 4275-B-7, Physical Characteristics . . . . . 129

A-4. System Calibration Parameters . . . . . . . . . . 133

A-5. Water Attenuation Coefficients, p (E)f2,, Actual and Calculated Values . 140

A-6. Water Attenuation Coefficients, p (E)oo
Calculated Values vs. Target Distance from the Detector . . . . 141

A-7. Representative Soil Elemental Concentrations . . . . . . . 144

A-8. Compton to Total Scatter Coefficients for Soil at 150 keV and 100 keV . 144

A-9. Average Compton to Total Scatter Ratio for Soil . . . . . . 145

B-1. Relative Sample Separation vs. Solution Matrix Condition . . . ... 151

B-2. Target-Detector Distance vs. Measured Peak Area . . . . . .. 154


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,~ X-Ray Spectral Peak . . . . . . . .... ... .25

4. Polynomial and Step Function X-Ray 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. ND-9900 Multichannel Analyzer, ADC, Amplifier, and Detector Power Supply . 82

12. Typical XRF K,1 Peak on MCA ................... ... 84

B-1. Relative Sample Separation vs. Solution Matrix Condition . . . . 152

B-2. Target-Detector 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


Edward Nicholas Lazo

December, 1988

Chairman: Genevieve S. Roessler
Major Department: Nuclear Engineering Sciences

Work with systems used to assay soil samples for U-238 and Th-232 indicated that the

need existed to more directly measure the concentrations of these radionuclides. An x-ray

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 x-ray 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 Co-57 gammas to induce fluorescent x rays which are

measured using an intrinsic Ge detector. The Co-57 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


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 x-ray 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.



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. Twenty-two 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 time-consuming

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 time-consuming 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, Co-57 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 Co-57 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 Co-57 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 Co-57. 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


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 U-238 and Th-232, 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 U-238 decay chain and the radiations emitted

by each member. Table 2 shows equivalent information for the Th-232 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.

U-238 Decay Chain
S Major Radiation Energies
MeV) and Intensities
Radionuclide IIalf-Life Alpha Beta Gamma

4.15 (25%)
4.20 (75%)













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%)

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%)

1.120 (17%)
1.764 (17%)

TABLE 1 (continued)
Major Radiation Energies
(MeV) and Intensities
Radionuclide Half-Life Alpha Beta Ganuna
Po-214 164.0 us 7.69 (100%) 0.799 (0.014%)

TI-210 1.3 min 1.3 (25%) 0.296 (80%)
(.02%) 1.9 (56%) 0.795 (100%)
2.3 (19%) 1.31 (21%)

Pb-210 21.0 a 3.72 (2E-6%) 0.016(85%) 0.047 (4%)

Bi-210 5.10 d 4.65 (7E-5%) 1.161 (100%)
(Branches) 4.69 (5E-5%)

Po-210 138.4 d 5.305 (100%) 0.803(0.0011%)

TI-206 4.19 min 1.571 (100%)

Pb-206 Stable
NOTES + Indicates those gamma rays that are commonly used to identify U-238. Equilibrium
must be assumed.

Th-323 Decay Chain
S Major Radiation Energies
(MeV) and Intensities
Radionuclide Half-Life Alpha Beta Gamma

3.95 (24%)
4.01 (76%)











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


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 Th-232. Equilibrium
must be assumed.


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%)


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 U-238, U-235, Th-232, and Th-230 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 non-destructive 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 x-ray 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
Ra-226 5 pCi/g averaged over the first 15 cm of
Ra-228 soil below the surface.
Th-232 15 pCi/g when averaged over any 15 cm
Th-230 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 Ra-226 from Th-230, and
Ra-228 from Th-232 and assume secular equilibrium. If either Th-230 and
Ra-226, or Th-232 and Ra-228 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
b: These guidelines represent unrestricted-use 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.

X-Ray Fluorescent Analysis (XRF)

Atoms can be ionized, i.e., have one or more electrons removed, via several processes. In

the case of x-ray fluorescence, gamma or x rays incident on an atom undergo photoelectric

reactions resulting in the ionization of the atom. The ionized atom then de-excites 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 x-ray emitting material. This technique can thus

be said to directly determine U and Th concentrations in soil samples. For a more detailed

description of x-ray 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 hyper-pure intrinsic Ge planar detector.

The areas under the Ka& x-ray peaks from U and Th are determined by a spectral analysis

system. The Ka peak was chosen because the K-shell lines are highest in energy, thus

minimizing attenuation effects, and the Kai line is the most predominant K-shell 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 x-ray 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.



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 x-ray 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 lower-energy 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 step-like 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 high-energy 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 electron-hole pairs
produced in the stopping process and random noise from the amplifying
electronics, the low energy tailing represents pulse-height 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.

Typical Gamma Ray Spectral Peak and Background



a 10000


* Spectral Data

Gaussian Fit

ERFC Background





Channel Number




The x-ray 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 732-739).

N(H)= R( ,E) x S(E)dE


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 434-440). Mo-

noenergetic gamma rays emitted by the de-excitation 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 397-403). This is

directly attributable to the Heisenberg uncertainty principle such that (Evans6 pp 397-403)

r (eV) =.66E 15 (eV s) /tm (8)


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 1E-15 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 434-440).

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 X-ray energy distributions must therefore be described by a Lorentzian distribution10

and an x-ray spectral peak must therefore be described by the convolution of a Gaussian

detector response function and a Lorentzian x-ray energy distribution.11 Mathematically,

this convolution is written as

C(E) = G (E') xL (E E') dE',
J -0


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),


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
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 x-ray 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 x-ray 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

Lorentzian X Ray as Seen Through
the Gaussian Response of a Detector

10000000 --





----- Lorentzian



960 980
Channel Number



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 x-ray spectrum in the vicinity of the Ka x-ray 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) ,


SB (X) = step background value at channel X,

A = amplitude,

Typical Th K1a Spectral Peak


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


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,
Y (Xi) = the sum of the gross channel counts from the
first peak channel to channel Xi, and
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


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.

Polynomial and Step Function X-Ray Peak Background


S_ \ D~c .grouLl.iU.
0 100000-

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, Co-57,

which emits gammas at 122 keV and 136 keV, was chosen because of the proximity of its

gamma energies to the K-shell absorption energy. At these energies, U and Th have high

cross sections for photoelectric reactions with K-shell electrons. Table 4 shows the absorp-

tion and emission energies for U and Th. Table 5 shows the emission energies and yields

for Co-57. Table 5 also lists emission energies and gamma yields for Eu-155. 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.

U and Th K-Shell Absorption and Emissionl7
K-Shell 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

Co-57 and Eu-155 Emission Energies and Yields18
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 Co-57. Europium-155 sources, therefore, are described by gamma
emission rates, Activity (Ci) x Yield (gammas/s). The listed yields are for
estimation purposes only.

Co-57 and Eu-155 Physical


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.

Co-57 Eu-155
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),


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)


Ao (E) = source gamma emission rate at energy E

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


(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),


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.


t (E) x p = (-1/x) x In(TF(E)),


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

Co-57 gamma rays used to induce x-ray 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


Since Co-57 is used to induce x-ray 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 Co-57 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,


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 Eu-155, 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 Co-57 gammas and two from Eu-155 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, x-ray energies can be calculated from the resulting curve fit. Table 7 shows typical

soil linear attenuation coefficients.


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 non-unique 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


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 x-ray 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 source-target-detector 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

Source Target Detector Physical Geometry





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 x-ray 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

FL(E) = 4ER x exp(-/p(E)poR2), (2)


FL (E) = excitation gamma flux at a point in the
target, (gammas/cm2s),

ER (E) = source emission rate at energy E,

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)


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,,


.037 = the number of disintegrations per second
per pCi of activity,

A = disintegration constant for U or Th,

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'),


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

x-ray fluorescent yield, can be described by

FDE' FY (E',E)x DA
FD (E) = DA x exp(-p (E')por2),


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),

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 Klein-Nishina differential scattering cross section (Evans6 pp 677-689) 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,


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),

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'),


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'),

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,


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


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),


DR (P) = fluorescent signal seen at the detector
from a target at position P,

C = contamination concentration in the
target, pCi/gm,

Source Target Detector Spatial Geometry

Target Cylinder


Detector Point Source
@ 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 x-ray 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 x-ray 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 x-ray 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),


FL1 (E) = excitation gamma flux at a point in the
target, (gammas/cm2s) ,

ER(E) = source emission rate at energy E,

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

Compton Scatter Spatial Geometry

R sin

in 0 6d



Klein-Nishina differential scatter cross section, in units of (cm2/electron) / (dl), is given

by (Evans6 pp 677-689)

doa = rx X do x [- -],


do = differential cross section,
(cm2/electron) ,

ro = classical electron radius, (cm),

d2 = sin (0) dOdo


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,


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 677-689)

,! mo c2
1 cos(0) + (/a)'


E' = energy of the scattered gamma, (keV),

0 = scatter angle,

mo c2 = electron rest mass,
= 511keV,

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

FL2 (E') = si( exp (-P Po X),
X2 sin (0) d~do


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,


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 x-ray 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 x-rays 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'),


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 x-ray 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')),


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 x-ray

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 follow-up

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"'


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


Fi"le = EDens x a K-N


EDens = soil electron density, (electrons/cm2),

,-N" = Klein Nishina Compton scatter cross
section, (cnm2/electron).


EDens- = -

where all terms are as previously defined.

Natural Production of Fluorescent X Rays

Since progeny of both U-238 and Th-232 emit U and Th x rays (see tables 1 and 2) the

natural x-ray production rate was also calculated. As will be seen in the results section,

this term contributed less than 1% to the total fluorescent x-ray production.

The decay chains of U-238 and Th-232 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 meta-stable

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, U-238 alpha decays to Th-234, which beta decays to Pa-234, which

beta decays to U-234. As the Pa-234 decays to U-234, 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, Th-232 alpha decays to Ra-228, which beta decays to Ac-228, which

beta decays to Th-228. As the Ac-228 decays to Th-228, 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 Th-232 is in equilibrium with Ac-228, and that U-238 is in equilibrium

with Pa-234 will introduce little error. As such, one pCi of Th-232 will be in equilibrium

with one pCi of Ac-228, and one pCi of U-238 will be in equilibrium with one pCi of Pa-234.

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 x-ray 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 U-238 and Th-232. 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 U-238 and Th-232

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 Th-232, 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 U-238 and Th-232 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.
Isotopic Concentrations: PPM vs. pCi/gm
Concentration Concentration
Isotope (ppm) (pCi/gm)
U-238 500 168.1
U-235 500 1.081E3
U-234 500 3.125E6
Th-232 500 54.65
Th-234 500 1.158E13
Th-230 500 1.009E7
Th-228 500 4.098E11

It is also reasonable to conclude that all U and Th seen by XRF is U-238 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 U-238 were in equilibrium

with Th-234, U-234, and Th-230, or if Th-232 were in equilibrium with Th-228, 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 U-238 or Th-232.

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 Th-230 to remove Ra-226 would be expected to show high Th-

230 levels and natural Th-232 levels. Again though, unless the Th-230 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 U-238 and Th-232. If it were known that the contamination in

question was primarily U-238 and/or Th-232, progeny equilibrium with U-238 and Th-232

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


The value of this technique is that it measures U-238 and Th-232 directly and without

relying on equilibrium with progeny. Neutron activation and its associated neutron source

was previously required to achieve this independence.

Error Analysis


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 chi-squared 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 x-ray 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 non-linear. 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,


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,


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


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,


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

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


[* ] = DP1 x d ) + DP2

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 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),


DF(m,n) =

/ dF(X,)


dF(X2 )

dF(X )o



. dPn
S. dPn

dF(X), P

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 by-product 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


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 x-ray 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

BKi =


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),


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


Em E= I" F.F (XY)a
ai = )H (a, i),
a. 0"-2


H-1 (i, i) = the covariance matrix,

ab = the standard deviation of
detector response b,

-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' = H-1 (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 Co-57 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 Co-57 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 Co-57 source is replaced with an Eu-155 source and step three is repeated, mea-

suring the transmitted fraction of 105 keV and 86 keV gammas from Eu-155. 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 X-Y-Z 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

X-Z 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 x-ray 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.

Exploded View of Target Holder Assembly

l I I

C~1 :~ :a

? .. l

i l

Target in Place Above Detector


l i

i JW

Target in Place Above Detector
Showing Laser Alignment System


ND-9900 Multichannel Analyzer, ADC,
Amplifier, and Detector Power Supply

Ai a

Typical XRF Ka Peak on MCA

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